State selective cooling of \mathrm{SU}(N) Fermi-gases
Aaron Merlin Müller, Miklós Lajkó, Florian Schreck, Frédéric Mila, Ji?í Miná?
SState selective cooling of
SU( N ) Fermi-gases
Aaron Merlin M¨uller,
1, 2
Mikl´os Lajk´o, Florian Schreck,
3, 4
Fr´ed´eric Mila, and Jiˇr´ı Min´aˇr
5, 4 Institute for Theoretical Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland Institute of Physics, ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, the Netherlands QuSoft, Science Park 123, 1098 XG Amsterdam, the Netherlands Institute for Theoretical Physics, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, the Netherlands (Dated: February 18, 2021)We investigate a species selective cooling process of a trapped SU( N ) Fermi gas using entropyredistribution during adiabatic loading of an optical lattice. Using high-temperature expansion ofthe Hubbard model, we show that when a subset N A < N of the single-atom levels experiencesa stronger trapping potential in a certain region of space, the dimple, it leads to improvement incooling as compared to a SU( N A ) Fermi gas only. We show that optimal performance is achievedwhen all atomic levels experience the same potential outside the dimple and we quantify the coolingfor various N A by evaluating the dependence of the final entropy densities and temperatures asfunctions of the initial entropy. Furthermore, considering Sr and
Yb for specificity, we providea quantitative discussion of how the state selective trapping can be achieved with readily availableexperimental techniques.
I. INTRODUCTION
In recent years, there has been a considerable effort inexperimental control of ultracold Fermi gases with theaim of realizing models of strongly interacting electrons,in particular the Hubbard model, upon loading the atomsinto a deep optical lattice [1]. Of particular interest areultracold quantum degenerate Fermi gases with nuclearspin I that is decoupled from the electronic spin, suchas Yb [2–4] or Sr [5–7], which feature N = 2 I + 1hyperfine states in the ground state manifold.The SU( N ) Fermi gases have attracted considerableattention as they allow for SU( N ) generalizations of theHubbard model [8] and can host a plethora of exoticphases including various spin orders and liquids [9–14],Mott insulator-metal transitions and crossovers [15, 16],valence bond solids and semimetals [17, 18], unconven-tional superconductors [19] or collective motional modes[20]. Remarkably, some of these scenarios have beenprobed also experimentally for N > N ) magnetism[26, 27], where the system can be effectively describedin terms of a Heisenberg model. This stimulated theo-retical investigations using representation theory [28–31],variational approaches [32] or large scale simulations atfinite temperature [33]. Furthermore, depending on N and the lattice geometry, the Heisenberg Hamiltonianscan be linked to Wess-Zumino-Witten models when at acritical point [34, 35], feature chiral spin liquids [36] andmagnetic orders such as generalized valence bond solids[37], plaquette [38, 39], N´eel and stripelike long-range[40, 41] or antiferromagnetic order [42].To observe these magnetic orders the atoms need tobe cooled to temperatures below the superexchange en-ergy 4 t /U , where t and U are the tunneling rate andinteraction strength of the parent Hubbard model. Here, a promising approach is based on an (adiabatic) entropyredistribution akin to the Pomeranchuk effect in solid he-lium [43]. For cold atoms in optical lattices this effect hasbeen studied theoretically by means of dynamical meanfield theory in [44], where the entropy was removed from acertain region - a dimple - by appropriately shaping thetrapping potential. In the context of SU( N ) fermions,Refs. [45, 46] have studied the enhancement of the cool-ing due to higher N (see also [47] for adiabatic coolingof interacting and [48, 49] of non-interacting fermions).Pomeranchuk and dimple cooling were experimentallydemonstrated in [50] and [51] respectively, leading to anantiferromagnetic order [23, 52] with [51] reporting thefinal temperature of T /t = 0 .
