PHYSICAL REVIEW LETTERS week ending PRL 95, 040403 (2005) 22 JULY 2005

Atomic Gases at Negative Kinetic

A. P. Mosk Department of Science & Technology, and MESA+ Research Institute, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands and Laboratoire Kastler-Brossel, Ecole Normale Supe´rieure, 24 rue Lhomond, 75231 Paris CEDEX 05, France (Received 26 January 2005; published 21 July 2005) We show that thermalization of the motion of at negative temperature is possible in an , 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 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, 05.70.2a

Amplification of a macroscopic degree of freedom, such ized in the vicinity of a band gap in a three-dimensional as a or maser field, or the motion of an object, is a (3D) optical lattice. The band gap is an energy range inside phenomenon of enormous technological and scientific im- which no single-particle eigenstates exist. If the atoms are portance. In classical thermodynamics, amplification is not constrained to energies just below the band gap the effec- possible in thermal equilibrium at positive temperature. In tive mass will be negative in all directions and the contrast, due to the fluctuation-dissipation theorems [1,2], of the ensemble will decrease with energy. This is the amplification is a natural phenomenon, and fully consistent condition for existence of a thermal ensemble with nega- with the second law of thermodynamics, in a negative- tive temperature [8]. temperature bath [3–6]. Negative were For thermalization to take place, the characteristic time introduced by Purcell and Pound [7] to describe scale for exchange of energy between particles must be systems, and further generalized theoretically by Ramsey shorter than time scales associated to heating and loss. In and Klein [8,9]. Useful introductions to thermodynamics at other terms, we need a high rate of intraband scattering, negative temperature are given by Landau and Lifshitz [10] with negligible interband scattering. Interband scattering is and by Kittel and Kroemer [11]. Negative temperatures are a collision process between two atoms in the first (‘‘va- used to describe distributions where the occupation proba- lence’’) band which causes one to be promoted to the bility of a quantum state increases with the energy of the second (‘‘conduction’’) band while the valence band atom state. The prerequisite for a negative-temperature en- loses energy. Additional losses may be caused by interband semble to be in internal equilibrium is that it occupies a tunneling, a one-body process in which an atom tunnels part of where the entropy decreases with through a classically forbidden region, to reappear in a . An ensemble in this part of phase space different band. cannot be in thermal equilibrium with any positive- In this Letter, we show that all the requirements for temperature system. An increasing occupation number thermalization at negative temperature can be met in a (and a decreasing entropy) can only exist up to a certain 3D optical lattice. First, we show that an optical lattice maximum energy. Unlike the magnetic energy of spins, can have a 3D band gap for atoms, at experimentally kinetic energy is not bounded from above in free space, feasible lattice intensities. Moreover, interband scattering which apparently rules out a negative kinetic temperature. can be completely suppressed, and confinement of the Here, we show that the band structure of a moderately deep negative-mass atoms can be easily accomplished using a optical lattice defines an upper bound to the kinetic energy magnetic trap for high-field seeking states. We calculate of atoms in the first band. Therefore, atoms in an optical rates of interband tunneling and find they can be made lattice form a simple system in which thermalization of the negligible. Finally, we show how a trapped gas at positive kinetic energy is possible at negative temperatures. temperature can be reversibly converted to a negative Optical lattices, which are periodic potentials created temperature. through optical standing waves [12], impose a band struc- The motion of atoms in a simple cubic 3D optical lattice ture on the motion of ultracold atoms. The kinetic energy with a harmonic confining potential is described by the of atoms in the lattice depends on their effective mass or Schro¨dinger equation (See, e.g., [12,15]), band mass, which, in analogy to the case of in a , can be strongly different from the normal @2 ÿ 2 2 2 2 E ˆ r ‡ Ulat ‡ x ‡ y ‡ z † rest mass. The effective mass can even become negative, as 2m (1) has been demonstrated in one-dimensional optical lattices 2 2 2 [13,14]. An isotropic negative-effective mass can be real- with Ulat ˆ U0 cos kx ‡ cos ky ‡ cos kz†:

0031-9007=05=95(4)=040403(4)$23.00 040403-1  2005 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 95, 040403 (2005) 22 JULY 2005

