Tunable coupling between a superconducting resonator and an artificial atom

Qi-Kai He1, 2 and D. L. Zhou1, 2, ∗

1Institute of Physics, Beijing National Laboratory for Condensed Matter Physics,Chinese Academy of Sciences, Beijing 100190, China

2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China(Dated: June 7, 2018)

Coherent manipulation of a quantum system is one of the main themes in current physics re-searches. In this work, we design a circuit QED system with a tunable coupling between an artificialatom and a superconducting resonator while keeping the cavity frequency and the atomic frequencyinvariant. By controlling the time dependence of the external magnetic flux, we show that it ispossible to tune the interaction from the extremely weak coupling regime to the ultrastrong cou-pling one. Using the quantum perturbation theory, we obtain the coupling strength as a function ofthe external magnetic flux. In order to show its reliability in the fields of quantum simulation andquantum computing, we study its sensitivity to noises.

PACS numbers: 03.67.Lx, 32.80.Qk, 74.50.+r, 03.65.Yz

I. INTRODUCTION

The light-matter interaction in circuit quantum elec-trodynamics (QED) finds lots of applications in manyquantum information processes, such as the simulation ofresonance fluorescence [1], an experimental proposal forboson sampling [2] and the single-photon scattering onan atom [3–5]. All these achievements show that circuitQED is an excellent platform for studying the physicsinduced by light-matter interaction [5–8].

In previous studies, the photon-photon interaction [9–11] and the atom-atom interaction [12] have been inves-tigated with elaborate superconducting Josephson junc-tion circuits, and the photon-atom interaction has beenstudied in all coupling regimes. Nevertheless, we notethat it is rarely mentioned how to control the light-matterinteraction during an adiabatic process [13], which is in-evitable and valuable in such experiments. To this end,it becomes an urgent need to design a superconductingcircuit for implementing the light-atom interaction witha continuously tunable coupling strength and with a fixedqubit frequency.

Designing such a circuit is equivalent to devising anunusual qubit. The widely used superconducting qubitsincludes phase qubit [14], transmon [15], and Xmon [16],which are different assemblies of Josephson junctions andother circuit components like capacitance, inductance.Because of different structures of these qubits, they canbe manipulated by different external signals and sensi-tive to diverse noise sources. In the circuit QED, thesingle mode cavity can be realized by a LC oscillator ora microwave transmission line. To construct a quantumnetwork, we need to couple qubits and photons, whichcan be implemented by the capacitance connection [17],the mutual inductance [18, 19] and the direct connec-tion [20]. In particular, it is worthy to point out that

∗ zhoudl72@iphy.ac.cn

the direct connection between a qubit and a microwavetransmission line has led the coupling strength into theultrastrong regime [5, 20].

Up to now, the qubit-resonator coupling strength canbe tuned from zero to a finite value [21–24]. This couplingstrength is controlled by the direct capacitance betweenthe upper island and the lower island, which can’t betuned in time. Other than directly changing the capaci-tance, the external signals like magnetic flux can also beused to modify the photon-atom interaction. In general,the qubit frequency will be changed with the variation ofthe coupling strength [5]. If we ignore the tiny variationof the qubit frequency, the dc-SQUID can be used as atunable coupler to modulate the coupling strength [25].However, the coupling strength controlled by this circuitdesign cannot be tuned in all coupling regimes.

Inspired by Refs. [17] and [20], we devise an artifi-cial atom based on the superconducting loop contain-ing two dc-SQUIDs and make the photon-atom couplingflux-controlled. To see clearly the underlying mechanism,we theoretically calculate the coupling strength betweentwo coupled states. As an application, we investigatethe adiabatic dynamics of the qubit in superconductingresonators in all coupling regimes. Furthermore, we dis-cuss the influence of the resistance of Josephson junc-tions on our circuit and show its extensive applicabilityin quantum simulations. For example, our system allowsthe adiabatic switching of the coupling strength from thestrong to the ultra strong coupling regime which makes ita reliable tool for implementing a quantum memory [26].Moreover, if fast-tuning is possible, our system can alsobe used for simulating the relativistic effects [27, 28].

The rest of this paper is organized as follows. In Sec. II,after introducing our design, we analytically study theHamiltonian of the artificial atom. With the help ofquantum perturbation theory, we obtain the explicit for-mula of the coupling strength and the energy differencebetween two lowest eigenstates of the atom, which are de-tailed derived in Appendix A. Based on the formula, wepropose the tunable coupling scheme, which makes our

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system a tunable coupling Rabi model. To test the ex-perimental feasibility of our circuit, we consider the dis-sipation of the artificial atom in Sec. III. With all theseanalytical results, we estimate the proper value of deviceparameters of our qubit in a circuit QED experiment inSec. IV. In Sec. V, we draw the conclusions.

II. TUNABLE COUPLING

In this section, we study the mechanism for the cou-pling between an artificial atom and the superconductingresonator. The artificial atom is based on superconduct-ing quantum interference devices (SQUIDs), which arewidely used in circuit QED. To manipulate the energysplitting of the system as well as the coupling strengthwith outsides, we control three time-dependent magneticfluxes. Since the coupling strength need to be tuned intothe ultrastrong regime, we directly connect the atom tothe transmission line. Based on the two previous ideas,we theoretically design the artificial atom schematicallyshown in Fig. 1. We will give its Hamiltonian and derivethe expression for the coupling strength.

A. System Hamiltonian

CJ

CJ

CJ

CJ

Φ1 Φ2

Lr

φ1

φ2

φ3

φ4

φr

Φ3

L0 L0

C0 C0

φaR φbL

φaL φbR

FIG. 1. (Color online). The schematic of the circuit layoutof an artificial atom sharing an inductance with a coplanartransmission line. In the lumped element approximation, thistransmission line resonator is constructed by two LC oscil-lators with the inductance L0 and capacitance C0. The ar-tificial atom in the dashed gray rectangle is composed of asuperconducting loop with two dc-SQUIDs. CJ representsthe capacitance of every Josephson junction. Three external

magnetic fluxes are denoted as Φ1,Φ2 and Φ3. φa(b)L and φ

a(b)R

are the reduced node flux for each side of the LC resonator.

