Unbroken PT Symmetry Research Articles

Coherent states of graphene layer with and without a [formula omitted]-symmetric chemical potential

In this paper we construct different classes of coherent and bicoherent states for the graphene tight-binding model in presence of a magnetic field, and for a deformed version where we include a PT-symmetric chemical potential V. In particular, the problems caused by the absence of a suitable ground state for the system is taken into account in the construction of these states, for V=0 and for V≠0. We introduce ladder operators which work well in our context, and we show, in particular, that there exists a choice of these operators which produce a factorization of the Hamiltonian. The role of broken and unbroken PT-symmetry is discussed, in connection with the strength of V.

Three perspectives on entropy dynamics in a non-Hermitian two-state system

Abstract A comparative study of entropy dynamics as an indicator of physical behavior in an open two-state system with balanced gain and loss is presented. To begin with, we illustrate the phase portrait of this non-Hermitian model on the Bloch sphere, elucidating the changes in behavior as one moves across the phase transition boundary, as well as the emergent feature of unidirectional state evolution in the spontaneously broken PT-symmetry regime. This is followed by an examination of the purity and entropy dynamics. Here we distinguish the perspective taken in utilizing the conventional framework of Hermitian-adjoint states from an approach that is based on biorthogonal-adjoint states and a third case based on an isospectral mapping. In this it is demonstrated that their differences are rooted in the treatment of the environmental coupling mode. For unbroken PT symmetry of the system, a notable characteristic feature of the perspective taken is the presence or absence of purity oscillations, with an associated entropy revival. The description of the system is then continued from its PT-symmetric pseudo-Hermitian phase into the regime of spontaneously broken symmetry, in the latter two approaches through a non-analytic operator-based continuation, yielding a Lindblad master equation based on the PT charge operator C. This phase transition indicates a general connection between the pseudo-Hermitian closed-system and the Lindbladian open-system formalism through a spontaneous breakdown of the underlying physical reflection symmetry.

Vorticity wave interaction, Krein collision, and exceptional points in shear flow instabilities.

We relate the model of vorticity wave interaction to Krein collision, PT-symmetry breaking, and the formation of exceptional points in shear flow instabilities. We show that the dynamical system of coupled vorticity waves is a pseudo-Hermitian system with nonreciprocal coupling terms. Krein signatures of the eigenvalues are illustrated as the signs of the action of the vorticity waves. Interaction between positive-action and negative-action vorticity waves then corresponds to the Krein collision between eigenvalues with opposite Krein signatures, the spontaneous breaking of PT symmetry, and the formation of exceptional points. The control parameter of the PT-symmetry-breaking bifurcation is the ratio between frequency detuning and coupling strength of the vorticity waves. The critical behavior near the exceptional points is described as a transition between phase-locking and phase-slip dynamics of the vorticity waves. The phase-slip dynamics correspond to nonmodal, transient growth of perturbations in the regime of unbroken PT symmetry, and the phase-slip frequency Ω∝|k-k_{c}|^{1/2} shares the same critical exponent with the phase rigidity of system eigenvectors.

Formations and dynamics of two-dimensional spinning asymmetric quantum droplets controlled by a PT-symmetric potential.

In this paper, vortex solitons are produced for a variety of 2D spinning quantum droplets (QDs) in a PT-symmetric potential, modeled by the amended Gross-Pitaevskii equation with Lee-Huang-Yang corrections. In particular, exact QD states are obtained under certain parameter constraints, providing a guide to finding the respective generic family. In a parameter region of the unbroken PT symmetry, different families of QDs originating from the linear modes are obtained in the form of multipolar and vortex droplets at low and high values of the norm, respectively, and their stability is investigated. In the spinning regime, QDs become asymmetric above a critical rotation frequency, most of them being stable. The effect of the PT-symmetric potential on the spinning and nonspinning QDs is explored by varying the strength of the gain-loss distribution. Generally, spinning QDs trapped in the PT-symmetric potential exhibit asymmetry due to the energy flow affected by the interplay of the gain-loss distribution and rotation. Finally, interactions between spinning or nonspinning QDs are explored, exhibiting elastic collisions under certain conditions.

