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arXiv:1904.11181 · Physical Review A

Interplay between nonclassicality and PT symmetry in an effective two-level system with open system effects

Javid Naikoo, Subhashish Banerjee, Anirban Pathak

Published Aug 22, 2019• 6 citations
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Abstract

A three-level atom in $\mathrm{\ensuremath{\Lambda}}$ configuration is reduced to an effective two-level system, under appropriate conditions, and its $\mathcal{PT}$ symmetric properties are investigated. This effective qubit system, when subjected to a beam-splitter type of interaction, provides the scope of directly (indirectly) probing the nonclassical properties of the output (input) state. Here, we study nonclassical properties of the output state by using some well-known measures of nonclassical correlations like the measurement-induced disturbance, concurrence, and negativity. The nonclassical features are found to enhance in the $\mathcal{PT}$ symmetric (PTS) phase compared to the $\mathcal{PT}$ symmetry broken (PTB) phase. Further, the output ports of the beam splitter are subjected to different quantum noise channels, both non-Markovian, e.g., random telegraph noise as well as Markovian, e.g., phase damping and amplitude damping noise. The application of noise channels is found to decrease the degree of nonclassicality, though continuing to exhibit distinct behavior in PTS and PTB phases, with the dominant behavior appearing in the former case. Further, the results are compared with the case when dynamics is governed by a Hermitian Hamiltonian. This allows one to demarcate the contributions to nonclassicality from different types of dynamics.

