RAPID COMMUNICATIONS PHYSICAL REVIEW A 93, 060301(R) (2016) Fock-state stabilization and emission in superconducting circuits using dc-biased Josephson junctions J.-R. Souquet and A. A. Clerk Department of Physics, McGill University, 3600 rue University, Montreal, Quebec, Canada H3A 2T8 (Received 1 February 2016; published 21 June 2016) We present and analyze a reservoir-engineering approach to stabilizing Fock states in a superconducting microwave cavity which does not require any microwave-frequency control drives. Instead, our system makes use of a Josephson junction biased by a dc voltage which is coupled to both a principal storage cavity and a second auxiliary cavity. Our analysis shows that Fock states can be stabilized with an extremely high fidelity. We also show how the same system can be used to prepare on-demand propagating Fock states, again without the use of microwave pulses. DOI: 10.1103/PhysRevA.93.060301 Introduction. The ability to prepare, stabilize, and, ulti- limitations of the single-cavity system and prepare single- mately, transmit nontrivial quantum states is crucial to a variety photon Fock states with an extremely high fidelity. In addition, of tasks in quantum information processing [1]. In the presence it can easily be adapted to prepare higher Fock states. The sec- of dissipation, state stabilization can be accomplished either ond cavity acts like an engineered dissipative reservoir which via measurement-plus-feedback schemes (see, e.g., [2–9]) or freezes the system dynamics in the desired Fock state. Similar autonomously, using quantum reservoir engineering (QRE) two-cavity-plus-junction systems have recently been realized techniques [10]. There has been considerable progress in im- experimentally [22] and have also been discussed theoretically plementing QRE ideas in superconducting circuits, including [28–31], largely in the context of generating cavity-cavity experiments which have stabilized qubit states [11–13]as correlations. Armour et al. [29] found numerically that such well as photonic Fock states [14] inside microwave frequency a system could generate cavity states with weakly negative cavities. These schemes are typically complex, requiring the Wigner functions; they did not, however, discuss Fock-state use of several high-frequency microwave control tones. stabilization or the particular mechanism we elucidate and In this work, we focus on the stabilization and emission of optimize. Fock states in superconducting quantum circuits. Such states In addition to efficient and high-fidelity Fock-state sta- have applications in quantum communication, cryptography, bilization, we show that our setup can also play another and information processing (see, e.g., [15]) and are a crucial crucial role: it can act as an efficient means for producing ingredient in linear optical quantum computation [16]. We take propagating Fock states on demand. In circuit QED setups, an alternate approach here to Fock-state stabilization, which propagating single-photon states are usually generated by requires no high-frequency microwave control tones and only driving microwave cavities coupled to a qubit with high- a dc voltage and a single Josephson junction. This leads to frequency control pulses (see, e.g., [32]). Among the many considerable simplicity over existing protocols and negates challenges in the standard approach is the requirement that the any issues involving the microwave control tones corrupting generated photon should be far detuned from the frequency the generated Fock state (a problem that is typically solved of the control pulse [17,18,33]. Our system is capable of by using elaborate fast frequency-tunable qubits [17,18]). Our on-demand Fock-state generation without any microwave approach also represents a fundamentally different approach to control pulses: one simply needs to pulse a dc control voltage. reservoir engineering which utilizes a highly nonlinear driving We note that recent work by Leppakangas¨ et al. studied an of a cavity, rather than nonlinear cavity-cavity interactions. alternative approach to generating propagating single photons The starting point for our scheme is a setup discussed in a using a biased Josephson junction [34,35]. Contrary to our number of recent theoretical studies [19–21] and experiments work, this setup is based on using Coulomb blockade physics [22–25], where a microwave cavity is coupled to a dc-biased and does not lead to the stabilization of an intracavity Fock Josephson junction. The driven junction does not act like state; it also does not allow for the production of higher Fock a qubit, but rather as a highly nonlinear driving element. states. That such a setup can produce nonclassical photonic states