Here m is the bare atomic mass, is the strength of the between the bands is Bragg forbidden; i.e., no propagating harmonic potential, U0 is the depth of the standing wave modes can exist here due to interference. potentials that form the lattice, and k ˆ 2=, where  is For the negative-temperature phase to be metastable, the wavelength of the lattice light. The above Schro¨dinger there must be a band gap between the top of the valence equation is trivially separable by setting E ˆ Ex ‡ Ey ‡ band, where the negative-mass states exist, and the bottom Ez. We now transform the equation to dimensionless units, of the conduction band, where the effective mass is posi- tive. In a 1D system (with the motion in the other two x;~ y;~ z~ ˆ kx; ky; kz; (2) directions confined) this band gap exists for jqj > 0. However, in a 3D optical lattice, the bands may overlap. ˆ 2† ~ = Erk ; (3) The maximum energy an atom in the valence band of a 3D 1 lattice can have is 3b1 q†, whereas the minimum energy in a ˆ E ÿ U =E ; (4) the conduction band is 2a q†‡a q†. The criterion for x x 2 0 r 0 1 the existence of a band gap is 3b1 q† < 2a0 q†‡a1 q†, which is fulfilled for q>0:559. q ˆ U0=4Er; (5) If the 3D band gap is narrow, collisions between two 2 2 with the lattice recoil energy Er ˆ @ k =2m. The one- valence band atoms can promote an atom to the conduction dimensional equation of motion takes the shape of a modi- band, where it has positive effective mass. This interband fied Mathieu equation, scattering will not only lead to loss of atoms, but also to 2 loss of energy, which is equivalent to heating (increasing d 2 ‡ ax ÿ 2q cos2x~† ‡ ~x~ ˆ 0: (6) j j dx~2 T ). We show now that interband scattering becomes energetically forbidden in two-body processes at high In the absence of a parabolic potential ( ~ ˆ 0), Eq. (6) is enough q. The maximum energy of the initial state, con- exactly the Mathieu equation. Its solutions are the Mathieu sisting of two valence band atoms, is 6b1 q†. The minimum functions, which are propagating Bloch-Flouquet waves energy of the final state, consisting of one valence and one for values of a and q in the bands, and evanescent functions conduction band atom, is ‰3a0 q†Š ‡ ‰2a0 q†‡a1 q†Š. in the band gaps. The relevant bands are depicted in Fig. 1. The interband scattering process is energetically forbidden Adopting the usual convention [16] we designate the ei- if 6b1 q† < 5a0 q†‡a1 q†, which is the case if q>0:936. genvalues corresponding to the band bottoms by This value of q corresponds to a lattice depth almost an a0 q†;a1 q†, etc., and the band tops by b1 q†, etc., (see order of magnitude less than the depth employed to create a Fig. 1). If 0 < j ~jq=x~, the lattice is locally only weakly Mott-insulator phase [17]. In such a moderately deep lat- perturbed by the parabolic potential, and the band structure tice, thermalization rates and inelastic collision rates are will remain essentially intact. The energy positions of the all of the same order of magnitude as the respective rates bands become position dependent, in the same way as in a at positive temperature in the absence of a lattice [15]. semiconductor in an applied electric field. A negative value Through elastic collisions, the ensemble will always re- of ~ corresponds to a confining potential for negative- lax to a distribution with the highest possible entropy at effective mass atoms, see Fig. 2. The area under the valence its given energy, in this case a negative-temperature band in Fig. 2 is classically forbidden as the kinetic energy distribution. of the atoms in this region is less than zero. The area To investigate how the gas thermalizes in the negative- temperature phase space, it is useful to consider the semi-

FIG. 1. Band structure of the one-dimensional Mathieu equa- tion. White area: Bands, gray area: gaps. Dotted line: q ˆ 0:559, FIG. 2. Valence and conduction bands of a 1D optical lattice at dashed line: q ˆ 0:936; see text. q ˆ 0:936. Grey area: Gaps, white area: Bands.

