We show that thermalization of the motion of atoms at negative temperature is
possible in an optical lattice, for conditions that are feasible in current
experiments. We present a method for reversibly inverting the temperature of a
trapped gas. Moreover, a negative-temperature ensemble can be cooled, reducing
abs(T), by evaporation of the lowest-energy particles. This enables the
attainment of the Bose-Einstein condensation phase transition at negative
temperature.Comment: 4 pages 5 figures; v4: Typo corrections. Accepted Phys. Rev. Let

Mosk, A. P.

English

arXiv

We show that thermalization of the motion of atoms at negative temperature is possible in an optical lattice, for conditions that are feasible in current experiments. We present a method for reversibly inverting the temperature of a trapped gas. Moreover, a negative-temperature ensemble can be cooled (reducing |T|) by evaporation of the lowest-energy particles. This enables the attainment of the Bose-Einstein condensation phase transition at negative temperature

Mosk, Allard

NARCIS 

Atomic Gases at Negative Kinetic Temperature

University of Twente Research Information

PRL 95, 040403 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending22 JULY 2005Atomic Gases at Negative Kinetic TemperatureA. P. MoskDepartment of Science & Technology, and MESA+ Research Institute, University of Twente,P.O. Box 217, 7500 AE Enschede, The Netherlandsand Laboratoire Kastler-Brossel, Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris CEDEX 05, France(Received 26 January 2005; published 21 July 2005)0031-9007=We show that thermalization of the motion of atoms at negative temperature is possible in an opticallattice, for conditions that are feasible in current experiments. We present a method for reversiblyinverting the temperature of a trapped gas. Moreover, a negative-temperature ensemble can be cooled(reducing jTj) by evaporation of the lowest-energy particles. This enables the attainment of the Bose-Einstein condensation phase transition at negative temperature.DOI: 10.1103/PhysRevLett.95.040403 PACS numbers: 03.75.Hh, 03.75.Lm, 05.70.2aAmplification of a macroscopic degree of freedom, suchas a laser or maser field, or the motion of an object, is aphenomenon of enormous technological and scientific im-portance. In classical thermodynamics, amplification is notpossible in thermal equilibrium at positive temperature. Incontrast, due to the fluctuation-dissipation theorems [1,2],amplification is a natural phenomenon, and fully consistentwith the second law of thermodynamics, in a negative-temperature heat bath [3–6]. Negative temperatures wereintroduced by Purcell and Pound [7] to describe spinsystems, and further generalized theoretically by Ramseyand Klein [8,9]. Useful introductions to thermodynamics atnegative temperature are given by Landau and Lifshitz [10]and by Kittel and Kroemer [11]. Negative temperatures areused to describe distributions where the occupation proba-bility of a quantum state increases with the energy of thestate. The prerequisite for a negative-temperature en-semble to be in internal equilibrium is that it occupies apart of phase space where the entropy decreases withinternal energy. An ensemble in this part of phase spacecannot be in thermal equilibrium with any positive-temperature system. An increasing occupation number(and a decreasing entropy) can only exist up to a certainmaximum energy. Unlike the magnetic energy of spins,kinetic energy is not bounded from above in free space,which apparently rules out a negative kinetic temperature.Here, we show that the band structure of a moderately deepoptical lattice defines an upper bound to the kinetic energyof atoms in the first band. Therefore, atoms in an opticallattice form a simple system in which thermalization of thekinetic energy is possible at negative temperatures.Optical lattices, which are periodic potentials createdthrough optical standing waves [12], impose a band struc-ture on the motion of ultracold atoms. The kinetic energyof atoms in the lattice depends on their effective mass orband mass, which, in analogy to the case of electrons in asemiconductor, can be strongly different from the normalrest mass. The effective mass can even become negative, ashas been demonstrated in one-dimensional optical lattices[13,14]. An isotropic negative-effective mass can be real-05=95(4)=040403(4)$23.00 04040ized in the vicinity of a band gap in a three-dimensional(3D) optical lattice. The band gap is an energy range insidewhich no single-particle eigenstates exist. If the atoms areconstrained to energies just below the band gap the effec-tive mass will be negative in all directions and the entropyof the ensemble will decrease with energy. This is thecondition for existence of a thermal ensemble with nega-tive temperature [8].For thermalization to take place, the characteristic timescale for exchange of energy between particles must beshorter than time scales associated to heating and loss. Inother terms, we need a high rate of intraband scattering,with negligible interband