Technische Universität München Fakultät für Physik Walther Meissner Institut für Tieftemperaturforschung

Transcription

1 WMI Technische Universität München Fakultät für Physik Walther Meissner Institut für Tieftemperaturforschung Abschlussarbeit im Bachelorstudiengang Physik Fast microwave beam splitters based on superconducting resonators Philipp Assum 20. Juli 2012

2 Erstgutachter (Themensteller): Prof. R. Gross Zweitgutachter: Prof. J. Greiner

3 Abstract Beam splitters are crucial components for quantum simulations and fundamental quantum experiments. In this work beam splitters based on microstrip resonators are investigated which are well-known components in microwave technology. We developed a model for so called distributed coupling, where the coupling length is similar to the resonator length. The interaction Hamiltonian of coupled resonators contains a beam splitter term and a fast rotating term. The relative strength of these terms depends on the location of the coupling region and therefore can be controlled. Our experiments prove that it is possible to achieve a pure beam splitter coupling for resonators with high coupling strength. That makes our coupled resonator a promising tool for circuit quantum electrodynamics experiments. iii

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5 Contents Abstract iii 1 Introduction Microwave fundamentals Microstrip transmission line Experimental procedure Microwave resonators Theoretical description of coupled resonators Point-like coupling Distributed coupling Results and discussion Transmission spectra of coupled resonators Discussion of simulation program Results of the simulation program Conclusion and outlook Conclusion Towards a beam splitter A Mathematica code B Result data Bibliography Acknowledgement v

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7 List of Figures 1.1 System for quantum simulation bunching as an example for a quantum rotation Microstrip box with sample, connectors and resistances Experimental setup with Cryostat Sketch of a resonator Transmission spectrum of a resonator Point-like coupling Distributed coupling Current and Voltage distribution in the resonator Resonator designs Micrograph of a sample with a coupled resonator Transmission spectrum with Lorentz fits Sketch of Lengths in a resonator Geometric parameter Relation between the two coupling terms Experimeantal results for the relation of the coupling strengths Strength of the beam splitter coupling Experimental results for the strength of the beam splitter coupling Sketch of a resonator Influence of the length shift Coupling strengths with a length shift Sketch of coupled resonators as beam splitter vii

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9 Chapter 1 Introduction Due to recent advances like single photon generation [10], two qubit gates [3, 22], three qubit gate [6, 15], proposed quantum error correction schemes [4] and quantum feedback control [23] it was made possible to experimentally investigate basic quantum computation algorithms based on solid state qubits. In digital quantum computing so called qubits serve as basic elements and mimic, combined with bus structures, a classical computer. Information in this system however is handled in a Hilbert space. An important success in digital quantum computing was to factorise the number 15 with a digital quantum system, using Shor s algorithm with eight qubits [14]. Digital quantum simulations of complex physical systems however would require a much larger number of qubits involved. In contrast analog quantum simulations represent an alternative working with a low number of qubits which is already accessible for some quantum simulations. Analog quantum simulations reproduce the quantum mechanical behaviour of physical systems because they can be described by the same quantum operators as the system of interest. Some aspects of this will be discussed in this work. For interesting applications of analog quantum simulation a combination of linear and non-linear components is required. Linear components like for example resonators or beam splitters will be investigated in this thesis. They have evenly distributed energy levels which means that any additional excitation adds the same energy to the system. In contrast qubits are non-linear components and their experimental realisations show two energy levels which are separated from the other levels and hence can be treated in good approximation as two level systems. Qubits in superconducting systems are realised by non-linear superconducting circuits with a characteristic energy gap equivalent to microwave frequencies about 5 GHz. Therefore it is necessary to develop suitable linear components in the microwave domain. Figure 1.1 shows an arbitrary system with qubits, beam splitters and resonators. If a certain state is put in, the system processes it and gives out another state. Components behaving in a controlled way allow to mimic various quantum mechanical operators, for example Hamiltonians. An important task in analog quantum computing is the creation of a toolbox containing these components. Since the 1

10 Chapter 1 Introduction Figure 1.1: System for quantum simulation, among others containing qubits and beam splitter. The system converts the initial state Ψ in into the state Ψ out and is represented by the operator Ô System. Hamiltonian determines the behaviour of a physical system, the procedure enables to simulate systems that are too complicated for classical numeric solutions, like the Bose Hubbard model for interacting bosons on a lattice [7]. Out of the several useful components for such simulations this work focuses on beam splitters. Beam splitter superpose propagating waves and apply quantum rotations. A propagating state put into the beam splitter rotates which leads to interesting quantum effects like bunching (figure 1.2). Figure 1.2: Illustration of bunching as an example for quantum rotation. On the left side each input channel carries one photon whereas on the right side the two photons are in an entangled state. Both are in the same output channel, either in the upper or the lower. Due to these features the beam splitter is a crucial device for interferometers or logic gates, like the CNOT gate [17]. For applications beam splitters have to 2

