Sai Li(李賽) Tao Chen(陳濤) Jia Liu(劉佳) and Zheng-Yuan Xue(薛正遠(yuǎn))

1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,and School of Physics and Telecommunication Engineering,South China Normal University,Guangzhou 510006,China

2Guangdong-Hong Kong Joint Laboratory of Quantum Matter,and Frontier Research Institute for Physics,South China Normal University,Guangzhou 510006,China

Keywords: non-reciprocity,quantum transport,superconducting quantum circuits

1. Introduction

Reciprocity, which means that the measured scattering does not change when the source and the detector are interchanged,[1]is a fundamental phenomenon in both classical and quantum regimes. Meanwhile, non-reciprocal devices, such as isolators and circulators, are also essential in both classical and quantum information processing. Especially, circulators can separate opposite signal flows, spanning from classical to quantum computation and communication systems.[2]Thus, circulators are vital for the design of full-duplex communication systems, which can transmit and receive signals through a same frequency channel, providing the opportunity to enhance channel capacity and reduce power consumption.[3]Therefore, many theoretical and experimental progresses have been made recently to build non-reciprocal devices in different quantum systems. Specifically, the conventional way of realizing non-reciprocity is achieved by adding magnetic field or using magnetic materials directly.[4]However, the external magnetic field would affect the transformation and magnetic materials hardly induce nonreciprocity.

Recently, realization of non-reciprocity has been proposed in many artificial quantum systems, such as in nonlinear systems,[5–7]synthetic magnetism systems,[8–14]non-Hermitian systems,[15–22]time modulated systems,[23–32,34,35]etc. Although these successful methods have been realized in nitrogen-vacancy centers systems,[14]cold atom systems,[22]superconducting circuits,[34,35]and optical systems,[36–38]tunable non-reciprocal process is still lacking for quantum manipulation. This is because previous schemes rely highly on special material properties, e.g., nonlinear property. In addition,for the experimental implementations of non-reciprocity induced by synthetic magnetism,e.g.,on superconducting circuits, they usually need cyclical interaction among at least three levels of a superconducting qubit device[35]or three coupled superconducting qubit devices in a two-dimensional configuration.[33,34]These are experimentally challenging for large scale lattices,as they require that quantum systems have cyclical transition or at least two-dimensional configuration for three-level non-reciprocal process. Therefore, proposals using tunable non-reciprocal process and its potential applications to achieve non-reciprocity are still highly desired theoretically and experimentally.

Here, we propose a general scheme on a three-level quantum system based on the conventional stimulated-Raman-adiabatic-passage (STIRAP) setup[39,40]to realize non-reciprocal operations by time modulation. The distinct merit of our proposal is that the realization only needs two time-modulation couplings, which removes the experimental difficulty of requiring cyclical interaction, and thus it is directly implementable in various quantum systems, for example, superconducting quantum circuits systems,[2,33,41,42]nuclear magnetic resonance systems,[43]a nitrogen-vacancy center in diamonds,[14]trapped ions,[44]hybrid quantum systems,[45]and so on. Meanwhile, the three-level nonreciprocal process can be implemented in a one-dimensional configuration instead of the two-dimensional system in previous proposals,which greatly releases the experiment difficulties for large lattices.

It is well known that the superconducting quantum circuits system is scalable and controllable,and thus attracts great attention in many researches. Different from the cold atoms and optical lattice systems, superconducting circuits possess good individual controllability and easy scalability. Following that,we illustrate our proposal on a chain of three coupled superconducting transmon devices with appropriate parameters,and achieve a non-reciprocal circulator with high fidelity.As our proposal is based on a one-dimension superconducting lattice, e.g., Refs. [46,47], and with demonstrated techniques there,thus it can be directly verified. Therefore,our proposal provides a new approach based on time modulation for engineering non-reciprocal devices,which can find many interesting applications in quantum information processing,including one-way propagation of quantum information,quantum measurement and readout,and quantum steering.

