We investigate the spectral and eigenstate properties of the quantum superexponential oscillator. Our focus is on the quantum signatures of the recently observed transition of the energy dependent period of the corresponding classical superexponential oscillator. We show that the ground state exhibits a remarkable metamorphosis of decentering, asymmetrical squeezing and the development of a tail. Analyzing the central moments up to high order a characteristic transition from exponentially decaying moments to increasing moments is unraveled. A corresponding spectral analysis shows that, surprisingly, to a good approximation the spectrum is equidistant. A closer look, however, reveals a spectral scaling behaviour below the transition point which is replaced by irregular oscillations above the transition energy. Excited bound states are analyzed up to the continuum threshold. We discuss future perspectives and possible experimental realizations of the superexponential oscillator.
We explore the spectral properties and behaviour of confining superexponential potentials. Several prototypes of these highly nonlinear potentials are analyzed in terms of the eigenvalues and eigenstates of the underlying stationary Schrödinger equation up to several hundreds of excited states. A generalization of the superexponential self-interacting oscillator shows a scaling behaviour of the spacing of the eigenvalues which turns into an alternating behaviour for the power law modified oscillator. Superexponential potentials with an oscillating power show a very rich spectral structure with…
We show that the Schrödinger equation for the quantum harmonic oscillator can be derived as an approximation to the Newtonian mechanics of a classical harmonic oscillator subject to a random force for time intervals $O( m / \hbar)$, when $\hbar / m \ll 1$. Conversely, every solution to the Schrödinger equation, including all the superposition states, arises this way. In other words, the quantum harmonic oscillator is approximately nothing but the classical harmonic oscillator, with the same mass and frequency as the quantum harmonic oscillator, subject to a random force. We generalize the resu…
The paper shows mechanisms of both the pumping and energy decay of an "isolated" oscillator. The oscillator is only non-resonantly coupled with the adjacent oscillator which resonantly interacts with the thermal bath environment. Under these conditions the "isolated" oscillator begins interacting with the thermal bath environment of the adjacent oscillator. The conclusion is based on the kinetic equation derived relative to anti-rotating terms of the initial Hamiltonian, with the latter being the Hamiltonian of two oscillators and environment of one of them.
States of nonlinear quantum oscillators (f-oscillators) are considered in the Weyl-Wigner-Moyal representation and the tomographic probability representation, where the states are described by standard probability distributions instead of wave functions or density matrices. The evolving integrals of motion for classical and quantum f-oscillators are found and the solution for the Liouville equation associated with the probability distribution on the phase space for this oscillator is obtained along with the solution of Moyal equation for quantum f-oscillator, which provide the solutions for pa…
In this paper, we study the quantum states generated from two and three linearly interacting quantum harmonic oscillators. We consider the possibility that one of the oscillators be under the influence of a classical external source and obtain the total and reduced density matrices related to the system. We show how the problem can be generalized to n linearly interacting oscillators straightforwardly.
This paper presents an application of white noise functional approach to derive the quantum propagator and the evolution of the reduced density matrix for an open quantum system consisting of coupled harmonic oscillators which are coupled to a bath consisting of a multimode harmonic oscillator. It is shown that the full quantum propagator is a product of three individual propagators. These propagators were obtained after two successive transformations of the coordinates for the system, and the coordinates for the coupling between the bath and one of the two harmonic oscillators in the system. …
We analyse the properties of a strongly-damped quantum harmonic oscillator by means of an exact diagonalisation of the full Hamiltonian, including both the oscillator and the reservoir degrees of freedom to which it is coupled. Many of the properties of the oscillator, including its steady-state properties and entanglement with the reservoir can be understood and quantified in terms of a simple probability density, which we may associate with the ground-state frequency spectrum of the oscillator.
Experimental investigation of the nonlinear dynamics of a quantum oscillator is a long standing goal of quantum physics. We propose a conditional method for inducing an arbitrary nonlinear potential on a quantum oscillator weakly interacting with light. Such an arbitrary nonlinear potential can be implemented by sequential repetition of an elementary conditional $X$-gate. To implement the $X$-gate, a single photon is linearly coupled to the oscillator and is subsequently detected by optical homodyne detection.
