For physical systems where the Hamiltonian is no longer a constant of motion, like oscillators with time-dependent frequency or dissipative systems, a dynamical invariant of the Ermakov type might still exist. This invariant depends on the classical position and velocity and an auxiliary variable that is proportional to position uncertainty. It is possible to construct this invariant via an algebraic method based on the classical Poisson bracket. A modified version for dissipative systems that also employs anti-Poisson brackets and is related to a description in an expanding coordinate system will be presented. The resulting invariant is identical to that obtained from a logarithmic nonlinear Schrödinger equation. The uncertainties of position and momentum, expressed in terms of the auxiliary variable, can be compared with the so-called moment-method for the description of the time-evolution of Bose–Einstein condensates, corresponding to a description in terms of the cubic nonlinear Gross–Pitaevskii equation. It will be shown how this method can be extended to also include dissipative damping effects into the Bose–Einstein dynamics.
For physical systems where the Hamiltonian is no longer a constant of motion, like oscillators with time-dependent frequency or dissipative systems, a dynamical invariant of the Ermakov type might still exist. This invariant depends on the classical position and velocity and an auxiliary variable that is proportional to position uncertainty. It is possible to construct this invariant via an algebraic method based on the classical Poisson bracket. A modified version for dissipative systems that also employs anti-Poisson brackets and is related to a description in an expanding coordinate system will be presented. The resulting invariant is identical to that obtained from a logarithmic nonlinear Schrödinger equation. The uncertainties of position and momentum, expressed in terms of the auxiliary variable, can be compared with the so-called moment-method for the description of the time-evolution of Bose–Einstein condensates, corresponding to a description in terms of the cubic nonlinear Gross–Pitaevskii equation. It will be shown how this method can be extended to also include dissipative damping effects into the Bose–Einstein dynamics.
In a former publication [D. Schuch and M. Moshinsky, Phys. Rev. A 73, 062111 (2006)] we have shown that the time evolution of a quantum system with at most quadratic Hamiltonian that can be described by different methods, namely, the time-dependent Schrödinger equation, the time propagator or Feynman kernel, and the Wigner function is connected via a dynamical invariant, the so-called Ermakov invariant. Since exact invariants of this type also exist for certain effective descriptions of dissipative quantum systems, we will show the relations between these descriptions and the corresponding invariants and will discuss how the results obtained for the conservative systems must be modified in the presence of a linear velocity dependent frictional force.
In the first part of the lectures dynamical invariants in classical mechanics and conventional quantum mechanics will be considered. In particular, we will begin with some remarks on classical mechanics and on quantization in order to establish the theory in the form that will be used later on. Starting from the time‐dependent Schrödinger equation, the dynamics of Gaussian wave packets and Ermakov invariants, the time‐dependent Green function/Feynman kernel, quantum‐classical connections, energetics and Lagrange—Hamilton formalism for quantum uncertainties, momentum space representation and the relation between the Wigner function and the Ermakov invariant will be discussed. The representation of canonical transformations in time‐independent and time‐dependent quantum mechanics, factorization of the Ermakov invariant and generalized creation/annihilation operators will be studied. Subsequently, the time‐independent Schrödinger equation, leading to nonlinear quantum mechanics related to Riccati/Ermakov systems as well as the occurrence of Riccati/Ermakov systems in the treatment of Bose—Einstein condensates via the so‐called moment method will be analyzed.In part two, irreversible dynamics of dissipative systems, classical and quantum mechanical descriptions and corresponding invariants will be treated. After some general remarks on classical and quantum mechanics with unitary time‐evolution and energy conservation, phenomenological Langevin and Fokker—Planck equations, master equations in classical and quantum mechanics and the system‐plus‐reservoir approach will be mentioned briefly. Then follows a more detailed discussion of modified Schrödinger equations and, particularly, of a nonlinear Schrödinger equation with complex logarithmic nonlinearity; its properties, solutions, invariants and energetics will be studied. Finally, a comparison with a classical description in expanding coordinates will lead to a non‐unitary connection between the logarithmic nonlinear Schrödinger equation, the Caldirola‐Kanai approach and the expanding coordinate approach.
In the first part of the lectures dynamical invariants in classical mechanics and conventional quantum mechanics will be considered. In particular, we will begin with some remarks on classical mechanics and on quantization in order to establish the theory in the form that will be used later on. Starting from the time‐dependent Schrödinger equation, the dynamics of Gaussian wave packets and Ermakov invariants, the time‐dependent Green function/Feynman kernel, quantum‐classical connections, energetics and Lagrange—Hamilton formalism for quantum uncertainties, momentum space representation and the relation between the Wigner function and the Ermakov invariant will be discussed. The representation of canonical transformations in time‐independent and time‐dependent quantum mechanics, factorization of the Ermakov invariant and generalized creation/annihilation operators will be studied. Subsequently, the time‐independent Schrödinger equation, leading to nonlinear quantum mechanics related to Riccati/Ermakov systems as well as the occurrence of Riccati/Ermakov systems in the treatment of Bose—Einstein condensates via the so‐called moment method will be analyzed.In part two, irreversible dynamics of dissipative systems, classical and quantum mechanical descriptions and corresponding invariants will be treated. After some general remarks on classical and quantum mechanics with unitary time‐evolution and energy conservation, phenomenological Langevin and Fokker—Planck equations, master equations in classical and quantum mechanics and the system‐plus‐reservoir approach will be mentioned briefly. Then follows a more detailed discussion of modified Schrödinger equations and, particularly, of a nonlinear Schrödinger equation with complex logarithmic nonlinearity; its properties, solutions, invariants and energetics will be studied. Finally, a comparison with a classical description in expanding coordinates will lead to a non‐unitary connection between the logarithmic nonlinear Schrödinger equation, the Caldirola‐Kanai approach and the expanding coordinate approach.
