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Chinese Physics B

Nonadiabatic geometric phase in a doubly driven two-level system

Weixin Liu, Tao Wang, Weidong Li

Published Aug 16, 2022• 2 citations
Publisher

Abstract

We study theoretically the nonadiabatic geometric phase of a doubly driven two-level system with an additional relative phase between the two driving modes introduced in. It is shown that the time evolution of the system strongly depends on this relative phase. The condition for the system returning to its initial state after a single period is given by the means of the Landau–Zener–Stückelberg–Majorana destructive interference. The nonadiabatic geometric phase accompanying a cyclic evolution is shown to be related to the Stokes phase as well as this relative phase. By controlling the relative phase, the geometric phase can characterize two distinct phases in the adiabatic limit.

DOI
10.1088/1674-1056/ac89e3

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We study theoretically the nonadiabatic geometric phase of a doubly driven two-level system with an additional relative phase between the two driving modes introduced in. It is shown that the time evolution of the system strongly depends on this relative phase. The condition for the system returning to its initial state after a single period is given by the means of the Landau–Zener–Stückelberg–Majorana destructive interference. The nonadiabatic geometric phase accompanying a cyclic evolution is shown to be related to the Stokes phase as well as this relative phase. By controlling the relative phase, the geometric phase can characterize two distinct phases in the adiabatic limit.
The generic two-level model with time-dependent matrix elements becomes soluble in the limit that its diagonal and off-diagonal terms vary along a flat ellipse, encircling the diabolical singular point in the parameter space. The time evolution of the state vector is explicitly obtained, and the condition for its evolution to form a closed circuit in the projective Hilbert space of rays is given as a result of destructive interference at level crossing. The Aharonov-Anandan geometrical phase is shown to be related to the Stokes phase for the Landau-Zener model, which is a natural extension of Berry's phase to nonadiabatic evolutions.
We theoretically investigate geometric Stückelberg interferometry in a doubly driven optical lattice clock (OLC). By tuning the relative phase between the two driving fields, we control the relative sign of the effective coupling strengths at the avoided crossings. Within the adiabatic-impulse model, we analyze the time evolution of the two-level system, where nonadiabatic transitions occur only near the crossing points and adiabatic evolution takes place between them. We show that, besides the usual dynamical phase and the Stokes phase, a gauge-invariant noncyclic geometric phase contributes to the final transition probability. This geometric contribution yields a stable π-phase shift in the Stückelberg interference fringes. Moreover, we demonstrate that, under realistic experimental conditions, this geometric Stückelberg interferometer remains insensitive to inhomogeneities in atom-light coupling arising from the finite temperature of the atomic ensemble. Our results provide a general framework for engineering and detecting geometric phases on the OLC platform.
Geometric phases, which accompany the evolution of a quantum system and depend only on its trajectory in state space, are commonly studied in two-level systems. Here, however, we study the adiabatic geometric phase in a weakly anharmonic and strongly driven multilevel system, realized as a superconducting transmon-type circuit. We measure the contribution of the second excited state to the two-level geometric phase and find good agreement with theory treating higher energy levels perturbatively. By changing the evolution time, we confirm the independence of the geometric phase of time and explore the validity of the adiabatic approximation at the transition to the nonadiabatic regime.
If the time evolution of a quantum state leads back to the initial state, a geometric phase is accumulated that is known as the Berry phase for adiabatic evolution or as the Aharonov-Anandan (AA) phase for nonadiabatic evolution. We evaluate these geometric phases using Floquet theory for systems in time-dependent external fields with a focus on paths leading through a degeneracy of the eigenenergies. Contrary to expectations, the low-frequency limits of the two phases do not always coincide. This happens as the degeneracy leads to a slow convergence of the quantum states to adiabaticity, resulting in a nonzero finite or divergent contribution to the AA phase. Steering the system adiabatically through a degeneracy provides control over the geometric phase as it can cause a π shift of the Berry phase. On the other hand, we revisit an example of degeneracy crossing proposed by AA. We find that, at suitable driving frequencies, both geometric-phase definitions give the same result and the dynamical phase is zero due to the symmetry of time evolution about the point of degeneracy, providing an advantageous setup for manipulation of quantum states.
Since the pioneering works by Landau, Zener, Stückelberg, and Majorana (LZSM), it has been known that driving a quantum two-level system results in tunneling between its states. Even though the interference between these transitions is known to be important, it is only recently that it became both accessible, controllable, and useful for manipulating a growing number of quantum systems. Here, we systematically study various aspects of LZSM physics and review the relevant literature, significantly expanding the review article in Ref. [504].
We present new developments in nonadiabatic geometric phases along two lines for systems undergoing changes of quantum state in intense fields. We first present a geometric representation of the non-Hermitian Schrodinger equation and introduce the notion of a complex multiphoton Aharonov-Anandan (AA) phase associated with dissipative two-level systems driven by periodic fields. The concept is further extended to include field modulation effects. We then develop the AA phase for spin-j systems in periodic fields and find conditions for cyclic evolution for general multi-level systems. In both cases, generalizations of the Floquet formalism lead to general analytical expressions for geometric phases that can be tested by experiments.