In Fig. 1, an artificial atom is galvanically attachedto the center of the coplanar waveguide transmission lineresonator which is described as a two-mode LC resonatorwith its inductance L0 and capacitance C0. As shownin the dashed gray rectangle of Fig. 1, this atom con-tains four Josephson junctions that are designed withthe same area and combined into two dc-SQUIDs. Simi-lar to the flux qubit, our atom adopts a superconducting

coil to connect dc-SQUIDs in series. In order to elimi-nate the induced currents flowing to the connecting loopwith self-inductance Lr, the penetrating fluxes of the twodc-SQUIDs have the same value (Φ1 = Φ2) but are in op-posite directions.

For simplicity, we neglect the additional flux generatedby the circulating loop current in the dc-SQUIDs, whichis equivalent to restricting our discussion to the screen-

ing parameter βL =LSIcϕ0

� 1. Here, LS is the loopinductance of the dc-SQUID, Ic is the critical current of

the Josephson junctions, and ϕ0 =~2e

is the reduced flux

quantum.As shown in Fig. 1, we divide the system into two

parts: the circuits inside and those outside of the dashedrectangle. The Lagrangian of our system is

L̂ = L̂′ + L̂′′ (1)

with

L̂′ =4∑i=1

[1

2CJϕ

20φ̇

2i − EJ(1− cosφi)

]− ϕ20

(φbL − φaR)22Lr

,

(2a)

L̂′′ =C02ϕ20[(φ̇

aL)

2 + (φ̇bR)2]− ϕ20

(φaL − φaR)2 + (φbL − φbR)22L0

,

(2b)

where L′ and L′′ are the Lagrangian of the inside partand that of the outside part respectively, φi (i = 1, 2, 3, 4)is the phase difference across the i-th junction, and

φa(b)L(R) =

1

ϕ0

∫ t−∞ V

a(b)L(R)(τ)dτ is the reduced node flux [29]

which corresponds to the electric potential Va(b)L(R) for each

side of the LC resonator.Following Ref. [20, 30], we introduce ϕ± = φbR ± φaL,

Φ+ = φbL+φ

aR and consider φr = φ

bL−φaR, then we rewrite

the Lagrangian (2a) and (2b) into

L̂′ =4∑i=1

[1

2CJϕ

20φ̇

2i − EJ(1− cosφi)

]− ϕ20

φ2r2Lr

, (3a)

L̂′′ =C04ϕ20(ϕ̇

2+ + ϕ̇

2−)− ϕ20

(ϕ+ − Φ+)2 + (ϕ− − φr)24L0

.

(3b)

Note that the conditions of fluxoid quantization alongthe independent loops in the circuit are given by

φ2 − φ1 = f, (4a)φ3 − φ4 = f, (4b)φ2 + φ4 − φr = f ′, (4c)

where

f =Φ1ϕ0

=Φ2ϕ0, (5a)

f ′ =Φ3ϕ0. (5b)

3

With the Lagrangian (3) and the relation (4), we writethe Hamiltonian of our system

Ĥ ′ =Ω2Jp

2+

8~ωr+

Ω2Jp2−

8~ωr+ 2~ωrφ2+ + 2EJ

{2− cos f

2

×[cos(φ+ + φ− −

f − f ′2

) + cos(φ+ − φ− +f + f ′

2)

]},

(6)

and

Ĥ ′′ =ω2cp

′2+

~ω0+

~ω0ϕ2+4

+~ω0Φ2+

4− ~ω0

2ϕ+Φ+

+ω2cp

′2−

~ω0+

~ω0ϕ2−4

+ ~ω0φ2+ − ~ω0ϕ−φ+,(7)

where ΩJ =1√LrCJ

, ~ωr =ϕ20Lr

, ωc =1√L0C0

,

~ω0 =ϕ20L0

, and p± = 4CJϕ20φ̇± is the conjugate

momentum with corresponding phase difference φ± =(φ2 − f ′/2)± (φ4 − f ′/2)

2, p′± =

1

2C0ϕ

20ϕ̇± is the canon-

ical momentum which corresponds to ϕ±.From Eqs. (6) and (7), we can indicate that Ĥ ′ rep-

resents the Hamiltonian of the artificial atom withoutoutside connections, Ĥ ′′ describes the two-mode LC res-onator with intrinsic frequency ωc. Since φ+ forms thepart of the qubit while Φ+ does not, we conclude thatthe qubit only couples with one mode of the resonatorand −~ω0ϕ−φ+ represents the qubit-resonator interac-tion. In this sense, we can treat the circuit shown in Fig. 1as a coupling system with one atom and a single-modecavity. In this simplified system, the atomic Hamiltonianshould consider not only the inside part of the dashedrectangle but also the renormalization ~ω0φ2+ from out-sides. Then the Hamiltonian of our system is

Ĥ = Ĥatom + Ĥcav + Ĥint, (8)

where

Ĥatom = Ĥ′ + ~ω0φ2+, (9a)

Ĥcav =ω2cp

′2−

~ω0+

~ω0ϕ2−4

, (9b)

Ĥint = −~ω0ϕ−φ+. (9c)

In the second quantization representation, the cavityHamiltonian (9b) is rewritten as

Ĥcav = (â†â+

1

2)~ωc, (10)

where â =

√ω04ωc

(ϕ̂− +

2iωc~ω0

p̂′−

)is the photon anni-

hilation operator for the cavity. Then the interactionHamiltonian (9c) can be changed to

Ĥint = −~√ω0ωcφ+(â

† + â), (11)

which implies that the tunable coupling to the cavity ismainly determined by φ+.