Mechanical squeezing in an active-passive-coupled double-cavity optomechanical system via pump modulation.

We focus on the generation of mechanical squeezing by using periodically amplitude-modulated laser to drive an active-passive-coupled double-cavity optomechanical system, where the coupled gain cavity and loss cavity can form into a parity-time (P T)-symmetry system. The numerical analysis of the system stability shows that the system is more likely to be stable in the unbroken-P T-symmetry regime than in the broken-P T-symmetry regime. The mechanical squeezing in the active-passive system exhibits stronger robustness against the thermal noise than that in the passive-passive system, and the so-called 3 dB limit can be broken in the resolved-sideband regime. Furthermore, it is also found that the mechanical squeezing obtained in the unbroken-P T-symmetry region is stronger than that in the broken-P T-symmetry region. This work may be meaningful for the quantum state engineering in the gain-loss quantum system that contributes to the study of P T-symmetric physics in the quantum regime.

Discrete spacetime symmetries, second quantization, and inner products in a non-Hermitian Dirac fermionic field theory

We extend to a non-Hermitian fermionic quantum field theory with PT symmetry our previous discussion of second quantization, discrete symmetry transformations, and inner products in a scalar field theory [arXiv:2006.06656]. For illustration, we consider a prototype model containing a single Dirac fermion with a parity-odd, anti-Hermitian mass term. In the phase of unbroken PT symmetry, this Dirac fermion model is equivalent to a Hermitian theory under a similarity transformation, with the non-Hermitian nature of the model residing only in the spinor structure, whereas the algebra of the creation and annihilation operators is just that of a Hermitian theory.

The operational foundations of PT-symmetric and quasi-Hermitian quantum theory

PT-symmetric quantum theory was originally proposed with the aim of extending standard quantum theory by relaxing the Hermiticity constraint on Hamiltonians. However, no such extension has been formulated that consistently describes states, transformations, measurements and composition, which is a requirement for any physical theory. We aim to answer the question of whether a consistent physical theory with PT-symmetric observables extends standard quantum theory. We answer this question within the framework of general probabilistic theories, which is the most general framework for physical theories. We construct the set of states of a system that result from imposing PT-symmetry on the set of observables, and show that the resulting theory allows only one trivial state. We next consider the constraint of quasi-Hermiticity on observables, which guarantees the unitarity of evolution under a Hamiltonian with unbroken PT-symmetry. We show that such a system is equivalent to a standard quantum system. Finally, we show that if all observables are quasi-Hermitian as well as PT-symmetric, then the system is equivalent to a real quantum system. Thus our results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory consistently.

Transmission properties and [formula omitted]-symmetry in a hybrid quantum electromechanical system

We theoretically study the transmission properties and the PT-symmetry in a hybrid quantum electromechanical system consisting of a coplanar-waveguide (CPW) microwave cavity, a nanomechanical resonator (NAMR) and a superconducting transmon qubit, where the qubit works in the large detuning condition with both the CPW microwave cavity and the NAMR. By adjusting the NAMR-qubit coupling, we find that the transmissivity and the phase shift of the input field passing through the system can be tuned effectively. We also find that the effective electromechanical coupling can be greatly enhanced by adjusting the NAMR-qubit coupling. Furthermore, with the effective electromechanical coupling, we can realize the transition from the broken PT-symmetry phase to the unbroken PT-symmetry phase when both the cavity loss and the mechanical gain are considered. This work presents an effective way to manipulate the propagation characteristics and to study the PT-symmetry in a hybrid quantum electromechanical system.

The geometrical properties of parity and time reversal operators in two dimensional spaces

The parity operator P and time reversal operator T are two important operators in the quantum theory, in particular, in the PT-symmetric quantum theory. By using the concrete forms of P and T, we discuss their geometrical properties in two dimensional spaces. It is showed that if T is given, then all P links with the quadric surfaces; if P is given, then all T links with the quadric curves. Moreover, we give out the generalized unbroken PT-symmetric condition of an operator. The unbroken PT-symmetry of a Hermitian operator is also showed in this way.

