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Theoretical investigation of dynamics and concurrence of entangled $${{\mathcal {P}}}{{\mathcal {T}}}$$ and anti-$${{\mathcal {P}}}{{\mathcal {T}}}$$ symmetric polarized photons

Javed Akram, Chao Zheng

Published May 26, 2023• 6 citations

Abstract

Non-Hermitian systems with parity-time [Formula: see text] symmetry and anti-parity-time [Formula: see text] symmetry have exceptional points (EPs) resulting from eigenvector co-coalescence with exceptional properties. In the quantum and classical domains, higher-order EPs for [Formula: see text] symmetry and [Formula: see text]-symmetry systems have been proposed and realized. Both two-qubits [Formula: see text]-[Formula: see text] and [Formula: see text]-[Formula: see text] symmetric systems have seen an increase in recent years, especially in the dynamics of quantum entanglement. However, to our knowledge, neither theoretical nor experimental investigations have been conducted for the dynamics of two-qubits entanglement in the [Formula: see text]-[Formula: see text] symmetric system. We investigate the [Formula: see text]-[Formula: see text] dynamics for the first time. Moreover, we examine the impact of different initial Bell-state conditions on entanglement dynamics in [Formula: see text]-[Formula: see text], [Formula: see text]-[Formula: see text] and [Formula: see text]-[Formula: see text] symmetric systems. Additionally, we conduct a comparative study of entanglement dynamics in the [Formula: see text]-[Formula: see text] symmetrical system, [Formula: see text]-[Formula: see text] symmetrical system, and [Formula: see text]-[Formula: see text] symmetrical systems in order to learn more about non-Hermitian quantum systems and their environments. Entangled qubits evolve in a [Formula: see text]-[Formula: see text] symmetric unbroken regime, the entanglement oscillates with two different oscillation frequencies, and the entanglement is well preserved for a long period of time for the case when non-Hermitian parts of both qubits are taken quite away from the exceptional points.

DOI: 10.1038/s41598-023-34516-x

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Theoretical investigation of dynamics and concurrence of entangled

{{\mathcal {P}}}{{\mathcal {T}}}

and anti-

{{\mathcal {P}}}{{\mathcal {T}}}

symmetric polarized photons

Non-Hermitian systems with parity-time [Formula: see text] symmetry and anti-parity-time [Formula: see text] symmetry have exceptional points (EPs) resulting from eigenvector co-coalescence with exceptional properties. In the quantum and classical domains, higher-order EPs for [Formula: see text] symmetry and [Formula: see text]-symmetry systems have been proposed and realized. Both two-qubits [Formula: see text]-[Formula: see text] and [Formula: see text]-[Formula: see text] symmetric systems have seen an increase in recent years, especially in the dynamics of quantum entanglement. However, to our knowledge, neither theoretical nor experimental investigations have been conducted for the dynamics of two-qubits entanglement in the [Formula: see text]-[Formula: see text] symmetric system. We investigate the [Formula: see text]-[Formula: see text] dynamics for the first time. Moreover, we examine the impact of different initial Bell-state conditions on entanglement dynamics in [Formula: see text]-[Formula: see text], [Formula: see text]-[Formula: see text] and [Formula: see text]-[Formula: see text] symmetric systems. Additionally, we conduct a comparative study of entanglement dynamics in the [Formula: see text]-[Formula: see text] symmetrical system, [Formula: see text]-[Formula: see text] symmetrical system, and [Formula: see text]-[Formula: see text] symmetrical systems in order to learn more about non-Hermitian quantum systems and their environments. Entangled qubits evolve in a [Formula: see text]-[Formula: see text] symmetric unbroken regime, the entanglement oscillates with two different oscillation frequencies, and the entanglement is well preserved for a long period of time for the case when non-Hermitian parts of both qubits are taken quite away from the exceptional points.

In the past years, many efforts have been made to study various noteworthy phenomena in both parity-time

(\mathcal{PT})

and anti-parity-time

(\mathcal{APT})

symmetric systems. However, entanglement dynamics in

\mathcal{APT}

-symmetric systems has not previously been investigated in both theory and experiments. Here, we investigate the entanglement evolution of two qubits in an

\mathcal{APT}

-symmetric system. In the

\mathcal{APT}

-symmetric unbroken regime, our theoretical simulations demonstrate the periodic oscillations of entanglement when each qubit evolves identically, while the nonperiodic oscillations of entanglement when each qubit evolves differently. In particular, when each qubit evolves near the exceptional point in the

\mathcal{APT}

-symmetric unbroken regime, there exist entanglement sudden vanishing and revival. Moreover, our simulations demonstrate rapid decay and delayed death of entanglement provided one qubit evolves in the

\mathcal{APT}

-symmetric broken regime. In this work, we also perform an experiment with a linear optical setup. The experimental results agree well with our theoretical simulation results. Our findings reveal novel phenomena of entanglement evolution in the

\mathcal{APT}

-symmetric system and opens a new direction for future studies on the dynamics of quantum entanglement in multiqubit

\mathcal{APT}

-symmetric systems or other non-Hermitian quantum systems.

