WMI

Technische

Universitat

Munchen

Walther -Meißner -

Institut fur Tief -

Temperaturforschung

Bayerische

Akademie der

Wissenschaften

Programmable control of

two non-linear resonators

Bachelor Thesis

Stefanie Grotowski

Supervisor: Dr. Frank Deppe

Munich, March 2019

Faculty of Physics

Technical University of Munich

Contents

1 Introduction 2

2 Flux-tunable resonators 4

2.1 Coplanar transmission line resonators . . . . . . . . . . . . . . . . . . 4

2.2 Superconducting quantum interference devices . . . . . . . . . . . . . 6

2.3 Tunable resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Mode function of a single resonator . . . . . . . . . . . . . . . 9

2.3.2 Coupled resonators . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Experimental techniques 14

3.1 Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Resonator control program . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Results 20

4.1 Automatic setting of currents for the resonance frequency . . . . . . . 20

4.2 Comparison of simulations and measurements . . . . . . . . . . . . . 22

4.2.1 Adjustment of current-to-flux conversion relation . . . . . . . 22

4.2.2 Unstable behavior of resonator 2 . . . . . . . . . . . . . . . . 28

4.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Summary and Outlook 32

A Extract of the calculation results 34

II

Chapter 1

Introduction

In 2012, the Nobel prize was awarded to D. Wineland and S. Haroche for three

decades of ground breaking work on the manipulation and measurement of individ-

ual quantum systems [1]. Nowadays, the community works on systems with several

tens of quantum elements. These e↵orts are mainly motivated by two grand goals:

universal quantum computing [2] and quantum simulation [3]. The latter was orig-

inally introduced in 1982, by Feynman who suggested to use quantum mechanics

itself rather than classical physics to e�ciently solve quantum mechanical problems

[4]. Since then, the interest in quantum simulation has grown rapidly because many

di�cult problems in physics, such as the understanding of many-body systems [5]

or the investigation of quantum phase transitions [6], could be studied by means

of su�ciently large quantum simulators. Among the di↵erent approaches pursued

towards quantum simulation, the most relevant ones are based on superconducting

quantum circuits [7, 8], ultra cold atoms [9, 10] and trapped ions [11, 12].

This thesis reports on an experimental study with superconducting circuits. In par-

ticular, we work towards a quantum simulator for Bose-Hubbard networks [13, 14]

in a bottom up-approach. Each network site is composed of a microwave resonator

galvaincally coupled to a flux-tunable Josephson nonlinearity. In this way, one min-

imizes the complexity and the amount of control needed to manipulate the sys-

tem [14]. Furthermore, superconducting chips are mostly fabricated with optical or

electron-beam lithography, thus they are very flexible in design and geometry [15].

Additionally, they have a good scalability and a direct control of the parameters

is possible. Especially when dealing with large systems, there is a great di�culty

in controlling the system, which is why at first, a simple system with few compo-

nents has to be understood before increasing the size of it. Thereupon the gained

knowledge can be applied to larger systems.

2

The ambition of this work is to gain control over a system of two coupled nonlinear

superconducting resonators. The Josephson nonlinearity of each resonator can be

controlled by means of a superconducting coil and an on-chip flux antenna. The

central issue is to set the two resonance frequencies automatically to their desired

values. Since coil and antenna always influence both resonators simultaneously, the

cross talk needs to be considered and modelled. Afterwards, this calibration is to

be verified by the successful setting of predefined frequencies.

The thesis is structured as described in the following. In chapter 2, the required

theoretical background is presented, as well as the components which are used in the

superconducting circuit. The next section (chapter 3) gives information about the

experiment itself, including the sample, the experimental conditions and the calcu-

lation of the current values required to tune the resonance frequency. In chapter 4,

the results of the experiment are presented and discussed. The first part is about

the evaluation of the calculation and the second about the measurements from the

experiment. Finally, a summary and conclusions can be found in chapter 5. Fur-

thermore, an outlook gives information about options for further experiments and

ideas of improvement.

3

Chapter 2

Flux-tunable resonators

In this chapter, the required fundamentals of circuit quantum electrodynamics (QED)

are explained. An overview on commonly used components in quantum circuits

such as waveguides, resonators and SQUIDs is given. Also, flux-tunable nonlin-

ear supercondcuting resonators are described. Besides, this section focuses on the

mathematical description of the tunability of this system.

2.1 Coplanar transmission line resonators

Generally, a transmission line is a structure formed of two or more parallel conduc-

tors, whose ends are connected either to a source or to a load. In superconducting

waveguides, electrical energy can be guided nearly lossless. The TEM mode, where

the electric and magnetic field components are perpendicular to the direction of

propagation, is the dominating mode for lines embedded in a homogeneous medium.

The ’coplanar waveguide’ (CPW) is a planar realization of a transmission line, in-

vented by Cheng P. Wen in 1969 at the RCA’s Sarno↵ Laboratories [16]. Unlike

other planar transmission lines, all conductors are placed on one side of the dielectric

substrate [17]. A thin metallic film is placed on a dielectric substrate forming two

ground planes and a center strip in between, separated by two narrow gaps. Since

the lines are surrounded by an inhomogeneous medium (air and dielectric substrate),

the CPW propagates quasi-TEM modes. The existing longitudinal components of

the electric and magnetic field in this mode are small enough to be neglected. A

cross-section of a CPW structure is presented in Fig. 2.1.

