Quantum bath engineering of a high impedance microwave mode through quasiparticle tunneling

Abstract

In microwave quantum optics, dissipation usually corresponds to quantum jumps, where photons are lost one by one. Here we demonstrate a new approach to dissipation engineering. By coupling a high impedance microwave resonator to a tunnel junction, we use the photoassisted tunneling of quasiparticles as a tunable dissipative process. We are able to adjust the minimum number of lost photons per tunneling event to be one, two or more, through a dc voltage. Consequently, different Fock states of the resonator experience different loss processes. Causality then implies that each state experiences a different energy (Lamb) shift, as confirmed experimentally. This photoassisted tunneling process is analogous to a photoelectric effect, which requires a quantum description of light to be quantitatively understood. This work opens up new possibilities for quantum state manipulation in superconducting circuits, which do not rely on the Josephson effect.

Fig. 1. Experiment principles.

a Schematic of the experimental circuit. A microwave mode at frequency ω ≈ 2π × 6 GHz, here represented by an LC resonator, is coupled to a superconducting tunnel junction with tunnel resistance R_T. The characteristic impedance Z_c of the mode is 4.5 kΩ, much larger than in a conventional superconducting resonator. A bias tee and a circulator are used to dc bias the sample while measuring the reflected microwave signal. b When the bias voltage is such that 2Δ − eV > lℏω, the tunneling of quasiparticles through the junction is allowed only if at least l photons (here two) are absorbed from the mode to provide the missing energy. c Microscope image of the sample realizing the circuit shown in a. The resonator consists of two grAl quarter wave resonators (red) connected by a wider Al wire (white). The tunnel junction (Al/AlO_x/Al) connects the resonator right end to the ground. The junction area is 150 × 150 nm², leading to a tunnel resistance R_T = 150 kΩ far above the gap. A microstrip Al line connects the resonator’s left end to the measurement circuit. All experiments are performed in a dilution fridge with a base temperature of 10 mK.

Fig. 2. Tunnel junction as a tunable quantum absorber.

a Quasiparticle tunneling induces an extra loss γ_n (Eq. (2)), which is different for each Fock state of the mode coupled to the junction. Dashed vertical lines show the onset of the l photon absorption process, given by (2Δ − lℏω)/e. The photon number n corresponding to the lowest Fock state with increased loss can be chosen with the bias voltage. The parameters correspond to the ones expected in the experiment. b Experimental spectroscopy of the 6 GHz mode as a function of the bias voltage V. The image shows the measured reflection coefficient ∣S₁₁∣² for an incident power of −115 dBm on the resonator. The onset of the different absorption processes is clearly visible up to l = 4.

Fig. 3. Quantum Zeno dynamics.

a Evolution of the squared mean amplitude ∣〈â〉∣² in the mode as a function of the pump amplitude η for two different bias voltages. In the absence of Zeno effect, ∣〈â〉∣² is quadratic with the pump amplitude (green data). When the voltage lies in the range where the quantum Zeno dynamics limits the dynamic to one of a two-level system, we observe a clear saturation (blue data). The solid red line shows the prediction of a master equation taking into account the different absorption rates for different Fock states (see SI). The dashed line shows the expectation for an ideal two-level system. The calibration of the measured intensity is detailed in the SI. b Evolution of the power broadening Γ as a function of the pump amplitude. The resonance spectra shown in the inset are fitted using the usual formula for a two-level system, which predicts a fwhm $\sqrt{{\kappa }^2+2{\mathrm{\Gamma }}^2}$ (see SI), where κ is the total loss rate. Power broadening is important in the case of Zeno dynamics (blue data) and negligible otherwise, except at very high power (green data). The solid and dashed lines show the result of the master equation simulation and the expectation for an ideal two-level system.

Fig. 4. One and two-photon spectroscopy as a function of voltage.

a Reflected signal measured with an injected microwave power of −140 dBm (η ≈ 2π × 2 MHz) as a function of frequency and bias voltage. The power is sufficiently low to mostly probe the 0 → 1 transition. The vertical lines are the same as in Fig. 1. b Reflected signal in the presence of a second microwave tone tuned at the frequency measured in a. This two-photon spectroscopy probes the 1 → 2 transition, which shifts differently than the 0 → 1 transition, showing the non-linearity induced by the Lamb shift. In both cases, the dashed blue lines correspond to the prediction of an ab initio quantum model. The dashed green line in a corresponds to the classical approximation, keeping only terms of order λ² in the expression of the frequency shift. The hatched area was not measured.