Circuit quantum electrodynamics with transmon qubits

Transcription

1 TECHNISCHE UNIVERSITÄT MÜNCHEN WMI WALTHER - MEISSNER - INSTITUT FÜR TIEF - TEMPERATURFORSCHUNG BAYERISCHE AKADEMIE DER WISSENSCHAFTEN Circuit quantum electrodynamics with transmon qubits Master s Thesis Javier Puertas Martínez Supervisor: Prof. Dr. Rudolf Gross Munich, February 2015

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3 TECHNISCHE UNIVERSITÄT MÜNCHEN WMI WALTHER - MEISSNER - INSTITUT FÜR TIEF - TEMPERATURFORSCHUNG BAYERISCHE AKADEMIE DER WISSENSCHAFTEN Circuit quantum electrodynamics with transmon qubits Master s Thesis Javier Puertas Martínez Supervisor: Prof. Dr. Rudolf Gross Munich, February 2015

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5 Contents Introduction 1 1 Theory Josephson junctions Josephson equations I-V curve for the Josephson junction DC-SQUID The transmon qubit The qubit as a two-level quantum system Dynamics of the qubit The Transmon qubit derived from the Cooper pair box Coplanar waveguide resonators Coupling of the transmon qubit to the CPW resonator Zero detuning Dispersive regime The Purcell effect Experimental techniques Nanofabrication Optical lithography Electron beam lithography Experimental setup He cryostat Dilution refrigerators Experimental results Measured samples The resonators The transmon qubit DC measurements Transmission measurements Two-tone spectroscopy Power calibration I

6 II Contents Flux dependence of the qubit frequency Qubit anharmonicity Time domain measurements Summary and outlook 55 Bibliography 57

7 Introduction Quantum computing was proposed by Feynman [1] as a way of efficiently studying quantum systems. It allows for an exponential speedup in the computation of certain problems with respect to a classical computer [2]. The key element of a quantum computer is the qubit. It is the analog to the classic bit. However, while a classical bit can only be in states 0 and 1, the qubit, due to its quantum nature, can be in any superposition of these states. The first step is to find in nature any two-level quantum system that might be used as a qubit. One of the first approaches was the use of two isolated levels of an atom. The atom can be placed inside an optical cavity where some photons are trapped. The interaction between the atom and the photons can be used for manipulating and transferring quantum information [3]. This field was called cavity quantum electrodynamics (QED). Another approach was the use of superconducting circuits instead of atoms and cavities. Cavities were replaced by superconducting resonators and atoms were replaced by superconducting qubits. Different circuits designs have been proposed for the superconducting qubits, mainly phase, flux and charge qubits [4]. They make use of Josephson junctions that can be seen as non-linear inductors. This non-linearity produces a non-equal spacing of the energy levels and, hence, makes the circuit suitable as a two-level system [5]. In 2004 the first measurement of a superconducting qubit coupled to a superconducting resonator was performed using a charge qubit and a coplanar waveguide resonator [6]. Due to the similarities with cavity QED, this field was called circuit QED. In this work, we couple a transmon qubit to a superconducting resonator. We study two samples that differ on the type of resonator used. From transmission measurements we obtain the coupling strength between the resonator and the qubit reaching the strong coupling limit. Using two-tone spectroscopy the energy levels of the qubit are measured. In addition the qubit coherence time is obtained from both spectroscopy and time domain measurements. The thesis is organized as follows. In Chapter 1 we introduce the theoretical concepts needed for the understanding of the performed measurements. In Chapter 2 we describe the fabrication process for both resonators and the transmon qubit. In Chapter 3 the measurements are shown and the main results are given. 1

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9 Chapter 1 Theory In this chapter we will introduce the theoretical concepts that are needed to understand the performed measurements. First the Josephson junction and the Josephson equations are explained. Then the transmon qubit and the coplanar waveguide resonator are described. Finally the coupling between the resonator and the qubit is studied using the Jaynes-Cummings model. 1.1 Josephson junctions Josephson junctions represent the key element in superconducting quantum bits (qubits). As superconductor devices, they have unique macroscopic quantum properties that can be exploited in the field of quantum computation. In this section we give the equations for the current and the voltage in a Josephson junction, the so called Josephson equations, and we use a simple model to describe its current-voltage characteristic curve. A Josephson junction consists on two superconductors separated by an insulator [Fig. 1.1(a)]. V (x) Superconductor Insulator Ψ A V 0 d Ψ B (a) (b) x Figure 1.1: (a) A schematic supercondcutor, insulator, superconductor (SIS) Josephson junction. The insulator is usually an oxide. (b) Sketch of the time-independent part of the macroscopic wavefunction (blue line) in an SIS Josephson junction. The red rectangle represents the insulation barrier. 3

10 4 Chapter 1 Theory Each superconductor can be described by a macroscopic wave function [7]. If we call them superconductor A and B we will have Ψ A ( r) = Ψ 0 e iφ A( r) Ψ B ( r) = Ψ 0 e iφ B( r) (1.1) where Ψ 0 2 = n( r,t) is the density of Cooper pairs in the superconductor. In a similar process to that of an electronic wave function, these macroscopic wave functions can tunnel through the potential barrier created by the insulator [Fig. 1.1(b)]. If we solve the Schrödinger equation for the three regions (superconductor A, insulator, superconductor B) and use the wave matching method, we arrive to the expression for the current density in the junction J = J c sin(φ A φ B ) (1.2) where J c is a material parameter. It depends on the characteristic decay constant κ = (2m(V 0 E)/ h 2 ) 1/2. For real junctions usually the barrier height V 0 is of the order of a few ev and therefore the decay length 1/κ less than a nanometer. In addition, the barrier thickness d is usually a few nm. Thus κd 1. In this regime J c exp( 2κd). It decays exponentially with increasing barrier width d Josephson equations From the tunneling described in the previous section we can derive the equations for the voltage and the current in a Josephson junction. They are the so called Josephson equations [7]. I = I c sinδ V = Φ 0 dδ 2π dt (1.3a) (1.3b) Here, δ is the gauge invariant phase difference between superconductor A and superconductor B [7]. The quantity I c is the critical current above which the Josephson junction shows a finite resistance. Two main properties of the junction can be derived from the above equations. First the current flow between the superconductors depends only on the relative phase of their macroscopic wave functions. Second there is only a voltage drop between both superconductors when this phase evolves in time. An important parameter than can be obtained from these equations is the Josephson inductance L J.

