Geometric Phase and Nonadiabatic Effects in an Electronic Harmonic Oscillator

Steering a quantum harmonic oscillator state along cyclic trajectories leads to a path-dependent geometric phase. Here we describe its experimental observation in an electronic harmonic oscillator. We use a superconducting qubit as a nonlinear probe of the phase, which is otherwise unobservable due to the linearity of the oscillator. We show that the geometric phase is, for a variety of cyclic paths, proportional to the area enclosed in the quadrature plane. At the transition to the nonadiabatic regime, we study corrections to the phase and dephasing of the qubit caused by qubit-resonator entanglement. In particular, we identify parameters for which this dephasing mechanism is negligible even in the nonadiabatic regime. The demonstrated controllability makes our system a versatile tool to study geometric phases in open quantum systems and to investigate their potential for quantum information processing.

Figure 1

(a) Adiabatically driven mechanical oscillator formed by a mass on a spring. Its displacement is proportional to the quickly oscillating force $F(t)$ with a slowly varying amplitude and phase. (b) The Lorentzian response function of the field amplitude $|\alpha |$ inside the resonator centered at frequencies ${\omega }_r$ and ${\omega }_r+2\chi$ for the qubit in $|g⟩$ and $|e⟩$, respectively. (c) Area of the coherent state path in the complex quadrature plane depending on the qubit state. The measured geometric phase difference is proportional to the area $\Delta A$ between the paths. Dashed circles with an area $A_{\mathrm{vac}}=\pi /2$ represent the size (rms) of vacuum fluctuations.

Figure 2

(a) Illustration of the qubit and resonator drive pulses. The maximum amplitude of the resonator drive is denoted by ${\epsilon }_0$ (corresponding to a mean photon number $n\approx 20$). (b) The four different shapes of the path used in the experiment and the corresponding geometric phases relative to the one of the circular path. (c) Path of the resonator drive components in the $IQ$ plane (left) and the corresponding time dependence (right) of the components ${\epsilon }_I$ and ${\epsilon }_Q$ (solid line) and the drive amplitude $({\epsilon }_I^2+{\epsilon }_Q^2{)}^{1/2}$ (dashed line) for the circular drive path (top row) and the straight path (bottom row) yielding the same dynamical phase.

Figure 3

(a) Measured total phase difference $\gamma$ for a resonator pulse of fixed maximum amplitude ${\epsilon }_0/2\pi \approx 370\mathrm{MHz}$, detuning $\delta /2\pi =40\mathrm{MHz}$, and varying duration $T$, tracing a counterclockwise circular (green circles), clockwise circular (orange squares), and straight trajectory (blue diamonds) with the same time dependence of the drive amplitude. Solid lines show theory. (b) Dependence of the phase difference $\delta {\gamma }_g$ on the drive pulse duration for the counterclockwise and clockwise trajectories. Solid lines show the adiabatic limit; dashed lines are fits with corrections proportional to $1/T$ and $1/T^2$. (c) The phase difference $\delta {\gamma }_g$ for a counterclockwise pulse and a clockwise circular pulse with fixed duration $T=300\mathrm{ns}$, detuning $\delta /2\pi =40\mathrm{MHz}$ and varying maximum amplitude, plotted as a function of the area $\Delta A$ enclosed between the coherent state trajectories, shown together with fitted linear functions (solid lines). Dashed lines represent the theoretical adiabatic dependence ${\gamma }_g=-2\Delta A$. (d) Measured geometric phase in the adiabatic limit as a function of the detuning $\delta$, compared with the theoretical dependence (solid line).

Figure 4

(a) Relative length $R$ of the $xy$ Bloch vector projection versus the maximum amplitude ${\epsilon }_0$ of the circular drive pulse for different lengths and orientations of the pulse, normalized to the projected Bloch vector length $R_0$ for ${\epsilon }_0=0$. Solid lines show fitted Gaussian functions. (b) Length $R$ of the $xy$ Bloch vector projection as a function of the resonator pulse length for a fixed pulse amplitude ${\epsilon }_0/2\pi \approx 190\mathrm{MHz}$ at $\delta =40\mathrm{MHz}$, corresponding to $n\approx 5$. The solid line is obtained from the analytically solvable evolution equation for the oscillator. (c) Trajectories of the resonator coherent state driven by a circular pulse for the ground (solid thick line) and excited (dashed thick line) qubit states in the nonadiabatic regime. The different final quadratures ${\alpha }_g$ and ${\alpha }_e$ are indicated by yellow circles at the end of the paths. A part of the trajectory for the opposite orientation of the drive pulse (dashed thin line), as well as the circular paths expected in the adiabatic limit (solid thin lines), are shown for comparison.