11 be fast enough. The time the beam splitter requires to process a quantum state has to be significantly smaller than the decoherence time of this state so that the quantum character of the process is maintained until the operation is finished. Since the existing realisations of beam splitters still have many problems, which will be discussed in chapter 5.1, this work presents a new design based on coupled resonators. The work is constructed the following way. Chapter 2 introduces the measurement setup and gives a brief overview of microstrip technology. In chapter 3 the theory of coupled resonators will be introduced. The quantum mechanical behaviour of point-like and distributed coupled resonators will be discussed. Chapter 4 presents and discusses all results. This starts with the transmission spectra and continues with the processing of the experimental results using a program. Finally the results for the coupling strengths will be presented. Chapter 5 contains a conclusion and describes how to use the results of this work to build a beam splitter. 3

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13 Chapter 2 Microwave fundamentals This chapter will introduce the basics of microwave technology which are required to understand the experimental procedure of this work. This implies descriptions of the measurement setup as well as a brief overview on theoretical topics. The topics are treated in a short and basic way. For further information suitable literature references will be given in the subsections. 2.1 Microstrip transmission line The microstrip transmission line is a popular concept in microwave engineering and is described in detail in Ref. [19]. Figure 2.1 a) shows a sketch of a microstrip transmission line. A strip of a conducting material is put on a dielectric substrate with relative permittivity ɛ r. The bottom side of the substrate is coated with a Figure 2.1: Microstrip, picture from Ref. [5]. a) Sketch of the geometry with dimensions. A conducting strip of niobium is put on a silicon substrate whose back side is coated with another layer of niobium, the so called ground plane. The specific dimensions lead to an impedance of Z 0 = 50 Ω. b) Sketch of the electric field E and the magnetic field H. ground plane. As dielectric we used silicon. The strip and the ground are made of niobium which becomes superconducting at approximately 9.5 K [12]. Transmission lines in microwave technology are usually characterised by their complex resistance, the so called impedance. The dimensions are shown in figure 2.1 a) and are chosen to achieve a characteristic impedance of the microstrip transmission line of 5

14 Chapter 2 Microwave fundamentals Z 0 = 50 Ω. The electric and magnetic field configurations are shown in figure 2.1 b). Simulations show that the electric field is wider extended than the magnetic field. [8] Microstrips are one of the most common types of transmission lines but there are several alternatives, e.g. the coplanar waveguide slotline. Since each type has different attributes it depends on the application which one is suitable. Microstrip transmission lines have some disadvantage: The field density is comparably low and the fields are not well localised. So microstrip structures on a chip can influence each other over millimetre-long distances. That causes hardly predictable effects that depend on the geometry of the chip structure. On the other side microstrips have some advantages which are of importance for this work. Firstly microstrips show few spurious resonances due to their geometry (cf. figure 2.1). Furthermore microstrip structures are convenient to design because of the possibility to use mitred bends (compare chapter 4.1.1) and because the coupling of two microstrip structures, which is essential for this work, can be easily realised by putting two microstrip lines in close vicinity without any change in the transmission line geometry. Finally microstrips are well accessible for simulation because all characteristic lengths are on a high scale in comparison to other microwave technologies. Due to these reasons all samples in this work use the microstrip transmission line design. 2.2 Experimental procedure The experimental procedure can be divided into two steps, the sample preparation and the actual measurement at helium temperature which are described in the following Sample preparation The samples used in this work were produced in a standard optical lithography process. Details can be found in Ref. [11]. To provide electromagnetic shielding the samples have to be mounted into copper boxes which is illustrated in figure 2.2. The box is coated with a gold layer to avoid corrosion. On each of the 4 sides a SMA connector is inserted into the box which allows to connect coaxial SMA cables. A copper plate with a thickness of 200 µm is put under the sample so that the distance between the SMA connector and the sample is kept small. The contact between the connector and the microstrip on the chip is established with silver glue. Silver glue is basically a solution of silver powder dissolved in toluene and dries out quickly being exposed to air. It is also used between the sample and the copper plate and 6

15 2.2 Experimental procedure Figure 2.2: box with sample, connectors and resistances. Picture from Ref. [2] the box. The silver glue connections are used to avoid additional unwanted cavities which would lead to spurious resonances. The shield of a connected coaxial cable is connected to the box and hence the sample s ground is matched to the cable s shield. The ports which are not supposed to be measured get terminated by a 50 Ω resistance. This avoids an impedance mismatch in these ports which would cause undesired reflections of the microwaves Measurement setup As a microwave signal is reflected in a resonator of our geometry about 1000 times, it is very important to avoid any attenuation effects. Therefore the microstrip transmission line is required to be superconducting. A detailed discussion of superconducting microstrip transmission lines can be found in the diploma thesis of Thomas Niemczyk [16]. Since niobium turns superconducting at 9.5 K [12] our experiments have to be done at low temperatures. We used a cryostat cooled with liquid Helium to 4.2 K (cf. figure. 2.3). Two ports of the box are connected with coaxial SMA cables to a vector network analyser (VNA). For more information about microwave network analysis compare chapter 4 in Ref. [19]. A Labview program on a control computer connected to the VNA then records the measured transmission spectrum. For accurate measurements of the sample it is necessary to eliminate the effects of the cable. Therefore the VNA compares the measured values to calibration 7