CPW structures segments with a capacitor at each end are called half-wavelength

’coplanar transmission line resonators’. These structures are often used in quantum

information processing and quantum optical experiments. They can detect radia-

tion in the optical range or higher [18], read out the quantum state of a qubit [19]

4

Figure 2.1: Cross-section view of a CPW structre. Two ground planes and a centerstrip out of a thin layer of metallic film (blue) are placed on a dielectricsubstate (gray). The distance between the center conductor and theground planes on each side is given by the width w.

or be used as magnetic field tunable resonators [20]. For small coupling strenghts

to the external feed lines, gap capacitors with a width of 10 µm to 50 µm are used,

whereas for larger coupling strengths finger capacitors with one or more pairs of fin-

gers are realized. Resonators are fabricated in optical or electron beam lithography

processes, thus they are very flexible in geometry and design. For instance, they

can be bent and their length is variable. Additionally, larger systems and networks

can be created because a high scalability is given and the resonators can be simply

attached to each other. Usually, the resonance frequency lies in a range of a few

gigahertz and is defined by the resonator length l, which is why this quality can

be well controlled during the fabrication process. In half-wavelength resonators, the

fundamental resonator mode corresponds to half the length of the resonator l = �/2.

When replacing a capacitor by a short circuit, the length l = �/4 determines the

fundamental mode and the circuit is called quarter-wavelength resonator. Since only

the boundary conditions are fixed, higher harmonics at integer (half-wavelength res-

onator) or odd integer (quarter-wavelength resonator) multiples of the fundamental

frequency are possible [21]. Further characteristics of CPW resonators are the in-

ductance L0 and capacitance C0 per unit length, which depend on the geometry

of the resonator. Equation (2.1) and Eq. (2.2) show the relation with the phase

velocity ⌫ph of the electromagnetic wave propagating through the resonator and the

characteristic impedance Z0.

⌫ph =1pC0L0

(2.1)

Z0 =p

L0/C0 (2.2)

5

2.2 Superconducting quantum interference devices

When dealing with superconducting circuits, the Josephson e↵ect is of great practi-

cal importance. A phenomenological understanding can be obtained by describing

the supercurrent as a macroscopic wave function with an amplitude representing

the square of the number of superelectrons (’Cooper pairs’) and a phase. The

Josephson e↵ect describes that a supercurrent can tunnel through a thin layer of

insulating material between two superconducting electrodes. Such a structure is

called a Josephson junction and is shown in Fig. 2.2 (a). The supercurrent across

a Josephson junction depends on the gauge-invariant phase di↵erence ' between

the macroscopic wave functions in the two electrodes. The current depends in a

nonlinear fashion on the phase di↵erence ' [22, 23].

Is = Ic sin(') (2.3)

This current-phase relation in Eq. (2.3) is also known as the first Josephson relation,

where Ic describes the maximum supercurrent through the junction. Furthermore,

Josephson junctions have an intrinsic shunt capacitance CJ and the supercurrent

branch can be interpreted as a nonlinear inductance [see Fig. 2.2 (b)] which is given

by

LJ =�0

2⇡Is. (2.4)

A dc-supercondcuting quantum interference device (SQUID) consists of two Joseph-

son junctions connected in parallel [23] and is shown schematically in Fig. 2.3.

Is(�) = 2Ic

����cos(⇡�

�0)

���� (2.5)

The maximum supercurrent Is through this parallel circuit is dependent on the

external flux � through the SQUID loop. Here, we assume that both junctions have

identical critical currents and the expression �0 denotes the flux quantum. Note,

that this relation is only valid for negligible inductance of the superconducting loop

formed by the junction electrodes, a condition well fulfilled throughout this work.

6

(a) Schematic view of a Josephson junction.The two superconductors (gray) are sep-arated by a thin layer of an insulatingmaterial (red).

(b) Circuit representa-tion of a Josephsonjunction by a Joseph-son inductance LJ

and an intrinsicshunt capacitanceCJ.

Figure 2.2: Schematic view and circuit representation of a Josephson junction.

Moreover, for an asymmetric SQUID with two di↵erent cirtical currents Ic1 and Ic2

for each junction, Eq. (2.3) has to be adjusted with an asymmetry parameter d.

d =Ic1 � Ic2Ic1 + Ic2

(2.6)

In this scenario, one obtains

Is(�) = 2Ic

����cos✓⇡�

�0

◆����

s

1 + d2 tan2

✓⇡�

�0

◆. (2.7)

The advantage of a SQUID over of a single Joesphson junction is that the critical

current can be controlled by changing the flux through the superconducting loop.

If the flux has the value of multiples of the flux quantum � = n · �0, the critical

current Is reaches its maximum as is displayed in Fig. 2.4.

7

Figure 2.3: Schematic view of a SQUID. The Josephson junctions are marked inred and the maximum current through the junction is given by Ic1 andIc2, respectively. The current Is through the junction can be varied bychanging the magnetic flux � through the SQUID loop.

-4 -2 0 2 4

0.0

0.5

1.0

1.5

2.0

Figure 2.4: Dependence of the maximum supercurrent Is of a dc-SQUID on themagentic flux �. It reaches its maximum at fluxes which correspond tomultiples of the magnetic flux quantum � = n · �0.

8

2.3 Tunable resonators

By placing a SQUID in the center of a coplanar transmission line resonator, a field-

controllable nonlinear tunable resonator can be fabricated. The integration of the

Josephson junction results in an unequal spacing of the energy levels, because it acts

as a nonlinear inductance [24]. A schematic representation of a tunable resonator

is shown in Fig. 2.5. In the following, the wave function and the ability to tune the

resonance frequency are examined in more detail.

Figure 2.5: Schematic view of a CPW tunable resonator of length l. There are fingercapacitors at both ends of the CPW structure and a SQUID placed inthe center of the transmission line. The white part is the conductorformed out of aluminium and the gray part is the silicon substrate.Red crosses indicate the Josephson junctions. The SQUID acts as atunable inductance, which, in turn, makes the resonator tunable in itsresonance frequency. Because of the nonlinear nature of the Josephsoninductance, the SQUID also introduces a tunable nonlinearity.

