We outline two general classes of examples of $\mathbb{P}\mathbb{T}$-symmetric quantum Liouvillian dynamics of open many-qubit systems, namely, interacting hard-core bosons (or more general $XYZ$-type spin-1/2 systems) having either (i) pure dephasing noise or (ii) solely single-particle (spin) injection (absorption) incoherent processes. The concept of $\mathbb{P}\mathbb{T}$ symmetry is defined following Prosen [Phys. Rev. Lett. 109, 090404 (2012)] as a formal quantum Liouville-space analog of the parity-time $\mathcal{PT}$ symmetry used in nonconservative classical systems with symmetrically distributed gain and loss.
We outline two general classes of examples of PT-symmetric quantum Liouvillian dynamics of open many-qubit systems, namely, interacting hard-core bosons (or more general XYZ-type spin-1/2 systems) having either (i) pure dephasing noise or (ii) solely single-particle (spin) injection (absorption) incoherent processes. The concept of PT symmetry is defined following Prosen [Phys. Rev. Lett. 109, 090404 (2012)] as a formal quantum Liouville-space analog of the parity-time PT symmetry used in nonconservative classical systems with symmetrically distributed gain and loss.
We investigate parity-time (PT) phase transitions in open quantum systems and discuss a criterion of Liouvillian PT symmetry proposed recently by Huber et al. [SciPost Phys. 9, 52 (2020)]. Using the third quantization, which is a general method to solve the Lindblad equation for open quadratic systems, we show, with a proposed criterion of PT symmetry, that the eigenvalue structure of the Liouvillian clearly changes at the PT-symmetry-breaking point for an open two-spin model with exactly balanced gain and loss if the total spin is large. In particular, in a PT-unbroken phase, some eigenvalues are pure imaginary numbers, while in a PT-broken phase, all the eigenvalues are real. From this result, it is analytically shown for an open quantum system including quantum jumps that the dynamics in the long-time limit changes from an oscillatory to an overdamped behavior at the proposed PT-symmetry-breaking point. Furthermore, we show a direct relation between the criterion of Huber et al. of Liouvillian PT symmetry and the dynamics of the physical quantities for quadratic bosonic systems. Our results support the validity of the proposed criterion of Liouvillian PT symmetry.
We investigate parity-time reversal (PT) phase transitions in open quantum systems and discuss a criterion of Liouvillian PT symmetry proposed recently by J. Huber, P. Kirton, S. Rotter and P. Rabl. Using the third quantization, which is a general method to solve the Lindblad equation for open quadratic systems, we show, with a proposed criterion of PT symmetry, that the eigenvalue structure of the Liouvillian clearly changes at the PT symmetry breaking point for an open 2-spin model with exactly balanced gain and loss if the total spin is large. Specially, in a PT unbroken phase, some eigenvalues are pure imaginary numbers while in a PT broken phase, all the eigenvalues are real. From this result, it is analytically shown for open quantum system including quantum jumps that the dynamics in the long time limit changes from an oscillatory to an overdamped behavior at the proposed PT symmetry breaking point. Furthermore, we show a direct relation between the criterion of Huber et al. of Liouvillian PT symmetry and dynamics of the physical quantities for quadratic bosonic systems. Our results support validity of the proposed criterion of Liouvillian PT symmetry.
The effect of \mathcal{PT} 𝒫 𝒯 -symmetry breaking in coupled systems with balanced gain and loss has recently attracted considerable attention and has been demonstrated in various photonic, electrical and mechanical systems in the classical regime. However, it is still an unsolved problem how to generalize the concept of \mathcal{PT} 𝒫 𝒯 symmetry to the quantum domain, where the conventional definition in terms of non-Hermitian Hamiltonians is not applicable. Here we introduce a symmetry relation for Liouville operators that describe the dissipative evolution of arbitrary open quantum systems. Specifically, we show that the invariance of the Liouvillian under this symmetry transformation implies the existence of stationary states with preserved and broken parity symmetry. As the dimension of the Hilbert space grows, the transition between these two limiting phases becomes increasingly sharp and the classically expected \mathcal{PT} 𝒫 𝒯 -symmetry breaking transition is recovered. This quantum-to-classical correspondence allows us to establish a common theoretical framework to identify and accurately describe \mathcal{PT} 𝒫 𝒯 -symmetry breaking effects in a large variety of physical systems, operated both in the classical and quantum regimes.
