Number Of Harmonic Oscillators Research Articles

APPROACH TO EQUILIBRIUM IN THE CALDEIRA–LEGGETT MODEL

The Caldeira–Leggett (CL) model describes a microscopic quantum system, represented by a harmonic oscillator, in interaction with a heat bath, represented by a large number of harmonic oscillators with a range of frequencies. We consider the case when the system oscillator starts out in the ground state and then thermalizes due to interactions with the heat bath, which is at temperature θ. We calculate the position autocorrelation function 〈x(t')x(t)〉 of the system oscillator at two different times and study its behavior in the small and large time limits. Our results show that the system oscillator thermalizes as expected. We also confirm by explicit calculation that the position autocorrelation function exhibits periodicity for imaginary values of the time difference t'-t = iτ at late-(real) times t.

A quantum coupler and the harmonic oscillator interacting with a reservoir: Defining the relative phase gate

In order to be able to study dissipation, the interaction between a single system and their environment was introduced in quantum mechanics. Master and quantum Langeving equations was derived. Moreover, decoherence has been studied using this approach. One of the most commonly used model in this research field is a single harmonic oscillator interacting with a reservoir. In this work we analytically solve this model, in the resonance case, using the evolution operator method. Then, we use this result to study a quantum coupler, which is a particular case where there are a finite number of harmonic oscillators interacting with the single one. After that, we focuses in the analisis of the conditional quantum dynamics of the quantum coupler (using the computational basis of two qubits) by choosing a proper interaction time. This conditional dynamics provides a realization of a new quantum phase gate, which we call the relative phase gate. To the best of our knowledge, this phase gate has not been previously defined or proposed.

A Sequence of Finite-Dimensional Controllers for Repetitive Control of Positive Real Systems

In this paper, we analyze the performance of positive real plants coupled with a progressively increasing number of harmonic oscillators. We prove that, as the number of oscillators tends to infinity thus covering the all frequency range, the steady-state output weakly converges to the periodic signal to be tracked. On the basis of this result, we next show that, under a suitable constraint on the signal to be tracked, strong convergence is actually achieved.

Two-centre harmonic oscillator integrals

Two-centre integrals involving displaced and distorted (or squeezed) simple harmonic oscillator number states, the DSN states, are often encountered in the molecular spectroscopy and in the quantum theory of wave packets and light. For example, the Franck–Condon factors are overlap integrals of the DSN states. In this work, finite, closed and algebraic formulas are derived for the overlap, exponential, trigonometric and polynomial integrals.

Discrete structures in fusion-barrier distributions for vibrational nuclei

We obtain a closed-form expression for the distribution of fusion barriers for vibrational nuclei using a generalization of Dasso, Landowne, and Winther's model, which represents the nuclear surface vibrations as a number of harmonic oscillators, and allows the excitation of an arbitrary number of phonons in the target and/or projectile. We find that this expression is in reasonable agreement with the average trends of the empirical distributions for the fusion of 16O with 92Zr, 144Sm and 208Pb, but fails to reproduce the double peaking of the distribution for the 144Sm target. Only when we restrict the number of excited phonons to a limited number, we are able to reproduce such discrete structures. We show that limiting the number of coupled channels, particularly in the case of strong coupling, increases the spacings between the channel eigenvalues that determine the positions of the peaks of the barrier distribution and modifies their heights.

Quantum twinkling: statistics, coherence and the high order moments of probability amplitudes for systems coupled to quantum baths

We generalize the path integral influence functional technique to the study of a quantum system with interferences between different histories of paths, in contact with a quantum bath. We examine high moments of squared wavefunctions and find that “quantum twinkling”, or the existence of nontrivial moments of the subsystem density matrix, depends strongly on interference between the paths and can not be completely characterized by the usual influence functional which treats environments as effective average bath through spectral density alone. One striking result is that the statistics depend on the number of harmonic oscillators in the bath. Thus for higher moments, one faces directly the question of the exact nature of the bath when using an effective harmonic description. Possible connections of our results to single molecule experiments are discussed.

Quantum Dissipation in Open Harmonic Systems: --Operator Solution and Application to Decay Process--

A finite number of harmonic oscillators coupled to infinitely many environment oscillators is fundamental to the problem of understanding quantum dissipation of a small system immersed in a large environment. Exact operator solution as a function of time is given to this problem, by using diagonalized dynamical variable of the entire system, the small system plus the environment. The decay law of prepared initial configuration is worked out in greatest detail. A clear separation of the exponential- and the power-law decay period is made possible by our method. Behavior of physical quantities at asymptotically late times can be understood in terms of the overlap probability of the system variable with the diagonal variable of the entire system.