16 Chapter 2 Microwave fundamentals Figure 2.3: Experimental setup with Cryostat, the measurements of the network analyser are recorded by a computer with the software "labview". Picture from Ref. [11] data which was obtained by measuring the transmission spectrum through the two cables connected by a female-female adapter. For all our experiments we set the input power to -10 dbm, which equals 0.1 mw. This is used as reference for all following plots. Furthermore we took 1601 points of measurement per sweep in the frequency range of interest and averaged 5 sweeps in order to reduce the noise. 2.3 Microwave resonators Transmission lines have a homogeneously distributed capacitance C and inductance L. This enables pieces of the transmission line to act as resonators. Further information can be found in the Master Thesis of Marta Krawczyk [13] or in chapter 6 of Ref. [19]. As one can see in figure 2.4, a resonator is a piece of a transmission line, surrounded by two capacitances, called coupling capacitors. The coupling capacitors are gaps in the transmission line and act like semi-permeable mirrors for the microwave photons. The interference of the initial and all reflected microwaves allows only discrete wave vectors and so standing waves with discrete wave lengths and frequencies can form in the resonator. Figure 2.4 shows the current for the fundamental mode and the first two higher harmonic frequencies. In the fundamental 8

17 2.3 Microwave resonators Figure 2.4: Sketch of a resonator, below that the current ground mode and the first two harmonics. mode the length of the resonator shown in figure 2.4 is half of the wavelength. Resonators show a characteristic transmission spectrum with Lorentzian peaks at the resonant frequencies as it is shown in figure 2.5. For this work only the fundamental modes are observed M a g n itu d e (d B ) F re q u e n c y (G H z ) Figure 2.5: Typical transmission spectrum of a resonator, the left peak represents to the fundamental mode. Each further peak is located at a multiple of the fundamental mode s frequency. 9

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19 Chapter 3 Theoretical description of coupled resonators In this chapter the coupling between resonators will be introduced and different types of coupling will be identified. For each coupling type the Hamiltonian of the coupled system will be discussed. 3.1 Point-like coupling A system of two coupled resonators can be realised by putting two microstrip resonators closely together, allowing the fields to interact with each other. A possible design for such a system and the respective transmission spectrum is shown in figure 3.1. As one can see in figure 3.1 a) the region where the resonators are closely together, respectively where the fields can interact, is small in comparison to the full resonator length. Therefore this type of coupling is called point-like coupling. The transmission spectrum of point-like coupled resonators, the blue curve in figure 3.1 b) features two peaks. The two frequencies of the peaks are located symmetrically around the peak of the single resonator, represented by the red curve in 3.1 b). The Hamiltonian of a quantum mechanical system determines the energy eigenstates and the time evolution of the system [21]. The Hamiltonian in the interaction picture of two point-like coupled resonators looks as follows: H = hω 0 (a a + b b) + hg(ab + a b) (3.1) Here ω 0 is the resonant frequency of the single resonator and g is called coupling strength. a, a and b, b are the ladder operators in the Fock space of each resonator. For example a creates a photon in resonator A: a n = n + 1 n + 1. (3.2) The operators a a and b b give the occupation number in each resonator [20]. In equation (3.1) vacuum energies are omitted because for this work only energy 11

20 Chapter 3 Theoretical description of coupled resonators (a) (b) Figure 3.1: Illustration of point-like coupling. a) Photograph of two point-like coupled resonators, picture from Ref. [24]. The red marks symbolise the coupling capacitors which surround each resonator. b) Transmission spectrum, single resonator in red, coupled resonators in blue. Plot from Ref. [2] differences matter. The first part of the Hamiltonian hω 0 (a a + b b) is independent of the coupling and therefore irrelevant for the theory of coupling. In the following only the coupling terms of the Hamiltonian, summarised to the interaction Hamiltonian H i, are discussed. The Hamiltonian 3.1 is a pure beam splitter Hamiltonian as it fundamentally describes the first order exchange of one excitation. A beam splitter Hamiltonian can only create symmetrical splitting. The coupling strength g appears in the coupled resonator spectrum in figure 3.1 and equals half the distance between the peaks. With a point-like coupling it is possible to achieve coupling strengths up to two percent of the resonant frequency ν 0 [24]. For many applications such as e.g. thermal entanglement [1] higher coupling strengths are requested because the time scale of operations in the coupled resonators is given by T = 1/g. 3.2 Distributed coupling Higher coupling strengths are achieved by increasing the length of the coupling region which is called distributed coupling. A chip design with a coupling length of 3 mm and the respective transmission spectrum is illustrated in figure 3.2. The splitting of the double peak is large compared to the point-like coupling, and the single peak (red) is not located symmetrically between the two coupled peaks (blue). 12

