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arXiv:1811.06045 · Physical Review A

Non-Abelian geometric phases in periodically driven systems

Viktor Novičenko, Gediminas Juzeliūnas

Published Jul 24, 2019• 11 citations
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Abstract

We consider a periodically driven quantum system described by a Hamiltonian which is the product of a slowly varying Hermitian operator $V(\mathbit{\ensuremath{\lambda}}\left(t\right))$ and a dimensionless periodic function with zero average. We demonstrate that the adiabatic evolution of the system within a fully degenerate Floquet band is accompanied by non-Abelian (noncommuting) geometric phases appearing when the slowly varying parameter $\mathbit{\ensuremath{\lambda}}=\mathbit{\ensuremath{\lambda}}\left(t\right)$ completes a closed loop. The geometric phases can have significant values even after completing a single cycle of the slow variable. Furthermore, there are no dynamical phases masking the non-Abelian Floquet geometric phases, as the former average to zero over an oscillation period. This can be used to precisely control the evolution of quantum systems, in particular for performing qubit operations. The general formalism is illustrated by analyzing a spin in an oscillating magnetic field with arbitrary strength and a slowly changing direction.

DOI
10.1103/physreva.100.012127

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We consider a periodically driven quantum system described by a Hamiltonian which is the product of a slowly varying Hermitian operator V(\mathbit\ensuremathλ(t))V(\mathbit{\ensuremath{\lambda}}\left(t\right))V(\mathbit\ensuremathλ(t)) and a dimensionless periodic function with zero average. We demonstrate that the adiabatic evolution of the system within a fully degenerate Floquet band is accompanied by non-Abelian (noncommuting) geometric phases appearing when the slowly varying parameter \mathbit\ensuremathλ=\mathbit\ensuremathλ(t)\mathbit{\ensuremath{\lambda}}=\mathbit{\ensuremath{\lambda}}\left(t\right)\mathbit\ensuremathλ=\mathbit\ensuremathλ(t) completes a closed loop. The geometric phases can have significant values even after completing a single cycle of the slow variable. Furthermore, there are no dynamical phases masking the non-Abelian Floquet geometric phases, as the former average to zero over an oscillation period. This can be used to precisely control the evolution of quantum systems, in particular for performing qubit operations. The general formalism is illustrated by analyzing a spin in an oscillating magnetic field with arbitrary strength and a slowly changing direction.
Holonomic quantum computing functions by transporting an adiabatically degenerate manifold of computational states around a closed loop in a control-parameter space; this cyclic evolution results in a non-Abelian geometric phase which may couple states within the manifold. Realizing the required degeneracy is challenging and typically requires auxiliary levels or intermediate-level couplings. One potential way to circumvent this is through Floquet engineering, where the periodic driving of a nondegenerate Hamiltonian leads to degenerate Floquet bands, and subsequently non-Abelian gauge structures may emerge. Here we present an experiment in ultracold Rb87 atoms where atomic spin states are dressed by modulated RF fields to induce periodic driving of a family of Hamiltonians linked through a fully tuneable parameter space. The adiabatic motion through this parameter space leads to the holonomic evolution of the degenerate spin states in SU(2) , characterized by a non-Abelian connection. We study the holonomic transformations of spin eigenstates in the presence of a background magnetic field, characterizing the fidelity of these single-qubit gate operations. Results indicate that while the Floquet engineering technique removes the need for explicit degeneracies, it inherits many of the same limitations present in degenerate systems. Published by the American Physical Society 2024
We consider a quantum system periodically driven with a strength which varies slowly on the scale of the driving period. The analysis is based on a general formulation of the Floquet theory relying on the extended Hilbert space. It is shown that the dynamics of the system can be described in terms of a slowly varying effective Floquet Hamiltonian that captures the long-term evolution, as well as rapidly oscillating micromotion operators. We obtain a systematic high-frequency expansion of all these operators. Generalizing the previous studies, the expanded effective Hamiltonian is now time dependent and contains extra terms appearing due to changes in the periodic driving. The same applies to the micromotion operators which exhibit a slow temporal dependence in addition to the rapid oscillations. As an illustration, we consider a quantum-mechanical spin in an oscillating magnetic field with a slowly changing direction. The effective evolution of the spin is then associated with non-Abelian geometric phases reflecting the geometry of the extended Floquet space. The developed formalism is general and also applies to other periodically driven systems, such as shaken optical lattices with a time-dependent shaking strength, a situation relevant to the cold-atom experiments.
