Engineered swift equilibration of a Brownian particle

Abstract

A fundamental and intrinsic property of any device or natural system is its relaxation time relax, which is the time it takes to return to equilibrium after the sudden change of a control parameter [1]. Reducing τrelax, is frequently necessary, and is often obtained by a complex feedback process. To overcome the limitations of such an approach, alternative methods based on driving have been recently demonstrated [2, 3], for isolated quantum and classical systems [4–9]. Their extension to open systems in contact with a thermostat is a stumbling block for applications. Here, we design a protocol, named Engineered Swift Equilibration (ESE), that shortcuts time-consuming relaxations, and we apply it to a Brownian particle trapped in an optical potential whose properties can be controlled in time. We implement the process experimentally, showing that it allows the system to reach equilibrium times faster than the natural equilibration rate. We also estimate the increase of the dissipated energy needed to get such a time reduction. The method paves the way for applications in micro and nano devices, where the reduction of operation time represents as substantial a challenge as miniaturization [10].

Fast driving between arbitrary states of a quantum particle by trap deformation

“…Applications of the method go beyond quantum mechanics, e.g. to determine potentials in a Fokker-Planck equation [33].…”

Engineered swift equilibration for Brownian objects: from underdamped to overdamped dynamics

“…Generically, when viscous friction is not high compared to the other characteristic frequencies of the problem, one should include the velocity degrees of freedom in the description in addition to positional ones; the overdamped approximation, on the other hand, assumes that the former are equilibrated at all times. To extend the ESE method proposed in [28] to the underdamped description of an object immersed in a thermal bath trapped in a confining potential, we introduce the probability density function K(x, v, t) of the position x and velocity v of the particle. It obeys the Kramers equation…”

“…Many strategies have been proposed to set up non-adiabatic routes to reach the same final state through the use of dynamical invariants [2], counter adiabatic driving [3][4][5], reverse engineering methods [6][7][8], fast-forward techniques [9,10], Lie algebraic approaches [11,12], and optimal control [13][14][15][16] to name but a few. Slow processes (adiabatic in quantum mechanics jargon) and thus STA are quite common to prepare the state of the system in a wide variety of domains including atomic and molecular physics [17,18], quantum transport [19][20][21], solid state [22], many-body physics [23][24][25], classical mechanics [26] and statistical physics [27][28][29]. STA also have applications in the design of optimal devices, as recently proposed in optics [30] and in internal state manipulation for interferometry [31].…”

“…More importantly, such an approach has been generalized to systems in contact with a thermostat which are ubiquitous in thermodynamics. The so-called engineered swift equilibration (ESE) protocols have been introduced to study the isothermal compression of a colloidal particle [28]. The idea was then adapted to encompass the shift of the cantilever of an atomic force microscope [29], and has been generalized in [42] to underdamped processes using a non-conservative driving force.…”

Shortcut to adiabaticity in a Stern-Gerlach apparatus

“…These methods enable one to guarantee a transitionless evolution faster than the time scale imposed by the adiabatic regime. The STA approach has been shown experimentally to efficently speed up the transport or manipulation of wave functions [10][11][12][13][14][15][16][17] and even thermodynamical transformations [18][19][20]. Concerning the transfer of quantum states, recent impressive implementations have been reported in cold atoms experiments [21], solid-state architectures [22] or in optomechanical systems [23].…”