The dynamics of time-dependent coupled oscillator model for the charged particle motion subjected to a time-dependent external magnetic field is investigated. We used canonical transformation approach for the classical treatment of the system, whereas unitary transformation approach is used when managing the system in the framework of quantum mechanics. For both approaches, the original system is transformed to a much more simple system that is the sum of two independent harmonic oscillators which have time-dependent frequencies. We therefore easily identified the wave functions in the transformed system with the help of invariant operator of the system. The full wave functions in the original system is derived from the inverse unitary transformation of the wave functions associated to the transformed system.
An unusual type of the exact solvability is reported. It is exemplified by the Coulomb plus harmonic oscillator in D dimensions after a complexification of its Hamiltonian which keeps the energies real. Infinitely many bound states are found in closed form which generalizes the even-parity harmonic-oscillator states at zero charge. Apparently, the model is halfway between exact and quasi-exact.
The master equation for a charged harmonic oscillator coupled to an electromagnetic reservoir is investigated up to fourth-order in the interaction strength by using Krylov averaging method. The interaction is in the velocity-coupling form and includes a diamagnetic term. Exact analytical expressions for the second, the third, and the fourth-order contributions to mass renormalization, decay constant, normal and anomalous diffusion coefficients are obtained for the blackbody type environment. It is found that, generally, the third and the fourth order contributions have opposite sign when thei…
A general treatment of the quantal time-dependent coupled oscillators in presence of the variable magnetic field is presented. The treatment is based on the use of an alternative canonical transformations, time-dependent unitary transformations and the invariant methods. Exact wave functions for Schrödinger equations of this system are constructed.We applied our theory to a particular case and, co,sequently, showed that our results recovers to the perviously known one.
In this study, we investigate the effects of noncommutative Quantum Mechanics in three dimensions on the energy levels of a charged isotropic harmonic oscillator in the presence of a uniform magnetic field in the z-direction. The extension of this problem to three dimensions proves to be non-trivial. We obtain the first-order corrections to the energy-levels in closed form in the low energy limit of weak noncommutativity. The most important result we can note is that all energy corrections due to noncommutativity are negative and their magnitude increase with increasing Quantum numbers and mag…
In some quantum chemical applications, the potential models are linear combination of single exactly solvable potentials. This is the case equivalent of the Stark effect for a charged harmonic oscillator (HO) in a uniform electric field of specific strength (HO in an external dipole field). We obtain the exact s-wave solutions of the Dirac equation for some potential models which are linear combination of single exactly solvable potentials (ESPs). In the framework of the spin and pseudospin symmetric concept, we calculate the analytic energy spectrum and the corresponding two-component upper- …
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.
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 system of two $Q$-deformed oscillators coupled so that the total Hamiltonian has the su$_Q$(2) symmetry is proved to be equivalent, to lowest order approximation, to a system of two identical Morse oscillators coupled by the cross-anharmonicity usually used empirically in describing vibrational spectra of triatomic molecules. The symmetry also imposes a connection between the self-anharmonicity of the Morse oscillators and the cross-anharmonicity strength, which can be removed by replacing the $Q$-oscillators by deformed anharmonic oscillators. The generalization to $n$ oscillators is stra…
We present a control-theoretic analysis of the system consisting of a two-level atom coupled with a quantum harmonic oscillator. We show that by applying external fields with just two resonant frequencies, any desired unitary operator can be generated.
We discuss the diagonalization of a general Hamiltonian operator for a set of coupled harmonic oscillators and determine the conditions for the existence of bound states. We consider the particular cases of two and three oscillators studied previously and show the conditions for bound states in the latter example that have been omitted in an earlier treatment of this model.
We consider a set of N linearly coupled harmonic oscillators and show that the diagonalization of this problem can be put in geometrical terms. The matrix techniques developed here allowed for solutions in both the classical and quantum regimes.
We investigate how to model Markovian evolution of coupled harmonic oscillators, each of them interacting with a local environment. When the coupling between the oscillators is weak, dissipation may be modeled using local Lindblad terms for each of the oscillators in the master equation, as is commonly done. When the coupling between oscillators is strong, this model may become invalid. We derive a master equation for two coupled harmonic oscillators which are subject to individual heat baths modeled by a collection of harmonic oscillators, and show that this master equation in general contain…
We show that, by using the quantum orthogonal functions invariant, we are able to solve a coupled of time dependent harmonic oscillators where all the time dependent frequencies are arbitrary. We do so, by transforming the time dependent Hamiltonian of the interaction by a set of unitary operators. In passing, we show that $N$ time dependent and coupled oscillators have a generalized orthogonal functions invariant from which we can write a Ermakov-Lewis invariant.
