Shaped Energy-Time Entangled Two-Photon States for Quantum Information

Transcription

1 Shaped Energy-Time Entangled Two-Photon States for Quantum Information Inauguraldissertation der Philosophisch-naturwissenschaftlichen Fakultät der Universität Bern vorgelegt von Bänz Bessire aus Péry (BE) Leiter der Arbeit: Prof. Dr. André Stefanov Prof. Dr. Thomas Feurer Institut für Angewandte Physik der Universität Bern

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3 Shaped Energy-Time Entangled Two-Photon States for Quantum Information Inauguraldissertation der Philosophisch-naturwissenschaftlichen Fakultät der Universität Bern vorgelegt von Bänz Bessire aus Péry (BE) Leiter der Arbeit: Prof. Dr. André Stefanov Prof. Dr. Thomas Feurer Institut für Angewandte Physik der Universität Bern Von der Philosophisch-naturwissenschaftlichen Fakultät angenommen. Bern, 29. November 2013 Der Dekan: Prof. Dr. Silvio Decurtins

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5 Abstract Entanglement between two spatially distant particles is one of the most intriguing features of quantum theory and reveals the non-local structure of nature at its fundamental level. The manifestation of entanglement can be observed by correlations between two systems with no classical analogue. Apart from its philosophical implications, entanglement exhibits a still unexploited potential in practical applications particularly in the domain of quantum information, the fusion between classical information theory and the laws of quantum mechanics. Among other facilities, quantum information allows the implementation of provably secure cryptographic protocols and enhanced computational efficiency for specific algorithms. While the bit is the building block of a classical information process, the qubit is the primary information carrier in the quantum domain. Unlike its classical counterpart, the qubit is a discrete superposition of two possible states a system can be in. In the past few years, however, it has become evident that quantum information can significantly benefit from the entanglement between d-dimensional states denoted as qudits. Entangled photon states have demonstrated their strong robustness against decoherence in a variety of experiments and have nowadays become a principal resource to perform quantum information science. We report on a new and versatile method for implementing and manipulating entangled qudits in the frequency spectrum of energy-time entangled photon pairs generated by spontaneous parametric down-conversion. The experimental setup takes advantage of an ultrafast detection scheme with femtosecond temporal resolution and allows for manipulating the phase and amplitude of each frequency component of the entangled photon spectrum. First of all, the spectral tuning curve of a type-0 entangled two-photon state is studied theoretically and experimentally. Numerical calculations of the Schmidt number and the entropy then show the existence of a potentially very high dimensional qudit state space if the underlying down-conversion process is induced by a quasi-monochromatic pump laser. By means of a spatial light modulator, we afterwards project the continuous space of frequency modes into a discrete subspace by applying various discretization schemes. Thereby, we observe the characteristic two-photon interferences of entangled qudits. Frequency bins are finally used to perform quantum state tomography for entangled qudits up to d = 4. The presence of entanglement is verified by measuring an extended Bell parameter for qubits and qutrits as a function of their degree of entanglement.

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7 Contents Contents 1 Introduction 3 2 Theoretical Framework Definition of a Bipartite Entangled State Generation of a Type-0 Entangled Two-Photon State Type-0 Spontaneous Parametric Down-Conversion Quasi-Phase Matching Type-0 Entangled Two-Photon State Propagation of a Field Operator through the Optical System Second-Order Correlation Function G (2) Perturbative Treatment of a SFG Process First-Order Correlation Function G (1) Heuristic Comparison between G (1) and G (2) Tuning Curve of Type-0 SPDC Formal Prerequisites Publication: Tuning Curve of Type-0 Spontaneous Parametric Down-Conversion Introduction Theory Experimental Setup Results and Discussion Conclusion Entanglement Quantification Formal Prerequisites Schmidt Number Analytical Approximation of K Numerical Approximation of K Comparison between Analytical and Numerical K Entropy Publication: Computing the Entropy of a Large Matrix Introduction Monte-Carlo Approximation Chebyshev Approximation and Clenshaw s Algorithm

8 CONTENTS Computing the Entropy Examples Conclusions Additional Results Experimental Setup Preparation Manipulation Point Spread Function Spatial Light Modulator Detection Amplitude and Phase Shaping of Energy-Time Entangled Photons Dispersion Cancellation Interferometric Autocorrelation Energy-Time Entangled Qudits From a Qubit to a Qudit Entangled Qudits Formal Prerequisites Publication: Versatile shaper-assisted discretization of energy-time entangled photons Introduction Theory Experimental setup Experimental results Conclusion and outlook Supplementary Information Single Photon Measurements in the Time-Bin Basis State Reconstruction in the Schmidt Basis Publication: Shaping frequency-entangled qudits Introduction Experimental setup Theory Results Conclusion and outlook Supplementary Information Procrustean Filtering: Generating Maximally Entangled States Tomographic Basis and Maximum Likelihood Estimation Bell Parameter Conclusion and Outlook Conclusion Outlook Setup Improvements Physics Projects A Derivation of a Type-0 Entangled Two-Photon State 121 B Derivation of G (2) 125 C Derivation of the SFG Coincidence State 127

9 CONTENTS 1 D Derivation of G (1) 131 E K with Matrix Functions 133 Acknowledgements 133 Declaration 135 Curriculum Vitae 137 List of Publications 139

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11 1 Introduction Classical electrodynamics is governed by Maxwell s equations and is capable to explain a wide variety of optical effects in particular the interference behaviour of electromagnetic waves. Nevertheless, the theory failed to answer the fundamental questions which gave rise to the birth and development of quantum mechanics. The radiation of a black-body and later, Einstein s explanation of the photoelectric effect enforced the concept of the photon. It was Dirac s rigorous quantization of the electromagnetic field who finally identified the photon as the quantum particle of light [1]. The invention and extended study of the laser at the beginning of the 1960s then allowed to experimentally establish the branch of optics which studies the non-classical phenomena of light - quantum optics. However, it seems to be inherent to quantum mechanics to predict results which are not compatible with the macroscopic phenomena we are faced with in our daily lives. One of the most astonishing features of quantum theory is entanglement. The notion of entanglement has emerged in 1935 by a Gedankenexperiment of Einstein, Podolsky, and Rosen (EPR) in which they cast doubt on the completeness of quantum theory [2]. Their scepsis about quantum mechanics being a complete theory stems from a particular form of a quantum state EPR introduced to describe a system of two-particles. This so-called EPR state has the peculiar feature that it describes the two-particle system as a whole and cannot be factorized into a product of states of the individual subsystems. This implies that the measurement on one particle instantaneously determines the outcome of a measurement on the second particle with absolute certainty. Moreover, the state itself sets no limit on the space-like separation between both particles to show up these strong correlations. In the same year Schrödinger, denoted these correlations with the term entanglement [3]. To argue that quantum mechanics is in fact an incomplete theory, EPR assumed the principle of locality which states that no matter what operation on one particle is made it does not influence the other particle. However, the EPR state allows to predict with certainty the outcome of the second particle as soon as the first particle has been measured. Since there should be no action at a distance, EPR concluded that the second particle must carry its predetermined outcome already as a specific property before it was measured. Because the EPR state does not include these properties, in the view of the authors, quantum mechanics has to be considered as incomplete and must be endowed with local hidden variables (LHV) to uphold locality. In 1964, J. S. Bell referred to the work of EPR and derived an inequality based on classical correlations which satisfies the principle of locality and includes LHV [4]. Bell further proved that the statistical predictions of quantum mechanics violate this inequality if they are determined by means of an entangled state. While Bell derived his inequality for spin 1/2 particles, the first reliable experimental violation of Bell s inequality was performed with polarization entangled photons generated by a two-level cascading process in calcium atoms [5]. Aspect et. al measured a violation of the CHSH theorem [6], a generalized version of Bell s result, by nine standard deviations and found good agreement with the quantum mechanical predictions. As a consequence, quantum theory is in fact a non-local theory and no LHV must be attributed to the involved particles. Since then, much effort has been put into closing the detection and locality loophole which both can cause misleading results in Bell-test measurements [7]. The strong robustness against decoherence in photonic entangled states 3

12 4 CHAPTER 1. INTRODUCTION was demonstrated in a CHSH inequality violation over a free-space distance of 144 km [8]. Apart from touching fundamental issues of quantum mechanics, entanglement has paved the way to potentially practical applications. Entanglement can occur between various systems [9, 10]. It is, however, the high resistance against decoherence that makes entangled photon states suitable for many promising tasks in applied research. The non-locality of entangled two-photon states can for instance be exploited in imaging experiments. Ghost imaging creates a spatially resolved image of an object carried by a photon which did not interact with the object itself [11, 12]. Quantum optical coherence tomography makes use of broadband entangled photons to enhance classical optical coherence tomography in view of resolution, visibility, and dispersion cancellation [13, 14]. In the same time period it has been theoretically proposed by Boto et. al that an entangled N-photon source at wavelength λ can obtain the same imaging resolution as a classical light source at wavelength λ/n [15, 16]. This proposal stimulated the research field of quantum lithography which today still owes a convincing laboratory demonstration since no lithographic material seems to be available which shows enough sensitivity for entangled N-photon absorption [17]. Entangled photon states can further be used to perform microscopy at reduced light intensity since the cross section for two-photon absorption is under certain conditions higher than for classical light [18 20]. Today, however, the most promising field in which entanglement plays a key role merges quantum physics and information technology - quantum information. Unlike the storage of classical information, which takes place in a system with a well-defined state 0 or 1, quantum information (QI) is encoded in the superposition of a quantum state. The smallest memory unit in QI is the qubit, a discrete linear combination of 0 and 1 [21]. The research field of QI has been launched by Bennett and Brassard in 1984 who first solved a fundamental issue of classical cryptography. In a conventional cryptography system, an eavesdropper can in principle always unnoticeably have access to a private key which is exchanged between two parties to decipher a secret message. The first quantum key distribution (QKD) protocol, labelled as BB84, prevents the eavesdropper from being undetected due to its disturbance on the key which is now encoded by means of qubits in a quantum channel [22]. An extension of the BB84 protocol was introduced by Ekert where an eavesdropper intervention can be detected by means of Bell-test measurements with entangled qubits [23]. Many of today s experimental realizations of QKD systems use, however, attenuated laser pulses instead of entangled two-photons sources [24]. Since photons can be distributed over long distances using optical fibres, they are preferably used to implement QKD protocols. Quantum information, at least in principle, has further shown to exhibit the potential to overcome the limits of classical computing for specific algorithms in terms of computational speed [25 28]. The most prominent example is Shor s algorithm which shows that a quantum computer could tackle the problem of factorizing large integers exponentially faster than a classical machine. Entanglement between qubits can occur during the computational process and is responsible for the enhanced speed-up [29]. The main challenge in building a quantum computer is, however, to protect the qubits from interactions with the environment which leads to decoherence in the quantum superposition. A variety of physical systems are actually under study as possible candidates for qubits in a quantum computer [30]. Although the mutual interaction of photons is too weak for direct gate operations, a scheme proposed by Knill, Laflamme, and Milburn allows for scalable quantum computing based on quantum interference with single photons [31]. It is a controversial debate to which extent quantum computers have already been built. The Canadian company D-Wave Systems, Inc. claimed to have constructed the first commercially available computer based on the laws of quantum mechanics [32]. Their D-Wave Two system comprises a chip of 512 superconducting qubits and exploits quantum annealing to speed up some computational processes. On the other hand, the D-Wave machine is not able to perform Shor s algorithm. There are ongoing research activities which investigate if the D-Wave computer in fact performs faster than a classical computer [33]. The previous paragraph has introduced the qubit as a basic ingredient in the context of QI processing tasks. However, in the past years it became apparent that fundamental tests of quantum mechanics and applications like QKD greatly benefit from the extension of qubits tod-dimensional states (qudits). A richer insight into the

13 5 nature of quantum correlations can be obtained by means of generalized Bell inequalities which allow to test the non-local properties of entangled qudits [34]. These inequalities have the property to be more robust against noise than their two-dimensional predecessors. At first sight, it seems to be intuitive that a maximally entangled state leads to the highest violation of a Bell inequality. However, for entangled qudits with d > 2 there exists theoretical evidence that non-maximally entangled qudits reach higher violations of a Bell-type inequality than maximally entangled states [35, 36]. To carry out reliable Bell measurements is albeit a challenge since real experiments always suffer from detection efficiencies smaller than one. This could lead to a Bell inequality violation which is nevertheless explainable by means of a LHV theory. Eberhard showed that a so called detection loophole free CHSH-test using entangled photons can be performed for a maximally entangled qubit and a detection efficiency of at least 82.8%. This efficiency can be lowered to 66.7% if a non-maximally entangled qubit is taken into account [37]. In Reference [38], it was proven that an efficiency of at least 61.8% is sufficient to perform a detection loophole free Bell measurement for entangled ququarts (d = 4). Furthermore, Sheridan and Scarani emphasize the importance of entangled d-dimensional quantum states for QKD [39]. The authors demonstrate that the secret key rate and the robustness against noise increases for QKD protocols which bear on qudits with d > 2. The importance of high-dimensional entangled states also slowly deeps into the area of quantum computing as recent publications demonstrate. On a pure theoretical level they show ad-dimensional extension of the quantum Fourier transformation [40] and enhanced efficiency in Grover s search algorithm involving qudits [41]. In many experiments photons have demonstrated their resistance against decoherence under standard experimental conditions. No cryogenic cooling is of need to uphold the coherence of photonic quantum states. Photons are thus an ideal choice to realize entangled qudits. It is nowadays a standard procedure to generate entangled photons using the process of spontaneous parametric down-conversion (SPDC) in a nonlinear crystal [42]. Based on conserved quantities, entanglement can occur in every degree of freedom a photon owns. Under suitable conditions the coherent interaction of SPDC can lead to entangled states in the finite dimensional Hilbert space of polarization. To go beyond qubits entanglement in infinite Hilbert spaces is achieved for transverse (momentum) or orbital angular momentum modes [43 48]. Entanglement in a continuous degree of freedom can also be realized in the frequency modes of the involved photons (energy-time entanglement) [49]. Depending on the spectral bandwidth of the pump laser used in the SPDC arrangement, the amount of entanglement in an energy-time entangled state can be substantially higher than for transverse (momentum) or orbital angular momentum degrees of freedom. This is particularly the case if a quasi-monochromatic pump field is used to induce SPDC [50]. Thus energy-time entanglement can provide a large resource to generate high-dimensional entangled qudits as we will show. In this thesis we demonstrate the creation, characterization, and manipulation of energy-time entangled qudits by discretizing the spectrum of broadband down-converted photons with a spatial light modulator. The thesis opens with a theoretical description of the generation, propagation, and detection of a type-0 entangled two-photon state (Chapter 2). Afterwards, the spectral tuning curve of the state effectively used in our experiments is studied numerically and experimentally for various parameters of the SPDC process (Chapter 3). The amount of entanglement in the two-photon state is then investigated with analytical and numerical methods by calculating the Schmidt number and the entropy in the long pump pulse regime (Chapter 4). After presenting the experimental setup (Chapter 5), we demonstrate the functionality of the spatial light modulator by measuring an interferometric autocorrelation function through amplitude and phase shaping in the spectral domain of the entangled photons (Chapter 6). The last chapter is devoted to quantum information (Chapter 7). We first demonstrate the ability of the spatial light modulator to discretize the spectrum of the entangled photons into various bases to encode qudits. By means of frequency bins we subsequently show quantum state tomography of entangled qudits up to d = 4 and measure a d-dimensional Bell parameter for qubits and qutrits as a function of their degree of entanglement. The thesis closes with a conclusion and an outlook onto future projects (Chapter 8).

14 6 CHAPTER 1. INTRODUCTION The following publications are in preparation, have been submitted, or are published as a part of this PhD thesis: B. Bessire, C. Bernhard, A. Stefanov, and T. Feurer, Shaper-assisted discretization of energy-time entangled photons, submitted to New Journal of Physics, (2013), arxiv: [quant-ph], (2013). A. Stefanov devised the concept of the experiment. Except the frequency-bin qutrit and ququart, which were measured by C. Bernhard, B. Bessire performed all experiments and did the corresponding data analysis. The article was written by B. Bessire with assistance of all the other authors. C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, These authors contributed equally to this work. Shaping frequency-entangled qudits, Physical Review A 88, , (2013), arxiv: [quant-ph], (2013). A. Stefanov devised the concept of the experiment. B. Bessire performed the qubit and qutrit measurement by means of a software implied by C. Bernhard who also did the maximum likelihood reconstructions. B. Bessire performed and evaluated all Bell-test measurements and wrote parts of the article. T. P. Wihler, B. Bessire, and A. Stefanov, Computing the Entropy of a Large Matrix, to be submitted, arxiv: [quant-ph] (2013). A. Stefanov and B. Bessire stimulated the collaboration with the Mathematics Institute of the University of Bern. B. Bessire generated the large density matrix and wrote the quantum optics section of the article. All the beautiful calculations were done by T. P. Wihler. S. Lerch, B. Bessire, C. Bernhard, T. Feurer, and A. Stefanov, Tuning curve of type-0 spontaneous parametric down-conversion, Journal of the Optical Society of America B 16, , (2013). The article is based on the Master thesis of S. Lerch which was supervised by B. Bessire with the help of A. Stefanov and T. Feurer. B. Bessire mainly contributed to the theoretical work. References [1] P. A. M. Dirac, Proc. Roy. Soc. A, 114, 243, (1927). [2] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, , (1935). [3] E. Schrödinger, Naturwissenschaften 23, , (1935). [4] J. S. Bell, Physics (Long Island City, N.Y.) 1, , (1964). [5] A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 47, , (1981). [6] J. F. Clauser, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, , (1969). [7] B. G. Christensen et. al, arxiv: [quant-ph], (2013). [8] A. Fedrizzi et. al, Nature Physics 5, , (2009). [9] K. C. Lee et. al, Science 223, 6060, (2011). [10] R. Blatt and D. Wineland, Nature 453, , (2008). [11] T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, Phys. Rev. A 52, R3429-R3432, (1995). [12] P. B. Dixon et. al, Phys. Rev. A 83, (R), (2011). [13] A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. A 65, , (2002).

15 7 [14] M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. Lett. 91, , (2003). [15] A. N. Boto et. al, Phys. Rev. Lett. 85, , (2000). [16] M. D Angelo, M. V. Chekhova, and Y. Shih, Phys. Rev. Lett. 87, , (2001). [17] R. W. Boyd and J. P. Dowling, Quantum Inf Process 11, , (2012). [18] H. B. Fei, B. M. Jost, S. Popescu, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 78, , (1997). [19] B. D. Dayan, A. Pe er, A. A. Friesem, and Y. Silbergerg, Phys. Rev. Lett. 93, 23005, (2004). [20] D. Lee and T. Goodson, Journ. of Phys. Chem. B 110, , (2006). [21] B. Schumacher, Phys. Rev. A 51, , (1989). [22] Ch. H. Bennett and G. Brassard, Quantum cryptography: public key distribution and coin tossing. Int. conf. Computers, Systems & Signal Processing, India, 10-12, , (1984). [23] A. K. Ekert, Phys. Rev. Lett. 67, , (1991). [24] D. Stucki, N. Brunner, N. Gisin, V. Scarani, and H. Zbinden, App. Phys. Lett. 87, , (2005). [25] D. Deutsch, Proc. R. Soc. Lond. A 400, 97, (1985). [26] P. W. Shor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, Los Alamitos, California, IEEE Computer Society Press, New York, , (1994). [27] L. K. Grover, Phys. Rev. Lett. 79, , (1997). [28] D. S. Simon, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, Los Alamitos, California, IEEE Computer Society Press, New York, , (1994). [29] Y. Shimoni, D. Shapira, and O. Biham, Phys. Rev. A 72, , (2005). [30] T. D. Ladd et. al, Nature 464, 45-53, (2010). [31] E. Knill, R. Laflamme, and G. J. Milburn, Nature 409, 46-52, (2001). [32] [33] S. Boixo et. al, arxiv: [quant-ph], (2013). [34] D. Collins, N. Gisin, N. Linden, S.Massar, and S. Popescu, Phys. Rev. Lett. 88, , (2002). [35] A. Acín, T. Durt, N. Gisin, and J. I. Latorre, Phys. Rev. A 65, , (2002). [36] S. Zohren and R. D. Gill, Phys. Rev. Lett. 100, , (2008). [37] P. H. Eberhard, Phys. Rev. A 47, R747-R750, (1993). [38] T. Vértesi, S. Pironio, and N. Brunner, Phys. Rev. Lett. 104, , (2010). [39] L. Sheridan and V. Scarani, Phys. Rev. A 82, (2010). [40] S. S. Ivanov, H. S. Tonchev, and N. V. Vitanov, Phys. Rev. A 85, , (2012). [41] Y. Cao, S.-G. Peng, C. Zheng, and G.-L. Long, Commun. Theor. Phys. 55, , (2011). [42] D. N. Klyshko, Photon and Nonlinear Optics, Gordon and Breach Science Publishers, New York, (1988).

16 8 CHAPTER 1. INTRODUCTION [43] A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, Nature Physics 7, , (2011). [44] H. Di Lorenzo Pires, C. H. Monken, and M. P. van Exter, Phys. Rev. A 80, , (2009). [45] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, , (2001). [46] M. Agnew, J. Leach, M. McLaren, F. Stef Roux and R. W. Boyd, Phys. Rev. A 84, , (2011). [47] R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, Science 338, 6107, (2012). [48] D. Giovannini, F. M. Miatto, J. Romero, S. M. Barnett, J. P. Woerdman, and M. J. Padgett, New J. Phys. 14, , (2012). [49] C. K. Law, I. A. Walmsley, and J. H. Eberly, Phys. Rev. Lett. 84, , (2000). [50] C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, Physical Review A, 88, , (2013).

17 2 Theoretical Framework This Chapter begins with a review of the theory of a type-0 entangled two-photon state as it is produced by spontaneous parametric down-conversion in a periodically poled nonlinear crystal. The derivation of the state makes use of first-order perturbation theory and allows the classical pump field to be pulsed [1]. We then derive field operators of the entangled photons which are modified by the transfer function of an optical setup. The detection process at the far end of the optical setup is analyzed in two ways. First, we assume that the detected signal can be described by a second-order correlation function. Second, we take into account coincidence detection through sum-frequency generation in a nonlinear crystal and we explicitly calculate the state of the up-converted photon [2]. The signal is then analyzed in terms of a first-order correlation function. The latter method has been established in [3] as an ultrafast coincidence detection process and is also used in our experiments. The last Section shows how the first-order correlation function of the up-converted photon inherits the properties of the detection crystal in comparison with a second-order correlation function of the entangled photons. To not keep the reader in the dark what entanglement means, we establish the theoretical framework with a definition of what an entangled state is [4]. 2.1 Definition of a Bipartite Entangled State We assume a bipartite quantum system composed of two non-interacting and arbitrary spatially separated subsystems A and B with associated Hilbert spaces H A = span{ a, a,...}, (2.1) H B = span{ b, b,...}. (2.2) Each Hilbert space is spanned by orthonormal basis vectors. A composed state Ψ H = H A H B is given by Ψ = f(a,b) a b, (2.3) where implies summation either over discrete or continuous variables a and b. 1 a,b 1 Throughout this thesis we use the abbreviation a b = a b. 9

18 10 CHAPTER 2. THEORETICAL FRAMEWORK The state Ψ is an entangled (non-separable) state if f(a,b) g(a)h(b) Ψ Ψ A Ψ B (2.4) with Ψ A = g(a) a HA and Ψ B = h(b) b HB. (2.5) a b Two methods how to quantify the amount of entanglement in a given state will be discussed in detail in Chapter Generation of a Type-0 Entangled Two-Photon State Type-0 Spontaneous Parametric Down-Conversion Nonlinear interactions in an appropriate medium occur if higher order terms in the polarization P i j,k,l,... χ (1) ij E j +χ (2) ijk E je k +χ (3) ijkl E je k E l +... (2.6) are excited through strong electric fields. Spontaneous parametric down-conversion (SPDC) appears when a nonlinear crystal is pumped by a laser beam strong enough to induce a second-order susceptibility χ (2) ijk process. In this case, a pump photon (p) may be annihilated and two new photons of lower frequencies, the idler (i) and signal (s), are created. We confine the derivation of the entangled two-photon state to the case in which the pump, the idler, and the signal photons have identical polarization (type-0) and thusχ (2) ijk χ(2). We treat here χ (2) as frequency independent since we assume the SPDC process far away from any resonance frequency of the nonlinear material. SPDC is governed by energy conservation and momentum conservation ω p = ω i +ω s (2.7) k p = k i +k s, (2.8) where ω j denotes the angular frequency and k j the wave vector with j {i,s,p}. Figure 2.1 schematically illustrates type-0 SPDC together with its conserved quantities. The idler and signal photon are then entangled Figure 2.1: Illustration of type-0 SPDC. A pump photon gets annihilated in a χ (2) -crystal while two photons, the idler and the signal, are created with lower frequencies. All involved photons are identically polarized. Energy and momentum are conserved. in wave vector (momentum) and frequency (energy-time). If the central frequencies ω cj of the idler, signal, and

19 2.2. GENERATION OF A TYPE-0 ENTANGLED TWO-PHOTON STATE 11 pump photons are related according to ω ci = ω cs = ω cp /2 then the SPDC process is said to be degenerated otherwise non-degenerated. In general, the idler and signal photon exit the crystal non-collinearly, however, under certain experimental conditions both photons may exit the crystal collinearly Quasi-Phase Matching In order to achieve the highest efficiency of the SPDC process momentum has to be conserved or, equivalently, the phase mismatch k z = k iz +k sz k pz along thez-dimension of the crystal has to be minimized. However, in a type-0 alignment, the birefringent property of the down-conversion crystal cannot be used to achieve best the phase matching condition k z = 0. Instead of birefringence, a technique called quasi-phase matching (QPM) can be used to minimize k z by means of a periodically poled crystal [5 7]. In this case, the secondorder susceptibility χ (2) periodically alternates along the z-axis of the SPDC crystal with a poling period G, i.e. χ (2) (z) = χ (2) 0 sign[cos(2πz/g)], (2.9) where χ (2) 0 is related to the effective nonlinearity of the crystal d eff through χ (2) 0 = 2d eff and determines the strength of the nonlinear interaction. Since χ (2) (z) is a periodic and piecewise continuous function of a real G Figure 2.2: Schematic view of a periodically poled nonlinear crystal. The optical axis alternates inz-direction with a poling periodicity G to achieve quasi-phase matching. variable z, it can be represented in a complex Fourier series χ (2) (z) = G m e ikmz, m= k m. = 2πm G. (2.10) The coefficients G m are then calculated to be G m = 1 G G 0 = 2χ(2) 0 π dz χ (2) (z) e ikmz sin(mπ/2) m (2.11) and therefore the susceptibility is expressed as χ (2) (z) = 2χ(2) 0 π m= sin(mπ/2) m e ikmz. (2.12) We consider here only QPM with a uniform poling period. Alternatively, QPM with a linearly chirped poling period can be used to generate ultrabroadband entangled photons [8].