25 (see also [53] for ex-perimental realization of short-ranged antiferromagneticorder, [22] for probing the Mott-insulator transition, and[25] for the thermodynamics of the interacting SU( N )Fermi gas).Motivated by these developments, in this work westudy the effect of adiabatically loading an initally har-monically trapped SU( N ) Fermi gas into a deep opticallattice in a species selective way: specifically, we con-sider a bi-partition of the atomic levels in two families, A and B such that N = N A + N B and an optical potentialwhich forms a dimple for only the A -family (hereafter werefer to the different atomic levels as colors). Using thehigh-temperature expansion of the Hubbard model wecompute the entropy density and show that this results infurther enhancement of the cooling of the Mott-insulatingstate of the A -family atoms in the dimple compared to aSU( N A ) Fermi gas only.The paper is structured as follows. In Sec. II we de-scribe the model and methodology, present the results inSec. III, discuss a possible experimental implementationin Sec. IV and conclude in Sec. V. a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b S i , T i S f = S i , T f Dimple V A < V B Reservoir V A = V B xV xV A , V B FIG. 1. Schematics of the experimental protocol. A harmon-ically trapped SU( N ) free Fermi gas with atoms belongingto families A (blue) and B (red) of initial total entropy S i and temperature T i is adiabatically loaded in a deep opticallattice with potentials V A , V B for the two families such that V A < V B in the dimple (blue shaded region) and V A = V B inthe reservoir (red shaded region). II. THE MODEL
Our main focus is to study the cooling of a SU( N )Fermi gas initially trapped in a harmonic potential. Thetrap is adiabatically transformed into a deep optical lat-tice, such that the system can be effectively described bya Hubbard model. We assume, that the final potential issuch that a number N A out of the N colors experiencea different potential in a certain region of space - a dim-ple - than the remaining N B = N − N A components, seeFig. 1.Specifically, we consider a SU( N ) Fermi gas of N i = (cid:80) Nα =1 N α particles, with N α the particle number of eachcolor α . We take the system to be initially a free gas ina harmonic potential V ( r ) = 1 / m (cid:80) dj =1 ω j x j , where m is the atom mass, r = ( x , . . . , x d ), d the dimensionalityof the system and ω j the trapping frequencies with thegeometric mean ¯ ω = ( ω . . . ω d ) /d . Denoting further thechemical potential of each color as µ α and taking thegas to be at an initial temperature T i , to first order in T i /µ α the particle number and the chemical potentialare related through (we use (cid:126) = k B = 1 throughout thearticle) [46] N iα = µ dα ¯ ω d d ! . (1)The initial entropy of color α is then given by S iα = T i µ d − α ¯ ω d ( d − π . (2) Taking now into account the chemical potentials of eachfamily, µ A , µ B , the total number of particles becomes N i = 1¯ ω d d ! (cid:0) N A µ dA + N B µ dB (cid:1) , (3)and, given that for a non-interacting gas, the total initialentropy is S i = (cid:80) α S iα . The entropy per particle is givenby S i N i = π d T i µ eff (4)with µ eff = N A µ dA + N B µ dB N A µ d − A + N B µ d − B . (5)Next, we assume that a deep optical lattice is loadedin an adiabatic, isentropic fashion, such that the systemis effectively described by a Hubbard Hamiltonian withtunneling rate t and isotropic on-site interaction strength U for all species [26] H = − t (cid:88) (cid:104) jk (cid:105) ,α c † α,j c α,k + (cid:88) j,α V α,j ˆ n α,j + U (cid:88) j ˆ n j (ˆ n j − (cid:88) j h j . (6)Here, c α,j are the fermionic annihilation operators fora particle of color α on site j with the usual anti-commutation relations { c α,j , c † β,k } = δ αβ δ jk , ˆ n α,j = c † α,j c α,j and ˆ n j = (cid:80) α ˆ n α,j . The sum in (6) runs over L sites, and (cid:104) jk (cid:105) denotes nearest neighbors.A crucial ingredient of the present work are the speciesand position dependent on-site potentials V α,j . Here, weconsider a different potential for each family, V F,j ≡ V α,j if α ∈ F, F = A, B . In particular, we consider box-like potentials, where V A < V B in a central region whichwe call a dimple ( D ). We denote the remainder of thesites as the reservoir ( R ). Assuming box-like potentialsis motivated by the fact, that in a quantum simulationof SU( N ) magnetism, one ideally wishes to create a flatoptical lattice to faithfully simulate the Hubbard model.There is indeed an ongoing effort to achieve this goal incurrent cold-atom experiments [51] as well as in creatingbox-shaped rather than harmonic potentials [54]. With-out loss of generality we choose the potentials as V A,j = (cid:40) j ∈ RV A for j ∈ D (7a) V B,j = 0 ∀ j (7b)with V A <