B. Theoretical analysis of the coupling strength

To give a theoretical analysis of the coupling strength,we resort to the quantum perturbation theory [31]. Nowwe consider the case of f = π − ∆ (∆ � π). For sim-plicity, we define the charging energy Ec =

e2

2CJand

~ω′r =ϕ20L′r

with L′r =2L0Lr

2L0 + Lr. Then we divide the

Hamiltonian (9a) into two parts, Ĥatom = Ĥ0 + V̂ , wherethe unperturbed Hamiltonian

Ĥ0 =Ec~2p̂2+ + 2~ω′rφ2+ +

Ec~2p̂2− + 4EJ , (12)

and a perturbation term

V̂ = −2∆EJ cos(φ+ +f ′

2) sin(φ− +

∆

2) (13)

with ∆ being a small parameter.In the unperturbed Hamiltonian (12), the conjugate

momentum p̂+ and its corresponding coordinate φ+ de-scribe a harmonic oscillator, and the other canonical mo-mentum p̂− = n̂−~ with n̂− being the relative cooper pairnumber operator between the two dc-SQUIDs. In otherwords, the unperturbed Hamiltonian given by Eq. (12)can be rewritten as

Ĥ0 = Eb(b̂†b̂+

1

2) + Ecn̂

2− + 4EJ , (14)

where Eb =√

8~ω′rEc, b̂ =1

2λ

(φ̂+ +

2iλ2

~p̂+

)is the

annihilation operator with λ =

√EcEb

. We solve the eigen

problem of Ĥ0:

Ĥ0|n;n−〉 = E(0)n;n− |n;n−〉, (15)

where b̂†b̂|n〉 = n|n〉, n̂−|n−〉 = n−|n−〉, and

E(0)n;n− = (n+1

2)Eb + Ecn

2− + 4EJ . (16)

It is easy to see that all excited states with n− 6= 0 of Ĥ0are two-fold degenerate. In our paper, we focus on theregion of Eb � Ec, where the lower energy eigen statessatisfies n = 0, e.g., |0; 0〉 is the ground state, and |0;±1〉are the lowest degenerate excited states.

To determine the eigenstates of Ĥatom, we first performthe symmetry analysis of the Hamiltonian. In fact, theHamiltonian is invariant under the transformation φ− →

4

f − φ−, which corresponds to the parity operator

P̂ =

∫dφ−|f − φ−〉〈φ−|

=∑n−

∑n′−

∫dφ−|n−〉〈n−|π − φ− −∆〉〈φ−|n′−〉〈n′−|

=1

2π

∑n−

∑n′−

∫dφ−|n−〉e−in−(π−φ−−∆)ein

′−φ−〈n′−|

=∑n−

|n−〉e−in−(π−∆)〈−n−|,

(17)

where |φ−〉 is the eigenstate of φ̂− with the eigenvalueφ−. Actually, it is easy to check that [P̂ , Ĥ0] = [P̂ , V̂ ] =[P̂ , Ĥatom] = 0. Therefore we can always choose the

eigenstate of Ĥatom with definite parity, whose zero-ordereigenstate has the same parity. The eigen problem of P̂is given by for any positive integer n−

P̂ |±n−〉 = ±|±n−〉, (18)

where

|±n−〉 = ein−∆/2|n−〉 ± P̂ |n−〉√

2

=ein−∆/2|n−〉 ± (−1)n−e−in−∆/2| − n−〉√

2. (19)

It is worthy to note that the state |n− = 0〉 is with evenparity: P̂ |0〉 = |0〉.

Using the parity operator P̂ , we write the zero-ordereigenstates of two lowest excited states as

|ψ(0)+ 〉 = |0; +1〉, (20a)|ψ(0)− 〉 = |0;−1〉 (20b)

which obeys P̂ |ψ(0)+ 〉 = |ψ(0)+ 〉 and P̂ |ψ(0)− 〉 = −|ψ(0)− 〉,that is, |ψ(0)+ 〉 and the zero-order ground state |ψ(0)0 〉 =|0; 0〉 are the states with even parity and |ψ(0)− 〉 is thatwith odd parity. Then the three lowest eigenstates ofĤatom, |ψ0〉, |ψ+〉, and |ψ−〉 have the same parity astheir zero-order eigenstates |ψ(0)0 〉, |ψ

(0)+ 〉, and |ψ(0)− 〉 re-

spectively.

Since [φ̂+, P̂ ] = 0, we have

〈ψ0|φ̂+|ψ−〉 = 〈ψ0|P̂ φ̂+P̂ |ψ−〉 = −〈ψ0|φ̂+|ψ−〉, (21a)〈ψ+|φ̂+|ψ−〉 = 〈ψ+|P̂ φ̂+P̂ |ψ−〉 = −〈ψ+|φ̂+|ψ−〉, (21b)

which implies that 〈ψ0|φ̂+|ψ−〉 = 〈ψ+|φ̂+|ψ−〉 = 0.Hence we can restrict the Hilbert space of the artificialatom into the subspace with the bases {|g〉 = |ψ0〉, |e〉 =

|ψ+〉}. In this subspace, the Hamiltonians

Ĥatom = Eg|g〉〈g|+ Ee|e〉〈e|, (22a)Ĥint = −~

√ω0ωc

(〈e|φ̂+|g〉|e〉〈g|+ 〈g|φ̂+|e〉|g〉〈e|

+〈g|φ̂+|g〉|g〉〈g|+ 〈e|φ̂+|e〉|e〉〈e|)

(↠+ â),

(22b)

where

Eg ≈1

2Eb+4EJ−2∆2E2Je−λ

2

cos2 f′

2Ec

+λ2 sin2

f ′

2Eb

,(23)

Ee ≈1

2Eb + Ec + 4EJ +

5

3∆2

E2JEc

e−λ2

cos2f ′

2

− 3∆2E2J

Ebλ2 sin2

f ′

2,

(24)

and

〈e|φ̂+|g〉 = (〈g|φ̂+|e〉)∗ ≈ i2√

2∆λ2EJEb

e−λ2/2 sin

f ′

2,

(25a)