11 1.1 Josephson junctions 5 di dt = I c cosδ 2π Φ 0 V L J = Φ 0 2πI c cosδ = Φ 0 2πI c 1 ( ) (1.4) 2 IIc As it can be seen, the Josephson inductance is not linear with current which makes it a fundamental element for superconducting qubits. The energy stored in the junction will be given by where U = t 0 IV dt = Φ t 0 I dφ 2π 0 dt dt = Φ t 0 I c sin(φ)dφ = E J (1 cosφ) (1.5) 2π 0 E J = I cφ 0 (1.6) 2π is the Josephson energy. Furthermore, we point out that due to the fact that a Josephson junction consists of two metals separated by an insulator, it will have some capacitance C. From this capacitance we can define the charging energy of the junction as E C = e2 2C Finally, from L J and C we define the Josephson junction plasma frequency (1.7) ω p = 1 LJ C = 1 h 8EJ E C (1.8) I-V curve for the Josephson junction In the previous section we treated the Josephson junction as a perfect tunneling junction with a capacitance C. In order to study its I-V curve we have to take into account its resistance for I > I c [Fig. 1.2]. Real junction C JJ R Figure 1.2: The junction is divided into three components: a perfect tunneling component, a capacitor and a resistor

12 6 Chapter 1 Theory The resistively and capacitively shunted junction model (RCSJ) takes into account both the capacitance and the resistance of the junction. RCSJ model The RCSJ model [8][9] can be used to describe the current voltage characteristic curve of a Josephson junction. The circuit diagram used in the RCSJ model is shown in Fig C JJ R V I C I J I R Figure 1.3: Circuit diagram for the RCSJ model. From this circuit we see that the total current I will be given by the current through the junction I J, the current through the resistor I R and the current through the capacitor I C. If we now insert Josephson equations [Eq. 1.3] we have I = I J + I R + I C = I J + V R +C V (1.9) 0 = I + I c sinδ + 1 Φ 0 R 2π δ +C Φ 0 2π δ (1.10) This expression can be compared to the one of a classical damped oscillator with U given in Eq. 1.12b. 0 = mẍ + Dẋ + U(x) x (1.11)

13 1.1 Josephson junctions 7 U(δ) δ = I cφ 0 2π ( I I c + sinδ) U(δ) = E J ( I I c δ cosδ) (1.12a) (1.12b) Here E J is the Josephson energy in Eq The potential described in Eq. 1.12b is the so called washboard potential and is plotted in Fig The phase δ behaves as a virtual particle in this potential. When I/I c < 1 the potential has local minima where the phase particle is trapped. This means that the phase does not evolve in time and therefore there is no voltage drop across the junction. When I/I c 1 there are no potential wells anymore. The phase particle moves continuously in time giving rise to a voltage drop according to the second Josephson equation [Eq. 1.3b]. 2 0 I I c = 0 U(δ)/EJ I I c = 0.5 I I c = 1 0 π 2 π 3 π 4 π Figure 1.4: Washboard potential U(δ)/E J for different I/I c values. The pink circle represents the phase particle. The black arrows indicate its movement in the potential. δ For measuring the I-V curve we increase the applied current and measure the voltage drop across the junction. When I < I c the particle is trapped in a minimum. Although it cannot escape directly from the well, it can tunnel out of it to an adjacent one. This can be seen in the I-V curve as a slope below I c. This effect is called thermal drift. Due to the thermal energy the phase particle can be excited to a virtual state by a phonon and then tunnel easily through the barrier. The energy of this virtual state is proportional to k B T which results in the temperature dependent slope mentioned above. When I I c the phase particle moves down the potential and a voltage

14 8 Chapter 1 Theory is measured. For higher applied currents the junction behaves as a resistor. Now, when we go in the opposite direction, decreasing the applied current to zero, we see a hysteretic behaviour. This can be explained using Fig Decreasing the current means tilting up the washboard potential to its horizontal position. When we reach the I = I c regime the virtual particle has enough kinetic energy to overcome the potential maxima and therefore there is still a voltage drop. For I = 0 it will be trapped again. In Fig. 1.5 a measured I-V curve is shown. 0.4 Intensity (µm) Voltage (mv) Figure 1.5: Measured I-V curve of a Josephson junction at T = 500 mk. The parameters are not the same as the used for the qubit. The values for the critical current are shown, dashed red lines DC-SQUID A DC superconducting quantum interference device (SQUID) allows us to tune circuit parameters in situ using a magnetic field. It consists on two junctions connected in parallel forming a loop as shown in Fig Each junction can be defined by its phase difference δ. If we now apply a magnetic flux to the loop, the phase difference δ 1 δ 2 between the two junctions will be given by [7] δ 1 δ 2 = 2π Φ ext Φ 0 (1.13) This will give an effective flux dependent critical current I c,eff, Eq

15 1.2 The transmon qubit 9 ( I c,eff = 2I c cos π Φ ) ext (1.14) Φ 0 where Φ 0 = h 2e is the flux quanta. I in I 1 I 2 Φ ext I out Figure 1.6: SQUID. The applied current I in is divided in the two branches with currents I 1 and I 2. The total outgoing current is I out. The magnetic flux inside the loop is Φ ext. When both junctions have different I c,1 and I c,2, Eq is rewritten as [10] ( I c,eff = I c,σ cos π Φ ) ( ext 1 + d Φ 2 tan 2 π Φ ) ext 0 Φ 0 (1.15) where I c,σ = I c,1 + I c,2 and d = I c,2 I c,1 I c,σ accounts for the asymmetry between the junctions. 1.2 The transmon qubit So far, the junction phase has been treated as a classical variable. In this section, we move one step beyond this picture and quantize our circuits by promoting phase (and charge) to operators. Specifically, we focus on the example of the transmon qubit, which is the main subject of study in this work. The transmon qubit is well-known for its low sensitivity to charge noise and its high coherence time that can reach up to 0.1 ms [11]. In this section we first describe two-level quantum systems in general and their dynamics. Then we introduce the transmon qubit as a type of superconducting qubit giving its principal parameters and characteristics.

16 10 Chapter 1 Theory The qubit as a two-level quantum system A qubit is a quantum two-level system. It is the quantum analog of a bit. As a classical bit, it has two states, 0 and 1, but due to its quantum nature a qubit can be in any superposition of this two states. Intuitively, we can think of the state of a qubit as any point in the surface of the Bloch sphere [Fig. 1.7]. Using angles θ and φ the state of a qubit will be Ψ = sin θ cos θ 2 eiφ 1 (1.16) 1 Ψ θ 0 i 1 φ 0 + i 1 0 Figure 1.7: The Bloch sphere. The red arrow represents an arbitrary qubit state Ψ. Angles θ and φ determine Ψ Dynamics of the qubit As we have seen, the state of a qubit is a quantum superposition of two states. This superposition depends on the angles θ and φ which in the end are related to the probability of finding the qubit in state 0 or 1 and their relative phase. In this section, we will briefly describe how does the qubit state Ψ evolve with time. First of all, as in any other quantum system, the qubit states will have a limited lifetime. After a time T 1 the qubit will relax to 0. This corresponds to the vector in the Bloch sphere going from one pole to the other and is related to angle θ. Secondly, after a time τ all the phase information

