Classical viscid media are quite common in our everyday life. However, we are not used to find such media in quantum mechanics, and much less to analyze their effects on the dynamics of quantum systems. In this regard, the Caldirola-Kanai time-dependent Hamiltonian constitutes an appealing model, accounting for friction without including environmental fluctuations (as it happens, for example, with quantum Brownian motion). Here, a Bohmian analysis of the associated friction dynamics is provided in order to understand how a hypothetical, purely quantum viscid medium would act on a wave packet from a (quantum) hydrodynamic viewpoint. To this purpose, a series of paradigmatic contexts have been chosen, such as the free particle, the motion under the action of a linear potential, the harmonic oscillator, or the superposition of two coherent wave packets. Apart from their analyticity, these examples illustrate interesting emerging behaviors, such as localization by "quantum freezing" or a particular type of quantum-classical correspondence. The reliability of the results analytically determined has been checked by means of numerical simulations, which has served to investigate other problems lacking of such analyticity (e.g., the coherent superpositions).
Based on the technique of integration within an ordered product (IWOP) of operators we introduce the Fresnel operator for converting Caldirola-Kanai Hamiltonian into time-independent harmonic oscillator Hamiltonian. The Fresnel operator with the parameters A,B,C,D corresponds to classical optical Fresnel transformation, these parameters are the solution to a set of partial differential equations set up in the above mentioned converting process. In this way the exact wavefunction solution of the Schrödinger equation governed by the Caldirola-Kanai Hamiltonian is obtained, which represents a squ…
We show that the ground state of the well-known pseudo-stationary states for the Caldirola-Kanai Hamiltonian is a generalized minimum uncertainty state, which has the minimum allowed uncertainty $Δq Δp = \hbar σ_0/2$, where $σ_0 (\geq 1)$ is a constant depending on the damping factor and natural frequency. The most general symmetric Gaussian states are obtained as the one-parameter squeezed states of the pseudo-stationary ground state. It is further shown that the coherent states of the pseudo-stationary ground state constitute another class of the generalized minimum uncertainty states.
The quadrature distribution for the quantum damped oscillator is introduced in the framework of the formulation of quantum mechanics based on the tomography scheme. The probability distribution for the coherent and Fock states of the damped oscillator is expressed explicitly in terms of Gaussian and Hermite polynomials, correspondingly.
We analytically study quantum dissipative dynamics described by the Caldirola-Kanai model with inter-particle interactions. We have found that the dissipative quantum dynamics of the Caldirola-Kanai model can be exactly mapped to a dissipationless quantum dynamics under a negative external harmonic potential, even when the particles are strongly interacting. In particular, we show that the mapping is valid for the unitary Fermi gas, which is relevant for cold atoms and nuclear matters.
Using the phenomenological quantum friction models introduced by Caldirola-Kanai, Kostin, and Albrecht, we study quantum tunneling of a one-dimensional potential in the presence of energy dissipation. To this end, we calculate the tunneling probability using a time-dependent wave packet method. The friction reduces the tunneling probability. We show that the three models provide similar penetrabilities to each other, among which the Caldirola-Kanai model requires the least numerical effort. We also discuss the effect of energy dissipation on quantum tunneling in terms of barrier distributions.
Dissipative quantum tunnelling through an inverted parabolic barrier is considered in the presence of an electric field. A Schrödinger-Langevin or Kostin quantum classical transition wave equation is used and applied resulting in a scaled differential equation of motion. A Gaussian wave packet solution to the resulting scaled Kostin nonlinear equation is assumed and compared to the same solution for the scaled linear Caldirola-Kanai equation. The resulting scaled trajectories are obtained at different dynamical regimes and friction cases, showing the gradual decoherence process in this open dy…
Dissipative backflow is studied in the context of open quantum systems. This theoretical analysis is carried out within two frameworks, the effective time-dependent Hamiltonian due to Caldirola-Kanai (CK) and the Caldeira-Leggett (CL) one where a master equation is used to describe the reduced density matrix in presence of dissipation and temperature of the environment. Two examples are considered, the free evolution of one and two Gaussian wave packets as well as the time evolution under a constant field. Backflow is shown to be reduced with dissipation and temperature but never suppressed. I…
We obtain the tomogram of squeezed correlated states of a quantum parametric damped oscillator in an explicit form. We study the damping within the framework of the Caldirola--Kanai model and chose the parametric excitation in the form of a very short pulse simulated by a delta-kick of frequency; the squeezing phenomenon is reviewed. The cases of strong and weak damping are investigated.
Thermal wave packets are used to analyze transmission probabilities and characteristic times through a parabolic repeller within the dissipative Bohmian mechanics. Thermal arrival, dwelling, transmission and reflection times are defined and calculated by using a Maxwell-Boltzmann distribution for the initial velocities of the incident particles. The dissipation is considered within the Caldirola-Kanai and Kostin approaches where a linear and nonlinear framework is used, respectively. The initial parameters are chosen to have only a dissipative tunnelling dynamics at zero temperature; at any no…
Using the Caldirola-Kanai Hamiltonian, we study the time evolution of the wave function of a particle whose classical motion is governed by the Langevin equation. We show, in particular, that if the initial wave function is Gaussian, then (i) it remains Gaussian for all times, (ii) its width grows, approaching a finite value when t->infinity, and (iii) its center describes a Brownian motion, and so the uncertainty in the position of the particle grows without limit.