〈g|φ̂+|g〉 ≈ −4∆2E2Jλ

2e−λ2

EbEcsin f ′, (25b)

〈e|φ̂+|e〉 ≈10∆2E2Jλ

2e−λ2

3EbEcsin f ′. (25c)

Finally we rewrite the Hamiltonian (8) as

Ĥ = (â†â+1

2)~ωc+

δE

2σ̂z+~(gσ̂y+g0σ̂0 +gzσ̂z)(â†+ â),

(26)where the Pauli operators σ̂z = |e〉〈e| − |g〉〈g|, σ̂y =−i|e〉〈g|+ i|g〉〈e|, σ̂0 = |e〉〈e|+ |g〉〈g|,

δE =Ee − Eg

≈Ec + ∆2E2Je−λ2

(11

3Eccos2

f ′

2− λ

2

Ebsin2

f ′

2

)(27)

is the energy level splitting of the atom,

g ≈√

8ω0ωc∆λ2EJEb

e−λ2/2 sin

f ′

2(28)

and

g0 ≈ g√

2∆EJ6Ec

e−λ2/2 cos

f ′

2, (29a)

gz ≈ −g11√

2∆EJ6Ec

e−λ2/2 cos

f ′

2(29b)

are the coupling strength which corresponds to σ̂y,σ̂0 and σ̂z respectively (see detailed derivations in Ap-pendix A).

5

0.490 0.492 0.494 0.496 0.498 0.500

0.6

0.8

1.0

0.490 0.492 0.494 0.496 0.498 0.5000.5

0.6

0.7

0.8

0.9

1.0

f/(2π)

f′ /(2π)

(a) f ′(f).

0.490 0.492 0.494 0.496 0.498 0.5004.24

4.25

4.26

4.27

4.28

0.490 0.492 0.494 0.496 0.498 0.5004.24

4.25

4.26

4.27

4.28

f/(2π)

E/E

J

EgEeE2E3E4

(b) E(f).

0.490 0.492 0.494 0.496 0.498 0.500

0.00

0.05

0.10

0.15

0.20

0.490 0.492 0.494 0.496 0.498 0.500

0.00

0.05

0.10

0.15

0.20

f/(2π)

couplingstrength

g/ωcgz/ωcg0/ωc

(c) g(f).

0.490 0.492 0.494 0.496 0.498 0.500

2.00

2.02

2.04

0.490 0.492 0.494 0.496 0.498 0.500

2.00

2.01

2.02

2.03

2.04

2.05

f/(2π)

(E−E

g)/h[G

Hz]

EeE2

(d) Zoom in of (b).

FIG. 2. (Color online). (a) The external flux f ′ as a function of f for a fixed δE. (b) The five lowest eigenenergies of Ĥatom asa function of f for a fixed δE. Results obtained via the numerical diagonalization are represented by lines with different colors.Eg and Ee obtained by Eqs. (23) and (24) are denoted as red and blue solid circles respectively. (c) The coupling strength asa function of f for a fixed δE. The triangles represent results obtained via the numerical diagonalization, and the solid linesrepresent results obtained by Eqs. (28), (29a) and (29b). (d) The energies Ee and E2 of two lowest excited states relative to

Eg. The black dashed line represents the value of the fixed atomic frequency δE/h. Here we takeEJEc

= 150, EJ/h = 300GHz

and δE/h =ωc2π

= 2.00005254655GHz, L0 = 0.06192867473nH, Lr = 12.29291953901nH.

From Eqs. (27) and (28), we can get

g2E2b8ω0ωcλ4E2J

= ∆2e−λ2

sin2f ′

2

=11Eb∆

2e−λ2

11Eb + 3λ2Ec− 3EbEc(δE − Ec)

(11Eb + 3λ2Ec)E2J.

(30)

In this article, we attempt to tune the coupling strengthg from zero to a finite value. In order to get an extremelysmall g, we should have δE > Ec according to Eq. (30).In this sense, f ′ needs to satisfy the condition that

f ′ > 2π − arccos(

3λ4 − 1111 + 3λ4

)(31)

for δE being fixed.As illustrated in Fig. 2a, we show how to change f ′

for different f to keep the atomic energy level splittingδE constant. With the same relation between f ′ and f ,

the five lowest eigenenergies of Ĥatom and the couplingstrength g as functions of f are given in Figs. 2b and 2crespectively. When we zoom in Fig. 2b, we observe thatEe −Eg is invariant with external flux f which can seenin Fig. 2d.

In Fig. 2c, we observe that g/ωc becomes smaller withthe increase of f , and it limits to zero when f tends toπ. We also note that |g0/g| and |gz/g| increase with thedecrease of f . When f is far below π, gz is comparablewith g which makes it nonnegligible. In this case, oursystem can simulate a tunable coupling generalized Rabimodel [32]. If we restrict the value range of f and f ′,we can get a tunable coupling Rabi model. For example,as shown in Fig. 2c, |gz/g| � 1 and |g0/g| � 1 whenf > 0.988π. Therefore, we can propose the tunable cou-pling scheme that changing the values of external flux fand f ′ to tune the coupling strength from the extremelyweak coupling regime to the ultrastrong coupling one andleave the energy splitting δE unchanged. Moreover, it is

6

worthy to point out that all the results in Fig. 2 obtainedby quantum perturbation theory agree well with thosefrom the numerical exact diagonalization, which verifiesthe validness of our calculations.

III. DISSIPATION OF THE QUBIT

In general, the coupling of a superconducting qubit toits environment will cause two different dissipative pro-cesses, relaxation and pure dephasing, each with theircharacteristic time constants T1 and Tϕ. In this section,we discuss the performance of the qubit in terms of thesetwo time constants in the tunable coupling scheme pro-posed above.