17 1.2 The transmon qubit 11 contained in angle φ will be lost and therefore we won t have a quantum superposition of states anymore. This process in which we lose phase information of our system is called dephasing. It corresponds to changes in angle φ. Of course, T 1 and τ are related. No matter the dephasing, after a time T 1 vector Ψ will be pointing at the south pole and the information in angle φ will be lost. Therefore a time T 2 is defined which accounts for both effects and is the one we measure in our experiments. It is given by 1 = T 2 2T 1 τ (1.17) The Transmon qubit derived from the Cooper pair box The transmon qubit has been described historically as an special case of the Cooper pair box [10]. The Cooper pair box consists of a Josephson junction coupled with a gate capacitance C g to a gate voltage V g [Fig. 1.8(a)]. Between the gate capacitor and the Josephson junction a superconducting island is formed. The number of Cooper pairs in the island n and the phase of the Josephson junction δ follow the uncertainty relation n δ 1 (1.18) This means that a well defined n implies a non defined δ and vice versa. The energy that is required to put a single charge e into the island is given by the charging energy E C = e2 2C Σ (1.19) Here, C Σ = C g +C J. For E C large compared to the Josephson energy E J [Eq. 1.6] the number of Cooper pairs in the island n is well defined. Therefore the energy levels of the Cooper pair box are defined by n. The energy difference between these levels can be tuned via V g. The transmon architecture is similar to the one of the Cooper pair box. The difference is that a shunt capacitance C s is connected to the junction [Fig. 1.8(b)]. This capacitance decreases the charging energy E C increasing the E J /E C ratio. Usually E J /E C 20 for a transmon qubit [10]. In this regime, δ is well defined. The energy levels of the transmon qubit are then given by the phase δ of the Josephson junction. The Hamiltonian describing a shunted junction is given by Eq Ĥ = 4E C ( n n g ) 2 E J cos δ (1.20) The operator n gives the number of Cooper pairs that are transferred between the islands and δ describes the gauge-invariance phase difference between the superconductors in the

18 12 Chapter 1 Theory C g C g C s V g C J V g C J (a) Cooper pair box (b) Transmon qubit Figure 1.8: (a) Circuit scheme of the Cooper pair box. A Josephson junction with intrinsic capacitance C J is coupled through a gate capacitance C g to a gate voltage V g. (b) Circuit scheme of the transmon qubit. The shunting capacitance C s in the transmon makes it insensitive to charge noise and reduces its charging energy E C. Josephson junction, which is now promoted to a quantum mechanical operator. The offset charge n g = C g V g /2e is the charge in units of Cooper pairs induced by the gate voltage. The first term in Eq is related to the kinetic energy of the system and the second term is related to its potential energy. The energy levels as a function of the offset charge n g are shown in Fig. 1.9(a) for different E J /E C ratios. As it can be seen, upon increasing the E J /E C ratio, the energy levels become flat. This means that the system is less sensitive to the ubiquitous charge noise. On the other hand, with an increasing E J /E C ratio the anharmonicity is reduced. According to Ref. [10] the flattening of the levels decreases much faster than the anharmonicity. In Fig. 1.9(b) the cosine potential with the energy levels for high E J /E C is shown. As seen in Fig. 1.9(b), the difference between energy levels is not constant. This is important when designing a qubit. If all the level transitions were equal, we would not be able to perform operations between just two levels. In Fig. 1.9(b) the two lowest levels are depicted in red. These are the levels used for the qubit. To obtain the frequency transition between this levels, we notice that the transmon qubit [Fig. 1.8(b)] is simply an LC oscillator with a non-linear inductance. The first transition frequency can be approximated by the plasma frequency For E J /E C 1, we can write the anharmonicity α as [10] ω q = 1 LJ C 1 h 8EJ E C (1.21) α = E 12 E 01 = E C (1.22)

19 1.2 The transmon qubit E J /E C = 0.1 E J /E C = 1 6 E/E 0,n g= E J /E C = 5 E J /E C = E/E 0,n g= n g (a) n g E E J 1 8EJ E C 0 π (b) δ π Figure 1.9: (a) Eigenenergies of the Hamiltonian in Eq as a function of the offset charge n g. Each color represents one eigenenergy. (b) Cosine potential for the transmon qubit. The two lowest levels, in red, are the ones that will be used for the qubit. The energy levels in the picture are not the exact ones.

20 14 Chapter 1 Theory where E i j is the energy difference between level i and j. If the line width of the transition is much smaller than α we can describe the transmon as an effective two-level system. Then we can rewrite the Hamiltonian in Eq using the Pauli matrix σ z and ω q H = h 2 ω q σ z (1.23) 1.3 Coplanar waveguide resonators The study of superconducting qubits is done using microwave signals whose frequencies usually are in the range of a few gigahertz. In order to propagate them we use a coplanar waveguide (CPW). The coplanar waveguide consists on a center strip surrounded by one or two ground planes. The electric field lines will go from the center strip to the ground planes being manly focused in the gap between them. In Fig a drawing of a coplanar strip line is given. Ground plane Center strip Substrate Figure 1.10: Plan view of a waveguide (left) with a horizontal cut (right). The green color represents the substrate and the gray color represents a metal. The electric field between the center strip and the ground plane of the line is depicted as black arrows. In order to enhance the light in the waveguide we will create a standing wave on the line using a resonator. We have used two types of resonators. The first type is a half-wavelength resonator. We introduce two capacitors in the line. The relation between the length of the resonator 2l and the wavelength λ of the standing wave is 2l = λ/2. For the microwave regime this means 2l in the mm range. The other type is the quarter-wavelength resonator. Instead of using two capacitors, we replace one of them by a short to ground. The standing wave will be formed between this short and the capacitor. The short reflects the wave with a 180 phase shift. In this

21 1.3 Coplanar waveguide resonators 15 case 2l = λ/4. In Eq the voltage and current first mode for the half-wavelength resonator are given [12]. The quarter-wavelength resonator has an additional amplitude factor of 2 due to the fact that we store the same ammount of energy in half of the length. h V λ2 = iω r 2ω r C (âeiω rt â e iωrt ) 2sin I λ2 = 1 π h dl l 2ω r C (âeiω rt + â e iωrt ) 2cos ( π x ) 2l ( π x ) 2l (1.24a) (1.24b) Here, ω r is the resonant frequency of the resonator, 2l is the resonator length, C is the total capacitance of the resonator and dl is the inductance per unit length. Both resonators, halfwavelength and quarter-wavelength, are shown in Fig. 1.11(a) and Fig. 1.11(b) and their current and voltage first modes in Fig. 1.11(c) and Fig. 1.11(d) respectively. λ 2 λ 4 (a) (b) Voltage Current x = 0 (c) x = 0 (d) Figure 1.11: (a) Half-wavelength and (b) quarter-wavelength resonators. Figures (c) and (d) show the first voltage and current modes for both resonators. As it can be seen, the voltage and current waves have antinodes and nodes at the capacitors respectively. Now, if we apply a microwave tone and measure the transmission of the resonator we will obtain a lorentzian peak. The peak is at the resonance frequency f r of the resonator, see Fig It depends on the resonator length. In the case of a quarter-wavelength resonator we measure the reflected signal so we will have a dip at the resonant frequency. The width of the peak (or dip) f is determined by signal decaying to the measurement line and signal decaying to the environment. The first one is related to the coupling capacitors and the second one to the

22 16 Chapter 1 Theory internal losses. In our case the first contribution is the main one. 100% Power transmission 50% f f r Frequency Figure 1.12: Power transmission as a function of frequency for a half-wavelength resonator. The peak is centered at the resonant frequency. The width f of the peak depends on the value of the coupling capacitors and the internal losses. Choosing between one resonator or another depends on the use we will give to it. In general there are two important parameters for a resonator, the full width at half maximum (FWHM) of its transmission peak f and the number of incoming ports. Regarding the FWHM, a narrow peak means that we can store photons in the cavity on a long time scale which is crucial for the design of a quantum memory [13]. On the other hand, a large FWHM allows fast measurements of the state of the qubit coupled to the resonator [14]. Regarding the number of incoming ports, using one port (quarter-wavelength) assures that while performing operations between the qubit and the resonator all the information about the state of the system will go to the same port. The Hamiltonian used to describe the resonators is the one for the quantum harmonic oscillator ( Ĥ = hω r â â + 1 ) 2 (1.25)