20 12 CHAPTER 2. THEORETICAL FRAMEWORK Type-0 Entangled Two-Photon State In this Section we briefly review the derivation of a type-0 entangled two-photon state. All considerations are mainly based on [1] whereas for mathematical details the reader is referred to Appendix A. The SPDC process is described by the second-order nonlinear interaction Hamiltonian Ĥ int (t) = 2ε 0 3 V 1 d 3 r χ (2) 1 (z) Êi(r,t)Ês(r,t)Êp(r,t), (2.13) whereε 0 denotes the electric permittivity of the vacuum andχ (2) 1 (z) is the nonlinear susceptibility of Eq. (2.9). The three-dimensional integration covers the crystal volume V 1. We assume that the transverse crystal dimensions are much larger than the waist of the pump beam and extend the integration in (x,y) to infinity. Note, that this approximation implies strict conservation of transverse momentum. In z-direction we integrate over the length of the SPDC crystal L 1. From the quantum point of view the pump source can be considered as a coherent state and thus as close to a classical field as possible. We thus treat the pump field as a classical field i.e. Êp E p. The pump field is linearly polarized and has a baseband signal around its central frequency ω cp E p (r,t) = E p (ϱ,z,t) = E + p (ϱ,z,t)+e p (ϱ,z,t) = E + p (ϱ,z,t) e iωcpt + c. c. (2.14) with a slowly varying pulse envelope E + p (ϱ,z,t) = 1 (2π) 3 d 2 q p dω p E + p (q p,ω p ) e i(ωpt kpr) (2.15) and a relative frequency Ω p = ω p ω cp. 2 Further, we introduce ϱ = (x,y) as the transverse part of r and q as the transverse part of k. With the dispersion relation we may write k pz = k p = k p = (ωp k 2 px +k2 py +k2 pz = ω p c n 1(ω p ) (2.16) ) 2 c n 1(ω p ) kpx 2 k2 py = (ωp ) 2 c n 1(ω p ) q 2 p (2.17) such that we implicitly integrate over dk pz by integrating over d 2 q p and dω p. The idler and signal fields are defined as operators acting in individual Hilbert spaces H i and H s. They are represented in Fourier decompositions Ê i (r,t) = Êi(ϱ,z,t) = Ê+ i (ϱ,z,t)+ê i (ϱ,z,t) 1 = (2π) 3 d 2 1 ( ) q i dω i e(ω i )â i (q i,ω i ) e i(ω it k i r) + H. c. n 1 (ω i ) (2.18) and with Ê s (r,t) = Ês(ϱ,z,t) = Ê+ s (ϱ,z,t)+ê s (ϱ,z,t) 1 = (2π) 3 d 2 1 ( ) q s dω s e(ω s )â s (q s,ω s ) e i(ωst ksr) + H. c. n 1 (ω s ) k j = k j = ω j c n 1(ω j ) k jz = (ωj 2 We substitute where no explicit integration boundaries are indicated. (2.19) ) 2 c n 1(ω j ) q 2 j, j {i,s}. (2.20)

21 2.2. GENERATION OF A TYPE-0 ENTANGLED TWO-PHOTON STATE 13 The field operator normalization function e(ω j ) is given by e(ω j ) = i ωj 2(2π) 3, ε 0 c j {i,s}, (2.21) wherecis the speed of light and1/(2π) 3 stems fromv Q (2π) 3, i.e. from the transition of a finite quantization volume V Q to the continuum. 3 In principle the refractive index n 1 of the SPDC crystal is a function of ω andk because the crystal may have a nontrivial symmetry. Here, we neglect the dependence on direction but include dispersion. Also note that the Ê+ j part corresponds to the annihilation operator â j and Ê j to the creation operator â j, which generates some single photon Fock states by acting on the corresponding vacuum state q i,ω i. = 1 qi,1 ωi = â i (q i,ω i ) 0 i and q s,ω s. = 1 qs,1 ωs = â s(q s,ω s ) 0 s. (2.22) Associated with a direct product, the single photon Fock states of Eq. (2.22) span the Hilbert spaceh DC of the SPDC state H DC = H i H s = span{ q i,ω i q s,ω s, q i,ω i q s,ω s,...}. (2.23) The interaction Hamiltonian of Eq. (2.13) can be rewritten by means of E p, Ê i and Ês with the aid of the rotating wave approximation to Ĥ int (t) = 2ε 0 3 V 1 d 3 r χ (2) 1 (z) Ê i (r,t)ê s (r,t)e + p (r,t)+h. c.. (2.24) The formal solution to the temporal evolution of a generic state Ψ(t) is given by time-dependent perturbation theory in the interaction picture t ) Ψ(t) = Texp ((i ) 1 dt Ĥ int (t ) Ψ(t 0 ), (2.25) t 0 where T denotes the time-ordering operator. Since the SPDC process is a weak interaction process in bulk crystals, it is sufficiently accurate to expand Eq. (2.25) up to first order t Ψ(t) =. Ψ(t 0 ) i dt Ĥ int (t ) Ψ(t 0 ). (2.26) t 0 As a special case, we consider the state at t 0 to be the two-photon vacuum state 0 =. 0 i 0 s of H DC. This implies Ψ(t) = 0 i t t 0 dt Ĥ int (t ) 0. (2.27) The integration over t is evaluated under the approximation that the interaction of the involved photons proceeds adiabatically within a finite region of time [10, 11]. The integration boundaries can thus be extended according to t 0 and t which imposes strict energy conservation. As a result, Ψ(t) becomes time independent. The state is then calculated by inserting the explicit expressions for the pump field and the field operators inĥint (Appendix A). The type-0 entangled two-photon state generated by SPDC finally reads Ψ = 0 + d 2 q i dω i d 2 q s dω s Λ(q i,ω i,q s,ω s ) q i,ω i q s,ω s, (2.28) 3 A thorough discussion of the generalization from a discrete to a continuous Fock space can be found in Reference [9].

22 14 CHAPTER 2. THEORETICAL FRAMEWORK with the joint spectral amplitude (JSA) Λ(q i,ω i,q s,ω s ) = 4iε 0χ (2) 1,0 L ( ) 1e(ω i )e(ω s ) 3 π(2π) 6 n 1 (ω i )n 1 (ω s ) E+ p (q kz1 L 1 i +q s,ω i +ω s ω cp ) sinc 2 ( exp i k ) z1l 1 2 (2.29) and the phase mismatch k z1 = (ωi ) 2 c n 1(ω i ) q 2 i + (ωs ) 2 c n 1(ω s ) q 2 s (ωi +ω s c ) 2 n 1 (ω i +ω s ) (q i +q s ) 2 + 2π. G 1 (2.30) Note, that the phase mismatch incorporates first-order quasi-phase matching with m = 1 in the Fourier series of Eq. (2.12). According to the definition in Section 2.1, the two-photon state of Eq. (2.28) is entangled in energy-time (frequency) and transverse momentum if Λ(q i,ω i,q s,ω s ) cannot be separated into a product Λ(q i,ω i,q s,ω s ) = g(q i,ω i )h(q s,ω s ) where the subsystems are assigned A i andb s. The JSA contains all the information about the degree of entanglement between the idler and the signal photon. How to quantify the amount of entanglement is discussed in Chapter 4. The JSA of Eq. (2.29) has been further used to investigate the spectra of the entangled photons under various experimental conditions in Chapter Propagation of a Field Operator through the Optical System In the previous Section we have used the idler and signal field operators formulated in Eq. (2.18) and Eq. (2.19) to derive the two-photon state in the framework of the SPDC process. At this point, these operators acted within a nonlinear material. However, on purpose to calculate a second-order correlation function G (2) (r 1,t 1 ;r 2,t 2 ), we need to propagate the operators from the exit face of the crystal through the optical setup to the space-time points (r 1,t 1 ) and (r 2,t 2 ). In order to do this we first review how a classical field distribution E 0 + (r,t) is altered by an optical system. Provided the system under consideration is linear, E 0 + (r,t) transforms according to E 1 + (r,t) = d 3 r dt E 0 + (r,t ) m(r,r,t,t ) (2.31) with an impulse response functionm(r,r,t,t ) of the optical setup [12]. By inserting the plane wave expansion of E 0 + (r,t ) and the Fourier transform ofm(r,r,t,t ) we find E 1 + (r,t) = 1 (2π) 7 d 2 q dω E 0 + (q,ω) d 3 k 1 dω 1 M(k 1,k,ω 1, ω) e iω 1t+ik 1 r. (2.32) Recall, that k = (q,k z ). In the following we assume that all optical elements are time-stationary filters which yields m(r,r,t,t ) = m(r,r,t t ). (2.33)

23 2.4. SECOND-ORDER CORRELATION FUNCTIONG (2) 15 The transfer function, i.e. the Fourier transform of Eq. (2.33), then reads Inserting Eq. (2.34) into Eq. (2.32) yields M(k,k,ω,ω ) = 2πM(k,k,ω) δ(ω +ω ). (2.34) E + 1 (r,t) = 1 (2π) 3 d 2 q dω E + 0 (q,ω) M(r,q,ω) eiωt, (2.35) where its one-time Fourier transform is E + 1 (r,ω) = 1 (2π) 2 d 2 q E 0 + (q,ω) M(r,q,ω). (2.36) We now apply the classical result of Eq. (2.31) to the idler and signal operators that is Ê + j1 (r,t) = d 3 r dt Ê + j0 (r,t ) m(r,r,t,t ). (2.37) That quantum field operators transform in the same manner as classical fields is justified in Reference [13]. By again inserting the plane wave expansion of Ê+ 0j (r,t ) and the two-times Fourier transform M(r,k,ω) we find Ê + j1 (r,t) = 1 (2π) 3 d 2 q j dω j e(ω j )â j (q j,ω j ) M j (r,q j,ω j ) e iω jt (2.38) with the field operator normalization function e(ω j ) given by e(ω j ) = i ωj 2(2π) 3, ε 0 c j {i,s}. (2.39) Since the field operators are assumed to propagate in free space we set n(ω j ) = 1 in Eq. (2.38). The negative frequency part of Eq. (2.38) is given by Ê j1 (r,t) = (Ê+ j1 (r,t)). (2.40) 2.4 Second-Order Correlation Function G (2) After having discussed the propagation of a photon field operator from the source to an arbitrary space-time point (r,t), we now evaluate the joint probability per unit (time) 2 that one photon is annihilated (detected) at (r 1,t 1 ) and the other at (r 2,t 2 ) by two ideal single photon detectors in a coincidence circuit. Here, ideal refers to a point-like detector with frequency-independent photon-absorption probability. According to [14], this probability is given by a second-order correlation function G (2) (r 1,t 1 ;r 2,t 2 ) = Ψ Ê s (r 1,t 1 )Ê i (r 2,t 2 )Ê+ i (r 2,t 2 )Ê+ s (r 1,t 1 ) Ψ = ψ i,s (r 1,t 1 ;r 2,t 2 ) 2, (2.41)

24 16 CHAPTER 2. THEORETICAL FRAMEWORK where ψ i,s (r 1,t 1 ;r 2,t 2 ) = 0 Ê+ i (r 2,t 2 )Ê+ s (r 1,t 1 ) Ψ (2.42) is denoted as the two-photon wave function (biphoton) [15]. The state vector Ψ is from Eq. (2.28). Subject to the calculation in Appendix B, the second-order correlation function with propagated idler and signal field operators reads G (2) (ϱ 1,t 1 ;ϱ 2,t 2 ) = d 2 q i dω i d 2 q s dω s η(ω i,ω s ) M i (ϱ 2,q i,ω i ) M s (ϱ 1,q s,ω s ) ( ) E + kz1 L 1 p (q i +q s,ω i +ω s ω cp ) sinc 2 ( exp i k ) z1l 1 2 exp(i(ω i t 2 +ω s t 1 )) 2 (2.43) with η(ω i,ω s ) = 4iε 0χ (2) 1,0 L 1 3 π(2π) 12 e(ω i)e(ω s ) e(ω i)e(ω s ) n 1 (ω i )n 1 (ω s ) (2.44) and the phase mismatch of Eq. (2.30). Note, that Eq. (2.43) is evaluated only in transverse direction, i.e.z = 0, where experimentally, ϱ 1 and ϱ 2 could be thought of as points on the photosensitive area of the first and the second detector. Eq. (2.43) shows a measured signal which is proportional to the Fourier transform of the JSA modified by the optical setup. 2.5 Perturbative Treatment of a SFG Process Instead of using two separated single photon detectors, coincidences of entangled photon pairs may be detected by sum-frequency generation (SFG) in a nonlinear crystal (detection crystal). In analogy to SPDC, the SFG process with entangled photons is sometimes also denoted as up-conversion [2]. Since SFG is the inverse process of SPDC, we again have to take into account the interaction Hamiltonian of order χ (2) 2, however, now integrating over the volume V 2 of the detection crystal Ĥ int (t) = 2ε 0 3 V 2 d 3 r χ (2) 2 (z)êi(r,t)ês(r,t)êc(r,t). (2.45) Analogue to Section 2.2.3, we incorporate first-order quasi-phase matching by χ (2) 2 (z) = 2χ(2) 2,0 π m= sin(mπ/2) m e ikmz (2.46) with m = 1 such that χ (2) 2 (z) = 2χ(2) 2,0 π e i 2π G 2 z. (2.47) With the rotating wave approximation, Eq. (2.45) becomes Ĥ int (t) = 2ε 0 d 3 r χ (2) 2 (z) 3 Ê c (r,t)ê+ s (r,t)ê+ i (r,t)+ H. c., (2.48) V 2

25 2.5. PERTURBATIVE TREATMENT OF A SFG PROCESS 17 where theê c Ê+ s Ê+ i term is responsible for the SFG process. The operator valued coincidence (c) photon field is given by Ê c (r,t) = Ê+ c (ϱ,z,t)+ê c (ϱ,z,t) 1 = (2π) 3 d 2 1 ( ) q c dω c e(ω c )â c (q c,ω c ) e i(ωct kcr) + H. c. n 2 (ω c ) (2.49) with a dispersion relation k c = k c = ω c c n 2(ω c ) (2.50) and a refractive index of the detection crystal denoted by n 2. Again, we identify the normalization function e(ω c ) = i ωc 2ε 0 c(2π) 3 (2.51) which guarantees the correct dimension of the field operator. Since the SFG process is governed by a combination of three annihilation and creation operators, we have to extend the Hilbert space H DC of Section with a subspace H c to obtain H SFG = H i H }{{} s H c = span{ q i,ω i q s,ω s q c,ω c, q i,ω i q s,ω s q c,ω c,...}. (2.52) =H DC, The commutation and orthogonality relations listed in Appendix A still hold, however, now with l, m {i, s, c}. The extended type-0 entangled photon state of Eq. (2.28) then reads Ψ = 0 + d 2 q i dω i d 2 q s dω s Λ(q i,ω i,q s,ω s ) q i,ω i q s,ω s 0 c (2.53) with the vacuum state now defined as 0. = 0 i 0 s 0 c. Because of the low SFG efficiency the incoming state Ψ is only weakly disturbed such that we are allowed to perform first-order perturbation theory to calculate the coincidence photon state denoted by Θ. The general formalism then specifies to Θ = Ψ i dt Ĥint(t) Ψ, (2.54) where Ψ is the state of Eq. (2.53). A derivation of the coincidence photon state taking into account the propagated field operators of Eq. (2.38) is carried out in Appendix C. There the state Θ is calculated to be Θ = Ψ + d 2 q c d 2 q s dω s d 2 q i dω i Ξ(q c,q s,ω s,q i,ω i ) q c,ω s +ω i 0 s 0 i. (2.55) The spectral amplitude of the coincidence photon is given by Ξ(q c,q s,ω s,q i,ω i ) = m(q c,q s,ω s,q i,ω i ) Γ(q s,ω s,q i,ω i ), (2.56) and is composed of

26 18 CHAPTER 2. THEORETICAL FRAMEWORK m(q c,q s,ω s,q i,ω i ) = d 2 ϱ M s (ϱ,q s,ω s ) M i (ϱ,q i,ω i ) e iqcϱ, (2.57) ( Γ(q s,ω s,q i,ω i ) = ) 2 χ (2) 1,0 χ(2) 4ε 0 3 π(2π) 7 2,0 L e(ω s +ω i ) e(ω s )e(ω i ) e(ω s )e(ω i ) 1L 2 n 2 (ω s +ω i ) n 2 (ω s )n 2 (ω i ) n 1 (ω s )n 1 (ω i ) ( ) ( ) E + kz1 L 1 kz2 L 2 p (q s +q i,ω s +ω i ω cp ) sinc sinc 2 2 ( exp i k ) ( z1l 1 exp i k ) z2l 2. (2.58) 2 2 The phase mismatch of the first and the second crystal reads k z1 = k z2 = (ωi ) 2 c n 1(ω i ) q 2 (ωs +ω i c i + (ωs n 2 (ω s +ω i )) 2 q 2 c ) 2 (ωi ) c n 1(ω s ) q 2 +ω 2 s s n 1 (ω i +ω s ) (q i +q s ) c 2 + 2π, G 1 (2.59) (ωs ) 2 (ωi ) 2 c n 2(ω s ) q 2 s c n 2(ω i ) q 2 i 2π. (2.60) G First-Order Correlation Function G (1) The coincidence photon is detected by a single photon counter at space-time point (r,t). Again, we assume the photon in Θ to be annihilated by an ideal detector. The probability per unit time that this process occurs is given by a first-order correlation function where G (1) (r,t) = Θ Ê c (r,t)ê+ c (r,t) Θ (2.61) = ψ c (r,t) 2, ψ c (r,t) = 0 Ê+ c (r,t) Θ (2.62) is the wave function of the coincidence photon [14]. Similar to Section 2.4 we consider G (1) only dependent on transverse coordinates ϱ. In Appendix D, the calculation of Eq. (2.61) is performed and we obtain G (1) (ϱ,t) = d 2 q c d 2 q s dω s d 2 q i dω i η(ω s,ω i ) E + p (q s +q i,ω s +ω i ω cp ) ( ) ( ) kz1 L 1 kz2 L 2 m(q c,q s,ω s,q i,ω i ) sinc sinc 2 2 ( exp i k ) ( z1l 1 exp i k ) z2l 2 2 exp(i(ω s +ω i )t iq c ϱ) 2 2, (2.63) ( η(ω s,ω i ) = 4ε 0 3 π(2π) 7 ) 2 χ (2) 1,0 χ(2) 2,0 L 1L 2 (2π) 3 e(ω s +ω i ) e(ω s +ω i ) n 2 (ω s +ω i ) e(ω s )e(ω i ) e(ω s )e(ω i ) n 2 (ω s )n 2 (ω i ) n 1 (ω s )n 1 (ω i )

27 2.7. HEURISTIC COMPARISON BETWEENG (1) ANDG (2) 19 with m(q c,q s,ω s,q i,ω i ) = d 2 ϱ M s (ϱ,q s,ω s ) M i (ϱ,q i,ω i ) e iqcϱ. (2.64) The phase mismatches k z1 and k z2 are given by Eq. (2.59) and Eq. (2.60). 2.7 Heuristic Comparison between G (1) and G (2) In the experimental setup described in Chapter 5, coincidences are detected by sending the photon pairs into a suitable nonlinear crystal and by measuring SFG photons generated in that crystal by a subsequent single photon counter. An idler-signal pair converts into a coincidence photon in the detection crystal if the time delay between the two constituents is of the order of s and if the spatial separation in transverse direction lies within the entanglement area of the photon pair. From an experimental point of view the first-order correlation function G (1) seems to be the appropriate description of the coincidence signal. On the other hand, there has been a variety of publications where SFG in a nonlinear crystal is described by a second-order correlation function [3, 16 18]. While the first approach takes into account specific properties of the SFG process, the latter assumes a perfect crystal with infinite acceptance bandwidth. The aim of this Section is to compare the two possible formalisms and to explicitly point out how the properties of the detection crystal are incorporated in G (1). Generally speaking, G (2) represents the correlation of two time- and space-separated photons measured by two point-like ideal detectors at (r 1,t 1 ) and (r 2,t 2 ). We assume these detectors to have a quantum efficiency one. In order to approximate best our detection method, we set t 1 = t 2 = t and ϱ 1 = ϱ 2 = ϱ. To simplify the discussion we use non-propagated field operators but the statements below also hold for propagated field operators. The second-order correlation function then reads G (2) (ϱ,t) = 4iε 0 χ (2) 1,0 L 1 3 π(2π) 12 d 2 q i dω i d 2 q s dω s e(ω i )e(ω s ) e(ω i)e(ω s ) n 1 (ω i )n 1 (ω s ) ( ) E + kz1 L 1 p (q i +q s,ω i +ω s ω cp ) sinc 2 ( exp i k ) z1l 1 2 exp(i{(ω i +ω s )t (q i +q s )ϱ}) 2, (2.65) where we explicitly inserted η(ω i,ω s ) according to Eq. (2.44). The first-order correlation function G (1) evaluated at a single space-time point (r,t) is a measure of the coincidence photons generated. Since G (1) formally takes into account the SFG process through the use of Θ, the first-order correlation function includes information of the detection crystal and is therefore more closely related to our experiment. With non-propagated field operators G (1) reads

28 20 CHAPTER 2. THEORETICAL FRAMEWORK G (1) (ϱ,t) = 4iε 0χ (2) 1,0 L 1 4iε 0χ (2) 2,0 L 2 3 π(2π) 12 3 π(2π) 3 d 2 q s dω s d 2 e(ω s )e(ω i ) q i dω i e(ω s +ω i ) n 1 (ω s )n 1 (ω i ) e(ω ( ) ( ) s +ω i ) e(ω s )e(ω i ) kz1 L 1 kz2 L 2 n 2 (ω s +ω i ) n 2 (ω s )n 2 (ω i ) E+ p (q s +q i,ω s +ω i ω cp )sinc sinc 2 2 ( exp i k ) ( z1l 1 exp i k ) z2l 2 2 exp(i{(ω i +ω s )t (q i +q s )ϱ}) 2 2. (2.66) Comparing Eq. (2.66) with Eq. (2.65) reveals three distinct differences. Firstly, the term e(ω s +ω i ) e(ω s )e(ω i ) n 2 (ω s +ω i ) n 2 (ω s )n 2 (ω i ), (2.67) which accounts for normalization and the dispersion properties of the detection crystal. Secondly, the term 4iε 0χ (2) 2,0 L π(2π) 3 = 4ε 0χ (2) 2,0 L π(2π) 3, (2.68) where χ (2) 2,0 R and, thirdly, an integration over sinc( k z2l 2 /2)exp(i k z2 L 2 /2). The latter two terms relate to the SFG process and include its strength, its spectral acceptance bandwidth and its phase mismatch conditions, respectively. Section 5.3 demonstrates on the basis of a simple measurement how accurate the properties of the detection crystal are inherited in Eq. (2.66). In conclusion, we find that the two correlation functions are related with respect to physical interpretation. That is to say, the second-order correlation function is a good description of a SFG coincidence signal if the phase matching conditions of the up-conversion process are adjusted accordingly. This statement has been experimentally verified within a certain range of experimental parameters in Reference [19]. References [1] T. E. Keller and M. H. Rubin, Phys. Rev. A 56, , (1997). [2] K. A. O Donnell and A. B. U Ren, Phys. Rev. Lett. 103, , (2009). [3] B. Dayan, A. Pe er, A. A. Friesem, and Y. Silberberg, Phys. Rev. Lett. 94, , (2005). [4] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, , (2009). [5] R. W. Boyd, Nonlinear Optics, Elsevier, Amsterdam, (2008). [6] K. Bencheikh, E. Huntzinger, and J. A. Levenson, J. Opt. Soc. Am. B 12, , (1995). [7] J. Svozilík and J. Perina, Phys. Rev. A 80, 1-9, (2009). [8] M. B. Nasr et al., Phys. Rev. Lett. 18, , (2008). [9] W. Vogel and D. G. Welsch, Lectures on Quantum Optics, p , Akademie Verlag, Berlin (1994).

29 2.7. HEURISTIC COMPARISON BETWEENG (1) ANDG (2) 21 [10] D. N. Klyshko, Photon and Nonlinear Optics, Gordon and Breach Science Publishers, New York, (1988). [11] E. Rebhan, Theoretische Physik II, Elsevier, Spektrum Akademischer Verlag, München, (2005). [12] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, John Wiley & Sons, New Jersey, 2nd Edition, (2007). [13] Y. Shih, An introduction to Quantum Optics, Taylor and Francis, Florida, (2011). [14] R. J. Glauber, Phys. Rev. 130, , (1963). [15] B. E. A. Saleh, A. F. Abourraddy, A. V. Sergienko, and M. C. Teich, Phys. Rev. A 62, , (2000). [16] A. Pe er, B. Dayan, A. A. Friesem, and Y. Silberberg, Phys. Rev. Lett. 94, , (2005). [17] B. Dayan, Phys. Rev. A 76, , (2007). [18] F. Zäh, M. Halder, and T. Feurer, Opt. Express 16, 16452, (2008). [19] C. Bernhard, Shaping of Energy-Time Entangled Photons, PhD Thesis, (2013).

30

31 3 Tuning Curve of Type-0 SPDC In this Chapter we present a detailed numerical and experimental study of the tuning curve obtained by type-0 SPDC in a periodically poled KTiOPO 4 (PPKTP) crystal. The spectrum of the entangled photons is investigated in view of various experimental parameters namely the pump beam waist, the poling period, the length, and the temperature of the SPDC source. The aim of this study is to experimentally and theoretically characterize the spectrum of the SPDC source latter used in the experiment within the context of quantum information theory (Chapter 7). 3.1 Formal Prerequisites Since the SPDC process examined in Section 3.2 is induced trough a quasi-monochromatic pump field with a spectral bandwidth of about 5 MHz, it is valid to approximate E + p with E + p (q i +q s,ω i +ω s ω cp ) E + p (q i +q s ) δ(ω i +ω s ω cp ). (3.1) The corresponding two-photon state of Eq. (2.28) then reads Ψ = 0 + d 2 q i d 2 q s dω s Λ(q i,ω cp ω s,q s,ω s ) q i,ω cp ω s q s,ω s, (3.2) with Λ(q i,ω cp ω s,q s,ω s ) e(ω cp ω s )e(ω s ) n(ω cp ω s )n(ω s ) E+ p (q i +q s ) sinc ( kz L 2 ), (3.3) where we omit the additional phase factor because it will cancel anyhow in the spectral density of Eq. (3.11). Note, that the subindex 1 which distinguished the SPDC crystal from the detection crystal in Chapter 2 is dropped since only one crystal is involved in the following experiment. In Section 3.2 it will be shown that the temperature T of the down-conversion crystal is a relevant parameter for the spectrum of the entangled photons. To account for T in Eq. (3.3), we explicitly introduce a temperature dependency in the length, the poling period and the dispersion properties of the source, i.e. L L(T), G G(T), n(ω j ) n(ω j,t). With this modifications andω cp ω p, Eq. (3.3) corresponds to Eq. (3.5) in Section 3.2 which is used throughout the publication to simulate the spectral content of a type-0 entangled two-photon state. 23

32 24 CHAPTER 3. TUNING CURVE OF TYPE-0 SPDC 3.2 Publication: Tuning Curve of Type-0 Spontaneous Parametric Down-Conversion This work has been published in the Journal of the Optical Society of America B 16, , (2013): Tuning Curve of Type-0 Spontaneous Parametric Down-Conversion Stefan Lerch, Bänz Bessire, Christof Bernhard, Thomas Feurer, and André Stefanov University of Bern, Institute of Applied Physics, Sidlerstr. 5, 3012 Bern, Switzerland Corresponding author: stefan.lerch@iap.unibe.ch Received November 20, 2012; revised January 7, 2013; accepted February 21, 2013; posted February 21, 2013 (Doc. ID ); published March 14, 2013 Abstract We study the tuning curve of entangled photons generated by type-0 spontaneous parametric down-conversion in a periodically poled KTP crystal. We demonstrate the X-shaped spatiotemporal structure of the spectrum by means of measurements and numerical simulations. Experiments for different pump waists, crystal temperatures, and crystal lengths are in good agreement with numerical simulations. OCIS codes: , , Introduction Entangled photons are a primary source for studying the properties of entanglement [1]. They allow to perform fundamental tests of quantum mechanics [2, 3] and to implement protocols which could not be realized within the framework of classical physics. These cover quantum cryptography [4] and more generally, quantum communication [5] and quantum computing [6]. Applications like quantum imaging [7] or quantum optical coherence tomography [8] revealed the potential of entangled photons to exceed classical resolution limits. The most common sources for entangled two-photon states are based on spontaneous parametric down-conversion (SPDC) occurring in nonlinear crystals [9]. This process allows to create entanglement with respect to polarization [10], angular orbital momentum [11], momentum [12] or energy [13]. Energy entanglement appears due to strong quantum correlations between the frequencies of the individual photons in a pair. Some applications require the spectrum of the individual photons to be broad [14]. For instance, a broad spectrum allows to achieve a high axial resolution in quantum optical coherence tomography [15] and the temporal properties of entangled photons can be measured with a spatial light modulator [16, 17]. Furthermore, the maximal flux of down-converted photons in the single photon regime is limited by the bandwidth of the entangled photon spectrum [18]. In general, the interplay between the energy and momentum degrees of freedom within a two-photon state shows a complex spatiotemporal structure of entanglement [19, 20]. Therefore, a detailed understanding of the SPDC process, as well as an accurate characterization of the down-converted spectrum is important. The joint spectrum of photon pairs has been measured for type-i [21,22] and for type-ii SPDC [23,24]. The type-i SPDC spectrum as a function of the pump waist is investigated in [25] and its transverse momentum dependency has been measured in the high gain regime [26 28]. Coherence properties of the down-converted beam are investigated in [29, 30].