0, see Fig. 1. In what follows, we analyze thetwo-family Hubbard model using its high temperature ex-pansion in the grand canonical setting [55] and local den-sity approximation (LDA), which is commonly adoptedfor deep optical lattices realizing the tight-binding mod-els [23, 46, 50] (we further comment on the applicability A / U0.000.250.500.751.001.251.501.752.00 | V A | n AD n BD n AR n BR FIG. 2. Single-site particle densities in the dimple ¯ n AD , ¯ n BD and in the reservoir ¯ n AR , ¯ n BR vs. µ A . The densities areplotted in the atomic limit, t = 0, for T = 0 (dashed lines)and T = U/
25 (solid lines), N A = 2 , N B = 8, V A = − . U ,cf. Eq. (7), and at fixed µ B = − . U . The arrow depictsthe offset | V A | between the dimple and the reservoir particledensities, see text for details. of LDA for the box potentials below). The particle andentropy densities at site j are given by ( F = A, B )¯ n F,j = − ∂ µ F Ω j (8) s j = − ∂ T Ω j , (9)where Ω j is the local contribution to the grand potential,cf. Eq. (14). Furthermore we define the entropy density per particle as ¯ s j = s j ¯ n A,j + ¯ n B,j . (10) The atomic limit.
We start our analysis by first con-sidering a single site in the atomic limit t = 0. The singlesite partition function is given by z ,j = tr e − βh j , where h j is a single site Hamiltonian in Eq. (6) and the traceis taken over a basis of single-site orbitals of h j . In thiscase, the single site partition function reads z ,j = N A (cid:88) n A =0 N B (cid:88) n B =0 (cid:18) N A n A (cid:19)(cid:18) N B n B (cid:19) e − β(cid:15) j ( n A ,n B ) , (11)where β = 1 /T and (cid:15) j ( n A , n B ) =( n A + n B )( n A + n B − U/ V A,j − µ A ) n A + ( V B,j − µ B ) n B . (12)It is instructive to consider further the limit of smalltemperatures and investigate the behavior of the particledensities (8) in the dimple and the reservoir as functionsof the chemical potentials µ F . For β (cid:29)
1, the partition function (11) is dominated by a single term, correspond-ing to the minimum of the energy (12), with the particu-lar combination of ( n A , n B ) such that n A = ¯ n A , n B = ¯ n B and (11) reduces to z ,j ≈ (cid:18) N A ¯ n A (cid:19)(cid:18) N B ¯ n B (cid:19) e − β(cid:15) j (¯ n A , ¯ n B ) . Consequently, the entropy density is given by s j = log (cid:18)(cid:18) N A ¯ n A (cid:19)(cid:18) N B ¯ n B (cid:19)(cid:19) . (13)For specificity, in what follows we seek to create a “clean”Mott-insulating state with ¯ n A = 1 and no B-particles,¯ n B = 0, in the dimple, a scenario we analyze in detail inSec. III. In this case, the value of V A has to be chosen inthe interval ( − U,
0) avoiding the proximity of the limitingvalues V A = − U,
0. This is to prevent possible doubleoccupancies (when V A = − U ) and to ensure ¯ n A = 1(avoiding too shallow dimple V A = − (cid:15), (cid:15) (cid:28)
1) at finitetemperature. We have found that these constraints arewell respected for V A = − . U which we consider in theremainder of the paper. We also note that ¯ n AR < ¯ n AD as a consequence of he dimple potential Eq. 7.Analogously, as discussed in detail in Appendix A, asuitable choice of the chemical potential for the B-familyis µ B < n BD = ¯ n BR = 0 at zerotemperature and ¯ n A undergoes changes in integer steps(0 → → . . . → N A ) as µ A is increased from −∞ topositive values, cf. the dashed lines in Fig. 2. The tran-sitions from ¯ n A to ¯ n A + 1 occur at µ A = V A + ¯ n A U inthe dimple and µ A = ¯ n A U in the reservoir, which differby V A , as indicated by the arrow in Fig. 2.The effect of the finite temperature is the characteristic“smearing” of the staircase profile of the particle densitiesas well as resulting in ¯ n B > µ A → −∞ limit, cf.the orange and red solid lines in Fig. 2. The precise valueof ¯ n AR , ¯ n BR can be further adjusted by µ A,B , which wetune in the vicinity of 0, cf. Fig. 2, such that the Mott-insulating state is achieved in the dimple, cf. Sec. III andAppendix A for further details.