2.3.1 Mode function of a single resonator

The mode functions can be determined by solving the linearized Euler-Lagrange

equations and using a separation ansatz

(x, t) =X

m

(m)(t) u(m)(x). (2.8)

The traveling mode function (x, t) can be split into an only time dependent part

(m)(t), which describes the oscillation at a frequency ! = k ·⌫, and an only position

dependent part u(m)(x) [25]. The boundary conditions force the currents to be equal

on both sides of the junction and to be zero at the ends of the transmission line

which leads to the following di↵erential equation

1

L0@x (x, t) = CJ� +

�

LJ, (2.9)

9

where � denotes the phase bias of the junctions. One ends up with two resonator

mode functions, one describing each side of the junction.

u(m)l (x) = Al cos

✓k(m)(x+

l

2)

◆(2.10)

u(m)r (x) = Ar cos

✓k(m)(x� l

2)

◆(2.11)

Index l describes the CPW section on the left side of the junction, whereas index r

represents the right part. The normalization coe�cients can be chosen in two dif-

ferent ways. The choice Al = Ar corresponds to the so called even or symmetric

mode, whereas Al = �Ar to an odd or asymmetric mode.

kodd tan

✓kodd +

l

2

◆=

2Z0

⌫CJ(!

2p � (kodd⌫)2) (2.12)

By inserting the functions in Eq. (2.9) one obtains an implicit equation for the wave

vector k. CJ is the shunt capacity, LJ the e↵ective inductance [see Eq. (2.4)] and

!p is denoted as the plasma frequency which is given by the following equation

!p =1pCJLJ

. (2.13)

When plotting the first three mode functions (see Fig. 2.6), we see that the SQUID

only a↵ects the odd modes of the resonator. The symmetric mode is una↵ected

because there is no flux drop at the junction, therefore no current is flowing [26].

Consequently, in the following, only the odd modes are considered. For these, at

the position of the junction, a flux-dependent di↵erence of the amplitude occurs.

The size of the jump can be interpreted as a change of the e↵ective length of the

resonator, leading to a tunability of the resonance frequency. Depending on the type

and design, a resonator can be tuned in a range of a hundred megahertz to a few

gigahertz.

2.3.2 Coupled resonators

Now two resonators are placed in a series with capacitive coupling as shown in

Fig. 2.7. They can be tuned almost independently, but if they are tuned to the

same resonance frequency, they couple and form two hybridized modes separated by

twice the coupling stength g between the resonators [27]. In Fig. 2.8, the transmis-

sion magnitude through such a two-resonator sample is displayed in dependency of

10

-0.4 -0.2 0.0 0.2 0.4

-1.0

-0.5

0.0

0.5

1.0

(a) Mode function at mag-netic flux � = 0

-0.4 -0.2 0.0 0.2 0.4

(b) Mode function at mag-netic flux � = �0/2

Figure 2.6: First three modes of the resonator with a junction at x = 0.(a) No flux is penetrating the SQUID loop.(b) The SQUID is penetrated by a magnetic flux of � = �0/2. Thejump is larger than without flux penetration.

the frequency. Both resonators are tuned to an uncoupled frequency of 7.133 GHz.

There are two peaks visible which correspond to the two modes of the coupled sys-

tem. They occur at 7.122 GHz and 7.144 GHz. It can be compared with two coupled

pendulums which also have two modes, one where the pendulums oscillate in phase

and another one in which they oscillate out of phase.

Figure 2.7: Schematic view of two capacitively coupled CPW tunable resonators.The white part is the conductor formed out of aluminium and the graypart is the silicon substrate. Red crosses indicate Josephson junctions.

11

Figure 2.8: The magnitude of the transmission through the resonators in depen-dency of the frequency. The two peaks correspond to the two eigen-modes of the coupled system at a lower frequency of flow = 7.122 GHzand a higher frequency fhigh = 7.144 GHz.

12

Chapter 3

Experimental techniques

In this chapter the details of the experiment are presented such as the sample design,

the experimental setup and conditions under which the experiments are performed.

The final part is about the approach to automate the process of bringing both

resonators to the same, arbitrarily chosen resonance frequency. For this purpose,

the determination of the right current values for the superconducting coil and the

antennas play an important role.

3.1 Sample

All components are fabricated by Michael Fischer in the context of his Ph.D. thesis.

The sample holder has a size of 31 mm ⇥ 27 mm ⇥ 10 mm, is made of copper and is

coated with gold. Inside this box, a superconducting chip with four ports is placed.

Two of these ports are connected to the resonator feed lines, while the other two

ports are connected to the feed lines of the antennas. Pictures of the actual sample

are shown in Fig. 3.1.

Superconducting chip

A topview of the superconducting chip is shown in Fig. 3.1 (a). The chip consists of

a silicon substrate of the size 10 mm ⇥ 6mm ⇥ 525 µm and a thin aluminium layer

on top of it, which is structured using eletron beam lithography. Two resonators are

coupled by a finger capacitor with a length of 15.6 µm [see Fig. 3.1 (c)]. The finger

capacitors which couple the resonators to the input and output lines both have a

length of 35.6 µm [see Fig. 3.1 (d)]. A SQUID with an area of 24 µm ⇥ 10 µm

is embedded in the middle of each resonator to induce a nonlinearity and make it

tunable.

14

On-chip flux antennas

The on-chip flux antennas are placed right next to each SQUID to apply a magnetic

field to the SQUID loop [see Fig. 3.1 (b)]. They induce a local magnetic field in order

to change the flux penetration through it. Note, that the antennas have to be placed

asymmetrically towards the SQUID, otherwise the field lines which penetrate the

SQUID loop cancel each other out. By varying the flux penetration of the SQUID,

the resonance frequency can be tuned as explained in section 2.3. Both antennas

are connected to a current source and can be controlled independently.