33 3.2. PUBLICATION: TUNING CURVE OF TYPE-0 SPONTANEOUS PARAMETRIC DOWN-CONVERSION 25 In this letter we study both theoretically and experimentally the full dependency of the type-0 SPDC spectrum on the transverse momentum [31] for a monochromatic pump focused into a periodically poled crystal in the single photon regime. In the first part we theoretically describe the spectrum generated by SPDC and we identify in particular four parameters which determine the spectrum. We present an experimental setup which allows to measure the tuning curves, i.e. the transverse momentum and frequency dependency of the SPDC emission, with respect to three of these parameters. Finally, we compare experimental results with numerical simulations Theory Quasi-phase-matched (QPM) parametric amplification has been computed in [32]. In the low gain regime the entangled two-photon state can equivalently calculated by first order perturbation theory as in [33] where the generation of entangled photon pairs in periodically poled crystals has been theoretically studied. We consider SPDC induced by an undepleted monochromatic pump beam of angular frequency ω p with a transverse field distribution E + p (q p), which propagates along the z-axis of a crystal with transverse momentum q p of the wave vector k p = (q p,k p,z ). The pump photon (p) is down-converted into the idler (i) and signal (s) photon with frequency ω i = ω p ω s and ω s, respectively. All involved photons are identically polarized, i.e. in type-0 configuration. Adapting the derivation of the entangled two-photon state in [34] for QPM type-0 configuration with finite crystal length yields to Ψ = 0 + d 2 q i d 2 q s dω s Λ(q i,ω p ω s,q s,ω s )â i (q i,ω p ω s )â s (q s,ω s ) 0, (3.4) ( ) Λ(q i,ω p ω s,q s,ω s ) = 2iɛ 0χ (2) 0 L(T)e(ω p ω s )e(ω s ) k z + 2π 3 (2π) 5 n(ω p ω s,t)n(ω s,t) E+ G(T) L(T) p (q i +q s )sinc, (3.5) 2 where 0 abbreviates the combined vacuum state 0 =. 0 i 0 s. The photon creation operator is denoted by â j (q j,ω j ) andλ(q i,ω p ω s,q s,ω s ) is the temperature dependent spectral amplitude function. Apart from the vacuum permittivity ɛ 0 and the reduced Planck constant, the strength of the SPDC process is governed by the second order susceptibility χ (2) 0 and the length L(T) of the nonlinear crystal. Dispersion properties are included through a frequency and temperature dependent refractive indexn(ω j,t) where we neglect its transverse wave vector dependencies. The phase mismatch k z = (ωp ω 2 s n(ω p ω s,t)) q 2 i c (ωs ) 2 + c n(ω s,t) q 2 s (ωp ) 2 c n(ω p,t) (q i +q s ) 2, (3.6) is compensated by a judicious choice of the poling period G(T) such that The normalization function k z + 2π G(T) = 0. (3.7)

34 26 CHAPTER 3. TUNING CURVE OF TYPE-0 SPDC originates from the field operator e(ω j ) = i ωj 2(2π) 3 ɛ 0 c, (3.8) Ê j (q j,ω j,z) = Ê+ j (q j,ω j,z)+ê j (q j,ω j,z) = e(ω j )â j (q j,ω j )e ik j,zz +h.c., (3.9) wherecis the speed of light in vacuum. The symmetry inλ(q i,ω p ω s,q s,ω s ) with respect toω p ω s ω s andq i q s causes the indistinguishability of signal and idler spectra. Therefore, the total spectrum measured by a spectrometer can be considered to be proportional to the signal spectrum. Glauber s first order correlation function for the signal photon without the normalization function in the field operators [35] S(q s,ω s,z) = G (1) s (q s,ω s,z) = Tr{ˆρ s Ê s,z(q s,ω s,z)ê+ s,z(q s,ω s,z)} (3.10) is proportional to the probability of measuring a photon with q s and ω s on a plane at the position z, i.e. a momentum dependent spectral photon count density, henceforth referred to as spectral density at z. The operator ˆρ s = Tr i {ˆρ} is the partial trace over the idler subsystem of the density operator ˆρ = Ψ Ψ. The assumption of a monochromatic pump beam and the field operator in Eq. (3.9) without normalization function yield a z-independent spectral density S(q s,ω s ) = d 2 q i Λ(q i,ω p ω s,q s,ω s ) 2. (3.11) For a Gaussian beam given by E + p (q i +q s ) exp ( w2 0 (q i +q s ) 2 4 ), (3.12) the spectral density depends only on the absolute value of q s. The poling period G, pump beam waist w 0, temperature T and crystal length L are the relevant parameters for the shape of S(q s,ω s ). As reported in [36] the temperature affects not only the refractive index but also G(T) and L(T) by thermal expansion G(T) = G 0 [ 1+α(T 25 C)+β(T 25 C) 2] with coefficients α andβ taken from [37]. This equation applies also for L(T). All numerically simulated tuning curves are done by calculating Eq. (3.11) for a given variable set (q s,ω s ). The integration boundaries forq i are limited to an empirically found range where the phase matching condition is fulfilled Experimental Setup The setup used to measure the spectral density of Eq. (3.11) is depicted in Fig The pump laser is a quasimonochromatic Nd:YVO 4 (Verdi) with central wavelengthλ p = 532 nm and a spectral bandwidth of5 MHz. A pump power of 5 W is focused by a lensl 1 with eitherf 1 = 300 mm orf 1 = 150 mm into a periodically poled

35 3.2. PUBLICATION: TUNING CURVE OF TYPE-0 SPONTANEOUS PARAMETRIC DOWN-CONVERSION 27 potassium titanyl phosphate (PPKTP) crystal where SPDC generates entangled photons centered around λ c = 1064 nm. The down-converted photon power is measured to be linear to the pump power and around P DC = 400 nw (parameter dependent) which correspond to a flux ofφ DC = photons/s. According to [18] the maximal flux of down-converted photons Φ max that can still be considered as composed of distinct photon pairs is approximately the down-converted bandwidth DC which is in our case greater than 50 nm. This leads to an experimental spectral mode density n DC = Φ DC /Φ max < 0.16 which confirms the single photon limit and legitimate the quantum mechanical description of our experiment. The parameterw 0 can be varied by changing the focal lengthf 1. The SPDC source is mounted in a copper block whose temperature is controlled to ±0.1 C. Heat transfer between copper and PPKTP is ensured by wrapping the crystal with an indium foil. Due to the low absorption coefficient of KTP at λ p, the temperature gradient between the location of the absorbed pump power within the crystal and the crystal surface is negligible as simulations have shown. Therefore, the measured parameter T of the copper block can be considered to be equal to the effective temperature of the crystal. Two different crystals are used to analyze the effect on Eq. (3.11) of different lengths L. The pump beam is filtered out by two dichroic mirrors (DM1&2) oriented such that the beam displacement is compensated. The wave vector-dependent spectrum is measured by raster-scanning a multi-mode fiber along the x-axis of a 2f imaging system with a lens L 2 of focal length f 2 = 50mm. The fiber position x is related to the transverse momentum by q s,x = ω s x/(f 2 c). Since S(q s,ω s ) = S( q s,ω s ) it is sufficient to scan the fiber along the x-axis. The direction of the remaining light from the pump beam after the two dichroic mirrors defines the optical axis and is used to align the fiber. The fiber is subsequently coupled into an optical spectrum analyzer (OSA) with 70 db dynamic range. Two typical spectral densities are indicated in Fig. 3.2 (a) and (b) for different fiber positions. The measured spectra are systematically broader than the simulated ones which is caused by limited resolution. Limiting factors are finite fiber core of 200 µm, chosen OSA resolution of 2 nm and imperfections in the 2f-image such as refraction at crystal surface, astigmatism ofl 2, and uncertainty about the distance betweenl 2 and fiber. The finite resolution smooths out the sinc structure given by Eq. (3.5). However for mostly collinear but non-degenerated cases the sinc structure is visible when measuring the total spectral density by replacing the fiber by a collimator as shown in Fig. 3.2 (c). Only in the non-degenerated case small side lobe contributions of the sinc structure remain after integration over q s as can be seen in Fig Laser L 1 -f 1 0 PPKTP f 2 L 2 DM1&2 x y Fiber OSA 2f2 z Figure 3.1: (Color online) Schematics of the experimental setup for measuring the spectral density Results and Discussion In the following the influence of the four parameters G, w 0,T, and L is investigated independently.

36 28 CHAPTER 3. TUNING CURVE OF TYPE-0 SPDC S(q s,x, ω s ) S( ω s ) ω [ ω /2] (a) (c) s p S(q s,x, ω s ) 1.0 (b) ω [ ω /2] ω [ ω /2] s Figure 3.2: (Color online) Typical normalized measured (solid curve) and simulated (dashed curve) spectral densities for two different fiber positions (x = 0 mm (a), x = 1 mm (b)). The narrow peak in (a) at ω p /2 comes from remaining light of the Verdi which is neither frequency doubled nor filtered. In (c) the fiber at x = 0 mm is replaced by a collimator. The measured (solid curve) spectral density shows the same structure at0.95 ω p /2 and 1.05 ω p /2 as the simulation (dashed curve), originating from the sinc term in Eq. (3.5). The asymmetry in the measured curve arises because a photon with frequency ω p /2 ω 0 (ω 0 > 0 rad/s) diverges more than its twin photon with frequency ω p /2+ω 0, giving rise to a lower coupling efficiency. p s p Poling Period G The poling period G 0 is not accessible experimentally and, therefore, we consider its influence on the spectral density in Eq. (3.11) only by numerical simulations with parameters taken from the experiment (see Fig. 3.3). The tuning curve S( q s,ω s ) shows an overall X-shaped structure with a maximum around ω p /2 and small q s values. The symmetry in the width of the branches with respect to a line at ω p /2 is not exact because the refractive index is frequency dependent. From Eq. (3.7) follows that spectral components propagate in different directions, giving rise to the X-shaped structure and thus colored emission rings. The emission angle relative to the optical axis can be calculated from ω s and q s by θ = arcsin( q s / k s ), which implies a larger emission angle for the photon with lower frequency. A change of G 0 affects the phase matching condition in Eq. (3.7) which leads to another propagation direction for the different spectral components. For G 0 = 9.00 µm very few photons propagate parallel to the optical axis, but most of the photons have frequency ω p /2, corresponding to a non-collinear but degenerate emission. With a slightly longer poling period (G 0 = 9.02 µm) the emission probability favors the collinear case and the overall down-conversion efficiency increases. For even longer poling periods (G 0 = 9.04 µm) the emission still favors the collinear case, however, becomes non-degenerate; thus a gap at ω p /2 in the total spectrum can be observed. Further enhancement of G 0 broadens the gap and leads to additional local maxima inside the gap

37 3.2. PUBLICATION: TUNING CURVE OF TYPE-0 SPONTANEOUS PARAMETRIC DOWN-CONVERSION 29 due to the sinc in Eq. (3.11). The simulations show that variations of G 0 in the nm-regime strongly affect the spectral density. Since G 0 is typically not specified to such precision, it is adjusted such that the simulations agree with the experimental results. S( ω s ) [a.u.] q s,x [1/ μ m] G = μ m G 0 = 9.02 μm G 0 = 9.04 μm S( q s,x, ω s ) [a.u.] ω [ ω /2] s (a) p ω [ ω /2] ω [ ω /2] s (b) (c) p s p 0.0 Figure 3.3: (Color online) Simulation of the spectral density Eq (normalized to the maximum of (b)) generated by a Gaussian beam in a PPKTP crystal. The parameters arew 0 = µm,t = 25 C,L 0 = 7.5 mm, and (a) G 0 = 9.00 µm, (b)g 0 = 9.02 µm, and (c)g 0 = 9.04 µm. The plots on the top result from integration along the q s,x -axis Pump Beam Waist w 0 The simulations for different beam waists in Fig. 3.4 show again a X-structure. Comparing the width of the branches illustrates that, if the pump beam is focused more tightly, each frequency is generated over a broader q s,x -range. For the utilized crystal we get best agreements between simulation and measurement for G 0 = µm. All numerical and experimental data in Fig. 3.4 are normalized to their maximum value since the fiber coupling efficiency varies for both measurements. In fact, the measured signal decreases with a bigger beam waist since w 0 appears in the exponent in Eq. (3.12). If the beam waist is too small the SPDC efficiency would decrease again since the beam divergence increases. The slight asymmetry in the width of the branches with respect to a line at ω p /2 is visible in both simulation and measurement and is again due to the frequency dependent refractive index. Apart from the efficiency the beam waist does not change the spectral density significantly as long as transverse wave vector dependencies in n(ω,t) can be neglected Crystal Temperature T By thermal expansion the temperature can change G(T) and L(T), and, in addition it influences the refractive index n(ω,t). Since n(ω,t)/ T is small, the refractive index in the denominator of Eq. (3.5) has negligible influence ons( q s,ω s ). Conversely,n(ω,T) affects the phase mismatch in Eq. (3.6) in a very sensitive manner. Figure 3.5 illustrates measured and simulated results for various temperatures T. For T = 15 C no spectral

38 30 CHAPTER 3. TUNING CURVE OF TYPE-0 SPDC 1.0 measurement simulation 1.0 q s,x [1/ μ m] q s,x [1/ μ m] (a) (c) (b) (d) ωs ω p ωs ω p S(q s,x, ω s) [a.u.] Figure 3.4: (Color online) Measurement, (a) and (c), and simulation, (b) and (d), of the normalized spectral density for different beam waists w 0. (a) and (b) f 1 = 150 mm and w 0 = µm, (c) and (d)f 1 = 300 mm and w 0 = µm. The other parameters arel 0 = 7.5 mm,t = 25 C, and G 0 = µm. component propagates parallel to the optical axis, corresponding to a non-collinear and degenerate emission. By increasing the temperature the propagation direction of frequencies around ω p /2 begins to approach the pump beam direction. At T = 25 C, the emission is mostly collinear and degenerate. Finally at T = 35 C, the phase mismatch for ω p /2 is too high and the efficiency decreases; the emission is still collinear but nondegenerate. When comparing Fig. 3.3 and Fig. 3.5 it can be observed, as expected, that the phase matching can be tuned either by varying the temperature or by changingg 0. Increasing the temperature by 10 C has the same effect as an elongation of the poling period of approximately 20 nm. Due to thermal expansion, G elongates about 0.6 nm and therefore contributes only 3% to changes in the phase matching condition (Eq. (3.7)). The main contribution comes from the temperature depending refractive index entering in k z. The variation of the spectrum due to a temperature dependent change of L can be completely neglected, as numerical simulations confirmed. The measured spectral density for T = 15 C is slightly asymmetric at ω p /2 with respect to a line at q s,x = 0 µm 1. The reason for this is the ring shaped spatial mode of the entangled photon beam in the noncollinear case which complicates the exact localization of the optical axis. Again experimental and numerical data are normalized to their maximal value. For simulations the temperature has to be known with a certainty of about 0.5 C Crystal LengthL The measurements for two crystals from the same manufacturer and with same nominal poling period G 0 but of different lengths indicate a clear difference in S(q s,x,ω s ) as shown in Fig In contrast, the simulations

39 3.2. PUBLICATION: TUNING CURVE OF TYPE-0 SPONTANEOUS PARAMETRIC DOWN-CONVERSION measurement simulation 1.0 q s,x [1/ μ m] (a) (b) q s,x [1/ μ m] q s,x [1/ μ m] (c) (e) ω s ω p (d) (f) ω s ω p S(q s,x, ω s) [a.u.] Figure 3.5: (Color online) Measurements, (a), (c) and (e), and simulations, (b), (d) and (f), of the normalized spectral density for T = 15 C ((a) and (b)), T = 25 C ((c) and (d)), and T = 35 C ((e) and (f)). The other parameters are w 0 = µm,l 0 = 7.5 mm, and G 0 = µm.

40 32 CHAPTER 3. TUNING CURVE OF TYPE-0 SPDC indicate that a change in length only could not cause the emission to change from collinear to non-collinear as measured, but has merely the effect of narrowing the spectrum. The discrepancy in the measurement is explained by slightly different G 0 for the two crystals. In order to achieve better agreement between measured and simulated spectrum, G 0 for the longer crystal has to be decreased by 2 nm in the simulation. The high sensitivity of S( q s,ω s ) on G 0 preclude an independent experimental investigation of the influence of L 0 on the spectral density. 1.0 measurement simulation 1.0 q s,x [1/ μ m] (a) (b) q s,x [1/ μ m] (c) ωs ω p q s,x [1/ μ m] (d) (e) ωs ω p S(q s,x, ω s) [a.u.] Figure 3.6: (Color online) Measurements, (a) and (c), and simulations, (b), (d), and (e), of the normalized spectral density for L 0 = 7.5 mm ((a) and (b)) and L 0 = 12 mm ((c), (d), and (e)). The other parameters are w 0 = µm,t = 25 C, and G 0 = µm. In (e) the poling period is changed to9.016 µm.

41 3.2. PUBLICATION: TUNING CURVE OF TYPE-0 SPONTANEOUS PARAMETRIC DOWN-CONVERSION Conclusion We have investigated the spectral photon count density of type-0 entangled photons in a PPKTP crystal depending on their transverse momentum. We further have characterized how different parameters influence the spectrum and we reported good agreement between measurement and simulated data. It turned out that the spectrum itself is most sensitive to the crystals poling period G 0. A change in the nm-range causes different emission. In practice, due to the equivalent behavior of the spectral density when changing the temperature T or G 0, the discrepancy between the nominal and effective poling periods can be compensated by temperature tuning. Increasing the temperature by 10 C has the same effect as an elongation of the poling period of approximately 20 nm. Thus, for simulations G 0 has to be known with nm accuracy and T up to 0.5 C. Low temperatures cause non-collinear and degenerate emission. Increasing T the emission gets mostly collinear. For even higher temperature the emission still favors the collinear case but becomes non-degenerate. Moreover, we have noticed that, apart from the total SPDC efficiency, changes of the pump waist or of the crystal length do not influence critically the emission spectra. Hence, the nominal value of the length and a beam waist estimation are precise enough for accurate simulations. By means of momentum dependent spectral measurements, we have shown a good agreement between computation and measurement. The understanding of the spatiotemporal structure of the spectrum is of importance for all applications where entanglement in energy is the relevant degree of freedom and allows to optimize the spectrum for specific applications. Acknowledgments This work was supported by the National Centre of Competence in Research - Molecular Ultrafast Sciences and Technology, and Swiss National Science Foundation grant PP00P2_ References [1] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, (2009). [2] A. Zeilinger, Experiment and the foundations of quantum physics, Rev. Mod. Phys. 71, S288 S297 (1999). [3] M. Genovese, Research on hidden variable theories: A review of recent progresses, Phys. Rep. 413, (2005). [4] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Quantum cryptography, Rev. Mod. Phys. 74, (2002). [5] N. Gisin and R. Thew, Quantum communication, Nature Photonics 1, (2007). [6] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear optical quantum computing with photonic qubits, Rev. Mod. Phys. 79, (2007). [7] L. A. Lugiato, A. Gatti, and E. Brambilla, Quantum imaging, J Opt. B: Quantum Semiclass. Opt. 4, S176 S183 (2002). [8] M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Demonstration of dispersion-canceled quantum-optical coherence tomography, Phys. Rev. Lett. 91, (2003). [9] D. N. Klyshko, Photons and nonlinear optics (Gordon and Breach, 1988). [10] P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Ultrabright source of polarizationentangled photons, Phys. Rev. A 60, R773 R776 (1999).

42 34 CHAPTER 3. TUNING CURVE OF TYPE-0 SPDC [11] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Entanglement of orbital angular momentum states of photons, Nature 412, (2001). [12] C. K. Law and J. H. Eberly, Analysis and interpretation of high transverse entanglement in optical parametric down conversion, Phys. Rev. Lett. 92, (2004). [13] M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, Theory of two-photon entanglement in type-ii optical parametric down-conversion, Phys. Rev. A 50, (1994). [14] M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion, Phys. Rev. Lett. 100, (2008). [15] A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Quantum-optical coherence tomography with dispersion cancellation, Phys. Rev. A 65, (2002). [16] A. Pe er, B. Dayan, A. A. Friesem, and Y. Silberberg, Temporal shaping of entangled photons, Phys. Rev. Lett. 94, (2005). [17] F. Zäh, M. Halder, and T. Feurer, Amplitude and phase modulation of time-energy entangled two-photon states, Opt. Express 16, (2008). [18] B. Dayan, A. Pe er, A. A. Friesem, and Y. Silberberg, Nonlinear interactions with an ultrahigh flux of broadband entangled photons, Phys. Rev. Lett. 94, (2005). [19] L. Caspani, E. Brambilla, and A. Gatti, Tailoring the spatiotemporal structure of biphoton entanglement in type-i parametric down-conversion, Phys. Rev. A 81, (2010). [20] A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. Lugiato, X Entanglement: The Nonfactorable Spatiotemporal Structure of Biphoton Correlation, Phys. Rev. Lett. 102, (2009). [21] W. Wasilewski, P. Wasylczyk, P. Kolenderski, K. Banaszek, and C. Radzewicz, Joint spectrum of photon pairs measured by coincidence Fourier spectroscopy, Opt. Lett. 31, (2006). [22] S.-Y. Baek and Y.-H. Kim, Spectral properties of entangled photon pairs generated via frequencydegenerate type-i spontaneous parametric down-conversion, Phys. Rev. A 77, (2008). [23] Y.-H. Kim and W. P. Grice, Measurement of the spectral properties of the two-photon state generated via type II spontaneous parametric downconversion. Opt. Lett. 30, (2005). [24] M. Hendrych, M. Mičuda, and J. P. Torres Tunable control of the frequency correlations of entangled photons, Opt. Lett. 32, (2007). [25] S. Carrasco, A. V. Sergienko, B. E. A. Saleh, M. C. Teich, J. P. Torres, and L. Torner, Spectral engineering of entangled two-photon states, Phys. Rev. A 73, (2006). [26] E. Lantz, L. Han, A. Lacourt, and J. Zyss, Simultaneous angle and wavelength one-beam noncritical phase matching in optical parametric amplification, Opt. Communications 97, (1993). [27] F. Devaux and E. Lantz, Spatial and temporal properties of parametric fluorescence around degeneracy in a type I LBO crystal, Eur. Phys. J. D 8, (2000). [28] O. Jedrkiewicz, J.-L. Blanchet, A. Gatti, E. Brambilla, and P. Di Trapani, High visibility pump reconstruction via ultra broadband sum frequency mixing of intense phase-conjugated twin beams. Opt. Express 19, (2011).

43 3.2. PUBLICATION: TUNING CURVE OF TYPE-0 SPONTANEOUS PARAMETRIC DOWN-CONVERSION 35 [29] M. Hamar, J. Peřina Jr., O. Haderka, and V. Michálek, Transverse coherence of photon pairs generated in spontaneous parametric down-conversion, Phys. Rev. A 81, (2010). [30] O. Jedrkiewicz, A. Picozzi, M. Clerici, D. Faccio, and P. Di Trapani, Emergence of X-shaped spatiotemporal coherence in optical waves, Phys. Rev. Lett. 97, (2006). [31] A. Joobeur, B. E. A. Saleh, and M. C. Teich, Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion, Phys. Rev. A 50, (1994). [32] K. Bencheikh, E. Huntziger, and J. A. Levenson, Quantum noise reduction in quasi-phase-matched optical parametric amplification, J. Opt. Soc. Am. B 12, (1995). [33] J. Svozilík and J. Peřina Jr., Properties of entangled photon pairs generated in periodically poled nonlinear crystals, Phys. Rev. A 80, (2009). [34] Y. Shih, Entangled biphoton source - property and preparation, Phys. Rep. Prog. Phys. 66, (2003). [35] R. J. Glauber, The quantum theory of optical coherence, Phys. Rev. 130, (1963). [36] M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, Quasi-phase-matched second harmonic generation: Tuning and tolerances, IEEE J. Quantum Electronics 28, (1992). [37] S. Emanueli and A. Arie, Temperature-dependent dispersion equations for KTiOPO 4 and KTiOAsO 4, Appl. Opt. 42, (2003).

44

45 4 Entanglement Quantification In Section 2.1, the reader was made familiar with a formal definition of entanglement. This definition is, however, unhelpful when it comes to the question of how much entanglement we have in a given bipartite quantum state. This question can only be answered by taking into account a quantitative description of entanglement commonly formulated in terms of entanglement measures. There exists a variety of entanglement measures for discrete and continuous quantum states while in the following we restrict ourselves to the Schmidt number K and the entropy E [1]. Both of these quantities can in principle be computed using the Schmidt decomposition of the corresponding JSA. Depending on the shape of the used JSA, the related numerics can, however, be very cumbersome. 1 This is especially the case when the SPDC process is induced by a spectrally narrow-band pump pulse since then, large matrices are involved in the calculation process. The previous Chapter gave a detailed analysis of the entangled photons spectra that will be latter used to encode qudits within the context of quantum information processing (Chapter 7). In the following Sections, we quantify the amount of entanglement in the two-photon state by calculating K and E. We first review an analytic approximation of the Schmidt number in the limit of a long pump pulse [3]. This result will then be compared with a numerical approximation of K performed for two different expressions of the JSA provided in Section 4.1. It turned out that a computation of the entropy for long pump pulses required a new numerical technique which stimulated a collaboration with the Mathematics Institute of the University of Bern. The obtained results for K and E demonstrate that energy-time entangled two-photon states can be a large resource of entanglement. 4.1 Formal Prerequisites This Section establishes two expressions for the JSA later used to compute the Schmidt number and the entropy of a type-0 entangled two-photon state. The experimental setup that will be presented in Chapter 5 is aimed at creating, manipulating, and detecting energy-time entangled photons. We are therefore interested to quantify the spectral degree of entanglement and thus restrict the following considerations to a collinear propagation of the entangled photons in which q i = q s = 0 q p = 0. For convenience we use relative frequencies Ω j = ω j ω cj, j {i,s,p}, such that ω i + ω s ω cp = Ω i + Ω s and we consider degenerated central frequencies ω ci = ω cs = ω cp /2. In Chapter 3, the pump field was assumed to be strictly monochromatic. Here, we allow the pump field to be of a Gaussian form and obtain with Eq. (2.29) 1 A profound discussion of this issue can be found in Reference [2]. 37

46 38 CHAPTER 4. ENTANGLEMENT QUANTIFICATION Λ(Ω i,ω s ) exp ( (Ω i +Ω s ) 2 τ 2 p 8ln(2) ) ( k1 L 1 sinc 2 ), (4.1) where the parameter τ p is the duration of the pump pulse. Apart from having made the restriction to longitudinal degrees of freedom we further replaced all constant factors in Eq. (2.29) by a proportionality symbol since we are only interested in the joint spectral amplitudes shape. Further, the functions e(ω j ) and n 1 (ω j ) are slowly varying quantities over the relevant ω j domain and therefore treated to be constant as well. To constrain the discussion to a real valued JSA we additionally neglect the phase factor in Eq. (2.29). In general, analytical calculations involving Eq. (4.1) are unfeasible since k 1 is a function of refractive indices expressed in the specific Sellmeier equations of the material under consideration. However, in order to obtain an analytic expression for the Schmidt number, Mikhailova et al. [3] simplified the above expression in a a first step by Taylor expanding the dispersion relation k j k j (Ω j ) = k j (ω cj )+Ω j +Ω 2 ω j j ωj =ω cj k j ω 2 j +O(Ω 3 j) (4.2) ωj =ω cj for the idler, signal, and pump photon. 2 The phase mismatch is then approximated by k 1 = k i (Ω i )+k s (Ω s ) k p (Ω p )+ 2π 1 c G 1 (A 1 (Ω i +Ω s ) B 1 (Ω i Ω s ) 2 /ω cp ) (4.3) where perfect phase-matching in 0th order k i (Ω ci )+k s (Ω cs ) k p (Ω cp )+ 2π G 1 = 0 (4.4) is assumed. 3 It is further used in Eq. (4.3) that contributions O(Ω 2 p) can be neglected which implied the additional simplification Ω 2 i +Ω2 s = 1/2[(Ω i +Ω }{{} s ) 2 +(Ω i Ω s ) 2 ] 1/2(Ω i Ω s ) 2. (4.5) =Ω p The dimensionless constant A 1 = c k p k i,s ω p ωp=ω ω i,s cp ωi,s =ω ci,cs = c(v 1 gp v 1 gi,gs ) (4.6) represents the temporal walk-off between the idler(signal) and the pump photon during their propagation within the SPDC crystal and v gj denotes the corresponding group-velocity. The constant B 1 = c 4 ω 2 k i,s cp ωi,s 2 (4.7) ωi,s =ω ci,cs denotes the group-velocity dispersion (GVD). The primary functional dependence on the Sellmeier equation in k 1 has now been substituted by A 1 and B 1. Replacing k 1 of Eq. (4.3) in Eq. (4.1) we end up with the following expression for the JSA 2 The here presented approximation of the JSA follows [3], however, is a standard procedure and not only used in the context of analytical Schmidt numbers. See e. g. [4]. 3 The authors in Reference [3] use an opposite sign convention such that k = k p k i k s. Both conventions, however, lead to the same result for the Schmidt number.