scattering. Interband scattering isa collision process between two atoms in the first (‘‘va-lence’’) band which causes one atom to be promoted to thesecond (‘‘conduction’’) band while the valence band atomloses energy. Additional losses may be caused by interbandtunneling, a one-body process in which an atom tunnelsthrough a classically forbidden region, to reappear in adifferent band.In this Letter, we show that all the requirements forthermalization at negative temperature can be met in a3D optical lattice. First, we show that an optical latticecan have a 3D band gap for atoms, at experimentallyfeasible lattice intensities. Moreover, interband scatteringcan be completely suppressed, and confinement of thenegative-mass atoms can be easily accomplished using amagnetic trap for high-field seeking states. We calculaterates of interband tunneling and find they can be madenegligible. Finally, we show how a trapped gas at positivetemperature can be reversibly converted to a negativetemperature.The motion of atoms in a simple cubic 3D optical latticewith a harmonic confining potential is described by theSchrödinger equation (See, e.g., [12,15]),E @22mr2 Ulat  x2  y2  z2with Ulat  U0cos2kx cos2ky cos2kz:(1)3-1  2005 The American Physical SocietyPRL 95, 040403 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending22 JULY 2005Here m is the bare atomic mass,  is the strength of theharmonic potential, U0 is the depth of the standing wavepotentials that form the lattice, and k  2=, where  isthe wavelength of the lattice light. The above Schrödingerequation is trivially separable by setting E  Ex  Ey Ez. We now transform the equation to dimensionless units,~x; ~y; ~z  kx; ky; kz; (2)~  =Erk2; (3)ax Ex 12U0=Er; (4)q  U0=4Er; (5)with the lattice recoil energy Er  @2k2=2m. The one-dimensional equation of motion takes the shape of a modi-fied Mathieu equation,d2d~x2 ax  2q cos2~x  ~~x2  0: (6)In the absence of a parabolic potential (~  0), Eq. (6) isexactly the Mathieu equation. Its solutions are the Mathieufunctions, which are propagating Bloch-Flouquet wavesfor values of a and q in the bands, and evanescent functionsin the band gaps. The relevant bands are depicted in Fig. 1.Adopting the usual convention [16] we designate the ei-genvalues corresponding to the band bottoms bya0q; a1q, etc., and the band tops by b1q, etc., (seeFig. 1). If 0 < j~j  q=~x, the lattice is locally only weaklyperturbed by the parabolic potential, and the band structurewill remain essentially intact. The energy positions of thebands become position dependent, in the same way as in asemiconductor in an applied electric field. A negative valueof ~ corresponds to a confining potential for negative-effective mass atoms, see Fig. 2. The area under the valenceband in Fig. 2 is classically forbidden as the kinetic energyof the atoms in this region is less than zero. The areaFIG. 1. Band structure of the one-dimensional Mathieu equa-tion. White area: Bands, gray area: gaps. Dotted line: q  0:559,dashed line: q  0:936; see text.04040between the bands is Bragg forbidden; i.e., no propagatingmodes can exist here due to interference.For the negative-temperature phase to be metastable,there must be a band gap between the top of the valenceband, where the negative-mass states exist, and the bottomof the conduction band, where the effective mass is posi-tive. In a 1D system (with the motion in the other twodirections confined) this band gap exists for jqj> 0.However, in a 3D optical lattice, the bands may overlap.The maximum energy an atom in the valence band of a 3Dlattice can have is 3b1q, whereas the minimum energy inthe conduction band is 2a0q  a1q. The criterion forthe existence of a band gap is 3b1q< 2a0q  a1q,which is fulfilled for q > 0:559.If the 3D band gap is narrow, collisions between twovalence band atoms can promote an atom to the conductionband, where it has positive effective mass. This interbandscattering will not only lead to loss of atoms, but also toloss of energy, which is equivalent to heating (increasingjTj). We show now that interband scattering becomesenergetically forbidden in two-body processes at highenough q. The maximum energy of the initial state, con-sisting of two valence band atoms, is 6b1q. The minimumenergy of the final state, consisting of one valence and oneconduction band atom, is 	3a0q  	2a0q  a1q.The interband scattering process is energetically forbiddenif 6b1q< 5a0q  a1q, which is the case if q > 0:936.This value of q corresponds to a lattice depth almost anorder of magnitude less than the depth employed to create aMott-insulator phase [17]. In such a moderately deep lat-tice, thermalization rates and inelastic collision rates areall of the same order of magnitude as the respective ratesat positive temperature in the absence of a lattice [15].Through elastic collisions, the ensemble will always re-lax to a distribution with the highest possible entropy atits given energy, in this case a negative-temperaturedistribution.To investigate how the gas thermalizes in the negative-temperature phase space, it