Implementation of an Impedance-MatchedΛSystem by Dressed-State Engineering

Abstract

In one-dimensional optical setups, light-matter interaction is drastically enhanced by the interference between the incident and scattered fields. Particularly, in the impedance-matched Λ-type three-level systems, a single photon deterministically induces the Raman transition and switches the electronic state of the system. Here we show that such a Λ system can be implemented by using dressed states of a driven superconducting qubit and a resonator. The input microwave photons are perfectly absorbed and are down-converted into other frequency modes in the same waveguide. The proposed setup is applicable to single-photon detection in the microwave domain.PACS numbers: 03.67. Lx, 85.25.Cp, 42.50.Pq In one-dimensional optical setups, radiation from a quantum emitter is guided completely to specified onedimensional propagating modes. We can realize such setups in a variety of physical systems, such as optical cavity quantum electrodynamics (QED) systems using atoms or quantum dots [1][2][3] and circuit QED systems using superconducting qubits [4][5][6]. When we apply a field to excite the emitter through the one-dimensional mode in these setups, the incident field inevitably interferes with the field scattered by the emitter due to the low dimensionality [7]. As a result, we can realize unique optical phenomena that are not achievable in three-dimensional free space. A classical example is the complete transmission of a resonant field through a two-sided cavity, in which reflection from the cavity is forbidden due to the destructive interference between the incident field and the cavity emission in the reflection direction. Such one-dimensional optical setups in which reflection from the emitter is forbidden are called impedance-matched, in analogy with properly terminated electric circuits [8,9]. Recently, perfect reflection of the incident field by a single emitter has been confirmed in both optical cavity QED and circuit QED systems [2,4]. Here, transmission is forbidden by the destructive interference occurring in the transmission direction.In this study, we investigate a three-level Λ system interacting with a semi-infinite one-dimensional field in a reflection geometry (Fig. 1). We denote the three levels of the Λ system by |g , |m and |e from the lowest. We assume that |m decays to |g with a finite lifetime and therefore that the system is in |g when stationary. When a single photon resonant to the |g → |e transition is input, there are three possible processes: (a) simple reflection without exciting the system, (b) elastic scattering, inducing the |g → |e → |g transitions, and (c) inelastic scattering, inducing the |g → |e → |m → |g transitions. Destructive interference occurs here between processes (a) and (b). In particular, they cancel each other completely when the two decay rates from the top level |e are identical (Γ eg = Γ em ) and the coherence length of the input photon is sufficiently long. As a result, the input photon is down-converted deterministically, inducing the Raman transition in the sy...

Microwave Down-Conversion with an Impedance-MatchedΛSystem in Driven Circuit QED

“…This can be observed in microwave spectroscopy as perfect absorption of the incident field and frequency down-conversion of the reflected field [10]. A schematic of the measurement setup at the 10 mK stage of a dilution refrigerator is shown in Fig.…”

“…Moreover, an artificial atom coupled directly with a microwave transmission line demonstrates near-perfect reflection of the incident field [8,9]. Recently, we have theoretically shown that an impedance-matched Λ system can be implemented by using the dressed states of a driven circuit-QED system [10]. Although Λ and Δ systems have been proposed theoretically [11][12][13][14] and implemented experimentally with a flux qubit by using the lowest three levels of its asymmetric double-well potential [15][16][17], realizing an impedance-matched Λ system has remained elusive.…”

“…Note that ω r and ω ge are not their bare frequencies but the renormalized ones including the dispersive shifts [25]. By choosing ω d within the range of ω ge − 2χ < ω d < ω ge , the system can be set into the "nesting regime," where the level structure becomes nested, i.e., ω jg;0i < ω je;0i < ω je;1i < ω jg;1i [10]. The qubit drive mixes the lower (higher) two levels in Fig.…”

“…The two radiative decay rates from each state ideally become identical at P d ¼ P d0 ≡ −80.6 dBm. In theory, the impedance matching is expected to occur at ðP d ; ω p Þ ¼ ðP d0 ;ω 41 Þ and ðP d0 ;ω 31 Þ, whereκ 41 ¼κ 42 andκ 31 ¼κ 32 [10]. In the actual system, however, inadvertent population of the jMi state due to the continuous probe field and the intrinsic loss of the resonator (κ 2 ) weaken the radiation from the Λ system, resulting in an imperfect destructive interference with the reflected wave.…”

“…For comparison, we theoretically calculate r and the radiative decay ratesκ ij for the jĩi → jji transition [Figs. 3(c) and 3(d)] [10]. Here, we use a decay rate of a resonator photon κ ¼ κ 1 þ κ 2 ¼ 2π × 16.4 MHz, where κ 1 ¼ 0.95κ and κ 2 ¼ 0.05κ are radiative and nonradiative components, respectively, and the qubit energy decay rate T −1 1 ¼ 2π × 0.227 MHz, all of which are determined from independent measurements [25].…”