040403-2 PHYSICAL REVIEW LETTERS week ending PRL 95, 040403 (2005) 22 JULY 2005 classical density of states (DOS). The DOS can usually be an attempt frequency of the order of the trap oscillation found by taking the energy derivative of the Wentzel- frequency (typically <1 kHz). The resulting tunneling Kramers-Brillouin (WKB) phase integral, a procedure re- time constant is of order 105 seconds at ~ ˆ 10ÿ3.For cently applied to a one-dimensional optical lattice with 10ÿ4 < <~ 10ÿ2 we checked Eq. (8) numerically by harmonic confinement (positive ) [18]. For negative high-precision quadrature of Eq. (6), and found agreement the WKB method cannot be used straightforwardly to within an order of magnitude. calculate wave functions, as the ‘‘Bragg reflection’’ bound- The conditions for existence of a negative-mass Bose- ary condition at the outer turning points behaves differently Einstein condensate (BEC) were recently studied by Pu from classical turning points. The WKB formula for the and coworkers [15]; they also investigated the possibility DOS does not depend on the boundary conditions and can of creating such a condensate by phase imprinting, and be straightforwardly calculated in the one-band model of found this likely to be an inefficient mechanism as many [18]. Within the one-band and WKB approximations, the atoms are transferred to other bands. Other mechanisms, DOS  at negative is related to the DOS calculated for such as accelerating the atoms uniformly to complete positive by one-half Bloch oscillation, may be more efficient, espe- cially since in a BEC nonlinearity due to the interactions  E†ˆÿ ÿE†; (7) counteracts dephasing [20,21]. Using these methods, the where in the case of negative , the top of the valence band negative-mass BEC can be created in a mechanically un- is taken as the energy zero. The 3D DOS crosses over stable state as the interatomic scattering length asc remains smoothly from a quadratic behavior at small negative positive [15]. energy to approximately square-root behavior at larger We propose a radically different mechanism for the negative energy, as shown in Fig. 3. The DOS is nonzero production of negative-temperature gases, based on the for any energy lower than the band edge at the trap center, reversible creation of a Mott-insulator phase from a BEC so in principle the negative temperatures are not bounded [17]. The idea is to ‘‘freeze’’ a positive-temperature gas in an infinite lattice. In fact, this asymmetric behavior of into a , by turning on a strong optical lattice. the DOS makes thermal equilibrium at a positive tempera- The Mott phase is then ‘‘molten’’ in a different trap con- ture impossible, which is in contrast to spin systems [7], figuration and with different interatomic interactions, to where thermal equilibrium at positive and negative tem- produce a negative-temperature gas. The procedure is out- perature can be studied in the same field configuration. lined in Fig. 4, we will describe it here for a pure BEC but We now examine the escape of the atoms through inter- there is no a priori reason why it would not work when band (Zener) tunneling. The trapped negative-mass va- some thermal excitations are present. We start with a lence band states are mixed with antitrapped conduction trapped BEC at >0 and asc > 0 4(a), and turn on a band states at the same energy, as the evanescent tails of strong optical lattice, as is done in [17]. A Mott-insulator these states extend through the Bragg forbidden region. phase will form in which the atoms are confined to their The Zener tunneling rate ÿ may be estimated using Zener local potential wells 4(b). We assume one atom is present [19] per lattice site. Now we remove the trap 4(c), which can be Z 1 ÿZener  !A exp ÿ2 Im‰kBloch x†Šdx ; (8) 0 where kBloch x† is the Bloch wave vector of the Mathieu 2 equation corresponding to the energy E ÿ x , and !A is

FIG. 4 (color online). Method for reversibly transforming a positive-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 without FIG. 3. Density of states  E† in a three-dimensional lattice trap, (d) a strong optical lattice with changed asc, (e) a strong with  ˆÿ10ÿ3. The energy E is taken with respect to the top optical lattice with negative trap, (f) an optical lattice at q ˆ of the first (valence) band. 0:936 with a negative-temperature gas.