is useful to consider the semi-FIG. 2. Valence and conduction bands of a 1D optical lattice atq  0:936. Grey area: Gaps, white area: Bands.3-2PRL 95, 040403 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending22 JULY 2005classical density of states (DOS). The DOS can usually befound by taking the energy derivative of the Wentzel-Kramers-Brillouin (WKB) phase integral, a procedure re-cently applied to a one-dimensional optical lattice withharmonic confinement (positive ) [18]. For negative the WKB method cannot be used straightforwardly tocalculate wave functions, as the ‘‘Bragg reflection’’ bound-ary condition at the outer turning points behaves differentlyfrom classical turning points. The WKB formula for theDOS does not depend on the boundary conditions and canbe straightforwardly calculated in the one-band model of[18]. Within the one-band and WKB approximations, theDOS  at negative  is related to the DOS calculated forpositive  byE  E; (7)where in the case of negative , the top of the valence bandis taken as the energy zero. The 3D DOS crosses oversmoothly from a quadratic behavior at small negativeenergy to approximately square-root behavior at largernegative energy, as shown in Fig. 3. The DOS is nonzerofor any energy lower than the band edge at the trap center,so in principle the negative temperatures are not boundedin an infinite lattice. In fact, this asymmetric behavior ofthe DOS makes thermal equilibrium at a positive tempera-ture impossible, which is in contrast to spin systems [7],where thermal equilibrium at positive and negative tem-perature can be studied in the same field configuration.We now examine the escape of the atoms through inter-band (Zener) tunneling. The trapped negative-mass va-lence band states are mixed with antitrapped conductionband states at the same energy, as the evanescent tails ofthese states extend through the Bragg forbidden region.The Zener tunneling rate Zener may be estimated using[19]Zener  !A exp2Z 10Im	kBlochxdx; (8)where kBlochx is the Bloch wave vector of the Mathieuequation corresponding to the energy E x2, and !A isFIG. 3. Density of states E in a three-dimensional latticewith    103. The energy E is taken with respect to the topof the first (valence) band.04040an attempt frequency of the order of the trap oscillationfrequency (typically <1 kHz). The resulting tunnelingtime constant is of order 105 seconds at ~  103. For104 < ~< 102 we checked Eq. (8) numerically byhigh-precision quadrature of Eq. (6), and found agreementwithin an order of magnitude.The conditions for existence of a negative-mass Bose-Einstein condensate (BEC) were recently studied by Puand coworkers [15]; they also investigated the possibilityof creating such a condensate by phase imprinting, andfound this likely to be an inefficient mechanism as manyatoms are transferred to other bands. Other mechanisms,such as accelerating the atoms uniformly to completeone-half Bloch oscillation, may be more efficient, espe-cially since in a BEC nonlinearity due to the interactionscounteracts dephasing [20,21]. Using these methods, thenegative-mass BEC can be created in a mechanically un-stable state as the interatomic scattering length asc remainspositive [15].We propose a radically different mechanism for theproduction of negative-temperature gases, based on thereversible creation of a Mott-insulator phase from a BEC[17]. The idea is to ‘‘freeze’’ a positive-temperature gasinto a Mott insulator, by turning on a strong optical lattice.The Mott phase is then ‘‘molten’’ in a different trap con-figuration and with different interatomic interactions, toproduce a negative-temperature gas. The procedure is out-lined in Fig. 4, we will describe it here for a pure BEC butthere is no a priori reason why it would not work whensome thermal excitations are present. We start with atrapped BEC at > 0 and asc > 0 4(a), and turn on astrong optical lattice, as is done in [17]. A Mott-insulatorphase will form in which the atoms are confined to theirlocal potential wells 4(b). We assume one atom is presentper lattice site. Now we remove the trap 4(c), which can beFIG. 4 (color online). Method for reversibly transforming apositive-temperature gas (a) into a negative-temperature gas (f).Each step is reversible; see text. (a) trapped gas at positive T.(b) a strong optical lattice, (c) a strong optical lattice withouttrap, (d) a strong optical lattice with changed asc, (e) a strongoptical lattice with negative trap, (f) an optical lattice at q 0:936 with a negative-temperature gas.3-3PRL 95, 040403 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending22 JULY 2005done rapidly without creation of entropy, as the atoms aretightly bound to their respective lattice sites. Then wechange the sign of asc 4(c) and 4(d), for example, by tuningthe magnetic bias field near a Feshbach resonance (See,e.g., [22]). Since the atoms are at different lattice sites anddo not interact, this can be done rapidly without causinglosses. The system passes through the point where both ascand  are zero, here the temperature of the