Superconducting coil

A superconducting coil is placed behind the superconducting chip and is control-

lable via a current source (’Keithley 6430’). When applying a current to the coil, it

induces a global magnetic field and tunes both resonators simultaneously. Coil and

current source are connected/disconnected via a persistent current switch [28].

15

Figure 3.1: Top view of the di↵erent components of the superconducting chip. Thealuminium is displayed as the light gray part, the substrate is the darkgray part.(a) The feed lines of the resonators are on the left and right, the portat the top is connected to the feed line of antenna 1, and the port atthe bottom is connected to the source of antenna 2.(b) Close-up of a SQUID loop and its on-chip flux antenna.(c) An enlargement shows the topview of the finger capacitor whichcouples both resonators to each other. The finger length is 15.6µm.(d) Finger capacitor which couples the resonator to the input and out-put line. The finger length is 35.6µm.

16

3.2 Experimental setup

The structure of two coupled nonlinear resonators corresponds to a Bose-Hubbard

chain with two lattice sites [14]. The system can be directly controlled, firstly during

the fabrication and secondly, during the experiment by changing the flux penetration

through the SQUID loop. By setting currents at the superconducting coil or the

antennas, a magnetic field is induced and as described in chapter 2.3, the resonance

frequency can be tuned by varying the flux through the SQUID. The coil and the

two antennas are connected to a current source as shown schematically in Fig. 3.2.

Therefore, their flux penetration is directly controllabe and so is the resonance

frequency. The input and output lines are separated by a circulator, and the ports

of the circulators are connected to the vector network analyzer (’KEYSIGHT PNA-

5222’) in the following configuration. Resonator 1 is connected with its input line

to port 1 and the output line to port 4. Further, the input line of resonator 2 is

connected to port 3, and the output line to port 2. Since the energy of microwave

photons is small, the sample is enclosed in a cryogenic setup, and the experiment

is performed at millikelvin temperature to avoid thermal excitations. The mean

temperature at the sample is approximately 27 mK. In order not to heat up the

inside of the cryostat, the currents which can be applied to the devices are limited.

The allowed currents are listet in Tab. 3.1. Icoil is the current applied at (’Keithley

6430’), Ia1 and Ia2 are the currents applied to the antenna 1 and antenna 2 current

source, repectively. The minimum value for the coil current is due to a hysteretical,

unstable behaviour, which was observed and investigated in previous measurements

[29]. Above a current of 8 µA, the behaviour is more stable.

Current Min. value Max. value

Icoil (µA) 8 40

Ia1 (µA) 0 2000

Ia2 (µA) 0 2000

Table 3.1: Minimum and maximum current values. Restriction criteria are to pre-vent heating up the inside of the cryostat and to avoid the hystereticbehavior of the superconducting coil at low coil currents.

17

Figure 3.2: Schematic setup of the experiment. The sample is enclosed into a cryo-genic setup, the mean temperature at the sample is 27 mK.

3.3 Resonator control program

A resonator control program should automate the process of setting the correct cur-

rent values at the coil and the antennas in order to tune the resonators into resonance

at a predefined frequency. The final program consists of two parts, firstly, one which

executes the calculation to find the correct current values for the superconducting

coil and the two antennas and secondly, one that sets the current at the devices.

The following calculation is based on a script written by Christian Besson who pro-

grammed the calculation of the wave vector k from the mode functions [29]. Using

the e↵ective inductance LJ and the plasma frequency !p, the implicit Eq. (2.12) can

be solved as a function of the flux. Together with the linear dispersion relation, in

which ⌫ is the phase velocity, the resonance frequency can be determined.

!res(�) = k(�) · ⌫ (3.1)

The currents set at the coil and the antennas can be converted linearly to the flux

through each SQUID loop by the following equations. It is assumed that the current

converts to the flux linearly. The currents Ia1 and Ia2 are the set at antenna 1 and

2, resepectively. The current applied at the coil is labeled with Icoil.

�res1(Ia1, Ia2, Icoil) = pa1, res1 · Ia1 + pa2, res1 · Ia2 + pcoil, res1 · Icoil +��res1 (3.2)

�res2(Ia1, Ia2, Icoil) = pa1, res2 · Ia1 + pa2, res2 · Ia2 + pcoil, res2 · Icoil +��res2 (3.3)

18

The parameters p (indices indicate flux source and a↵ected resonator in this order)

are the current-to-flux conversion parameters, which were determined by previous

straightforward calibration measurements, performed by Michael Fischer, in which

a sweep of each control current is performed while the other two are set to zero. The

values for the parameters can be found in Tab. 3.2 [30]. ��res1 and ��res2 denote

the background fluxes at each SQUID position. Note, that all conversion parameters

and the o↵sets are given in the unit of the magnetic flux quantum.

Parameter Value

Ic1 (µA) 1.60

pcoil, res1 (�0/µA) 0.069

pa1, res1 (�0/µA) 0.52 · 10�3

pa2, res1 (�0/µA) 1.05 · 10�3

��res1 (�0) 0.38

(a) Parameters for resonator 1

Parameter Value

Ic2 (µA) 1.85

pcoil, res2 (�0/µA) 0.153

pa1, res2 (�0/µA) 0.06 · 10�3

pa2, res2 (�0/µA) 0.6 · 10�3

��res2 (�0) 0.85

(b) Parameters for resonator 2

Table 3.2: Current to flux conversion parameters determined in previous calibrationmeasurements [30].

For both resonators, the angular resonance frequency given by Eq. (3.1) needs to be

converted into the frequency.

fset =!res1(�res1)

2⇡and fset =

!res2(�res2)

2⇡(3.4)

Afterwards, Eq. (3.4) are solved numerically for �res1 and �res2. The � dependence

comes from the implicit function for the wave vector k [see Eq. (2.12)]. In the

following step, the system of equations containing Eq. (3.2) and Eq. (3.3) is solved.