Quantum work relations and response theory in parity-time-symmetric quantum systems.

In this work, we show that a universal quantum work relation for a quantum system driven arbitrarily far from equilibrium extends to a parity-time- (PT-) symmetric quantum system with unbroken PT symmetry, which is a consequence of microscopic reversibility. The quantum Jarzynski equality, linear response theory, and Onsager reciprocal relations for the PT-symmetric quantum system are recovered as special cases of the universal quantum work relation in a PT-symmetric quantum system. In the regime of broken PT symmetry, the universal quantum work relation does not hold because the norm is not preserved during the dynamics.

Pseudo-Hermitian Systems with P T $\mathcal {P}\mathcal {T}$ -Symmetry: Degeneracy and Krein Space

We show in the present paper that pseudo-Hermitian Hamiltonian systems with even PT-symmetry admit a degeneracy structure. This kind of degeneracy is expected traditionally in the odd PT-symmetric systems which is appropriate to the fermions as shown by Jones-Smith and Mathur [1] who extended PT-symmetric quantum mechanics to the case of odd time-reversal symmetry. We establish that the pseudo-Hermitian Hamiltonians with even PT-symmetry admit a degeneracy structure if the operator PT anticommutes with the metric operator {\eta} which is necessarily indefinite. We also show that the Krein space formulation of the Hilbert space is the convenient framework for the implementation of unbroken PT-symmetry. These general results are illustrated with great details for four-level pseudo-Hermitian Hamiltonian with even PT-symmetry.

Solitons stabilization in PT symmetric potentials through modulation the shape of imaginary component

Stability and dynamics of PT symmetric fundamental bright solitons supported by localized potentials in a focusing Kerr medium are investigated numerically. How the shape and magnitude of the imaginary component affect soliton stability is addressed when fixed real part of the potentials. The unbroken PT symmetry in linear case and stable region in nonlinear case are discussed. Numerical simulations proved that solitons can propagate stably when the loss or gain distribution become narrower. So we can stabilize the solitons through modulation the shape of imaginary component, and can use the method for solitons formation and control in other nonlinear media with PT potentials.

Parity-time symmetry-breaking mechanism of dynamic Mott transitions in dissipative systems

We describe the critical behavior of electric field-driven (dynamic) Mott insulator-to-metal transitions in dissipative Fermi and Bose systems in terms of non-Hermitian Hamiltonians invariant under simultaneous parity (P) and time-reversal (T) operations. The dynamic Mott transition is identified as a PT symmetry-breaking phase transition, with the Mott insulating state corresponding to the regime of unbroken PT symmetry with a real energy spectrum. We establish that the imaginary part of the Hamiltonian arises from the combined effects of the driving field and inherent dissipation. We derive the renormalization and collapse of the Mott gap at the dielectric breakdown and describe the resulting critical behavior of transport characteristics. The obtained critical exponent is in an excellent agreement with experimental findings.

Peregrine rogue wave dynamics in the continuous nonlinear Schrödinger system with parity-time symmetric Kerr nonlinearity

In this work, we have studied the peregrine rogue wave dynamics, with a solitons on finite background (SFB) ansatz, in the recently proposed (Ablowitz and Musslimani, (2013) [31]) continuous nonlinear Schrödinger system with parity-time symmetric Kerr nonlinearity. We have found that the continuous nonlinear Schrödinger system with PT-symmetric nonlinearity also admits Peregrine soliton solution. Motivated by the fact that Peregrine solitons are regarded as prototypical solutions of rogue waves, we have studied Peregrine rogue wave dynamics in the c-PTNLSE model. Upon numerical computation, we observe the appearance of low-intense Kuznetsov–Ma (KM) soliton trains in the absence of transverse shift (unbroken PT-symmetry) and well-localized high-intense Peregrine rogue waves in the presence of transverse shift (broken PT-symmetry) in a definite parametric regime.