We study a single two-level system coupled resonantly to an oscillator mode or a large spin. By adiabatically turning on a linear driving term on the oscillator or the spin, the eigenstates of the system change character and its ground state evolves into squeezed states of the oscillator or the spin. The robust generation of such states is of interest in many experimental systems with applications for sensing and quantum information processing.
Applying the Milburn equation to describe intrinsic decoherence, we study the interaction of three-coupled quantum harmonic oscillators or quantized fields. We give an explicit solution for the complete equation, i.e., beyond the usual second order approximation used to arrive to the Lindblad form. Then we calculate the expectation value of the number operator of each oscillator or mode for one of the modes given in an initial coherent
We have applied a collocation approach to obtain the numerical solution to the stationary Schrödinger equation for systems of coupled oscillators. The dependence of the discretized Hamiltonian on scale and angle parameters is exploited to obtain optimal convergence to the exact results. A careful comparison with results taken from the literature is performed, showing the advantages of the present approach.
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.…
For $N$-coupled generalized time-dependent oscillators, primary invariants and a generalized invariant are found in terms of classical solutions. Exact quantum motions satisfying the Heisenberg equation of motion are also found. For number states and coherent states of the generalized invariant, the uncertainties in positions and momenta are obtained.
The propagator for a certain class of two time-dependent coupled and driven harmonic oscillators with time-varying angular frequencies and masses is evaluated by path integration. This is simply done through suitably chosen generalized canonical transformations and without presupposing the knownledge of any auxiliary equation. The time-dependent oscillators system with an exponentially growing masses and coupling coefficient in time may be considered as particular case.
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. …
Macedo and Guedes showed recently how to solve a system of coupled harmonic oscillators with time dependent parameters [{ J. Math. Phys.} {\bf 53}, 052101 (2012)]. We show here that the way in which they get rid of the time dependent masses is incorrect and some terms lack in the transformed Hamiltonian. We also show a correct way of eliminating from the Hamiltonian the time dependent masses.
The purpose of this paper is the discussion of a pair of coupled linear oscillators that has recently been proposed as a model of a system of two optical resonators. By means of an algebraic approach we show that the frequencies of the classical and quantum-mechanical interpretations of the optical phenomenon are exactly the same. Consequently, if the classical frequencies are real, then the quantum-mechanical eigenvalues are also real.
For a system of coupled anharmonic oscillators we compare the convergence rate of the variational collocation approach presented recently by Amore and Fernandez (2010 Phys.Scr.81 045011) with the one obtained using the optimized Rayleigh-Ritz (RR) method. The monotonic convergence of the RR method allows us to obtain more accurate results at a lower computational cost.
We comment on several incorrect results given in a recent paper by Lo and Wong. In particular, it is pointed out that their evaluation of the propagator for two coupled general driven time-dependent oscillators is not satisfactory. The correct expression can be obtained by applying an appropriate time-dependent canonical transformation.
We describe the construction and characterisation of a nano-oscillator formed by a Paul trap. The frequency and temperature stability of the nano-oscillator was measured over several days allowing us to identify the major sources of trap and environmental fluctuations. We measure an overall frequency stability of 2 ppm/hr and a temperature stability of more than 5 hours via the Allan deviation. Importantly, we find that the charge on the nanoscillator is stable over a timescale of at least two weeks and that the mass of the oscillator, can be measured with a 3 % uncertainty. This allows us to …
We explore the possibility of engineering quantum states of a charged mechanical oscillator by coupling it to a stream of atoms in superpositions of high-lying Rydberg states. Our scheme relies on the driving of a two-phonon resonance within the oscillator by coupling it to an atomic two-photon transition. This approach effectuates a controllable open system dynamics on the oscillator that permits the dissipative creation of squeezed and other non-classical states which are central to applications such as sensing and metrology or for studies of fundamental questions concerning the boundary bet…
In this study, we investigate the effects of noncommutative Quantum Mechanics in three dimensions on the energy levels of a charged isotropic harmonic oscillator in the presence of a uniform magnetic field in the z-direction. The extension of this problem to three dimensions proves to be non-trivial. We obtain the first-order corrections to the energy-levels in closed form in the low energy limit of weak noncommutativity. The most important result we can note is that all energy corrections due to noncommutativity are negative and their magnitude increase with increasing Quantum numbers and mag…
In this work, we expose the role of environment, confinement and external magnetic field ($B$) in determining the low temperature thermodynamic behaviour in the context of cyclotron motion of a charged oscillator with anomalous dissipative coupling involving the momentum instead of the much studied coordinate coupling. Explicit expressions for different quantum thermodynamic functions (QTF) are obtained at low temperatures for different quantum heat bath characterized by spectral density function, $μ(ω)$. The power law fall of different QTF are in conformity with third law of thermodynamics. B…
The dynamics of a qubit coupled with a quantum oscillator is re-studied in the region of strong coupling. The non-degenerate perturbation is added to the usual degenerate one and new results are given.
