Time-dependent Harmonic Potential Research Articles

Quantum uncertainty as an intrinsic clock

In quantum mechanics, a classical particle is raised to a wave-function, thereby acquiring many more degrees of freedom. For instance, in the semi-classical regime, while the position and momentum expectation values follow the classical trajectory, the uncertainty of a wave-packet can evolve and beat independently. We use this insight to revisit the dynamics of a 1d particle in a time-dependent harmonic well. One can solve it by considering time reparameterizations and the Virasoro group action to map the system to the harmonic oscillator with constant frequency. We prove that identifying such a simplifying time variable is naturally solved by quantizing the system and looking at the evolution of the width of a Gaussian wave-packet. We further show that the Ermakov-Lewis invariant for the classical evolution in a time-dependent harmonic potential is actually the quantum uncertainty of a Gaussian wave-packet. This naturally extends the classical Ermakov-Lewis invariant to a constant of motion for quantum systems following Schrödinger equation. We conclude with a discussion of potential applications to quantum gravity and quantum cosmology.

Evolution of the wave function's shape in a time-dependent harmonic potential

Abstract An effective operational approach to quantum mechanics is to focus on the evolution of wave packets, for which the wave function can be seen in the semi-classical regime as representing a classical motion dressed with extra degrees of freedom describing the shape of the wave packet and its fluctuations. These quantum dressing are independent degrees of freedom, mathematically encoded in the higher moments of the wave function. We review how to extract the effective dynamics for Gaussian wave packets evolving according to the Schrödinger equation with time-dependent potential in a 1 + 1-dimensional spacetime, and derive the equations of motion for the quadratic uncertainty. We then show how to integrate the evolution of all the higher moments for a general wave function in a time-dependent harmonic potential.

Optimal performance of the stochastic thermodynamic engine with a periodic heat bath

Stochastic thermodynamics provides a conceptual framework for describing the fluctuating behavior of small systems like colloids or biomolecules far from thermodynamic equilibriums but still contacted with a heat bath. In contrast to most literature focusing on the classical paradigm of Carnot engines, we herein study the optimal performance of the thermodynamic heat engine with a heat bath that periodically changes temperature, which is outside controllable by a time-dependent harmonic potential. Under reasonable assumptions on the control actuation, we derive the achievable upper bound for the maximal power and also the optimal control protocol. In addition, we also obtain the corresponding efficiency at maximal power, which only depends on the ratio of the minimal and maximal value of the temperature profile.

Theoretic impetuses and lengths of Feinberg-Horodecki equation

The exact solution of the Feinberg-Horodecki equation for time-dependent harmonic vector potential has been investigated under a one-dimensional system. The quantized momentum and its corresponding un-normalized wave functions were explicitly obtained. The Fisher information (for time and momentum) and variance (for time and momentum) were calculated using expectation values of time and momentum via Hellman-Feynman Theory (HFT). The time and momentum Shannon entropy were obtained using an existing formula. Numerical results were for time and momentum Fisher information to confirm the Cramer-Rao inequality. Another numerical results were obtained for time and momentum Shannon entropy to confirm Bialynick-Birula, Mycielski (BBM) inequality. The effects of the potential parameters such as mass of the spring and the frequency on the theoretic quantities were fully examined. The new variance inequality was established using the inequalities of Fisher information. The established inequality was confirmed by numerical results which also satisfied the popular Cramer-Rao inequality. The theoretic impetuses for Fisher information, variance and Shannon entropy respectively, were calculated and their variation with some potentials were studied.

Low-dissipation engines: Microscopic construction via shortcuts to adiabaticity and isothermality, the optimal relation between power and efficiency.

We construct a microscopic model of low-dissipation engines by driving a Brownian particle in a time-dependent harmonic potential. Shortcuts to adiabaticity and shortcuts to isothermality are introduced to realize the adiabatic and isothermal branches in a thermodynamic cycle, respectively. We derive an analytical formula of the efficiency at maximum power with explicit expressions of dissipation coefficients under the optimized protocols. When the relative temperature difference between the two baths in the cycle is insignificant, this expression satisfies the universal law of efficiency at maximum power up to the quadratic term of the Carnot efficiency. For large relative temperature differences, the efficiency at maximum power tends to be 1/2. Furthermore, we analyze the issue of power at any given efficiency for general low-dissipation engines and then obtain the supremum of the power in three limiting cases, respectively. These expressions of maximum power at given efficiency provide the optimal relations between power and efficiency which are tighter than the results in previous references.

