Fast Optimal Frictionless Atom Cooling in Harmonic Traps: Shortcut to Adiabaticity

Abstract

A method is proposed to cool down atoms in a harmonic trap without phase-space compression as in a perfectly slow adiabatic expansion, i.e., keeping the populations of the instantaneous initial and final levels invariant, but in a much shorter time. This may require that the harmonic trap becomes an expulsive parabolic potential in some time interval. The cooling times achieved are also shorter than previous minimal times using optimal-control bang-bang methods and real frequencies.PACS numbers: 37.10. De, 37.10.Vz A fast adiabatic expansion in a short finite time looks like a contradiction in terms. An "adiabatic" process in quantum mechanics is a slow process where the system follows at all times the instantaneous eigenvalues and eigenstates of the time-dependent Hamiltonian. This is in a sense maximally efficient as the populations do not change, i.e. there is no heating or friction, but the price to pay is that the long times needed may render the process useless or even impossible to implement. Thus, a highly desirable goal is to prepare the same final states and energies of the adiabatic process in a given finite time t f , without necessarily following the instantaneous eigenstates along the way. We would also like the process to be robust with respect to arbitrary initial states. If fulfilled, this old goal [1] has important implications. In particular, cooling without phase-space compression, which is all that is needed for many applications other than Bose Einstein condensation, could be performed in fast cycles increasing, for example, the flux of cold atoms produced and the signal to noise ratio in an atomic clock [2], or in cold-atom pulsed beam experiments and related technology [3]. This goal also includes as a particular case a long standing question in the fields of optimal control theory and finite time thermodynamics, namely, to optimize the passage between two thermal states of a system [4,5,6,7]. For time-dependent harmonic oscillators, minimal times have been established using "bang-bang" real-frequency processes believed up to now to be optimal [6], in which the frequencies are changed suddenly at certain instants but kept constant otherwise. In this letter we shall describe a robust solution to the stated general goal for atoms trapped in a time-dependent harmonic oscillator which applies both to equilibrium and non-equilibrium states. In particular we describe cooling processes performed in a time interval smaller than the minimal time of the bang-bang methods considered so far. We shall for simplicity describe our method for states representing single atoms of mass m, but the same results are immediately applicable to N -body non-interacting fermions or to a Tonks-Girardeau gas [8], and generalizations will be relevant for other driving processes, such as cold atom launching or the transport of ultracold atoms with optical tweezers [9].We consider an effectively one dimensional time dependent harmonic oscillator, H =p 2 /2m + mω(t) 2q2 /2, with an initial angular frequency ω(0) > 0 a...

Transitionless quantum drivings for the harmonic oscillator

“…In this section we describe a different method for transitionless dynamics of the harmonic oscillator [6]. A harmonic oscillator such as H 0 (t) in Eq.…”

“…Once the a j are determined from (40), ω(t) is obtained from the Ermakov equation (37), and one gets directly a transitionless Hamiltonian H II (t) = H 0 (t) with a local, ordinary, harmonic potential, but note that ω(t) 2 may become negative for some time interval, making the potential an expulsive parabola [6,18]. The II method is thus clearly distinct from from TT and implements a different Hamiltonian.…”

“…We shall term it "transitionless-tracking" approach, or TT for short; the other one [6,7] engineers the Lewis-Riesenfeld invariant [8] by an inverse method [9] to satisfy the desired boundary conditions; we shall call this method "inverse-invariant", or II for short. In the basic version of TT the dynamics is set to follow at all intermediate times the adiabatic path defined by an auxiliary Hamiltonian H 0 (t) (in our case a regular harmonic oscillator with frequency ω(t) and boundary conditions ω(0) = ω 0 and ω f = ω(t = t f )), and its instantaneous eigenvectors |n(t) , up to phase factors.…”

Deterministic entanglement generation between a pair of atoms on different Rydberg states via chirped adiabatic passage

“…In the future, we will further explore the ways to improve the fidelity of entanglement with more appropriate Rydberg states, e.g. involving longer lifetimes or larger Rabi frequencies, or by using other approaches such as shortcut to adiabaticity [65], adiabatic rapid passage [66] and so on. Besides, we will extend our model to a N-atom system for realizing a many-body entanglement by the adiabatic tools.…”

Robust coherent superposition of states using quasiadiabatic inverse engineering

“…Shortcut to adaibaticity (STA) has been proposed as a set of techniques to speed up the slow adiabatic processes, while keeping or enhancing robustness [19,20,21,22]. Also, a similar technique called designer evolution of quantum systems by inverse engineering (DEQSIE) [23] has also been developed for synthesizing Hamiltonians for the desired quantum state evolution.…”