Caldirola-Kanai Hamiltonian Research Articles

The performance of a dissipative electro-optomechanical system using the Caldirola–Kanai Hamiltonian approach

By combining all optical, electronic, and mechanical capabilities, electro-optomechanical systems, along with the use of mechanical displacement, the conversion of low-frequency electrical signals into much higher-frequency optical signals is possible. This process provides several capabilities that have very important applications in various fields, including classical, as well as quantum communications. In this regard, the inevitable effect of dissipation on the performance indicators of such systems is one of the important issues affecting its efficiency. One approach for investigating the effect of dissipation is based on a time-dependent Hamiltonian has been known as the Caldirola-Kanai (CK) Hamiltonian, wherein the exponentially increasing mass, i.e. enters the mechanical oscillator system. Utilizing this approach shows that, the performance of the system under the influence of dissipation with CK Hamiltonian formalism, in which the dissipation terms are embedded in the Hamiltonian, are more logical, analytical, and reliable than considering the effect of dissipation with the phenomenological method (including the dissipation effects in the equations of motion by hand), which has been recently used in Bagci et al. (Nature, 507, 81, 2014). In our method, it is also possible to monitor the performance of the system via the adjustment of dissipation components, i.e., resistance and reactance (and CK damping parameter ) in the electrical (mechanical membrane) part of the system.

Adiabatic and nonadiabatic evolution of a generalized damped harmonic oscillator

In this work we present a simple and elegant approach to study the adiabatic and nonadiabatic evolution of a generalized damped harmonic oscillator which is described by the generalized Caldirola–Kanai Hamiltonian, in both classical and quantum contexts. Based on time-dependent dynamical invariants, we find that the geometric phase acquired when the damped oscillator evolves adiabatically in time provides a direct connection between the classical Hannay’s angle and the quantum Berry’s phase. In addition, we solve the time-dependent Schrödinger equation for this system and calculate various quantum properties of the damped generalized harmonic one, such as coherent states, expectation values of the position and momentum operators, their quantum fluctuations and the associated uncertainty product.

London superconductor and time-varying mesoscopic LC circuits

In this work we study the classical and quantum dynamics of a London superconductor and of a time-dependent mesoscopic or nanoscale LC circuit by assuming that the inductance and capacitance vary exponentially with time at constant rate. Surprisingly, we find that the behavior of these two systems are equivalent, both classically and quantum mechanically, and can be mapped into a standard damped harmonic oscillator which is described by the Caldirola-Kanai Hamiltonian. With the aid of the dynamical invariant method and Fock states, we solve the time-dependent Schr\"odinger equation associated with this Hamiltonian and calculate some important physical properties of these systems such as expectation values of the charge and magnetic flux, their variances and the respective uncertainty principle.

Relation between a relaxation time and the friction coefficient used in the theory of damped motion

The paper deals with the Caldirola–Kanai Hamiltonian in which the potential energy is defined as a linear function of the position variable. The exact solution of the Schrödinger equation using this Hamiltonian is presented. It is shown that this solution can be utilized in the theory of electric conduction of metals and semiconductors. The main idea of this paper is that if the Caldirola–Kanai Hamiltonian is applied to conduction electrons , it is possible to calculate successfully the drift velocity of the electrons in a constant electric field. The alternative calculation of the drift velocity in the framework of the density-matrix theory which is also presented leads to the same result. A useful lesson is drawn: if the friction coefficient , the parameter defining the Caldirola–Kanai Hamiltonian, concerns the conduction electrons, it is precisely equal to the reciprocal value of the relaxation time used in the kinetic theory of the electron transport in solids.