The t/U expansion at finite temperature.
Next, we turnto the t (cid:54) = 0 regime. Since we assume a box-shapedpotential, the LDA is satisfied everywhere but at theboundary between the dimple and the reservoir, wherethe potential V A changes in a step-like fashion. For largeenough reservoir and dimple, we expect the thermody-namic properties of the Fermi gas far from the boundarybetween the two regions to be still well captured by theLDA. Under this approximation, the grand-canonical po-tential of the two-family Hubbard model Eq. (6), up tosecond order in t/U for t (cid:28) T (cid:28) U , reads [55]Ω = L (cid:88) j =1 Ω j = − β − L (cid:88) j =1 log( z ,j ) + L (cid:88) j =1 Ω ,j , (14)where L = L D + L R , L D,R being the number of sites inthe dimple and the reservoir respectively and (see Ap-pendix B for derivation)Ω ,j = − β − t c (cid:96) z − ,j (cid:88) F = A,B (cid:34) N F N F (cid:88) n F =1 N ¯ F (cid:88) n F =0 N F − (cid:88) n F =0 N ¯ F (cid:88) n F =0 × e − β ( (cid:15) j ( n A ,n B )+ (cid:15) j ( n A ,n B )) (cid:18) N F − n F − (cid:19)(cid:18) N F − n F (cid:19)(cid:18) N ¯ F n F (cid:19)(cid:18) N ¯ F n F (cid:19) I ( U ( n A + n B − n A − n B − . (cid:35) (15) n AR n B R (a)0.70 1.70 2.69 3.69 4.68 s i n AR , max n B R , m a x (b) N A = 2 3 4 5 6 7 8 9 L D / L R [10 ]0.000.050.10 n R , m a x (c) N A = 2 N B = 8 n AR , max n BR , max n AD = 2 n AD = 1 FIG. 3. (a)
Isolines of the initial entropy density per particle¯ s i as a function of the particle densities in the reservoir at fixed T f = 4 t . The cross indicates the location (¯ n AR, max , ¯ n BR, max )of maximum of ¯ s i . The dashed (solid) lines correspond to theatomic limit (second order high temperature expansion) of theHubbard model respectively. The inset shows a larger rangeof reservoir particle densities, with a black dashed line delim-iting the Mott-insulating regions ¯ n AD = 1 , (b) ¯ n AR, max vs. ¯ n BR, max for various N A . The data points cor-respond to various dimple/reservoir sizes L D /L R indicated inpane (b). (c) ¯ n AR, max and ¯ n BR, max as a function of the rela-tive size of the dimple and the reservoir L D /L R . Parametersused: U/t = 100, V A = − . U , and L D /L R = 1 /
50. In (a,c): N A = 2 , N B = 8. Here, ¯ F denotes the complement of the family F , i.e.either F = A, ¯ F = B or vice versa, c (cid:96) is the coordinationnumber of the lattice, the energies (cid:15) j ( n A , n B ) are givenby (12) and the function I is given by I (∆) = (cid:40) β , ∆ = 0 ( e β ∆ − β ∆ − , ∆ (cid:54) = 0 . (16) III. RESULTS
For the present simulations, we consider a two-dimensional square lattice with coordination number c (cid:96) = 4. Motivated by possible applications in ongoingexperiments with Sr atoms, we also fix N = 10 [5–7]. Particle densities in the dimple and the reservoir.
