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Ultracold Alkali Atoms in Optical Lattices: A New Type of Quantum Crystals? Alexander E. Meyerovich Department of Physics, University of Rhode Island, Kingston, RI 02881, USA Similarities between alkali gases in optical lattices with non-integer occupa-tion of the lattice sites and quantum crystals are explored. The analogy withthe vacancy liquid provides an alternative explanation to the Mott transi-tion for the recent experiment on the phase transition in the lattice. Thevacancy liquid can undergo BEC with Tc within experimental reach. Directand vacancy-assisted mechanisms of the band motion for hyperÞne impuritiesare discussed. The presence of vacancies can result in the spatial decompo-sition of the system into pure hyperÞne components. Below BEC for thevacancies, the impurity component resembles 3He in 3He − HeII mixtures.
PACS numbers: 03.75.Fi, 05.30.Jp, 66.35.+a, 67.80.-s.
Experimental discovery of Bose condensation made the study of ultra- cold alkali gases a focal point in atomic, low temperature, and condensedmatter physics. Experiments revealed some of the phenomena that havebeen discussed earlier only within theoretical models1 and that combine fea-tures inherent to diverse condense matter and low temperature systems2.
For example, BEC in trapped gases resembles, but is not quite the sameas the transition in other superßuid systems. Another example is the dy-namics of the hyperÞne components which resembles the spin dynamics inspin-polarized quantum gases3.
One more example is an ultracold alkali gas in an optical lattice in which the atoms are located in periodic potential wells induced by the Starkeffect of interfering laser beams4. The depth V0 and width a of the wellsand the rate of tunneling between them are determined by the intensity and the wavelength λ of the beams (a = λ/2 = π/k). By adjusting theintensity, one can scan a wide range of the tunneling frequency t, effectivemass m∗ ∼ 2/ta2, and repulsion U of atoms inside the same well from afree gas with a periodic perturbation to a well-localized "solid".
According to the Hubbard model, the system undergoes5 the Mott metal-insulator transition at U/t ≈ 5.8z (z is the number of the nearestneighbors). At U À t, the system is an "insulator" without interwell tran-sitions (the tunneling increases the on-site energy by U and is energeticallyprohibitive). At U ¿ t, the interaction does not restrict tunneling and theatoms are in the "metal" phase. At low temperature, such "metal" under-goes BEC into a lattice superßuid. The energies U , t, and V0 are oftenmeasured in units of the recoil energy Er = 2k2/2m. A typical example5 ist/Er ∼ 0.07 and U/Er ∼ 0.15 for V0/Er = 15.
This transition was reported in Ref. 6 for the lattice with a ∼ 426 nm and Er ∼ 1 kHz. At low beam intensity (large t), there was a BEC peakin the center of the trap. The peak disappeared at small t (V0/Er between13 and 22). The identiÞcation of this new phase as the Mott insulator(MI) was not direct. The MI exists only when the average population ofa lattice site is integer. If the population is fractional, the highest on-sitestates are not fully occupied. The tunneling of a particle from the occupiedto an unoccupied state cannot be banned by the on-site interaction sincesuch tunneling is not energetically prohibitive. This opens the way to theband motion of the "excessive" particles leaving the lattice with non-integeroccupation in the "metal" or "semiconductor" state even at large U . Thoughthere is experimental evidence6 that the conduction band is separated fromthe Þlled band by a gap, it is difficult to conclude whether the conductiongap is empty or not. Below we suggest an alternative interpretation for Ref.
6 based on the analogy with the quantum crystals (QC).
VACANCIES AND IMPURITIES IN OPTICAL LATTICES There is a similarity between the particles in optical lattices and atoms in QC, such as solid helium, in which the interstate tunneling ensures theband motion of atoms unless prohibited by large on-site repulsion (see review7).