23 1.4 Coupling of the transmon qubit to the CPW resonator 17 where ω r = 1 LC is the resonance frequency of the resonator. 1.4 Coupling of the transmon qubit to the CPW resonator In previous sections we have presented first the transmon qubit as a two level system and the waveguide resonators as the propagating media for the microwaves. Now we are going to treat the interaction between both systems. A circuit diagram of the qubit capacitively coupled to the resonator is shown in Fig Coupling capacitors Resonator Qubit Figure 1.13: Circuit diagram of the qubit coupled to the resonator. The resonator is depicted as an harmonic LC oscillator. The qubit is an anharmonic LC oscillator where the inductance L has been replaced by a Josephson junction. The coupling of the two systems is done via coupling capacitors. The resonator is represented as a linear LC oscillator whereas the qubit is represented as a non linear one. The coupling between the qubit and the resonator is represented as the two capacitors between both circuits. This coupling is therefore called capacitive coupling. The Hamiltonian of the circuit can be written as Ĥ = Ĥ res + Ĥ qubit + Ĥ int (1.26) Here, Ĥ res describes the resonator [Eq. 1.25], Ĥ qubit describes the qubit [Eq. 1.23] and Ĥ int

24 18 Chapter 1 Theory describes the interaction between the qubit and the resonator. The interaction will be given by the dipole operator and the electric field operator. Ĥ int = d E (1.27) Taking into account that d ( σ + + σ ) and E ( â + â ) we can rewrite the interaction Hamiltonian as Ĥ int = hg ( σ + + σ )( ) â + â (1.28) Here g is the coupling constant. From this expression we can neglect the terms proportional to σ + â and σ â because they violate energy conservation (rotating wave approximation) [15]. All the terms together give the Jaynes-Cummings Hamiltonian [16]. ( Ĥ = hω r â â + 1 ) + h ( 2 2 ω q σ z + hg σ + â + σ â ) (1.29) In the coupling term we have two terms. The first term σ + â is the annihilation of a photon in the resonator and the excitation of the qubit. The second term σ â represents the opposite, the relaxation of the qubit and the creation of a photon in the resonator. The coupling constant g gives the hopping rate of a photon between the resonator and the qubit when ω r = ω q. Next, we study the Hamiltonian in Eq as a function of the detuning between the qubit and the resonator. = ω q ω r (1.30) We distinguish two regimes, zero detuning ( = 0) and dispersive regime (g/ 1) Zero detuning Using the basis shown in Eq where g and e are the ground and excited state of the qubit and n is the Fock state of the resonator, the Jaynes-Cummings Hamiltonian can be written in a matrix form [Eq. 1.32]. ψ n+1,g = n + 1,g ψ n,e = n,e (1.31)

25 1.4 Coupling of the transmon qubit to the CPW resonator 19 Ĥ = hω r ( n ) h 2 ω q hg n + 1 hg n + 1 hω r ( n ) + h 2 ω q (1.32) For zero coupling g = 0 and zero detuning = 0, the eigenstates of the Hamiltonian are the ones of Eq As shown in Fig. 1.14, the states n + 1,g and n,e have the same energy (black levels). Now, if we couple both systems g 0, the eigenstates of the Hamiltonian for zero detuning are an equal superposition of n + 1,g and n,e n,+ = 1 2 ( n + 1,g + n,e ) (1.33a) n, = 1 2 ( n + 1,g n,e ) (1.33b) with eigenenergies E n± = hω r (n + 1) ± hg n + 1 (1.34) As it can be seen, due to the coupling, the degeneracy of the energy levels is lifted, blue levels in Fig. 1.14(a). The splitting of this levels depends on the coupling constant g and the number n of photons in the resonator Dispersive regime When the qubit frequency is far away from the resonator frequency, we are in the dispersive regime. It is characterized by g 1 (1.35) In this limit, we can eliminate the photon exchange term of the Hamiltonian in Eq via a unitary transformation U = exp [ g (âσ + + â σ )]. We obtain

26 20 Chapter 1 Theory (a) = 0 (b) g n, + 2g n + 1 n + 1 n, n E χ g χ ω r χ 0 g e g e Figure 1.14: Energy levels for the system qubit plus resonator. In black the levels for zero coupling. (a) For zero detuning each level is split in two due to the coupling, blue lines. (b) In the dispersive regime the levels can be seen as shifted in frequency depending on the state of the qubit, blue and red lines. Ĥ eff = ÛĤÛ = h[ω r + χ σ z ]â â + h 2 [ ωq + χ ] σ z (1.36a) Where χ = g2 (1.37) We neglected terms of order g 2 / 2. We can see in Eq that the frequency of the resonator is shifted depending on the state of the qubit [Fig. 1.14(b)]. The black levels represent the state of the system without coupling. When the qubit is in the ground state g, the levels of the system are shifted to lower frequencies by χ (blue levels). When the qubit is in the excited state e, the levels are shifted to higher frequencies by the same quantity (red levels). In Fig the transmission of the resonator is shown for the qubit in the ground state g and the qubit in the excited state e [17]. This means that by measuring the transmitted signal through the resonator we can measure the state of the qubit without destroying it. This is the so called dispersive readout [17]. The Hamiltonian in Eq can be rewritten to highlight the influence of the resonator on the qubit frequency.

27 1.4 Coupling of the transmon qubit to the CPW resonator 21 g e Power transmission ω r χ ω r + χ Frequency Figure 1.15: Power transmission of a half-wavelength resonator for the qubit in the ground (blue) and in the excited state (red). Ĥ eff = hω r â â + h [ ] ω q + 2χâ â + χ σ z (1.38) 2 Now it can clearly be seen that the qubit frequency depends linearly on the number of photons in the resonator. This will be used in Sec to calibrate the photon number in the resonator The Purcell effect When placed inside a resonator, the qubit spontaneous emission rate is altered. This is due to the so called Purcell effect. In the case of our resonator-transmon system, depending on the detuning, the qubit state has a resonator part that affects its line width. Therefore, the Purcell effect contributes with an induced relaxation rate γ P given by [10] γ P = κ g2 2 (1.39) where κ is the linewidth of the resonator and is related to its average photon loss. g and are respectively the coupling constant and the detuning between the resonator and the qubit.