DOI
10.1103/physreva.100.023836

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A three-level atom in \ensuremathΛ\mathrm{\ensuremath{\Lambda}}\ensuremathΛ configuration is reduced to an effective two-level system, under appropriate conditions, and its PT\mathcal{PT}PT symmetric properties are investigated. This effective qubit system, when subjected to a beam-splitter type of interaction, provides the scope of directly (indirectly) probing the nonclassical properties of the output (input) state. Here, we study nonclassical properties of the output state by using some well-known measures of nonclassical correlations like the measurement-induced disturbance, concurrence, and negativity. The nonclassical features are found to enhance in the PT\mathcal{PT}PT symmetric (PTS) phase compared to the PT\mathcal{PT}PT symmetry broken (PTB) phase. Further, the output ports of the beam splitter are subjected to different quantum noise channels, both non-Markovian, e.g., random telegraph noise as well as Markovian, e.g., phase damping and amplitude damping noise. The application of noise channels is found to decrease the degree of nonclassicality, though continuing to exhibit distinct behavior in PTS and PTB phases, with the dominant behavior appearing in the former case. Further, the results are compared with the case when dynamics is governed by a Hermitian Hamiltonian. This allows one to demarcate the contributions to nonclassicality from different types of dynamics.
The interplay between the nonclassical features and the parity-time (PT\mathcal{PT}PT) symmetry (or its breaking) is studied here by considering a PT\mathcal{PT}PT-symmetric system consisting of two cavities with gain and loss. The conditions for PT\mathcal{PT}PT invariance are obtained for this system. The behavior of the average photon number corresponding to the gain and loss modes for different initial states (e.g., vacuum, NOON, coherent, and thermal states) has also been obtained. With the help of the number operators, quantum Zeno and anti-Zeno effects are studied, and the observed behavior is compared in PT\mathcal{PT}PT-symmetric (PTS) and PT\mathcal{PT}PT-symmetry-broken (PTSB) regimes. It has been observed that the relative phase of the input coherent fields plays a key role in the occurrence of these effects. Further, some nonclassicality features are witnessed using criteria based on the number operator(s). Specifically, intermodal antibunching, sum, and difference squeezing are investigated for specific input states. It is found that the various nonclassical features, including the observed quantum Zeno and anti-Zeno effects, are suppressed when one goes from the PTS to PTSB regime. In other words, the dominance of the loss or gain rate in the field modes over the coupling strength between them diminishes the nonclassical features of the system.
If a single-mode nonclassical light is combined with the vacuum on a beam splitter, then the output state is entangled. As proposed in [Phys. Rev. Lett. 94, 173602 (2005)], by measuring the output-state entanglement for a balanced lossless beam splitter, one can quantify the input-state nonclassicality. These measures of nonclassicality (referred to as entanglement potentials) can be based, in principle, on various entanglement measures, leading to the negativity (NP) and concurrence (CP) potentials, and the potential for the relative entropy of entanglement (REEP). We search for the maximal nonclassicality, which can be achieved by comparing two entanglement measures for arbitrary two-qubit states and those which can be generated from a photon-number qubit via a balanced lossless beam-splitter, where the qubit basis states are the vacuum and single-photon states. Surprisingly, we find that the maximal nonclassicality, measured by the REEP for a given value of the NP, can be increased (if NP<0.527) by using either a tunable beam splitter or by amplitude damping of the output state of the balanced beam splitter. We also show that the maximal nonclassicality, measured by the NP for a given value of the REEP, can be increased by dephasing. The entanglement itself is not increased by these local losses, but the possible ratios of different measures are affected. Moreover, we show that partially-mixed states can be more nonclassical than both pure states and completely-mixed states, by comparing the NP for a given value of the REEP. This implies that not all entanglement measures can be used as entanglement potentials. Alternatively, a single balanced lossless beam splitter is not always transferring the whole nonclassicality of its input state into the entanglement of its output modes. Applying a lossy beam splitter can solve this problem at least for the cases analyzed in this paper.
A two-mode optical parity-time (PT) symmetric system, with gain and damping, described by a quantum quadratic Hamiltonian with additional small Kerr-like nonlinear terms, is analyzed from the point of view of nonclassical-light generation. Two kinds of stationary states with different types of (in)stability are revealed. Properties of one of these are related to the presence of semiclassical exceptional points, i.e., exotic degeneracies of the non-Hermitian Hamiltonian describing the studied system without quantum jumps. The evolution of the logarithmic negativity, principal squeezing variances, and sub-shot-noise photon-number correlations, considered as entanglement and non-classicality quantifiers, is analyzed in the approximation of linear-operator corrections to the classical solution. Suitable conditions for nonclassical-light generation are identified in the oscillatory regime, especially at and around exceptional points that considerably enhance the nonlinear interaction and, thus, the non-classicality of the generated light. The role of quantum fluctuations, inevitably accompanying attenuation and amplification in the evolution of quantum states, is elucidated. The evolution of the system is analyzed for different initial conditions.
If a single-mode nonclassical light is combined with the vacuum on a beam splitter, then the output state is entangled. As proposed in [Phys. Rev. Lett. 94, 173602 (2005)], by measuring this output-state entanglement for a balanced lossless beam splitter, one can quantify the input-state nonclassicality. These measures of nonclassicality (referred to as entanglement potentials) can be based, in principle, on various entanglement measures, leading to the negativity (NP) and concurrence (CP) potentials, and the potential for the relative entropy of entanglement (REEP). We search for the maximal relative nonclassicality, which can be achieved by comparing two entanglement measures for (i) arbitrary two-qubit states and (ii) those which can be generated from a photon-number qubit via a balanced lossless beam splitter, where the qubit basis states are the vacuum and single-photon states. Surprisingly, we find that the maximal relative nonclassicality, measured by the REEP for a given value of the NP, can be increased (if NP<0.527\mathrm{NP}<0.527NP<0.527) by using either a tunable beam splitter or by amplitude damping of the output state of the balanced beam splitter. We also show that the maximal relative nonclassicality, measured by the NP for a given value of the REEP, can be increased by phase damping (dephasing). Note that the entanglement itself is not increased by these losses (since they act locally), but the possible ratios of different measures are affected. Moreover, we show that partially dephased states can be more nonclassical than both pure states and completely dephased states, by comparing the NP for a given value of the REEP. Thus, one can conclude that not all standard entanglement measures can be used as entanglement potentials. Alternatively, one can infer that a single balanced lossless beam splitter is not always transferring the whole nonclassicality of its input state into the entanglement of its output modes. The application of a lossy beam splitter can solve this problem, at least for the cases analyzed in this paper.
A two-mode optical parity-time (PT\mathcal{PT}PT) symmetric system, with gain and damping, described by a quantum quadratic Hamiltonian with additional small Kerr-like nonlinear terms, is analyzed from the point of view of nonclassical-light generation. Two kinds of stationary states with different types of (in)stability are revealed. Properties of one of these are related to the presence of semiclassical exceptional points, i.e., exotic degeneracies of the non-Hermitian Hamiltonian describing the studied system without quantum jumps. The evolution of the logarithmic negativity, principal squeezing variances, and sub-shot-noise photon-number correlations, considered as entanglement and nonclassicality quantifiers, is analyzed in the approximation of linear-operator corrections to the classical solution. Suitable conditions for nonclassical-light generation are identified in the oscillatory regime, especially at and around exceptional points that considerably enhance the nonlinear interaction and, thus, the nonclassicality of the generated light. The role of quantum fluctuations, inevitably accompanying attenuation and amplification in the evolution of quantum states, is elucidated. The evolution of the system is analyzed for different initial conditions.
Dynamics of atom–field correlations and single-mode nonclassicalities present in the resonant Jaynes–Cummings model are investigated using negativity and entanglement potential for a set of initial states. The study has revealed the interplay among three different types of nonclassicalities present in the model and established that the nonclassicality is continuously exchanged between the field and atom through atom–field correlations. Further, it is observed that the entanglement potential does not capture all single-mode nonclassicality, and there exists some residual nonclassicality in the reduced single-mode states at the output of the beam splitter that is not captured by the entanglement in which single-mode nonclassicality is quantitatively mapped in Asboth’s criterion. Additional layers of beam splitters are added to deplete all the nonclassicality and to reveal that almost all residual nonclassicality is captured with three layers of beam splitters. Further, the reduced states of the atom and field have zero (non-zero) quantum coherence in the Fock basis when the atom–field correlations are maximum if the field (or atom) has zero (non-zero) quantum coherence initially.