DOI 10.1007/s10946-016-9597-1

Journal of Russian Laser Research, Volume 37, Number 5, September, 2016

LASER-INDUCED CLIMBING OF COLD ATOMS AGAINST THE GRAVITY Sergey V. Prants Laboratory of Nonlinear Dynamical Systems Pacific Oceanological Institute of the Russian Academy of Sciences Vladivostok 690041, Russia ∗ E-mail:

[email protected] Abstract

We demonstrate climbing of cold atoms against the gravity in a one-dimensional vertical laser standing wave. At an appropriately chosen laser–atom detuning, we show that freely falling atoms change the direction of motion and move upward. The effect is due to a tiny interplay between internal and external atomic degrees of freedom in a rigid optical lattice.

Keywords: atomic climbing, vertical optical lattice, chaotic walking.

1.

Introduction

The mechanical action of light upon neutral atoms, placed in a laser standing wave, is caused by dispersion (gradient) and dissipative forces [1, 2]. The dissipative force arises due to absorption and emission of photons. The dispersion force is proportional to a synchronized component of the atomic electric dipole moment and to the electric-field gradient; it is zero in a running laser wave. At a blue laser–atom detuning, ωf − ωa > 0, the dispersion force “pulls” atoms to nodes of the standing wave. At a red one, ωf − ωa < 0, it “pushs” them to antinodes [3]). Thus, sufficiently cold atoms can be trapped in a standing-wave field and oscillate around a node or antinode [4]. The dispersion force disappears at zero detuning. In contrast to the dissipative force, it does not saturate with increasing Rabi frequency and does not disappear in the absence of spontaneous emission. The tremendous progress in cooling and trapping of atoms, the tailoring of optical potentials of a desired form and dimension, and controlling the level of dissipation and noise make possible impressive experiments to study fundamental principles of the atom–field interaction with a number of applications in physics, chemistry, and biology [5–7]. Cold atoms confined in an optical lattice are perspective quantum objects to study the intriguing phenomenon of quantum chaos (see, e.g., [8, 9]). In the semiclassical and Hamiltonian limits, if one treats atoms as point-like particles and neglects spontaneous emission and other losses of energy, the atom in a standing-wave laser field may be treated as a nonlinear dynamical system with coupled internal (electronic) and external (mechanical) degrees of freedom [10, 11]. A number of nonlinear dynamical effects have been found in these limits including chaotic Rabi oscillations [10, 11], Hamiltonian chaotic atomic transport and dynamical fractals [12–14], L´evy flights, and anomalous diffusion [15]. Detailed theories of Hamiltonian and dissipative chaotic transport of atoms in a standing wave have been developed in [16] and [17], respectively. The atomic chaotic transport is caused by local instability of the center-of-mass motion in a standing wave. It has much in common with Manuscript submitted by the authors in English on July 25, 2016. c 2016 Springer Science+Business Media New York 1071-2836/16/3705-0459 

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transport phenomena in very different disciplines, from underwater acoustics [18,19] to chaotic advection in fluids [20, 21]. In this paper, we study analytically and numerically the motion of cold atoms in a one-dimensional vertical optical lattice taking into account the Earth’s acceleration. A small sample of laser-cooled atoms is released from a trap located above the cavity with a standing-wave field. We show that varying only one parameter, the detuning between the frequencies of a working atomic transition ωa and of the laser field ωf , it is possible to create conditions under which a large part of the initially free falling atoms does not just slow fall but even changes the direction of motion and moves upward against the gravity. It becomes possible with a rigid vertical optical lattice without any amplitude and frequency or phase modulation.

2.

Climbing of Atoms in a Vertical Optical Lattice

Let us consider a one-dimensional optical lattice created by the interference pattern of two vertical counter-propagating laser beams with wavelength λf and frequency ωf . Cooled atoms are released from a trap located above the cavity with a standing-wave field. Atoms effectively see a periodic potential with a depth dictated by the laser intensity. We write the Hamiltonian of a two-level atom in such a lattice in the frame rotating with the laser frequency as follows: 2 ˆ = P +  (ωa − ωf )ˆ H σz − Ω (ˆ σ− + σ ˆ+ ) cos kf X + F X, 2ma 2