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29 Chapter 2 Experimental techniques In this chapter we introduce the experimental techniques used for the fabrication and measurement of the samples. In Sec. 2.1 we describe the fabrication of the resonators with optical lithography and that of the transmon qubits with electron beam lithography. We include the dose test performed for writing the interdigitals of the transmon qubit. The shadow evaporation technique is also explained. In Sec. 2.2 we briefly describe the experimental setup and the refrigerators used for the measurements. 2.1 Nanofabrication The sample consists of a coplanar waveguide resonator made from niobium and a transmon qubit made from aluminum capacitively coupled to it. The fabrication process is divided into two parts, we first make the resonator using optical lithography and then we write the transmon in the resonator using e-beam lithography Optical lithography Optical lithography is a technique that uses UV light to write structures in the micrometer range. Our circuit is made out of niobium sputtered on a silicon substrate with a 50 nm SiO 2 layer on top. The different steps for the optical lithography are shown in Fig First we cover the sample with a thin layer of resist Fig. 2.1 panel 1. To do so we use a spin coater that spins the sample with the resist to obtain an homogeneous layer. The used angular speed is 4000 rpm. We use a mask that contains the structure we want to fabricate. Using a mask aligner we can align the mask and the sample. Now we expose the sample to UV light through this mask [Fig. 2.1 panel 2]. After the exposure [Fig. 2.1 panel 3] we introduce the sample in a glass with a developer. Only the exposed parts of the resist are soluble in the developer and therefore are removed [Fig. 2.1 panel 4]. This leaves some parts of the niobium layer uncovered. In the next step a reactive ion etching (Ar + SF 6 ) process is performed to remove the exposed niobium parts [Fig. 2.1 panel 5]. Finally we carry out the lift off introducing the 23

30 24 Chapter 2 Experimental techniques Resist Niobium Silicon oxide Figure 2.1: Optical lithography steps. The substrate is represented in dark green, the nioubium in gray and the resist in pink. The blue arrows in 2 represent the UV light. The pink arrows in 5 represent the ion etching. See text for further details. sample in acetone to remove the remaining resist [Fig. 2.1 panel 6 and 7]. In order to improve the resolution of the structure we first do steps 2 and 3 exposing only the edges of the chip with a high dose. The fabrication parameters are summarized in Table 2.1. Resist 5214E Developer AZ 726 MIF Dose for the edges( mj ) 100 cm 2 Dose for the structure( mj ) 36 cm 2 Development time (s) 110 Table 2.1: Parameters for the fabrication of the resonators using optical lithography Electron beam lithography To fabricate the Josephson junctions of the qubit we need a resolution in the nm range. The electron beam lithography (EBL) allows us to work in this regime. It relies on electrons instead of light for writing the structure. We make the whole transmon using this technique. The

31 2.1 Nanofabrication 25 principal steps are shown in Fig Upper resist Lower resist Silicon oxide δ Figure 2.2: Electron beam lithography steps. The substrate is represented in dark green, the lower resist in blue and the upper resist in red. The orange and yellow colors in 2 represent the beam of electrons. The undercut is depicted as δ in 3. See text for further details. As in optical lithography, we first spin coat the sample with resist. The difference is that the resist is deposited directly on the silicon substrate. In this case we use two resists [Fig. 2.2 panel 1] in order to obtain some undercut δ [Fig. 2.2 panel 3]. This undercut will be crucial for the fabrication of the junctions using shadow evaporation. The first resist is spin coated at 2000 rpm to obtain a thicker layer and the second one at 4000 rpm. After the spin coating we introduce the sample in the EBL for writing. As it can be seen in Fig. 2.2 panel 2, the incoming electrons reach the first resist with a high kinetic energy. The beam used is not perfectly focused, mainly due to electrons backscattered from the substrate. This means that regions next to the writing position also get exposed with a lower dose. In addition the two resists have different sensitivity. Only the lower resist is fully exposed by background electrons. After the writing we develop the sample in two steps. First we introduce it in the developer AR and then in isopropanol. Both resists have different sensitivity to isopropanol, the lower one has a higher sensitivity than the upper one. These two effects lead after the development [Fig. 2.2 panel 3] to an undercut δ as mentioned before. Using this undercut, resist bridges necessary for the fabrication of nanoscale Josephson junctions are created at suitable locations. For a detailed description of the EBL process see Ref. [18]. The fabrication parameters for the transmon qubit are shown in Table 2.2. Finally we carry out the evaporation of aluminum on the sample. We use the shadow evaporation technique [19][20] pictured in Fig. 2.3.

32 26 Chapter 2 Experimental techniques Lower resist PMM/MA 33% Upper resist PMMA 950K Dose for the SQUID ( µc ) 680 cm 2 E-beam voltage (kev) 30 Developer AR Development time (s) 60 Time in isopropanol (s) 120 Table 2.2: Fabrication parameters for the transmon qubit using electron beam lithography. Al θ AlOx θ Al 1 2 Junction 3 Figure 2.3: Shadow evaporation steps. The substrate is represented in dark green, the aluminum in gray and the aluminum oxide in dark red. The bridge is represented in the upper part in blue for the lower resist and red for the upper resists. The evaporation angle θ is given. In 3 the junction is marked in red. The shadow evaporation technique allows the fabrication of the junctions in one step. After preparing a resist bridge as described above, we evaporate aluminum under an angle θ [Fig. 2.3 panel 1]. The shadow mask of the resist bridge manifests as a gap in the evaporated aluminum. Then we oxidize the aluminum [Fig. 2.3 panel 2]. This oxide plays the role of the insulator in the junction. Finally we evaporate aluminum again using the angle θ. Due to the shifted position of the gap in this upper aluminum layer we obtain an SIS Josephson junction [Fig. 2.3 panel 3]. The evaporation parameters are shown in Table 2.3. Aluminum thickness 1 (nm) 40 Angle 1 (θ) 17 Aluminum thickness 2 (nm) 70 Angle 2 ( θ) +17 Oxygen pressure (mbar) Oxidation time (min) 40 Table 2.3: Evaporation parameters

33 2.2 Experimental setup Experimental setup The measurement of the interaction between the resonator and the qubit requires low temperatures, not only because we need the metals to be in the superconducting state but also because we need to avoid thermal excitations of the system. The frequency transitions in our system are of the order of a few gigahertz, therefore we need to work at temperatures below 100 mk. To do so we install the sample in a dilution refrigerator. The sample is introduced in a gold-plated copper box with microwave connectors for the measurement. A photo of a sample with a half-wavelength resonator in the sample box is shown in Fig Three ports are used, two for the resonator and one for an antenna inductively coupled to the qubit. Input port Output port Antenna Figure 2.4: Sample inside a gold-plated copper box for measurement. Three ports are shown, two for the resonator and one for the antenna. Regarding the measurements, we apply microwave signals to the sample and measure the output. All the microwave generators are at room temperature. This means that we will have some thermal noise coming form the outside in both our input and output ports. To reduce it we apply a large signal at room temperature with a good signal to noise ratio. We attenuate this signal in the cryostat preserving this ratio. In the outgoing line we make use of circulators that act as diodes for MW signals. They avoid thermal noise coming from the output line reaching the sample. Finally we make use of amplifiers to be able to read the signal. One amplifier is located at 4K where the amplified thermal noise is much lower than at room temperature. The experimental setup is shown in Fig. 2.5.

