Introduction to Cavity QED

Transcription

1 Introduction to Cavity QED Fabian Grusdt March 9, 2011 Abstract This text arose in the course of the Hauptseminar Experimentelle Quantenoptik in WS 2010 at the TU Kaiserslautern, organized by Prof. Ott and Prof. Widera. I want to give a short introduction - also from the theoretical point of view - to the fascinating field of cavity quantum electrodynamics (CQED). Modern experiments (QND measurements on photons and photon blockade) are discussed and basic results derived. Contents I. Introduction 1 II. The Jaynes-Cummings Model 1 III. Strong- and Weak- coupling regime 2 IV. Basic experimental techniques 2 First proofs of strong coupling Atom traps inside cavities V. QND measurements on single photons 3 VI. Photon blockade 5 Photon blockade experiment VII. Acknowledgements 7 interacting with a single electromagnetic field mode. This setup is shown in figure 1. The total Hamiltonian is given by where H = H field + H atom + H int, (1) H field = ω C â â (2) describes the longitudinal quasi-mode in the cavity and H atom = ω Aˆσ eg ˆσ ge (3) the non-interacting two-level atom. The spin-flip operators are defined as ˆσ ge = g e. The operators â, â are given by the decomposition of our singlemode field 1 I. Introduction Cavity QED investigates the interaction of single atoms with single electromagnetic field modes. To achieve the experimental goal of realizing such a system, effort was made and nowadays it is achievable even for optical transitions. This paves the way for many interesting physical applications. Two of the most interesting ones are for sure the use of cavity QED for the construction of a quantum network (with the aim to use it for quantum computational tasks) and on the other hand its usefulness for elementary verifications of quantum mechanics. II. The Jaynes-Cummings Model In this section we want to introduce the important Jaynes-Cummings Hamiltonian describing a two-level atom (consisting of states g, e separated by ω A ) Ê = E sin(kx)(â + â ), E = ωc 2ɛ 0 V. (4) Using this quantity one obtains the interaction energy in the dipole approximation H int = ˆd Ê = E sin(kx)(ˆσ eg + ˆσ ge )(â + â ) RWA = g (âˆσ eg + â ˆσ ge ), where in the last step the rotating wave approximation (RWA) was made. The coupling constant g is called single-photon Rabi frequency and given by g = E sin(kx), = e eˆ r g (5) The Hamiltonian (1) in the rotating wave approximation (RWA) is then called Jaynes-Cummings- Hamiltonian, and it is analytically solvable. That can easily be seen when restricting ourselves on the invariant subspace N = { n + 1 g, n e }: HN N, (6) 1 We have chosen our x-axis in such a way that there is a standing-wave node at x = 0, thus the sine appears in the decomposition. 1

