Dispersive Readout of Adiabatic Phases

We propose a protocol for the measurement of adiabatic phases of periodically driven quantum systems coupled to an open cavity that enables dispersive readout. It turns out that the cavity transmission exhibits peaks at frequencies determined by a resonance condition that involves the dynamical and the geometric phase. Since these phases scale differently with the driving frequency, one can determine them by fitting the peak positions to the theoretically expected behavior. For the derivation of the resonance condition and for a numerical study, we develop a Floquet theory for the dispersive readout of ac driven quantum systems. The feasibility is demonstrated for two test cases that generalize Landau-Zener-Stückelberg-Majorana interference to two-parameter driving.

Figure 1

Reflection of the resonantly excited cavity as a function of the detuning and the amplitude for one-parameter driving (a) and two-parameter driving (b) of the qubit with frequency $\mathrm{\Omega }=0.1\mathrm{\Delta }$. Cavity frequency and line width are ${\omega }_0=0.95\mathrm{\Delta }$ and $\kappa ={\omega }_0/1000$, while the coupling constants to the qubit and the bath read $g=0.005\mathrm{\Delta }$ and $\alpha =0.01$. Crosses and dashed lines mark the values used in Figs. 2 and 3, respectively.

Figure 2

(a) Cavity reflection as a function of the driving frequency for one-parameter (dashed) and two-parameter driving (solid) with $A=\epsilon =2\mathrm{\Delta }$. All other parameters are as in Fig. 1. (b,c) Corresponding reciprocal peak positions as a function of their index with an unknown offset $k_0$. The scaling factor $C_{10}$ stems from fitting the data to Eq. (8) with ${\gamma }_{10}$ as a second fit parameter. The vanishing Berry phases for one-parameter driving correspond to integer values marked by horizontal lines. Lines connecting the crosses are a guide to the eye.

Figure 3

(a) Adiabatic phases as a function of the driving amplitude $A$ for the detuning $\epsilon =A$ marked in Fig. 1 by dashed lines. The symbols depict the values reconstructed from the positions of the first 10 resonance peaks with $\mathrm{\Omega }<0.05\mathrm{\Delta }$. The lines correspond to ${\gamma }_{10}^{\mathrm{ad}}$ computed with the adiabatic eigenstates of $H_{\mathrm{qb}}(t)$. Inset: deviation $|{\gamma }_{10}-{\gamma }_{10}^{\mathrm{ad}}|$ as a function of the largest ${\mathrm{\Omega }}_k$ used for the fitting at $A=B=\epsilon =2\mathrm{\Delta }$. (b) Corresponding plot for the alternative Hamiltonian (10) for constant detuning $\epsilon$ as a function of the driving amplitude $A$ with the tunneling $\mathrm{\Delta }=A/1.2$.