(1)

where σ ˆ±,z are the Pauli operators for the internal atomic degrees of freedom, X and P are the classical atomic position and momentum, ωa and Ω are the atomic transition and the maximum Rabi frequencies, respectively, and F is the gravity force. In the semiclassical approximation, the atom with quantized internal dynamics is treated as a pointlike particle to be described by the Bloch–Hamilton equations of motion without relaxation terms [16] x˙ = ωr p,

p˙ = −u sin x − κ,

u˙ = Δv,

v˙ = −Δu + 2z cos x,

z˙ = −2v cos x,

(2)

where x ≡ kf X, p ≡ P/kf , u and v are synchronized with the laser field and quadrature components of the atomic electric dipole moment, respectively, z is the atomic population inversion, and the dot denotes differentiation with respect to the dimensionless time τ ≡ Ωt. The set of equations (2) has the three control parameters ωr ≡ kf2 /ma Ω,

Δ ≡ (ωf − ωa )/Ω,

κ ≡ F/kf Ω,

(3)

which are the normalized recoil frequency ωr , atom–field detuning Δ, and the gravity force or normalized acceleration κ, which is chosen to be directed in the negative direction of the optical axis x. The system has two integrals of motion, namely, the total energy H≡

Δ ωr 2 p + κx − u cos x − z 2 2

(4)

and the length of the Bloch vector u2 + v 2 + z 2 = 1. The center-of-mass motion is described by the first two equations in the set of equations (2). The normalized gravity force κ accelerates the atoms. The dispersion force u sin x changes while atoms move

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through the lattice. These changes are dictated not only by the laser-field gradient but the behavior of the u-component of the atomic dipole moment governed by the other three equations in (2). Far from resonance with |Δ|  1, atoms practically do not absorb and emit photons, and they just fall with the gravity acceleration. At exact resonance, atoms absorb and emit photons effectively. The optical potential is absent at Δ = 0, and cold atoms fall under the Earth’s gravity with a small modulation of their velocity due to the photon recoil effect. The interesting situation arises at small and medium values of the detuning, where the interplay between internal and external atomic degrees of freedom provides rich and strongly nonlinear atomic dynamics. It has been found in [16] that the Bloch-vector component u and, therefore, the dispersion force in rigid horizontal optical lattices without any external force change abruptly when atoms cross nodes of the standing wave. This jump-like behavior can be described by the stochastic map [16] um = sin(Θ sin φm + arcsin um−1 ),

(5)

 u just after the mth where Θ ≡ |Δ| π/ωr pnode is the angular amplitude of the jump, um is the value of node crossing, φm are random phases to be chosen in the range [0, 2π], and pnode ≡ 2H/ωr is the value of the atomic momentum when the atom crosses a node, which is the same with a given value of the energy H for all the nodes. Such random-like jumps may cause chaotic motion of atoms in absolutely deterministic horizontal optical lattices. Depending on the lattice and atomic parameters, the regimes of the chaotic motion can be different including chaotic oscillations in wells of the optical potential, chaotic ballistic transport, and chaotic walking when atoms change the direction of motion with alternating trappings in the wells of the optical potential and flights over its hills [15, 16]. The jump-like behavior of the dispersion force should occur in a vertical lattice as well. However, this force competes with the gravity of freely falling atoms. The point is: Is it large enough and oriented upwards in order to turn atoms back against the gravity and force them to move upward? The set of equations (2) is a nonlinear Hamiltonian autonomous system with two and a half degrees of freedom. Existence of two integrals of motion reduces it to an effective dynamical system with one and a half degree of freedom. It is a minimum requirement for the existence of chaotic motion in the sense of an exponential sensitivity to small variations in the initial conditions and/or control parameters. Chaotic motion is known to be characterized by a positive Lyapunov exponent that is the mean rate of the exponential divergence of initially close trajectories and serves as a quantitative measure of dynamical chaos. We computed the finite-time Lyapunov exponent for Eqs. (2) by the method developed in [23, 24] and found it to be positive at the detunings in the range |Δ| < 0.5 and under reasonable values of κ. Therefore, we expect chaotic walking of atoms with the detuning values in those range. In our numerical experiments, we assume that two-level atoms are initially prepared in the ground states u0 = v0 = 0, z0 = −1 at fixed value of the normalized recoil frequency ωr = 10−3 . All the control parameters are normalized to the Rabi frequency Ω; so, we can vary all the parameters just varying the laser intensity. A small sample of N = 103 cooled atoms are released from a trap and fall downward. At τ = 0, the laser standing wave is turned on. At this moment, all the atoms are assumed, for simplicity, to have the same momentum p0 = −10, but their positions at τ = 0 are homogeneously distributed in the space −π/10 ≤ x0 ≤ π/10. The trajectories of all the atoms are computed till a fixed time moment T. In Fig. 1, we illustrate our main result, climbing of atoms against the gravity in a vertical standing wave. The distribution of 103 atoms is shown on the phase plane x − p at the moment of time T = 103