Quantum manipulation through finite fluctuations for a generalized parametric oscillator using a lie algebra representation

In this paper, we focus on the effects of temporally compact fluctuations on the parameters m(t) (mass), (frequency), and (spatial dilation coefficient), for the generalized Caldirola-Kanai Hamiltonian. We propose an algorithm such that given control over the timing of repeated compact fluctuations, one can produce any time-evolution associated with complete parameter control of the generalized Caldirola-Kanai Hamiltonian. Computational simplicity is achieved in this endeavor using a faithful representation of the Hamiltonian’s associated Lie algebra. Further simplification occurs as a result of using unitarity conditions for the time-evolution operator. We apply our analysis to the effects of Dirac delta fluctuations in mass and frequency, both separately and simultaneously. We also numerically demonstrate control of the generalized Caldirola-Kanai system for the case of timed Gaussian fluctuations in the mass term. Furthermore, we use the faithful Lie algebra representation to show that any unitary evolution of this generalized Caldirola-Kanai system is given by the squeeze operator multiplied with a phase factor. We also show that the classical evolution of a probabilistic Gaussian phase space distribution and the evolution of the Wigner function of a coherent state in the quantum Caldirola-Kanai system are identical.

On the Quantization of the London Superconductor

We present a simple and direct phenomenological quantization scheme for the London superconductor. Using Maxwell and continuity equations, we show that this quantization can be mapped into a quantum damped harmonic oscillator which is described by the Caldirola-Kanai Hamiltonian. With the help of the dynamic invariant method, we solve the time-dependent Schrodinger equation associated with this Hamiltonian and write its solutions in terms of a particular solution of the Milne-Pinney equation. Based on the Fock states representation, we also demonstrate that our quantization model is similar to that of the standard mechanical harmonic oscillator. Finally, we evaluate some important physical quantities of this superconductor such as expectation values of the charge and magnetic flux and their variances and the respective uncertainty principle.

Quantum Light and Coherent States in Conducting Media

We present a simple description of classical and quantum light propagating through homogeneous conducting linear media. With the choice of Coulomb gauge, we demonstrate that this description can be performed in terms of a damped harmonic oscillator which is governed by the Caldirola-Kanai Hamiltonian. By using the dynamical invariant method and the Fock states representation we solve the time-dependent Schrodinger equation associated with this Hamiltonian and write its solutions in terms of a special solution of the Milne-Pinney equation. We also construct coherent states for the quantized light and show that they are equivalent to the well-known squeezed states. Finally, we evaluate some important properties of the quantized light such as expectation values of the amplitude and momentum of each mode, their variances and the respective uncertainty principle.

Quantifying continuous-variable realism

The debate instigated by the seminal works of Einstein, Podolsky, Rosen, and Bell, put the notions of realism and nonlocality at the core of almost all philosophical and physical discussions underlying the foundations of quantum mechanics. However, while experimental criteria and quantifiers are by now well established for nonlocality, there is no clear quantitative measure for the degree of reality associated with continuous variables such as position and momentum. This work aims at filling this gap. Considering position and momentum as effectively discrete observables, we implement an operational notion of projective measurement and, from that, a criterion of reality for theses quantities. Then, we introduce a quantifier for the degree of irreality of a discretized continuous variable which, when applied to the conjugated pair position-momentum, is shown to obey an uncertainty relation, this meaning that quantum mechanics prevents classical realism for conjugated quantities. As an application of our formalism, we study the emergence of elements of reality in an instance where a Gaussian state is submitted to the dissipative dynamics implied by the Caldirola-Kanai Hamiltonian. In particular, at the equilibrium, we make some links with the measurement problem and identify aspects that can be taken as the quantum counterpart for the notion of rest.