A. Estimates for the relaxation time T1

As the de-excitation process of a qubit, the relaxationis originated from a perturbation which couples the qubitwith its noise sources. In this perturbation, the qubit

operator is defined as∂Ĥatom∂µ

where µ is the external

parameter in the qubit’s Hamiltonian which correspondsto the noise source [33]. For weak noise sources, we follow

Ref. [34] and give T(µ)1 by Fermi’s golden rule

Γ(µ)1 =

1

T(µ)1

=1

~2

∣∣∣∣∣〈e

∣∣∣∣∣∂Ĥatom∂µ∣∣∣∣∣ g〉∣∣∣∣∣

2

Sµ(ω), (32)

where Sµ(ω) is the spectral density of the bath noise. Tothis end, we can obtain the relaxation time with Eq. (32).

Compared to other solid-state qubits [15, 16, 35–38],our qubit is remarkably sensitive to flux noise due to itsunique structure. Therefore, it is reasonable to investi-gate the qubit’s dissipation caused by the flux noise atfirst.

Reviewing the qubit architecture in Fig. 1, we findthat the coupling of our qubit to three external magneticflux biases opens up an additional channel for energy re-laxation, which is the internal coupling between the cir-cuit and the flux biases through mutual inductance Mi(i = 1, 2, 3). After introducing the flux noise power spec-trum

SΦi(ω) = M2i SIi(ω) = 2M

2i Θ(ω)~ω/ZR (33)

with ω =δE

~and the environmental impedance ZR for

low temperature kBT � ~ω, we get

T(Φ)1 ≈

~2ϕ20ZRE2JδE

eλ2

[M21 cos

2 ∆ + f′

2+M22 cos

2 ∆− f ′2

+∆2M23 sin2 f′

2

]−1.

(34)

0.490 0.492 0.494 0.496 0.498 0.5000.0

0.2

0.4

0.6

0.8

1.0

1.2

f/(2π)

T(Φ

)1

[s]

(a) T(Φ)1 (f).

0.490 0.492 0.494 0.496 0.498 0.5000.0

0.1

0.2

0.3

0.4

0.5

0.6

f/(2π)

T(Φ

)ϕ

[ms]

(b) T(Φ)ϕ (f).

FIG. 3. (Color online). The characteristic time T(Φ)1 and

T(Φ)ϕ as a function of the external flux f . Here we take the

same parameters as those of Fig. 2 and M1 = M2 = 40Φ0/A,M3 = 35Φ0/A, ZR = 50Ω, AΦ1 = AΦ2 = AΦ3 = 10

−6Φ0.In Fig. (b), the red triangles represent results obtained viathe numerical diagonalization, and the solid lines representresults obtained by Eq. (37).

With the same qubit for Fig. 2 and realistic deviceparameters, we plot the relation between the relaxation

time T(Φ)1 and f . As shown in Fig. 3a, T

(Φ)1 reaches its

maximum (1.06893s) near f = 0.998487π. Obviously,

T(Φ)1 is much longer than the time unit (ns) of a clock

cycle used in experiments, which indicates that the re-laxation induced by flux coupling is unlikely to limit themanipulation of our qubit.

Similar to the flux noise, the charge noise is an im-portant noise source which limits the applications ofcharge type superconducting qubits. To our qubit, the

charge noise corresponds to the qubit operator∂Ĥatom∂n̂−

=

2Ecn̂−.Since n̂−P̂ + P̂ n̂− = 0, we have

〈e|n̂−|g〉 = 〈e|n̂−P̂ P̂ |g〉 = −〈e|P̂ n̂−P̂ |g〉 = −〈e|n̂−|g〉,(35)

which implies that 〈e|n̂−|g〉 = 0. With Eqs. (32) and

7

(35), we can show that the relaxation transition rate ofour qubit induced by the charge noise is zero. In thissense, the charge noise will not affect the relaxation pro-cess of our qubit in the tunable coupling scheme.

B. Estimation of the pure dephasing time Tϕ

The coupling with the environment results not onlyin relaxation but also in pure dephasing. As we know,the origin of dephasing can be interpreted as the qubittransition frequency fluctuations induced by noises fromoutside. In order to study the dephasing of our qubit, wedefine Tϕ as the characteristic time for the decay of theoff-diagonal density matrix element. For sufficiently lowfrequencies, we assume that the environment provides1/f noise [39] to our qubit. In this sense, we have [15]

T (s)ϕ '~As

∣∣∣∣∂δE∂s∣∣∣∣−1 , (36)

where As is the 1/f amplitude corresponding to theexternal parameter represented by s for different noisesources.

According to Eq. (27), δE is dominated by the externalflux and the Josephson energy EJ , which implies that theflux noise and the critical current noise are the main noisesources for our qubit. By Eq. (36), we have

T (Φ)ϕ =2~ϕ0

∆AΦE2Jeλ

2

[(11

3Ec− λ

2

Eb

)+

(11

3Ec+λ2

Eb

)×(

cos f ′ − 3∆2

sin f ′)

+

∣∣∣∣( 113Ec − λ2

Eb

)+

(11

3Ec+λ2

Eb

)(cos f ′ +

∆

2sin f ′

)∣∣∣∣]−1(37)

with AΦ = AΦ1 = AΦ2 = AΦ3 . For AΦ = 10−6Φ0 [40],

the same qubit of Fig. 2 yields a dephasing time of the

order of T(Φ)ϕ ∼ 10µs. Fig. 3b shows the variation of T (Φ)ϕ

with f in our tunable coupling scheme.For the critical current noise, we choose AIc =

10−6Ic [41]. Then Eq. (36) gives

T (Ic)ϕ '~AIc

∣∣∣∣∂δE∂Ic∣∣∣∣−1 = ~2(AIc/Ic)(δE − Ec) ≈ 1.51442s

(38)for our qubit in Fig. 2.