Asymptotic behavior of solutions to a dissipative nonlinear Schrödinger equation with time-dependent harmonic potentials

We analyzed the Cauchy problem of a dissipative nonlinear Schrödinger equation with a time-dependent harmonic potential. We identified a critical situation that determines whether the L2-norm of dissipative solutions exhibits decay, which depends on a nonlinear power and time decay order of harmonic potential.

Dynamics of dipolar quantum droplets in an extended Gross–Pitaevskii equation in the presence of time-dependent harmonic trapping potential and a damping term

The purpose of this paper is to study the dynamics of solutions to an extended Gross–Pitaevskii equation that models the formation of droplets in a dipolar Bose–Einstein condensate (BEC). The formation of these droplets has been recently discovered by driving the BEC into the strongly dipolar regime. Surprisingly, instead of collapsing, the system formed stable droplets. So far, no rigorous mathematical explanation has been proved. To the best of our knowledge, only experimental results have been obtained. The goal of this paper is to validate this breakthrough discovery. Many predictions/ conjectures properties of these droplets have been stated by some research groups in physics and engineering. In particular, it has been claimed that the stability of these droplets is a consequence of the presence of the damping term in the extended Gross–Pitaevskii equation under study. This term describes the three-body loss process. To accurately model the dynamics of formation of these droplets, it is necessary to consider a time-dependent harmonic trapping potential as well as other terms with different types of nonlinearity among them that describe the Lee–Huang–Yang (LHY). This presents some challenges that will be solved in this paper.

Reversibility in nonequilibrium steady states as a measure of distance from equilibrium.

From the detailed balance-like relation, we propose a measure, K^{*}, of a nonequilibrium steady-state (NESS) distance from equilibrium. We investigate in particular the NESS of a particle confined in a time-dependent harmonic potential of constant stiffness but with an ON-OFF state following a telegraph process. Experimental results coupled with simulations show that K^{*} increases at slow switching rates (far from equilibrium) and approaches to zero at equilibrium conditions. Thus, the steady-state distribution together with K^{*} fully characterizes a NESS.

Fast-forward approach to stochastic heat engine.

The fast-forward (FF) scheme proposed by Masuda and Nakamura [Proc. R. Soc. A 466, 1135 (2010)1364-502110.1098/rspa.2009.0446] in the context of conservative quantum dynamics can reproduce a quasistatic dynamics in an arbitrarily short time. We apply the FF scheme to the classical stochastic Carnot-like heat engine which is driven by a Brownian particle coupled with a time-dependent harmonic potential and working between the high- (T_{h}) and low- (T_{c}) temperature heat reservoirs. Concentrating on the underdamped case where momentum degree of freedom is included, we find the explicit expressions for the FF protocols necessary to accelerate both the isothermal and thermally adiabatic processes and obtain the reversible and irreversible works. The irreversible work is shown to consist of two terms with one proportional to and the other inversely proportional to the friction coefficient. The optimal value of efficiency η at the maximum power of this engine is found to be η^{*}=1/2{1+1/2(T_{c}/T_{h})^{1/2}-5/4T_{c}/T_{h}+O[(T_{c}/T_{h})^{3/2}]} and η^{*}=1-(T_{c}/T_{h})^{1/2}, respectively, in the cases of strong and weak dissipation. The result is justified for a wide family of time-scaling functions, making the FF protocols very flexible. We also revealed that the accelerated full cycle of the Carnot-like stochastic heat engine cannot be conceivable within the framework of the overdamped case, and the power and efficiency can be evaluated only when the momentum degree of freedom is taken into consideration.

Asymptotic behavior of solutions to nonlinear Schrödinger equations with time-dependent harmonic potentials

In this study, we examine the asymptotic behavior of solutions to nonlinear Schr\"{o}dinger equations with time-dependent harmonic oscillators and prove the time-decay property of solutions in the case of a long range power type nonlinearity.