34 28 Chapter 2 Experimental techniques Microwave source VNA 20 db 300 K -50 db -50 db 30 db 4 K -10 db -10 db 40 db Still-800 mk -10 db -10 db 50 Ω Coil-360 mk -10 db -10 db Mixing chamber-60 mk -10 db -30 db 50 Ω Sample Figure 2.5: Experimental setup showing the different temperature stages. The attenuators are depicted as empty rectangles and the amplifiers as triangles. The circulators are depicted as circles. They are connected to 50 Ω loads to dissipate the thermal noise form the temperature stage above He cryostat In order to test the structure we perform DC measurements. The critical temperature of aluminum is 1.2K so we need to measure below this value. To do so we perform the measurement in a 3 He evaporation cryostat. This cryostat consists of several stages. The first one is an isolation vacuum that reduces the heat transfer from the environment. A second stage contains liquid- 4 He at 4.2K. Through a Joule-Thompson process some 4 He is cooled down to 1.5K. Now 3 He can be condensed in a third stage. Using evaporation cooling on the 3 He a temperature of 500 mk can be reached. A more detailed explanation of the working principle of the cryostat is given in Ref. [21] Dilution refrigerators To perform the AC measurements we introduce the sample in a dilution refrigerator. They are based on a 3 He/ 4 He mixture. This mixture undergoes a phase separation at approximately 870

35 2.2 Experimental setup 29 mk. One phase has a high concentration of 3 He, the rich phase, and the other one has only 6,6% of 3 He, the diluted phase. Due to gravity, the rich phase will be on top of the diluted phase. Liquid 3 He is pumped into the mixture from above and removed from below in a cycle. The process of 3 He going from the rich to the diluted phase is endothermic and therefore it takes energy from the system cooling down the environment. A detailed description of the process is given in Ref. [22]. A photo of the cryostat is shown in Fig Figure 2.6: Dilution fridge used for the measurements.

36

37 Chapter 3 Experimental results In this chapter we include a description of the fabricated samples and the performed measurements. In Sec. 3.1 we give the design and parameters for both resonators, half-wavelength and quarter-wavelength, and the design and dose test for the transmon qubits. In Sec. 3.2 the DC measurements used to test the qubit junctions are shown. Transmission measurements of the resonator coupled to the qubit are included in Sec Two-tone spectroscopy measurements are shown in Sec Finally, an exemplary time domain measurement is included in Sec Measured samples In this work we study two different samples, sample A and sample B. In sample A we couple the transmon qubit to a half-wavelength resonator. In sample B we use a quarter-wavelength resonator. In this section we describe first the resonators and then the transmon qubits The resonators Sample A has three ports, two for the input and output signal and the third one for an antenna inductively coupled to the qubit. This antenna is meant for tuning the qubit frequency. In sample B we have two quarter-wavelength resonators each one with an input port, a transmon qubit coupled to them and a similar antenna. We only perform measurements on one of the qubits in sample B. Parameter λ/2 λ/4 Resonant frequency (GHz) f (MHz) 7 20 Finger width (µm) Finger spacing (µm) Table 3.1: Resonators parameters 31

38 32 Chapter 3 Experimental results The different parameters for both resonators are listed in Table 3.1. The width f depends on the coupling capacitance. The half-wavelength resonator has a lower coupling capacitance and therefore a smaller f in contrary to the quarter-wavelength resonator. For both samples we use resonators with f larger than the qubit line width. This is done in order to potentially use the design as a single photon source. The design for both samples is shown in Fig. 3.1 and an optical microscope image in Fig mm 12 mm 10 mm 12 mm 100µm 100µm 100µm 100µm (a) Sample A (b) Sample B Figure 3.1: Layout of the two samples studied in this work: (a) Sample A and (b) Sample B. Blue areas are covered with niobium. White areas represent the substrate. The resonators are depicted in yellow and their coupling capacitors in red. The antennas are depicted in green. In orange the transmon is shown The transmon qubit We can divide the qubit design in two parts. One part consists on the shunt capacitance which is needed to get an E C on the order of 100 MHz. We decide here to take the commonly used interdigital capacitors. The second part consists of two Josephson junctions arranged in a DC SQUID geometry. Depending on the SQUID loop area, the sensitivity of the transmon to magnetic flux threading the SQUID loop changes. A large area means a higher sensitivity. The transmon is sample A has a small loop in order to avoid huge flux noise sensitivity. However, we increased the loop in the transmon in sample B in order to be able to tune its frequency via the antenna. A drawing of the transmon with the small loop is shown in Fig. 3.3.

39 3.1 Measured samples µm 100µm 100µm 100µm 100µm (a) Sample A (b) Sample B Figure 3.2: Pictures of both samples. Both transmon qubits, in orange, and one of the coupling capacitors, in red, are shown. Figure 3.3: Transmon qubit design. It consists on a SQUID, red rectangle, and two groups of interdigitals, blue rectangle. A detailed draw of the SQUID is shown with the two junctions marked in orange. Parameter optimization Regarding the interdigitals, due to the fact that they are closed to each other, we have to carry out a dose test to obtain the proper EBL dose for the structure. Pictures of the test are shown in Fig A low dose leads to a non defined structure. Some resist is not sufficiently exposed and therefore after the development it is not totally removed. On the other hand, if we use too much dose we might overexpose the structure. This means that some areas might get undesired dose because they are affected by background electrons in the EBL system. In Fig. 3.4 two effects related to the EBL are shown. The blue circle shows a shift in the structure. It is due to a bad write field alignment. The EBL divides the structure in squares ( µm 2 ). It goes to the center of one square and deflecting the beam writes that part of the structure. Then it moves to the next square. The write field alignment consists in joining all these squares together. If it is not optimal the consequence is the defect shown in Fig In this figure, also the proximity effect can be observed. The dose each finger of the interdigitals receives is affected by the surrounding structures. Therefore the last ones receive less dose. To compensate for this effect we increase

40 34 Chapter 3 Experimental results (a) 100 (b) 180 (c) 240 (d) 280 (e) 300 Figure 3.4: Dose test for the interdigitals on one side of the structure. The dose is given in each picture in µc/cm 2. In blue the effect of a bad write field alignment is shown. The proximity effect is depicted in red. See text for more details. the dose at this fingers in later samples. The final dose used for all transmons in both samples is shown in Fig Figure 3.5: Final dose for each part of the transmon qubit design. Pale colors mean less dose according to the legend. Dose in µc/cm 2. Both qubits, the one with the big loop and the one with the small loop, are shown in Fig. 3.6 after evaporation. Some of the fingers in the interdigitals in Fig. 3.6(a) are not connected to the transmon qubit. This problem arises because when writing the structure, the sample gets

41 3.1 Measured samples 35 Antenna 100µm 100µm (a) Sample A (b) Sample B Figure 3.6: Transmon after evaporation in (a) sample A and (b) sample B. In (a) a detailed image of the interdigitals is given in blue. In (b) the bigger loops allows for the tuning of the qubit via the on-chip antenna. 991 nm 176 nm 499 nm 937 nm 266 nm 508 nm Figure 3.7: SEM image of the SQUID and the junctions of a transmon in a half-wavelength resonator. It was fabricated with the same parameters as the one in sample A.