Furthermore, the fluctuation of our qubit transitionfrequency can also be induced by the fluctuation of therelative cooper pair number between the two dc-SQUIDs.Assuming that the relative cooper pair number n̂c can bedecomposed into the noiseless charge number n̂− and asmall noise term, i.e., n̂c = n̂− + δn− with δn− � n−.Then a Taylor expansion of Ĥatom yields

Ĥatom → Ĥ0 + V̂ + 2Ecn̂−δn−. (39)

Considering the relation between n̂− and the parityoperator P̂ , we introduce the odd-parity excited state|ψ−〉 with its corresponding eigenenergy

E2 =Eb2

+Ec+4EJ−∆2E2Je−λ2

cos2 f′

23Ec

+λ2 sin2

f ′

2Eb + 3Ec

.(40)

Therefore, we use

〈ψ−|n̂−|g〉 ≈ −i√

2∆EJe−λ2/2 cos

f ′

2Ec

, (41a)

〈ψ−|n̂−|e〉 ≈ 1 + 2∆2E2Je−λ2

cos2 f′

29E2c

−λ2 sin2

f ′

2(Eb + 3Ec)2

(41b)

to get the modification of the energy level splittingδE(n−) from the coupling between |ψ−〉 and |e〉, |g〉 bythe charge noise term 2Ecn̂−δn−

δ[δE] =δE(n− + δn−)− δE(n−)

=4E2c (δn−)2

( |〈ψ−|n̂−|g〉|2E2 − Eg

− |〈ψ−|n̂−|e〉|2

E2 − Ee

),

(42)

which is proportional to the square of δn−. In otherwords, the second order contributions of the charge noisewill dominate the pure dephasing process.

Based on the above consideration, we generalizeEq. (36) to the second order

T (c)ϕ '∣∣∣∣π2A2c~ ∂2δE∂(eδn−)2

∣∣∣∣−1=

~e2

π2A2c

∣∣∣∣ limδn−→0[δE(n− + 2δn−)− δE(n− + δn−)

(δn−)2

−δE(n− + δn−)− δE(n−)(δn−)2

]∣∣∣∣−1=

~e2

8π2A2cE2c

∣∣∣∣ |〈ψ−|n̂−|g〉|2E2 − Eg − |〈ψ−|n̂−|e〉|2

E2 − Ee

∣∣∣∣−1 ,(43)

where Ac is the amplitude of the charge 1/f noise.With the same parameters in Fig. 2 and Ac =

10−4e [42], we find that T (c)ϕ reaches its minimum1.00293ms near f = 0.99951π which can be seen in Fig. 4.In this sense, our qubit has an excellent performance sup-pressing the charge noise.

IV. ESTIMATION OF DEVICE PARAMETERS

In order to apply our design to experiments, we shouldchoose proper value of device parameters such as L0, Lr.

8

0.490 0.492 0.494 0.496 0.498 0.5000

500

1,000

1,500

2,000

2,500

f/(2π)

T(c)

ϕ[s]

(a) T(c)ϕ (f).

0.49974 0.49975 0.49976 0.49977

0.00

0.05

0.10

0.15

0.20

f/(2π)

T(c)

ϕ[s]

(b) Zoom in of (a).

FIG. 4. (Color online). The characteristic time T(c)ϕ as a func-

tion of the external flux f . Here we take the same parametersas those of Fig. 2 and Ac = 10

−4e.

In this paper, we attempt to tune the atom-photon cou-pling strength from weak coupling regime into the ultrastrong one while keeping the cavity frequency and theatomic frequency invariant. With Eq. (28), we can writethe formula of the coupling strength with respect to thecavity frequency

g

ωc=

√8ω0ωc

∆λ4EJEc

e−λ2/2 sin

f ′

2, (44)

which indicates that λ, the ratioEJEc

andω0ωc

are the main

factors influencing its value.During the derivation process of δE and g, we assume

that ∆ � π and λ � 1 (Eb � Ec) to ensure the estab-lishment of quantum perturbation theory. In this sense,we have

1

2L0+

1

Lr=

1

L′r� Ec

8ϕ20, (45)

which means that L0 and Lr are both far more less than8ϕ20Ec

. In Fig. 2, we take Ec/h = 2GHz, then Eq. (45)

gives {L0, Lr} � 0.653983µH which implies the reason-ableness of our choice.

Generally, the relaxation time of a qubit in a circuitQED experiment should be long enough. According toEqs. (34), (37), (38) and (43), the characteristic time

T1 and Tϕ are proportional to the ratioEJEc

once EJ is

fixed. It seems that we need to takeEJEc� 1 to get a suf-

ficiently large coupling strength. However, as expressed

in Eqs. (29a) and (29b), the ratiogzg

andg0g

are inversely

proportional toEJEc

. Therefore the value ofEJEc

should

be restricted by

∣∣∣∣gzg∣∣∣∣� 1 if we want to simulate the Rabi

model.Besides, focusing on Eq. (44), we can find that the

transverse coupling strength is proportional to the square

root ofω0ωc

. This indicates that we need to take a suffi-

ciently small L0 to obtain a relatively largeg

ωc.

V. CONCLUSION

In this article, we present a theoretical proposal witha tunable coupling between an artificial two-level atomand a waveguide transmission line resonator by control-ling the external magnetic fluxes. In our scheme, the cou-pling can be continuously tuned from zero to the ultra-strong regime while keeping fixed atomic level splitting.We also investigate the performance of our qubit underthe influences of the environment, and find that our sys-tem operates well against the main noises. Our analyticalresults are based on quantum perturbation theory withthe parity symmetry, which are verified by the numericalsimulations. In order to apply our qubit design to exper-iments, we discuss how to choose the device parametersby our analytical results. We hope that our work willstimulate the coherent manipulation of the circuit QEDsystem in the fields of quantum simulation and quantumcomputing.

ACKNOWLEDGMENTS

This work is supported by NSF of China (Grant Nos.11475254 and 11775300), NKBRSF of China (Grant No.2014CB921202), the National Key Research and Devel-opment Program of China (2016YFA0300603).