21 3.2 Distributed coupling f (g i, g c,) M a g n itu d e (d B ) (a) F re q u e n c y (G H z ) (b) Figure 3.2: Illustration of distributed coupling. a) Chip with two distributed coupled resonators. b) Transmission spectrum of the single resonator in red and of coupled resonators in blue. Accordingly the distributed coupling cannot be described by the beam splitter Hamiltonian (3.1). A more appropriate theory for the distributed coupling by David Zueco [18] will be shortly introduced in the following. Two forms of coupling occur, the capacitive coupling and the inductive coupling. The capacitive coupling depends on the overlap integral of the voltage distribution of both resonators, the inductive coupling on the current distribution. The distributions of current and voltage are illustrated in figure 3.3. The green areas represent the amplitude a.u Figure 3.3: Current and Voltage distribution in the resonator. Red: current, blue: voltage. a) Point-like coupling, overlap integral of voltages can be neglected. b) Distributed coupling, contribution of the voltage overlap integral adds capacitive coupling. c) Distributed coupling location moved away from the middle, overlap integrals of voltage and current are effected. 13

22 Chapter 3 Theoretical description of coupled resonators length and the position of the resonator coupling. Figure 3.3 a) shows the situation for a point-like coupling. The voltage is approximately zero in the coupling region and can therefore be neglected. The coupling is purely inductive. But for larger coupling lengths (cf. figure 3.3 b) ) the voltage differs from zero and causes an additional capacitive coupling. More detailed derivations, starting with a Lagrangian of the lumped element representation of the transmission line, lead to the following Hamiltonian: H i / h = g i (a + a )(b + b ) g c (a a )(b b ) (3.3) In contrast to a pure beam splitter (3.1) this Hamiltonian contains the inductive coupling strength g i and the capacitive coupling strength g c. The splitting in the coupled resonator spectrum in figure 3.2 b) is a complex function of g i, g c and ω. In the derivation of (3.3) two fields with a mutual phase shift of π/2 are assumed. They can be multiplied with any global phase. Here the phase is chosen in a way that the coupling coefficients are positive. The Hamiltonian (3.3) can also be written in the form H i / h = (g i + g c )(ab + a b) + (g i g c )(ab + a b ) (3.4) with a beam splitter term of the strength (g i + g c ) and a fast rotating term of the strength (g i g c ). The second term would oscillate with the double eigenfrequency of the single resonator in the Schrödinger representation and is therefore called fast rotating term. The overlap integrals of voltage and current and hence the capacitive and inductive coupling depend on the location of the coupling in the resonators, compare figure 3.3 c). Therefore the relative strength of the two types of coupling, the beam splitter term and the fast rotating term, can be modified by varying the coupling location. This was experimentally done and will be discussed in the following chapter. 14

23 Chapter 4 Results and discussion This chapter presents all results of this work. It will start by discussing the design and the spectra of the measured samples. Furthermore it will be described how we derived the relevant parameters from the measured spectra. For a better understanding of this procedure the Mathematica simulation which is essential for this work will be introduced. Lastly the results will be shown. 4.1 Transmission spectra of coupled resonators The experimental procedure, as described in chapter 2, produces a transmission spectrum for each measured sample. In the following the sample design will be explained as well as the procedure of obtaining the relevant parameters out of each samples transmission spectrum Introduction of the sample design The main goal of the experimental work was to achieve different relations between the beam splitter coupling strength g i + g c and the strength of the fast rotating term g i g c by varying the location of the coupling region while keeping the geometry as constant as possible. The desired resonant frequency of approximately 5 GHz determines the length of the resonator. According to these demands two different groups of samples were designed. The designs with coupling lengths of 3 respectively 2 mm can be seen in figure 4.1. A photo of a sample with design 1 and close-up views on the coupling region and the coupling capacitors are shown in figure 4.2. The gaps at the beginning and the end of the resonators which act as coupling capacitors have a width of 10 µm. The gap between the two resonators in the coupling region is 2 µm wide. The 90 angles with the cut off corners are called mitred bends. They have suitable geometric properties for matching the impedance. [2] For each design several samples with different location of the closing capacitances but equal total resonator length were measured. The gaps were displaced like it is 15

24 Chapter 4 Results and discussion Figure 4.1: Design of the coupled resonators. The red bars symbolise coupling capacitors. a) Design 1 with a coupling length of 3 mm. b) Design 2 with a coupling length of 2 mm. Figure 4.2: Micrograph of a sample with a coupled resonator. The purple square provides a close-up view on the coupling region. The green square shows a coupling capacitor. illustrated in figure 4.1 by the red arrows. Thereby it is important not to change the geometry of the chip, for example by adding more mitred bends, because this would influence the geometric parameters (compare section 4.2.1). Only with constant parameters the correlation between the coupling strengths and the coupling location can be investigated. As reference for each coupled resonator we measured a respective single resonator. To do that one resonator was stripped from the coupled resonator designs Discussion of transmission spectra All transmission spectra in this work are plotted in the logarithmic db scale. The number x of the unit db is correlated with the input power P in and the output power 16