42 36 Chapter 3 Experimental results charged and this charge deflects the beam. To avoid this we superpose the fingers to the body of the transmon in the design file of the second qubit. This means that the interdigital capacitance for the transmon in sample A is smaller than in sample B leading to an increase in the charging energy E C. This affects some qubit parameters. An scanning electron microscope (SEM) image of the SQUID and the junctions for a transmon in a half-wavelength resonator is shown in Fig DC measurements We perform DC measurements on the SQUID of a test sample in order to verify our junction process. We contact transmons for four point measurements. First we measure the I-V curve. The current is increased from 0 to 200 na and the voltage drop across the structure is measured. The test is performed at T = 500mK. The result is shown in Fig. 3.8(a) and the measurement setup in Fig. 3.8(b). I + I V + V 100µm (a) (b) Figure 3.8: (a) Measured I-V curve of the test qubit structure. The x-axis is the measured voltage and the y-axis the applied current. (b) On-chip part of the four point measurement setup. When I < I c we see no voltage drop because the virtual phase particle is trapped in the washboard potential, see Sec When I > I c the virtual phase particle moves down the potential and a voltage drop V g is measured. For aluminum V g 0.4 mv. Theory predicts the following relation

43 3.3 Transmission measurements 37 between the critical current I c and the gap voltage V g [7] I c R n = π 4 V g (3.1) Here R n is the normal resistance of the junction. An estimation from Fig. 3.8(a) gives R n = 5.7 kω. This gives a theoretical I c of 55 na. In a second measurement we apply a magnetic field to the SQUID and measure the critical current as a function of the magnetic field [Fig. 3.9]. We use a solenoid to apply the field. The critical current changes periodically with the magnetic flux according to Eq Figure 3.9: Critical current of the SQUID as a function of the applied magnetic flux of the test transmon qubit structure. From this measurement we obtain a maximum critical current of 18 na. This value is lower than expected. One reason for this is the thermal noise. Thermal energy allows the virtual phase particle to escape from one of the washboard potential wells, see Sec This gives a lower effective potential barrier. In our case, we measured at T = 500 mk which implies I 21 na. In addition the used current source may have introduced some noise in the measurement. 3.3 Transmission measurements In this section we investigate the transmon qubit coupled to the resonator. The samples are mounted inside a dilution refrigerator and cooled to the base temperature of 60 mk. Using the Vector Network Analyzer (VNA) we measure the transmission of the resonator as a function

44 38 Chapter 3 Experimental results of the input microwave frequency. In order to observe the interaction between the resonator and the qubit we need to sweep the qubit frequency. To do so we apply a magnetic field to the sample using an external coil. As we saw in Sec , the magnetic flux will change the I c of the SQUID in the transmon qubit. A change in I c will lead to a change in the qubit transition frequency ω q. We perform a transmission measurement for several magnetic flux values in order to observe how the qubit influences the resonator transmission. Frequency (GHz) n Magnitude (db) Frequency (GHz) n = Magnitude (db) Flux (mφ 0 ) Flux (mφ 0 ) (a) Sample A (b) Sample B Figure 3.10: Transmission of the resonator as a function of magnetic flux. The x-axis is the applied magnetic flux in units of mili flux quanta. The y-axis is the frequency of the applied microwave signal. The color code is the transmission in db. The mean value of the population of photons n in the resonator during the measurement is given. In Fig the transmission for different flux values is shown for both samples. The average photon population in the resonator n during the measurement is given. As it can be seen the transmission shows a periodic behaviour with the applied magnetic flux. This is related to the transmon qubit. When the qubit frequency matches the resonator frequency an anticrossing with a clearly double-peak stucture is measured. For zero applied flux the qubit frequency has its maximum value above the resonator frequency. We are in the dispersive regime, see Sec The transmission measurement shows single peaks at this regime. A closer view of one anticrossing is shown in Fig As it can be seen in this closer view we obtain two resonant modes in our samples. From left to right first a non flux dependent mode is seen. This mode is related to the resonator. With increasing flux a second mode at lower frequencies appears. This mode is flux dependent and is related to the qubit. At the point where both modes meet there is an avoided crossing. At this point, = 0 and both peaks represent an equal superposition of qubit and resonator states. We are in the zero detuning regime, see Sec The separation between these peaks is given by 2g/2π where g is the coupling constant between the resonator and the qubit. We introduce a 2π factor due to the fact that we are working with

45 3.3 Transmission measurements 39 Frequency (GHz) n = g π Flux (mφ 0 ) Magnitude (db) Frequency (GHz) n = Flux (mφ 0 ) Magnitude (db) (a) Sample A (b) Sample B Figure 3.11: Detailed resonator transmission measurement showing an avoided crossing. frequencies f not angular frequencies ω. In Fig cuts of Fig. 3.11(a) for different flux values are shown. First a single peak is measured, Fig. 3.12(a). With increasing flux a second peak appears, Fig. 3.12(b). This peak is related to the qubit. When = 0 both peaks have the equal height, Fig. 3.12(c). They are an equal superposition of resonator and qubit states. It can be seen that the separation between the two peaks is almost 30 times larger than the line width. This shows that we reach the strong coupling limit [15]. We fit each peak in Fig to a lorentzian. Taking the central frequency of each lorentzian we obtain the frequency difference between each peak as a function of magnetic flux. We fit this difference using Eq. 3.2 to obtain g. f 2 f 1 = 1 4g 2π 2 + (Φ) 2 (3.2) The result of the fit is shown in Fig The obtained g values are g A 2π = 71 MHz g B 2π = 67 MHz (3.3) We assumed that is linear with flux for this flux range. From Eq. 3.2 we see that the minimum distance between the peaks is obtained for = 0. This distance is given by 2g/2π. From lorentzian fits at Φ = nφ 0, where the qubit is far detuned, we extract the resonators linewidth as κ A = 7 MHz κ B = 20 MHz (3.4)

46 40 Chapter 3 Experimental results (a) Φ ext = 368 mφ 0 (b) Φ ext = 342 mφ 0 g π (c) Φ ext = 338 mφ 0 Figure 3.12: Resonator transmission for different flux values, black dots. They correspond to vertical cuts of Fig. 3.11(a). Each peak is fitted to a lorentzian, blue and red curves. Peak difference (MHz) g π Peak difference (MHz) Flux (mφ 0 ) (a) Sample A Flux (mφ 0 ) (b) Sample B Figure 3.13: Frequency difference between the two modes, blue dots, as a function of the applied magnetic flux. The fit using Eq. 3.2 is shown in red.

47 3.4 Two-tone spectroscopy 41 Finally, we test the tunability of the qubit frequency via the on-chip antenna in sample B. To do so we apply a DC signal to the antenna and measure the resonator transmission as a function of the applied current [Fig. 3.14]. We obtain an avoided crossing as the one shown in Fig. 3.12(a). The different orientation is due to the different current direction for the antenna and for the coil. From this measurement we estimate the current applied to the antenna line corresponding to one mili flux quantum mφ 0 1 mφ µa (3.5) Only a small fraction of the applied current flows through the on-chip antenna. Most of the current is flowing to ground via the attenuators intersecting the microwave line in the cryostat. n = Frequency (GHz) Magnitude (db) Current (ma) Figure 3.14: Transmission of the resonator in sample B for different current values. The current was applied via the on-chip antenna. 3.4 Two-tone spectroscopy In the previous section we have observed the interaction of the qubit and the resonator measuring the transmission of the resonator as a function of the applied magnetic flux. In this section we in-