The stochastic projected Gross-Pitaevskii equation (SPGPE) is a finite-temperature theory of Bose-Einstein condensate dynamics, utilizing a classical-field approach to describing the dynamics of a low-energy coherent subspace in contact with an incoherent thermal reservoir. Interactions with the reservoir are encapsulated in two distinct processes, known as number damping and energy damping. Historically the energy-damping process has received little attention, at least in part due to difficulty in finding a sufficiently efficient and accurate numerical algorithm for solving the SPGPE. Rooney et al. [Phys. Rev. E 89, 013302 (2014)] developed such an algorithm, providing new opportunities to investigate the energy-damping process. In this work we provide an analytic treatment of the energy-damping process for a range of systems using functional Ito calculus.\n\nWe perform a linear fluctuation analysis on the SPGPE for a homogeneous system. We find the dispersion relation to be the usual Bogoliubov dispersion with an additional momentum dependent damping rate. The damping rate is a combination of number- and energy-damping terms with distinct momentum dependence, revealing that each damping process is dominant for excitations of different length scales. We also find the spectra of density and phase fluctuations in momentum space, again with distinct momentum dependence associated with the two damping processes. These results demonstrate that there always exists a regime of length scales where one can expect energy-damping to be the dominant process.\n\nWe derive stochastic Ehrenfest relations (SER) from the SPGPE using projected functional change of variables. The SER take the form of stochastic differential equations for matter wave moments of the system and include drift and diffusion terms corresponding to the number- and energy-damping processes. We find that for a well-chosen cutoff the SER can often admit approximate analytic solutions. Analytic solutions of the SER for a quasi-1D Thomas-Fermi system show good agreement with simulations of the 1D SPGPE.\n\nNext, we characterise the centre of mass equilibrium properties of a 3D harmonically trapped finite temperature system in the Thomas-Fermi regime using the SER for position, momentum, and particle number. We find that the energy-damping process is generally dominant over the number-damping process when considering position and momentum fluctuations. Considering the stochastic evolution of particle number in this system, we find that the steady-state fluctuation properties depend only on the number-damping process. This suggests that the effects of each reservoir interaction can be measured individually in the same system, allowing for experimental detection of the energy-damping reservoir interaction.\n\nWe finish by considering the motion of a singly-charged quantum vortex confined to a disc trap in an energy-damped system. We find a stochastic differential equation for the dissipative motion of the vortex. It is revealed that the presence of noise can cause a stabilizing effect on the vortex, extending its lifetime. We also consider the presence of a rotating thermal cloud, a scenario that leads to the vortex becoming the energetically favourable state. This allows for analysis of steady-state stochastic properties of the vortex motion.
The dissipative dynamic stability is investigated of dark solitons in elongated Bose–Einstein condensates that can be described by the Gross–Pitaevskii equation including an additional term. Based on the direct perturbation theory for the nonlinear Schrodinger equation, the dependence of the soliton velocity on time is explicitly given, and the shape of dark solitons remaining unchanged under the dissipative condition is confirmed theoretically for the first time. It is found that the dynamically stable dark solitons turn out to be thermodynamically unstable.
The time-evolution of the maximum and the width of exact analytic wave packet (WP) solutions of the time-dependent Schrödinger equation (SE) represents the particle and wave aspects, respectively, of the quantum system.The dynamics of the maximum, located at the mean value of position, is governed by the Newtonian equation of the corresponding classical problem.The width, which is directly proportional to the position uncertainty, obeys a complex nonlinear Riccati equation which can be transformed into a real nonlinear Ermakov equation.The coupled pair of these equations yields a dynamical invariant which plays a key role in our investigation.It can be expressed in terms of a complex variable that linearizes the Riccati equation.This variable also provides the time-dependent parameters that characterize the Green's function, or Feynman kernel, of the corresponding problem.From there, also the relation between the classical and quantum dynamics of the systems can be obtained.Furthermore, the close connection between the Ermakov invariant and the Wigner function will be shown.Factorization of the dynamical invariant allows for comparison with creation/annihilation operators and supersymmetry where the partner potentials fulfil (real) Riccati equations.This provides the link to a nonlinear formulation of timeindependent quantum mechanics in terms of an Ermakov equation for the amplitude of the stationary state wave functions combined with a conservation law.Comparison with SUSY and the time-dependent problems concludes our analysis.