25 4.1 Transmission spectra of coupled resonators P out with the equation: x = 10 log 10 (P out /P in ) (4.1) The resonant peaks in the resonator s transmission spectra can be fitted with a Lorentzian function: κ y(x) 4(x ν c ) 2 + κ 2 (4.2) The interesting properties of the transmission spectra are the centre frequencies ν c of the peaks. In the following the centre frequency of the single resonator will be called ν 0, the centre frequency of the lower coupled resonator peak ν and the frequency of the higher one ν +. These frequencies are required to calculate coupling strengths in the referring interaction Hamiltonian. The parameter κ equals the full width half maximum (FWHM) and indicates how fast a photon leaves the resonator. This is important for the production of a beam splitter and will be further discussed in the outlook (chapter 5). κ is necessary to calculate an common attribute of a resonator, the quality factor Q which is defined as follows: Q = ν c (4.3) κ 0 C o u p le d R e s o n a to r s S in g le R e s o n a to r L o r e n tz fits 0 M e a s u r e m e n t L o r e n tz fit M a g n itu d e (d B ) a ) F r e q u e n c y (G H z ) M a g n itu d e (d B ) b ) c F r e q u e n c y (G H z ) Figure 4.3: Transmission spectrum with Lorentz fits. a) Spectra of a sample with design 1. Single resonator in blue, coupled resonators in black, Lorentz fits of the peaks in red. b) Detailed view on the single resonator peak in a). Measured data in black, Lorentz fit in red. The loss rate κ equals the FWHM and is illustrated in green. The center frequency ν c is shown in blue. Figure 4.3 a) shows the spectra of a coupled resonator (black) and the respective single resonator (blue). All peaks are fitted with Lorentz functions (red). Figure 4.3 b) provides a detailed view of the single resonator peak and the parameter κ which equals the FWHM. As one can see in the plots Lorentz functions fit the peaks very well. 17

26 Chapter 4 Results and discussion For all measured samples the frequencies ν 0, ν and ν + are given by the Lorentz fits. 4.2 Discussion of simulation program The process of deriving coupling strengths from the measured frequencies is complex. For this work the computational software program Mathematica was used to implement the formulas of David Zueco s theory [18] to a program which is included in appendix A. The program consists of several functions. The functions "calculatebend", "calculatecnorm"and "calculatelrat"obtain geometric parameters from the measured data. The function "Simulation" simulates the coupling strengths as a function of the coupling location. "Experiment" calculates the actual coupling strengths from the measured data. The function "EvaluationPlot" contains all other functions and provides a comparison between the theory and the measured results. The workflow and the functions will be discussed in detail in the following Introduction of parameters in the simulation program The major difficulty is that the theoretical prediction of the coupling strengths (with "Simulation") as well as their derivation from measured data (with "Experiment") require the knowledge of some geometric parameter which depend on the specific geometry and can hardly be predicted. Those are: the effective relative permittivity ɛ. It can be obtained from finite element simulations [8]. the electric length of a mitred bend L bend. This is an important property because it determines the total resonator length and the position of the coupling. Two equally shaped mitred bends can have different electric length if they are in two resonators with different geometry. the mutual inductances L rat. It is a dimensionless parameter that contains information about the relation between the resonators mutual inductance per length and its self inductance. the mutual capacitance c norm which is similar to L rat, but describes capacitances instead of inductances. Furthermore it is weighted by the overlap integral of the square voltage in the calculations. So the first step of the Mathematica program is to derive these geometric parameters (except ɛ, see below). The input for the program can be divided into two 18

27 4.2 Discussion of simulation program groups, the design specific and the sample specific parameters. The design specific parameters are equal for all samples of a design. This includes: the total length of the resonator exclusive the length of the mitred bends L tot1. It is the sum of all straight sections of the resonator. the length of the coupling region L c. the number of bends on the left side of the coupling n l. the number of bends on the right side of the coupling n r. The sample specific parameter are: the length of the resonator section left of the coupling without mitred bends L l1. the measured data, ν 0, ν and ν Introduction of functions and their application in the simulation program The effective relative permittivity ɛ is obtained from finite element microstrip simulations and is set in the program to However, a variation of ɛ leads only to small changes in the results. The other geometric parameters are calculated by the Mathematica program. The function "calculatebend" gives L bend for each sample. Using the frequency of the single resonator ν 0, "calculatebend" calculates the electric length of the resonator. The difference of this length and the length L tot1 of the straight resonator sections divided by the number of mitred bends gives the length one mitred bend. All values L bend of samples with common design are averaged and the mean is appended to the list of design specific parameters. The effective length of the mitred bends determines the relative position of the coupling region in the resonator and the total resonator length (cf. figure 4.4). With this information the functions "calculatebend" and "calculatecnorm" calculate L rat and c norm for each sample of a design. The derivation of L rat and c norm is comparably complex and involves all other design specific parameters and all samples specific parameters. Again the average is taken and appended to the list of 19

