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Deviations from Generalized Equipartition in Confined, Laser Deviations from generalized Equipartition in Confined, Laser Cooled Atoms Gadi Afek,∗ Alexander Cheplev, Arnaud Courvoisier, and Nir Davidson Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel We observe a significant steady-state deviation from the generalized equipartition theorem, one of the pivotal results of classical statistical mechanics, in a system of confined, laser cooled atoms. We quantify this deviation, measure its dynamics and show that its steady state value quantifies the departure of non-thermal states from thermal equilibrium even for anharmonic confinement. In particular, we find that deviations from equipartition grow as the system dynamics becomes more anomalous. We present numerical simulations that validate the experimental data and reveal an inhomogeneous distribution of the kinetic energy through the system, supported by an analytical analysis of the phase space. The 100 year-old generalized equipartition theorem [1], In this Paper, we present a detailed experimental in- one of the cornerstones of classical statistical physics, re- vestigation of the steady-state deviation from equiparti- mains of great interest throughout various fields of re- tion for trapped, laser cooled atoms in contact with a search to this day [2–4]. It states that for a system in non-thermal heat bath, implemented by a 1D, dissipa- thermal equilibrium with a heat bath of temperature T , tive optical lattice. For completion, we also investigate for any generalized coordinate qi and Hamiltonian H: numerically the effect of anharmonicity of the confining potential on the dynamics and magnitude of this devia- ∂H q = δ k T (1) tion. Finally, as a basis for further work, we present a i ∂q ij B novel prediction of the position dependence of the local j kinetic energy for such confined Sisyphus cooled atoms, where kB is the Boltzmann constant, δij is a Kronecker supported by analytics and numerics. Delta and h...i denotes ensemble averaging. An immedi- ate result of this theorem is the well known equipartition theorem [5, 6], valid for degrees of freedom which ap- pear quadratically in the Hamiltonian, in which case the relation implies equipartition of energy among those de- grees of freedom. Other significant extensions have been Using the equipartition theorem to quantify departure from thermal equilibrium shown for finite sized systems [7], generalized Canoni- cal ensembles [8] and non-extensive thermodynamics [9]. While the equipartition theorem is strictly correct only Any departure of a 1D confined system with coordi- in thermal equilibrium, it was extended and applied to nates (x, p) and Hamiltonian H from thermal equilibrium other thermodynamic systems and observed to hold even can be parametrized using the Equipartition parameter outside of thermal equilibrium [10]. χ [34, 35]: Ultracold atomic systems have been pushing the un- derstanding of statistical physics for several decades, ∂H ∂H and have recently began to explore aspects of non- χ ≡ p x (2) ∂p ∂x equilibrium physics [11–15]. An especially interesting s system for probing non-equilibrium statistical mechanics is that of ultracold atomic ensembles in dissipative one- which depends explicitly on the details of the confining dimensional (1D) optical lattices, where the heat bath is potential. Given that χ = 1 for systems in thermal equi- implemented by the field of the lattice lasers. The main librium, deviations of χ from unity imply non-thermal advantage of such a system is the unique control over distributions and a possible break of energy-probability experimental parameters, allowing fine-tuning of the dy- equivalence. For the simple case of harmonic confine- arXiv:1909.07584v4 [cond-mat.stat-mech] 9 Feb 2020 namics. In addition to it being an experimentally and ment, one can derive the harmonic equipartition param- theoretically well established test bed for anomalous dy- eter χH from Eq. 2, namics [16–33], it has recently been linked with the no- σv tion of non-thermal equilibrium [34, 35]. Through exten- χH ≡ , (3) ωσx sive analysis of the phase space dynamics of such a system confined in a harmonic potential, a prediction has been where σx and σv are the respective standard deviations put forth of a violation, under certain conditions, of the of the position and velocity distributions (both of which equipartition theorem. are Gaussian for a harmonic potential and thermal equi- librium), and ω is the harmonic trap frequency. χH is a leading order approximation of χ for any continuous po- ∗ Present address: Wright Laboratory, Department of Physics, tential with a minimum, and is, in practice, considerably Yale University, New Haven, Connecticut 06520, USA easier to access experimentally than χ for many systems. 