2 g y x e ω A g ω C Γ κ Figure 1: The cavity QED parameters. i.e. H is block-diagonal. Exact diagonalization of the 2 2 blocks yields the eigenstates (dressed-states [15]): with +, n = sin θ n n e + cos θ n n + 1 g,, n = cos θ n n e sin θ n n + 1 g cos θ n = Ωn δ Ωn + δ, sin θ n =, 2Ω n 2Ω n Ω n = 4g 2 (n + 1) + δ 2, δ = ω A ω C. main goals of modern experiments is to trap single atoms long enough to perform quantum-logical operations with them [3,4]. But there is also a second, reverse way of using cavity QED, where the field in high-q cavities itself is measured via beams of Rydberg atoms[1]. These experiments (and also their methods) will be discussed in the following chapter. First proofs of strong coupling The eigenenergies of these dressed states are E ±,n = 1 2 ω A + (n + 1) ω C ± 1 2 Ω n. (7) They are compared to the case of vanishing coupling g = 0 in figure 2. III. Strong- and Weak- coupling regime The Jaynes-Cummings Hamiltonian does not include any coupling to an environment leading to decay. In order to include the two loss-mechanisms, namely spontaneous decay into the vacuum modes from e (rate Γ) and decay of the field mode 2 (rate κ), one has to use a master equation(see e.g. [5]). It should - by physical understanding - be clear that large losses, i.e. Γ, κ g lead to strong decay and thus no coherent evolution is expected. This case is referred to as weak coupling regime. In the opposite case Γ, κ g coherent evolution dominates for a very long time, until dephasing destroys it. This case is referred to as strong coupling regime. After some analysis [5] it turns out that the cavity coupling leads to a modified decay rate of the atom. Both extreme cases can be found, that of enhanced and that of inhibited spontaneous decay. IV. Basic experimental techniques In this section I want to discuss the basic experimental techniques employed in cavity QED. One of the 2 In the experiment this decay is usually caused by mirror imperfections. The first aim when dealing with cavity QED was to proof operation in the strong coupling regime. Therefore it was tried to measure the single-atom vacuum Rabi splitting. A basic problem in the first experiments was that one couldn t trap atoms in cavities but only beams of atoms where available. The first experiment [7] that measured single atom transits shall be presented in the following. The experimental setup is shown in figure 3 (a): The cold atoms are provided by a Cesium-MOT some millimeters above the cavity. At the beginning of each measurement the MOT is released and the atoms can escape. Some of them then pass through the cavity and interact with the field. The energy levels themselves are recorded via the transmission of a probe beam incident on the cavity: The transmittance is high when the probe laser is on resonance with a cavity mode. When an atom enters, it acts like a dispersive medium and thus the mode energies are effectively shifted. The shift is given by the vacuum Rabi splitting g (provided there are no (e.g. thermal) photons inside the cavity, which is very unlikely for optical frequencies). In order to make sure that only a single atom is inside the cavity at a time, a second probe laser is used to monitor the transit of single atoms. Therefore the second probe laser is held at zero detuning (relative to ω C ). When an atom enters, the transmittance of this second laser decreases dramatically and the presence of a single atom is shown. 2

3 2 g 1 e 1 g 0 e ω C = ω A +, 1, 1 +, 0, 0 2g g 0 g ω C = ω A 0 Figure 2: The Jaynes-Cummings energy levels for zero detuning δ = 0. Left: g = 0, right: g 0. (a) (b) Figure 3: Experimental setup (a) and single atom vacuum Rabi splitting (b) as main result of [7]. measurement in explained in the text below. The Atom traps inside cavities It is very desirable to trap a single atom for a long time inside an optical cavity. There exist proposals how one can then couple many such cavity-atom systems to implement quantum networks [3], or how to perform quantum computation with the neutral atom ground states as qubits [4]. For trapping single atoms dipole traps are standard. In order to not disturb the cavity mode used for the CQED setup one uses far off-resonance traps (FORTs). The needed additional laser beam can be coupled into a TEM 00 cavity mode besides a standard probe beam for example. The central experimental problem is the finite trapping time of the atom. It is limited by several heating mechanisms (discussed e.g. in [8,9,10]). Additionally the two levels needed e and g experience different AC-Stark shifts, dephasing the coherent evolution of the system. However the mentioned problems could be solved by using a specific trapping wavelength called magic wavelength. It depends on the sort of atoms used. In [10] a trapping mechanism based on the magic wavelength λ Cs = 935 nm for Cesium is reported where the authors reach an atomic trapping time of 2 3s. V. QND measurements on single photons The group of Haroche at ENS in Paris succeeded in performing non-destructive measurements on single photons trapped inside a high-q cavity [1,2] where the photon is not lost in the process. These processes are expected to be described by the quantum mechanical postulates of measurements. They used the dispersive interaction of single atoms with the trapped intra-cavity-photons in the strong coupling regime to perform a measurement on the field via atoms. Therefore they need atomic states with a long enough lifetime for the detection, provided by circular Rydberg atoms. These are atoms in states with a high principle quantum number n (n = 50 in the experiment) and the highest possible angular and magnetic quantum numbers: l = n 1 m l = n 1 (8) These states have radiative lifetimes of about a factor 1000 larger than their corresponding low-l states, and in the experiment they were of the order τ = 30 ms. The atomic transition used (n = 50 n = 51) is in the microwave regime and thus a microwave cavity is used. It is constructed of superconducting niobium mirrors (separated 27 mm apart from each other) and reaches a photon storage time T 0.13 s. Together 3