28 Chapter 4 Results and discussion design specific parameters. Using the meanwhile complete list of design specific parameters, the function "Simulation" calculates the desired properties, for example the strength of the beam splitter coupling g i + g c, as a function of the coupling location. The coupling location, which represents the x-axis in all plots, is expressed by the dimensionless value x = L l L tot L c. (4.4) L l is the effective length on the left-hand side of the coupling and L tot the total length, both inclusive mitred bends. This is illustrated in figure 4.4. (a) (b) (c) Figure 4.4: Lengths in the resonator. a) The coupling region is located in the middle of L the resonator. The value of the dimensionless coupling location x is l L tot L c = 0.5. b) The resonators are coupled at the left end of the lower resonator. With L l = 0 the value of the dimensionless coupling location is x = L l L tot L c = 0. c) The coupling region is put to the right end of the lower resonator which implies x = L l L tot L c = 1. The function "Experiment" uses the design specific and the sample specific parameters to calculate the coupling strengths of a certain sample and gives out the desired properties, similar to "Simulation". The major difference to "Simulation" is that "Simulation" creates a function of x = L l /(L tot L c ) using only the design specific parameter whereas "Experiment" accesses the sample specific parameters which contain the experimental data. The next step is to compare the experimental results of "Experiment" with the theoretical prediction of "Simulation". Therefore the function "EvaluationPlot" automatises all previous steps and gives a plot containing the outputs of "Experiment" and "Simulation. Additionally "EvaluationPlot" offers the possibility to manipulate the parameter L l1 in the sample specific parameter lists. The use of this is explained in section

29 4.3 Results of the simulation program 4.3 Results of the simulation program In this work seven samples of design 1 and three samples of design 2 were measured with the respective single resonator samples (so all in all 20 samples). A table containing all parameters and results can be found in appendix B. The Mathematica program discussed above was applied to all experimental results. The program s output will be presented in the following Geometric parameters As discussed in the last section the geometric parameters are derived from the measured data. The results for design 1 are shown in table 4.1, for design 2 in table 4.2 L bend L rat c norm Mean µm Standard deviation 1.4 µm Standard deviation Mean Table 4.1: Geometric parameters for design 1 Parameter L bend L rat c norm Mean µ m Standard deviation 1.7 µ m Standard deviation Mean Table 4.2: Geometric parameters for design 2 Figure 4.5 shows each parameter normalized to its mean, plotted against the coupling location of the sample. According to the model the geometric parameters should be constant because the geometry of the different samples with common design hardly changes. However, a large variation in the geometric parameters would be an indicator for an error in the theoretical description underlying the simulation program. The tables 4.1 and 4.2 and figure 4.5 support this assumption because all parameters feature small standard deviations. Both designs show similar trends. The length of the mitred bends L bend is the most constant parameter. Nevertheless its value for design 1 differs significantly from its value for design 2. That shows that electric lengths strongly depend on the full resonator geometry. 21

30 Chapter 4 Results and discussion L bend c norm L rat Parameter Mean Parameter Mean l res c (a) l res c (b) Figure 4.5: Geometric parameter normalised to mean, plotted against the coupling location. a) Geometric parameters for design 1. b) Geometric parameters for design 2. The mutual capacitance c norm fits worst to the assumption of constant geometric parameters. This can be explained by the field distribution of a microstrip, mentioned in chapter 2.1. Capacitances are connected to the electric field which has a wider range in a microstrip geometry than the magnetic field. So it is plausible that the mutual capacitance c norm reacts more sensitively to small changes in the resonator geometry Coupling strength parameters The knowledge of the geometric parameters enables an investigation of the coupling strengths. The relation between the beam splitter coupling g i + g c and the fast rotating terms g i g c can be simulated with the function "Simulation" in the Mathematica program. The result is shown in figure 4.6. As one can see the relative strength of the coupling terms can be manipulated in a wide range by adjusting the coupling location. For both designs the relation between the coupling terms, (g i g c )/(g i + g c ), becomes zero for certain coupling locations. Coupled resonators with these coupling locations feature a pure beam splitter Hamiltonian. A comparison between theory and experiment for both designs is shown in figure 4.7. The plots show the section which is highlighted in figure 4.6 by the green squares. For both designs the experimental results differ only little from the theory. The experimental results of design 1 lie slightly closer to the theory curve than the 22

31 4.3 Results of the simulation program g i g c g i g c Measured samples g i g c g i g c Pure beam splitter coupling L l L res L c L l L res L c (a) Design 1 (b) Design 2 Figure 4.6: Theoretical expectation for the relation between the two coupling terms. In both plots the green square indicates the area of the measured samples. Both curves intersect the x-axis. At these points the coupling is a pure beam splitter coupling. This is highlighted by the brown circles. results of design 2. Furthermore the results of design 1 are more reliable because they consist of seven points of measurement in contrast to design 2 with only three points of measurement. g i g c g i g c MHz g i g c g i g c MHz l res c (a) Design l res c (b) Design 2 Figure 4.7: Experimental results for the relation of the coupling strengths. The blue curve shows the theoretical expectation, the red points represent the results obtained from the measured data. For a discussion of figure 4.7 some detail of the Mathematica program introduced above must be considered. Both, the theory curve (blue) and the experimental results (red) access the same list of geometric parameters which was derived from the experimental data. So the fact that the average position of the experimental 23