47 4.2. SCHMIDT NUMBER 39 Λ(Ω i,ω s ) exp ( (Ω i +Ω s ) 2 τ 2 p 8ln(2) ) ( L1 [ ] ) sinc A 1 (Ω i +Ω s ) B 1 (Ω i Ω s ) 2 /ω cp. (4.8) 2c 4.2 Schmidt Number The Schmidt number in the case of continuous entanglement was first introduced in Reference [5] to quantify the measure of electron-electron correlation in atomic physics. There, the Schmidt number is calculated without referring on an explicit expansion of the state into a Schmidt decomposition [6, 7]. This strategy has been resumed in Reference [3] to analytically approximate K in the case of energy-time entangled two-photon states for various lengths of the pump pulse duration τ p. We first briefly review the analytical result for the long pump pulse case in Section which is based on the JSA of Eq. (4.8). Afterwards, we confirm the obtained result by a numerical approximation of K using a discretized expression of Eq. (4.8). To go beyond the analytic prediction, we additionally perform numerical calculations of the Schmidt number incorporating Eq. (4.1) which contains the full dispersion relation of the nonlinear material through its Sellmeier equation Analytical Approximation ofk In Reference [3], the authors carry out an analytical approximation of K in the limit of a short and a long pump pulse duration. To distinguish the regime of a short pump pulse from the regime of long pump pulses, Mikhailova et al. introduced the control parameter η = 2cτp A 1 L. The pump pulses are considered to be short if η 1 and long for η 1. A full width at half maximum (FWHM) of 5 MHz in the spectral intensity of the pump used in our SPDC arrangement corresponds to τ p = ns. Together with L 1 = mm and A 1 = 0.19 for PPKTP we obtainη = and thus the condition for the long pulse approximation is de facto fulfilled. We therefore restrict ourselves to outline the calculation only for the long pump case where the reader is referred to Reference [3] for mathematical details. Let us consider a type-0 two-photon state in collinear configuration Ψ = 0 + dω i dω s Λ(Ω i,ω s ) Ω i Ω s (4.9) and its assigned density operator ˆρ = Ψ Ψ with the JSA of Eq. (4.8). The state of the individual idler(signal) subsystem is given by the reduced density matrix ˆρ i,s = Tr s,i (ˆρ) (4.10) which for the idler(signal) photon results in a partial trace over the signal(idler) degrees of freedom. Represented in the corresponding frequency basis, the reduced density operators read ρ i (Ω i,ω i ) = Ω i ˆρ i Ω i = dω s Λ(Ω i,ω s )Λ(Ω i,ω s), (4.11) ρ s (Ω s,ω s) = Ω s ˆρ s Ω s = dω i Λ(Ω i,ω s )Λ(Ω i,ω s), (4.12) where we have used that Λ(Ω i,ω s ) = Λ(Ω i,ω s ). As an aside, the reduced density operators ˆρ i and ˆρ s are in general mixed states whereas the density operator ˆρ of the two-photon state is pure. Further, since Λ(Ω i,ω s ) = Λ(Ω s,ω i ), the density operator for the idler and signal photon is identical and we thus define the reduced density operator ρ r. = ρi = ρ s. The Schmidt number which will be further defined in Eq. (4.19) is now

48 40 CHAPTER 4. ENTANGLEMENT QUANTIFICATION equal to K = N 2[ ] 1 Tr(ρ 2 r) [ = N 2 dω i dω i dω s dω sλ(ω i,ω s )Λ(Ω i,ω s )Λ(Ω i,ω s)λ(ω i,ω s)] 1, (4.13) where N is the normalization of the JSA N = dω i dω s Λ(Ω i,ω s ) 2. (4.14) Despite using the already approximated Eq. (4.8), the integrals in Eq. (4.13) cannot be solved analytically. Therefore, further approximations are needed to compute K. In the case of a long pump pulse Eq. (4.8) is dominated by a narrow pump spectral function whereas the sinc function is broad. SinceΩ i +Ω s = ω cp ω p 0 for a narrow-band pump, the linear term in the argument of the sinc function can be neglected and therefore ( L1 [ ] ) ( ) sinc A 1 (Ω i +Ω s ) B 1 (Ω i Ω s ) 2 L1 B 1 /ω cp sinc (Ω i Ω s ) 2. (4.15) 2c 2cω cp Using this approximation and inserting Eq. (4.8) in Eq. (4.13) expressions of the form exp ( τ2 p 8ln(2) [(Ω i +Ω s ) 2 +(Ω i +Ω s) 2 ] ) = exp ( τ2 p 4ln(2) [ ( Ω i + (Ω s +Ω s ) ) 2 + (Ω s Ω ]) s )2 2 4 (4.16) appear in the integrals. This function has its maximum at Ω i,max (Ω s,ω s) = (Ω s + Ω s)/2. To perform the integration over Ω i the sinc function is made independent of Ω i by replacing Ω i with Ω i,max in the argument. An exactly similar approximation is carried out to make the integration over Ω i. The residual integration over Ω s and Ω s ( τ contains a narrow exponential term exp p 2/(8ln(2))[Ω s Ω s ]2). Since the sinc function is still much wider than the exponential function, Ω s is now substituted with Ω s in the argument. One can then perform the remaining integrations without further approximations to obtain the Schmidt number in the long pump pulse limit K long = 105 π 72 cτ p 2ln(2)( (4.17) 1) B1 L 1 λ cp Numerical Approximation ofk The following Sections introduce two approaches to numerically evaluate the Schmidt number. The obtained results are then compared to Eq. (4.17) Coupled Integral Equations Approach To quantify the amount of entanglement in a pure and bipartite two-photon state it is common to expand the state in a Schmidt decomposition. The problem of how to decompose a two-dimensional and continuous function into a superposition of separated basis functions has been first discussed by Schmidt in the pure mathematical framework of linear integral equations [6]. The Schmidt decomposition has then been put into the context of continuous frequency entanglement [8]. We adopt the strategy to the type-0 two-photon state of Eq. (4.9) where the JSA can now be either Eq. (4.1) or Eq. (4.8). To test if the state of Eq. (4.9) is separable we represent Λ(Ω i,ω s ) in a Schmidt decomposition

49 4.2. SCHMIDT NUMBER 41 Λ(Ω i,ω s ) = j=0 λ j f i j (Ω i)f s j (Ω s). (4.18) A specific property of Eq. (4.18) is that only one summation index is used since each Schmidt mode of the idler photon f i j (Ω i) is uniquely correlated with a Schmidt modef s j (Ω s) of the signal photon. The Schmidt number, quantifying the degree of entanglement inλ(ω i,ω s ), is then defined as K = 1 λ 2 j j=0, (4.19) where j=0 λ j = 1 provided that dω i dω s Λ(Ω i,ω s ) 2 = 1. (4.20) From Eq. (4.18) it becomes obvious that Λ(Ω i,ω s ) is separable if and only if K = 1. Otherwise, the state is entangled. The solutions for λ j and f i,s j (Ω i,s ) are then given by a coupled set of linear and homogeneous Fredholm integral eigenvalue equations [9] dω iρ i (Ω i,ω i)f i j(ω i) = λ j f i j(ω i ), (4.21) dω s ρ s(ω s,ω s )fs j (Ω s ) = λ jf s j (Ω s), (4.22) by means of the reduced density operators of Eq. (4.11) and Eq. (4.12). The coupling between the above equations is due to the identical eigenvalue distributions for the idler and signal photon. Since ρ i (Ω i,ω i ) and ρ s (Ω s,ω s) are Hermitean functions ρ i,s (Ω i,s,ω i,s ) = ρ i,s (Ω i,s,ω i,s) all λ j R and the Schmidt modes {f i,s j (Ω i,s )} are linearly independent, complete, and orthonormal [10]. Due to the symmetry of the JSA, the reduced density operator for the idler and signal photon is identical and thus the set of eigenvalue equations is reduced to dω ρ r (Ω,Ω )f j (Ω ) = λ j f j (Ω). (4.23) However, solving this integral equation analytically with the reduced density matrix ρ r (Ω,Ω ) considered here is hardly possible. Therefore, we approximate ρ r (Ω,Ω ) in a uniform discretization on a lattice with a compact support C =. [Ω min,ω max ] [Ω min,ω max ] and step size δω =. Ωmax Ω min m to obtain a symmetric matrix A R m m. 4 The continuous equation of Eq. (4.23) is then replaced by a discrete eigenvalue problem of the form 4 The matrix is denoted with A to be consistent with the notation used in the publication in Section 4.4.

50 42 CHAPTER 4. ENTANGLEMENT QUANTIFICATION Af j = λ j f j, (4.24) where{f j } are the eigenvectors of the matrix A. Since A is the density operator of a physical state its eigenvalue spectrum σ(a). = {λ 1,λ 2...,λ m } 0 because the λ j are to be interpreted as classical probabilities. We therefore conclude that A has to be positive semidefinite. How to choose C and δω now strongly depends on the shape of the JSA. For short pump pulses, where τ p is of the order of fs, σ(a) can be calculated with MATLAB R2011b 5 since only small grid sizes (m 800) are required to accurately resolve ρ r (Ω,Ω ) on a lattice [8]. Unfortunately, the grid size needed to discretize ρ r (Ω,Ω ) can be very large in the case of long pump pulses where the JSA is dominated by a narrow Gaussian function (Figure 4.1). For τ p of the order of nanoseconds grid sizes up to m = are used to obtain convergent results for K as we will demonstrate in Section The attempt to compute σ(a) for a smaller matrix on a 12 core Intel Xeon X5650 (2.66 GHz) processor with 96 GB RAM using the MATLAB built in function eigs( ) had to be terminated after four days without result. Since σ(a) has not been accessible for large m we used a basic procedure of matrix functions theory to compute K without having to diagonalize A. Figure 4.1: Left: The discretized density operator A for the idler(signal) subsystem is shown. For a pump pulse duration of τ p = ns its shape is dominated by a narrow Gaussian function which implies a large discretization scheme. Right: Excerpt from the m m = matrix used to calculate K. The matrix has a band structure with 37 diagonals to obtain convergent results Matrix Functions Approach The theory of matrix functions can be used to relate a scalar function u(σ(a)) to a function u(a) which has the same dimension as the matrix A. A thorough exposition of matrix functions can be found in Reference [11]. We use this technique to numerically calculate the Schmidt number since σ(a) itself is not accessible due to a large discretization scheme. The Schmidt number is then expressed in the matrix elements a ij of A according to 5 MATLAB is a trademark of The MathWorks, inc.

51 4.2. SCHMIDT NUMBER 43 K 1 = = 1 ( λ σ(a) λ)2 1 Tr(A) 2 m i,j=1 λ σ(a) λ 2 a 2 ij. (4.25) The proof of Eq. (4.25) can be found in Appendix E. We make use of Tr(A) = λ σ(a) λ to normalize the density operator such that Tr(A) = Comparison between Analytical and Numerical K To determine the Schmidt number for conditions as close as possible to the experiment, we allow for an additional temperature dependence in the JSA. The temperature T 1 affects the length and the poling period of the SPDC crystal according to X 1 (T 1 ) = X 01 [1+α(T 1 25 C)+β(T 1 25 C) 2 ], X {L,G}, (4.26) where α = and β = for a PPKTP crystal as per Reference [12]. Moreover, we introduce a temperature dependency in the PPKTP refractive index such that n 1 (Ω) n 1 (Ω,T 1 ) and therefore also the GVD constant B 1 in the expression of K long becomes a function of T 1. Table 4.1 summarizes the parameters that we have used to calculate K long and K using the matrix functions approach. Table 4.1: Experimental parameters used to analytically and numerically calculate the Schmidt number. T 1 [ C] L 01 [mm] G 01 [µm] Table 4.2: Parameters of the discretized density matrix A used to obtain convergent results for K. τ p [ns] m Ω max [rad/fs] # diagonals The corresponding results are depicted in Figure 4.2 for various τ p in the long pump pulse regime where τ p = ns is related to the FWHM of 5 MHz of the pump field used in the experimental SPDC arrangement. Discretizing the approximated JSA of Eq. (4.8) we obtain convergent results for K (blue dots) for a matrix A with the parameters of Table 4.2. In the case ofτ p = ns the discretized density operator A has the form of a band matrix with 37 diagonals andm m = entries (Figure 4.1). A band matrix is a sparse matrix which has its nonzero elements arranged on a small number of diagonals compared to the size of the matrix. The Schmidt number is obtained by evaluating Eq. (4.25) which leads to a relative deviation 1 K/K long on the order of 10 3 for τ p = 10 ns and τ p = 50 ns. The relative deviation for τ p [86,96] ns is on the order of Equation (4.17) can thus be considered as a reliable expression for the Schmidt number under the approximations in Section applied to evaluate the integrals. The parameter values of Table 4.2 are then used to numerically calculate K with the non-approximated JSA of Eq. (4.1) (orange dots). The obtained

52 44 CHAPTER 4. ENTANGLEMENT QUANTIFICATION results demonstrate that in this case the analytical approximation K long underestimates the Schmidt number. Further, one can observe an increasing difference between the numerical and analytical solution for increasing values of τ p. Figure 4.2: Schmidt number K of energy-time entangled photons - Analytical vs. numerical results in the long pump pulse limit. The red line shows the analytical approximation K long of Eq. (4.17). Numerical results using the approximated JSA of Eq. (4.8) are indicated with blue dots. Orange dots represent K values based on the non-approximated JSA of Eq. (4.1) where the connecting line is indicated to guide the eye. All numerical values have been calculated with Eq. (4.25) and the parameters listed in Table 4.1 and Table 4.2. The vertical green dashed line indicates the pump pulse duration for a pump field with a spectral FWHM of 5 MHz. The storage of a matrix with m of the order of 10 6 in a common dense matrix format is unfeasible in terms of memory space. However, for long pump pulses, the shape of the JSA is such that only a small amount of entries in A are significantly nonzero. This allows to generate the matrix A in a sparse format where only the indices (i,j) of the nonzero elements together with its values a ij are stored by MATLAB. A matrix with m m = entries then requires approximately 13 GB memory space whereas the same matrix in a common dense format overflowed the available storage space of 96 GB. All numerical simulations of K were computed in MATLAB R2011b on a 12 core Intel Xeon X5650 (2.66 GHz) processor with 96 GB RAM. The computational time for a single value of K is about 5 h in the case of the approximated k 1 and about 13 h using the JSA of Eq. (4.1). There is, however, room for improvement with regard to the efficiency of the code in the second case. The results in Figure 4.2 demonstrate that energy-time entangled photons provide a large resource of entanglement if the SPDC process is induced with a long pump pulse. We make use of this resource in Chapter 7 where we demonstrate the encoding of qudits in the frequency domain. The calculations based on Eq. (4.1) are to the best of our knowledge the first numerically computed values for K in the long pump pulse limit which honour the full dispersion relation of the nonlinear material. Section 7.4 and 7.6 additionally indicate values for the Schmidt number which are calculated taking into account the properties of the SFG crystal which modifies

53 4.3. ENTROPY 45 Eq. (4.1) with an additional sinc function. It will also turn out that K will dramatically decrease under the influence of a finite spectral resolution of the experimental setup discussed in the forthcoming Chapter. The subsequent Section, however, first introduces the entropy as a further measure of entanglement. 4.3 Entropy In 1989 Barnett and Phoenix introduced [13, 14] the von Neumann entropy [15] of the reduced density matrix as a quantitative measure of entanglement. The entropy is defined as E = λ j log(λ j ) (4.27) j=0 and is also known as logarithmic negativity [1]. 6 Theλ j are again given by the corresponding Schmidt decomposition. That E indeed fulfills all the mathematical requirements to be a valid quantifier of entanglement was later shown in Reference [16]. The entropy can be interpreted from an information theory point of view as the amount of uncertainty contained in the density operator of a physical state [13]. Or, equivalently, the entropy corresponds to the amount of information one gains by measuring the state of a system. If we consider a system in a pure state, then we have a maximal knowledge of it within the laws of quantum mechanics. A measurement on this system will not lead to an increase of information. Consequently, the entropy of a pure state is zero. We have, however, already pointed out, that the density matrix of the idler or signal subsystem represents in general a mixed state and is only pure if the state is separable. The full knowledge of a mixed state requires more information than of a pure state and thus E > 0 for a mixed density operator. The entropy of a subsystem is maximal if the state is maximally entangled. The maximal entropy can be finite or infinite depending on the degrees of freedom the photons are entangled in. If we consider two photons entangled in polarization Ψ = 1 2 ( H A H B + V A V B ) (4.28) then the Hilbert space of each subsystem has dimension two. In the case of the maximally entangled state of Eq. (4.28) one obtains E = log(2). In general, E = log(d) holds for maximally entangled states where d denotes the Hilbert space dimension of the individual subspace. On the contrary, if the photons are entangled in their frequencies, the corresponding subspaces are infinite dimensional and the entropy is limited by the bandwidth of the pump field. In the limit of a strictly monochromatic pumpe + p (Ω i,ω s ) δ(ω i +Ω s ) also the entropy will be infinite. It is now tempting to again use matrix functional theory to numerically calculate E. Analogous to the Schmidt number, the entropy can be expressed according to [13]. ( ) 1 λ E = λ σ(a) λ λlog λ σ(a) λ σ(a) λ = 1 m ( ) A a ij log Tr(A) Tr(A) i,j=1 ij (4.29) 6 According to the existing literature, the entropy can be either defined withlog( ) log 2 ( ) [8] or withlog( ) ln( ). = log e ( )

54 46 CHAPTER 4. ENTANGLEMENT QUANTIFICATION now with a u(a) log(a) which is a way more difficult operation than just taking the square of a matrix. In principle the MATLAB built in function logm( ) is able to take the logarithm of a matrix but, unsurprisingly, fails to tackle the problem for the large size matrices we consider here. As a consequence, a collaboration with Prof. Dr. Thomas Wihler from the Mathematics Institute of the University of Bern emerged in order to find a new way to calculate E for a large matrix. The outcome of this collaboration is presented in the subsequent publication. References [1] M. B. Plenio and S. Virmani, Quantum Information & Computation 7, , (2007). [2] M. Mauerer, On Colours, Keys, and Correlations: Multimode Parametric Downconversion in the Photon Number Basis, PhD Thesis, (2009). [3] Yu. M. Mikhailova, P. A. Volkov, and M. F. Fedorov, Phys. Rev. A 78, , (2008). [4] T. E. Keller and M. H. Rubin, Phys. Rev. A 56, , (1997). [5] R. Grobe, K. Rza `zewski, and J. H. Eberly, J. Phys. B: At. Mol. Opt. Phys. 27, L503, (1994). [6] E. Schmidt, Math. Ann. 63, 433, (1906). [7] A. Ekert and P. L. Knight, Am. J. Phys. 63, 5, (1995). [8] C. K. Law, I. A. Walmsley, and J. H. Eberly, Phys. Rev. Lett. 84, , (2000). [9] R. Courant and D. Hilbert, Methoden der Mathematischen Physik, Springer Verlag Berlin, (1968). [10] S. Parker, S. Bose, and M. B. Plenio, Phys. Rev. A 61, , (2000). [11] N. J. Higham, Functions of Matrices, Society for Industrial and Applied Mathematics, (2008). [12] S. Emanueli and A. Arie, Appl. Opt. 42, , (2003). [13] S. M. Barnett and S. J. D. Phoenix, Phys. Rev. A 40, , (1989). [14] S. M. Barnett and S. J. D. Phoenix, Phys. Rev. A 44, , (1991). [15] J. Von Neumann, Göttinger Nachrichten, 1: , (1927). [16] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett. 78, , (1997).

55 4.4. PUBLICATION: COMPUTING THE ENTROPY OF A LARGE MATRIX Publication: Computing the Entropy of a Large Matrix This article is in preparation for submission and is available on arxiv: [math.na]: Computing the Entropy of a Large Matrix Thomas P. Wihler 1, Bänz Bessire 2, and André Stefanov 2 1 Mathematics Institute, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland 2 Institute of Applied Physics, University of Bern, Sidlerstr. 5, 3012 Bern, Switzerland Abstract Given a large real symmetric, positive semidefinite matrix, the goal of this paper is to show how a numerical approximation of the (von Neumann) entropy, given by the sum of the entropies of the individual eigenvalues, can be computed in an efficient way. An application from quantum-optics dealing with the entanglement of photons illustrates the new algorithm Introduction In quantum mechanics the most general description of a state is given by a Hermitean, positive semidefinite operator commonly referred to as density operator. The density operator allows to quantify the amount of disorder in a quantum system in terms of the (von Neumann) entropy [19]. The concept of entropy can be extended to a valid measure of non-classical correlations between the subsystems of an entangled state each described by its own density matrix [2, 3, 18]. More specifically, focusing on real symmetric, positive semidefinite matrices A R m m, i.e., the entropy of A may be defined by A = A, v Av 0 v R m, E(A) = λ σ(a) L(λ). (4.30) Here, σ(a) [0, ) signifies the spectrum of A, and L is a continuous function on the real interval [0, ) which is given by { xlog(x) if x > 0 L : [0, ) R, x 0 if x = 0. (4.31) Computing the entropy of a matrix By the symmetry of A, the spectral theorem implies that A can be diagonalized by means of an orthogonal matrix Q R m m, Q 1 = Q, i.e. A = QDQ, where D = diag(λ 1,λ 2,...,λ m ), and λ 1,λ 2,...,λ m 0 denote the eigenvalues of A. Then, we may define the matrix function L : R m m R m m induced by (4.31) (and denoted with the same letter) in a standard way (see, e.g., [10]) by L(A) = QL(D)Q,

56 48 CHAPTER 4. ENTANGLEMENT QUANTIFICATION where Furthermore, we see that L(D) = diag(l(λ 1 ),L(λ 2 ),...,L(λ m )). E(A) = tr(l(d)) = tr(q L(A)Q). Moreover, since the trace of a matrix is invariant with respect to similarity we arrive at E(A) = tr(l(a)). (4.32) At first glance, computing the entropy is possible by either applying (4.30), i.e., by computing the full spectrum of A, or by using formula (4.32) which involves the computation of the matrix logarithm. Evidently, for large matrices, both approaches are prone to be computationally unfeasible due to their high degree of complexity. A new algorithm The goal of this paper is to calculate the entropy of a matrix without the need of finding the eigenvalues of A or the necessity of computing the matrix logarithm of A explicitly. To this end, two key ingredients will be taken into account: The function L will be approximated by a polynomial p; in so doing the term L(A) p(a) can be expressed approximately as a sum of powers of A. This avoids the computation of the matrix logarithm. Using the relations E(A) = tr(l(a)) tr(p(a)), the entropy of A can be found approximately by computing the trace ofp(a). This quantity, in turn, may be determined numerically by appropriately combining a Monte-Carlo procedure and a Clenshaw type scheme. In this way, the explicit computation of the matrix powers occurring in p(a) can be circumvented. Applying these ideas, we will obtain a low-complexity algorithm for the matrix entropy which is still able to generate accurate computational results. As a practical application we will consider a two-photon state entangled in frequency. More precisely, we consider large density matrices resulting from suitable discretizations of the related continuous operators. We will use the new algorithm developed in this paper to demonstrate entanglement quantification by means of the entropy for a large real-valued density matrix A R m m, with a matrix size of the order m = O(10 8 ). The article is organized as follows: In Section we will recall a Monte-Carlo procedure proposed in [1] to compute the trace of a matrix function. Subsequently, a Chebyshev approximation polynomial of the function L will be derived in Section 4.4.3, together with a sharp error estimate with respect to the supremum norm. Furthermore, Section contains the new algorithm and a probabilistic error analysis. Next, we present some numerical examples including a traditional finite element matrix and the above-mentioned quantum optics application in Section Finally, we draw some conclusions in Section Throughout the paper, 2 denotes the Euclidean norm. Furthermore, tr( ) signifies the trace of a matrix, i.e., the sum of its diagonal entries. We note the fact that tr(a) = λ σ(a) λ for any A Rm m. We also notice that, for any γ 0 > 0, there holds that E(A) = λ σ(a) [ γ 0 L ( λ γ0) +log(γ 0 )λ ] = γ 0 tr ( L ( γ 1 0 A )) log(γ 0 )tr(a). (4.33)

57 4.4. PUBLICATION: COMPUTING THE ENTROPY OF A LARGE MATRIX 49 The appealing property of this identity is that it allows to compute the entropy by means of the function L as restricted to the interval [0, λ /γ 0 ]. In particular, since we approximate L by a Chebyshev polynomial, the interval of approximation can be limited from [0,max λ σ(a) λ] to the smaller interval [0,γ0 1 max λ σ(a) λ], if γ 0 > Monte-Carlo Approximation The following proposition, see, e.g., [6, 12], motivates a Monte-Carlo procedure for the computation of the trace of a symmetric matrix. Throughout the paper, we let Ω = { 1,+1}, and Ω m = { 1,+1} m R m. Proposition Consider a symmetric matrix A R m m with tr(a) 0. Furthermore, let X be a random variable that takes values 1 and 1 with probability 1 /2 each. Moreover, let ω Ω m be a vector of m independent samples generated by X. Then, where E denotes the expected value. E(ω Aω) = tr(a), From a practical point of view, this result allows for the computation of a numerical approximation of the trace of a symmetric matrix A by taking the mean of a finite number N of sample computations, tr(a) 1 N N ωi Aω i, i=1 where ω i Ω m are random vectors like defined above. Thence, recalling (4.33), we find, for anyγ 0 > 0, that E(A) γ 0 N N i=1 ω i L ( γ 1 0 A )ω i log(γ 0 )tr(a). (4.34) ( In order to provide bounds for terms of the form ω L γ0 )ω, 1 A for ω Ω m, we note the following lemma. It is based on an elementary analysis of the graph of L. Lemma Let x 0 > 0. Then, there hold the estimates min(l(x 0 ),e 1 sign(e 1 x 0 )) L(x) max(0,l(x 0 )), for any x [0,x 0 ]. Here, is the sign function. 1 if x < 0 sign(x) = 0 if x = 0 1 if x > 0 We continue by deriving some upper and lower bounds for v L(A)v. Corollary Let A R m m be real symmetric, and positive semidefinite, and σ(a) [0,x 0 γ 0 ], for some x 0,γ 0 > 0. Furthermore, consider v Ω m. Then, the estimates hold true for any v Ω m. mγ 0 min(l(x 0 ),e 1 sign(e 1 x 0 )) γ 0 v L(γ 1 0 A)v mγ 0max(0,L(x 0 ))

58 50 CHAPTER 4. ENTANGLEMENT QUANTIFICATION d(x0) Figure 4.3: Graph of the function d(x 0 ) from (4.35) in[0,5]. x 0 Proof. Letv Ω m. We choose an orthogonal matrixqsuch that there holdsq AQ = D = diag(λ 1,λ 2,...,λ m ). Then, m v L(γ0 1 A)v = (Q v) L(γ0 1 D)Q v = (Q v) 2 il(γ0 1 λ i). Then, using the upper bound from Lemma and the identity Q v 2 = v 2 = m, it follows that m v L(γ0 1 A)v max(0,l(x 0)) (Q v) 2 i = mmax(0,l(x 0)). i=1 The proof of the lower bound is completely analogous. Remark Suppose that λ max > 0 is the maximal eigenvalue of A. Then, we choose x 0,γ 0 > 0 such that x 0 γ 0 = λ max. Evidently, σ(a) [0,x 0 γ 0 ]. Hence, by the previous Corollary 4.4.3, we see that λ max mx 1 0 min(l(x 0 ),e 1 sign(e 1 x 0 )) γ 0 v L(γ 1 0 A)v λ maxmx 1 0 max(0,l(x 0 )). We may ask the question of how to choosex 0 andγ 0 such that the upper and lower bound in the above estimates are as close as possible to each other. In other words, we seek x opt 0 > 0 such that the function [ ] d(x 0 ) = x 1 0 max(0,l(x 0 )) min(l(x 0 ),e 1 sign(e 1 x 0 )) (4.35) is minimal. It turns out that this is the case for x opt 0 = 1, i.e.,γ 0 = λ max ; see Figure Chebyshev Approximation and Clenshaw s Algorithm Let us recall the Chebyshev polynomials of the first kind i=1 T n ( x) = cos(narccos( x)), n 0, on the reference interval Î = [ 1,1]. They satisfy the three term recurrence relation T n+1 ( x) = 2 x T n ( x) T n 1 ( x), n 1,

59 4.4. PUBLICATION: COMPUTING THE ENTROPY OF A LARGE MATRIX 51 with T 0 ( x) = 1, T 1 ( x) = x. These polynomials are orthogonal with respect to the inner product (f,g) = 1 1 f( x)g( x) 1 x 2 d x. More precisely, ( T m, T π if m = n = 0, n ) = δ mn π if m+n > 0 2 where δ mn is Kronecker s delta; see, e.g., [7]. Then, for x 0 > 0, the affine transformation F : [ 1,1] [0,x 0 ], F( x) = x 0 2 ( x+1), F 1 (x) = 2 x 0 x 1, (4.36) allows to define the Chebyshev polynomials {T n } n 0 on an interval I = [0,x 0 ]: T n = T n F 1, x I = [0,x 0 ]. Proposition Let x 0 > 0, and n N 0. Then the function L on the interval I = [0,x 0 ] is approximated by the polynomial function p n (x) = a 0 n 2 + a k T k (x), (4.37) where the coefficients {a k } n k=0 are given by a 0 = x 0 (log k=1 ( ) ) x0 +1, a 1 = x ( 0 2log 4 4 ( ) ) x0 +3, (4.38) 4 a k = ( 1)k x 0 k(k 2, k 2. (4.39) 1) Furthermore, for n 1, there holds the error estimate L p n,(0,x0 ) with,(0,x0 ) denoting the supremum norm on I. x 0 2n(n+1), (4.40) Proof. We begin by defining the function L = L F : x x 0 2 ( x+1)log ( x0 2 ( x+1) ) on Î = [ 1,1]. Using standard Fourier theory and affine scaling, the function L can be represented by the infinite series with L(x) = a a k T k (x) = a a k ( T k F 1 )(x) = ( L F 1 )(x), k=1 a k = 2 π Then, we define the polynomial p n by truncation: 1 1 k=1 L( x) T k ( x) 1 x 2 d x. p n (x) = a 0 n 2 + a k T k (x). k=1

60 52 CHAPTER 4. ENTANGLEMENT QUANTIFICATION n = 0 n = 1 n = 2 n = 3 xlog(x) x 10 3 n = 0 n = 1 n = 2 n = x Figure 4.4: Polynomials p n, for n = 0,1,2,3, and x 0 = 3. Left: Graphs of p n. Right: Approximation errors L(x) p n (x). The coefficients {a k } n k=0 can be computed by employing the substitution x = cos(t), cf. [17]. This implies that a k = 2 π L(cost)cos(kt)dt, k 0. π 0 For k = 0,1 we find the formulas (4.38) by direct calculation. In addition, noting the identity cost = 1 2 (eit + e it ) and switching to complex variables z = e it, dz = izdt, the formula (4.39) follows from the residual theorem. As for the error estimate, we notice that T k,(0,x0 ) = T k,( 1,1) = 1. Then, for n 1, L p n,(0,x0 ) = Noticing the telescope sum completes the proof. k=n+1 k=n+1 1 k(k 2 1) = 1 2 a k T k,(0,x0 ) k=n+1 a k = x 0 ( ) 1 k(k 1) 1 = (k +1)k k=n+1 k=n+1 1 k(k 2 1). 1 2n(n+1) Remark The polynomials p n, for n = 0,1,2,3, and x 0 = 3 are shown, together with the moduli of the approximation errors, in Figure 4.4. The above result implies the following estimates: Corollary LetA R m m be symmetric and positive semidefinite, withσ(a) [0,x 0 γ 0 ], for somex 0,γ 0 > 0. Furthermore, consider v Ω m. Then, for n 1, it holds the bound v (L(γ 1 0 A) p n(γ 1 0 A))v mx 0 2n(n+1), where p n is the Chebyshev approximation polynomial of the function L from (4.37).

