[Paper Review] Effect of quantum jumps on non-Hermitian system

Xiangyu Niu, Jianning Li+2 more|arXiv (Cornell University)|Feb 25, 2022

Quantum Mechanics and Non-Hermitian Physics55 references4 citations

TL;DR

This paper develops a perturbative framework to study the effect of quantum jumps on non-Hermitian systems by treating them as corrections to an effective non-Hermitian Hamiltonian derived from the Lindblad master equation. Using biorthonormal basis expansion and perturbation theory up to second order, the authors analyze how quantum jumps modify energy levels, eigenstates, and dynamics in open quantum systems, demonstrating their impact on phase transitions and fidelity in a two-level system and a dissipative BCS superfluid.

ABSTRACT

One among the possible realizations 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 quantum jump on the dynamics and the nonequilibrium phase transition is demonstrated and discussed.

Motivation & Objective

• To investigate the validity and consequences of neglecting quantum jump terms in non-Hermitian systems derived from open quantum systems.
• To develop a perturbative method that systematically includes quantum jumps as corrections to the effective non-Hermitian Hamiltonian.
• To analyze how quantum jumps affect the energy spectrum, eigenstates, and dynamics of open quantum systems, particularly in the long-time regime.
• To examine the influence of quantum jumps on nonequilibrium phase transitions and state fidelity in specific models like the two-level system and dissipative BCS superfluid.
• To establish a formal framework combining effective Hamiltonian mapping with non-Hermitian perturbation theory for open systems.

Proposed method

• Derive an effective non-Hermitian Hamiltonian from the Lindblad master equation by mapping the density matrix evolution to a pure state evolution in a composite system of system and ancilla.
• Use a biorthonormal basis to represent the system's dynamics and enable perturbative expansion in the quantum jump terms.
• Treat the quantum jump terms as perturbations and expand eigenstates and eigenenergies up to second order in the jump strength.
• Apply the effective Hamiltonian approach to transform the master equation into a Schrödinger-like equation with a non-Hermitian Hamiltonian acting on a composite Hilbert space.
• Utilize the mapping rule $|\psi_\rho(t)\rangle = \sum_{mn} \langle l_m|\rho|l_n\rangle |r_m\rangle \otimes |R_n\rangle^{A*}$ to relate the density matrix to the pure state evolution.
• Formally derive higher-order approximate solutions for the density operator, energy, and state fidelity using perturbation theory within the effective Hamiltonian framework.

Experimental results

Research questions

• RQ1 How do quantum jumps, neglected in standard non-Hermitian approximations, affect the long-time dynamics of open quantum systems?
• RQ2 What is the quantitative impact of quantum jumps on the eigenenergies and eigenstates of a non-Hermitian system?
• RQ3 How do quantum jumps modify the critical behavior and nonequilibrium phase transitions in dissipative quantum systems?
• RQ4 To what extent does the effective Hamiltonian approach remain valid when quantum jumps are included as perturbations?
• RQ5 How do quantum jumps influence the fidelity of the initial state in dissipative two-level and BCS superfluid systems?

Key findings

• Quantum jumps significantly alter the long-time dynamics of non-Hermitian systems, invalidating the standard approximation that omits them.
• The perturbative expansion up to second order in the jump strength reveals corrections to eigenenergies and eigenstates that are non-negligible in the long-time regime.
• In the dissipative two-level system, quantum jumps reduce the fidelity of the initial state, indicating decoherence effects not captured by the non-Hermitian Hamiltonian alone.
• For the dissipative BCS superfluid, quantum jumps modify the critical behavior near the phase transition, altering the stability and structure of the superfluid order parameter.
• The effective Hamiltonian approach successfully maps the master equation to a pure-state evolution, enabling analytical treatment of open systems with perturbative jump corrections.
• The hybrid-Liouvillian superoperator framework connects non-Hermitian and full Liouvillian dynamics, showing that exceptional points differ qualitatively between systems with and without quantum jumps.