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Electromagnetic-Wave Propagation Through an Array of Superconducting Qubits: Manifestations of Nonequilibrium Steady States of Qubits

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Electromagnetic-Wave Propagation Through an Array of Superconducting Qubits: Manifestations of Nonequilibrium Steady States of Qubits PHYSICAL REVIEW A 100, 023844 (2019) Electromagnetic-wave propagation through an array of superconducting qubits: Manifestations of nonequilibrium steady states of qubits M. V. Fistul1,2 and M. A. Iontsev2 1Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea 2National University of Science and Technology “MISIS,” Russian Quantum Center, 119049 Moscow, Russia (Received 2 March 2019; published 27 August 2019) We report a theoretical study of the propagation of electromagnetic waves (EWs) through an array of superconducting qubits, i.e., coherent two-level systems, embedded in a low-dissipation transmission line. We focus on the near-resonant case as the frequency of EWs ω ωq,whereωq is the qubit frequency. In this limit we derive the effective dynamic nonlinear wave equation allowing one to obtain the frequency-dependent transmission coefficient of EWs, D(ω). In the linear regime and a relatively wide frequency region we obtain a strong resonant suppression of D(ω) in both cases of a single qubit and chains composed of a large number of densely arranged qubits. However, in narrow frequency regions a chain of qubits allows the resonant transmission of EWs with greatly enhanced D(ω). In the nonlinear regime realized for a moderate power of applied microwave radiation, we predict and analyze various transitions between states characterized by high and low values of D(ω). These transitions are manifestations of nonequilibrium steady states of an array of qubits achieved in this regime. DOI: 10.1103/PhysRevA.100.023844 I. INTRODUCTION and an inductive or capacitive coupling of such networks to an external low-dissipation transmission line allows one The propagation of electromagnetic waves (EWs) in to experimentally access the frequency-dependent transmis- metamaterials—artificially prepared media composed of a sion coefficient of EWs, D(ω). The interaction of EWs with network of interacting lumped electromagnetic circuits— quantum networks of qubits results in a large number of has attracted recently enormous attention due to the vari- coherent quantum phenomena on a macroscopic scale, e.g., ety of physical phenomena occurring in such systems, e.g., collective quantum states [16–19], the observation of magnet- electromagnetically induced transparency (reflectivity) [1–3], ically induced transparency [2], and coherent propagation of “left-handed” metamaterials [4,5], and dynamically induced electromagnetic pulses [20,21], and the nonclassical states of metastable states [6,7], just to name a few. These networks photons [22] have been theoretically predicted and studied. have been fabricated from metallic, semiconducting, mag- Therefore, in this quickly developing field a natural question netic, or superconducting materials. arises [19–23]: How does the coherent quantum dynamics The last case of networks based on superconducting el- of a network of superconducting qubits influence the EW ementary circuits presents a special interest because of an propagation? extremely low dissipation, a great tunabilty of the microwave In this paper we present a systematic study of propagation resonances, and a strong nonlinearity [7–9]. In most of studied systems these superconducting electromagnetic circuits con- of EWs through an array of qubits embedded in a low- tained by one or a few Josephson junctions can be precisely dissipation transmission line (see Fig. 1). We will focus on the ω ω ω described as classical nonlinear oscillators, and the interaction resonant case, i.e., q, where q is the qubit frequency, ω of propagating EWs with a network of such superconducting and the transmission coefficient D( ) will be theoretically an- lumped circuits is determined by a set of classical nonlinear alyzed. In this paper we neglect completely the direct coupling dynamic equations [7,10,11]. between qubits and take into account the coupling between However, it has been well known for many years that qubits and transmission line only. Such great simplification small superconducting circuits can be properly biased in the allows one to characterize the dissipation and decoherence of coherent macroscopic quantum regime, and in a simplest case a whole quantum network with a single parameter, i.e., the the dynamics of such circuits is equivalent to the quantum relaxation time of a single qubit, T . With this assumption dynamics of two-level systems, i.e., qubits [12–18]. A surfeit we reduce the low-dissipation dynamics of a qubits network of different types of superconducting qubits has been realized, to the dynamics of independent qubits exposed to the space- e.g., dc-voltage-biased charge qubits [Fig. 1(a)][12], flux and time-dependent electromagnetic field. To obtain D(ω)we qubits weakly [17] [Fig. 1(b)], and strongly [Fig. 1(c)][2]in- derive the effective nonlinear EW equation that is applicable teracting with a low-dissipation transmission line, transmons in both limits, the low and high power of applied microwave [13,18], etc. radiation. As a next step these qubits are organized in different In the linear regime and a relatively wide frequency region arrays or lattices forming quantum electromagnetic networks, near the resonance we obtain a strong suppression of D(ω)in 2469-9926/2019/100(2)/023844(7) 023844-1 ©2019 American Physical Society M. V. FISTUL AND M. A. IONTSEV PHYSICAL REVIEW A 100, 023844 (2019) II. MODEL, LAGRANGIAN, AND DYNAMIC EQUATIONS A. Model Let us consider a regular one-dimensional array of N lumped superconducting quantum circuits embedded in a low- dissipation nondispersive transmission line (see Fig. 1). As the amplitude of propagating EWs is not too low, i.e., in the regime of a large number of photons, the electromagnetic field in the transmission line is characterized by coordinate- and time-dependent classical variables—the charge distribution, Q(x, t ). Different types of lumped superconducting quantum circuits have been realized (the schematics of arrays com- posed of charge [Fig. 1(a)] and flux [Figs. 1(b) and 1(c)] qubits are shown), and the quantum dynamics of such circuits is characterized by quantum variables—the Josephson phases, ϕn. An artificially prepared potential U (ϕn ) allows one to vary the circuits’ resonant frequencies in a wide region. The dy- namics of a whole system in the classical regime is described by a total Lagrangian, which consists of three parts: the Lagrangian of an electromagnetic field LEF, the Lagrangian of an array of lumped superconducting quantum circuits (qubits) Lqb, and the interaction Lagrangian Lint describing the interac- tion between qubits and an electromagnetic field: L = LEW + Lqb + Lint. (1) B. Lagrangian and dynamic equation The electromagnetic field Lagrangian LEF is written as ∂ 2 ∂ 2 L0 Q 2 Q FIG. 1. Schematic of qubit arrays coupled to a low-dissipation LEF = − c , (2) 2 ∂t 0 ∂x transmission line: voltage-biased charge qubits (a), weakly coupled flux qubits (b), and strongly coupled flux qubits (c). Josephson junc- √ where c = 1/ L C is the velocity of EWs propagating in tions are illustrated by crosses; input and output of EWs are shown 0 0 0 the transmission line, and L0 and C0 are the inductance and by arrows. The classical, Q, and quantum ϕn dynamic variables are shown. The properties of transmission line are characterized by two capacitance of the transmission line per length, respectively; is the length of the system. parameters: the capacitance C0 and inductance L0 per length. The D is the transmission coefficient of propagating EWs. The Lagrangian of an array of lumped quantum circuits is written as N E L = J (˙ϕ − φ˙ )2 − U (ϕ ), (3) both cases of a single qubit and chains composed of a large qb ω2 n 0n n 2 p number of densely arranged qubits. However, in a narrow n=1 frequency region for chains of qubits we obtain the resonant where EJ , ωp are the Josephson energy and the plasma transmission of EWs with a greatly enhanced D(ω). As we frequency, respectively; the parameter φ˙0n is proportional to turn to the nonlinear regime realized for a moderate power of the gate voltage, and it allows one to vary the frequency of applied microwave radiation, we predict and analyze various charge qubits [such gate circuits are shown by dashed arrows transitions between states characterized by high and low val- in Fig. 1(a)]. For charge qubits the potential U (ϕn )iswritten ues of D(ω). We argue that these transitions are fingerprints explicitly as U (ϕn ) = EJ (1 − cos ϕn ), whereas for flux qubits of nonequilibrium steady states of an array of qubits. [see Figs. 1(b) and 1(c)] the double-well potential reads as The paper is organized as follows: In Sec. II we present U (ϕn ) =−EJ [2 cos ϕn − κ cos(2ϕn )], where κ is determined our model for a qubits array embedded in a low-dissipation by an external magnetic flux and the critical currents of transmission line, introduce the Lagrangian, and derive the Josephson junctions. effective nonlinear wave equation for the electromagnetic The interaction part of the Lagrangian is derived by making field interacting with an array of qubits. In Sec. III we analyze use of a standard method [22,24,25], i.e., we start with the the coherent quantum dynamics of a single qubit subject to an discrete model of the transmission line coupled to the array applied electromagnetic field in both limits of low and high of qubits (see Fig. 1), write the dynamic equations (the Kirch- power. In Sec. IV we apply the effective nonlinear wave equa- hoff’s current and voltage laws) for voltages Vn and currents tion derived in Sec. II to a study of the frequency-dependent In flowing in the nth cell of the transmission line taking into transmission coefficient, D(ω), for a chain of densely arranged account the Josephson current flowing through the Josephson qubits.
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