Qubit interference at avoided crossings: The role of driving shape and bath coupling

We derive the structure of Landau-Zener-Stückelberg-Majorana (LZSM) interference patterns for a qubit that experiences quantum dissipation and is additionally subjected to time-periodic but otherwise general driving. A spin-boson Hamiltonian serves as the model, which we treat with a Bloch-Redfield master equation in the Floquet basis. It predicts resonance peaks whose form depends significantly on the operator through which the qubit couples to the bath. The Fourier transforms of the LZSM patterns exhibit arc structures which reflect the shape of the driving. These features are captured by an effective time-independent Bloch equation which provides an analytical solution. Moreover, we determine the decay of these arcs as a function of dissipation strength and temperature.

Figure 1

(a)–(c) Adiabatic energies $E_{\pm }\left(t\right)$ (red and blue solid lines) of the Hamiltonian (1) in units of $\hslash \Omega$ for vanishing static detuning, ${\epsilon }_0=0$, and the driving shapes $f_1\left(t\right)$–$f_3\left(t\right)$ in Eq. (2). The dashed black line marks the integral of the driving $F\left(t\right)$ in units of $1/\Omega$. (d)–(f) Resulting nonequilibrium populations in ${\epsilon }_0-A$ space. (g)–(i) Two-dimensional (2D) Fourier transform $W({\tau }_{\epsilon },{\tau }_A)$ of the interference patterns, defined in Eq. (23). The dashed lines in the upper half plane mark the analytic expressions for the arc structure derived in Sec. 4. The patterns are computed with the stationary solution of the Bloch-Redfield master equation for the tunnel matrix element $\Delta =0.5\Omega$, dissipation strength $\alpha ={10}^{-3}$, temperature $1/\beta =0.1\hslash \Omega$, and transverse qubit-bath coupling, i.e., $X={\sigma }_x$ in Eq. (3).

Figure 2

Nonequilibrium population $P_{\mathrm{ex}}$ as a function of the detuning ${\epsilon }_0$ and the driving amplitude $A$ for $f\left(t\right)=cos\left(\Omega t\right)$. The qubit-bath coupling $H_{\mathrm{int}}$ is determined by $X={\sigma }_x$ (a) and $X={\sigma }_z$ (b), while $\Delta =0.5\Omega ,\alpha ={10}^{-3}$, and $1/\beta =0.1\hslash \Omega$.

Figure 3

Overlap of the interference pattern for the mixed coupling (12) with the patterns for the coupling operators ${\sigma }_x$ (squares) and ${\sigma }_z$ (triangles) as a function of the mixing angle. All other parameters are as in Fig. 2.

Figure 4

Nonequilibrium population $P_{\mathrm{ex}}$ shown in Fig. 2 as a function of the detuning ${\epsilon }_0$ for the driving amplitude $A=10\Omega$. (a) Comparison between numerical result with ${\sigma }_x$ coupling obtained with the Bloch-Redfield master equation (solid red) and the analytical solution (20) for the resonances with $n=7,8$ (dotted blue). (b) Comparison between numerical result with ${\sigma }_z$ coupling (dashed black) and the analytical solution (22) for $n=2,3$ (dotted blue). (c) Numerical results for ${\sigma }_x$ and ${\sigma }_z$ coupling plotted together with the analytical result for the off-resonant background predicted by Eq. (15).

Figure 5

Determination of the “nongeneric” arcs for the driving shapes $f_2$ (a) and $f_3$ (b). The color code in the lower panels depicts $G(t,{\tau }_{\epsilon })$, while the horizontal dashed lines mark the generic solutions of Eq. (32) at multiples of $T/2$. The solid lines represent numerical solutions of Eq. (32). Significant contributions to $W({\tau }_{\epsilon },{\tau }_A)$ are determined by the solutions of the transcendental equations (32) and (33), i.e., the cuts of $G(t,{\tau }_{\epsilon })$ along the solid and dashed lines. Projection of these solutions on the ${\tau }_{\epsilon }$ axis (hinted by vertical dotted lines) results in the arc structures plotted in the upper panels and in Figs. 1 and 1.

Figure 6

Analysis of the principal arc for the driving $f\left(t\right)=cos\left(\Omega t\right)$ and system-bath coupling with $\Delta =0.5\Omega$. (a) Fourier transform of the interference pattern, $W({\tau }_{\epsilon },{\tau }_A)$, along the principal arc ${\tau }_A=2F({\tau }_{\epsilon }/2)$ for ${\sigma }_x$ coupling and $\alpha =0.05$. The symbols show numerical results for different temperatures $1/\beta$, while the straight lines are fits to an exponential decay. (b) Decay rate $\lambda$ for temperature $1/\beta =0.5\hslash \Omega$ as a function of the dissipation strength $\alpha$. The error bars are determined by slightly varying the fit range. (c) Decay rate as a function of the temperature for dissipation strengths $\alpha =0.05$.