4 Figure 4: The setup of the experiment described in the text, [1,2]. e δ ω C g C R 1 R 2 D π 2 Φ π 2 Figure 5: The working principle of the Ramsey interferometer used to measure the state of the intra-cavity field. with the cavity resonance used at ω C /2π = 51.1 GHz this yields a Q-value 3 of Q In order to avoid thermal radiation the whole setup us cooled down to 0.8 K. The maximal coupling strength is given by g 0 /2π = 51 khz while the detuning between cavity resonance and atomic transition is δ/2π = 67 khz. Thus δ g 0 which is required for the Ramsey interferometer that shall be discussed now. In order to detect the coherent atom-light interaction, the high-q cavity C is sandwiched by two additional cavities (R 1, R 2 ) forming an atomic Ramsey interferometer, see figure 4. Its working principle is shown figure 5. Atoms optically pumped into g are prepared in a superposition state by applying a π/2-pulse in R 1. They fly through the cavity where they interact with the field-mode. In their first QND experiment [1] the Paris group considered only field states 5 of at maximum one photon: Ψ field = c c 1 1. (9) The atom-cavity interaction can be understood as an adiabatic following of the Jaynes-Cummings states and when properly adjusted the result is the following: n = 0 Φ = π n = 1 Φ = 2π where Φ stands for the rotation angle of the atomic state vector around the Bloch-sphere z-axis. This result is obtained assuming adiabatic following of the atomic states when passing C. The adiabaticity condition then gives the restriction mentioned above, δ g 0. After this process another classical π/2- pulse is applied in R 2, leading to the final atomic states Ψ atom : n = 0 Ψ atom = g n = 1 Ψ atom = e. The state-sensitive detectors D 1 and D 2 can thus (indirectly) measure the state of the intra-cavity field. To be precise, they only measure whether the photon number is odd or even since within the experimental errors 0 and 2 photon states yield the same probability for detecting the atom in e. In [2] the experimental setup was then refined such that up to n = 7-photon-states can be subsequently measured in the same way. In this case, of course, several atoms are needed to gain complete information about the field-state. After having understood how the measurement works we will now discuss some of the results. The main goal of the experiment was to confirm the quantum mechanical postulates. In [1] two kinds of experiments 3 The Q-value is used to characterize the quality of a cavity. It is defined as Q = ω C κ. 4 This is the highest Q-value ever achieved with a cavity so far. 5 The atomic states are not to be mistaken for the field states. Here n, for n an integer, denotes field states while e, g denote the atomic ones. 4