Non-Hermitian systems with parity-time (

\mathcal{PT}

) symmetry give rise to exceptional points (EPs) with exceptional properties that arise due to the coalescence of eigenvectors. Such systems have been extensively explored in the classical domain, where second- or higher-order EPs have been proposed or realized. In contrast, quantum information studies of

\mathcal{PT}

-symmetric systems have been confined to systems with a two-dimensional Hilbert space. Here, by using a single-photon interferometry setup, we simulate the quantum dynamics of a four-dimensional

\mathcal{PT}

-symmetric system across a fourth-order exceptional point. By tracking the coherent, nonunitary evolution of the density matrix of the system in

\mathcal{PT}

-symmetry unbroken and broken regions, we observe the entropy dynamics for both the entire system, and the gain and loss subsystems. Our setup is scalable to the higher-dimensional

\mathcal{PT}

-symmetric systems, and our results point towards the rich dynamics and critical properties.

Abstract Non-Hermitian parity-time (

{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}

P T ) and anti-parity-time (

{{{{{{{\mathcal{APT}}}}}}}}

APT )-symmetric systems exhibit novel quantum properties and have attracted increasing interest. Although many counterintuitive phenomena in

{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}

P T - and

{{{{{{{\mathcal{APT}}}}}}}}

APT -symmetric systems were previously studied, coherence flow has been rarely investigated. Here, we experimentally demonstrate single-qubit coherence flow in

{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}

P T - and

{{{{{{{\mathcal{APT}}}}}}}}

APT -symmetric systems using an optical setup. In the symmetry unbroken regime, we observe different periodic oscillations of coherence. Particularly, we observe two complete coherence backflows in one period in the

{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}

P T -symmetric system, while only one backflow in the

{{{{{{{\mathcal{APT}}}}}}}}

APT -symmetric system. Moreover, in the symmetry broken regime, we observe the phenomenon of stable value of coherence flow. We derive the analytic proofs of these phenomena and show that most experimental data agree with theoretical results within one standard deviation. This work opens avenues for future study on the dynamics of coherence in

{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}

P T - and

{{{{{{{\mathcal{APT}}}}}}}}

APT -symmetric systems.

Non-Hermitian systems with parity-time (

\mathcal{PT}

) symmetry and anti-

\mathcal{PT}

symmetry give rise to exceptional points (EPs) with important properties due to the coalescence of eigenvectors. Because of the limitations on the Hermiticity of the Hamiltonian, these kinds of non-Hermitian systems have recently attracted extensive attention and emerged as a new way to engineer the response of open quantum systems. We propose an efficient and easily computable tool, based on the Hilbert-Schmidt speed (HSS), which does not require the diagonalization of the evolved density matrix, to characterize non-Markovian effects in (anti-)

\mathcal{PT}\!-

symmetric systems. Our proposed witness of non-Markovianity can also detect exactly the EPs and hence the critical behavior of the non-Hermitian systems, especially high-dimensional qudits. Our theoretical predictions are completely consistent with results extracted from recent experiments studying the non-Markovianity and criticality in (anti-)

{{{{{{{\mathcal{APT}}}}}}}}

symmetric systems.

In this paper we study exceptional-point (EP) effects and quantum sensing in a parity-time (PT)-symmetric two-qubit system with the Ising-type interaction. We explore EP properties of the system by analyzing degeneracy of energy eigenvalues or entanglement of eigenstates. We investigate entanglement dynamics of the two qubits in detail. In particular, we demonstrate that the system can create the steady-state entanglement in the PT-broken phase and collapse-revival phenomenon of entanglement in the PT-symmetric phase during the long-time evolution. We show that entanglement can be generated more quickly than the corresponding Hermitian system. Finally, we prove that the sensitivity of eigenstate quantum sensing for the parameters exhibits the remarkable enharncement at EPs, and propose a quantum-coherence measurement to witness the existence of EPs.

The exploration of the effect of non-Hermitian (NH) in quantum systems is gaining renewed momentum as a result of the recent progress in experiments. It has been indicated that the ability to engineer the orders of exceptional points (EPs) can be fundamental to its applications in quantum control, such as accelerating entanglement generation; however, an initial survey of the interactions between different orders of EPs is still missing. In this work, we try to partially address this issue by employing a non-reciprocally coupled cavity system which can be experimentally realizable with a two-dimensional reconfigurable NH gauged laser array. With deliberate design, our systems display both second and third orders of EPs. The sign of a novel, purely quantum continuous entanglement phase transition is observed, which reveals how multi-EP interactions alter entanglement dynamics across different regimes while maintaining accelerated generation. Our findings reveil the genuinely quantum NH physics of higher-order EPs, which has opened up a new avenue for future investigations of the higher-order degeneracy application and the dynamic phase transitions of quantum systems.