Biased junction as a nonlinear drive. To set the stage for our was first pointed out in Ref. [20], which showed that states two-cavity system, we start by quickly reviewing the physics could be produced having suppressed number fluctuations and of the one-cavity version, as studied in [19,20]. The system a vanishing g(2) intensity correlation function. While these consists of a Josephson junction in series with a LC resonator states violate a classical Cauchy-Schwarz inequality, they (frequency ωc, impedance Z) and a dc voltage source Vdc.We are mixtures of the vacuum and the one-photon Fock state focus exclusively on temperature and voltages small enough and have a limited fidelity with a pure Fock state [26]. As that superconducting quasiparticles are never excited. The † a result, they do not exhibit any negativity in their Wigner Hamiltonian has the form Hˆ = ωc(aˆ aˆ + 1/2) − EJ cos(φˆ), functions [27]. where aˆ is the cavity photon annihilation operator, EJ is the Here, we show how optimally coupling the junction to a Josephson energy, and φˆ is the phase difference across the second cavity (as depicted in Fig. 1) lets one transcend the junction; unless otherwise noted, we set = 1 throughout. 2469-9926/2016/93(6)/060301(6) 060301-1 ©2016 American Physical Society RAPID COMMUNICATIONS J.-R. SOUQUET AND A. A. CLERK PHYSICAL REVIEW A 93, 060301(R) (2016) FIG. 1. Schematic of the proposed setup. A dc-biased Josephson junction of energy EJ is in series with two cavities. Each cavity has 2 an impedance close to RK = h/e and is damped via a coupling to a transmission line. The voltage drop across the junction dφ/dtˆ is given by the ˆ difference of Vdc and the cavity voltage√Vcav. This leads † FIG. 2. Matrix elements w + [λ] for junction-driven cavity to φˆ = 2[eVdct + λ(aˆ + aˆ )], where λ = πZ/RK sets the n l,n magnitude of the cavity’s zero-point flux fluctuations and Fock-state transitions as a function of the zero-point voltage fluc- 2 tuation amplitude λ for different values of n and l. These functions RK = h/e is the resistance quantum. The cavity is also are highly nonlinear with respect to λ and can cancel for specific coupled to a transmission line, which is treated (as standard) ˜ as a Markovian reservoir and which gives rise to an energy values. Roots of wn+l,n[λ], denoted λn+l,n, are indicated. damping rate κ. Working in an interaction picture at the cavity frequency, the Hamiltonian of the cavity plus biased junction We note that in comparison to the Bessel function expansion takes the form [19,20,30,36–38] used in Ref. [20], the decomposition in Eq. (3) makes it easier to identify values of λ yielding zeros and also highlights the EJ − † Hˆ =− (e2ieVdct Dˆ [α(t)] + e 2ieVdct Dˆ [α(t)]), (1) direct connection to Franck-Condon physics. 2 The route to preparing single Fock states now seems clear: where Dˆ [α(t)] is the cavity displacement operator [corre- by tuning the value of both Vdc and λ (via the cavity impedance sponding to a time-dependent displacement α(t)]. It is defined Z), one can arrange for the effective driving of the cavity by in terms of the photon annihilation operator aˆ as the junction to shut off when a given Fock state is reached = † ∗ [14]. For concreteness, suppose we chose Vdc k/2e (k an α(t)aˆ −α (t)aˆ iω t Dˆ [α(t)] = e ,α(t) = 2λe c . (2) integer), so that to leading order, Cooper pairs can only tunnel Note that we consider the case where intrinsic phase fluctua- by emitting or absorbing k cavity photons. If we take ωc tions of the junction (due to charging effects or noise from a EJ ,κ, we can restrict attention to these processes and make a shunt resistor) play no role. For more details on the derivation rotating-wave approximation (RWA) to our full Hamiltonian. of this Hamiltonian, see, e.g., Refs. [19,20]. Within the RWA we have From the point of view of the cavity, Hˆ describes a highly ∞ k+1 EJ Hˆ = (−1) w + [λ]|n + kn|+H.c. (6) nonlinear (but coherent) cavity drive [20]: each term tunnels RWA 2 n k,n a Cooper pair and also displaces the cavity state by ±α(t). To n=0 see clearly how these displacements can result in Fock-state Consider the simplest case where k = 1, and each resonantly generation, we follow a different route from [20] and express tunneling Cooper pair√ emits or absorbs a single cavity photon. ˆ D[α(t)] directly in the Fock basis (see, e.g., [39]): If we then set λ = 1/ 2 ≡ λ˜ 2,1 (see Fig. 2, left panel), there ∞ ∞ is no matrix element in Eq. (6) for a transition from one to two −ilωct Dˆ [α(t)] = wn,n+l[λ] |nn + l|e cavity photons. As first discussed in Ref. [20], the junction- n=0 l=0 induced drive now can add at most one photon to the cavity, ∞ implying that the system effectively acts like a driven two-level l ilωct system.