Dynamic phase fluctuations in potential-driven Bose–Einstein condensate

We report on the dynamics of a Bose–Einstein condensate in one and two dimensions driven by the time-dependent harmonic trapping potential. Without the inter-particle interaction, the condensate exhibits the coherent behavior with the time-oscillating density distribution. When the inter-particle interaction is taken into consideration, the phase fluctuations, the multi-peak structure of the density distribution, and the coherence revival phenomenon, apart from the well-studied density oscillation behavior, can be observed in the condensate. Furthermore, it is demonstrated that due to the dimensional restriction these effects are more stable in the two and three-dimensional system, if compared to the one-dimensional case.

Work fluctuation and its optimal extraction with time dependent harmonic potential from a non-Markovian bath

We study the optimal work extraction protocol for a Brownian particle described by generalized Langevin dynamics from a single reservoir with the aid of position measurement. The particle is confined in a time dependent harmonic potential. In the overdamped situation, we show that memory effects typically reduce the extracted work, and full information-work conversion can only be achieved by manipulating both the stiffness and center of the potential quasistatically. In the non-Markovian underdamped regime, we investigate the exponential tail of work distribution using the path integral formalism, memory effects on the characteristic work production is also revealed.

Vector solitons for the coupled Gross-Pitaevskii equations with a time-dependent external harmonic potential in the Bose-Einstein condensation

In this paper, we study the nonautonomous vector solitons for the coupled Gross-Pitaevskii equations with a time-dependent external harmonic potential, which can be used to describe the properties of the two-component Bose-Einstein condensation in two hyperfine states. Based on the Kadomtsev-Petviashvili hierarchy reduction, we construct the dark-dark, bright-dark and bright-bright soliton solutions in terms of the Gramian. When the atomic scaled s-wave scattering length a>0, which indicates that the interparticle interaction is repulsive, the N-dark-dark and N-bright-dark solitons exist; when a<0, which indicates that the interparticle interaction is attractive, the N-bright-bright and N-bright-dark solitons exist. With the help of the analytic and graphic analysis, interactions between the two solitons with different forms of the external harmonic potential are studied. Interaction between the two dark-dark solitons is elastic. However, interactions between the two bright-dark solitons and between the two bright-bright solitons may be inelastic under certain parameter constraints. Two bright-dark-soliton and two bright-bright-soliton bound states with varying amplitudes, velocities and widths are presented.

Investigation of relativistic bosons in the presence of two-dimensional time-dependent harmonic interaction

The Duffin–Kemmer–Petiau (DKP) equation has been exactly solved for the spin-one particle in the presence of time-dependent harmonic potential in a two dimensional space using the Lewis–Riesenfeld dynamical invariant and unitary transform methods. The dynamical invariant has been constructed and its eigen functions have been obtained. The total wave function as well as the evolution operator have been derived.

Retainability of canonical distributions for a Brownian particle controlled by a time-dependent harmonic potential

The retainability of canonical distributions for a Brownian particle controlled by a time-dependent harmonic potential is investigated in the overdamped and underdamped situations, respectively. Because of different time scales, the overdamped and underdamped Langevin equations (as well as the corresponding Fokker-Planck equations) lead to distinctive restrictions on protocols maintaining canonical distributions. Two special cases are analyzed in details: First, a Brownian particle is controlled by a time-dependent harmonic potential and embedded in medium with constant temperature; Second, a Brownian particle is controlled by a time-dependent harmonic potential and embedded in a medium whose temperature is tuned together with the potential stiffness to keep a constant effective temperature of the Brownian particle. We find that the canonical distributions are usually retainable for both the overdamped and underdamped situations in the former case. However, the canonical distributions are retainable merely for the overdamped situation in the latter case. We also investigate general time-dependent potentials beyond the harmonic form and find that the retainability of canonical distributions depends sensitively on the specific form of potentials.