2. General framework

Now, we start from a general three-level quantum system labeled in the Hilbert space{|A〉,|M〉,|B〉}. As shown in Fig. 1(a), considering|A〉and|B〉simultaneously coupled to|M〉resonantly. Assuming=1 hereafter, the interaction Hamiltonian in the interaction picture can be written as

whereg1,2(t) are the time-modulation coupling strength.Based on the Hamiltonian?(t),even if there is no direct coupling between bare states|A〉and|B〉,these two bare states are both coupled to state|M〉, then, the transition between bare states|A〉and|B〉can be realized via the middle state|M〉,i.e.,STIRAP.[39,40]Especially,when the pulse shapes ofg1(t)andg2(t) are different from each other, the symmetry of the system can be broken naturally. Thus, with appropriate designed time-modulation coupling strengthsg1,2(t),a quantum circulator with one-direction flow can be achieved through a period evolution with timeτ, i.e.,|A〉 →|B〉 →|M〉 →|A〉illustrated in Fig. 1(b). This means, on the one hand, transition|A〉→|B〉is allowed, not vice versa. Meanwhile, the processes|B〉→|M〉and|M〉→|A〉can be simultaneously realized, which means that|M〉can simultaneously receive the quantum information from sender|B〉and send information to receiver|A〉. These quantum processes are very important for quantum information transformation and processing.

Fig.1. Schematic diagram. (a)Initial picture for generating non-reciprocal devices: two subspaces|A〉and|B〉are resonantly coupled to subspace|M〉simultaneously with time modulated coefficients g1(t)and g2(t). (b)Quantum circulator with one direction flow through a period τ. (c)|M〉simultaneously serve as receiver R1 and sender S2 for receiving information from sender S1 and send information to receiver R2 through a period τ.

3. Construction

where in the Hilbert space{|A〉,|M〉,|B〉},

are the eigenstates of the invariant[49]

whereμis an arbitrary constant with unit of frequency to keepI(t) with dimensions of energy,γ(t) andβ(t) are auxiliary parameters, which satisfy the von-Neumann equation?I(t)/?t+i[?(t),I(t)]=0, andθn(τ) is the LR phase withθ0(τ)=0 andθ?(τ)=?θ+(τ), which can be addressed by auxiliary parametersγ(t)andβ(t). To induce non-reciprocal transition evolution process,we set the boundary conditions as

After that, the final evolution operator in the Hilbert space{|A〉,|M〉,|B〉}can be determined as

To understand the result clearly, for the caseθ+(τ) =π, the final evolution operatorU[π]=?|M〉〈M|?|A〉〈B|?|B〉〈A| represents a normal two-direction transition. Especially, for another caseθ+(τ)=3π/2, the evolution processU[3π/2]= i|A〉〈M|+i|M〉〈B|?|B〉〈A|shows non-reciprocal transitions, that means transition|A〉 →|B〉is allowed and transition|B〉→|A〉is forbidden for the same process. Obviously, the evolution operatorU[3π/2] also means a cyclic transportation. To sum up, the evolution operatorU[3π/2]induces a cyclic chiral transportation, which exactly realizes a quantum circulator, from pure time-modulation of the interaction.

4. Illustrative scheme with transmons

Here, we propose a scheme on superconducting quantum circuits. For a transmon,[51]there are three lowest levels, which can be resonantly driven by two microwave fields to induce the Hamiltonian?(t) in Eq. (1). In this case, only nonreciprocal state transfer within a transmon can be obtained,and the implementation is straightforward,i.e.,letting|0〉,|1〉,and|2〉take the role of|A〉,|M〉,and|B〉.