32 Chapter 4 Results and discussion points agrees with the theory curve is implied by the data processing in the Mathematica program and provides no information about the validity of the theoretical description. But the theory is confirmed by the good agreement of the slopes and by the fact that each experimental point deviates only by a small amount from the theory curve. Figure 4.7 proves that the coupling terms can be tuned by the coupling location. Another interesting property is the absolute strength of the beam splitter coupling g i + g c. Usually high coupling strengths are desired. In this work coupling strengths are expressed in normal frequencies, not in angular frequencies which would include an additional factor of 2π. The results for the beam splitter coupling are illustrated in figure 4.8. The beam splitter coupling is approximately constant for couplings in g i g c MHz Measured samples g i g c MHz Measured samples L l L res L c (a) Design L l L res L c (b) Design 2 Figure 4.8: Theoretical expectation for the strength of the beam splitter coupling. In both plots the green square indicates the area of the measured samples. the middle of the resonator and decreases for a coupling location moving towards the resonator ends. Due to the longer coupling the beam splitter coupling in design 1 is stronger than in design 2. A comparison between experiment and theory is shown in figure 4.9. The experimental results for the beam splitter coupling do not fit well to the slope of the theory curve. In contrast to the theoretical expectation the points representing the experiment are approximately constant for both designs. An explanation for this behaviour will be given in the next subsection The achieved coupling strengths are much larger compared to the point-like coupling (compare chapter 3.1). Design 1 features a beam splitter coupling strength 24

33 4.3 Results of the simulation program g i g c MHz g i g c MHz l res c (a) Design l res c (b) Design 2 Figure 4.9: Experimental results for the strength of the beam splitter coupling. The measured data is represented by the red points, the theoretical expectation by the blue curve. normalized to the resonator frequency of g i + g c f 18% (4.5) which equals about 10 times the point-like coupling strength. Hence the coupled resonators work on a time scale of In Design 2 the coupling strength is which implies a characteristic time of T 1.6 ns. T = 1 g i + g c 1.2 ns. (4.6) g i + g c f 13% (4.7) Explanation of the discrepancy between theory and experiment In the following a possible explanation for the discrepancy between the coupling strength derived from the measured data and the theoretical prediction which appeared in figure 4.9 will be given. The Mathematica program assumes that the electric length of straight resonator sections and mitred bends is equal on both sides of the coupling region. But due to the different geometry of the two sides this assumption might be incorrect. On the right-hand side of the coupling in figure 4.10 there are much more corners. Hence the local fields can interact stronger with each 25

34 Chapter 4 Results and discussion other than on the left-hand side what might reduce the electric length of the right side in comparison to the left side. This would cause the coupling location assumed by the program to differ from the actual coupling location. Figure 4.10: Sketch of resonator with design 1. The resonator section left of the coupling is marked by the green ellipse, the section on the right-hand side by the purple. The right-hand side includes more corners and gives the local fields more possibilities to interact with each other. To eliminate the effect of different electric lengths the coupling location in the program can be adjusted. Therefore a length shift L shi f t was added to the length left of the coupling L l while keeping the total length L tot constant. The length shift has two implications which are shown in figure Firstly the value of the relative coupling location of the measured points increases. This causes the green square in figure 4.11 to move to the right. Secondly it changes the values of the geometric parameters and therefore the theory curve. The beam splitter coupling strength g i + g c (figure 4.11 a)) changes more than the relation (g i g c )/(g i + g c ) (figure 4.11 b)). The desired result of the length shift is shown in figure 4.12 a). The measured values for the beam splitter coupling of design 1 fit well to the theory for a length shift of L l = 0.6 mm. The relative strength of the coupling is also in agreement with the theoretical expectation (cf. figure 4.12 b)). A reason therefore might be that the slope of the relative coupling strength as a function of the coupling location, which is shown in figure 4.11 b), is approximately constant in the area of the measured samples (green square). So a small shift in the coupling location has no influence on the results. The assumed length shift of 0.6 mm is arbitrary, but it allows to correct the inconsistency in the beam splitter coupling (cf. figure 4.9). For design 2 a correction with a length shift was not done because the low number of only three measurements is does not allow to give a reliable information about the needed length shift L shift. 26