Mott phase isnot defined, but the entropy is. This passage transforms theMott insulator from the lowest-energy many-particle stateto the highest energy in the valence band. The confiningpotential is reconfigured to change the sign of  4(e). Thiscan be done either by changing the confinement field, or bychanging the internal state of the atoms. A magnetic fieldminimum will trap negative-mass atoms in their lowestZeeman state [15]. No change in the spatial distributionof atoms will take place due to the extremely low tunnelingrate in the Mott insulator, so this step can proceed rapidly.When the lattice is slowly reduced in strength 4(f), theMott insulator will melt into high energy states in thevalence band, i.e., a negative-mass, negative-temperaturegas. This approach seems practicable for all ultracoldbosonic atoms which have suitably broad Feshbach reso-nances, such as 7Li, 23Na, 85Rb, and 133Cs. A detailedanalysis of the reversibility criteria for each step will bepublished elsewhere.The atomic gas can be cooled by forced evaporation. Inthe case of a magnetic trap, the evaporation can proceedthrough radio frequency (rf) induced spin flips to an un-trapped (magnetic quantum number m  0) state. Oneshould tune the rf to selectively remove atoms withlower-than-average energy, so that the average energy peratom will increase. This leads to an increase of T, and areduction of the entropy per particle S=N. A reduction ofthe entropy per particle is equivalent to an increase of thephase space density. Therefore we use the term cooling forany process that lowers jTj. At low enough jTj the systemcan be described by a (negative) power law density ofstates, and the theory of evaporative cooling in a powerlaw density of states [23] can be applied. (One should takecare to reverse the signs of m, E, and T in Ref. [23]wherever applicable.) The condition for BEC in a 3D sys-tem is n)3  2:61, where n is the number density of atomsand )2  2@2m1eff k1B T1, where kB is Boltzmann’s con-stant. Since both the effective mass meff and T are negative,) is a real number. Achieving the BEC phase transition atnegative temperature seems possible, either by heating(extracting energy), starting from a pure BEC, or by cool-ing, starting from a negative-thermal gas.The negative value of asc, which is necessary to cre-ate the negative-mass BEC from a Mott insulator, isalso a condition of stability for the negative-mass BEC.04040In fact, the negative-mass BEC is likely to collapse ifasc is positive [15]. It is very interesting to consider thepossibility of superfluidity at negative asc in a negative-temperature system. One expects the superfluid phase toallow for dissipationless motion of objects, unless a criticalvelocity is exceeded, in which case negative dissipationsets in. Further properties of negative-temperature super-fluidity are hitherto unexplored theoretically as well asexperimentally.In conclusion, in an optical lattice combined with aparabolic potential, a large phase space in which atomscan thermalize to negative kinetic temperatures is open forexploration.I have greatly benefited from inspiring discussions withClaude Cohen-Tannoudji, Ad Lagendijk, and Willem Vos.3-4[1] A. Einstein, Ann. Phys. (Leipzig) 17, 549 (1905).[2] R. Kubo, Rep. Prog. Phys. 29, 255 (1966).[3] M. W. P. Strandberg, Phys. Rev. 106, 617 (1957).[4] J. E. Geusic, E. O. Schultz-Dubois, and H. E. D. Scovil,Phys. Rev. 156, 343 (1967).[5] T. Nakagomi, J. Phys. A 13, 291 (1980).[6] W. Hsu and R. Barakat, Phys. Rev. B 46, 6760 (1992).[7] E. M. Purcell and R. V. Pound, Phys. Rev. 81, 279 (1951).[8] N. F. Ramsey, Phys. Rev. 103, 20 (1956).[9] M. J. Klein, Phys. Rev. 104, 589 (1956).[10] L. D. Landau and E. M. Lifshitz, Statistical Physics(Pergamon, New York, 1980), 3rd ed., p. 221.[11] C. Kittel and H. Kroemer, Thermal Physics (Freeman, SanFrancisco, 1980), 2nd ed., Appendix E.[12] P. S. Jessen and I. H. Deutsch, Advances in Atomic,Molecular and Optical Physics, edited by B. Bendersonand H. Walther (Academic Press, New York, 1996),Vol. 37, p. 95.[13] M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, andC. Salomon, Phys. Rev. Lett. 76, 4508 (1996).[14] B. Eiermann et al., Phys. Rev. Lett. 91, 060402 (2003).[15] H. Pu, L. O. Baksmaty, W. Zhang, N. P. Bigelow, andP. Meystre, Phys. Rev. A 67, 043605 (2003).[16] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals,Series, and Products (Academic Press, San Diego,1994), 5th ed..[17] M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, andI. Bloch, Nature (London) 415, 39 (2002).[18] C. Hooley and J. Quintanilla, Phys. Rev. Lett. 93, 080404(2004).[19] C. Zener, Proc. R. Soc. A 145, 523 (1934).[20] Y. Zheng, M. Koštrun, and J. Javanainen, Phys. Rev. Lett.93, 230401 (2004).[21] J. Brand and A. Kolovsky, cond-mat/0412549.[22] S. Inouye et al., Nature (London) 392, 151 (1998).[23] O. J. Luiten, M. W. Reynolds, and J. T. M. Walraven, Phys.Rev. A 53, 381 (1996).

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Atomic Gases at Negative Kinetic Temperature

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