Quantum dynamics of two XX interacting PT-symmetric non-Hermitian qubits: enhancement of quantum annealing
Quantum information platforms enable analog quantum simulations, such as quantum annealing, offering a promising route to solving complex combinatorial optimization problems. Here, we propose a quantum information architecture based on networks of interacting parity-time (PT)-symmetric non-Hermitian qubits. While the dynamics of individual PT-symmetric qubits have been experimentally demonstrated across multiple platforms including NV centers, superconducting circuits, and trapped-ion systems yet coherent dynamics in interacting systems remain largely unexplored. To address this issue we theoretically investigate stationary and time-dependent Hamiltonians relevant to quantum annealing using a minimal model of two interacting XX-coupled PT-symmetric non-Hermitian qubits. We analyze both symmetry-preserving and symmetry-broken regimes and demonstrate that adding even tiny PT-symmetric non-Hermitian terms in the qubits Hamiltonian allows to greatly enhance the probability of reaching the ground state after annealing.
Analysis of fundamentally open systems which exhibit both spatial reflection (parity) and time-reversal symmetry, called PT-symmetric [1] systems, has played an important role in understanding the dynamics of a broad class of non-Hermitian systems. Such systems support distinct regimes of stability and instability respectively, and the transition between these plays an important role in the system dynamics. We investigate a variety of non-Hermitian systems which exhibit a PT-symmetry breaking transition in both classical and quantum contexts using an array of theoretical and numerical approaches which are directly amenable to physical experiments. Organizationally, we divide these studies into those done on classical and quantum systems. Specifically for classical systems we analyze several interesting phenomena in the context of optical waveguide arrays. These provide an important testing ground for much of the discrete, PT-symmetric theory. In our case, we analyze PT-symmetric extensions of several interesting lattice models. Firstly, we explore single gain/loss pair, PT-symmetric extensions of the Su-Schreiffer-Heeger (SSH) model [2] and the Aubry-Andre-Harper (AAH) model [3,4]. In both cases, we find a veiled symmetry which is responsible for the PT-symmetry. Secondly we expand upon the Rudner-Levitov (RL) model [5,6], a non-Hermitian extension of the SSH model with multiple gain/loss pairs, and we explore the dependence of topological phenomena on changes in the initial state as well as their robustness in the presence of gain-induced nonlinearity. For quantum systems, we explore the rich PT-symmetric phase diagram resulting from time-periodic, Floquet PT-symmetry [7,8]. This is experimentally implementable in ultracold atomic systems [9], and features a reentrant PT-symmetric phase. Secondly, we examine quantum multi-photon statistics for a simple non-Hermitian model which is transformed and mapped to a system with unitary evolution via a unitary dilation, and is fully realizable on a small-scale quantum simulator [10].
The concept of parity-time (PT) symmetry originates from the framework of quantum mechanics, where if the Hamiltonian operator satisfies the commutation relation with the parity and time operators, it shows real energy spectrum. Recently, PT symmetry was introduced into various systems, such as optics, electronic circuits, acoustics, and other classical fields to further study the dynamical behaviors of the Hamiltonian and the energies. To focus on the dynamical evolution of the quantum state, here we experimentally studied the dynamical evolution of a two-level quantum system under the PT symmetric Hamiltonian using single-photon system. By enlarging the system using ancillary qubit and encoding the subsystem under the non-Hermitian Hamiltonian with post-selection, the evolution of the state can be characterized with a high fidelity. Owing to the effectively operation of the dilation method, our work provides a route for further exploiting the exotic properties of PT symmetric Hamiltonian for quantum simulation and quantum information processing.