Control of power in parity-time symmetric lattices

We investigate wave transport properties of parity–time (PT) symmetric lattices that are periodically modulated along the direction of propagation. We demonstrate that in the regime of unbroken PT-symmetry, the system Floquet–Bloch modes may interfere constructively leading to either controlled oscillations or power absorption and unlimited amplification occurring exactly at the phase-transition point. The differential power response is affected by the overlap of the gain and loss system distribution with wave intensity pattern that is formed through Rabi oscillations engaging the coupled Floquet–Bloch modes.

Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarff-II potentials.

We present a unified theoretical study of the bright solitons governed by self-focusing and defocusing nonlinear Schrödinger (NLS) equations with generalized parity-time- (PT) symmetric Scarff-II potentials. Particularly, a PT-symmetric k-wave-number Scarff-II potential and a multiwell Scarff-II potential are considered, respectively. For the k-wave-number Scarff-II potential, the parameter space can be divided into different regions, corresponding to unbroken and broken PT symmetry and the bright solitons for self-focusing and defocusing Kerr nonlinearities. For the multiwell Scarff-II potential the bright solitons can be obtained by using a periodically space-modulated Kerr nonlinearity. The linear stability of bright solitons with PT-symmetric k-wave-number and multiwell Scarff-II potentials is analyzed in detail using numerical simulations. Stable and unstable bright solitons are found in both regions of unbroken and broken PT symmetry due to the existence of the nonlinearity. Furthermore, the bright solitons in three-dimensional self-focusing and defocusing NLS equations with a generalized PT-symmetric Scarff-II potential are explored. This may have potential applications in the field of optical information transmission and processing based on optical solitons in nonlinear dissipative but PT-symmetric systems.

Jarzynski Equality in PT-Symmetric Quantum Mechanics.

We show that the quantum Jarzynski equality generalizes to PT-symmetric quantum mechanics with unbroken PT symmetry. In the regime of broken PT symmetry, the Jarzynski equality does not hold as also the CPT norm is not preserved during the dynamics. These findings are illustrated for an experimentally relevant system-two coupled optical waveguides. It turns out that for these systems the phase transition between the regimes of unbroken and broken PT symmetry is thermodynamically inhibited as the irreversible work diverges at the critical point.

Coherent Perfect Absorption in 𝓟𝓣 Symmetric Optical Lattice

In this work we study scattering in a 𝓟𝓣 symmetric periodic optical lattice, when two coherent beams are incident on it from two opposite directions. We focus our attention on both 𝓟𝓣 broken as well as 𝓟𝓣 unbroken phase, with special emphasis on spectral singularity. This is another example of an analytically solvable model, apart from the 𝓟𝓣 symmetric Scarf II potential discussed in ref. Ahmed (Phys. Rev. A 81, 022102 (2014)), which exhibits the phenomenon of Coherent Perfect Absorption (CPA) with lasing, in the domain of broken 𝓟𝓣 symmetry. At the same time, CPA is neither possible for unbroken 𝓟𝓣 symmetry, nor at the transition point.

Real discrete spectrum in the complex non-PT-symmetric Scarf II potential

Abstract Hitherto, it is well known that complex PT-symmetric Scarf II has real discrete spectrum in the parametric domain of unbroken PT-symmetry. We reveal new interesting complex, non-PT-symmetric parametric domains of this versatile potential, V ( x ) , where the spectrum is again discrete and real. Showing that the Hamiltonian, p 2 / 2 μ + V ( x ) , in the new cases is pseudo-Hermitian could be challenging, if possible.

Resonant mode conversion in the waveguides with unbroken and broken PT symmetry

We study resonant mode conversion in parity time (PT)-symmetric multimode waveguides, where symmetry breaking manifests itself in the sequential destabilization (i.e., the appearance of complex eigenvalues) of the pairs of adjacent guided modes. We show that efficient mode conversion is possible even in the presence of the resonant longitudinal modulation of the complex refractive index. The distinguishing feature of the resonant mode conversion in the PT-symmetric structure is a drastic growth of the width of the resonance curve when the gain/losses coefficient approaches a critical value, at which symmetry breaking occurs. We found that in the system with broken symmetry, the resonant coupling between the exponentially growing mode and the stable higher-order one effectively stabilizes dynamically coupled pairs of modes and remarkably diminishes the average rate of the total power growth.