5 (A) (B) Figure 6: (A) Birth and death of thermal intra-cavity photons observed with the method explained in the text. (B) Loss of one prepared intra-cavity photon (upper case) and average over 904 such measurements (lower case). All from [1]. were performed on the field subspace consisting of 0 and 1. In the first, thermal photons were continuously observed, an example is shown in figure 6 (A). One can clearly distinguish between the one photon state and vacuum although there are several unexpected events caused by experimental imperfections. In the second experiment the field state 1 was prepared in a controlled way by sending through the cavity an atom in e. Its interaction time inside the cavity was adjusted such that the atom exits the cavity in g and a photon is left in the cavity field. Afterwards the decay of these photons is observed like before, see figure 6 (B). In the upper diagram a single measurement is shown while in the lower the average over 904 such single ones was calculated. The expected quantum-jump behavior for single events as well as the smooth exponential decay of the ensembleaverage can clearly be seen. VI. Photon blockade In this last section we will discuss how one can utilize cavity QED for strong photon-photon interactions. These can be used to create non-classical states of light. I will present an experiment [12] based on a theoretical proposal [11] for creating such states. Photon blockade experiment The experiment reported here [12] was performed at the CALTECH in 2005 by Birnbaum et. al. The basic idea was to use the Jaynes-Cummings anharmonicity of a single trapped atom in a high-q optical cavity to reach strong enough photon-photon interaction. If 6 For details about photon statistics the reader is referred to e.g. [5]. these interactions are large enough (i.e. if the additional energy needed to place a second photon into the cavity besides ω C is large enough) only one or no photon can populate the cavity mode due to energy conservation. This phenomenon is referred to as photon blockade. The required large non-linearity is achieved here by strong atom-field coupling, i.e. large g. Photon blockade can be observed by examination of the photon statistics 6 of light transmitted trough the cavity mirrors: sub-poissonian and anti-bunched light is expected. The experimental parameters are: (g max, Γ, κ) /2π = (34, 2.6, 4.1)MHz. (10) Thus the setup is far in the strong coupling regime. The photon-photon interaction for a two-level system can be deduced from figure 2. We assume a pumpbeam with photon energy ω 0 = E,0 E 0 and zero detuning δ = 0 between cavity-mode and atomic transition. Thus 0, can be populated. If an additional photon shall populate the cavity, the lowest possible energy state is 1,. This yields an estimate for the photon-photon interaction energy γ E,1 2 ω 0 = (2 2)g 20MHz 2π, which is large compared to the typical line width which is of the order κ. From the latter it is clear that photon-photon interactions give a substantial contribution. The assumption of one cavity mode and a simple two-level system however is not justified for a realistic system. In the actual experiment two orthogonally polarized cavity modes l y,z are present and atomic cesium was used. The atomic transition is the Cs D2-line 6S 1/2, F = 4 6P 3/2, F = 5 and the manifold of hyperfine sub-levels has to be taken into 5

6 Figure 7: Energy levels of the full hyperfine-manifold dressed states of the Cs D2-line. ω 0 ω C = ω A was assumed. From [12]. (a) (b) Figure 8: (a) Experimental setup. (b) Theoretical calculation (steady-state solution to the full master equation) for a two-level system (left) and the actual Cs D2-transition manifold in the experiment. Both from [12]. (a) (b) Figure 9: (Measurement result around τ = 0, (a), and on longer time scales (b). Both from [12]. 6