It is known that the dynamics and geometric phase of a quantum system could be mapped mathematically into a classical system of coupled oscillators without loss of physics. In this work, we show that this quantum-classical mapping can also be used to simulate non-Abelian dynamics and phase of the quantum system with energy degeneracy by interpreting the quantum degeneracy as the resonance between the eigenfrequencies of the classical oscillators. The quantum evolution and Wilczek-Zee phase in the degenerate subspace can also be described by the classical non-Abelian evolution and a non-Abelian geometric angle in the resonant subspace. By studying the adiabatic evolution in this classical non-Abelian case, our results can also be used to generalize the concept of averaging principle and Hannay's angle into the classical system with resonant eigenfrequencies. The quantum-classical mapping of a tripod-scheme Hamiltonian with two degenerate dark states is employed to illustrate the classical non-Abelian dynamics and geometric angles.
Periodically driven quantum systems have emerged as one of the most versatile tools for quantum simulation, enabling the engineering of effective Hamiltonians, topological phases, and inherently nonequilibrium phenomena such as discrete time crystals. Yet fundamental challenges persist: from a spectral ambiguity that precludes a unique Floquet ground state to an inability to manipulate Floquet eigenstates beyond the adiabatic regime. This thesis demonstrates that counterdiabatic driving—a technique from adiabatic quantum control—provides a unifying framework to address these challenges. We first show that the periodic lab-frame Hamiltonian generates counterdiabatic driving for the Floquet eigenstates, revealing Floquet theory as an inverse counterdiabatic driving problem. Exploiting this equivalence, we develop geometric Floquet theory: a reformulation based on the parallel-transport gauge, which uniquely decomposes Floquet dynamics into a geometric contribution, encoded in Berry phases, and a dynamical contribution, generated by the average-energy operator. The average-energy operator provides an unambiguous sorting of the quasienergy spectrum, identifies a unique Floquet ground state at all drive frequencies, and serves as a sensitive diagnostic for heating and symmetry-breaking transitions. Inherently nonequilibrium phenomena—the 𝜋-quasienergy splitting of discrete time crystals and the anomalous edge modes of Floquet topological insulators—are traced to geometric phases. On the practical side, we exploit the intrinsic robustness of prethermal discrete time crystals for a highly frequency-selective DTC-based AC magnetic field sensing scheme and demonstrate it experimentally on a nuclear spin quantum simulator. We then generalize counterdiabatic driving to periodically driven systems. In particular, we derive a non-perturbative variational principle for the Floquet adiabatic gauge potential; this yields local, experimentally feasible control protocols for transitionless manipulation of Floquet states. We demonstrate this formalism on Floquet state preparation in an ultracold atom quantum simulator and many-body state transfer across a quantum phase transition, with significant improvements over unassisted protocols in both cases.
We study the slow crossing of non-equilibrium quantum phase transitions in periodically driven systems. We explicitly consider a spin chain with a uniform time-dependent magnetic field and focus on the Floquet state that is adiabatically connected to the ground state of the static model. We find that this Floquet ground state undergoes a series of quantum phase transitions characterized by a non-trivial topology. To dynamically probe these transitions, we propose to start with a large driving frequency and slowly decrease it as a function of time. Combining analytical and numerical methods, we uncover a Kibble-Zurek scaling that persists in the presence of moderate interactions. This scaling can be used to experimentally demonstrate non-equilibrium transitions that cannot be otherwise observed.
Quantum adiabaticity is the evolution of a quantum system that remains close to an instantaneous eigenstate of a time-dependent Hamiltonian. Using Floquet formalism, we derive a rigorous sufficient condition for adiabaticity in closed, finite-dimensional periodically driven systems that is valid for arbitrarily many driving periods. The condition is stroboscopic and geometric, depending only on single-cycle information: the Fubini--Study length of the instantaneous eigenray and a quasienergy-separation measure extracted from the Floquet operator. We also formulate a state-targeted refinement that reduces conservativeness when only one adiabatic branch is relevant. Rather than synthesizing control pulses, the result provides a certification criterion for a given periodic protocol. We illustrate the criterion and contrast it with conventional instantaneous-gap conditions in three representative examples.