All three currents are calculated for a chosen range and stepsize and as a result,

one obtains a file with all possible frequency values. To each frequency value a coil

current and two antenna currents are assigned by the calculation. If those values are

set at the devices, the flux through the SQUID is regulated such that the resonators

are in resonance. As an example, if a resonance frequency of 7.0 GHz is chosen, a

coil current of Icoil = 22.498µA and two antenna currents of Ia1 = 1024.075µA and

Ia2 = 361.061µA, have to be set at the devices.

19

Chapter 4

Results

In this chapter, the results of the calculation are evaluated. Furthermore, the sim-

ulations are compared with the experimental results. The data which is presented

is used to test the program above. The measured signals are magnitude and phase

of the transmission (S21, S43) and the reflection (S41, S23) of resonator 1 and 2 with

the vector network analyzer.

4.1 Automatic setting of currents for the resonance

frequency

In order to automate the process of setting an arbitray resonance frequency for both

resonators, a subprogram for the home-made measurement program ’DeepThought’

of the Walther-Meißner-Institut (programmed in ’LabVIEW’ by ’National Instru-

ments’ ) is implemented. As an input it requires a frequency value and a file which

contains the current values for the coil and the two antennas. An extract of a current

parameter list can be found in the appendix A. The data from the file is read out,

then the current values assigned to the resonance frequency are set at each device

in the order shown in Fig. 4.1.

The calculation presented in section 3.3 provides all three currents to each frequency

for a given frequency range. The system is described by two equations, each de-

scribing one SQUID, and three variables, which correspond to the coil and antennas

currents, therefore one variable can be chosen freely.

fcal1 =!(�res1(Ia1,calc, Ia2,calc, Icoil,calc))

2⇡(4.1)

fcal2 =!(�res2(Ia1,calc, Ia2,calc, Icoil,calc))

2⇡(4.2)

20

Figure 4.1: Flow diagram of the resonator control program. The current valuescorresponding to the input resonance frequency are read out and thecurrents are set firstly at the source of the coil current, then at thesource of antenna 1. Finally, antenna 2 is set to its value.

To confirm that the calculated values yield to the correct resonance frequency, the

calculated current values are inserted into the primary Eq. (3.1) and the frequency

is recalculated.

error1,2 = fcal1, cal2 � fset (4.3)

An error is determined by forming the di↵erence between the set value and the

recalculated frequency. A plot of this error as a function of the set frequency can

be found in Fig. 4.2. The reason for the linear increase (decrease) on the lower

(upper) end of the graph is that the frequency tuning is limited by the maximum of

resonator 1 which has a lower critical current, and the minimum of the resonator 2

with a higher critical current (see Tab. 3.2). Thus, the resonators can be tuned in

an intervall from fmin = 5.604 GHz to fmax = 7.123 GHz.

The enlargement of Fig. 4.2 shows that a numerical accuracy of 5 · 10�15 GHz is

reached. The small errors are probability due to an insu�cient number of approx-

imation steps of the numerical algorithm which is used by the calculation program

to solve the equations.

As a conclusion, the calculations provide a high accuracy and make it possible to

find valid values for the whole tuning range. The achieved numerical accuracy is

more than su�cient, because the results for the currents are more accurate than the

accuracy which can be reached with the current devices. An arbitrary frequency

can be chosen and valid results for the currents are found.

21

Figure 4.2: Error of the recalculation dependent from the set frequency. The en-largement shows that a numerical accuracy of 5 · 10�15 GHz is reached.

4.2 Comparison of simulations and measurements

Measurements are taken in order to confirm the results predicted by the model. For

the evaluation, the results are overlayed with the simulations. In case the model

does not describe the tuning correctly, the model is improved by adjusting the

parameters or adding some if neccessary. All measurements are sweeps of either the

coil or one antenna current where, in each step, the phase and the magnitude of

the transmission (S21, S43) and the reflection (S41, S23) are measured with a vector

network analyzer.

4.2.1 Adjustment of current-to-flux conversion relation

The parameters used up to now are determined in measurements performed by

Michael Fischer where a single control current was sweeped while the other two

were set to zero. However, simple test runs reveal that the linear relation between

the current and the flux is an insu�cient description. Although an antenna induces

22

only a local magnetic field, a cross talk can be observed and the other resonator

is a↵ected as well. Until now, this is considered by a the cross talk conversion

parameters (pa1, res2 and pa2, res1) which take into account that the magnetic field

induced by the antennas is not locally confined. In the following measurement, this

cross talk and the periodicity of the tuning are investigated in more detail. The

measurement set is devided in three parts, starting with a coil sweep while both

antennas currents are set to zero. In each following step one antenna current is

added to the system. By this procedure, the same order as in the program (see

Fig. 4.1) is followed and the influence can be included in the evaluation. As we

will see later, the periodicy of the tuning changes when the total flux is increased.

Therefore, the model described by Eq. (3.2) and Eq. (3.3) needs to be improved.

From now on the following two functions with additional parameters in the exponent

of each control current are used for the conversion of applied current to the flux at

the resonators.

