Landau-Zener interference at bichromatic driving

We investigate experimentally and theoretically the interference at avoided crossings which are repeatedly traversed as a consequence of an applied ac field. Our model system is a charge qubit in a serial double quantum dot connected to two leads. Our focus lies on effects caused by simultaneous driving with two different frequencies. We work out how the commensurability of the driving frequencies affects the symmetry of the interference patterns both in real space and in Fourier space. For commensurable frequencies, the symmetry depends sensitively on the relative phase between the two modes, whereas for incommensurable frequencies the symmetry of monochromatic driving is always recovered.

Figure 1

(a) Scanning electron micrograph of the wafer surface. The GaAs surface is dark gray, gold gates are shown in yellow and cobalt gates in blue. Red filled circles sketch the approximate quantum dot positions in the two-dimensional electron system 85 nm beneath the surface. The voltage on the lower right gate is radio frequency modulated with the function $f\left(t\right)$. (b) Avoided crossing of the eigenstates (solid lines) composed of the singlets (dashed lines) with charge configuration $(1,1)$ and $(2,0)$ (visualized in double well sketches), which mix via the interdot tunnel coupling $\mathrm{\Delta }$. (c) Measured LZSM pattern $I({\epsilon }_0,A)$ for monochromatic driving with frequency $6\mathrm{GHz}$. (d) Corresponding Fourier transformed LZSM pattern (plotted with a logarithmic color scale).

Figure 2

(a) Shape of the ac driving, $f\left(t\right)=sin\left(\mathrm{\Omega }t\right)+1.25sin(2\mathrm{\Omega }t+\varphi )$, for various phases $\varphi$. For $\varphi =\pi /2$ and $3\pi /2,f\left(t\right)$ possesses reflection symmetry at specific times, for $\varphi =0$ and $\pi ,f\left(t\right)$ is antisymmetric in time, while for other phases in the range $0\le \varphi <2\pi$ symmetry is lost. (b) Phase dependent LZSM interference for $f\left(t\right)$ as in (a), constant amplitude $A=57\mu \mathrm{eV},\eta =1.25$, and $\mathrm{\Omega }/2\pi =4\mathrm{GHz} \left({\mathrm{\Omega }}^{\prime }=2\mathrm{\Omega }\right)$. (c) Interference patterns $I\left({\epsilon }_0\right)$ at the phases of enhanced symmetry along horizontal lines in (b) [color coded]. The gray line in the lower panel of (c) is the blueish curve after reflection at the ${\epsilon }_0=0$ axis. (d) and (e) Theory data corresponding to the measurements in (b) and (c) for interdot and dot-lead tunnel couplings $\mathrm{\Delta }=8\mu \mathrm{eV}$ and ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=0.002\mu \mathrm{eV}$, decoherence rate $\gamma =0.001\mu \mathrm{eV}$, and inhomogeneous broadening ${\gamma }^{*}=5\mu \mathrm{eV}$. Colorscales in panels (b) and (d) as in Fig. 1.

Figure 3

Measured and computed LZSM pattern in real space (a) and (c) and in Fourier space (b) and (d) for the phase $\varphi =\pi /2$ and the amplitude ratio $\eta =2$. All other parameters are as in Fig. 2, the color scales are as in Fig. 1. The enhanced resolution of the theory data in Fourier space is achieved by considering data beyond the range shown in (c).

Figure 4

Theoretical LZSM patterns in Fourier representation for bichromatic driving with commensurable frequencies for various relative phases. The driving frequencies are $\mathrm{\Omega }/2\pi =4\text{GHz}$ and ${\mathrm{\Omega }}^{\prime }/2\pi =8\text{GHz}$, while the amplitude ratio is $\eta =1.25$. The gray lines are the solutions of Eqs. (22) and (23). The phases $\varphi =\pi /2$ and $\varphi =3\pi /2$ correspond to symmetric driving for which part of the structure is given by the analytic expression (25).

Figure 5

Experiment [(a) and (b)] and theory [(c) and (d)] for the driving frequencies $\mathrm{\Omega }=3.708\text{GHz},{\mathrm{\Omega }}^{\prime }=6\text{GHz}$, and the amplitude ratio $\eta =0.9$. The frequency ratio approximates the golden mean with a precision of ${10}^{-4}$. All other parameters are as in Fig. 2, the color scales are as in Figs. 1 and 1. Dashed lines visualize the analytical prediction in Eq. (28).

Figure 6

Measured LZSM pattern in real space and in Fourier space for commensurable frequencies $\mathrm{\Omega }/2\pi =4$ and $8\text{GHz}$ and the phases and amplitude ratios displayed in the graphics. The gray lines indicate the arcs predicted by Eqs. (22) and (23).

Figure 7

Measured LZSM pattern in real space and in Fourier space for the incommensurate frequencies with ratios $g$ (the golden mean) and $2g$ and amplitude ratios displayed in the graphics. The gray lines indicate the arcs predicted by Eq. (28).