61 4.4. PUBLICATION: COMPUTING THE ENTROPY OF A LARGE MATRIX 53 Proof. LetQ R m m be orthogonal such that Q AQ = diag(λ 1,λ 2,...,λ m ). Then, v (L(γ0 1 A) p n(γ0 1 A))v = (Q v) (L(γ0 1 D) p n(γ0 1 D))Q v m = (Q v) 2 i(l(γ0 1 λ i) p n (γ0 1 λ i)). Hence, since Q v 2 = v 2 = m, we arrive at i=1 v (L(γ 1 0 A) p n(γ 1 0 A))v m max 1 i m L(γ 1 0 λ i) p n (γ 1 0 λ i). Finally, using the error estimate (4.40), yields the desired result. In practical applications, Chebyshev series can be evaluated using the Clenshaw algorithm, see, e.g., [5, 7]: Starting from functions b n+2 (x) = b n+1 (x) = 0, and applying affine scaling from [ 1,1] to[0,x 0 ], see (4.36), we define ( ) 2 b k (x) = a k +2 x 1 b k+1 (x) b k+2 (x), x0 k = n,n 1,...,0, where{a k } are the coefficients from (4.38) (4.39). Then, it can be shown thatp n from (4.37) can be represented in the form p n (x) = 1 2 (a 0 +b 0 (x) b 2 (x)). For a symmetric matrix A R m m the polynomial γ 0 p n (γ0 1 A) can be computed in a similar way (indeed, this is possible since all terms involved consist of commuting sums of powers of A): SettingB n+2 = B n+1 = 0 R m m we define a finite sequence {B k } n+2 k=0 Rm m of matrices by the reverse recurrence relation ( ) 2 B k = a k I +2 A I B k+1 B k+2, k = n,n 1,...,0, x 0 γ 0 where I signifies the identity matrix inr m m. Then, p n (γ 1 0 A) = 1 2 (a 0I +B 0 B 2 ). From the above relation we find for a vector v Ω m that B k v = a k v + 4 x 0 γ 0 AB k+1 v 2B k+1 v B k+2 v, k = n,n 1,...,0. Therefore, introducing the variable y k = B k v, we see that y k = a k v + 4 x 0 γ 0 Ay k+1 2y k+1 y k+2, for k = n,n 1,...,0, starting from y n+2 = y n+1 = 0. Finally, we obtain v p n (γ0 1 A)v = 1 ( ) ma 0 +v (y 0 y 2 ). 2 Consequently, evaluating the above product essentially amounts to n matrix-vector multiplications.

62 54 CHAPTER 4. ENTANGLEMENT QUANTIFICATION Algorithm Let A R m m be a symmetric matrix, and v Ω m a vector. Then, for any γ 0 > 0, the quantity γ 0 v p n (γ0 1 A)v, where p n, n 1, is the polynomial from (4.37) can be computed by means of the following procedure: 1. Set y n+2 = y n+1 = 0 R m. 2. Fork = n,n 1,...,0 do ) 3. Output γ 0 2 (ma 0 +v (y 0 y 2 ). y k = a k v + 4 x 0 γ 0 Ay k+1 2y k+1 y k+2. Here, {a k } n k=0 are the coefficients from (4.38) (4.39) Computing the Entropy We now return to the idea of computing a numerical approximation of the entropy of a matrix by means of (4.34). In order to avoid the computation of the matrix logarithm, however, we will use the approximation Ẽ(A) E(A), where, for some γ 0 > 0 to be specified later, Ẽ(A) = γ 0 N N ωi p n i=1 ( γ 1 0 A )ω i log(γ 0 )tr(a). (4.41) Here p n is the approximation polynomial of degree n for L from (4.37), and ω i Ω m are random vectors. More precisely, we propose the following basic algorithm: Algorithm Let A R m m be a real symmetric, positive semidefinite matrix, and n,n N. Furthermore, choose γ 0 > 0. Then: 1. Compute N random vectors ω i Ω m, i = 1,2,...,N, with entries ±1 occurring with the same probability 1/2. 2. Determine the scalars ξ i = γ 0 ω i p n(γ 1 0 A)ω i using Algorithm Output 1 N Ni=1 ξ i log(γ 0 )tr(a). The approximation provided by the above algorithm has two essential error sources: Firstly, the use of the Monte-Carlo approach (4.34) brings about a certain randomness, and, secondly, replacing the function L byp n in (4.41) leads to an approximation error. The latter point has been addressed already in Corollary In order to deal with the issue of randomness, we provide a confidence interval analysis for the numerical approximation (4.41) following the approach presented in [1]. To this end, we recall a special case of Hoeffding s inequality [11]: Proposition Let X 1,X 2,...,X N be independent random variables with zero means and bounded ranges α i X i α + i, i = 1,2,...,N. Then, for any η > 0, there holds the probability bound ( 2η 2 ) P( X 1 +X X N η) 2exp Ni=1 (α + i α. i )2

63 4.4. PUBLICATION: COMPUTING THE ENTROPY OF A LARGE MATRIX 55 In order to apply the previous result, we define, for i = 1, 2,..., N, the random variables X i = γ 0 ω i L(γ 1 0 A)ω i log(γ 0 )tr(a) E(A), where ω i Ω m are random vectors with entries ±1 appearing with equal probability of 1/2. Using (4.33), we conclude that [ ] X i = γ 0 ωi L(γ 1 0 A)ω i +tr(l(γ0 1 A)). According to Proposition 4.4.1, we have that E(X i ) = 0, i = 1,2,...,N, provided that tr(l(a)) 0. Furthermore, we have that X i = γ 0 [ ω i p n (γ 1 0 A)ω i +tr(l(γ 1 0 A)) ]+γ 0 ω i [ L(γ 1 0 A)+p n(γ 1 0 A) ]ω i, and thus, with the aid of Corollary 4.4.7, with for i = 1,2,...,N. Then, setting α i X i α + i, α ± i = γ 0 [ ω i p n (γ 1 0 A)ω i +tr(l(γ 1 0 A)) ] ± mx 0γ 0 2n(n+1), α min = min 1 i N α i, α max = max 1 i N α+ i, and δ = α max α min = max 1 i N γ 0ω i p n (γ 1 0 A)ω i min 1 i N γ 0ω i p n (γ 1 0 A)ω i + mx 0γ 0 n(n+1), (4.42) we obtain the uniform bounds α min X i α max, 1 i N, and hence, by Hoeffding s inequality, Proposition 4.4.1, we find, for any η > 0, the probability estimate ( 1 P N N X i η ) ( [ = P N γ 0 N N i=1 2exp ωi L(γ0 1 A)ω i i=1 ( 2Nδ 2 ( η /N) 2). Using the approximation (4.41), and recalling again Corollary results in E(A) Ẽ(A) = γ [ 0 N N γ 0 N + γ 0 N ωi p n i=1 N ωi i=1 [ N ( γ 1 0 A ) ] log(γ 0 )tr(a) E(A) η ) N ω i ] log(γ 0 )tr(a) E(A) ( ( (p n γ0 ) L 1 A γ0 ))ω 1 A i ωi L(γ 1 0 A)ω i i=1 [ N mx 0γ 0 2n(n+1) + γ 0 N ] log(γ 0 )tr(a) E(A) ] log(γ 0 )tr(a) E(A). ωi L(γ0 1 A)ω i i=1 (4.43)

64 56 CHAPTER 4. ENTANGLEMENT QUANTIFICATION Thus, with (4.43), we obtain ( E(A) Ẽ(A) η P N + mx ) 0γ 0 2n(n+1) ( [ N ] P γ 0 ωi L(γ0 1 N A)ω i log(γ 0 )tr(a) E(A) η ) N i=1 ( 2exp 2Nδ 2 (η/n) 2), and therefore, P ( E(A) Ẽ(A) η < N + mx ) ( 0γ 0 > 1 2exp 2Nδ 2 ( 2n(n+1) η /N) 2). (4.44) Now, fixing an error tolerance τ > 0, we select η/n > 0 such that η N = τ mx 0γ 0 2n(n+1). Hence, ( E(A) Ẽ(A) ) ( P < τ > 1 2exp ( 2Nδ 2 τ mx ) ) 2 0γ 0. 2n(n+1) We thus have proved the following result: Theorem Consider a real symmetric, positive semidefinite matrix A R m m, and constants γ 0,x 0 > 0 such that σ(a) [0,γ 0 x 0 ]. Moreover, let τ > 0 be a prescribed error tolerance, and n N a polynomial degree such that τ > mx 0γ 0 2n(n+1). (4.45) Then, computing N N sample vectors ω i Ω m, i = 1,2,...,N, with entries ±1 of equal probability 1 /2, the output of Algorithm 4.4.9, denoted byẽ(a), satisfies E(A) Ẽ(A) < τ, (4.46) with probability at least p = 1 2exp Here, δ = α max α min is defined in (4.42). ( ( 2Nδ 2 τ mx ) ) 2 0γ 0. (4.47) 2n(n+1) Remark The above theorem shows that, in order to achieve a certain prescribed accuracy τ in the computations, the polynomial degree n of p n from (4.37) needs to be sufficiently large in accordance with (4.45). In addition, we see that the probability p of satisfying the error estimate (4.46) can be increased by adding more samples in the Monte-Carlo approach. In addition, from (4.47), it follows that N = 1 ( 2 δ2 τ mx ) 2 ( ) 0γ 0 2 log. 2n(n+1) 1 p Therefore, noticing that δ = O(m) (cf. Corollary 4.4.3) may imply that the theorem could require N to be unfeasibly large. Consequently, again following [1, Section 4.2], it may often be more practical to fix the

65 4.4. PUBLICATION: COMPUTING THE ENTROPY OF A LARGE MATRIX 57 number N of samples, or in this paper, a polynomial degree n beforehand, and to provide an error bound for a given probability p. Indeed, withpfrom (4.47) we solve for τ to arrive at τ = mx ( ) 0γ n(n+1) +δ 2N log, (4.48) 1 p where we have obeyed (4.45) in choosing the sign in front of the square root. We notice that τ is a sum of two independent error contributions. Thus, for given polynomial degree n it is reasonable to choose the number of samples N such that ( ) mx 0 γ n(n+1) = δ 2N log, 1 p i.e., This observation leads to Algorithm below. ( ) N = 2n2 (n+1) 2 δ 2 log 2 1 p m 2 x 2. (4.49) 0 γ2 0 Algorithm Let A R m m be a real symmetric, positive semidefinite matrix, and n N a prescribed polynomial degree. Furthermore, choosex 0,γ 0 > 0 withσ(a) [0,x 0 γ 0 ], and a probability p (0,1). Then: 1. Set i = 0, N = 1, ξ max =, ξ min =. 2. Whilei < N do i = i+1. Find a random vector ω i Ω m with entries ±1 occurring with the same probability 1 /2. Determine the scalar ξ i = γ 0 ω i p n(γ 1 0 A)ω i using Algorithm Compute ξ min = min(ξ min,ξ i ), ξ max = max(ξ max,ξ i ) Find and N from (4.49). δ = ξ max ξ min + mx 0γ 0 n(n+1), End do. 3. Output the approximate entropy Ẽ(A) = 1 N N ξ i log(γ 0 )tr(a), i=1 and the error tolerance from (4.48). Remark In accordance with Remark it is sensible to choosex 0 = 1 andγ 0 = λ max, where λ max > 0 is an upper bound on the spectrum σ(a). Incidentally, λ max could be determined, for example, by means of the Gerschgorin circle theorem Examples We shall now illustrate the method developed in this paper by means of two examples.

66 58 CHAPTER 4. ENTANGLEMENT QUANTIFICATION matrix polyn. number of exact abs. err. rel. err. comput. size m deg. n samples N entropy err % % % % % % Table 4.3: Entropy of a finite element matrix for various sizes m and p = A Finite Element Matrix We consider the classical stiffness matrix A = R m m, which appears in the discretization of the one-dimensional boundary value problem u (x) = f(x), x (0,1), u(0) = u(1) = 0, by uniform linear finite element; see, e.g., [13, Chap. 1]. It can be shown that the eigenvalues ofaare given by ( ) iπ λ i = 4sin 2, 1 i m, 2m+2 and therefore, m ( iπ E(A) = 4 sin 2 2m+2 i=1 4(2m+2) π/2 π 0 ) ( ( )) iπ log 4sin 2 2m+2 ( ) sin 2 (x)log 4sin 2 (x) dx. Now, using the fact that π/2 0 sin 2 (x)log ( 4sin 2 (x) ) dx = π /4, we see that E(A) 2m. In particular, the entropy decreases asymptotically linearly as m. In Table 4.3 we present numerical results for a prescribed probability p = 0.95 and several polynomial degrees n. The latter quantity has been chosen by hand with moderate growth as the matrix size m is increasing. We clearly see that the algorithm generates quite accurate results already for a low number of samples. Indeed, the relative errors are (except for m = 10) below 1%, and the computed errors based on (4.48) are very reasonable as compared to the magnitude of the exact entropy An Application in Quantum-Optics Entangled photons have become a widely used non-classical light source to investigate fundamental aspects of entanglement [8, 20]. Their unique properties have further paved the way to potentially practical applications

67 4.4. PUBLICATION: COMPUTING THE ENTROPY OF A LARGE MATRIX 59 in quantum communication and quantum computing [9, 15]. In recent years spontaneous parametric downconversion (SPDC) has become the standard procedure to generate entangled photon states. SPDC occurs when a noncentrosymmetric crystal is pumped by a laser beam strong enough to induce nonlinear interactions. In this case, a pump photon with angular frequency ω p may be annihilated and two new photons of lower frequencies ω i and ω s, denoted as the idler and the signal, are created. Energy conservation always demands ω i +ω s = ω p. If the experimental configuration of the three involved photons is further restricted to the case where they propagate collinearly, the resulting two-photon state, given by, Ψ = 0 + dω i dω s f(ω i,ω s ) â i (ω i)â s(ω s ) 0, (4.50) describes entanglement in the frequency domain [14]. We consider here identically polarized photon states created by the action ofâ j (ω j), j {i,s}, on the combined vacuum state 0 =. 0 i 0 s. The state in (4.50) is an entangled state if the joint spectral amplitude ( f(ω i,ω s ) exp (ω i +ω s ω cp ) 2 τp 2 ) ( ) k(ωi,ω s )L sinc (4.51) 8log(2) 2 cannot be separated into a product f(ω i,ω s ) = g(ω i )h(ω s ). The pump pulse with center frequency ω cp is represented by the exponential term in (4.51) and its duration is given by τ p. The parameter L denotes the length of the crystal. The efficiency of the SPDC process is dominated by k(ω i,ω s ) = k i (ω i ) + k s (ω s ) k p (ω i +ω s )+ 2π G, wherek j(ω) is the frequency-dependent propagation constant of a periodically poled crystal with poling period G. Using the corresponding Schmidt decomposition, the amount of entanglement in (4.50) can now be quantified by the entropy (4.33) of either idler or signal subsystem [4]. The state of each subsystem is described by its corresponding continuous density matrix, given by, A i (ω,ω ) = dω s f(ω,ω s )f (ω,ω s ), A s (ω,ω ) = dω i f(ω i,ω)f (ω i,ω ). Due to the symmetry off(ω i,ω s ) in (4.51) we define A i (ω,ω ) = A s (ω,ω ) =. A(ω,ω ) R. In order to calculate (4.33), the continuous function A(ω,ω ) has to be discretized on a lattice, i.e. A(ω,ω ) A R m m, with A = A. Since A is the density operator of a physical state, its eigenvalues are further distributed such that λ 0 for allλ σ(a). For short pump pulses, whereτ p is of the order of fs, the exact E(A) can be calculated by means of σ(a) since only small grid sizes (m 800) are needed to sufficiently resolve A(ω,ω ) [16]. Unfortunately, the grid sizes required to discretize A(ω,ω ) for long pump pulses, e.g., forτ p on a timescale of ns, are very large since in this case f(ω i,ω s ) is dominated by a narrow Gaussian function. Diagonalization of A is then practically unfeasible. However, a numerical approximation of the entropy according to Algorithm 4.5 is still possible. We have calculated the entropy Ẽ(A/tr(A)) for a pump pulse duration of τ p = 8.8 ns and a L = 11.5 mm long potassium titanyl phosphate crystal with G = µm. In the frequency domain, this specific choice of τ p corresponds to a pump pulse with a narrow spectral bandwidth of 5 MHz. Notice, that the normalization A/tr(A) results from the fact that a physical state needs to be normalized; indeed, since tr(a) = λ σ(a) λ we have that tr(a/tr(a)) = 1. In order to save memory space we made use of the fact that only a small amount of entries in A are significantly nonzero which allows to store the matrix in sparse format. This procedure results in a band matrix withm = and 37 diagonals. Figure 4.5 shows convergence of Ẽ(A/tr(A)) and the error tolerance from (4.48). For polynomial degree n = 20 and error probability p = 0.95 we obtain Ẽ(A/tr(A)) = ± Up to a polynomial degree of n = 20, the discretization error for m = is still smaller than the computational error τ. For all n the number N of sample vectors is N = 8. Our computations where performed in MATLAB 7 on a 12 core Intel Xeon X5650 (2.66 GHz) processor with 96 GB RAM. It is remarkable that for a matrix sizem = the computational 7 MATLAB is a trademark of The MathWorks, inc.

68 60 CHAPTER 4. ENTANGLEMENT QUANTIFICATION Ẽ(A/tr(A)) n Figure 4.5: Quantum optics application, for matrix size m = : Approximate entropy ẽ(a/tr(a)) based on Algorithm with p = 0.95 for various polynomial degrees n; the vertical bars indicate the computational error ranges according to (4.48). time for the entropy only took about 25 minutes. This clearly underlines the high efficiency of Algorithm 4.5 for this example. In the case of a maximally entangled, discrete bipartite system of finite dimension m 2, the entropy increases according toẽ = O(log(m)). Due toτ p being of the order of ns the state under consideration exhibits a very high degree of entanglement and is therefore almost equivalent to a maximally entangled system with m exp(14.934) Conclusions In this article, we have derived a new algorithm for the computation of the entropy of a large real symmetric, positive semidefinite matrix. The proposed procedure does neither require the computation of the spectrum nor of the matrix logarithm. Indeed, it is based on the following two main ideas: Approximation of the entropy function L by a reasonably accurate Chebyshev polynomial. Computation of the entropy by combining a Monte-Carlo type sampling procedure and a Clenshaw algorithm for matrix polynomials. The new algorithm is parallelizable and straightforward to implement. It was tested for a classical finite element matrix as well as for a large discretization matrix originating from a quantum-optics application. In both cases, our algorithm is able to achieve accurate results in a very efficient way. An interesting extension of this research constitutes the computation of the matrix entropy in the context of complex discrete Hermitean operators. Here, an important ingredient is the appropriate redefinition of the function L for complex input values and the corresponding approximation by polynomial functions for both the real as well as the imaginary part.

69 4.4. PUBLICATION: COMPUTING THE ENTROPY OF A LARGE MATRIX 61 References [1] Z. Bai, M. Fahey, and G. Golub. Some large-scale matrix computation problems. J. Comput. Appl. Math., 74(1-2):71 89, TICAM Symposium (Austin, TX, 1995). [2] S. M. Barnett and Phoenix S. J. D. Entropy as a measure of quantum optical correlation. Phys. Rev. A, 40(5): , [3] S. M. Barnett and Phoenix S. J. D. Information theory, squeezing, and quantum correlations. Phys. Rev. A, 44(1): , [4] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher. Concentrating partial entanglement by local operations. Phys. Rev. A, 53: , [5] C. W. Clenshaw. A note on the summation of Chebyshev series. Math. Tables Aids Comput., 9: , [6] S. Dong and K. Liu. Stochastic estimation withz 2 noise. Phys. Lett B, 328(1 2): , [7] L. Fox and I. B. Parker. Chebyshev polynomials in numerical analysis. Oxford University Press, London, [8] M. Genovese. Research on hidden variable theories: A review of recent progresses. Physics Reports, 413(6): , [9] N. Gisin and R. Thew. Quantum communication. Nature Photonics, 1: , [10] N. J. Higham. Functions of matrices. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, Theory and computation. [11] W. Hoeffding. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc., 58:13 30, [12] M. F. Hutchinson. A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Comm. Statist. Simulation Comput., 18(3): , [13] C. Johnson. Numerical solution of partial differential equations by the finite element method. Dover Publications Inc., Mineola, NY, Reprint of the 1987 edition. [14] T. E. Keller and M. H. Rubin. Theory of two-photon entanglement for spontaneous parametric downconversion driven by a narrow pump pulse. Phys. Rev. A, 56(2): , [15] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn. Linear optical quantum computing with photonic qubits. Rev. Mod. Phys., 79: , [16] C. K. Law, I. A. Walmsley, and J. H. Eberly. Continuous frequency entanglement: Effective finite hilbert space and entropy control. Phys. Rev. Lett., 84(23): , [17] L. N. Trefethen. Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev., 50(1):67 87, [18] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight. Qantifying Entanglement. Phys. Rev. Lett., 78(12), [19] J. Von Neumann. Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik. Göttinger Nachrichten, 1: , [20] A. Zeilinger. Experiment and the foundations of quantum physics. Rev. Mod. Phys., 71:S288 S297, 1999.

70 62 CHAPTER 4. ENTANGLEMENT QUANTIFICATION 4.5 Additional Results Figure 4.6 shows further numerical results for the entropy in dependence of the pump pulse duration τ p. All calculations were performed by applying the algorithm of Section on the same density matrices A already used to calculate the Schmidt number (Table 4.1 and Table 4.2). According to the paper, we use alog( ) ln( ) definition of the entropy. Figure 4.6: Entropy E of energy-time entangled photons - Numerical results in the long pump pulse limit. Numerical results using the approximated JSA of Eq. (4.8) are depicted with blue dots. Orange dots represent E values based on the non-approximated JSA of Eq. (4.1) where in both cases the connecting lines are only present to guide the eye. The error probability is p = 0.95 and the polynomial degree is n = 20. The vertical bars indicate the computational error ranges according to Eq. (4.48). The vertical green dashed line shows the pump pulse duration for a pump field with a spectral FWHM of 5 MHz.

71 5 Experimental Setup The experimental setup described in this Chapter has been first introduced in order to study SFG with a high flux of broadband entangled photons [1]. At the same time, the arrangement was used to modulate the spectral phase of the entangled pairs by a spatial light modulator to investigate their temporal properties. Due to the ultrafast SFG detection, a direct measurement of the modified second-order correlation function was performed which is in fact not possible using an electronic coincidence detection method [2]. With the aim at manipulating the spectrum of the entangled photons in its phase and amplitude, the same setup has then been rebuilt during a PhD thesis at the Institute of Applied Physics in Bern [3, 4]. In Reference [5], a modified version of the experiment was demonstrated where the optical paths of the idler and signal photon are spatially separated through non-collinear emission to examine the influence of additional dispersion on the coincidence signal or to perform non-local dispersion cancellation using two prism compressors [6]. Figure 5.1 shows a schematic overview of the experimental arrangement. The setup can be subdivided into three parts: The SPDC source crystal prepares entangled photon states (Preparation) which are manipulated (Manipulation) by a standard pulse shaping apparatus and detected through SFG in a second nonlinear crystal (Detection). In the following Sections the individual parts are presented in more detail where Figure 5.1 serves as a reference for the individual setup components. Preparation (SPDC) Manipulation (SLM) Detection (SFG) BD P2 P3 1 E 2 L0 0 L1 P1 y x z P4 L2 2 L3 DM BF L4 SPCM Figure 5.1: Overview of the experimental setup. Preparation: L0 pump beam lens with focal length f = 150 mm, PPKTP nonlinear crystal for SPDC. Manipulation: BD beam dump, SLM spatial light modulator, symmetric two-lens (L1, L2) imaging arrangement (f = 100 mm) with a (de)magnification factor of six, P1-P4 four-prism compressor. Detection: PPKTP nonlinear crystal for SFG, BF bandpass filter, SPCM single photon counting module with a two-lens (L3, L4) imaging system, DM dichroic Mirror, E 2 measurement with a powermeter. The three parts of the setup are discussed in the main text. 5.1 Preparation 63

72 CHAPTER 5. EXPERIMENTAL SETUP 64 To prepare type-0 entangled photon pairs degenerated at. λc = λci = λcs = 1064 nm through SPDC, we pump a 11.5 mm long, positive uniaxial and periodically poled nonlinear KTiOPO4 crystal with a poling periodicity of G1 = 9 µm (Figure 5.2). The pump laser is a quasimonochromatic Nd:YVO4 (Verdi) laser with λcp = 532 nm featuring a narrow spectral FWHM of about 5 MHz. The collinear pump beam is focused (L0) into the middle of the PPKTP crystal (Σ0 ) with a power of 5 W leading to a focal intensity of 0.6 MW/cm2. The downconversion crystal is mounted in a temperature stabilized copper block and surrounded with indium foil to guarantee heat transfer. The temperature stabilization itself is performed by a water circulation system whose tempera- Figure 5.2: Preparation (SPDC). Shown is the lens ture can be controlled to ±0.1 C. The operating temper- L0 focussing the pump beam into a KTiOPO crys4 ature of the PPKTP crystal is chosen for collinear emis- tal mounted in a copper block to generate type-0 ension and according to type-0, all involved photons are tangled photon pairs. identically polarized (in x-direction). This configuration leads to a spectral width of the down-converted photons of λdc 105 nm around λc. According to Reference [1], the maximal flux of down-converted photons in which photon pairs are well distinguishable from each other scales linearly with the down-converted bandwidth Φmax νdc. (5.1) The flux Φmax itself sets a boundary between the quantum low power regime and the classical high power regime and corresponds to a mean spectral mode density n = 1 [1]. Equation (5.1) implies that broadband entanglement allows for a high flux of distinct photon pairs. In fact, with λdc 105 nm, Eq. (5.1) induces a maximal flux of Φmax = photons per second with a corresponding power of Pmax = 5.2 µw. The measured photon flux power P in our experiment is 1 µw which corresponds to a spectral mode density of n = P/Pmax = 0.2 < 1. This assures that we are below the single photon limit and thus in the quantum regime where the entangled pairs are temporally well separated from each other. 5.2 Manipulation It became apparent that the detected coincidence rate is very sensitive to second-order dispersion of the idler and signal photon. To compensate for group-velocity dispersion, mainly induced by the crystals, the entangled photons are therefore imaged from the SPDC crystal through a four-prism compressor (P1-P4) to the plane Σ2. The prism compressor consists of four equilateral N-SF 11 (Schott) prisms arranged in minimum deviation geometry. Prism P2 and P3 are mounted on translation stages and can be manually placed within the beam. The residue of the pump is distracted into a beam dump after P1. The lens L1 images the plane Σ0 to Σ1 such that the spectral components are spatially dispersed along the x-direction in order to form a quasi-fourier plane. Within the actual setup, the spectrum is aligned along 5 mm in x-direction before lens L2 recombines the spectral components while imaging Σ1 to the middle of the detection crystal at Σ2. Since any imaging system suffers from finite resolution it is indispensable to introduce the concept of a point spread function Point Spread Function If we consider the case of an ideal single-lens imaging system a point in an object plane is imaged to a point in an imaging plane. In real systems, however, there is no point-to-point correspondence between the object and the image. Due to diffraction effects of the finite aperture of the lens and further optical elements in the beam

73 5.2. MANIPULATION 65 path, a point is in fact mapped into a patch in the imaging plane. The size of this patch determines the resolution of the optical system. This effect is described by the point spread function (PSF) of the imaging system. In the actual setup the central plane Σ 0 is imaged with lens L1 through the prisms P1 and P2 to the plane Σ 1 where the imaging distances are chosen to obtain a six-fold magnification. To determine the resolution at Σ 1 we used the built in function "FFT PSF" in Zemax (Version May 4, 2009) to compute the intensity distribution of the diffraction image at Σ 1 for a single point source. Figure 5.3 shows the simulated images at Σ 1 for three different wavelengths. The corresponding PSF for λ c = 1064 nm is depicted in Figure 5.4 where we used Figure 5.3: Zemax spot diagram. This picture shows the image atσ 1 of a point source atσ 0 for three different wavelengths. The orientation is such that the vertical direction corresponds to the x-axis and the horizontal direction to the y-axis. k c Ω j (x) = fγ(m+1) x (5.2) to relate the x-axis to the frequency axis. The space-to-frequency mapping in Eq. (5.2) is obtained by means of Fourier optics [3] and Ω j is the relative frequency introduced in Section 4.1. The linearized dispersion coefficient γ 2λ c dn c dλ (5.3) λc takes into account the presence of prism P1 and P2 where m is the magnification factor obtained by L1 with focal length f. Further, k c = 2π/λ c. To incorporate the finite spectral resolution at Σ 1 we approximate the output of Zemax with a Gaussian function to obtain a FWHM of Ω PSF = rad/fs (Figure 5.4). However, in the simulations Ω PSF has to be treated as a quasi-free parameter adjusted to fit the experimental data. The PSF is then defined as a two-dimensional Gaussian function PSF(Ω i,ω s ) = exp ( (Ω2 i +Ω2 s )2ln(2) Ω 2 PSF ), (5.4) where we assume Ω PSF to be equal for all wavelengths. The JSA expressed in collinear geometry and relative frequencies

74 66 CHAPTER 5. EXPERIMENTAL SETUP Figure 5.4: PSF-Zemax-Gaussian. The PSF according to Zemax (blue curve) is approximated with a Gaussian function (red curve) to obtain a FWHM of Ω PSF which serves as a measure for the resolution at Σ 1. Γ(Ω i,ω s ) exp ( (Ω i +Ω s ) 2 2ln(2) ω 2 p ) ( k1 L 1 sinc 2 ) sinc ( ) k2 L 2 2 (5.5) used to calculate G (1) is then modified by a convolution with Eq. (5.4) to Γ(Ω i,ω s ) Γ PSF (Ω i,ω s ) (Γ PSF)(Ω i,ω s ). (5.6) In Eq. (5.5) we have neglected the additional phase factors which appear in Eq. (7.17) since we assume the prism compressor to be adjusted such that all spectral phases are compensated. Note, thatγ PSF still represents a pure state since the modification through a PSF does not add statistical noise to the state. However, as can be seen in Figure 5.5, the strong correlation between the idler and signal photon generated by a quasi monochromatic pump is reduced. ForΓ PSF we obtain values for the Schmidt number and the entropy ofk 4.9 ande 2.6 by direct diagonalization of the reduced density matrix.

