Spectroscopy of Quantum Phase Slips: Visualizing Complex Real-Time Instantons

TL;DR

This work addresses phase slips in parametrically driven oscillators as real-time instantons that limit qubit coherence, even at zero temperature. It develops a unified Keldysh path-integral framework to connect classical activation and quantum activation and proposes spectroscopy with a weak auxiliary drive to reveal the instanton dynamics through the logarithmic susceptibility. The authors derive phase-slip rates in both classical and quantum regimes, compute the spectral features of the LS across temperatures and detunings, and analyze the behavior near the bifurcation point where the intrawell frequency vanishes. The findings provide a practical spectroscopic route to visualize complex real-time instantons and offer insights for improved qubit control in Floquet-based bosonic systems.

Abstract

Parametrically driven oscillators can emerge as a basis for the next generation of qubits. Classically, these systems exhibit two stable oscillatory states with opposite phases. Upon quantization, these states turn into a pair of closely spaced Floquet states, which can serve as the logical basis for a qubit. However, interaction with the environment induces phase-slip events which set a limit on qubit coherence. Such phase slips persist even at zero temperature due to a mechanism known as quantum activation \cite{QuantumActivation}. In contrast to conventional tunneling, the quantum activation is described by a {\em real-time} instanton trajectory in the complexified phase space of the system. In this work, we show that the phase-slip rate is exponentially sensitive to weak AC perturbations. The spectrum of the system's response -- captured by the so-called logarithmic susceptibility (LS) -- enables a direct observation of characteristic features of real-time instantons. Studying this spectrum suggests new means of efficient qubit control.

Figures (8)

• Figure 1: Rotating wave quasi-potential $E=H_0(\bar{\phi},\phi)$ of the PDO. The two minima correspond to the two out-of-phase period-two oscillations.
• Figure 2: Examples of the quantum LS shown for two different values of detuning, $\Delta\propto \omega_0-\omega_p$, scaled by the modulation amplitude $\propto \lambda$; fainter lines correspond to smaller temperatures; $n_\mathrm{B}$ is the thermal occupation number of the oscillator in the absence of modulation. The top plot shows $\Delta/2\lambda=0.6$ for $T=0$, $n_\mathrm{B}=0.5$, and the classical limit. The bottom plot shows $\Delta/2\lambda=-0.6$ for $n_\mathrm{B}=0.2$, $n_\mathrm{B}=0.5$, and the classical limit. The $T=0$ result from the top plot is obtained using the $T=0$ instanton from Eq. (\ref{['pinstT=0']}), while all finite temperature results are obtained by solving Eq. (\ref{['K0quantum']}) numerically for $K(I,p)=0$. The classical limits show Eq. (\ref{['LSharmonicClassical']}) rescaled by $2T/\omega_\mathrm{p}$. In the bottom plot, the vanishing of $\chi_n$ for $n<0$ at frequencies larger than $n\omega_\mathrm{min}$ can be observed for all temperatures; this is discussed in detail below, see Fig. \ref{['fig:Classical_Log_Susc']}.
• Figure 3: Phase portrait of the effective Hamiltonian $K_0$ from Eq. (\ref{['K0classical']}) plotted with $\Delta/2\lambda=0.3$ Bold blue lines show the separatrix trajectories, which are the $K_0=0$ lines. They connect the fixed points, which are shown by dots. The instanton action Eq. (\ref{['SwitchingRate0Classical']}) is indicated by the shaded region.
• Figure 4: Phase portraits of the quantum Hamiltonian $K_0(I,p)$, Eq. (\ref{['K0quantum']}) for $\Delta/2\lambda=0.3$ (a) and $\Delta/2\lambda=-0.6$ (b). Bold blue lines show the separatrix trajectories in the $T\to 0$ limit. They are the $K_0(I,p)=0$ lines, which connect the fixed points, shown by bold dots. The instanton action Eq. (\ref{['SwitchingRate0Quantum']}) is indicated by the shaded region. The red dashed lines show the path $p_<(I)$ and the yellow dashed lines show the $T=0$ instanton. Note that this instanton trajectory diverges in panel (b) at $I=I_\mathrm{D}(\Delta/2\lambda)$. This is a generic feature of the $T=0$ instanton when $\Delta/2\lambda\leq0$. The singularity of $p_\mathrm{inst}(I)$ is integrable, resulting in a finite switching rate. This singularity is no longer present for $T\to0$ because $I_\mathrm{D}>I_\mathrm{F}$ for all values of $\Delta/2\lambda$.
• Figure 5: Classical phase slip rate logarithmic susceptibility as a function of reduced spectroscopic frequency, $\nu =\omega_\mathrm{d}-\omega_\mathrm{p}$ according to Eq.(\ref{['LSharmonicClassical']}) for $\Delta/2\lambda = -0.4$. For $\Delta<0$ and $n<0$, at $n\omega(I_\mathrm{D}) = \nu$, see Fig. \ref{['fig:quantumInstanton']}b, the corresponding $\chi_n$ becomes zero, here highlighted for $\chi_{-1}$ in the inset.