�res1(Ia1, Ia2, Icoil) = pa1, res1 · Iza1, res1

a1 + pa2, res1 · Iza2, res1

a2 + pcoil, res1 · Izc, res1

coil +��res1

(4.4)

�res2(Ia1, Ia2, Icoil) = pa1, res2 · Iza1, res2

a1 + pa2, res2 · Iza2, res2

a2 + pcoil, res2 · Izc, res2

coil +��res2

(4.5)

There are many possible e↵ects which could cause the observed change in flux pe-

riodicity. One possibility is that the applied fields cause a change in the magnetic

properties of the materials surrounding the sample, which, in turn, change the flux

at the sample position. Our model does not rely on a specific mechanism, but is

useful for a weak and continuous change in flux periodicity. The current-to-flux

parameters remain unchanged, but a set of additional parameters in the exponent

of the actual control current is identified. Thus, in the first step the parameters

zcoil, res1 and zcoil, res2 are determined. In the following measurement this parameter

is included and exponents for the antenna 1 current za1, res1 and za1, res2 are deter-

mined. Finally, za2, res1 and za2, res2 for the antenna 2 current are specified. All

identified additional parameters are listed in Tab. 4.1. Hence, in order to calculate

the resonance frequency [Eq. (3.1)] the external flux at each resonator is determined

by Eq. (4.4) and Eq. (4.5). Note, that the o↵set of the current-to-flux conversion

di↵ers between the measurements and has to be determined new for each evaluation.

The corresponding timescales are discussed in more detail in chapter 4.2.3.

23

Parameter Value

Ic1 (µA] 1.60

pcoil, res1 (�0/µA) 0.069

pa1, res1 (�0/µA) 0.52 · 10�3

pa2, res1 (�0/µA) 1.05 · 10�3

zc, res1 1.0305

za1, res1 1.012

za2, res1 0.983

��res1 (�0) 0.28

Parameter Value

Ic2 (µA) 1.85

pcoil, res2 (�0/µA) 0.153

pa1, res2 (�0/µA) 0.06 · 10�3

pa2, res2 (�0/µA) 0.6 · 10�3

zc, res2 1.0565

za1,res2 1.021

za2,res2 1

��res2 (�0) 0.85

Table 4.1: Updated conversion parameters for the adjusted flux-to-current conver-sion equations [see Eq. (4.4) and Eq.(4.5)] which are valid for coil currentsabove 20 µA and correspond to the red overlay function.

Step 1: Coil

Figure 4.3 shows the phase of the reflection of resonator 1 (S41) and resonator 2

(S23), respectively. A coil sweep from 10 µA to 40 µA with antenna currents set

to Ia1 = 0µA and Ia2 = 0µA is performed. The signatures of resonator 1 and

resonator 2 are clearly visible and the graph can be split into two areas, one above

and one below 20 µA. Both areas are overlayed with the function for the resonance

frequency [Eq. (3.1)] using Eq. (4.4) and Eq. (4.5) for the current-to-flux conversion.

The part above 20 µA is overlayed with the red function and the part below with the

purple one. The overlay shows that the conversion from current to flux is not linear

[see Eq. (4.4) and Eq. (4.5)] for both areas and resonators. By overlaying the graph

with the model, the parameters zc, res1 = 1.0305 and zc, res2 = 1.0565 as exponents of

the coil current for the area above 20 µA are determined (see Tab. 4.1). Below 20 µA,

the instable behavior already mentioned in chapter 3 increases drastically. The o↵set

�� of both resonators have to be adjusted, as well as zc, res2. The parameters used

for the current-to-flux conversion below 20 µA are listed in Tab. 4.2. In addition,

a clear jump in the resonance frequency can be observed. It occurs at 38.5 µA

where the resonator frequency is discontinuous and drops from a value close to its

maximum value to a lower value.

Step 2: Antenna 1 sweep with constant coil current

In the second step, the coil current is set to Icoil = 22.778µA, while antenna 2 is

still set to zero. Both measurements are again overlayed with the function for the

24

Parameter Value

zc, res1 1.0305

��res1 (�0) 0.4

Parameter Value

zc, res2 1.0105

��res2 (�0) 0.46

Table 4.2: Updated conversion parameters for the adjusted flux-to-current conver-sion equations [see Eq. (4.4) and Eq. (4.5)] which are valid for coil cur-rents below 20 µA (purple overlayresonance frequency function).

resonance frequency [Eq. (3.1)] using the improved description for the current-to-

flux conversion [see Eq. (4.4) and Eq. (4.5)]. In this step the parameters zc, res1 and

zc, res2 remain unchanged and the parameters za1,res1 = 1.012 and za1,res2 = 1.021 are

identified. In contrast to resonator 1, which is tuned strongly by the applied current,

resonator 2 changes only slightly. Also from the small conversion parameter pa1, res2

one can see that resonator 2 is only weakly coupled to antenna 1. Since the induced

magnetic field by the antenna is typically local, this behavior is as expected.

Step 3: Antenna 2 sweep with constant antenna 1 and coil current

Finally, a sweep of antenna 2 is performed while the coil current is set to Icoil =

22.778µA and the antenna 1 current is set to Ia1 = 611.479µA. The parameters

determined until now stay constant and the last set of exponents for the antenna 2

current (za2,res1 and za2,res2) are identified. Since za2,res2 = 1, this parameter could

also be neglected. Comparing resonator 2 in Fig. 4.4 with resonator 1 in Fig. 4.5,

resonator 1 is tuned clearly stronger by antennna 2 than resonator 2 by antenna 1.

Parasitic paths and strong cross talks are possible causes for these di↵erences. To

get back to the calculation, both resonators should have the same uncoupled reso-

nance frequency of 7.031 GHz, when the currents are set as in Tab. 4.3. Having a

look again at resonator 2 in Fig. 4.5 the result can be confirmed at an antenna 2

current of Ia2 = 448.239µA which is marked in the graph with a white vertical line.