48 42 Chapter 3 Experimental results troduce a different measurement technique, the two-tone spectroscopy. A two-tone spectroscopy consists in measuring the resonator transmission using two microwave signals, see Fig. 3.15(b). The first one, the probe tone, comes from the VNA and has a constant frequency. Its frequency is the one of the resonator when the qubit is in the ground state [Fig. 3.15(a)]. The second one, the drive tone, comes from a microwave source. It varies in frequency. Whenever the microwave tone frequency matches the qubit transition frequency it changes the qubit state to a classical mixture of states. This changes the response of the resonator leading to a dip in transmission and a change in the transmitted phase. Transmission g e ω d constant MW Source VNA Sample ω r χ ω r + χ Frequency (a) (b) Figure 3.15: Two-tone spectroscopy. (a) Resonator transmission for the qubit in the ground and excited state, blue and red curves respectively. The blue arrow marks the VNA probe tone with constant frequency ω r χ. (b) Measurement setup. The VNA and the microwave (MW) source are connected to the sample. A beam splitter is used to separate both signals, tilted line. In green the applied MW signal, drive tone, with frequency ω d. The blue arrow represents the probe tone applied with the VNA. An example of a two-tone measurement sweep for sample B is shown in Fig It is performed at Φ ext = 0. The transmission is constant while the drive tone does not match the qubit frequency. However when it matches the qubit frequency a dip occurs. The central frequency of this dip is the qubit frequency ω q and its line width the qubit line width γ. For the measurement shown in Fig the qubit frequency is at ω q = GHz and the width is γ = 0.9 MHz. The probe tone is always applied to the input port of the resonator. The drive tone can be applied to the resonator or to the antenna. As we mentioned before in Sec , the antenna is inductively coupled to the transmon qubit. This means that the mutual inductance between the SQUID and the antenna has to be large enough to induce a level transition in the qubit. In the

49 3.4 Two-tone spectroscopy 43 case of sample A the loop is too small and the drive tone has to be applied to the resonator, too. However, in sample B a bigger loop is used and we can therefore use the antenna to excite and tune the qubit. ω q γ Figure 3.16: Two tone measurement of the qubit in sample B, black dots, with lorentzian fit in red. The x-axis is the frequency of the drive tone and the y-axis the transmission through the resonator Power calibration Two-tone spectroscopy can be used to calibrate the average photon population in the resonator n as a function of the applied microwave power. This is important because we want to make transitions only between the first two levels of the transmon qubit. In order to extract the proper Hamiltonian parameters from the spectrum we have to work with n < 1. In addition, we avoid many photon transitions from the ground to other excited states. The microwave power applied to the resonator is proportional to the average photon population in the resonator [23]. According to Eq. 1.38, the qubit frequency depends linearly on the photon population in the resonator. Therefore if we measure the qubit transition frequency as a function of the probe tone power we can obtain the relation between power and n. To do so we perform two-tone spectroscopy varying the power of the probe tone (VNA) which is applied to the input port of the resonator at its resonant frequency. This will populate the resonator with in average n photons. With the qubit far detuned from the resonator we apply a drive tone and measure the transmission of the resonator. The result is shown in Fig We clearly see a shift in the qubit transition to lower frequencies. A broadening of the transmission peak is observed too. At low power the qubit transition is a lorentzian peak with its central frequency shifted due to the AC-Stark effect. At high power, the transition peak becomes a gaussian with a line width proportional to n [23]. We fit only the peaks at low power with a lorentzian. Then we do a linear fit of the center frequency as a function of power. The slope α of this fit gives the number of photons n = a a as a function of the applied power P

50 44 Chapter 3 Experimental results Frequency (GHz) Magnitude (db) Frequency (GHz) Magnitude (db) VNA Power (dbm) VNA Power (dbm) (a) Sample A (b) Sample B Figure 3.17: Transmission of the resonator as a function of the applied VNA power. The y-axis is the frequency of the drive tone. g 2 a a = αp (3.6) The fit is shown in Fig and the result in Table 3.2 where the power is the output power of the VNA. We have 90 db of attenuation between the sample A and the VNA and 110 db in sample B. Frequency (GHz) Frequency (GHz) Power (dbm) (a) Sample A Power (dbm) (b) Sample B Figure 3.18: Linear fit (red line) of the measured qubit frequency (blue dots) as a function of the VNA microwave power. The difference between both values might be related to the different microwave cables used in each setup. We can make an estimation of the power needed to have n = 1 in the resonator. If

51 3.4 Two-tone spectroscopy 45 Parameter Symbol Sample A Sample B Photon population for 1 mw n of microwave power at room P(mW) temperature Power needed to populate the resonator with a photon on average P (mw) Measured qubit line width γ (MHz) Table 3.2: Obtained parameters from the power calibration. we assume an overcoupled resonator this power is given by Eq. 3.7 [24]. P = π hω r f mw (3.7) Taking into account the attenuation for each sample we obtain the power P needed to put on average one photon in the resonator. For sample A the attenuation including the microwave cables is -107 db. For sample B it is -118 db. Using these values we obtain the results in Table 3.2. It can be seen that for sample B the result is only one order of magnitude larger than the estimation we did. However for sample A the difference is of two orders of magnitude. Sample A was measured in a gold-plated copper box using silver glue for the connections. This silver glue may attenuate the incoming signal up to 4 db [25][26]. In addition, when working at low temperatures, the attenuators change their attenuation ratio and their impedance creating impedance mismatches that cause reflections. On the other hand sample B is placed in a box with a printed circuit board which gives a better matching with the incoming lines [27][28] and the used attenuators show a good impedance matching at low temperatures. When g we can use the dip in transmission to estimate the qubit line width. We need low microwave power so the width of the peak is not affected by the photons in the resonator. Two transmission measurements obtained at low power are shown in Fig We fit both dips with a lorentzian. From the fit we obtain the line width of the qubit transition for both samples, Table 3.2. If we compare these values with the coupling constant g obtained in Sec. 3.3 and the resonators widths κ we see that we reach the strong coupling regime where g κ,γ [15] Flux dependence of the qubit frequency In order to measure the qubit transition frequency for different flux values we perform two-tone spectroscopy sweeping the magnetic flux. To do so we use an external coil. The magnetic flux changes the qubit transition frequency according to

52 46 Chapter 3 Experimental results n 1 ω A n 1 ω B γ A γ B (a) Sample A (b) Sample B Figure 3.19: Measured spectroscopic lines for large detuning and low power, black dots. The y-axis is the transmission of the resonator and the x-axis is the frequency of the drive tone. A lorentzian fit is shown in red. ω q (Φ ext ) = 1 h 8EJ (Φ ext )E C = 1 h 8 Φ 0I c,eff (Φ ext ) E C (3.8) 2π where I c,eff is given by Eq This equation is only valid for a large detuning. Exemplary measurements for both qubits are shown in Fig n = n = Frequency (GHz) Magnitude (db) Frequency (GHz) Magnitude (db) Flux (mφ 0 ) Flux (mφ 0 ) (a) Sample A (b) Sample B Figure 3.20: Flux dependence of the qubit frequency. The color indicates the resonator transmission. The x-axis is the applied magnetic flux and the y-axis is the drive tone frequency. In the case of sample A, only a part of the whole measurement is shown. We fit each peak with a lorentzian. The central frequency of the peaks can be fitted using Eq. 3.8 and Eq All the measurement range in sample A is used. In Fig the fit is shown. We remove the points