7 account. Exact diagonalization of the corresponding Hamiltonian yields the dressed-states energy levels given in figure 7. The photon-photon interaction is still of the order of γ calculated for the two-level model above. Next, the experimental setup (see figure 8 (a)) shall be discussed. A probe beam E p (y,z) incident on mirror M 1, polarized in y- or z-direction respectively, drives the cavity. Inside the latter a single Cs-atom is trapped in a FORT and there are further laser beams for cooling and testing (for details, see [12]). Behind the second mirror M 2 only the z-polarized light can pass the polarizing beam splitter PBS. Then g (2) (τ) of the transmitted field Et z is measured via coincidences of D1 and D 7 2. For this setup the steady-state solution of the full master-equation is shown in figure 8 (b), once for the simple two-level model and for the actual Cs-atom. In the latter case T zz (T yz ) refers to the transmittance of light from l z (l y ) 8 into l z and analogously g zz (2) (g yz (2) ) refers to the second-order correlation function of the transmitted l z -light for incident l z (l y ) light. One sees that the realistic system qualitatively behaves like the two-level system. The transmittance has just got more structures around the vacuum-rabi peaks due to the additional states. The correlation functions can at least partly be understood. Around ω p ω 0 = g max g 0 all plots exhibit sub-poissonian statistics. This can most clearly be seen at g yz (2) (τ). These statistics can be understood as a direct consequence of photon-blockade: Only one photon at a time can populate the cavity-mode. So we also expect photon anti-bunching for this frequency. One also clearly sees peaks at about ω p = ω 0 ±g 0 / 2, at two-photon resonance. Here the field shows superpoissonian statistics g (2) (0) 1. For more details, [1 ] Sébastien Gleyzes, Stefan Kuhr, Christine Guerlin, Julien Bernu, Samuel Deléglise, Ulrich Busk Hoff, Michel Brune, Jean-Michel Raimond, Serge Haroche, Nature 446, (2007) [2 ] Christine Guerlin, Julien Bernu, Samuel Deléglise, Clément Sayrin, Sébastien Gleyzes, Stefan Kuhr, Michel Brune, Jean-Michel Raimond, Serge Haroche, Nature 448, (2007) [3 ] T. Pellizzari, S.A. Gardiner, J.I. Cirac, P. Zoller, Phys.Rev.Lett. 75, 3788 (1995) see literature given in [12]. The aim of the experiment was to measure subpoissonian statistics and photon anti-bunching for the light emitted from the cavity. The theoretical calculations (figure 8 (b)) suggest measuring g yz (2) (τ) at ω p = ω 0 ±g 0. Here we expect the most significant signal. Measuring at this ω p has the additional benefit that motional effects of the trapped atom are suppressed. Such motion causes the coupling g to decrease, and thus the transmittance T yz (ω 0 g 0 ) also decreases. This yields a smaller contribution to the total counts. In figure 9 the measurement result is shown. Around τ = 0 (see (a)) one clearly sees photon-anti-bunching and sub-poissonian statistics. One finds: g (2) yz (0) = 0.13 ± 0.11 < 1 (11) and the field is clearly non-classical here. The width of the dip ( τ = 45ns) at τ = 0 is consistent with the lifetime of 0, given by τ = 2/(Γ + κ) = 48ns. In (b) the same curve but over longer timescales is shown. One recognizes a periodic modulation that can be explained by atomic motion in the FORT potential. The authors in [12] used this modulation to gain more insight into the trapping mechanisms (for more details, see [12]). VII. Acknowledgements I thank all the participants of the seminar for many nice talks and discussions, and our professors H. Ott and A. Widera for the organization. I am also thankful for the help preparing my talk by Prof. Widera. [5 ] P. Meystre, M. Sargent III: Elements of Quantum Optics, 4th Edition, Springer (2007) [6 ] R Miller, T E Northup, K M Birnbaum, A Boca, A D Boozer, H J KimbleJ. Phys. B: At. Mol. Opt. Phys. 38, 551 (2005) [7 ] C.J. Hood et. al, Phys. Rev. Lett. 80, 4157 (1998) [8 ] J. Ye et. al, Phys.Rev.Lett.83, 4987 (1999) [9 ] T. A. Savard, K. M. O Hara, J. E. Thomas, Phys. Rev. A 56, R1095 (1997) [4 ] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Phys.Rev.Lett. 78, 3221 (1997) [10 ] J. McKeever et. al, Phys.Rev.Lett. 90, (2003) 7 These coincidences yield g (2) directly because the beamsplitter BS forces two-photon packets to split up and leave in distinct paths, see [5]. 8 l α stands for linear polarized light in α-direction. 7

8 [11 ] A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys.Rev.Lett. 79, 1467 (1997) [12 ] K.M. Birnbaum, A. Boca, R. Miller, A.D. Boozer, T.E. Northup, H.J. Kimble, Nature 436, (2005) [13 ] M. Fleischhauer, A. Imamoglu, J.P. Marangos, Rev.Mod.Phys 77, 633 (2005) [14 ] Boca A, Miller R, Birnbaum K M, Boozer A D, McKeever J and Kimble H J, Phys. Rev. Lett. 93, (2004) [15 ] C.N. Cohen-Tannoudji, Nobel Lecture, December 8,