A network of driven nonlinear oscillators without dissipation has recently been proposed for solving combinatorial optimization problems via quantum adiabatic evolution through its bifurcation point. Here we investigate the behavior of the quantum bifurcation machine in the presence of dissipation. Our numerical study suggests that the output probability distribution of the dissipative quantum bifurcation machine is Boltzmann-like, where the energy in the Boltzmann distribution corresponds to the cost function of the optimization problem. We explain the Boltzmann distribution by generalizing t…
A particle bounded in a potential with finite range is described by using an $f$-deformed quantum oscillator approach. Finite range of this potential can be considered as a controllable deformation parameter. The non-classical quantum statistical properties of this deformed oscillator can be manipulated by nonlinearities associated to the finite range.
The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a dam…
An exact invariant operator of time-dependent coupled oscillators is derived using the Liouville-von Neumann equation. The unitary relation between this invariant and the invariant of two uncoupled simple harmonic oscillators is represented. If we consider the fact that quantum solutions of the simple harmonic oscillator is well-known, this unitary relation is very useful in clarifying quantum characteristics of the original systems, such as entanglement, probability densities, fluctuations of the canonical variables, and decoherence. We can identify such quantum characteristics by inversely t…
Glauber coherent states of quantum systems are reviewed. We construct the tomographic probability distributions of the oscillator states. The possibility to describe quantum states by tomographic probability distributions (tomograms) is presented on an example of coherent states of parametric oscillator. The integrals of motion linear in the position and momentum are used to explicitly obtain the tomogram evolution expressed in terms of trajectories of classical parametric oscillator
In this paper we study some basic quantum confinement effects through investigation of a deformed harmonic oscillator algebra. We show that spatial confinement effects on a quantum harmonic oscillator can be represented by a deformation function within the framework of nonlinear coherent states theory. We construct the coherent states associated with the spatially confined quantum harmonic oscillator in a one-dimensional infinite well and examine some of their quantum statistical properties, including sub-poissonian statistics and quadrature squeezing.
Quantum control of engineered mechanical oscillators can be achieved by coupling the oscillator to an auxiliary degree of freedom, provided that the coherent rate of energy exchange exceeds the decoherence rate of each of the two sub-systems. We achieve such quantum-coherent coupling between the mechanical and optical modes of a micro-optomechanical system. Simultaneously, the mechanical oscillator is cooled to an average occupancy of n = 1.7 \pm 0.1 motional quanta. Pulsed optical excitation reveals the exchange of energy between the optical light field and the micromechanical oscillator in t…
The classical self-oscillations can collapse merely due to their mutual couplings. We investigate this oscillation collapse in quantum van der Pol oscillators. For a pair of quantum oscillators, the steady-state mean phonon number is shown to be lower than in the corresponding classical model with a Gaussian white noise that mimics quantum noise. We further show within the mean-field theory that a number of globally coupled oscillators undergo a transition from the synchronized periodic motion to the collective oscillation collapse. A quantum many-body simulation suggests that the increase in …
We recycle Cruz et al.'s (Phys. Lett. A 369 (2007) 400) work on the classical and quantum position-dependent mass (PDM) oscillators. To elaborate on the ordering ambiguity, we properly amend some of the results reported in their work and discuss the classical and quantum mechanical correspondence for the PDM harmonic oscillators. We use a point canonical transformation and show that one unique quantum PDM oscillator Hamiltonian (consequently, one unique ordering-ambiguity parametric set j=l=-1/4 and k=-1/2) is obtained. To show that such a parametric set is not just a manifestation of the quan…
We analyze an optomechanical system that can be used to efficiently transfer a quantum state between an optical cavity and a distant mechanical oscillator coupled to a second optical cavity. We show that for a moderate mechanical Q-factor it is possible to achieve a transfer efficiency of $99.4\%$ by using adjustable cavity damping rates and destructive interference. We also show that the quantum mechanical oscillator can be used as a quantum memory device with an efficiency of $96\%$ employing a pulsed optomechanical coupling. Although the mechanical dissipation slightly decreases the efficie…
This work is mainly based on some theoretical surveys on two dimensional quantum gravitational well, considering harmonic oscillator potential causes an effective plank constant. We find that there is a similarity between two different cases, potential free and oscillator potential in noncommutative phase space in commutative momentum, but in noncommutative, non zero space and momentum The Hamiltonian would a little bit different in energy levels.