Self-accelerating matter waves

The free particle Schrödinger equation admits a nontrivial self-accelerating Airy wave packet solution. Recently, the Airy beams that freely accelerate in space was experimentally realized in photonics community. Here, we present self-accelerating waves for the Bose–Einstein condensate in a time-dependent harmonic oscillator potential. We show that parity and time reversal symmetries for self-accelerating waves are spontaneously broken.

Analytical solitonlike solutions and the dynamics of ultracold Fermi gases in a time-dependent three-dimensional harmonic potential

We present a theoretical study of solitonlike solutions and their dynamics of ultracold superfluid Fermi gases trapped in a time-dependent three-dimensional (3D) harmonic potential with gain or loss. The 3D analytical solitonlike solutions are obtained without introducing any additional integrability constraints used elsewhere. The propagation of both bright- and dark-soliton-like solutions is investigated. We show that the amplitudes of dark-soliton-like solutions exhibit periodic oscillation, whereas those of the bright-soliton-like ones do not show such behavior. Moreover, we highlight that the oscillation periods of dark-soliton-like solutions predicted by our approach are matched very well with those observed in a recent experiment carried out by Yefsah etal. [T. Yefsah, A. T. Sommer, M. J. H. Ku, L. W. Cheuk, W. Ji, W. S. Bakr, and M. W. Zwierlein, Nature (London) 499, 426 (2013)NATUAS0028-083610.1038/nature12338] in both Bose-Einstein condensation and unitarity regimes.

Controlling rogue wave triplets in Bose–Einstein condensate

We present the coherent control of rogue wave triplets in Bose–Einstein condensate (BEC) for the parametric choices which support their occurrence experimentally. This analysis has been done for two cases involving the trapping of BEC in (i) time-independent (ii) time-dependent harmonic potential. We have found that these waves no longer lie on the apex of the equilateral triangle as in the case of homogeneous system but can be manipulated by modulating the trap. This enables us to control the relative distance among the three rogue wave peaks and their amplitude, for a given nonlinearity parameter. These results can be helpful to stimulate the study of rogue wave triplets experimentally in BEC.

Phase space path integral approach to harmonic oscillator with a time-dependent force constant

The quantum statistical mechanical propagator for a harmonic oscillator with a time-dependent force constant, mω2(t), has been investigated in the past and was found to have only a formal solution in terms of the solutions of certain ordinary differential equations. Such path integrals are frequently encountered in semiclassical path integral evaluations and having exact analytical expressions for such path integrals is of great interest. In a previous work, we had obtained the exact propagator for motion in an arbitrary time-dependent harmonic potential in the overdamped limit of friction using phase space path integrals in the context of Lévy flights - a result that can be easily extended to Brownian motion. In this paper, we make a connection between the overdamped Brownian motion and the imaginary time propagator of quantum mechanics and thereby get yet another way to evaluate the latter exactly. We find that explicit analytic solution for the quantum statistical mechanical propagator can be written when the time-dependent force constant has the form ω2(t)=λ2(t)−dλ(t)dt, where λ(t) is any arbitrary function of t and use it to evaluate path integrals which have not been evaluated previously. We also employ this method to arrive at a formal solution of the propagator for both Lévy flights and Brownian subjected to a time-dependent harmonic potential in the underdamped limit of friction.

Single-particle stochastic heat engine.

We have performed an extensive analysis of a single-particle stochastic heat engine constructed by manipulating a Brownian particle in a time-dependent harmonic potential. The cycle consists of two isothermal steps at different temperatures and two adiabatic steps similar to that of a Carnot engine. The engine shows qualitative differences in inertial and overdamped regimes. All the thermodynamic quantities, including efficiency, exhibit strong fluctuations in a time periodic steady state. The fluctuations of stochastic efficiency dominate over the mean values even in the quasistatic regime. Interestingly, our system acts as an engine provided the temperature difference between the two reservoirs is greater than a finite critical value which in turn depends on the cycle time and other system parameters. This is supported by our analytical results carried out in the quasistatic regime. Our system works more reliably as an engine for large cycle times. By studying various model systems, we observe that the operational characteristics are model dependent. Our results clearly rule out any universal relation between efficiency at maximum power and temperature of the baths. We have also verified fluctuation relations for heat engines in time periodic steady state.