In the optomechanical cooling of a dispersively coupled oscillator, it is only possible to reach the oscillator ground state in the resolved sideband regime, where the cavity-mode line width is smaller than the resonant frequency of the mechanical oscillator being cooled. In this paper, we show that the dispersively coupled system can be cooled to the ground state in the unresolved sideband regime using an ancillary oscillator, which is coupled to the same optical mode via dissipative interaction. The ancillary oscillator has a resonant frequency close to that of the target oscillator; thus, t…
We illustrate the geometric phase associated with the cyclic dynamics of a classical system of coupled oscillators. We use an analogy between a classical coupled oscillator and a two-state quantum mechanical system to represent the evolution of the oscillator on an equivalent Hilbert space, which may be represented as a trajectory on the surface of a sphere. The cyclic evolution of the system leads to a change in phase, which consists of a dynamic phase along with an additional phase shift dependent on the geometry of the evolution. A simple experiment suitable for advanced undergraduate stude…
System of the quantum Langevin equations for two quantum coupling oscillators within independent heat baths of quantum oscillators are obtained using a model Hamiltonian and corresponding Heisenberg equations of motion. Expressions for mean energy of coupled oscillators and their mean energy of interaction are derived and analyzed. Nonmonotonic dependence of the interaction energy versus a coupling constant is demonstrated and explained. Nontrivial dependence of the quantities as a consequence of the difference in temperatures of heat baths is shown.
We calculate the reduced density matrix for a system of coupled harmonic oscillators in a bosonic heat bath using the Born-Markov approximation and show that the expectation values of position and momentum evolve like classical quantities. We consider the cases where every oscillator is coupled to its own reservoir and where all oscillators are coupled to the same heat bath, pointing out the differences between these two models. In particular, for systems coupled to a common reservoir, we present an uncommon dissipation mechanism, which disappears, when the dynamics of the system are gouverned…
This work addresses the problem of relaxation of open systems to quasi-equilibrium states. Time-dependent density matrix of two arbitrary coupled quantum oscillators of arbitrary properties interacting with separate reservoirs is derived based on path integration. Temporal dynamics of spatial variances and covariances of the oscillators from any given time up to quasi-equilibrium steady states is studied. It is demonstrated for general case that asymptotic spatial variances of two arbitrary oscillators and their covariances achieve stationary values in the long-time limit. A comparison of stea…
Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two-by-two matrices. If two oscillators are coupled, the problem combines both two-by-two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. Couple…
Optically coupled optomechanical oscillators has turned out to be a versatile experimental resource for exploring optomechanical synchronizations and correlations. In this work, we investigate the phenomena of quantum synchronization and quantum correlations in two optically coupled optomechanical oscillators with two different topologies. In one case the oscillators are coupled with optical photons in a reversible manner, termed as bidirectional coupling, while in the other photons are allowed to enter to the other oscillator but not allowed to exchanged back in the opposite direction, termed…
The Hamiltonian for a PT-symmetric chain of coupled oscillators is constructed. It is shown that if the loss-gain parameter $γ$ is uniform for all oscillators, then as the number of oscillators increases, the region of unbroken PT-symmetry disappears entirely. However, if $γ$ 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 p…
We theoretically describe the weak measurement of a two-level system (qubit) and quantify the degree to which such a qubit measurement has a quantum non-demolition (QND) character. The qubit is coupled to a harmonic oscillator which undergoes a projective measurement. Information on the qubit state is extracted from the oscillator measurement outcomes, and the QND character of the measurement is inferred by the result of subsequent measurements of the oscillator. We use the positive operator valued measure (POVM) formalism to describe the qubit measurement. Two mechanisms lead to deviations fr…
In this paper it is studied the influence of a minimal thermal environment on the dynamics of a quantum harmonic oscillator (labelled A), prepared in a coherent state. The environment itself consists of a second oscillator (labelled B), initially in a thermal state. Two types of interaction Hamiltonians are considered, and the time-evolution of the reduced density operator of oscillator A is compared to the one obtained from the usual master equation approach, i.e., assuming that oscillator A is coupled to a large reservoir. An analysis of the linear entropy evolution of oscillator A shows tha…
We investigate the quantum entanglement of systems of coupled harmonic oscillators on the basis of thermo-field dynamics (TFD). For coupled harmonic oscillators at equilibrium, the extended entanglement entropy is derived using the TFD method, and it is demonstrated to be controlled by temperature and coupling parameters. For non-equilibrium systems, in addition to temperature and coupling parameters, the time dependence of the extended entanglement entropy is calculated in accordance with the dissipative von Neumann equation, and the dissipative dynamics of the systems of coupled harmonic osc…