Quantum entanglement has been actively sought for in optomechanical and electromechanical systems. The simplest such system is a mechanical oscillator interacting with a coherent beam, while the oscillator also suffers from thermal decoherence. For this system, we show that quantum entanglement is always present between the oscillator and continuous outgoing fields -- even when the environmental temperature is high and the oscillator is highly classical. Such universal entanglement is also shown to be able to survive more than one oscillation cycle if characteristic frequency of the optomechan…
Although people have already artificially formed parity--time ($\mathcal{PT}$) symmetry with gain and loss in a balanced manner, it is still a defect that the gain is restricted to semi--classical but not full quantum. Here we propose and analyze a theoretical scheme to realize full quantum oscillator $\mathcal{PT}$--symmetry. The quantum gain is provided by a dissipation optical cavity with blue detuned laser field. After adiabatically eliminating the cavity modes, we give an effective master equation, which is a more complete quantum description compared with non--Hermitian Hamiltonian, to r…
Uncertainties $(Δx)^2$ and $(Δp)^2$ are analytically derived in an $N$-coupled harmonic oscillator system when spring and coupling constants are arbitrarily time-dependent and each oscillator is in an arbitrary excited state. When $N = 2$, those uncertainties are shown as just arithmetic average of uncertainties of two single harmonic oscillators. We call this property as "sum rule of quantum uncertainty". However, this arithmetic average property is not generally maintained when $N \geq 3$, but it is recovered in $N$-coupled oscillator systems if and only if $(N-1)$ quantum numbers are equal.…
We use the Fourier operator to transform a time dependent mass quantum harmonic oscillator into a frequency dependent one. Then we use Lewis-Ermakov invariants to solve the Schrödinger equation by using squeeze operators. Finally we give two examples of time dependencies: quadratically and hyperbolically growing masses.
The paper deals with the problem of dynamics of externally driven open quantum systems. Using the path integral methods we found an analytical expression for time-dependent density matrix of two externally driven coupled quantum oscillators interacting with different baths of oscillators. It is shown that at the zeroing of external forces the density matrix becomes identical to the previously obtained one for freely developing coupled oscillators. Mean values of observables are computed by using the Hermitian part of the matrix. All elements of the covariance matrix composed by coordinates and…
The dynamics of nonlinear systems qualitatively change depending on their parameters, which is called bifurcation. A quantum-mechanical nonlinear oscillator can yield a quantum superposition of two oscillation states, known as a Schrödinger cat state, via quantum adiabatic evolution through its bifurcation point. Here we propose a quantum computer comprising such quantum nonlinear oscillators, instead of quantum bits, to solve hard combinatorial optimization problems. The nonlinear oscillator network finds optimal solutions via quantum adiabatic evolution, where nonlinear terms are increased s…
We study two-level q-deformed angular momentum states and us- ing q-deformed harmonic oscillators, we provide a framework for con- structing qubits and quantum gates. We also present the construction of some basic quantum gates including CNOT, SWAP, Toffoli and Fredkin.
We propose the simple quantum model of nonlinear autonomous oscillator in hard excitation regime. We originate from classical equations of motion for similar oscillator and quantize them using the Lindblad master equation for the density matrix of this system. The solution for the populations of the stationary states of such oscillator may be explicitly found in the case when nonlinearity parameters of the problem are small. It was shown that in this situation there are three distinct regimes of behavior of the model. We compare properties of this model with corresponding ones of close open sy…
In this paper, we study the dynamical properties of two coupled quantum harmonic oscillators coupled with bosonic non-Markovian environment both in position and momentum. We deduce the exact analytical master equation using Quantum State Diffusion method and give the quantum trajectory description when the control is added to the system by applying interaction between two harmonic oscillators. With numerical simulation, we compare the evolution of entanglement under different controlling effects. At last, we use nonlinear Quantum State Diffusion method to strengthen our above results by gettin…
We propose a new witness operation for the non-classical character of a harmonic oscillator state. The method does not require state reconstruction. For all harmonic oscillator states that are classical, a bound is established for the evolution of a qubit which is coupled to the oscillator. Any violation of the bound can be rigorously attributed to the non-classical character of the initial oscillator state.
