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IEEE Conference on Decision and Control (CDC)

Time-optimal adiabatic-like expansion of Bose-Einstein condensates

Dionisis Stefanatos, Jr-Shin Li

Published Dec 1, 2012• 0 citations
Publisher

Abstract

In this paper we study the fast adiabatic-like expansion of a one-dimensional Bose-Einstein condensate (BEC) confined in a harmonic potential, using the theory of time-optimal control. We find that under reasonable assumptions suggested by the experimental setup, the minimum-time expansion occurs when the frequency of the potential changes in a bang-bang form between the permitted values. We calculate the necessary expansion time and show that it scales logarithmically with large values of the expansion factor. This work is expected to find applications in areas where the efficient manipulations of BEC is of utmost importance. As an example we present the field of atom interferometry with BEC, where the wavelike properties of atoms are used to perform interference experiments that measure with unprecedented precision small shifts induced by phenomena like rotation, acceleration, and gravity gradients.

DOI
10.1109/cdc.2012.6426290

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•Jan 1, 2012•0 citations
In this paper we study the fast adiabatic-like expansion of a one-dimensional Bose-Einstein condensate (BEC) confined in a harmonic potential, using the theory of time-optimal control. We find that under reasonable assumptions suggested by the experimental setup, the minimum-time expansion occurs when the frequency of the potential changes in a bang-bang form between the permitted values. We calculate the necessary expansion time and show that it scales logarithmically with large values of the expansion factor. This work is expected to find applications in areas where the efficient manipulations of BEC is of utmost importance. As an example we present the field of atom interferometry with BEC, where the wavelike properties of atoms are used to perform interference experiments that measure with unprecedented precision small shifts induced by phenomena like rotation, acceleration, and gravity gradients.
In this paper we study a fast adiabatic-like expansion of a one-dimensional Bose-Einstein condensate (BEC) in the Thomas-Fermi regime confined in a harmonic potential, using the theory of time-optimal control. We find that under reasonable assumptions suggested by the experimental setup, the minimum-time expansion occurs when the frequency of the potential changes in a bang-bang form between the permitted values. We calculate the necessary expansion time and compare it with that obtained using the state-of-the-art inverse engineering method under the same constraints. We also show that this minimum time scales logarithmically with the expansion factor for large values of the latter. This work is expected to find applications in areas where efficient manipulation of a BEC is of utmost importance, for example, in atom interferometry with a BEC.
Shortcuts to adiabatic expansion of the effectively one-dimensional Bose-Einstein condensate (BEC) loaded in the harmonic-oscillator (HO) trap are investigated by combining techniques of variational approximation and inverse engineering. Piecewise-constant (discontinuous) intermediate trap frequencies, similar to the known bang-bang forms in the optimal-control theory, are derived from an exact solution of a generalized Ermakov equation. Control schemes considered in the paper include imaginary trap frequencies at short time scales, i.e., the HO potential replaced by the quadratic repulsive one. Taking into regard the BEC's intrinsic nonlinearity, results are reported for the minimal transfer time, excitation energy (which measures deviation from the effective adiabaticity), and stability for the shortcut-to-adiabaticity protocols. These results are not only useful for the realization of fast frictionless cooling, but also help us to address fundamental problems of the quantum speed limit and thermodynamics.
Frictionless atom cooling in harmonic traps is formulated as a time-optimal control problem and a synthesis of optimal controlled trajectories is obtained. This work has already been used to determine the minimum time for transition between two thermal states and to show the emergence of the third law of classical thermodynamics from quantum thermodynamics. It can also find application in the fast adiabatic-like expansion of Bose-Einstein condensates, with possible applications in atom interferometry. This paper is based on our recently published article in SIAM J. Control Optim. [1].
Fast adiabatic cooling procedures have important implications for the attainability of absolute zero. While traditionally adiabatically cooling a system is associated with slow thermal processes, for the parametric quantum harmonic oscillator fast frictionless processes are known, which transfer a system from an initial thermal equilibrium at one temperature into thermal equilibrium at another temperature. This makes such systems special tools in analyzing the bounds on fast cooling procedures. Previous discussions of those systems used frictionless cooling assuming real frequencies of the oscillator. Using a control with imaginary frequencies (repulsive potential) revises previous implications for the possible operation of a quantum refrigerator. Here we discuss these requisite revisions in the context of the third law of thermodynamics. In addition to minimum time controls, which are always of the bang-bang form, fast frictionless processes with a continuous variation of the frequency have been presented previously in the literature. Such continuous variation controls have been experimentally verified by cooling a Bose-Einstein condensate, while minimum time controls still await verification. As some implementations may indeed not be able to implement the instantaneous jumps in frequency required by bang-bang controls, constraining the rate of change in the frequency calls for ramped bang-bang solutions. We present such solutions and compare their performance to the continuous controls used in the experiment.
Interferometry with ultracold atoms promises the possibility of ultraprecise and ultrasensitive measurements in many fields of physics, and is the basis of our most precise atomic clocks. Key to a high sensitivity is the possibility to achieve long measurement times and precise readout. Ultracold atoms can be precisely manipulated at the quantum level and can be held for very long times in traps; they would therefore be an ideal setting for interferometry. In this paper, we discuss how the nonlinearities from atom–atom interactions, on the one hand, allow us to efficiently produce squeezed states for enhanced readout and, on the other hand, result in phase diffusion that limits the phase accumulation time. We find that low-dimensional geometries are favorable, with two-dimensional (2D) settings giving the smallest contribution of phase diffusion caused by atom–atom interactions. Even for time sequences generated by optimal control, the achievable minimal detectable interaction energy Δ E min is of the order of 10 −4 μ, where μ is the chemical potential of the Bose–Einstein condensate (BEC) in the trap. From these we have to conclude that for more precise measurements with atom interferometers, more sophisticated strategies, or turning off the interaction-induced dephasing during the phase accumulation stage, will be necessary.
In this thesis we use optimal control theory to control a BoseEinstein condensate. We focus on an experiment conducted at the Vienna University of Technology where a Bose-Einstein condensate is produced and trapped on an atom chip, and two entangled matter waves are produced. Such a twin atom beam has many technical applications and is necessary for matter wave interferometry, for example. One critical step in this experiment is the excitation of the condensate, which is achieved by displacing the minimum of the confinement potential (“shaking”). In order to find a time dependent potential that excites the condensate, optimal control theory is necessary and able to produce satisfying results. However, theoretical predictions and experimental data do not match perfectly. This is mainly due to the impulse response filter, causing a difference between the calculated and the experimentally accessible control field. To solve this problem in this thesis, we incorporate the effect of this impulse response filter in the optimal control algorithm. The new system is then not local in time anymore and there are new boundary conditions to meet. These problems are overcome and the obtained control fields are much closer to the experiment than without considering the effects of the impulse response filter. It is even possible to find working control fields with a transition time of only two milliseconds or to transfer the condensate in a higher excited state.