Parameter Value

Icoil (µA) 22.778

Ia1 (µA) 611.479

Ia2 (µA) 448.239

Table 4.3: Current values so that both resonators have an uncoupled resonancefrequency of 7.031 GHz

25

��

���

���

-�

-�

-�

�

�

�

��� ��� ��� ��� ��� ��� ���

��

���

���

-�

-�

-�

�

�

�

Figure 4.3: A coil sweep with Ia1 = 0µA and Ia2 = 0µA is performed. The up-per graph shows the phase of the reflection (S41) of resonator 1 and thelower graph shows the result for resonator 2 (S23). The graphs are over-layed with the simulation of the resonance frequency given by Eq. (3.1),using the current-to-flux conversion function in Eq. (4.4) and Eq. (4.5),respectively. Furthermore, the updated parameters of Tab. 4.1 are usedfor the area above 20 µA and the parameters of Tab. 4.2 fur the areabelow 20 µA. The discontinuity in resonator frequency of the second res-onator occuring at 38.5 µA is marked in the graph by a white verticalline.

All parameters found by this calibration are summarized in Tab. 4.1. Keep in mind,

that the parameters listed in the table above are only valid for coil currents greater

than 20 µA (red overlay resonance frequency function). For lower coil currents the

parameters of Tab. 4.2 have to be considered. As already mentioned in section 3.2,

previous measurements showed unstable, strong hysteretic behavior for low coil cur-

26

Figure 4.4: Antenna 1 sweep with Icoil = 22.778µA and Ia2 = 0µA. The current-to-flux convesion functions given by Eq. (4.4) and Eq. (4.5) are usedwith the parameters of Tab. 4.1 to calculate the overlay function of theresonance frequency [Eq. (3.1)].

Figure 4.5: Antenna 2 sweep with current values Ia1 = 611.479µA and Icoil =22.778µA and the reflection of both resonators is clearly visible. In or-der to describe the graph accurately, the resonance frequency [Eq. (3.1)]using the current-to-flux conversion Eq. (4.4) and Eq. (4.5) with the pa-rameters listed in Tab. 4.1.

27

rents. On this account low currents should be avoided anyway. Moreover, longer

pauses between the measurements and filling processes of the helium tank showed

that they change the o↵set behavior. Since the helium tank needs to be refilled every

five to six days, at least after the filling the o↵set-parameters have to be updated.

Still, the o↵set-value is stable for a few days for several measurements which are

performed right after another.

4.2.2 Unstable behavior of resonator 2

To confirm that the behavior is reproducable, the coil sweep of section 4.2.1 with

antenna currents set to zero is performed twice. Although the settings are identical,

the results di↵er from the previous ones (see Fig. 4.3) presented in section 4.2.1.

A more detailed look reveals that the parameter in the exponent (zcoil, res1) of res-

onator 1 stays constant for currents above 20 µA (red overlay resonance frequency

function) and only changes for lower values (purple overlay resonance frequency

function). On the contrary, comparing the second resonator in Fig. 4.3 and Fig. 4.6,

the parameter zcoil, res2 di↵ers for both areas from the ones listed in Tab. 4.1 and

Tab. 4.2. The parameters used to calculate the flux at the resonators using Eq. (4.4)

and Eq. (4.5) are summarized in Tab. 4.4. It has to be mentioned that between these

two measurements the helium dewar of the cryostat is refilled. Thus, a likely cause

for the di↵erent behavior would be a disturbance by trapped fluxes in other parts of

the sample, which could occur during filling proccesses of helium into the cryogenic

system. In general, resonator 2 seems to be a↵ected more strongly by such changes

in the magnetic environment.

As a consequence, calibration would be necessary at least after every refill of the

helium dewar. A whole coil sweep in the coil range of interest should be performed

as a calibration procedure. Since the tuning is periodically, one can narrow the coil

range and choose the one with the best results, for instance a coil current between

20 µA and 35 µA.

4.2.3 Discussion

As already discussed, Eq. (3.2) and Eq. (3.3) are successfully improved by adding

an exponent to each current. Now, an accurate description of the tuning is given

by Eq. (4.4) and Eq. (4.5), which is fundamental for further calculations and sim-

ulations. A system calibration routine in the form of a three step measurement is

established and all parameters are identified. Nevertheless, more measurements are

28

Parameter Res. 1 above 20 µA

zc, res1 1.0305

��res1 (�0) 0.28

Parameter Res. 1 below 20 µA

zc, res1 0.8550

��res1 (�0) 0.75

Parameter Res. 2 above 20 µA

zc, res2 1.0305

��res2 (�0) 0.3

Parameter Res. 2 below 20 µA

zc, res2 1.0005

��res2 (�0) 0.64

Table 4.4: Updated conversion parameters for the adjusted flux-to-current conver-sion equations [see Eq. (4.4) and Eq. (4.5)] for the second measurement.

neccessary to improve the level of control over the two coupled resonators. A trial

and error of randomly chosen frequencies from the mathematical calculation should

performed in order to confirm the model.

Still, the major issue is the tuning behavior of the superconducting coil. Because

of the fact that the coil induces a global magnetic field, one would expect both

resonators to be tuned in a very simliar way. The contrary experimental observations

indicate an inhomogeneous field over the sample chip. Furthermore, time-varying

flux o↵sets and nonperiodic behavior complicate the predictions. Additionally, the

magnetic field of the coil does not only depend on the current applied to the coil, but

also on its history [31]. If higher currents are applied to the coil previously, currents

can be trapped inside the superconducting coil. To minimize irregularities caused by

history, at any setting an up sweep with a su�ciently slow rate starting at current

zero is performed. However, it can not be excluded with certainty that no current is

trapped inside the superconducting coil because setting the current at the device to

zero only means that no additional current is applied. An enclosing of fluxes could

for instance occur during the filling process of the helium dewar which has to be

performed every five to six days. In our measurements, drastic changes of the o↵sets

appear in both coil sweeps at Icoil = 38.5µA (see Fig. 4.3) or Icoil = 38.8µA (see

Fig. 4.6). These flux jumps occur at very similar current values which indicates a

reproducable behavior. Consequently, coil currents above 35 µA should be avoided.