9

Appendix A: Derivation of the energy splitting and the coupling strength

Because Eb � Ec, we can restrict ourselves in the subspace with the bases {|n = 0〉, |n = 1〉}. In this subspace, theterm

cos(φ+ +f ′

2) = cos

f ′

2cosφ+ − sin

f ′

2sinφ+

= cosf ′

2(〈0| cosφ+|0〉|0〉〈0|+ 〈1| cosφ+|1〉|1〉〈1|)− sin

f ′

2(〈0| sinφ+|1〉|0〉〈1|+ 〈1| sinφ+|0〉|1〉〈0|)

= cosf ′

2

(e−λ

2/2|0〉〈0|+ (1− λ2)e−λ2/2|1〉〈1|)− sin f

′

2λe−λ

2/2 (|0〉〈1|+ |1〉〈0|) ,

(A1)

where we use the following expressions:

〈0| cosφ+|0〉 =1

2

(〈0|eiλ(b̂+b̂†)|0〉+ 〈0|e−iλ(b̂+b̂†)|0〉

)=e−λ

2/2

2

(〈0|eiλb̂†eiλb̂|0〉+ 〈0|e−iλb̂†e−iλb̂|0〉

)= e−λ

2/2, (A2)

〈1| cosφ+|1〉 =1

2

(〈1|eiλ(b̂+b̂†)|1〉+ 〈1|e−iλ(b̂+b̂†)|1〉

)=e−λ

2/2

2

(〈1|eiλb̂†eiλb̂|1〉+ 〈1|e−iλb̂†e−iλb̂|1〉

)= (1− λ2)e−λ2/2,

(A3)

〈1| sinφ+|0〉 =1

2i

(〈1|eiλ(b̂+b̂†)|0〉 − 〈1|e−iλ(b̂+b̂†)|0〉

)=e−λ

2/2

2i

(〈1|eiλb̂†eiλb̂|0〉 − 〈1|e−iλb̂†e−iλb̂|0〉

)= λe−λ

2/2, (A4)

〈0| sinφ+|1〉 = (〈1| sinφ+|0〉)∗ = λe−λ2/2. (A5)

In addition, in the subspace with even parity {|0〉, |+〉n}, the term

− 2 sin(φ− +

∆

2

)= i(ei(φ−+

∆2 ) − e−i(φ−+ ∆2 )

)= i

∑n>0

(|+n+1〉〈+n| − |+n〉〈+n+1|) + i√

2(|+1〉〈0| − |0〉〈+1|). (A6)

Thus we obtain the expression for the perturbation V̂ in the relative subspace for our problem. Using the pertur-bation theory, we obtain

|g〉 =|0; 0〉 − 〈0; +1|V̂ |0; 0〉Ec

|0; +1〉 −〈1; +1|V̂ |0; 0〉Eb + Ec

|1; +1〉+〈0; +2|V̂ |0; +1〉〈0; +1|V̂ |0; 0〉

4E2c|0; +2〉

+〈0; +2|V̂ |1; +1〉〈1; +1|V̂ |0; 0〉

4Ec(Eb + Ec)|0; +2〉+

〈1; +2|V̂ |0; +1〉〈0; +1|V̂ |0; 0〉(Eb + 4Ec)Ec

|1; +2〉

+〈1; +2|V̂ |1; +1〉〈1; +1|V̂ |0; 0〉

(Eb + 4Ec)(Eb + Ec)|1; +2〉+

〈1; 0|V̂ |0; +1〉〈0; +1|V̂ |0; 0〉EbEc

|1; 0〉

+〈1; 0|V̂ |1; +1〉〈1; +1|V̂ |0; 0〉

Eb(Eb + Ec)|1; 0〉,

(A7)

and

|e〉 =|0; +1〉 −〈0; +2|V̂ |0; +1〉

3Ec|0; +2〉+

〈0; 0|V̂ |0; +1〉Ec

|0; 0〉 − 〈1; +2|V̂ |0; +1〉Eb + 3Ec

|1; +2〉 −〈1; 0|V̂ |0; +1〉Eb − Ec

|1; 0〉

+〈0; +3|V̂ |0; +2〉〈0; +2|V̂ |0; +1〉

24E2c|0; +3〉+

〈0; +3|V̂ |1; +2〉〈1; +2|V̂ |0; +1〉8Ec(Eb + 3Ec)

|0; +3〉

+〈1; +3|V̂ |0; +2〉〈0; +2|V̂ |0; +1〉

3Ec(Eb + 8Ec)|1; +3〉+

〈1; +3|V̂ |1; +2〉〈1; +2|V̂ |0; +1〉(Eb + 8Ec)(Eb + 3Ec)

|1; +3〉

+〈1; +1|V̂ |0; +2〉〈0; +2|V̂ |0; +1〉

3EbEc|1; +1〉 −

〈1; +1|V̂ |0; 0〉〈0; 0|V̂ |0; +1〉EbEc

|1; +1〉

+〈1; +1|V̂ |1; +2〉〈1; +2|V̂ |0; +1〉

Eb(Eb + 3Ec)|1; +1〉+

〈1; +1|V̂ |1; 0〉〈1; 0|V̂ |0; +1〉Eb(Eb − Ec)

|1; +1〉,

(A8)

10

where

〈0; +1|V̂ |0; 0〉 = i√

2∆EJe−λ2/2 cos

f ′

2, (A9a)

〈1; +1|V̂ |0; 0〉 = 〈0; +1|V̂ |1; 0〉 = −i√

2∆EJλe−λ2/2 sin

f ′

2, (A9b)

〈0; +2|V̂ |0; +1〉 = 〈0; +3|V̂ |0; +2〉 = i∆EJe−λ2/2 cos

f ′

2, (A9c)

〈1; +2|V̂ |0; +1〉 = 〈0; +2|V̂ |1; +1〉 = 〈0; +3|V̂ |1; +2〉 = 〈1; +3|V̂ |0; +2〉 = −i∆EJλe−λ2/2 sin

f ′

2, (A9d)