75 5.2. MANIPULATION 67 Ω i [rad/fs] Ω s [rad/fs] Ω s [rad/fs] Figure 5.5: Left: This figure showsγ(ω i,ω s ) for a spectral bandwith ν p = 5 MHz of the pump,l 1 = L 2 = 11.5 mm and G 1 = G 2 = 9µm. The narrow JSA implies a high degree of entanglement between the idler and signal photon. Right: The figure depicts Γ PSF (Ω i,ω s ) taking into account the finite spectral resolution of the optical setup through a PSF with Ω PSF = rad/fs Spatial Light Modulator Placing a spatial light modulator (SLM) at Σ 1 allows to individually manipulate the spectral amplitude and phase of the entangled photons similar to pulse shaping techniques applied to classical femtosecond laser pulses (Figure 5.6) [7]. The SLM used in the experiments is a SLM-S640d commercially available at Jenoptik and applicable within a wavelength range from 430 to 1500 nm [8]. At the heart of its functionality are two coupled but individually selectable nematic liquid crystal displays (LCD) with 640 pixels each (Figure 5.7). The orientation of the molecules within each single pixel can be changed by applying a specific voltage. To explain the main principle of the SLM we consider two extreme cases. If no voltage is applied, the molecules within a pixel tend to orient themselves perpendicular to the direction of the linearly polarized incident light field. In this case the molecules act similar to a birefringent material. The incoming light field will be divided into two perpendicular polarization directions namely along the axis of the ordinary (o) and the axis of the extraordinary (e) refractive index. Since the phase velocities for a wave Figure 5.6: Manipulation (SLM). The SLM is located at Σ 1 between the prisms P2 and P3. polarized along the o-axis is different compared to a wave polarized along the e-axis, birefringence implies a phase shift φ. On the other hand, if a maximal voltage is applied to the pixel, the molecules are oriented along the direction of the light propagation. The birefringent property has disappeared and thus φ = 0. The phase shift is therefore dependent on the applied voltage U and, in addition, φ is a function of ω since the involved

76 68 CHAPTER 5. EXPERIMENTAL SETUP refractive indices are frequency dependent. Explicitly, the phase shift reads φ(u,ω) = φ e φ o = ω c n(u,ω)d LC, (5.7) where n(u,ω) is the difference in the index of refraction of the ordinary and the extraordinary wave and d LC denotes the thickness of the LCD. It can be demonstrated by means of the Jones matrix formalism that Figure 5.7: LCD of the SLM. Shown are the dimensions of a LCD togheter with its Indiumzinnoxid (ITO) electrodes. two LCD s together with a polarization dependent detection process allow to independently manipulate the amplitude A and phase φ of the transmitted frequencies at each pixel [8]. The corresponding expression for A and φ read ( ) φ1 (U,ω) φ 2 (U,ω) A(U,ω) = cos (5.8) 2 and φ(u,ω) = φ 1(U,ω)+ φ 2 (U,ω), (5.9) 2 where φ 1 and φ 2 now refer to the phase shift induced by LCD 1 and LCD 2. For a pure amplitude modulation φ 1 = φ 2 whereas for a pure phase modulation φ 1 = φ 2 holds. In general, the spectral components are, however, manipulated in amplitude and phase at the same time. To guarantee the correct functionality of the SLM, two calibration procedures have to be performed in advance. In the pixel array aligned along the x-direction each pixel has to be assigned to the specific wavelength at its position. A corresponding calibration results in a unique ω pixel attribution. According to the pixel size of 100 µm each, about 50 pixels are actually illuminated by the SPDC spectrum. Due to the PSF discussed in the previous Section a specific frequency is blurred over about two pixels. A second calibration procedure is needed for each display individually to assign a defined phase shift to an applied voltage, i.e. φ 1,2 U. How to perform this procedure is outlined in Reference [8]. Reduced to the essential steps, a transfer function M i,s (Ω) is programmed on the SLM via LabVIEW. Corresponding sub vi s then decompose M i,s (Ω) into A(Ω) = M i,s (Ω) and φ(ω) = arg{m i,s (Ω)}. With the inverse relations of Eq. (5.8) and Eq. (5.9) φ 1 (Ω) = φ(ω)+arccos(a(ω)), (5.10) φ 2 (Ω) = φ(ω) arccos(a(ω)), (5.11)

77 5.2. MANIPULATION 69 the phase shifts φ 1 (Ω) and φ 2 (Ω) are evaluated for each display individually. Here, Ω is related to each pixel of the SLM through theω pixel calibration. For given φ 1,2 (Ω) and calibration φ 1,2 U each pixel on each LCD is then supplied with the corresponding voltage to induce the correct phase shift on the transmitted spectral component. To obtain amplitude modulation the polarization dependent detection is given trough the SFG process since only photon pairs polarized in x-direction are up-converted in the detection crystal. In the experimental arrangement of Figure 5.1 there is, in contrast to e.g. [5, 6], no spatial separation between the optical paths of the idler and signal photon. This implies that both photons are indistinguishably affected by the same transfer function M i,s (Ω) where the formal description of a coincidence signal depends on a total transfer function M(Ω i,ω s ) = M i (Ω i )M s (Ω s ). (5.12) However, as long as Ω PSF is not too large it is nevertheless possible to assign, with respect to λ c, the lower frequency part of the spectrum to the idler and the higher frequency part to the signal photon. 1 1 That this assignment is valid will be justified in the context of quantum state tomography (Section 7.6).

78 70 CHAPTER 5. EXPERIMENTAL SETUP 5.3 Detection Figure 5.8: Detection (SFG). From right to left: The PPKTP crystal for SFG is mounted in a copper block. Further shown is the lens L3 followed by the dichroic mirror, the bandpass filter, and lens L4 in front of the SPCM. In orthogonal direction a collimator is placed to guide the remainig IR photons in a powermeter or a spectrometer. Instead of using two spatially separated single photon detectors, coincidences of the entangled photon pairs are measured through SFG in a second PPKTP crystal (Figure 5.8). Currently, the SFG efficiency is such that each 10 9 th photon pair is up-converted in the detection crystal. However, there are ongoing attempts to enhance this rather low efficiency (Chapter 8). The recombined 532 nm photons are then imaged by a telescope arrangement consisting of L3 (f=60 mm) and L4 (f=11 mm) onto the photosensitive area of a SPCM (IDQ id100-50). To avoid stray light from being detected by the SPCM, the preparation stage and the detection stage are shielded with an optically opaque box and a black towel respectively. The measurements are performed in a dark laboratory where all light sources are eliminated. Under these conditions, the remaining dark counts are about 11 Hz and mainly stem from the SPCM itself. In order to exclude the detection of residual infrared (IR) photons a dichroic mirror is installed after L3 to deflect the IR photons into a powermeter (Ophir Laserstar). In addition, a 4 mm thick BG 18 bandpass filter is mounted in front of L4. If required, the E 2 detector can be replaced by a spectrometer (Avantes NIR ). In order to prove that there are indeed no accidental IR photons measured by the SPCM we altered the temperature of the detection crystal such that the SFG efficiency went to zero. At the same time also the count rate went to zero which strongly indicates that in fact all IR photons are filtered out by the DM/BF combination. Figure 5.9 shows the SFG rate for various temperatures T 2 of the detection crystal whereas the temperature of the SPDC crystal T 1 is kept fixed. The theoretical signal is calculated by including T 1 and T 2 in the length and the poling period of the crystals according to X k (T k ) = X 0k [1+α(T k 25 C)+β(T k 25 C) 2 ], X {L,G}, k {1,2}, (5.13) where α = and β = compliant with Reference [9]. Since the poling periods have now become a function of the temperature also k 1,2 (Ω i,ω s ) k 1,2 (Ω i,ω s,t 1,2 ), (5.14) where additionally, we introduce a temperature dependency in the refractive index n k (Ω) n k (Ω,T k ). The coincidence signal is then expressed as G (1) (T 2 ) dω i dω s Γ(Ω i,ω s,t 2 ) 2 (5.15) by means of Eq. (5.5). We have used that ω i + ω s ω cp for a narrow-band pump to eliminate the time dependence in G (1). Figure 5.9 demonstrates that the phase matching condition and thus the up-conversion rate of the SFG process strongly depends on the temperature of the detection crystal. The maximal count rate is achieved if the phase matching conditions of the SPDC crystal and the detection crystal coincide. In Section 5.1 we demonstrated that the current flux of down-converted photons implies a mean spectral mode density of n < 1. The coincidence counts in Figure 5.9 are therefore due to SFG of well-separated pairs. In

79 5.3. DETECTION 71 Section 2.7 it has been discussed how the phase matching properties of the SFG process are included in a firstorder correlation function. Figure 5.9 indeed shows good agreement between the simulated signal according to Eq. (5.15) and the detected coincidence counts. (1) G (SFG signal) [Hz] T [ºC] 2 Figure 5.9: SFG signal versus T 2. The SFG coincidence signal is depicted for various temperatures of the detection crystal (red dots) and a fixed temperature of the SPDC crystal (T 1 ). Shown are dark count subtracted net counts. The blue curve shows the theoretical prediction according to Eq. (5.15). The here presented setup allows to have control over group-velocity dispersion and inherits an ultrafast optical coincidence method. It is therefore ideally suited to study broadband energy-time entangled photons. Together with the versatility of a SLM it can be used to implement quantum information protocols within the photons frequency degree of freedom (Chapter 7). In parallel, this setup has been used to study the scattering behaviour of entangled photons in a turbid medium [10]. References [1] B. Dayan, A. Pe er, A. A. Friesem, and Y. Silberberg, Phys. Rev. Lett. 94, , (2005). [2] A. Pe er, B. Dayan, A. A. Friesem, and Y. Silberberg, Phys. Rev. Lett. 94, , (2005). [3] F. Zäh, Shaping of entangled two-photon states and applications in quantum lithography, PhD Thesis, (2009). [4] F. Zäh, M. Halder, and T. Feurer, Opt. Express 16, 16452, (2008). [5] K. A. O Donnell and A. B. U Ren, Phys. Rev. Lett. 103, , (2009). [6] A. K. O Donnell, arxiv: v1 [quant-ph], (2011). [7] A. M. Weiner, Rev. Sci. Instrum. 71, 1929, (2000). [8] SLM-S640d, Technische Dokumentation, Jenoptik, Jena, (2004).

80 72 CHAPTER 5. EXPERIMENTAL SETUP [9] S. Emanueli and A. Arie, Appl. Opt. 42, , (2003). [10] C. Bernhard, Shaping of Energy-Time Entangled Photons, PhD Thesis, (2013).

81 6 Amplitude and Phase Shaping of Energy-Time Entangled Photons Shaping femtosecond light pulses in the spectral domain is nowadays a well established procedure to generate customized pulse forms applicable in a variety of fields from coherent control of quantum systems to optical communication [1]. Thereby it has turned out, that tailoring the pulses with a SLM is particularly user friendly since no movable parts are involved in the shaping process and the device can be programmed with almost any form of a transfer function. To gain insight why an energy-time entangled two-photon state can be shaped analogous to a classical light pulse, it is illustrative to compare the expression for the latter E(t) dωe(ω)e iωt (6.1) with the two-photon wave function of Eq. (2.42) in collinear form and M i,s (Ω) = 1, i.e. ψ i,s (t 1,t 2 ) dω i dω s Λ(Ω i,ω s )e i(ω it 1 +Ω st 2 ). (6.2) It can be seen that ψ i,s (t 1,t 2 ) has exactly the same formal structure as a classical pulse except that we have in general an non-separable JSA such that no individual pulse can be assigned to the idler and signal photon. However, due to its similar form, the two-photon wave function as a whole can be modulated in the spectral domain like its classical counterpart. This has been demonstrated for spectral phase shaping only in Reference [2] and extended to simultaneous amplitude and phase modulation in Reference [3]. In the following, we first discuss how dispersion is controlled in the experimental setup. Afterwards, an interferometric autocorrelation measurement is reviewed as an example for simultaneous amplitude and phase shaping of an energy-time entangled two-photon state. 6.1 Dispersion Cancellation To investigate the effect of dispersion onto a coincidence signal we consider the first-order correlation function G (1) dω i dω s Γ PSF (Ω i,ω s )e iφ(ωi) e 2, iφ(ωs) (6.3) where φ(ω i,s ) =. φ i (Ω i ) = φ s (Ω s ) is the additional phase picked up by the idler and signal photon while propagating through the same dispersive media. We further assume that φ(ω i,s ) can be expanded in a Taylor series such that φ(ω i )+φ(ω s ) = k=0 1 k! k φ ωi,s k ωi,s =ω ci,cs }{{}. =c k 73 (Ω k i +Ω k s). (6.4)

82 74 CHAPTER 6. AMPLITUDE AND PHASE SHAPING OF ENERGY-TIME ENTANGLED PHOTONS With regard to the odd phase contributions, there is an interesting observation to made provided that SPDC is stimulated by a narrow-band pump. In this case, the pump field can be approximated by E + p (Ω i + Ω s ) δ(ω i +Ω s ) and Eq. (6.4) then reads φ(ω s )+φ( Ω s ) = = 2 k=0 l=0 c k k! (Ωk s +( 1)k Ω k s ) c 2l (2l)! Ω2l s. (6.5) The strong correlation between the idler and signal photon leads to a cancellation of the odd phase contributions. This feature is important from an experimental point of view since the prism compressor itself may only compensate for even-order dispersion. Although the narrow spectral FWHM of the pump field is finite in the experiment, the insensitivity of the coincidence signal to odd phases can still be measured by applying specific transfer functions on the SLM [4]. Note, that the phase functions k 1,2 L 1,2 /2, which arise from the SPDC and the SFG process respectively, can be absorbed in the second-order of the Taylor expansion since they are of a quadratic form. To compress the two-photon wave function to its Fourier limit, it is therefore sufficient to probe the setup to even-order phase contributions only whereas fourth-order dispersion is already strongly suppressed compared to second-order dispersion. The actual amount of GVD in the setup is then determined by programming the SLM with the transfer function M i,s c 2 (Ω) = e ic 2 2 Ω 2 (6.6) and scanning the GVD coefficient over a suitable range. The coincidence signal is then detected in dependence of c 2 and is given by 2 G (1) (c 2 ) dω i dω s Γ PSF (Ω i,ω s )e ic 2 2 (Ω 2 i +Ω2 s) (6.7) with a maximum at the net GVD value c 2net of the setup. Figure 6.1 shows a corresponding measurement with c 2net = 189 fs 2. This amount of GVD can now be compensated either by symmetrically adjusting the prisms P2 and P3 or by applying c 2net on the SLM. 1 The setup is considered to be free of GVD if a renewed measurement is symmetric around c 2net = 0 fs 2. 1 Since the prism compressor adds additional negative GVD to the setup, the net value of c 2net = 189 fs 2 can be cancelled by pulling P2 and P3 out of the beam.

83 6.2. INTERFEROMETRIC AUTOCORRELATION 75 c 2net = -189 fs 2 [Hz] Figure 6.1: GVD measurement. Coincidence counts in dependence of a varied GVD value c 2 (blue dots). Errors are based on Poisson statistics. The red line shows the simulation by means of Eq. (6.7). The resulting net GVD value c 2net is located at the peak of the coincidence signal. 6.2 Interferometric Autocorrelation The measurement of an interferometric autocorrelation (IAC) is a common procedure to characterize ultrashort laser pulses. A pulse of a coherent light source is separated into two replicas which then propagate along different optical paths within an interferometer. One replica undergoes an additional time delay τ usually applied by a mechanical delay stage. Finally, the two pulses are recombined at a beam splitter before they are sent into a nonlinear material to perform SFG. For a classical light pulse, the intensity of the up-converted signal I(τ) dt [E + (t)+e + (t τ)] 2 2 (6.8) is then detected in dependence ofτ and is sensitive to the phase present in the initial light pulse [5 7]. An unbalanced interferometer (Michelson or Mach-Zehnder) is mimicked on a SLM by the transfer function M i,s IAC (Ω) = 1 (1+e i[(ω+ωcp 2 )τ+φ] ) (6.9) 2 where the additional phaseφallows to switch between the two output ports (Figure 6.2). In contrast to the pure phase modulation of Eq. (6.6), the transfer function for an IAC requires the ability of the SLM to modulate the spectral amplitude and phase simultaneously. The signal of the coincidence photon depends on M IAC (Ω i,ω s ) = 1 4 (1+e i[(ω i+ ωcp 2 )τ+φ] )( 1+e i[(ωs+ωcp 2 )τ+φ] ) (6.10)

84 76 CHAPTER 6. AMPLITUDE AND PHASE SHAPING OF ENERGY-TIME ENTANGLED PHOTONS and reads G (1) (τ) Ω i +Ω s 0 = 2 dω i dω s Γ PSF (Ω i,ω s )M IAC (Ω i,ω s ) ( 2 τ dω i dω s Γ PSF (Ω i,ω s )e iω iτ e iωcp + dω i dω s Γ PSF (Ω i,ω s )e iωsτ ) 2 + dω i dω s Γ PSF (Ω i,ω s )(1+e iωcpτ ) = 2 e iωcp 2 τ dω i dω s Γ PSF (Ω i,ω s )e iω iτ }{{}. =A 0 + dω i dω s Γ PSF (Ω i,ω s ) }{{}. =A e iωcpτ dω i dω s Γ PSF (Ω i,ω s ) }{{}. =A 2 = 2A 0 +A 1 +A 2 2, (6.11) where we considered the beam splitter output port φ = 0. The total signal of Eq. (6.11) is a superposition of the amplitudes A 0,A 1, anda 2 representing all possible processes which occur if both photons of a two-photon state enter an unbalanced interferometer in a collinear way (Figure 6.2). The first process to happen is that the idler and signal photon travel along different paths in the interferometer before they recombine at the second beam splitter. The corresponding state of this process is 1 a 1 b after the first beam splitter. One of the two indistinguishable photons then undergoes a time delay τ which in the spectral domain corresponds to a linear phaseω i,s τ. Interference between the idler and signal photon occurs if the two photons meet at the second beam splitter within their coherence time. A subsequent coincidence signal can then be measured if both photons exit the beam splitter in the same direction. Due to the indistinguishability of the idler and signal photon this process contributes with a factor of two in Eq. (6.11). The amplitudes A 1 and A 2 rely on the processes in which both photons either go along path a or path b with equal probability. The interference signal is expected to oscillate with ω cp and is thus governed by the coherence length of the pump photon. Right after the input beam splitter, the process attributed toa 1 +A 2 generates an entangled state of the form Ψ 2002 = 1 2 ( 2 a 0 b + 0 a 2 b ) (6.12) which belongs to the class of N 00N-states first introduced in [8]. These non-classical states are of special importance in the field of quantum metrology where they serve to measure physical parameters with a precision unattainable by classical light states [9]. Figure 6.3 (b) shows the measurement of an IAC together with the theoretically predicted result in (a). Experimentally, the time delay τ has been varied in steps of 0.15 fs. Since the IAC is symmetric with respect to τ τ the measurement is performed only for τ 0. The windowed Fourier transformations (spectrograms) in Figure 6.3 (c) (theory) and (d) (experiment) give some indication of the time scales which can be attributed to the processes behind A 0 and A 1 + A 2. In fact, the signal close to

85 6.2. INTERFEROMETRIC AUTOCORRELATION 77 SFG = SFG + SFG + SFG SFG SFG SFG SFG Figure 6.2: Pictorial illustration of an IAC measurement at the single photon level. If an entangled two-photon state enters an unbalanced interferometer, the resulting coincidence signal is a coherent superposition of the following two processes: The idler and signal photon propagate in an indistinguishable manner along different paths before they interfere at the second beam splitter. The amplitude for this process is given by A 0. Or, both photons propagate along pathaandbin a superposition 2 a 0 b + 0 a 2 b before they overlap at the output beam splitter. The resulting process occurs with probability A 1 +A 2 2. (a) (b) (c) (d) Figure 6.3: SLM based realization of an IAC and intensity AC measurement. Simulated data using G (1) with Eq. (6.10) (IAC) and Eq. (6.14) (AC) are shown in (a) together with a windowed Fourier transformation in (c). Corresponding experimental data are depicted in (b) and (c). The windowed Fourier transforms reveal oscillation frequencies at ω cp /2 and ω cp. τ = 0 fs is dominated by oscillations at ω cp /2 within the coherence time τi,s c of the idler and signal photon.2 Since the spectrum of the entangled photons has a FWHM of about 105 nm their coherence time is rather short compared to the coherence time of a pump photon which is about τp c 88 ns for a corresponding bandwidth of 5 MHz. Consequently, the oscillations at ω cp continue to about 350 fs. Note, that the two-photon state used in the experiment is assumed to be close to its Fourier transform limit since all second-order dispersion has been compensated for by a proper adjustment of the prism-compressor. Figure 6.4 shows three excerpts of the theoretical and experimental data forτ < τi,s c (a),τ τc i,s (b) andτ > τc i,s (c). It is now obvious to ask why the oscillations with ω cp do not persist until τp c which one would intuitively expect. The reason for this behaviour lies in the finite spectral resolution we established in the form of a PSF in Section Figure 6.5 depicts a simulated IAC trace for Γ PSF (Ω i,ω s ) of Eq. (5.6) and Γ(Ω i,ω s ), the latter assuming a perfect resolution. The PSF leads to a continuously decreasing visibility of theω cp oscillations with increasing time delayτ which explains the corresponding observation in the experiment. On the other hand, if the setup is assumed to have a 2 The coherence time of the entangled photons was experimentally determined to be 44±0.5 fs [4], however, depends on the given experimental conditions especially the temperatures of the SPDC and the SFG crystal.

86 78 CHAPTER 6. AMPLITUDE AND PHASE SHAPING OF ENERGY-TIME ENTANGLED PHOTONS (a) (b) (c) Figure 6.4: Three excerpts of an IAC measurement. Forτ < τi,s c (a) the signal oscillates withω cp/2 and stems from interference at the second beam splitter between the idler and signal photon after having travelled along different paths in the interferometer. In (b), the time delay reached about τi,s c and thus the signal starts to oscillate withω cp. Figure (c) depicts interference fringes atω cp because of both photons propagating simultaneously along either paths of the interferometer. perfect frequency resolution, the rapid oscillations would last until a time delay of the order of τ c p 400 fs. The PSF has further influence on the contrast of the IAC which is defined as G (1) (0)/G (1) (± ) according to the definition for classical pulses [7]. Since lim τ A ωcp/2 = 0 due to a fast oscillatory integrand, we obtain G (1) (0) G (1) (± ) = 16 1+e iωcpτ 2 = 8 1 (6.13) in the case of a deactivated PSF which corresponds to the contrast one obtains for a classical IAC. Note, that signifies an average value. For a finite spectral resolution, however, 1+e iωcpτ 2 1 for τ due to the damped ω cp oscillations. Thus, we observe a 16:1 contrast in the experiment. Figure 6.5: Simulated IAC signal using Γ PSF and Γ. The red line shows the IAC signal for an experimental arrangement assumed to have perfect spectral resolution. The blue line incorporates the finite resolution by means of a PSF with a width Ω PSF = rad/fs to best fit the measured data in Figure 6.3.

87 6.2. INTERFEROMETRIC AUTOCORRELATION 79 An intensity like autocorrelation measurement can be realized on the SLM by subtracting the central frequencies ω cp /2 in the exponential functions of Eq. (6.9). The resulting transfer function reads M AC (Ω i,ω s ) = 1 4 ( 1+e i[ω iτ+φ] )( 1+e i[ωsτ+φ]) (6.14) and implies a time delay only on the slowly varying envelope of the entangled photons whereas the carrier frequencies remain unaffected. The theoretical and experimental results are shown in Figure 6.3 for φ = 0 and φ = π and show no oscillatory behaviour. The use of a SLM allows to switch between measurements at two different interferometer output ports without moving the SFG crystal. In addition, the SLM guarantees a high interferometric stability compared to measurements involving movable parts. Due to its non-trivial oscillatory structure, an IAC measurement is particularly suited to fix experimental parameters within the theoretical model [4]. References [1] A. M. Weiner, Rev. Sci. Instrum. 71, 1929, (2000). [2] A. Pe er, B. Dayan, A. A. Friesem, and Y. Silberberg, Phys. Rev. Lett. 94, , (2005). [3] F. Zäh, M. Halder, and T. Feurer, Opt. Exp. 16, 16452, (2008). [4] C. Bernhard, Shaping of Energy-Time Entangled Photons, PhD Thesis, (2013). [5] J. A. Armstrong, Appl. Phys. Lett. 10, 16, (1967). [6] K. Sala et al., IEEE J. Quantum Electron. 16, 990, (1980). [7] T. F. Feurer, G. Roberts, Femtosecond Light: Optics and Interactions, Vers , to be published. [8] H. Lee et al., J. Mod. Opt. 49, , (2002). [9] V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics 5, , (2011).

