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We consider the problem of understanding the basic features displayed by quantum systems described by parametric oscillators whose time-dependent frequency parameter ω ( t ) varies continuously during evolution so to realise quenching protocols of different types. To this scope we focus on the case where ω ( t ) 2 behaves like a Morse potential, up to possible sign reversion and translations in the ( t , ω 2 ) plane. We derive closed form solution for the time-dependent amplitude of quasi-normal modes, which is the very fundamental dynamical object entering the description of both classical and quantum parametric oscillators, and highlight its significant characteristics for distinctive cases arising based on the driving specifics. After doing so, we provide an insight on the way quantum states evolve by paying attention on the position-momentum Heisenberg uncertainty principle and the statistical aspects implied by second-order correlation functions over number-type states.

The upside-down $-x^4$, $-x^6$, and $-x^8$ potentials with appropriate PT-symmetric boundary conditions have real, positive, and discrete quantum-mechanical spectra. This paper proposes a straightforward macroscopic quantum-mechanical scattering experiment in which one can observe and measure these bound-state energies directly., Comment: 5 pages, 13 figures

International audience; An original method able to fully characterize high-Q resonators in an add-drop configuration has been implemented. The method is based on the study of two cavity ringdown (CRD) signals, which are produced at the transmission and drop ports by wavelength sweeping a resonance in a time interval comparable with the photon cavity lifetime. All the resonator parameters can be assessed with a single set of simultaneous measurements. We first developed a model describing the two CRD output signals and a fitting program able to deduce the key parameters from the measured profiles. We successfully validated the model with an experiment based on a fiber ring resonator of known characteristics. Finally, we characterized a high-Q, home-made, MgF2 whispering gallery mode disk resonator in the add-drop configuration, assessing its intrinsic and coupling parameters.

It is now shown that coupled optical microcavities bear all the hallmarks of parity–time symmetry; that is, the system’s dynamics are unchanged by both time-reversal and mirror transformations. The resonant nature of microcavities results in unusual effects not seen in previous photonic analogues of parity–time-symmetric systems: for example, light travelling in one direction is resonantly enhanced but there are no resonance peaks going the other way.

We address generalized measurements of linear multimode operators and discuss some aspects relevant to constructing angle operators for arbitrary quadratic Hamiltonian systems via Weyl-ordered expansions in terms of position and momentum operators.

We study a simple open quantum system with a $\mathcal{PT}$-symmetric defect potential as a prototype in order to illustrate a number of general features of $\mathcal{PT}$-symmetric open quantum systems; however, the potential itself could be mimicked by a number of $\mathcal{PT}$ systems that have been experimentally studied quite recently. One key feature is the resonance in continuum (RIC), which appears in both the discrete spectrum and the scattering spectrum of such systems. The RIC wave function forms a standing wave extending throughout the spatial extent of the system and in this sense represents a resonance between the open environment associated with the leads of our model and the central $\mathcal{PT}$-symmetric potential. We also illustrate that as one deforms the system parameters, the RIC may exit the continuum by splitting into a bound state and a virtual bound state at the band edge, a process which should be experimentally observable. We also study the exceptional points appearing in the discrete spectrum at which two eigenvalues coalesce; we categorize these as either EP2As, at which two real-valued solutions coalesce before becoming complex-valued, and EP2Bs, for which the two solutions are complex on either side of the exceptional point. The EP2As are associated with $\mathcal{PT}$-symmetry breaking; we argue that these are more stable against parameter perturbation than the EP2Bs. We also study complex-valued solutions of the discrete spectrum for which the wave function is nevertheless spatially localized, something that is not allowed in traditional open quantum systems; we illustrate that these may form quasibound states in continuum under some circumstances. We also study the scattering properties of the system, including states that support invisible propagation and some general features of perfect transmission states. We finally use our model as a prototype for the construction of scattering states that satisfy $\mathcal{PT}$-symmetric boundary conditions; while these states do not conserve the traditional probability current, we introduce the $\mathcal{PT}$ current which is preserved. The perfect transmission states appear as a special case of the $\mathcal{PT}$-symmetric scattering states.