Path integral for the relativistic particle and harmonic oscillators

The action for a massive particle in special relativity can be expressed as the invariant proper length between the end points. In principle, one should be able to construct the quantum theory for such a system by the path integral approach using this action. On the other hand, it is well known that the dynamics of a free, relativistic, spinless massive particle is best described by a scalar field which is equivalent to an infinite number of harmonic oscillators. We clarify the connection between these two—apparently dissimilar—approaches by obtaining the Green function for the system of oscillators from that of the relativistic particle. This is achieved through defining the path integral for a relativistic particle rigorously by two separate approaches. This analysis also shows a connection between square root Lagrangians and the system of harmonic oscillators which is likely to be of value in more general context.

Iterative Bogoliubov transformations and anharmonic oscillators.

An algebraic method for studying quantum anharmonic oscillators is proposed. It consists in the iterative application of generalized (squeezing) Bogoliubov transformations to the usual creation and annihilation operators. The convergence of this procedure is analyzed for quartic, cubic plus quartic, and symmetric-double-well Hamiltonians. For these systems, we show the usefulness of the resulting harmonic-oscillator number states in calculating the energy spectra.

The Weyl Quantization of Phase Angle

We suggest a candidate for a non-canonical Hermitian operator corresponding to phase angle, based on the Weyl correspondence rule. The matrix elements of this operator, for harmonic oscillator number states, coincide, in the correspondence limit, with those of a phase operator proposed by Barnett and Pegg, when the dimension of their defining space becomes infinite.

Heisenberg's equations of motion for dissipative tunneling.

The quantum-mechanical problem of tunneling of a particle coupled to a heat bath that consists of a large number of harmonic oscillators is studied in this paper. Here the equation of motion for the particle which is a nonlinear integro-differential equation is found by eliminating the degrees of freedom of the oscillators. The nonlinear operator equation may be solved by noting that the position operator at the time t can be expanded as a power series in terms of the initial position and momentum with time-dependent c-number coefficients. These coefficients also satisfy nonlinear integro-differential equations. For the problem of dissipative tunneling of a particle in a double-well potential, the first few terms of expansion have been calculated numerically. The expansion parameter in this formulation is a small dimensionless parameter which is inversely proportional to the product of the width and the square root of the height of the barrier.

Extended projector method: Application to Jahn-Teller and pseudo-Jahn-Teller systems.

Within the Trotter-Suzuki approximations on e^–βH, the extended projector method previously introduced by us is shown to be a valuable technique in investigating the general class of few-level systems coupled both linearly and quadratically to a selected number of harmonic oscillators. As an illustration we compare eigenvalues and Ham-reduction factors with the literature for the cubic linear Jahn-Teller systems Txe and Txτ2 as well as for the system Exe for which a quadratic warping interaction is included. For the latter system a symmetry projector is additionally developed, dealing with the transformation properties of nonlinear combinations of representational basis functions. As another illustration, the low-energy eigenstates of a two-level system quadratically coupled to a single harmonic oscillator are investigated. Within the ground state, the system becomes unstable at high coupling strengths and through the evaluation of time- and frequency-dependent correlation functions we found substantially different ground-state dynamic properties compared with those of a linearly coupled system.

Method of time-ordered products in polaron theory

The method of time-ordered products is used to investigate the equilibrium thermodynamic properties of the Frolich model in polaron theory. The polaron free energy at finite temperatures is calculated on the basis of Bogolyubov's variational principle. The trial functional is chosen in the most general form corresponding to an arbitrary number of harmonic oscillators interacting with the electron. An upper bound for the polaron ground-state energy is obtained and studied in the case of weak coupling and low temperatures. It is shown that the accuracy of the bound increases with increasing number of oscillators.

Coherent states and the irreversible motion of a particle coupled to a system of oscillators

A many-body system consisting of a central elastically bound particle coupled to a large number of harmonic oscillators is studied. The co-ordinate of this particle satisfies the equation of motion of a damped harmonic oscillator. To get the quantal description of this damped motion, the many-body wave function for the complete system is written in the coherent-state basis,i.e. as a product of a real stationary wave function and a co-ordinate- and time-dependent phase factor. In order to isolate the motion of the central particle, the stationary wave function is approximated by a product of single-particle wave functions. By calculating the contribution of the variable phase factor to the single-particle wave function, it is shown that the resulting equation for the damped motion of an oscillator is the Schrodinger-Langevin equation.

On the Theory of Brownian Motion and Spin Relaxation

The Brownian motion of a harmonic oscillator is treated quantum-mechanically and the same method is applied to the relaxation of a spin. The method is quite general and applicable to a system displaced from the equilibrium state by an arbitrary amount initially. The oscillator or the spin is interacting with a thermostat composed of a great number of harmonic oscillators. The equation of motion is solved in operator form taking average as to the thermostat to the second order of the interaction parameter. The Langevin equation, the Kramers-Chandra-sekhar equation for a harmonic oscillator and the Bloch equation for a static field are derived.

On the Brownian Motion of a Classical Oscillator

The Brownian motion of a classical harmonic oscillator is treated mechanically throughout. The oscillator is coupled with thermostat which is composed of a great number of harmonic oscillators. By use of a transformation which eliminates the coupling to the first order, the second order quantities like the friction force in obtained. Correlation of random force is also discussed.