Moreover, the overlay function of the simulation does not describe the tuning with a

single parameter set for all coil currents. On this account the evaluation is realized

by separating the sweep into two areas above and below 20 µA. Possibly, more flux

jumps occur at coil currents below 20µA and at frequencies lower than 6.9 GHz

which are not visible in the measurement results. In the end, the o↵set parameter

is stable for a set of measurement performed right after another, hence a few days,

29

��

���

���

-�

-�

-�

�

�

�

��� ��� ��� ���

��

���

���

-�

-�

-�

�

�

�

Figure 4.6: Coil sweep with Ia1 = 0µA and Ia2 = 0µA. The course of resonator1 is overlayed with the simulation of the resonance frequency given byEq. (3.1), using the same parameters as in the first measurement. Tooverlay the lower graph, which shows resonator 2, the parameters haveto be updated. The updated parameters are summarized in Tab. 4.4.The course is discontinued by flux jump occuring at 38.8 µA which ismarked by a white vertical line.

but after the filling process of the helium dewar a calibration is neccessary.

A longterm improvement could be expected by replacing the superconducting coil

by a normal conducting coil. Then, however, a heating inside the cryostat would

occur, but the flux trapping inside the coil completely excluded. In addition, the

history of the coil would be less relevant. Both aspects would improve the stability

of the parameters enormously. Another approach would be to tune both resonators

only with the two antennas and use only constant coil currents. That would lead

30

to a more stable and therefore predictable behavior. The downside of this approach

is that the frequency tuning is limited to a smaller frequency range. In practice,

such a reduced range could be su�cient, because the antennas only have to cover

one period of the maximum supercurrent of the SQUID (see Fig. 2.4). This scenario

could be realized with the current setup by setting a constant coil current and the

single resonators could be adjusted by using only the antennas. Even though not any

arbitrary resonance frequency can be chosen because the calculation does not provide

valid solutions for all frequencies, valid solutions close to the desired frequency can

be found. Since the current sources are limited to set the values with an accuracy of

10�4 µA, an accuracy of 1 MHz can be reached for the frequency, which is completely

su�cient for our purpose. Because of the instable behavior below 20µA and the

discontiunous jumps around 38µA, choosing a smaller frequency range is favoured.

One should focus on a small range where the course of the tuning is more steady.

Coil currents within the range of 20 µA to 35 µA would provide a good basis.

Nevertheless, the determined parameters are only valid until the time of a filling

process of the helium dewar. On this account, a calibration process would be nec-

cessary, to simplify and speed up the evaluation of the results. A possible action

to determine the o↵set �� could be structured as follows. As a reference the first

maximum above 20 µA of the coil sweep is chosen. By comparing the reference cur-

rent to the coil current of the maximum in the uncalibrated setup, the o↵set can be

determined and inclued in further measurements. By running the same procedure

with the other two parameters a complete calibration can be performed.

To summarize, with the mathematical description of the tuning behavior for both

resonators in dependency of three parameters, the coil current and two antenna

currents, the resonators can be set to arbitrary degenerate frequencies within their

tuning range. The best approach is to choose a constant coil current to bring both

resonators close to the frequency of interest. Because the most stable behavior is

observed in a range of 20 µA to 35 µA, a coil current within this range should be

chosen. Afterwards the fine-tuning can be achieved by using only the antennas.

Since the parameters are only valid for only a few days until the next filling process,

a calibration should be performed after each filling procedure of the helium dewar.

By following these proposals, the results are most predictable. Therefore, further

measurements at di↵erent frequencies and after several helium refills should be taken

in order to confirm the results.

31

Chapter 5

Summary and Outlook

The primary ambition of this work is to automate the process controlling the res-

onance frequencies of two coupled nonlinear resonators. For this purpose two pro-

grams are created. One calculates the right current values, based on a mathematical

description of the tuning. The description uses linear relations to convert the ap-

plied currents into the flux through the SQUID plane [see Eq. (3.2) and Eq. (3.3)].

The second program allows to set the current value for each device by just inserting

the desired resonance frequency. The accuracy of the calculation program is satis-

fyingly high and the subprogram works as expected. To confirm accordance of the

simulations with the experimental results, two sets of measurements are taken to

investigate the behavior. It turns out that the so far used mathematical description

is insu�cient, which is why additional parameters in the exponent to each current

are identified to describe the system more accurately [see Eq. (4.4) and Eq. (4.5)].

Moreover, the tests reveal that the o↵set of the current-to-flux dependence is a

strongly changing parameter and is only stable between two refilling processes of

the helium dewar. On this account a calibration process is suggested to bring more

regularity into the tuning procedure. Furthermore, the behavior of the supercon-

ducting coil is not completely reproducible which is why a working range should

be chosen. The coil current it set to a constant value and only the two antennas

are used to tune the resonators. By following this structure, the history-dependent

behavior and hysteretic e↵ects of the superconducting coil are reduced to a mini-

mum. A convenient current range for the coil would be between 20 µA and 35µA

because in that range a quite stable behavior is observed. If these two aspects, the

calibration and the restriction of the coil current range are considered, the system

is controllable with a, for our purposes, su�cient accuracy.

To look ahead, as soon as the control of the two coupled resonators is functional,

the system can be expanded to a larger system with more components. Because of

32

the good scalability and the flexibility in design which is aready mentioned in the

beginning, larger and more complex systems can be constructed. First attempts

are already made by creating a network of coupled resonators [14] or a network of

pairwise coupled qubits [32] which already show successes. In the end, the approach

of using superconducting quantum circuits is very promising for the field of quantum

simulation.

33

Appendix A

Extract of the calculation results

34

35

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