〈1; +2|V̂ |1; +1〉 = 〈1; +3|V̂ |1; +2〉 = i∆EJ(1− λ2)e−λ2/2 cos

f ′

2, (A9e)

〈1; 0|V̂ |1; +1〉 = −i√

2∆EJ(1− λ2)e−λ2/2 cos

f ′

2. (A9f)

Therefore

Eg =1

2Eb + 4EJ − 2∆2

E2JEc

cos2f ′

2e−λ

2 − 2∆2 E2J

Eb + Ecsin2

f ′

2λ2e−λ

2

≈12Eb + 4EJ − 2∆2E2Je−λ

2

cos2 f′

2Ec

+λ2 sin2

f ′

2Eb

, (A10)

Ee =1

2Eb + Ec + 4EJ +

5

3∆2

E2JEc

cos2f ′

2e−λ

2 − 2∆2 E2J

Eb − Ecsin2

f ′

2λ2e−λ

2 −∆2 E2J

Eb + 3Ecsin2

f ′

2λ2e−λ

2

≈12Eb + Ec + 4EJ + ∆

2E2Je−λ2

5 cos2 f′

23Ec

−3λ2 sin2

f ′

2Eb

. (A11)

Then the energy splitting is

δE =Ee − Eg = Ec +11

3∆2

E2JEc

cos2f ′

2e−λ

2 − 4∆2 EcE2J

E2b − E2csin2

f ′

2λ2e−λ

2 −∆2 E2J

Eb + 3Ecsin2

f ′

2λ2e−λ

2

≈Ec + ∆2E2Je−λ2

(11

3Eccos2

f ′

2− λ

2

Ebsin2

f ′

2

).

(A12)

The approximation is valid due to the assumption Eb � Ec.Now we can calculate the operator φ̂+ up to the second order:

〈g|φ̂+|g〉 ≈λ(〈0; 0|V̂ |0; +1〉〈1; +1|V̂ |0; 0〉

Ec(Eb + Ec)+〈1; 0|V̂ |0; +1〉〈0; +1|V̂ |0; 0〉

EbEc+〈1; 0|V̂ |1; +1〉〈1; +1|V̂ |0; 0〉

Eb(Eb + Ec)+ h.c.

)

=− 2∆2E2Jλ2e−λ2

sin f ′[

1

Ec(Eb + Ec)+

1

EbEc+

1− λ2Eb(Eb + Ec)

]=− 2∆2E2Jλ2

2Eb + (2− λ2)EcEbEc(Eb + Ec)

e−λ2

sin f ′ ≈ −4∆2E2Jλ

2e−λ2

sin f ′

EbEc,

(A13)

11

〈e|φ̂+|e〉 ≈λ(〈1; +2|V̂ |0; +1〉〈0; +1|V̂ |0; +2〉

3Ec(Eb + 3Ec)− 〈1; 0|V̂ |0; +1〉〈0; +1|V̂ |0; 0〉

Ec(Eb − Ec)+〈1; +1|V̂ |0; +2〉〈0; +2|V̂ |0; +1〉

3EbEc

−〈1; +1|V̂ |0; 0〉〈0; 0|V̂ |0; +1〉EbEc

+〈1; +1|V̂ |1; +2〉〈1; +2|V̂ |0; +1〉

Eb(Eb + 3Ec)+〈1; +1|V̂ |1; 0〉〈1; 0|V̂ |0; +1〉

Eb(Eb − Ec)+ h.c.

)

=2∆2E2Jλ2e−λ

2

[sin f ′

(5

6EbEc+

1

Ec(Eb − Ec)− 3(1− λ

2)Ec + Eb6EbEc(Eb + 3Ec)

)− λ(1− cos f

′)Eb(Eb − Ec)

]=2∆2E2Jλ

2e−λ2

[λ(cos f ′ − 1)Eb(Eb − Ec)

+10E2b + (26 + 3λ

2)EbEc − 3(4 + λ2)E2c6EbEc(Eb + 3Ec)(Eb − Ec)

sin f ′]

≈103

∆2E2JEbEc

λ2e−λ2

sin f ′,

(A14)

and

〈e|φ̂+|g〉 ≈ −λ〈1; +1|V̂ |0; 0〉

Eb + Ec− λ(〈1; 0|V̂ |0; +1〉)

∗

Eb − Ec=i2√

2∆EJEbλ2e−λ

2/2

E2b − E2csin

f ′

2≈ i2

√2∆EJλ

2e−λ2/2

Ebsin

f ′

2. (A15)

With Eqs. (A13), (A14) and (A15), we can rewrite Eq. (22b) as

Ĥint = ~(gxσ̂x + gσ̂y + g0σ̂0 + gzσ̂z)(↠+ â), (A16)

where

gx =−√ω0ωc

〈e|φ̂+|g〉+ 〈g|φ̂+|e〉2

= 0, (A17a)

g =− i√ω0ωc〈e|φ̂+|g〉 − 〈g|σ̂+|e〉

2≈√

8ω0ωc∆EJλ2e−λ

2/2

Ebsin

f ′

2, (A17b)

g0 =−√ω0ωc

〈e|φ̂+|e〉+ 〈g|φ̂+|g〉2

≈√ω0ωc∆

2E2Jλ2e−λ

2

3EbEcsin f ′, (A17c)

gz =−√ω0ωc

〈e|φ̂+|e〉 − 〈g|φ̂+|g〉2

≈ −11√ω0ωc∆

2E2Jλ2e−λ

2

3EbEcsin f ′. (A17d)

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Tunable coupling between a superconducting resonator and an artificial atomAbstractI IntroductionII Tunable couplingA System HamiltonianB Theoretical analysis of the coupling strength

III Dissipation of the qubitA Estimates for the relaxation time T1B Estimation of the pure dephasing time T

IV Estimation of device parametersV Conclusion AcknowledgmentsA Derivation of the energy splitting and the coupling strength References