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89 7 Energy-Time Entangled Qudits Amplitude and phase shaping of energy-time entangled photons will now be used to generate, characterize and manipulate entangled qudits in the frequency modes of photon pairs. At the core of this Chapter are two publications entitled with Versatile shaper-assisted discretization of energy-time entangled photons Shaping frequency-entangled qudits followed by individual Supplementary Information Sections where additional results or explanations are provided. However, we first introduce the concept of a qudit beginning with it simplest form, the qubit. 7.1 From a Qubit to a Qudit The notion of a qubit was first introduced by Schumacher (1989) within the framework of quantum information theory [1]: The quantum bits, or "qubits", are the fundamental units of quantum information. To distinguish the classical bit from a qubit we first consider a system which is macroscopic enough to be described by classical physics. This system, named bit, is prepared in one of two possible states conventionally denoted with 0 and 1. A concrete realization of such a system could be a coin. The information that can be retrieved by measuring the actual state of the system is given by the Shannon entropy and is maximal one bit if the probabilities to find the system in state 0 or 1 are p 0 = p 1 = 1/2 [2]. Therefore, one bit corresponds to the maximum information capacity a classical two-level system can have. In contrast to a classical system, the state of an unmeasured quantum system is in general a linear superposition of all possible states the system can be in. An arbitrary quantum two-level system is therefore described by Ψ = c 0 0 +c 1 1, (7.1) where the classical probabilities p j are now replaced by the complex amplitudes c 0 and c 1. The probability to measure the system in state j is given by c j 2 and normalization of the state demands c c 1 2 = 1. The state of Eq. (7.1) is denoted as a qubit and is an element of a two dimensional Hilbert space H = span{ 0, 1 } C 2. In contrast to its classical counterpart an arbitrary amount of information can be encoded in the amplitudes c j of a qubit. However, the maximal amount of information that can be gained from the system by a measurement is still one bit [1,3]. Qubits are the primary building blocks for quantum computation processing [4]. A direct product state of n-qubits can form a superposition of 2 n states denoted as a quantum register. Each computational operation is then performed simultaneously on each individual constituent of the superposition. This behaviour is known as quantum parallelism and is exploited by quantum algorithms [5 8] to solve specific problems exponentially faster than a classical computer can do. 81

90 82 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS In general, a system is not limited to only two levels. One can think about a classical d-level system realized in the form of a dice withd-faces. The corresponding quantum d-level system is just a natural extension of the qubit and reads d 1 Ψ = c j j. (7.2) j=0 This state is then generically denoted as a qudit and is an element of H = span{ 0,..., d 1 } C d. Alternatively, a qudit can be represented with a density operator according to ˆρ = 1 d d 2 1 j=0 r jˆλj, (7.3) where the{λ j } are the generators of a SU(d) system together with ad-dimensional identity operator. Ford = 2 the {λ j } are given by the well-known Pauli spin operators. The state is fully characterized by the parameters r j R [9]. 7.2 Entangled Qudits The previous Section laid the focus on single qudit states. However, if we consider a bipartite quantum system with subsystems A and B, the most general two-qudit state reads d 1 d 1 Ψ = c jk j A k B (7.4) j=0k=0 and belongs toh = H A H B C d d. The peculiar feature of Eq. (7.4) is that this state can be entangled. If the amplitudes are such that c jk = 1/ dδ jk we obtain a maximally entangled qudit whose Schmidt number and entropy are given by d 1 K = λ 2 j j=0 Ψ = 1 d 1 j A j B (7.5) d j=0 1 d 1 1 = d j=0 2 1 = d, (7.6) d 1 d 1 1 E = λ j log(λ j ) = d log(d 1 ) = log(d) (7.7) j=0 j=0 since the eigenvalue spectrum of the reduced density operator ˆρ A = ˆρ B is λ j = 1/d. For two qudits, the density matrix has dimension d 2 and is a linear combination of all tensor product combinations of the SU(d) generators and ad 2 -dimensional identity operator ˆρ = 1 d 2 1 d 2 j=0 d 2 1 k=0 where Eq. (7.8) represents an entangled state if ρ ρ A ρ B. r jkˆλj ˆλ k, (7.8)

91 7.3. FORMAL PREREQUISITES Formal Prerequisites In Section 2.5, we have derived the state Θ of the coincidence photon after the SFG process which is subsequently annihilated by a SPCM. It was then experimentally demonstrated in Section 5.3 that the corresponding first-order correlation function G (1) is a good approximation of the detection process since all properties of the up-conversion crystal are taken into account. Such a detection scheme can, however, formally not be distinguished from an arrangement in which SPDC is assumed to generate a two-photon state Ψ(Γ) with a JSA Γ(Ω i,ω s ) = Λ(Ω i,ω s )Φ SFG (Ω i,ω s ) ( ) k2 L 2 = Λ(Ω i,ω s )sinc 2 ( exp (Ω ) i +Ω s ) 2 ( ) ( ) 2ln(2) k1 L 1 k2 L 2 ωp 2 sinc sinc 2 2 (7.9) afterwards detected by two single photon counters in coincidence (Figure 7.1). Equation (7.9) comprises the SPDC SFG SPDC Figure 7.1: Pictorial illustration between two formally equivalent detection schemes. In the upper case a SFG photon is annihilated by one single photon detector. In the lower case a two-photon state is measured in coincidence by two ideal SPCM, however, includes the properties of the SFG crystal through a modified joint spectral amplitude Γ. joint spectral amplitude Λ(Ω i,ω s ) modified by Φ SFG (Ω i,ω s ), the latter function being responsible for the acceptance bandwidth of the SFG crystal. 1 The JSA of Eq. (7.9) includes the necessary information of the SFG process while still maintaining the physics of two correlated subsystems required to generate entangled qudits. From now on we therefore consider the entangled two-photon state in the form of Ψ = dω i dω s Γ(Ω i,ω s ) Ω i Ω s, (7.10) where we have omitted the vacuum state and use Eq. (7.9). Due to its discrete nature an entangled qudit state belongs to a countable Hilbert space. The state of Eq. (7.10) is, however, entangled in a continuous degree of freedom namely its frequency. In order to encode qudits in the frequency domain we thus have to discretize Ψ by a projection into a discrete d 2 -dimensional subspace. The 1 Again, we assume the JSA to be Fourier limited through an adjustable dispersion control in the experimental setup.

92 84 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS publication in Section 7.4 makes use of the experimental setup introduced in Chapter 5 to project the state of Eq. (7.10) into various bases to generate entangled qudits. The subsequent article is restricted to frequency bins and shows quantum state tomography of entangled qudits up to d = 4. For qubits and qutrits we additionally measured the dependency of ad-dimensional Bell parameter for various degrees of entanglement. References [1] B. Schumacher, Phys. Rev. A 51, , (1989). [2] C. E. Shannon and W. W. Weaver, The Mathematical Theory of Communication, University of Illinois Press, (1949). [3] A. S. Holevo, Probl. Inf. Transm. 9, 177, (1973). [4] Collection of review articles, Science 339, 6124, , (2013). [5] D. Deutsch, Proc. R. Soc. Lond. A 400, 97, (1985). [6] P. W. Shor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, Los Alamitos, California, IEEE Computer Society Press, New York, , (1994). [7] L. K. Grover, Phys. Rev. Lett. 79, , (1997). [8] D. S. Simon, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, Los Alamitos, California, IEEE Computer Society Press, New York, , (1994). [9] R. T. Thew, K. Nemoto, A. G. White, and W. J.Munro, Phys. Rev. A 66, , (2002).

93 7.4. PUBLICATION: VERSATILE SHAPER-ASSISTED DISCRETIZATION OF ENERGY-TIME ENTANGLED PHOTONS Publication: Versatile shaper-assisted discretization of energy-time entangled photons This work has been submitted to New Journal of Physics (2013), arxiv: [quant-ph], (2013): Versatile shaper-assisted discretization of energy-time entangled photons B Bessire, C Bernhard, A Stefanov and T Feurer Institute of Applied Physics, University of Bern, CH-3012 Bern, Switzerland andre.stefanov@iap.unibe.ch Abstract We demonstrate the capability to discretize the frequency spectrum of broadband energy-time entangled photons by means of a spatial light modulator to encode qudits in various bases. Exemplarily, we implement three different discretization schemes, namely frequency bins, time bins and Schmidt-modes. Entangled qudits up to dimension d = 4 are then revealed by two-photon interference experiments with visibilities violating ad-dimensional Bell inequality Introduction Entanglement [1] is a unique feature of quantum theory having no analogue in classical physics. Spontaneous parametric down-conversion (SPDC) has been used as a source of entangled photon pairs for more than two decades [2] and provides an efficient way to generate non-classical states of light for fundamental tests of nature [3,4], for quantum information processing [5 7] or for quantum metrology [8]. Entanglement between two photons emitted by SPDC can occur in one or several (hyperentanglement [9]) possible degrees of freedom of light, namely polarization, transverse momentum and energy. Polarization entanglement [10] has been used in many experiments even involving multiple pair states [11]. However, because light supports only two polarization modes, the Hilbert space of each photon is limited to a dimension of two (qubits). On the other hand, entangling d-dimensional states denoted as qudits requires multi-mode states of light with d 2. This can be for instance achieved using a specific discretization scheme of the transverse momentum degree of freedom. Experiments have been performed where entanglement appears in a discrete set of orbital angular momentum (OAM) modes [12 17], pixel modes [18] and slit modes [19]. The manipulation and detection of transverse entanglement mainly rely on the ability to experimentally address the transverse momentum modes with the help of holograms or spatial light modulators. The entanglement content is theoretically quantified by the Schmidt number K, and is, for usual parameters of the pump laser and of the SPDC crystal, on the order of 10 to 50 for transverse wave vector entanglement [17, 20]. Similar Schmidt numbers are in reach for energy-time entanglement generated by a short pump pulse [21 23] but considerably higher values of K can be obtained for SPDC driven by a quasi-monochromatic pump laser. While high dimensional entanglement in the transverse momentum modes has been extensively studied, there is still a lack of experiments exploiting the energy degree of freedom to generate qudits for d > 2. Thus far, qudits with d = 3, 4 have been demonstrated via two-photon interferences in [24 26]. In these experiments, the entanglement was encoded in a time-bin basis realized by interferometers with multiple arms. Time-bins have been preferentially used as a basis for entangled qudits because they can be coherently manipulated by interferometers. However, the scalability to higher dimensions becomes prohibitively complex in view of interferometric stability.

94 86 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS By directly manipulating the photon spectrum, we demonstrate the implementation of various discretization schemes to realize entangled qudits. For this purpose we coherently address selected spectral components of the entangled photons by a spatial light modulator (SLM) and make use of an ultrafast optical coincidence detection, i.e. sum-frequency generation (SFG) [27, 28]. As compared to the aforementioned experiments using interferometric methods, this procedure is intrinsically phase stable and potentially scalable to very high dimensions. It has been used to investigate frequency-bin entangled qudits by quantum state tomography and Bell measurements [29]. In this paper, we demonstrate the versatility of the experiment and discretize the continuous frequency space not only in frequency bins but in different other relevant orthogonal bases. Specifically, in the time bin basis and a basis obtained by a Schmidt decomposition. Moreover, we show that the presented method is well suited to gain physical insights, e.g. with respect to the coherence time of the entangled photons in the time-bin basis. The projection into Schmidt modes further lays the groundwork for a reconstruction of the SPDC state in terms of the Schmidt basis in the frequency domain. The paper is organized as follows: In section 2 we introduce the general theoretical framework to discretize a continuous frequency space in a countable subspace together with three specific realizations: Frequency bins, time bins and Schmidt modes. Subsequently, we describe the experimental setup in section 3. In section 4, we experimentally demonstrate and quantify entanglement in the three bases by means of projective measurements equivalent to two-photon interferometry [31] in the time-bin case. Entangled qudits up to d = 4 and their visibilities are analyzed within the context of a d-dimensional Bell inequality. Finally, we conclude this paper by discussing the limitations of the setup and the possible improvements in view of higher dimensions Theory Theoretical framework In the following, the theoretical framework of the experimental results presented hereafter is discussed according to the schematic shown in figure 7.2. Preparation Manipulation Detection Physical process SPDC SLM SFG Mathematical expression Figure 7.2: The three main parts of the experiment: The preparation of energy-time entangled photons through SPDC; the subsequent manipulation of their spectra using a SLM; and the detection through SFG. Mathematical expressions for Λ(ω i,ω s ),M(ω i,ω s ), Γ(ω i,ω s ) and S are derived in the corresponding subsections Preparation A coherent superposition of energy-time entangled idler (i) and signal (s) photon pairs occurs through vacuum fluctuations if a pump photon (p) is annihilated in the SPDC process. For a configuration where all involved photons are mutually collinear and identically polarized [30] the two-photon state reads ψ = dω i dω s Λ(ω i,ω s )â i (ω i)â s (ω s) 0 i 0 s. (7.11)

95 7.4. PUBLICATION: VERSATILE SHAPER-ASSISTED DISCRETIZATION OF ENERGY-TIME ENTANGLED PHOTONS 87 The operators â i,s (ω i,s) act on the combined vacuum state 0 i 0 s to create the idler and signal photon with corresponding relative frequencies ω i,s. The normalized joint spectral amplitude Λ(ω i,ω s ) α(ω i,ω s )Φ DC (ω i,ω s ) (7.12) is written in terms of the pump envelope function α(ω i,ω s ) and the phase matching function Φ DC (ω i,ω s ) explicitly given by ( α(ω i,ω s ) = exp (ω ) i +ω s ) 2 2ln2 ωp 2, (7.13) Φ DC (ω i,ω s ) = sinc exp i ( k DC (ω i,ω s)+ 2π 2 ( k DC (ω i,ω s)+ 2π 2 G DC )L DC G DC )L DC (7.14) with a pump pulse that has a full width at half maximum of ω p in the spectral intensity. When the nonlinear crystal with length L DC is periodically poled with poling period G DC to achieve quasi-phase matching, then the efficiency for SPDC is optimal if k DC 2π/G DC. The phase mismatch k DC (ω i,ω s ) = k i (ω i ) + k s (ω s ) k p (ω i +ω s ) includes the dispersion properties of the SPDC crystal through its corresponding Sellmeier equations Manipulation The spectrum of the entangled photons is manipulated in amplitude and phase by the SLM. The modulator action on each photon is described by a complex transfer function M i,s (ω). The two-photon spectrum is transformed by the SLM to with Λ(ω i,ω s ) = Λ(ω i,ω s )M(ω i,ω s ) (7.15) M(ω i,ω s ) = M i (ω i )M s (ω s ), (7.16) where additional restrictions on M(ω i,ω s ) are time-stationarity and M i,s (ω) Detection In general, coincidence detection is indispensable to reveal entanglement. The state Λ(ω i,ω s ) could be experimentally detected by a combination of narrow-band frequency filters and single pho- ton counters. However, this would yield a signal proportional to Λ(ω i,ω s ) 2 which is insensitive to any phase modulation inm(ω i,ω s ). To circumvent this problem we seek for a detection scheme that yields a signal which 2 is proportional to F{ Λ(ωi,ω s )}, the 2D Fourier transform of the joint spectral amplitude. Such a scheme requires a time resolution better than the inverse of the photon s spectral bandwidth, which in our experiment is on the order of a few femtoseconds. This is about five orders of magnitudes smaller than the time resolution of the best single photon counters actually available. Therefore, we resort to an optical coincidence method that relies on SFG in a second nonlinear crystal. To account for its acceptance bandwidth we define the modified joint spectral amplitude Γ(ω i,ω s ) Λ(ω i,ω s )Φ SFG (ω i,ω s ) (7.17) with Φ SFG (ω i,ω s ) = sinc exp i ( k SFG (ω i,ω s) 2π ( k SFG (ω i,ω s) 2π 2 2 G SFG )L SFG G SFG )L SFG. (7.18)

96 88 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS Analogous to Φ DC (ω i,ω s ), the length and the poling period of the SFG crystal are denoted by L SFG and G SFG with a phase mismatch k SFG (ω i,ω s ) = k p (ω i +ω s ) k i (ω i ) k s (ω s ). The temporal resolution of the SFG-based detection process is governed by the inverse width of Φ SFG (ω i,ω s ) and is sufficiently short. In the following we neglect the additional phase factors in (7.14) and (7.18) since they can not be distinguished from other dispersion contributions in the setup and are assumed to be compensated in the experiment. The detected signal after the SFG process is given by 2 S dω i dω s Γ(ω i,ω s )M(ω i,ω s ) (7.19) and is sensitive to a phase in the transfer function Finite spectral resolution Because of the finite spectral resolution of the optical setup at the position of the SLM, one given frequency component illuminates several pixels of the SLM. To describe this effect, we convolve Γ(ω i,ω s ) with the point spread function to obtain PSF(ω i,ω s ) = exp ( (ω2 i +ω2 s)2ln2 ω 2 PSF ) (7.20) Γ PSF (ω i,ω s ) (Γ PSF)(ω i,ω s ). (7.21) The width ω PSF is determined by the imaging distances and the optical elements within the experimental setup. Γ PSF is in particular used to determine the Schmidt basis functions in section Figure 7.3 depicts Γ(ω i,ω s ) andγ PSF (ω i,ω s ) showing that the effect of the PSF is a considerable broadening of the joint spectral amplitude along the diagonal direction Entanglement quantification In order to quantify the degree of entanglement between the idler and signal photon we use the von Neumann entropy E = Tr(ˆρ i,s log 2 ˆρ i,s ). The entropy is commonly referred to be a valid quantifier of entanglement between two subsystems of a pure entangled state with individual density operators ˆρ i,s [32]. Through a numerical approximation method [33] we calculated the entropy of Λ(ω i,ω s ) and Γ(ω i,ω s ) for a pump spectral bandwidth of ν p = 5 MHz and further experimental parameters of the preparation and the detection crystals. For Λ(ω i,ω s ) we obtain E = (21.8±0.1) ebits. The entropy is calculated to be E = (21.1±0.2) ebits using Γ(ω i,ω s ) i.e. we observe almost no influence of the detection process on the degree of entanglement. This amount of entropy is the same as in a maximally entangled qudit of dimension d 2 with d = 2 E This demonstrates that SPDC driven by a spectrally narrow-band pump field offers a potentially very high dimensional state space to encode qudits in frequency modes. Accordingly, we calculate by numerical computation a Schmidt number K = 1/Tr(ˆρ 2 i,s ) of K Figure 7.3 shows that the effect of the PSF leads to an effective loss in the correlation between the two photons. Consequently, the values for E, d and K are reduced to E 2.6, as obtained by direct diagonalization of the reduced density matrix, d 6 and K Discretization of the frequency space The state (7.11) is a continuous superposition of frequency modes. To encode quantum information in the form of qudits, we project ψ = dω i dω s Γ(ω i,ω s )â i (ω i)â s (ω s) 0 i 0 s (7.22) into a discrete d 2 -dimensional subspace spanned by orthonormal product states j i k s with multi-mode states j i,s dωfi,s j (ω)â i,s (ω) 0 i,s and j = 0,...,d 1. The projected state then reads d 1 d 1 ψ (d) = c jk j i k s (7.23) j=0k=0

97 7.4. PUBLICATION: VERSATILE SHAPER-ASSISTED DISCRETIZATION OF ENERGY-TIME ENTANGLED PHOTONS 89 Figure 7.3: Left: Γ(ω i,ω s ) for ν p = 5 MHz, L DC = L SFG = 11.5 mm and G DC = G SFG = 9 µm. Inset: The narrow joint spectral amplitude implies a high degree of entanglement between idler and signal photon. Right: Γ PSF (ω i,ω s ) taking into account the PSF with ω PSF = rad/fs. with coefficients c jk = and the orthonormality condition Given (7.23), the probability to measure the direct product state reads dω i dω s f i j (ω i )f s k (ω s)γ(ω i,ω s ) (7.24) dωf i,s j (ω)f i,s k (ω) = δ jk. (7.25) ( d 1 d 1 ) χ = u i j j i u s k k s j=0 k=0 S (d) = χ ψ (d) d 1 2 = j,k=0 u i j us k c jk If we decompose the transfer function of the SLM into the same basis, i.e. d 1 M i,s (ω) = j=0 u i,s d 1 j fi,s j (ω) = j=0 2 (7.26). (7.27) u i,s j eiφi,s j f i,s j (ω), (7.28) we obtain S (d) = S of (7.19). Therefore, the experimentally measured signal S is given by the projection of the state ψ (d) onto χ, and the SLM together with a SFG coincidence detection performs a projective measurement. Because u i,s j andφi,s j can be adjusted independently, any state χ can be implemented provided the conditions in section are fulfilled.

98 90 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS Through a judicious choice of f i,s j (ω), various discretization schemes can be realized with the SLM using (7.28). Here, we present three different basis functions f i,s j (ω) to encode qudits in the frequency domain Frequency-bin basis An intuitive method to discretize the frequency space is to subdivide the spectrum into frequency bins through amplitude modulation (figure 7.4). The corresponding f i,s j (ω) are defined [ [ Figure 7.4: Measured SPDC spectrum overlaid with a schematic frequency-bin pattern. The transmitted amplitude u i,s j (white bars) of each bin can be adjusted through amplitude modulation by means of the SLM. according to { f i,s 1/ ωj for ω ω j < ω j j (ω) = 0 otherwise, (7.29) where we impose ω j ω k > ( ω j + ω k )/2 for allj,k to guarantee that adjacent bins do not overlap. If we further assume a continuous wave pump we restrict (7.23) to its diagonal form d 1 ψ (d) = c j j i j s. (7.30) j= Time-bin basis Frequency entangled photons are typically manipulated in the time domain by interferometers with variable optical path lengths [24 26]. Up to now, time-bins have been preferentially used to encode quantum information but they are also used to analyze the temporal properties of the down-converted photons, a concept that was first proposed by Franson [31]. In his interferometric scheme, each photon of a pair enters an unbalanced Mach-Zehnder interferometer where both photons undergo the same time delay t 10 when traveling along the long path (figure 7.5). For t 10 τp coh the state generated by the first pair of beam splitters is a coherent superposition 1 1 ψ (2) = c jk j i k s, (7.31) j=0k=0 where we associate 0 i,s with the short and 1 i,s with the long path of the interferometer. Characteristic qubit interference fringes can be observed by a coincidence detection between the output ports A and B while varying the phases φ i and φ s. Franson s original experiment can be extended to higher dimensional qudits by endowing each interferometer with additional arms. Two four-arm interferometers were used in Reference [26]

99 7.4. PUBLICATION: VERSATILE SHAPER-ASSISTED DISCRETIZATION OF ENERGY-TIME ENTANGLED PHOTONS 91 SPDC Figure 7.5: Scheme to analyze energy-time entangled two-photon states proposed by Franson [31]. A photon pair is generated by SPDC within the coherence time τp coh of a pump photon. Both photons are injected into two separate unbalanced Mach-Zehnder interferometers with a time delay of t 10 imprinted on the photon in the longer arm. A qubit state is then measured by varying the phases φ i and φ s while performing coincidence measurements between output ports A and B. to experimentally demonstrate energy-time entangled ququarts. Instead of using interferometers, we discretize the time domain of idler and signal photons into time bins (figure 7.6) f i,s j (t) = 2π t j for t t j < t j 0 otherwise (7.32) analogous to (7.29) in the frequency domain with the help of the SLM. The coefficients in (7.31) are now related Figure 7.6: Franson s original scheme adapted to time bins implemented by a SLM. Under specific conditions for t 10 and t j (see text) projections onto a superposition of 0 i 0 s and 1 i 1 s can be measured. to the joint temporal amplitude Υ(t i,t s ) of the SPDC photons through c jk = tj + t j 2 t j t j 2 tk + t k 2 t k t k 2 dt i dt s Υ(t i,t s ), (7.33) where Υ(t i,t s ) is the Fourier transform of Γ(ω i,ω s ). Since the SLM manipulates the frequency spectrum of the entangled photons, the time bins of (7.32) are Fourier transformed to obtain f i,s j (ω) = t j 2π e iωt j sinc ( ω tj 2 ), (7.34)

100 92 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS where orthonormality holds for any j and k provided that t j t k > ( t j + t k )/2. In order to obtain an entangled qubit, the time delay t 10 = t 1 t 0 has to exceed the coherence time τi,s coh of the idler and signal photon to avoid single photon interference. A coincidence window for SFG detection is typically on the order of a few femtoseconds, which itself corresponds to τi,s coh for broadband SPDC emission. It is guaranteed for t 10 > τi,s coh that no 0 i 1 s and 1 i 0 s events contribute to the coincidence signal. Further, to prevent (7.34) acting as a filter on the entangled photons spectra, t j is restricted to t j τi,s coh. Given the constraints on t 10 and t j are satisfied, the event that both photons of a pair, down-converted at a given time, pass time bin f i,s 1 (t) (long arm) can not be distinguished from two photons created after a time delay t 10 and traveling through bin f i,s 0 (t) (short arm). The state generated by two time bins then reads ψ (2) = c 0 0 i 0 s +c 1 1 i 1 s. (7.35) To demonstrate the equivalence between Fransons (FR) interferometric scheme and the implementation of time-bins in the frequency space we consider the transfer function for a single photon in a Mach-Zehnder interferometer M i,s FR (ω) = T +Rei φ i,s (ω), (7.36) where both beam splitters have transmission and reflection coefficientst andr. The total phase shift φ i,s (ω) = ω t 10 +φ i,s is the sum of the phase differences between the long and the short arm ω t 10 and an additional absolute phase φ i,s. The total transfer function of the scheme depicted in figure 7.5 then reads M FR (ω i,ω s ) = MFR i (ω i)mfr s (ω s). If we restrict in (7.34) the width of the bins to t 0 = t 1 = t and put t 0 = 0 fs such that t 10 = t 1, the general transfer function from (7.28) transforms to ( ) M i,s t ω t ( ) (ω) = 2π sinc u i,s ui,s 1 ei φ i,s (ω) (7.37) for d = 2 with φ i,s (ω) = ωt 1 + φ i,s. In the limit t j 0, the time-bins in (7.32) are reduced to f i,s j (t) = 2πδ(t tj ). Equation (7.37) then takes the form M i,s (ω) = 1/ 2π( T +Re i φ i,s (ω) ) and is equal to M i,s FR (ω) witht = ui,s 0 and R = ui,s 1 up to a normalization constant Schmidt mode basis The Schmidt decomposition for continuous variable systems and its application to quantify entanglement has been extensively studied in [21,34]. To decompose a state which is as close as possible to the state at the position of the SLM we have to consider the effective joint spectral amplitude Γ PSF taking into account the effect of the finite spectral resolution. Because of being bipartite and pure, the two-photon state of (7.22) can then be represented in a Schmidt decomposition Γ PSF (ω i,ω s ) = j=0 βj f i j (ω i)f s j (ω s) d 1 j=0 βj f i j (ω i)f s j (ω s) (7.38) where the real valued functionsf i,s j (ω) are the eigenvectors or Schmidt modes of the reduced density operators and β j the corresponding eigenvalues. The Schmidt modes itself are orthogonal and form a complete basis. As can be seen in figure 7.7 only a fewβ j are significantly nonzero if we decomposeγ PSF (ω i,ω s ). By substituting Γ PSF (ω i,ω s ) in (7.22) and using (7.38) one obtains d 1 ψ ψ (d) = c j j i j s (7.39) with c j = β j. The Schmidt basis thus provides a direct way to discretize the state (7.11) into an entangled qudit state whose dimensionality is only bound by the number of nonzero β j. For a symmetric joint spectral amplitudeγ PSF (ω i,ω s ) (figure 7.3) we findfj i(ω) = fs j (ω). The Schmidt decomposition has been performed numerically by discretizing the continuous function Γ PSF (ω i,ω s ) on a lattice with size j=0

101 7.4. PUBLICATION: VERSATILE SHAPER-ASSISTED DISCRETIZATION OF ENERGY-TIME ENTANGLED PHOTONS 93 Figure 7.7: Left: Exemplary, the first three Schmidt modes of (7.38). Right: Shown are the eigenvalues β j up to j = Experimental setup The experimental setup shown in figure 7.8 is composed of three parts: The entangled state preparation by SPDC, the spectral manipulation with the SLM and the detection by SFG. Preparation (SPDC) Manipulation (SLM) Detection (SFG) SLM Figure 7.8: Schematic of the experimental setup. Preparation: L0 pump beam focusing lens (f = 150 mm), PPKTP nonlinear crystal for SPDC. Manipulation: BD beam dump, SLM spatial light modulator, symmetric two lens (L1, L2) imaging arrangement (f = 100 mm) to magnify the spectral resolution by 1:6, four-prism compressor. Detection: PPKTP nonlinear crystal for SFG, BF bandpass filter, SPCM single photon counting module with a two lens (L3, L4) imaging system. Energy-time entangled photons are created by SPDC in a periodically poled KTiOPO 4 (PPKTP) crystal with

102 94 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS length L DC = 11.5 mm and a poling periodicity of G DC = 9 µm. The pump is a single mode 5 W Nd:YVO 4 (Coherent Verdi V5) laser operating at 532 nm with a spectral bandwidth of about ν p = 5 MHz. According to type-0 phase matching, the created idler and signal photon have the same polarization as the pump photon. The operating temperature of the PPKTP crystal is optimized for almost degenerate and collinear emission with a spectral width of the down-converted photons of λ DC 105 nm centered around 1064 nm. To control the down-converted spectrum it is dispersed in a symmetric four-prism compressor consisting of equilateral N-SF 11 prisms in minimum deviation geometry. At the same time, the prism compressor serves to compensate for the total accumulated group velocity dispersion in the setup and to deflect the residue of the pump into a beam dump. At the symmetry axis of the prism compressor the dispersed spectrum passes two identical nematic liquid crystal arrays of a programmable SLM (Jenoptik, SLM-S640d). Both arrays consist of 640 pixels each 100 µm wide and separated by a gap of 3 µm from its nearest neighbors. The transmitted frequencies at each pixel are manipulated independently in amplitude and phase by adjusting the orientation of the nematic molecules with a specific voltage. A second, identical, PPKTP crystal is phase matched to detect entangled photons in coincidences through SFG [27]. This provides a coincidence time window with femtosecond temporal resolution. The sum-frequency photons are detected by a single photon counting module (ID Quantique, id uln) and the remaining IR photons are filtered by a bandpass filter (4 mm BG18). According to Reference [35], the maximal allowed flux of down-converted photons at the single photon limit is given byφ max ν DC. A spectral bandwidth of λ DC 105 nm corresponds to a maximal flux ofφ max = photons per second or a maximal power ofp max = 5.2µW. That is, for the actual power of 1µW we find a spectral mode density ofn = P/P max = 0.2 which assures that we are below the single photon limit and therefore no coincidences are measured between photons of different pairs Experimental results CGLMP inequality In order to show entanglement without performing full quantum state tomography it is sufficient to demonstrate two-photon interference by projecting (7.23) onto χ = 1 ( d 1 d 1 ) e ijφ i j i e ikφs k s (7.40) d j=0 with φ i = φ s = (φ + φ 0 ) and φ 0 being introduced as a fitting parameter as discussed below. We assume the qudits in our experiment to be described by a symmetric noise model k=0 ˆρ (d) = λ d ψ (d) (d) ψ +(1 λ d )½ d 2/d 2, (7.41) where deviations from a pure state are quantified by λ d and ½ d 2 denotes the d 2 -dimensional identity operator. Here, ψ (d) is presumed to be maximally entangled and the coincidence signals S (2) ) λ (ˆρ (φ) = Tr (2) χ χ and S (3) ) λ (ˆρ (φ) = Tr (3) χ χ S (4) ) λ (ˆρ (φ) = Tr (4) χ χ 1+λ 2 cos2(φ+φ 0 ), (7.42) 3+2λ 3 [2cos2(φ+φ 0 )+cos4(φ+φ 0 )], (7.43) 4+2λ 4 [3cos2(φ+φ 0 )+2cos4(φ+φ 0 )+cos6(φ+φ 0 )] (7.44)