Westart our investigations by discussing the role of the par-ticle densities. It follows from the form of the poten-tial for family A , Eq. (7a), and the discussion in Sec. II, that as µ A is increased, particles of family A willaccumulate in the dimple until they reach unit filling.Upon further increase of µ A , they will start to populatethe reservoir, see Fig. 2. Subsequently, when increas-ing µ B , for µ B < U , particles of family B will startto populate only the reservoir as they will be repelledfrom the dimple by particles A present therein. Focusingspecifically on the range of chemical potentials resultingin ¯ n A,D = 1 (cf. the inset of Fig. 3a), in Fig. 3a weshow the dependence of the entropy density per particle¯ s i = S i / ( N A + N B ) = ( L R s R + L D s D ) / ( N A + N B ) ata given final temperature ( T f = 4 t ) as a function of theparticle densities. Ultimately, we seek conditions whichminimize the entropy density per particle ¯ s D in the dim-ple, which we analyze in the subsequent section. Alterna-tively, one can invert the question and ask, given the finaltemperature T f , what parameter set maximizes the (to-tal) initial entropy density per particle ¯ s i . It is apparentfrom Fig. 3a, that there is a unique combination of theparticle densities ¯ n AR, max , ¯ n BR, max , denoted by a cross,which maximizes ¯ s i . Two comments are in order – first,the fact that n B, max > B . Intuitively,this is an expected result, since the presence of family B increases the number of degrees of freedom in the reser-voir which are able to absorb the entropy from the dim-ple. Second, starting from the partition function in theatomic limit (11) in the regime ¯ n A , ¯ n B , <
1, in the Ap-pendix C we show that (¯ n AR, max , ¯ n BR, max ) correspondsto the symmetric point µ A = µ B restoring the SU( N )Hubbard model in the reservoir. In Fig. 3b we show thedependence of ¯ n BR, max on ¯ n AR, max for various N A and L D /L R denoted by the data points in Fig. 3c. This de-pendence can be understood by considering the atomiclimit, in which N B n BR, max = N A n AR, max , which followsdirectly from the properties of the partition function (11)[see Appendix C].Next, in Fig. 3c we show the dependence of ¯ n F R, max vs. L D /L R . This is motivated by the requirement thatwithin the finite amount of space available to the ex-periment, one has a trade off between the size of thedimple and the reservoir. In order to optimize the cool-ing, one has to adjust the particle densities in the reser-voir. In particular, in the limit of infinite reservoirsize L D /L R → n F R, max → B with the situation when it is absent, the lattercorresponding to the SU( N A ) Hubbard model only. s D (c) N A = 9, N B = 1 s i n A D (d) 510 T f / t T i [ K]3.9 4.2 4.5 4.81.1001.1251.1501.175 s D (a) N A = 3, N B = 7 s i n A D (b) 2.55.07.510.0 T f / t T i [ K] with Bw/o B s D T f T f , t = 0 FIG. 4. Entropy density per particle in the dimple ¯ s D vs. theinitial entropy density ¯ s i for (a) N A = 3 and (c) N A = 9. Theblue (orange) curves correspond to situations with (without)family B. The solid, dash-dotted and dashed lines correspondto the entropy density ¯ s D and the final temperature T f tosecond order expansion Eq. (15) and in the atomic limit re-spectively. For illustration we show on the top horizontal axisof panels (a) and (c) the initial temperature obtained usingEq. (4) with the experimental parameter ¯ ω/ π = 115 . n AD . Dimple cooling.