Quantum dissipation is studied within two model oscillators, the Caldirola-Kanai (CK) oscillator as an open system with one degree of freedom and the Bateman-Feshbach-Tikochinsky (BFT) oscillator as a closed system with two degrees of freedom. Though these oscillators describe the same classical damped motion, the CK oscillator retains the quantum coherence, whereas the damped subsystem of the BFT oscillator exhibits both quantum decoherence and classical correlation. Furthermore the amplified subsystem of the BFT oscillator shows the same degree of quantum decohernce and classical correlation…
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 to visualize the effects of damped oscillations in the classical (ray optics) limit and wave packet collapse in the quantum (wave optics) regime.
We associate the stationary harmonic oscillator with time-dependent systems exhibiting non-Hermiticity by means of point transformations. The new systems are exactly solvable, with all-real spectrum, and transit to the Hermitian configuration for the appropriate values of the involved parameters. We provide a concrete generalization of the Swanson oscillator that includes the Caldirola-Kanai model as a particular case. Explicit solutions are given in both, the classical and quantum pictures.
A nonlinear quantum-classical transition wave equation is proposed for dissipative systems within the Caldirola-Kanai model. Equivalence of this transition equation to a scaled Schrödinger equation is proved. The dissipative dynamics is then studied in terms of what we call scaled trajectories following the standard procedure used in Bohmian mechanics. These trajectories depend on a continuous parameter allowing us a smooth transition from Bohmian to classical trajectories. Arrival times and actual momentum distribution functions are also analyzed. The propagation of a Gaussian wave packet in …
A definition of quantum mechanical work is introduced in this dissertation, preserving the mathematical structure of the Classical Mechanics concept of work without, however, in any way invoking the notion of trajectory. By use of Gaussian states and the Caldirola-Kanai model, a case study is conducted through which the proposed quantum work is compared with another Quantum Thermodynamics well-known definition, both in quantum and semiclassical regimes, showing promising results. Finally the new definition is investigated in comparison with a classical-statistical approach for superposition an…
Integrals of motion and statistical properties of quantized electromagnetic field (e.-m. field) in time-dependent linear dielectric and conductive media are considered, using Choi-Yeon quantization, based on Caldirola-Kanai type Hamiltonian. Eigenstates of quadratic and linear invariants are constructed, the solutions being expressed in terms of a complex parametric function that obeys classical oscillator equation with time-varying frequency. The time evolutions of initial Glauber coherent states and Fock states are considered. The medium conductivity and the time-dependent electric permeabil…
We model a quantum system coupled to an environment of damped harmonic oscillators by following the approach of Caldeira-Leggett and adopting the Caldirola-Kanai Lagrangian for the bath oscillators. In deriving the master equation of the quantum system of interest (a particle in a general potential), we show that the potential is modified non-trivially by a new inverted harmonic oscillator term, induced by the damping of the bath oscillators. We analyze numerically the case of a particle in a double-well potential, and find that this modification changes both the rate of decoherence at short t…
A novel functional integral formulation of quantum mechanics for non-Lagrangian systems is presented. The new approach, which we call "stringy quantization," is based solely on classical equations of motion and is free of any ambiguity arising from Lagrangian and/or Hamiltonian formulation of the theory. The functionality of the proposed method is demonstrated on several examples. Special attention is paid to the stringy quantization of systems with a general A-power friction force $-κ[\dot{q}]^A$. Results for $A = 1$ are compared with those obtained in the approaches by Caldirola-Kanai, Batem…
In this work, dissipative quantum backflow is studied for a superposition of two stretched Gaussian wave packets and two identical spinless particles within the Caldirola-Kanai framework. Backflow is mainly an interference process and dissipation is not able to suppress it in the first case. For two identical spinless particles, apart from interference terms, the symmetry of the wave function seems to be crucial in this dynamics. The combined properties make bosons display this effect, even for the dissipative regime but, for fermions, backflow is not exhibited in any regime, dissipative and n…
Interference and diffraction of two-identical-particles are considered in the context of open quantum systems. This theoretical study is carried out within two approaches, the effective time-dependent Hamiltonian due to Caldirola-Kanai (CK) and the Caldeira-Leggett (CL) one where a master equation for the reduced density matrix is used under the presence of dissipation and temperature of the environment. Two simple but very illustrative examples are considered, diffraction by a single and two Gaussian slits by analyzing the mean square separation between particles, single-particle probability …
Decoherence is a well established process for the emergence of classical mechanics in open quantum systems. However, it can have two different origins or mechanisms depending on the dynamics one is considering, speaking then about intrinsic decoherence for isolated systems and environmental decoherence due to dissipation/fluctuations for open systems. This second mechanism can not be considered for backflow since no thermal fluctuation terms can be added in the formalism in order to keep an important requirement for the occurrence of this effect: only contributions of positive momenta along ti…