103 7.4. PUBLICATION: VERSATILE SHAPER-ASSISTED DISCRETIZATION OF ENERGY-TIME ENTANGLED PHOTONS 95 are used to fit the experimental data with the free parameters φ 0 and λ d. The latter accounts for white noise as well as for the point spread function and for couplings between frequency and transverse modes due to a possible misalignment in the experimental setup. The parameter λ d serves to analyze the non-local features of the generated qudits. In Reference [36], Collins et al. (hereafter referred to as CGLMP) introduced a new family of Bell inequalities to study the non-local properties of d-dimensional bipartite quantum states. Correlations between two separated systems can be explained by local realism if I d 2 for a dimensional dependent Bell parameter I d. In our experiment, the coincidence detection is not performed by measuring the two sub-systems remotely with space-like separation. Nevertheless, ) if we observe I d > 2, the two systems involved are correlated by entanglement. Because of Tr(ˆBˆρ (d) = λ d Id max for ad 2 -dimensional Bell operator ˆB [37], the aforementioned inequality can be reformulated in terms of the noise inherent in the state λ d > 2 I max d. = λ c d. (7.45) The parameter Id max denotes the strongest possible violation of Bell s inequality under the assumption that two parties share a maximally entangled state. The critical values for the noise parameter λ c d up to d = 4 are listed in table 7.1. For increasing d one finds a decreasing λ c d and thus the violation of the Bell inequality is more robust against noise for higher dimensional qudits [36]. Since we experimentally demonstrate qudits in terms of interference patterns, we make use of the fact that λ d can be related to a visibility [25] according to V d = Entanglement between idler and signal photon is then present if dλ d 2+λ d (d 2). (7.46) V d > V d (λ c d ). = V c d. (7.47) The values for the critical visibility Vd c are listed in table 7.1. To relate CGLMP s inequality to the visibility of interference fringes has the advantage that a possible phase shift φ 0 present in the experiments, e.g. due to non perfectly compensated dispersion, has no influence on the violation of a Bell inequality. This would be the case if single projection measurements at fixed phase settings were used to determine the value of a Bell parameter Frequency-bin basis With the goal to maximize the entanglement in d 1 ψ (d) = c j j i j s, (7.48) j=0 we make use of the Procrustean method of entanglement concentration [32]. In general, this method equalizes the amplitudes in a partially entangled state through local operations where contributions with higher probabilities are diminished by appropriate filtering. One is then left with a maximally entangled state ψ (d) = 1 d 1 j i j s (7.49) d with all amplitudes being equal. For a bin structure according to (7.29) we experimentally equate the c j by performing single projection measurements onto χ k = u k 2 k i k s fork = 0,...,d 1 with corresponding coincidence signals S k = u k 2 c k 2.. The amplitude of the k-th frequency bin u k = u i k = us k is then adjusted with the SLM such that all S k are equal to S min = min {S k} within their errors using Poisson k=0,...,d 1 statistics. The measurement time for each S k was 300 s with a SPCM background coincidence rate of about j=0

104 96 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS Figure 7.9: Frequency-bin basis: Two-photon interferences for a maximally entangled qubit (blue, top left), qutrit (green, top right) and ququart (red, bottom). Shown are net counts with 1σ standard deviations using Poisson statistics and the solid curves are fits to the data. Table 7.1: Frequency-bin basis: Critical and fitted values for the noise parameter (λ c d,λ d) and the visibility (V c d,v d) for different dimensions d. The1σ standard deviations are based on Poisson statistics. d λ c d λ d V c d ± ± ± ± ± ±0.008 V d 11 Hz. Since the prism compressor is adjusted such that any dispersion in the setup is compensated, we have arg(c j ) = 0, i.e. all probability amplitudes are assumed to be real valued. Figure 7.9 depicts the measured two-photon interference fringes for qudits up to d = 4. Equations (7.42), (7.43) and (7.44) are used to fit the data. The fitting parameters λ d and the visibilities V d are summarized in table 7.1 and we find λ d > λ c d and V d > Vd c for all d which demonstrates the existence of frequency-bin entanglement Time-bin basis Due to the finite spectral resolution of the optical setup the time-bin basis supports qubits only. To make the individual bins as narrow as possible, all measurements are performed with t 0 = t 1 = 0 fs. In order to obtain maximally entangled states, the transfer function on the SLM reads M i,s (ω) = u i,s 0 + ui,s 1 ei(ωt 1+φ i,s ) (7.50) with u i,s 0 = ui,s 1 = 1/2 which is equivalent to using beam splitters with T = R = 1/2 in the Franson experiment (figure 7.5). Although we choose t 0 = t 1 = 0 fs in the actual experiment, the time bins,

105 7.4. PUBLICATION: VERSATILE SHAPER-ASSISTED DISCRETIZATION OF ENERGY-TIME ENTANGLED PHOTONS 97 however, are always of finite width since the transfer function of (7.50) is limited inω due to the finite aperture of the SLM. Note, that in the present experiment the coherence time of the entangled photons is always larger than t j and is thus the limiting factor for the minimal t 10. In figure 7.10 we depict the interference traces for various qubits where the time-bin f i,s 0 (t) is fixed at t 0 = 0 fs and t 10 = t 1 and therefore bin f i,s 1 (t) is scanned in constant time increments. If both bins completely overlap, i.e. t 1 = 0 fs, the coincidence signal Figure 7.10: Time-bin basis: Normalized two-photon interferences (net counts) for a qubit encoded in timebins. The width of the bins is chosen to be t 0 = t 1 = 0 fs and t 0 is fixed to t 0 = 0 fs. For t 1 = 0 fs the experimental data are described by (7.51). Since the visibility of the measured signal is equal to one no additional fitting parameter is needed. All other measurements are fitted to (7.54). consists of a product of two single photon interference rates S (2) (φ) = χ ψ (2) 2 1+e i(φ+φ 0) 2 1+e i(φ+φ 0 ) 2 cos 4 ( φ+φ0 2 ) (7.51) with ψ (2) = 1 2 ( 0 i + 1 i )( 0 s + 1 s ). (7.52) Between t 1 25 fs and t 1 50 fs the contribution to the coincidence rate due to single photon interference decreases since t 1 begins to exceed the coherence time of the entangled photons. Consequently, the signal in figure 7.10 approaches the interference pattern of a maximally entangled qubit. The transformation from a non-entangled to a maximally entangled state can only be measured because sum frequency generation in a nonlinear crystal offers a coincidence window on the same time scale as τi,s coh. Note, that all measurements in

106 98 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS figure 7.10 are performed for t 1 τp coh 88 ns. To model the transition from a non-entangled qubit to a maximally entangled qubit we consider the state ψ (2) = 1 ( 0 i 0 s +γ 1 [ 0 i 1 s + 1 i 0 s ]+γ 2 1 i 1 s ) (7.53) 1+2γ1 2 +γ2 2 with parameters γ 1 and γ 2 quantifying the contribution of the one-photon and two-photon interferences to the coincidence rate χ ψ (2) 2 S (2) γ 1,γ 2 (φ) = The signal of (7.54) has the property that 1+2γ 1 e i(φ+φ 0) +γ 2 e i2(φ+φ 0) 2. (7.54) lim γ 1 0 S(2) γ 1,γ 2 (φ) = S (2) (φ) (7.55) with S (2) λ (φ) of (7.42) and λ 2 = V 2 = 2γ 2 /(1 + γ 2 2 ). In the limit of γ 1 0 it is therefore indistinguishable if a measured signal stems from a non-maximally entangled state given by (7.53) with γ 2 < 1 or a maximally entangled state with reduced visibility λ 2 < 1 related to (7.41). The behaviour of γ 1,2 for t 1 25 fs is shown in figure It can be seen that γ 1 0 for t 1 50 fs. In this regime, t 1 largely exceeds the coherence time of the idler and signal photon and thus maximally entangled qubits with reduced visibilities can be observed. The theoretical curves are based on (7.19) with Γ(ω i,ω s ) and Γ PSF (ω i,ω s ). The result implies that the reduction in the visibility of the qubits can be to some extent attributed to the finite spectral resolution at the SLM. The experimental parameters, including ω PSF, used in the simulations are determined by other measurements [38]. Since the states incorporated to calculate the theoretical curves are pure, the deviation between theory and data points is therefore caused by impurities in the experimentally realized state in combination with imperfections in the alignment of the optical setup. For maximally entangled states, the CGLMP inequality can be related to the visibility V 2 of the interference fringes as discussed in section On the other hand, the visibility is not a well-defined quantity for the state of (7.53) with γ 1 > 0. To study whether the generated qubits reveal entanglement we thus consider the Bell parameter I 2. This parameter is commonly determined by a series of projective measurements onto (7.40) for specific angles (φ i,φ s ) [29,36]. We calculate I 2 (figure 7.12) by a combination of single projection signals expressed as S (2) γ 1,γ 2 (φ i,φ s ) λ 1+γ 1 e i(φ i+φ s) +γ 2 (e iφ i +e iφs ) 2. (7.56) and the fitted values for γ 1 and γ 2 The corresponding values fori 2 are in fact underestimated since the settings for(φ i,φ s ) used to calculate the Bell parameter are only optimal in the case of maximally entangled qubits. Up to t 1 = 50 fs figure 7.11 reveals a strong decrease in γ 1 whereas γ 2, responsible for the visibility of the qubit, only weakly decreases. This distinct behavior of γ 1 and γ 2 leads to an increase of I 2 for 25 fs t 1 50 fs. Towards larger t 1 we observe a further decrease inγ 2 due to the PSF of the setup. This reduces the visibility of the maximally entangled qubits and hence the CGLMP inequality I 2 2 becomes satisfied. Consistently, the values of V 2 are below V c for t fs (table 7.2) Schmidt mode basis To encode qudits in the Schmidt decomposition of the idler and signal photon, the transfer function (7.28) consists of a linear combination of the Schmidt modes f i,s j (ω). Figure 7.13 shows the corresponding twophoton interference fringes where (7.42) and (7.43) are used to fit the data points. Due to the combination of even and odd Schmidt modes an additional phase shift of π/2 can be observed in the measured curves (top

107 7.4. PUBLICATION: VERSATILE SHAPER-ASSISTED DISCRETIZATION OF ENERGY-TIME ENTANGLED PHOTONS 99 Figure 7.11: The fitting parameters γ 1 (green dots) and γ 2 (blue diamonds) of (7.54) together with the 1σerrors as a function of t 1. Theoretical curves for γ 1 andγ 2 are calculated withγ(ω i,ω s ) (γ 1 : green dashed, γ 2 : blue dashed) and Γ PSF (ω i,ω s ) (γ 1 : green straight, γ 2 : blue straight) using (7.19). Table 7.2: Time-bin basis: Fitted values of the visibility V 2 for t 1 50 fs where maximally entangled qubits are observed. The critical visibility for a qubit isv c t 1 [fs] V ± ± ± ± ±0.011 row). The corresponding fitting parameters are summarized in table 7.3 and table 7.4. In all measurements, the critical values for a Bell violation are exceeded. The shape of the measured qudits, however, strongly depends on the deviations between the calculated Schmidt modesf i,s j (ω) and the experimentally realized Schmid modes. Together with a low count rate this leads to the small discrepancies between the data points and the fits of (7.42) and (7.43) Experimental limitations Using a frequency-bin discretization, we have demonstrated maximally entangled qudits up to d = 4. The actual limitation to reach higher dimensions is the finite spectral resolution at the position of the SLM. If we increase the density of bins, frequencies from adjacent bins will overlap and the orthonormality condition of (7.25) begins to fail. This leads to a decrease in the entanglement due to non-vanishing 0 i 1 s, 1 i 0 s

108 100 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS Figure 7.12: The red dots show the Bell parameter I 2 in dependence of the experimentally evaluated fitting parameters γ 1,2 for various t 1 with 1σ uncertainties. The dashed black line indicates the local realism limit. Theoretical predictions are calculated using (7.17) (red dashed) and (7.21) (red solid) taking into account the finite spectral resolution of the experimental setup. Table 7.3: Schmidt mode basis. Critical and fitted values for the visibility (V c 2,V 2) dependent on the combination of Schmidt modes. The1σ standard deviations are based on Poisson statistics. { } f i,s { 0,fi,s 1 } f i,s 0,fi,s 2 V c 2 V ± ±0.028 contributions in (7.23). For time bins, the reduction of the Bell parameter for large time delays between the two bins can also be attributed to the finite spectral resolution. Here, it constrains even more the maximally accessible dimension. Both of these methods have, however, the advantage that a perfect knowledge of the two-photon state is not required in order to discretize a maximally entangled state. In contrast, the Schmidt decomposition is very sensitive to the form of the state generated by SPDC. It is likely, that Γ PSF (ω i,ω s ) deviates from the joint spectral amplitude realized in the experiment due to a lack in the precise knowledge of all the physical parameters used to compute the Schmidt modes. This may lower the quality of the qudit states obtained by that method Conclusion and outlook We have presented a thorough theoretical exposition how to encode qudits in the frequency domain of broadband energy-time entangled photons whose coincidences are detected through SFG in a nonlinear crystal. Al-

109 7.4. PUBLICATION: VERSATILE SHAPER-ASSISTED DISCRETIZATION OF ENERGY-TIME ENTANGLED PHOTONS 101 Figure { 7.13: Schmidt } mode { basis: Left: Two-photon } interferences for an entangled qubit. Schmidt modes f i,s 0 (ω),fi,s 1 (ω) (top) and f i,s 0 (ω),fi,s 2 (ω) (bottom) are used. Right: Two-photon interferences for an { } { } entangled qutrit. Schmidt modes f i,s 0 (ω),fi,s 1 (ω),fi,s 2 (ω) (top) and f i,s 0 (ω),fi,s 2 (ω),fi,s 4 (ω) (bottom) are used. Table 7.4: Schmidt mode basis. Critical and fitted values for the noise parameter (λ c 3,λ 3) and the visibility (V3 c,v 3) dependent on the combination of Schmidt modes. The 1σ standard deviations are based on Poisson statistics. { } f i,s { 0,fi,s 1,fi,s 2 } f i,s 0,fi,s 2,fi,s 4 λ c 3 λ 3 V c 3 V ± ± ± ±0.024 though applied to the specific case of a spatial light modulator, the discretization procedure of the frequency space is general and describes a unified framework for different detection schemes. It has been discussed, that the entanglement content in a two-photon state generated by continuous wave parametric down-conversion is very high, however, is reduced if the finite spectral resolution of the experimental setup is accounted for. The flexibility of a spatial light modulator has been exploited in order to project the energy space of the entangled photons in different basis. In particular, we implemented frequency and time bins to measure maximally entangled qudits through two-photon interference fringes. All qudits have been investigated in view of their entanglement properties by means of a generalized Bell-type inequality. The time bin scheme allowed to show the transition from a separable to a maximally entangled qubit state taking advantage of an ultrafast detection method with femtosecond temporal resolution. In addition, we expressed the two-photon wave function in a Schmidt decomposition to use the resulting modes as a further basis for qudits. As the current limitation in the quality and dimension of the generated states, we identified the finite spectral

110 102 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS resolution at the position of the SLM in combination with the size and number of its pixels. A way to improve the spectral resolution is to replace the prisms with gratings. This would allow to spatially disperse the spectrum along a wider range of the SLM display. The achievable dimension becomes then only limited by the number of pixels of the shaping device. The low efficiency coincidence detection method using SFG therefore constitutes a further bound on the dimension of the qudits. A higher coincidence rate could be achieved by generating SFG in a waveguide instead of a bulk crystal [39] or using enhanced detection schemes [40]. That a state can be precisely characterized in terms of its Schmidt decomposition has been shown in [41] for transverse momentum entangled photons. The here presented experimental setup allows for projective measurements on single Schmidt modes f i,s j (ω) in the frequency domain. A state reconstruction similar to [41] can then be performed in the entangled photons energy-time degrees of freedom provided the correct eigenfunctions are known. The obtained β j can then further be used to calculate the Schmidt number to quantify the entanglement in the state. The above mentioned improvements will allow to encode qudits in dimensions inaccessible by standard interferometry. Energy-time entangled photon states are an essential tool of future quantum communication networks. The ability to increase the dimension of the encoded states would substantially increase the rate of quantum communication. The method presented here allows to manipulate and characterize this entanglement in a very flexible way. Acknowledgements This research was supported by the grant PP00P2_ and by the NCCR MUST, both funded by the Swiss National Science Foundation. References [1] Horodecki R, Horodecki P, Horodecki M and Horodecki K 2009 Rev. Mod. Phys [2] Ghosh R and Mandel L 1987 Phys. Rev. Lett [3] Zeilinger A 1999 Rev. Mod. Phys [4] Genovese M 2005 Phys. Rep [5] Gisin N, Ribordy G, Tittel W and Zbinden H 2002 Rev. Mod. Phys [6] Gisin N and Thew R 2007 Nature Photonics [7] Kok P, Munro W J, Nemoto K, Ralph T C, Dowling J P and Milburn G J 2007 Rev. Mod. Phys [8] Giovannetti V, Lloyd S and Maccone L 2011 Nature Photonics [9] Barreiro J T, Langford N K, Peters N A and Kwiat P G 2005 Phys. Rev. Lett [10] Kwiat P G, Waks E, White A G, Appelbaum I and Eberhard P H 1999 Phys. Rev. A 60 R773-R776 [11] Bouwmeester D, Pan J-W, Daniell M, Weinfurter H and Zeilinger A 1999 Phys. Rev. Lett [12] Dada A C, Leach J, Buller G S, Padgett M J and Andersson E 2011 Nature Physics [13] Di Lorenzo Pires H, Monken C H and van Exter M P 2009 Phys. Rev. A [14] Mair A, Vaziri A, Weihs G and Zeilinger A 2001 Nature [15] Agnew M, Leach J, McLaren M, Stef Roux F and Boyd R W 2011 Phys. Rev. A

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112 104 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS 7.5 Supplementary Information Single Photon Measurements in the Time-Bin Basis We showed interference fringes of time-bin entangled qubits by coincidence detection in a nonlinear crystal in Section Here, additional results are presented where we performed the measurement depicted in Figure 7.10 by detecting the intensity of the single photon signal with a powermeter. The powermeter was positioned according to Figure 5.1 together with a linear polarizer which guarantees a polarization dependent detection. The expected intensity signal is then given by I (2) γ 1,γ 2 (φ) = Tr(ˆρ i,s χ i,s i,s χ ) 1+V i,s 2 (γ 1,γ 2 )cos(φ+φ 0 ) (7.57) with a reduced density operator ˆρ i,s = Tr s,i (ˆρ) equal for both subsystems and ˆρ = ψ (2) (2) ψ by means of Eq. (7.53). Since we perform a measurement on only one of the two photons, the corresponding projection state reads χ i,s = 1 1 e ijφ i,s j i,s (7.58) 2 j=0 with φ i,s = φ + φ 0. It is now interesting to consider the behaviour of Eq. (7.57) in the limiting case of a maximally entangled qubit (γ 1 = 0,γ 2 = 1) and a separable qubit (γ 1 = 1,γ 2 = 1). Because the visibility of the single photon signal is calculated to be V i,s 2 (γ 1,γ 2 ) = 2γ 1(1+γ 2 ) 1+2γ 1 +γ 2 2 (7.59) we obtain I (2) γ 1,γ 2 (φ) { 1 forγ1 = 0,γ 2 = 1 1+cos(φ+φ 0 ) forγ 1 = 1,γ 2 = 1. (7.60) For a maximally entangled qubit with a visibility of one we therefore expect the corresponding intensity signal to be constant whereas for a non-entangled state, Eq. (7.60) predicts a sinusoidal form with unit visibility. Figure 7.14 depicts the experimental results for selected positions of the time bin f i,s 1 (t) where fi,s 0 (t) is kept fix at t 0 = 0 fs. Equation (7.57) is used to fit the data points of the single photon signal (magenta). For t 1 = 0 fs (a), the coincidence signal (blue) is fitted with Eq. (7.51) derived for γ 1 = γ 2 = 1, i.e. a fully separable state. According to Eq. (7.59) this, however, implies a visibility of one for the single photon signal which the measurement does not exhibit. The reduced visibility could be explained through a misalignment of the polarizer. For increasing t 1 ((b)-(e)), the single photon signal keeps its sinusoidal behaviour, however, with decreasing visibility. In this regime, the coincidence signal starts to approach the form of a maximally entangled qubit since t 1 exceeds the coherence time of the entangled photons. Consequently γ 1 0 which leads to a decreasing V i,s 2 (γ 1,γ 2 ). Nevertheless, since γ 1 = 0 does not exactly hold in the experiment, we do not observe the constant signal predicted by Eq. (7.60). The data in Figure 7.14 shows a slightly different behaviour compared to the experimental outcomes in Figure 7.10 for the same values of t 1. This is due to the fact that the experiments were performed on different days where a small realignment of the setup led to different experimental conditions State Reconstruction in the Schmidt Basis Straupe et al. reported a proof-of-principle experiment which demonstrates the reconstruction of a transverse momentum correlated JSA in the form of a Schmidt decomposition [41]. The JSA itself is approximated by a double Gaussian model to obtain an analytical expression for the Schmidt modes and the corresponding eigenvalues. The double Gaussian model has the advantage that the resulting JSA is of particularly simple form

113 7.5. SUPPLEMENTARY INFORMATION 105 S/I [a.u.] (a) (b) S/I [a.u.] (c) (d) S/I [a.u.] (e) Figure 7.14: Time-bin basis. Single photon signal (magenta) and coincidence signal (blue) for various positions t 1 of the second bin. (a): t 1 = 0 fs, (b): t 1 = 50 fs, (c): t 1 = 100 fs, (d): t 1 = 150 fs, (e): t 1 = 200 fs. The signal of the powermeter has been averaged over 30 s for a single data point whereas coincidences were averaged over 120 s. The subtracted background coincidence counts are about 11 Hz. Equation (7.57) is used to fit the data points in the case of the single photon signal. The two-photon interferences are fitted with Eq. (7.51) for (a) and Eq. (7.54) for (b)-(e). with only two experimentally adjustable parameters. The transverse modes of the idler and signal photon are then independently projected into selected Schmidt modes by means of a SLM and a phase-step plate. The resulting coincidence counts are related to the eigenvalues whose knowledge is sufficient to reconstruct the two-photon state in the a priori given Schmidt basis. We experimentally demonstrated the projection of the two-photon state onto single Schmidt modes in Here, we formally show that the coincidence signal obtained by such projective measurements can be directly related to the eigenvalues β j in the Schmidt decomposition Γ PSF (ω i,ω s ) = j=0 βj f i j (ω i)f s j (ω s) d 1 j=0 βj f i j (ω i)f s j (ω s). (7.61) It is worth to notice that for a symmetric JSA the Schmidt modes for the idler and signal photon are identical and thus f j (ω) =. f i,s j (ω) R. This property is crucial from an experimental point of view since our setup does actually not allow to individually project the idler and signal onto different Schmidt modes. By means of the Schmidt decomposition and the transfer function M(ω i,ω s ) = d 1 k,l=0 u k u l f k (ω i )f l (ω s ) (7.62)

114 106 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS with u k =. u i k = us k. The corresponding coincidence signal reads 2 S = dω i dω s Γ PSF (ω i,ω s )M(ω i,ω s ) 2 d 1 = β j u k u l dω i f j (ω i )f k (ω i ) dω s f j (ω s )f l (ω s ) j,k,l=0 }{{}}{{} =δ jk =δ jl d 1 2 = β j u k u l δ jk δ jl j,k,l=0 d 1 2 = β j u j 2. (7.63) j=0 The coincidence signal of a single projection measurement onto Schmidt modef j (ω) with is given by The normalization condition d 1 j =0 β j = 1 implies M(ω i,ω s ) = u j u j f j (ω i )f j (ω s ) (7.64) S j = β j u j 4. (7.65) d 1 j =0 S j u j 4 = 1 (7.66) and thus the normalized eigenvalue of thej -th Schmidt mode is related to the corresponding coincidence count rate by β j = S j d 1 j =0 S j, (7.67) where we have chosen the amplitude u j = 1. It remains to determine the correct Schmidt modes with as few experimental parameters as possible. Similar to the case of transverse momentum entanglement, there exists a double Gaussian model for a frequency correlated JSA which seems expendable to approximate Γ(ω i,ω s ) in the case of a long pump pulse. 2 The analytic f j (ω) could then be used as Schmidt basis functions to perform first experimental attempts. 2 M. V. Fedorov, Y. M. Mikhailova, and P. A. Volkov, J. Phys. B. 42, , (2009).

115 7.6. PUBLICATION: SHAPING FREQUENCY-ENTANGLED QUDITS Publication: Shaping frequency-entangled qudits The article has been published in Physical Review A, 88, , (2013) and is available on arxiv: [quant-ph]: Shaping frequency-entangled qudits Christof Bernhard, Bänz Bessire, Thomas Feurer, and André Stefanov Institute of Applied Physics, University of Bern, 3012 Bern, Switzerland These authors contributed equally to this work To whom correspondence should be addressed. Abstract We demonstrate the creation, characterization, and manipulation of frequency-entangled qudits by shaping the energy spectrum of entangled photons. The generation of maximally entangled qudit states is verified up to dimension d = 4 through tomographic quantum-state reconstruction. Subsequently, we measure Bell parameters for qubits and qutrits as a function of their degree of entanglement. In agreement with theoretical predictions, we observe that for qutrits the Bell parameter is less sensitive to a varying degree of entanglement than for qubits. For frequency-entangled photons, the dimensionality of a qudit is ultimately limited by the bandwidth of the pump laser and can be on the order of a few millions. DOI: /PhysRevA PACS number(s): Bg, Ud, Wj, Dv Introduction Entanglement [1] is one of the most intriguing features of quantum theory and is a fundamental resource for quantum information processing. It was experimentally revealed by the observation of correlations with no classical origins. Through Bell inequalities, the nonlocality of nature was tested by numerous experiments using entangled two-dimensional states (qubits) [2]. Both, fundamental tests of quantum theory and applications benefit greatly from entanglement in higher dimensions. Entangling d-dimensional states denoted as qudits allows to formulate generalized Bell inequalities, which are more resistant to noise than their two-dimensional predecessors [3, 4]. In loophole-free Bell experiments the detection efficiency threshold can be lowered [5]. Finally, both the effective bit rate of quantum key distribution and the robustness to errors can be increased [6]. These examples, among others, stimulated research towards different schemes to generate and manipulate photonic qudits in high dimensions. Due to their low decoherence rate, photons are used in many experiments as a robust carrier of entanglement. Photonic entangled states are usually produced by the nonlinear interaction of spontaneous parametric down conversion (SPDC). The coherence of this process, together with conservation rules, can generate entanglement in the finite Hilbert space of polarization modes [7]. Entanglement in infinite spaces can be realized for transverse (momentum) or orbital angular momentum (OAM) modes [8 14] and for energy-time states [15]. The amount of entanglement is commonly quantified by the Schmidt number K. For transverse-wave-vector entanglement, K is on the order of 10 for perfect SPDC phase-matching conditions [16], approximately 400 for specific nonperfect phase-matching conditions [9], and approximately 50 for OAM entanglement [14]. Similar Schmidt numbers can be achieved in energy-time entanglement generated by short pump pulses [15, 17], but much larger K numbers are obtained for a quasimonochromatic pump laser. In practice, the infinite Hilbert

116 108 CHAPTER 7. ENERGY-TIME ENTANGLED QUDITS space is projected onto a finite space of, for example, discrete time or frequency bins. In the time-bin subspace two-photon interferences for d = 3, 4 have been observed by interferometers with multiple arms [18, 19]. This, however, requires interferometric stability and becomes very complex for higher dimensions. In the frequency-bin subspace, interferences between two entangled photons, each in an effective two-dimensional space, have been observed by manipulating the spectra with a combination of narrow-band filters and electrooptic modulators [20]. Some of the aforementioned experiments are very useful for quantum key distribution because frequency is the most suitable degree of freedom of light to distribute entanglement over a large distance through optical fibers [21]. However, these experiments do not provide sufficient control of the phase and amplitude of qudits in the frequency domain to extensively study the properties of d-dimensional states with d > 2. In this article, we demonstrate a methodology that allows for full control over entangled qudits through coherent modulation of the photon spectra. It is derived from a classical pulse-shaping arrangement and contains a spatial light modulator (SLM) as a reconfigurable modulation tool. This technique is widely used in ultrafast optics [22] and has been adapted to manipulate the wave function of energy-time entangled two-photon states [23,24]. The flexibility of the experimental setup allowed to reconstruct the density matrices of maximally entangled qudits up to d = 4. Moreover, we demonstrate the presence of energy-time entanglement by measuring a Bell parameter above the local variable limit for maximally and certain nonmaximally entangled qubit and qutrit states. Figure 7.15: (Color online) Schematic of the experimental setup and frequency-bin structure. L0 pump beam focusing lens (f = 150 mm); PPKTP, nonlinear crystal; BD, beam dump; SLM, spatial light modulator; L1 and L2, two lens symmetric imaging arrangement (f = 100 mm) to enhance the spectral resolution with a magnification of 1:6 at the symmetry axis of the four-prisms compressor; BF, bandpass filter; SPCM, singlephoton counting module with a two-lens (L3, L4) imaging system. The inset shows the measured downconverted spectrum overlaid with a schematic illustration of the frequency bins for a ququart. Each of the gray shaded areas represents a single bin whose amplitude and phase can be manipulated individually Experimental setup Figure 7.15 depicts a schematic of the experimental setup. The entangled photons are generated in a type-0 SPDC process where all involved photons, the pump, the created idler (i), and signal (s), are identically po-