2 practically unaffected. The Rayleigh length of the beam is > 4 cm, much larger than any other relevant length (a) scale. Extra care is taken to avoid reflections that may cause interference affecting the dynamics. The > 2 sec long evaporation, much longer than the ∼ 100 msec collision time leaves the atoms in thermal equilibrium with the confining potential [36]. The atoms are then coupled for a duration t to a non-thermal heat bath, 20 implemented using a 1D dissipative Sisyphus lattice, where they may exhibit anomalous dynamics, depending on the modulation depth U0 [25, 33]. Set by the power 0 and detuning of the lattice beams from the relevant (b) atomic transition, U0 is the main control parameter of -20 the experiment. The other experimental parameters are 0 2 4 6 8 10 12 similar to those in [33]. The trap depth is ∼ 3 MHz, small compared to the ∼ 60 MHz detuning of the lattice, 0.15 32 (c) (d) rendering trap-induced differential AC stark shifts negli- 30 gible. Each measurement of the equipartition parameter 0.1 28 χH (U0,t) is comprised of three separate experiments: 0.05 trap oscillations, time of flight and extrapolation to zero 26 density of an in situ image of the cloud, giving access 0 24 to the information needed to calculate the equipartition 0 50 0 0.5 1 parameter of Eq. 3. The in situ absorption images are taken as the cloud is released from the trap, and capture both the atoms trapped in the focus, those held in the FIG. 1. Measurement scheme for the equipartition parameter, area of the beams removed from the overlapped focii. χH ≡ σv/ωσx. (a) A sketch of the experimental setup. Laser To ensure we do not wrongly include these atoms in our cooled 87Rb atoms are trapped in a crossed dipole trap (light analysis, we fit the data to a sum of two Gaussians and red) overlapped with a strong, single beam tube trap (bold use the narrower one [37]. red), and coupled to a non-thermal heat bath implemented by a set of 1D Sisyphus cooling lattice beams (orange) through Fig. 1 (b) shows a typical trap oscillation experiment, dichroic mirrors. The atoms propagate in this combined po- tential and are subsequently imaged. (b) The trapped cloud in which center of mass oscillations are excited using a is kicked with a short, directional pulse of near-resonant light short, near resonant light pulse. The atoms are sequen- to excite subsequent center of mass oscillations, and the cen- tially imaged as a function of the time elapsed after the ter of mass position is extracted (circles). Trap frequency pulse. The measured frequency is ω =2π × (332 ± 2) Hz ω is calculated by fitting the data to an exponentially de- (used throughout the Paper) with approximately 1.6 os- caying sine. The decay of the center of mass oscillations is cillations before 1/e decay of the contrast, attributed to attributed to anharmonicity of the confining potential. (c) dephasing of the ensemble-averaged oscillations due to Typical result of a time of flight experiment (circles), with a 2 2 2 2 the anharmonicity of the confining potential. The trap fit to σx(τ) = σx(τ = 0)+ σvτ (solid line). (d) Size of the frequency itself is unaffected by anharmonicity for atoms atomic cloud. Obtained by scanning the number of atoms, much colder than the trap depth. and hence density, in the trap and extrapolating to zero den- sity using microwave pulses (blue circles, see text). Solid line The width of the velocity distribution is measured us- is a linear fit. The data presented in (b-d) corresponds to ing time of flight. The cloud is released and allowed to measurements at thermal equilibrium, yielding the value of expand in one dimension for a time τ within the tube ± 2 2 χH = 0.82 0.02. trap. Its size is fitted with the relation σx(τ) = σx(τ = 2 2 0) + σv τ between the standard deviation of the spatial distribution σx(τ) and that of the velocity distribution Experimental σv. Fig. 1 (c) shows the result of such a measurement, giving σv = 42±1 mm/sec. We verify that scattered light In the experiment [Fig. 1 (a)], a cloud of 87Rb atoms from the unidirectional tube trap does not affect the dy- is magneto-optically trapped and then cooled down namics substantially by allowing the atoms to expand in to a temperature of ∼ 20µK. The final cooling step is it for a relatively long time (> 100 msec, much longer optical evaporation in a far detuned, 1064 nm crossed than the duration of the experiment) and verifying that dipole trap focused down to a waist of 60µm overlapped the center of mass of the cloud does not shift due to scat- with a strong, ∼ 180W, single 1070 nm beam tube trap tering of trap photons by more than 1% compared to its (YLR-200-LP-AC-Y14, IPG photonics), loosely focused initial position. to a waist of 120µm to provide strong confinement in Measuring the in situ cloud size is challenging, mostly the radial direction while leaving the axial dynamics due to the optical density of the clouds, biasing the 3 output of our absorption imaging.
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