In helium crystals, the atomic band motion is impossible, as for all MI, whenthe lattice sites are occupied by identical particles with the occupancy equalto one. If some of the lattice sites are empty, nothing prohibits tunnelingof atoms from the occupied onto the vacant sites leading to the formationof peculiar band quasiparticles - vacancy waves. Similar quasiparticles areformed when atoms occupy interstitial sites or when some of the lattice sites Optical Lattices: A New Type of Quantum Crystals? are occupied by atoms of a different kind (impurities). The impurities alsotunnel through the QC even if each site has the occupancy equal to one. Thetunneling rate for impurities is smaller than for the vacancies since the ex-change of places between the impurity and host atoms involves high-energyintermediate states with either double on-site occupancy or the atom in aninterstitial position. Often, a more efficient mechanism of impurity motionis the vacancy-assisted diffusion. The vacancy and impurity waves in QC arewell understood7. However, the most exciting effect - superßuidity and BECin the system of vacancy waves - has not been observed for "classical" QC,namely, solid 4He despite two decades of intensive efforts8. The reason is theabsence of the zero-temperature vacancies in 4He: with lowering tempera-ture, the concentration of vacancies drops exponentially always remaininginsufficient for BEC.
Atoms in optical lattices resemble QC with an appealing difference: the BEC for vacancy or "impurity" waves could be within reach. Whenthe occupancy of the individual wells is close to an integer K, the systemresembles QC with either a small density of vacancies nv = KN − n ¿ Nor "excessive" atoms ne ≡ N − nv = n − NInt [n/N] ¿ N (n and N arethe densities of atoms and lattice sites). The tunneling frequencies are thesame for vacancies and "excessive" atoms, tv = te, since, in both cases, anatom tunnels to an empty site. As a result, many properties of the systemare symmetric with respect to the vacancies and excessive atoms. [In usualQC the environments for a vacancy and an "excessive" atom are different].
Below we consider the situation with large on-site interaction U when the lattice system with the integer site occupation is a Mott insulator.
In the tight binding approximation for vacancies in a cubic lattice, , where ∆ = 12t v is the bandwidth. The chemical potential µv is Þnite in contrast to µv = 0 for thermally-activatedvacancies in helium. When nv (or ne) is small, the BEC transition tempera-ture in the vacancy gas is determined by the equations for lattice gases withlow band Þlling. A good extrapolation between these limiting cases is (N − nv)2/3 .
With the above values of Er and a, the estimate for (1) is Tc ∼ 3 ×10−7 (tv/Er) x2/3 (1 − xv)2/3 K where xv = a3nv.
The density of participating particles (vacancies or excessive atoms in the highest on-site state) is lower than the overall density. In the experimentof Ref. 6 with the occupancy between 2 and 3, this leads to a factor 5−2/3in Tc with respect to a free gas and even stronger lowering of Tc in systems close to an integer occupancy. Second, the effective mass m∗ = m Er/π2tcould be much larger than the mass of the free atoms m. One has limited control over the vacancy concentration. On the other hand, the exponentialdependence of m∗ on the intensity of the laser beams can make Tc in thevacancy system observable.
This suggests an alternative to the Mott transition for the experiment of Ref. 6. At large t, the experiment showed the BEC, probably, in the "free"gas rather than in the vacancy liquid. At small t, there was no condensate.
However, the experiment, by design, cannot distinguish between the MI andthe vacancy liquid. The likely transition was between the superßuid and thevacancy liquid rather than between the superßuid and the MI.
The analogy with QC leads to other predictions. The role of "impurity waves" can be played by atoms in the different hyperÞne states which canbe studied by means similar to NMR for 3He diffusion in solid 4He. WhenU is large, the tunneling frequency ti for direct impurity-host exchanges isnegligible, ti ∼ t2v/U ¿ tv, and the vacancy-assisted processes can dominateimpurity motion with an effective tunneling rate ti ∼ tva3nv À t2v/U (in thiscontext, the asymmetry of the vacancy-assisted motion10 is not important).
The impurity mean free path at nv → 0 is atomic while at nv → N it is largeand is determined by the scattering by other impurities or remaining upper-state host atoms. At T > Tc, the vacancy-assisted processes are responsiblefor impurity diffusion with After the vacancy system undergoes the superßuid transition (1), the impurity becomes a completely delocalized quasiparticle in the vacancy su-perßuid background similar to 3He impurities in superßuid 4He11. Theeffective mass of such quasiparticles at T = 0 is m∗i ∼ 2/2tva5nv and goesup with temperature with a decrease in density of the vacancy condensate.