In this paper, we consider quantization of powers of the ratio between the Hamiltonian coordinates for position and momentum in one-dimensional systems. The domain of the operators consists of square integrable functions over a finite real interval to ensure boundedness and self-adjointness. The spectral problems for the operators that result from using Weyl-ordering are discussed by introducing Fredholm integral operator forms in position representation, and the symmetry of the actions of the parity and time reversal operators on the kernels is discussed. Finally, the general structure and properties of the eigenfunctions and eigenvalues are also derived and analyzed.

We address the problem of integrating operator equations concomitant with the dynamics of non autonomous quantum systems by taking advantage of the use of time-dependent canonical transformations. In particular, we proceed to a discussion in regard to basic examples of one-dimensional non-autonomous dynamical systems enjoying the property that their Hamiltonian can be mapped through a time-dependent linear canonical transformation into an autonomous form, up to a time-dependent multiplicative factor. The operator equations we process essentially reproduce at the quantum level the classical integrability condition for these systems. Operator series form solutions in the Bender-Dunne basis of pseudo-differential operators for one dimensional quantum system are sought for such equations. The derivation of generating functions for the coefficients involved in the \emph{minimal} representation of the series solutions to the operator equations under consideration is particularized. We also provide explicit form of operators that implement arbitrary linear transformations on the Bender-Dunne basis by expressing them in terms of the initial Weyl ordered basis elements. We then remark that the matching of the minimal solutions obtained independently in the two basis, i.e. the basis prior and subsequent the action of canonical linear transformation, is perfectly achieved by retaining only the lowest order contribution in the expression of the transformed Bender-Dunne basis elements.

The Hamiltonian for a PT-symmetric chain of coupled oscillators is constructed. It is shown that if the loss-gain parameter $\gamma$ is uniform for all oscillators, then as the number of oscillators increases, the region of unbroken PT-symmetry disappears entirely. However, if $\gamma$ is localized in the sense that it decreases for more distant oscillators, then the unbroken-PT-symmetric region persists even as the number of oscillators approaches infinity. In the continuum limit the oscillator system is described by a PT-symmetric pair of wave equations, and a localized loss-gain impurity leads to a pseudo-bound state. It is also shown that a planar configuration of coupled oscillators can have multiple disconnected regions of unbroken PT symmetry., Comment: 18 pages, 9 figures

We report the first demonstration of PT-symmetry and its breaking in a system of two directly-coupled on-chip whispering-gallery-mode microresonators[1]. We observed strong nonreciprocity due to strong-field localization and enhanced nonlinearity in the broken PT-symmetry phase.

The inspiration for this theoretical paper comes from recent experiments on a PT-symmetric system of two coupled optical whispering galleries (optical resonators). The optical system can be modeled as a pair of coupled linear oscillators, one with gain and the other with loss. If the coupled oscillators have a balanced loss and gain, the system is described by a Hamiltonian and the energy is conserved. This theoretical model exhibits two PT transitions depending on the size of the coupling parameter \epsilon. For small \epsilon the PT symmetry is broken and the system is not in equilibrium, but when \epsilon becomes sufficiently large, the system undergoes a transition to an equilibrium phase in which the PT symmetry is unbroken. For very large \epsilon the system undergoes a second transition and is no longer in equilibrium. The classical and the quantized versions of the system exhibit transitions at exactly the same values of \epsilon., Comment: 9 pages, 14 figures

The study of PT-symmetric physical systems began in 1998 as a complex generalization of conventional quantum mechanics, but beginning in 2007 experiments began to be published in which the predicted PT phase transition was clearly observed in classical rather than in quantum-mechanical systems. This paper examines the classical PT phase transition in dynamical-system models that are moderately accurate representations of antigen-antibody systems. A surprising conclusion that can be drawn from these models is that it might be possible treat a serious disease in which the antigen concentration grows out of bounds (and the host dies) by injecting a small dose of a second ( different) antigen. In this case a PT-symmetric analysis shows there are two possible favorable outcomes. In the unbroken-PT-symmetric phase the disease becomes chronic and is no longer lethal, while in the appropriate broken-PT-symmetric phase the concentration of lethal antigen goes to zero and the disease is completely cured.

In a remarkable paper Chandrasekar et al. showed that the (second-order constant-coefficient) classical equation of motion for a damped harmonic oscillator can be derived from a Hamiltonian having one degree of freedom. This paper gives a simple derivation of their result and generalizes it to the case of an nth-order constant-coefficient differential equation.