Discussing the extension and applications of a variational approach with deformed derivatives

One has recently presented an extension of the standard variational calculus to include the presence of deformed derivatives, both in the Lagrangian of systems of particles and in the Lagrangian density of field-theoretic models. Classical Euler-Lagrange equations and the Hamiltonian formalism have been reassessed in this approach. Whenever applied to a number of physical systems, the resulting dynamical equations come out to be the correct ones found in the literature, especially with mass-dependent and with nonlinear equations for classical and quantum-mechanical systems. In the present contribution, one extends the variational approach, including a piecewise form of deformed derivatives to study higher-order dissipative systems and to obtain, as an option, deformed equations as well. Applications to concrete situations are contemplated, such as an accelerated point charge—this is the problem of the Abraham-Lorentz-Dirac force—stochastic dynamics like the Langevin, the advection-convection-reaction and Fokker-Planck equations, the Korteweg-de Vries equation, the Landau-Lifshitz-Gilbert problem, and the Caldirola-Kanai Hamiltonian and heat transfer equation of the Fourier and non-Fourier types. By considering these different examples, it is shown that the formulation proposed in this paper may be a simple, but promising, path for dealing, for example, with dissipative, nonlinear, stochastic systems and the anomalous heat transfer problem, by adopting a variational approach.

The damped interaction between a single-mode cavity field with Caldirola–Kanai Hamiltonian and a three-level atom

In this paper, we studied the damped interaction between a Λ-type three-level atom and a quantized single-mode cavity field, where the cavity field features a damped harmonic oscillator with a time-dependent mass that is described by the so-called Caldirola–Kanai (CK) Hamiltonian. Assuming that, initially, the atom is prepared in a superposition state and the field is prepared in coherent states, the temporal behavior of the atomic population inversion, von Neumann entropy, fidelity and Husimi function Q have been investigated by numerically solving the master equation.

Damping in the Interaction of a Two-Photon Field and a Two-Level Atom Through Quantized Caldirola-Kanai Hamiltonian

Following the lines of the recent papers (Daneshmand and Tavassoly, Int. J. Theor. Phys. 56, 1218 (2017)), we study quantum mechanical treatments of an interaction between a two-level atom with a single-mode field in the two-photon Jaynes-Cummings model, where the Hamiltonian of the field is considered to be the quantized Caldirola-Kanai (CK) Hamiltonian. As a result, we would expect that the quantum dynamics of the two-photon JCM in terms of the CK Hamiltonian is qualitatively different from that of the usual one-photon case. We analytically calculate the explicit form of the atom-field entangled state and numerically evaluate the dynamics of its physical properties. The degree of entanglement, atomic population as well as sub-Poissonian statistics and quadrature squeezing of the field are analyzed. We adjust the latter evolved parameters by appropriately tuning the damping parameter within the CK Hamiltonian and detuning factor. Finally, we report a field detuning asymmetry in the collective statistical behavior.

A further study of the matte black-body

A deduction of the radiation intensity at see level is deduced by using a semiclassical approach by means of the quantum energy spectrum deduced from the driven damped harmonic oscillator, the ergodic theorem and the mean energy obtained by using the Abraham equation of motion for a charged particle. The method is based on using the quantum energy values of a modified Caldirola–Kanai Hamiltonian calculated by Um et al., (2002) with a particular driven force.

Damping in the Interaction of a Field and Two Three-Level Atoms Through Quantized Caldirola–Kanai Hamiltonian

We investigate the damped interaction between two Λ-type three-level atoms and a quantized single-mode cavity field, for which the Hamiltonian of the field is rewritten in Caldirola–Kanai form. We obtain the wave functions for the case where the two atoms are initially prepared in arbitrary pure states and the field is initially prepared in the coherent state. We investigate numerically the influence of the damping parameter on the temporal behavior of the Mandel Q-parameter, linear entropy, and normal squeezing. We find the damping parameter and initial atomic states to play central roles in the nonclassical features and the degree of entanglement.

The one-dimensional harmonic oscillator damped with Caldirola-Kanai Hamiltonian

In this paper, the solution to the Hamilton-Jacobi equation for the one-dimensional harmonic oscillator damped with the Caldirola-Kanai model is presented. Making use of a canonical transformation, we calculate the Hamilton characteristic function. It was found that the position of the oscillator shows an exponential decay similar to that of the oscillator with damping where the decay is more pronounced when increasing the damping constant γ. It is shown that when γ = 0, the behavior is of an oscillator with simple harmonic motion. However, unlike the damped harmonic oscillator where the linear momentum decays with time, in the case of the oscillator with the Caldirola-KanaiHamiltonian, the momentum increases as time increases due to an exponential growth of the mass m(t) = meγt.