35 4.3 Results of the simulation program g i g c MHz g i g c g i g c MHz L l L res L c L l L res L c (a) (b) Figure 4.11: Influence of the shift in the coupling location. The dashed curve and the dashed square represent the situation without the shift. The solid blue curve and the solid green square represent the theoretical expectation and the area of the measured samples in case of a length shift of L shift = 0.6 mm. a) Strength of th beam splitter coupling term. b) Relative strengths of the coupling terms. g i g c MHz g i g c g i g c MHz l res c (a) l res c (b) Figure 4.12: Coupling strengths with a length shift of 0.6 mm. a) Beam splitter coupling, the shift of 0.6 mm makes the experimental results fit better to the theory. b) Relative strength of the two coupling terms, not significantly effected by the length shift. 27

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37 Chapter 5 Conclusion and outlook This chapter will start with a short summary of this bachelor thesis. A overview on possible applications and a brief comparison to other beam splitters will be given. At last it will be pointed out which further efforts have to be taken in order to actually realise a fast beam splitter from coupled resonators. 5.1 Conclusion In this work the coupling between two superconducting microstrip resonators was investigated in order to develop a microwave beam splitter. According to the theory [18] the interaction Hamiltonian of two coupled resonators contains two terms, a beam splitter term and a fast rotating term. The strength of these terms depends on the relative location of the coupling region. Therefore several samples of coupled resonators were designed with equal global geometry but different locations of the coupling capacitors and hence different relative coupling locations. The coupling strengths are obtained from the transmission spectra of the single and coupled resonators. We achieved high coupling strengths of 18% of the resonator frequency. Furthermore the results show that it is possible to fabricate a beam splitter out of a coupled resonator system. The coupled resonators investigated in this work enable two promising applications. Firstly a fast beam splitter can be constructed. The required adjustments will be discussed in the next section 5.2. Secondly the coupled resonators are a tool with tunable beam splitter and fast rotating terms. This allows to construct components which apply certain quantum operations to an input propagating state. So the coupled resonators might provide new possibilities for analog quantum simulations. A beam splitter made of coupled resonators might have some advantages in comparison to other existing designs. A common design for microwave beam splitters on the quantum level are hybrid rings. For more information on this compare [9, 11]. Hybrid rings have some problems and inconveniences. Firstly their geometric dimensions are large compared to coupled resonators. Secondly hybrid rings demand a given port configuration which might be inconvenient for certain 29

38 Chapter 5 Conclusion and outlook applications whereas the port configuration of coupled resonators is arbitrary. Lastly hybrid rings may turn out problematic when working with short pulses as they are based on interference of continuous waves. A beam splitter based on coupled resonators can process in principle short pulses as well as continuous signals. All in all coupled resonators are a well understood system and offer high design flexibility so that their attributes like coupling strength or bandwidth can be adjusted according to the demands. 5.2 Towards a beam splitter The investigation of the distributed coupling between two resonators in this work supplied the fundamentals towards building a beam splitter. But for an actual beam splitter some adjustments remain to be done. First of all the fast rotating terms in the Hamiltonian have to be completely suppressed in order to achieve a pure beam splitter coupling. This can be achieved by choosing a suitable location for the coupling. The only remaining coupling strength g i + g c will be summarised to g in this section. Besides that the loss rate κ and the coupling strength g have to have a certain correlation. This is illustrated in figure 5.1. The time a photon needs to switch Figure 5.1: Sketch of coupled resonators as beam splitter. The photons rotate with the frequency g between the two resonators and leave the resonators via the capacitances on the right-hand side with the frequency κ. The coupling capacitors on the left-hand side differ from the right ones in order to prevent the photons from leaving the resonator to the left. from the lower resonator to the upper and back again is given by 1/g. The photon leaves the resonator on average after the time 1/κ. A beam splitter which equally distributes an incoming signal on the two outgoing ports requires a probability of 50 % for a photon to leave the beam splitter via the lower resonator and the same for the upper one. This can be achieved by setting the relation between κ and g to 1 κ = 2n 1 4g (5.1) 30

39 5.2 Towards a beam splitter where n is an arbitrary integer. With other relations beam splitter with arbitrary distribution of the incoming signal can be developed, not only a 50:50 beam splitter. The relation (5.1) can be realised by adjusting g or κ. The coupling strength g can be set by the length of the coupling region. The loss rate κ depends on the coupling capacitors at the ends of the resonator. For the relation 1 κ = 1 4g (5.2) which equals equation (5.1) with n = 1 a loss rate of κ = 3.3 GHz would be required. This would implies a quality factor of Q = 1.4. In a resonator with such a low quality factor standing waves would smear out and the system could not be considered as resonator any more. Quality factors about 10 would be a reasonable compromise to keep the resonator behaviour and achieve a good bandwidth. The quality factors of the samples measured in this work range from 148 to 665. A reduction of the quality factor can be achieved by increasing the capacitance of coupling capacitors. More information on coupling capacitors can be found in Ref. [16]. Photons should leave the beam splitter exclusively via the two destined ports (the ports on the right-hand side in figure 5.2). Otherwise a part of the input signal would disappear due to backscattering. Therefore the coupled resonators need two different types of coupling capacitors. In this way the most likely output port for the photons can be determined. 31