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at the positive detuning Δ = 0.15. At κ = 0.001, this distribution is practically Gaussian with a slight asymmetry in favor of falling atoms (Fig. 1 a). At κ = 0.01. the atomic distribution is more asymmetric as expected (Fig. 1 b). The points with positive values of x and p mean that those atoms changed their direction of motion and moved upward climbing against the gravity. Fixing the atoms with positive values of momentum in a real experiment would mean that one observe not only the effect of climbing of atoms against the gravity but the effect of chaotic walking as well because just that effect is a single and ultimate reason for the climbing. The maximum Lyapunov exponent was computed to be equal to 0.1 at the detuning value Δ = 0.15 and under the chosen initial conditions. After releasing from the trap, atoms start to move in the negative x direction. In the regime of chaotic walking, they may change the direction of motion when crossing nodes of the standing wave. As a result, some atoms in the cloud are able to move in the positive x direction for a time long enough to climb upward for a large number of the laser wavelength. Increasing the integration time ten times, we obtain the picture shown in Fig. 2. The atomic distribution is again

Fig. 1. Distribution of atoms on the phase plane at time moment T = 103 at κ = 0.001 (a) and κ = 0.01 (b), with Δ = 0.15. The points with x > 0 and p > 0 show atoms climbing against the gravity.

Fig. 2. The same as in Fig.1 but at time moment T = 104 at κ = 0.001 (a) and κ = 0.01 (b). An asymmetry in atomic distribution appears at increased value of the normalized gravity force, κ = 0.01.

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practically Gaussian at κ = 0.001 (Fig. 2 a). At κ = 0.01, it is much more asymmetric with a tail of accelerating atoms due to increasing effective gravity (Fig. 2 b). However, there is a large number of climbing atoms even in this case. From the expression for the total energy (4), we can estimate the conditions under which atoms continue to move in the same direction after crossing a node or change the direction of motion not reaching the nearest antinode. The Bloch-vector component u changes its values after crossing a node in a random-like manner (5) taking a value that dictates to the atom either to prolong its motion in the same direction or to turn back. The conditions for different regimes of motion depend on whether the crossing number m is even or odd. Motion in the same direction occurs at (−1)m+1 um < H and turns at (−1)m+1 um > H because even values of m correspond to cos x > 0, whereas odd values to cos x < 0. Climbing of atoms may occur if the atomic energy is in the range 0 < H < 1. At H < 0, the atoms cannot reach even the nearest node and oscillate in the first potential well in a regular manner. At H > 1, the values of u always satisfy the flight condition, and the atoms just fall. Since the atomic energy should be positive in the regime of chaotic transport, the corresponding conditions for climbing of atoms can be summarized as follows: at |u| < H, the atom always falls, whereas at |u| > H, the atom either falls or turns back and move upward depending on the sign of cos x within a given interval of motion.

3.

Conclusions

In conclusion, we showed in numerical experiments that freely falling cold atoms in a rigid vertical optical lattice can be turned back to move against the gravity under appropriately chosen detuning. We explained this effect by a specific behavior of the electric-dipole component u of atoms after crossing nodes of the standing wave, namely, by its random-like jumps. The effect can be observed in a real experiment with the freely falling atomic cloud fixing the atoms with positive momentum. We used some approximations. As to 1D, one-dimensional vertical optical lattices are now routine created (see, e.g., [25–27]). As to chosen values of the control parameters, they are defined mainly by the value of the Rabi frequency Ω. The chosen values correspond to real atoms and to Ω in the range 105 − 107 rad·s−1 . We neglected spontaneous emission. After emitting a photon spontaneously, the atom goes to the ground state, u0 = v0 = 0, z0 = −1. Its momentum changes abruptly to a quantity in the range −1 ≤ δp ≤ 1. A trajectory of a specific atom would change, of course, after many events of spontaneous emission as compared to its relaxation-free trajectory. From the statistical point of view, the behavior of trajectories for a large number of atoms is not expected to change significantly in the sense that, instead of given chaotically walking relaxation-free atoms, we obtain another chaotically walking spontaneously-emitting atoms. Thus, the effect of climbing of atoms against the gravity should be observable with spontaneously emitting atoms as well.

Acknowledgments The author acknowledges Dr. M. Yu. Uleysky for his help in preparing figures.

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