Furthermore, we consider a more interesting case, that is, three coupled transmons implementation, with the lowest two levels|0〉and|1〉in superconducting quantum circuits. As shown in Fig. 2(a), we label three transmons withA,M, andBwith frequenciesωA,M,Band anharmonicitiesαA,M,B. Here,we introduce qubit frequency drivesf(?(t)),[52]which can be determined experimentally by the longitudinal field?(t)=f?1(˙F(t)),whereF(t)=η(t)sin(νt)is intentionally chosen withνbeing the frequency of the longitudinal field?(t), and two qubit-frequency drivesf(?j(t)) (j=A,B)are added in transmonsAandBrespectively to induce timemodulation resonant interaction with transmonM. Then, the coupled system can be described by?T(t)=?f(t)+?int(t),where?f(t) and?int(t) are free and interaction Hamiltonians,respectively. For the free part,

where?(t)=f?1(˙F(t))withF(t)=η(t)sin(νt). For the interaction term,

in the single-excitation subspace{|100〉,|010〉,|001〉}, where|amb〉≡|a〉A(chǔ) ?|m〉M ?|b〉Blabels the product states of three transmons, after neglecting the high order oscillating terms,the Hamiltonian can be written as

Fig. 2. Illustration of our scheme with three transmon devices. (a) Two qubit-frequency driven transmons A and B with the respective longitudinal field ?A,B(t) resonantly coupled to the transmon M. (b) Effective resonant coupling architecture in the single-excitation subspace{|100〉,|010〉,|001〉}.

With the LR invariant method,[49]according to the von-Neumann equation?I(t)/?t+i[?eff(t),I(t)]=0,the form ofg＇j(t)can be given as

Considering the boundary conditions Eq.(6),the commutation relations[H(0),I(0)]=[H(τ),I(τ)]=0,and the experimental apparatus restriction, the valuesg＇j(t) can be set as zeros at timet=0 andτ, thus, a set of auxiliary parametersγ(t) andβ(t)can be selected in a proper form[50]as

whereλis a tunable time-independent auxiliary parameter,which directly determines the LR phaseθ+(τ) concerned in our proposal shown in Fig. 3(a). Furthermore, the effective coupling strengthg＇j(t) can be carried out according to Eq. (13). Then, we realize the final evolution operatorU[θ+(τ)].

In the following, we choose appropriate experimental parameters[53]and show how to realize our protocol to achieve non-reciprocal operations on superconducting quantum circuits. As the anharmonicity of transmon qubits is relatively small, thus the second excited state will contribute harmfully to the quantum process. To numerically quantify this effect,we set the anharmonicity of three transmons asαA=2π×220 MHz,αM=2π×210 MHz, andαB=2π×230 MHz.Meanwhile, we set the frequency of the longitudinal fieldνjequal to the corresponding frequency difference?jasνA=?A=2π×345 MHz andνB=?B=2π×345 MHz respectively to induce time-modulation resonant interaction in the single-excitation subspace. Furthermore, we set the decoherence rates of the transmons asΓA= 2π×3 kHz,ΓM=2π×4 kHz, andΓB= 2π×5 kHz, the coupling strengths for transmonsA,BtoMasgA=gB=2π×10 MHz, and the quantum evolution periodτ=145 ns. Then, to realize the quantum circulatorU[3π/2], we modify auxiliary parameterλ=0.4974 to makeθ+(τ)=3π/2 and naturally determine the time-modulation coupling strengthg＇j(t), whose pulse shapes are plotted in Fig.3(b),which is smooth and easily experimentally realized.

We numerically simulate the performance of the quantum circulatorU[3π/2]by using Lindblad master equation as

Fig.3. Numerical performance. (a)The LR phase θ+(τ)with respect to auxiliary parameter λ,where black square represents θ+(τ)=3π/2. (b)The pulse shapes of asymmetrical time-modulation coupling strength g＇A,B(t) with θ+(τ)=3π/2 in a period τ. (c) The fidelity of the quantum circulator U[3π/2]for simultaneously sending and receiving the quantum information in a period τ. (d)–(f)The state populations and fidelity of the quantum circulator U[3π/2]in a period τ with the initial state being|100〉,|001〉,and|010〉,respectively.

5. Conclusion

In summary,we propose a general scheme based on time modulation to realize non-reciprocal operations. Our proposal can be easily realized in many quantum systems. We illustrate our proposal on superconducting quantum circuits with two driving transmons simultaneously coupled to the middle transmon. Considering the scalability and controllability of the superconducting quantum circuits, our scheme provides promising candidates for non-reciprocal quantum information processing and devices in the near future.