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arXiv:1803.04411 · Physical Review A

Non-Hermitian dynamics of slowly varying Hamiltonians

Hailong Wang, Li-Jun Lang, Y. D. Chong

Published Jul 16, 2018• 26 citations
arXivHTMLPDF

Abstract

We develop a theoretical description of non-Hermitian time evolution that accounts for the breakdown of the adiabatic theorem. We obtain closed-form expressions for the time-dependent state amplitudes, involving the complex eigenenergies as well as interband Berry connections calculated using basis sets from appropriately chosen Schur decompositions. Using a two-level system as an example, we show that our theory accurately captures the phenomenon of ``sudden transitions,'' where the system state abruptly jumps from one eigenstate to another.

DOI
10.1103/physreva.98.012119

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We develop a theoretical description of non-Hermitian time evolution that accounts for the breakdown of the adiabatic theorem. We obtain closed-form expressions for the time-dependent state amplitudes, involving the complex eigenenergies as well as interband Berry connections calculated using basis sets from appropriately chosen Schur decompositions. Using a two-level system as an example, we show that our theory accurately captures the phenomenon of ``sudden transitions,'' where the system state abruptly jumps from one eigenstate to another.
The adiabatic theorem states that an initial eigenstate of a slowly varying Hamiltonian remains close to an instantaneous eigenstate of the Hamiltonian at a later time. We show that a perfunctory application of this statement is problematic if the change in eigenstate is significant, regardless of how closely the evolution satisfies the requirements of the adiabatic theorem. We also introduce an example of a two-level system with an exactly solvable evolution to demonstrate the inapplicability of the adiabatic approximation for a particular slowly varying Hamiltonian.
Abstract We study state conversion in a parity-time (PT) symmetry broken non-Hermitian two level system. We construct a theory and explain the underlying mechanism for state conversion and define adiabatic evolutions in non-Hermitian systems. The adiabatic theorem can be used if the initial state is an instantaneous eigenstate in Hermitian systems. We show that the system can adiabatically follow the modulationally stable instantaneous eigenstate sooner or later, regardless of the initial state if the system parameters are varied slowly in non-Hermitian systems. We discuss the topological feature of dynamical circling in the parameter space.
Non-Hermitian Hamiltonians with nonreciprocal coupling can achieve amplification of initial states without external gain due to a kind of inherent source. We discuss the source and its effect on time evolution in terms of complex eigenenergies and nonorthogonal eigenstates. Demonstrating two extreme cases of Hamiltonians, namely one having complex eigenenergies with orthogonal eigenstates and one having real eigenenergies with nonorthogonal eigenstates, we elucidate the differences between the amplifications from complex eigenenergies and from nonorthogonal eigenstates.
In many classical and quantum systems described by an effective non-Hermitian Hamiltonian, spectral phase transitions, from an entirely real-energy spectrum to a complex spectrum, can be observed as a non-Hermitian parameter in the system is increased above a critical value. A paradigmatic example is provided by systems possessing parity-time (PT\mathcal{PT}PT) symmetry, where the energy spectrum remains entirely real in the unbroken PT\mathcal{PT}PT phase while a transition to complex energies is observed in the broken PT\mathcal{PT}PT phase. Such spectral phase transitions are universally sharp. However, when the system is slowly and periodically cycled, the phase transition can become smooth, i.e., imperfect, owing to the complex Berry phase associated to the cyclic adiabatic evolution of the system. This remarkable phenomenon is illustrated by considering the spectral phase transition of the Wannier-Stark ladders in a PT\mathcal{PT}PT-symmetric class of two-band non-Hermitian lattices subjected to an external dc field, revealing that a nonvanishing imaginary part of the Zak phase---the Berry phase picked up by a Bloch eigenstate evolving across the entire Brillouin zone---is responsible for imperfect spectral phase transitions.
We introduce a new approach for describing nonstationary quantum systems with a discrete energy spectrum. The essence of this approach is that we describe the evolution of a quantum system in a time-dependent basis. In a sense, this approach is similar to the description of the system in the interaction representation. However, the time dependence of the basic states of the representation is determined not by the evolution operator with a time-independent Hamiltonian but by the eigenstates of the time-dependent Hamiltonian defined at the current time. The time dependence of the basic states of the representation leads to the appearance of an additional term in the Schrödinger equation, which in the case of slowly changing parameters of the Hamiltonian can be considered as a small perturbation. The adiabatic representation is suitable in cases where it is impossible to apply the standard interaction representation. The application of the adiabatic representation is illustrated by the example of two spins connected by a magnetic dipole–dipole interaction in a slowly varying external magnetic field.
One possible realization of non-Hermitian systems is based on open quantum systems by omitting quantum jumping terms in the master equation. This is a good approximation at short times where the effects of quantum jumps can be ignored. However, the jumps can affect the long-time dynamics of the system, motivating us to take the jumps into account in these studies. In this paper, by treating the quantum jumps as perturbations, we examine the effect of the quantum jumps on the non-Hermitian system. For this purpose, we first derive an effective Hamiltonian to describe the dynamics of the open quantum system based on the master equation, then expand the eigenstates and eigenenergies up to the first and second order in the quantum jumps. Finally, we apply our theory to a dissipative two-level system and dissipative fermionic superfluids. The effect of the quantum jump on the dynamics and the nonequilibrium phase transition is demonstrated and discussed.