040403-3 PHYSICAL REVIEW LETTERS week ending PRL 95, 040403 (2005) 22 JULY 2005 done rapidly without creation of entropy, as the atoms are In fact, the negative-mass BEC is likely to collapse if tightly bound to their respective lattice sites. Then we asc is positive [15]. It is very interesting to consider the change the sign of asc 4(c) and 4(d), for example, by tuning possibility of superfluidity at negative asc in a negative- the magnetic bias field near a Feshbach resonance (See, temperature system. One expects the superfluid phase to e.g., [22]). Since the atoms are at different lattice sites and allow for dissipationless motion of objects, unless a critical do not interact, this can be done rapidly without causing velocity is exceeded, in which case negative dissipation losses. The system passes through the point where both asc sets in. Further properties of negative-temperature super- and are zero, here the temperature of the Mott phase is fluidity are hitherto unexplored theoretically as well as not defined, but the entropy is. This passage transforms the experimentally. Mott insulator from the lowest-energy many-particle state In conclusion, in an optical lattice combined with a to the highest energy in the valence band. The confining parabolic potential, a large phase space in which atoms potential is reconfigured to change the sign of 4(e). This can thermalize to negative kinetic temperatures is open for can be done either by changing the confinement field, or by exploration. changing the internal state of the atoms. A magnetic field I have greatly benefited from inspiring discussions with minimum will trap negative-mass atoms in their lowest Claude Cohen-Tannoudji, Ad Lagendijk, and Willem Vos. Zeeman state [15]. No change in the spatial distribution of atoms will take place due to the extremely low tunneling rate in the Mott insulator, so this step can proceed rapidly. When the lattice is slowly reduced in strength 4(f), the [1] A. Einstein, Ann. Phys. (Leipzig) 17, 549 (1905). Mott insulator will melt into high energy states in the [2] R. Kubo, Rep. Prog. Phys. 29, 255 (1966). valence band, i.e., a negative-mass, negative-temperature [3] M. W. P. Strandberg, Phys. Rev. 106, 617 (1957). gas. This approach seems practicable for all ultracold [4] J. E. Geusic, E. O. Schultz-Dubois, and H. E. D. Scovil, bosonic atoms which have suitably broad Feshbach reso- Phys. Rev. 156, 343 (1967). nances, such as 7Li, 23Na, 85Rb, and 133Cs. A detailed [5] T. Nakagomi, J. Phys. A 13, 291 (1980). [6] W. Hsu and R. Barakat, Phys. Rev. B 46, 6760 (1992). analysis of the reversibility criteria for each step will be [7] E. M. Purcell and R. V. Pound, Phys. Rev. 81, 279 (1951). published elsewhere. [8] N. F. Ramsey, Phys. Rev. 103, 20 (1956). The atomic gas can be cooled by forced evaporation. In [9] M. J. Klein, Phys. Rev. 104, 589 (1956). the case of a magnetic trap, the evaporation can proceed [10] L. D. Landau and E. M. Lifshitz, Statistical Physics through radio frequency (rf) induced spin flips to an un- (Pergamon, New York, 1980), 3rd ed., p. 221. trapped (magnetic quantum number m ˆ 0) state. One [11] C. Kittel and H. Kroemer, Thermal Physics (Freeman, San should tune the rf to selectively remove atoms with Francisco, 1980), 2nd ed., Appendix E. lower-than-average energy, so that the average energy per [12] P.S. Jessen and I. H. Deutsch, Advances in Atomic, atom will increase. This leads to an increase of T, and a Molecular and Optical Physics, edited by B. Benderson reduction of the entropy per particle S=N. A reduction of and H. Walther (Academic Press, New York, 1996), the entropy per particle is equivalent to an increase of the Vol. 37, p. 95. [13] M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and phase space density. Therefore we use the term cooling for C. Salomon, Phys. Rev. Lett. 76, 4508 (1996). any process that lowers jTj. At low enough jTj the system [14] B. Eiermann et al., Phys. Rev. Lett. 91, 060402 (2003). can be described by a (negative) power law density of [15] H. Pu, L. O. Baksmaty, W. Zhang, N. P. Bigelow, and states, and the theory of evaporative cooling in a power P. Meystre, Phys. Rev. A 67, 043605 (2003). law density of states [23] can be applied. (One should take [16] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, care to reverse the signs of m, E, and T in Ref. [23] Series, and Products (Academic Press, San Diego, wherever applicable.) The condition for BEC in a 3D sys- 1994), 5th ed.. tem is n3 ˆ 2:61, where n is the number density of atoms [17] M. Greiner, O. Mandel, T. Esslinger, T. W. Ha¨nsch, and 2 @2 ÿ1 ÿ1 ÿ1 I. Bloch, Nature (London) 415, 39 (2002). and  ˆ 2 m kB T , where kB is Boltzmann’s con- eff [18] C. Hooley and J. Quintanilla, Phys. Rev. Lett. 93, 080404 stant. Since both the effective mass meff and T are negative,  is a real number. Achieving the BEC phase transition at (2004). [19] C. Zener, Proc. R. Soc. A 145, 523 (1934). negative temperature seems possible, either by heating [20] Y. Zheng, M. Kosˇtrun, and J. Javanainen, Phys. Rev. Lett. (extracting energy), starting from a pure BEC, or by cool- 93, 230401 (2004). ing, starting from a negative-thermal gas. [21] J. Brand and A. Kolovsky, cond-mat/0412549. The negative value of asc, which is necessary to cre- [22] S. Inouye et al., Nature (London) 392, 151 (1998). ate the negative-mass BEC from a Mott insulator, is [23] O. J. Luiten, M. W. Reynolds, and J. T. M. Walraven, Phys. also a condition of stability for the negative-mass BEC. Rev. A 53, 381 (1996).

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