We discuss conditions giving rise to stationary position-momentum correlations among quantum states in the Fock and coherent basis associated with the natural invariant for the one-dimensional time-dependent quadratic Hamiltonian operators such as the Kanai-Caldirola Hamiltonian. We also discuss some basic features such as quantum decoherence of the wave functions resulting from the corresponding quantum dynamics of these systems that exhibit no timedependence in their quantum correlations. In particular, steady statistical momentum averages are seen over well-defined time intervals in the evolution of a linear superposition of the basis states of modified exponentially damped mass systems.

We consider time-dependent quadratic Hamiltonians in the supersymmetric quantum mechanical framework. We study the quantum mechanical properties of some basic states in the fermionic sector of the Hilbert space. To understand the differences between the bosonic and fermionic sectors better, we focus on periodic quadratic Hamiltonians of the Paul-trap type.

In 1980 Englert examined the classic problem of the electromagnetic self-force on an oscillating charged particle. His approach, which was based on an earlier idea of Bateman, was to introduce a time-reversed (charge-conjugate) particle and to show that the two-particle system is Hamiltonian. Unfortunately, Englert?s model did not solve the problem of runaway modes, and the corresponding quantum theory had ghost states. It is shown here that Englert?s Hamiltonian is symmetric, and that the problems with his model arise because the symmetry is broken at both the classical and the quantum level. However, by allowing the charged particles to interact and by adjusting the coupling parameters to put the model into an unbroken -symmetric region, one eliminates the classical nonrelativistic runaway modes and obtains a corresponding nonrelativistic quantum system that is in equilibrium and ghost free.

We show that a pair of coupled nonlinear oscillators, of which one oscillator has positive and the other one negative damping of equal rate, can form a Hamiltonian system. Small-amplitude oscillations in this system are governed by a -symmetric nonlinear Schrodinger dimer with linear and cubic coupling. The dimer also represents a Hamiltonian system and is found to be exactly solvable in elementary functions. We show that the nonlinearity softens the -symmetry breaking transition in the nonlinearly-coupled dimer: stable periodic and quasiperiodic states with large enough amplitudes persist for an arbitrarily large value of the gain–loss coefficient.

The operator in -symmetric quantum mechanics satisfies a system of three simultaneous algebraic operator equations, , , and . These equations are difficult to solve exactly, so perturbative methods have been used in the past to calculate . The usual approach has been to express the Hamiltonian as H = H0 + ϵH1, and to seek a solution for in the form , where Q = Q(q, p) is odd in the momentum p, even in the coordinate q, and has a perturbation expansion of the form Q = ϵQ1 + ϵ3Q3 + ϵ5Q5 + ⋅⋅⋅. (In previous work it has always been assumed that the coefficients of even powers of ϵ in this expansion would be absent because their presence would violate the condition that Q(p, q) is odd in p.) In an earlier paper it was argued that the operator is not unique because the perturbation coefficient Q1 is nonunique. Here, the nonuniqueness of is demonstrated at a more fundamental level: it is shown that the perturbation expansion for Q actually has the more general form Q = Q0 + ϵQ1 + ϵ2Q2 + ⋅⋅⋅ in which all powers and not just odd powers of ϵ appear. For the case in which H0 is the harmonic-oscillator Hamiltonian, Q0 is calculated exactly and in closed form and it is shown explicitly to be nonunique. The results are verified by using powerful summation procedures based on analytic continuation. It is also shown how to calculate the higher coefficients in the perturbation series for Q.

In quantum mechanics the time operator $\Theta$ satisfies the commutation relation $[\Theta,H]=i$, and thus it may be thought of as being canonically conjugate to the Hamiltonian $H$. The time operator associated with a given Hamiltonian $H$ is not unique because one can replace $\Theta$ by $\Theta+ \Theta_{\rm hom}$, where $\Theta_{\rm hom}$ satisfies the homogeneous condition $[\Theta_{\rm hom},H]=0$. To study this nonuniqueness the matrix elements of $\Theta$ for the harmonic-oscillator Hamiltonian are calculated in the eigenstate basis. This calculation requires the summation of divergent series, and the summation is accomplished by using zeta-summation techniques. It is shown that by including appropriate homogeneous contributions, the matrix elements of $\Theta$ simplify dramatically. However, it is still not clear whether there is an optimally simple representation of the time operator., Comment: 13 pages, 3 figures