Dynamics of Nonclassicality of Time- and Conductivity-Dependent Squeezed States and Excited Even/Odd Coherent States

Recently, the quantum description of electromagnetic waves in conducting media has been performed. It has been demonstrated that in particular case, the Hamiltonian of the corresponding field can be expressed by Caldirola–Kanai Hamiltonian. In this paper, using the associated annihilation and creation operators of the above-mentioned quantized field, the time- and conductivity-dependent squeezed vacuum and one-photon squeezed states as well as their superpositions, and also the time- and conductivity-dependent excited even and odd coherent states are produced. Also, using a few well-known nonclassicality criteria, the time evolution of nonclassicality features of the above classes of obtained states, in addition to the influence of medium conductivity on them are demonstrated, numerically. It has been shown that the nonclassicality indicators may be adjusted by tuning the conductivity of media.

Description of Atom-Field Interaction via Quantized Caldirola-Kanai Hamiltonian

In this paper we outline an approach to the study of atom-field interacting systems, where the Hamiltonian of the field is simply inspired from the quantized Caldirola-Kanai Hamiltonian. As a simple physical realization of the model, the interaction between a two-level atom with such a single-mode field is studied. The explicit form of the atom-field entangled state associated with the considered system is analytically deduced and the dynamics of a few of its physical properties is numerically evaluated. To achieve the latter purposes, the temporal behavior of the degree of entanglement, atomic population inversion as well as sub-Poissonian statistics and quadrature squeezing of the field are evaluated. Moreover, the effects of the intensity of initial field and the damping parameter within the Caldirola-Kanai Hamiltonian on the above-mentioned criteria are investigated. As is shown, by adjusting the latter evolved parameters one can appropriately tune the discussed physical quantities.

Optical realization of the dissipative quantum oscillator.

An optical realization of the damped quantum oscillator, based on transverse light dynamics in an optical resonator with slowly-moving mirrors, is theoretically suggested. The optical resonator setting provides a simple implementation of the time-dependent Caldirola-Kanai Hamiltonian of the dissipative quantum oscillator and enables the visualization of the effects of damped oscillations in the classical (ray optics) limit and wave packet collapse in the quantum (wave optics) regime.

On the exact solutions of the damped harmonic oscillator with a time-dependent damping constant and a time-dependent angular frequency

In this paper, we consider the damped harmonic oscillator with a time-dependent damping constant and a time-dependent angular frequency, which is actually the generalized Caldirola-Kanai Hamiltonian. For some choices of the time-dependent damping constant and the time-dependent angular frequency, we investigate the exact solutions. We also obtain the corresponding conserved quantities and finally discuss the quantum damped harmonic oscillator with a time-dependent damping constant and a time-dependent angular frequency.

The effects of nonextensivity on quantum dissipation

Nonextensive dynamics for a quantum dissipative system described by a Caldirola-Kanai (CK) Hamiltonian is investigated in SU(1,1) coherent states. To see the effects of nonextensivity, the system is generalized through a modification fulfilled by replacing the ordinary exponential function in the standard CK Hamiltonian with the q-exponential function. We confirmed that the time behavior of the system is somewhat different depending on the value of q which is the degree of nonextensivity. The effects of q on quantum energy dissipation and other parameters are illustrated and discussed in detail.

Quasi-coherent states for damped and forced harmonic oscillator

In this study, first we construct the quasi-coherent states for a damped and forced harmonic oscillator and discuss the transition of the system from the damped oscillations to forced steady state oscillations. Second, we generalize the Caldirola-Kanai Hamiltonian into the new systems such as the frequency and the mass parameters are time dependent and discuss three examples of them.