The interaction effects in this quasiparticle gas are negligible. At low tem-peratures, the impurities with density ni also undergo BEC with Tci ' 6.62tva5nvn2/3 = a3n1/3 where Tc is the temperature for the vacancy BEC (1). The emerging systemwith two condensates should exhibit properties similar to those of liquid3He - 4He mixtures with two condensates below the 3He transition12. Sincethis BEC is based on the vacancy-assisted tunneling, the corresponding two-condensate system is different from the one considered in Ref. 9.
The band motion of the vacancies is possible when the impurity concen- tration xi = a3ni is low. At higher xi, the vacancy motion is accompaniedby the host-impurity permutations suppressing the band motion. This is Optical Lattices: A New Type of Quantum Crystals? similar to the vacancy motion in solid 3He with disordered spins. Then thevacancies autolocalize within the homogeneous domains of the size 2m∗N T [(xi − 1) ln (1 − xi) − xi ln xi] which are Þlled by particles in one hyperÞne state (Nagaoka polarons). At large density of vacancies, n1/3 R & 1, this leads to the decomposition of the system into macroscopic hyperÞne domains. In contrast to the vacancy-driven spin polarization of solid 3He13, this transition takes place at Þxedconcentration of the zero-point vacancies and "polarization". This stationarydecomposition is also different from the transient domains in Ref. 3.
Another issue is the sensitivity of the narrow-band particles to external Þelds. Since the energy cannot change by more than the bandwidth ∆,the external Þeld Ω (r) (e.g., the overall trapping potential Ω = αr2/2)localizes the particles in an area of size δr ∼ aσ, σ = ∆/ (∂Ω/∂r) a. In themomentum representation, 1 αr2 = 1 p, the problem reduces to that for a particle with "mass" 1/α in potential 4 (p). Near the band minimathe motion is harmonic with the characteristic frequency ω∗0 = (2tα)1/2 a/ ;ω∗ is (m/m∗)1/2 times lower than for the free particles. The quantization in potential Ω (r) is usually not important and the quasiclassical motion isunrestricted at large σ. When σ → 1, even the classical motion becomescompressed towards the multiwell shells of thickness σa around the centerof the trap. In experiment6, ω∗0 ∼ 75 (t/Er)1/2 Hz, σ & 100 (t/Er) and theshells are narrow when t/Er . 0.01.
The inhomogeneity of the trap also leads to the non-uniform redistri- bution of particles5,14. If the change of the trapping potential from well towell is large, ∆/T σ À 1, the shells with the lower energy are fully Þlled,have integer population, and become the MI. The rest of the shells will havenon-integer population and resemble the vacancy liquid with a rather largedensity of vacancies. In experiment6, ∆/T σ ∼ 10−10/T (with T in K) seemsto be small, meaning insigniÞcant redistribution of particles between theshells. Even if this parameter were large, the system would represent a thickshell with σ & 100 (t/Er) of coupled well layers in the quasi-2D vacancyliquid state with the rest of the shells being the MI. This may actually in-crease the BEC temperature for the vacancy liquid since this liquid, thoughrestricted to a lower number of shells, now has a higher density of vacancies.
In summary, we explored the analogy between the gases in optical lat- tices with non-integer occupation and large on-site interaction with QC.
This explains experiment6 as a transition between the BEC and vacancyliquid states. The BEC for the vacancy liquid seems to be within experi-mental reach. The presence of unoccupied states provides a vacancy-assisted mechanism for diffusion of hyperÞne impurities and can lead to a spatial de-composition of the system into pure hyperÞne components. The propertiesof the hyperÞne mixture strongly depend on whether the system is above orbelow BEC for the vacancy liquid. At low temperatures one can observe thetransition to the state with two - vacancy and impurity - condensates.
The work is supported by NSF grants DMR-0077266 and INT—9724728.
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