Using the analysis described above, foreach T f we find maximum ¯ s i and evaluate the entropydensity per particle in the dimple ¯ s D . The dependenceof ¯ s D and T f on ¯ s i is shown in Fig. 4a and Fig. 4cfor N A = 3 and N A = 9 respectively. Figs. 4b,d showthe corresponding particle densities in the dimple. Forillustration we also show the corresponding initial tem-peratures T i evaluated using Eq. (4) and specific exper-imental parameters, see caption for details. It is appar-ent from the figures that the improvement in cooling,i.e. achieving the same ¯ s D for a larger initial entropydensity, increases with increasing N B . We further notethat the atomic limit predictions (dashed lines in Figs.4a,c) saturate for a certain ¯ s i at ¯ s D = log N A signalingthe necessity to include higher order terms Eq. (15) tocapture the behavior of the entropy in the dimple. Therelatively small change in ¯ s D can be attributed to thefact that for high temperatures T f (cid:29) t considered herethe entropy density is only weakly dependent on the tem-perature [57].Addressing quantitatively the regime of small finaltemperatures T f (cid:46) t relevant for the superexchangephysics would require different theoretical tools as we dis-cuss in Sec. V. IV. EXPERIMENTAL CONSIDERATIONS
In this section we briefly discuss a possible implemen-tation of the proposed scheme. We seek parametersthat satisfy the following constraints: ( i ) a deep opti-cal lattice with potential amplitude V latt ≈ O (10 E r ),where E r = ( (cid:126) k latt ) / (2 m ) is the recoil energy, such thatthe tight binding approximation holds, ( ii ) the latticeband gap, which for the deep lattice we estimate as asingle lattice site harmonic oscillator frequency E gap ≈ (cid:112) V latt k latt /m , to be much larger than the interactionenergy to neglect higher band excitations, E gap (cid:29) U ,and ( iii ) a negligible off-resonant scattering rate with re-spect to the Hamiltonian energy scales. For the sakeof concreteness, in the following we specifically focus onfermionic Sr [5–7, 58] and provide a quantitative exam-ple restoring the dimensionful quantities using (cid:126) .In the far-detuned regime, the optical potential andoff-resonant scattering rate are given by the classical for-mulas V = − πc / (2 ω ) γ [1 / ( ω − ω ) + 1 / ( ω + ω )] I and γ sc = 3 πc / (2 (cid:126) ω )( ω/ω ) γ [1 / ( ω − ω ) + 1 / ( ω − ω )] I ,where ω , ω , γ and I are the atomic transition frequency,the laser light frequency, the atomic excited state decayrate and the laser intensity respectively [59].We shall consider the dimple potential to be cre-ated by a laser light on the | S (cid:105) − | P (cid:105) transitions, where | S (cid:105) ≡ | S , F = 9 / (cid:105) , | P (cid:105) ≡ | P , F (cid:48) = 11 / (cid:105) for brevity[60]. The choice of the P manifold is motivated by the factthat the main optical lattice wavelength λ latt = 900 nmis approximately magic for the | S (cid:105) − | P (cid:105) transition [61]which ensures a position independent frequency selectionof the individual m F states. To this end, a laser inten-sity of the lattice I latt = 5 kW / cm yields V latt /E r ≈ U = 5 kHz we get E gap ≈
160 kHz (cid:29) U as de-sired. We also anticipate that the dominant scatteringrate corresponds to the scattering of the lattice light onthe | S (cid:105) − | P (cid:105) transition and evaluates to γ sc ≈ ω − ω and requiring that | ∆ | (cid:29)| V A | such that the far-detuning approximation holds, wefind that the desired V A ≈ − U is achieved for I ≈
20 W / cm and ∆ = 50 kHz. This value of ∆ is com-patible with the single m F -level addressability using theZeeman splitting of the P -manifold with the energy shiftbetween adjacent m F states of 0 .
255 MHz / G giving, say,25 MHz for a magnetic field of 100 G [62], cf. also [3] forexperimental demonstration using
Yb.Importantly, the dimple light gives rise to additionalcontribution to the dimple potential δV ≈ N A × all m F states stemming from the | S (cid:105) − | P (cid:105) transition,which is of the order comparable to the target dimpleoffset U . Here the factor N A accounts for the N A dimplelaser beams. In principle one could mitigate this addi-tional potential by reducing further ∆ (while modifyingthe dimple laser intensity I to keep | V A | ≈ U ), howeverthis is precluded by the requirement | ∆ | (cid:29) | V A | so thatone remains in the far-detuned regime to prevent detri-mental light scattering. A possible remedy is to com-pensate for the additional dimple potential δV with adipole laser beam in the dimple that is blue detuned tothe | S (cid:105) − | P (cid:105) transition or, alternatively, a red-detunedone in the reservoir region.Finally, we note that using Yb instead might pro-vide further improvement in reducing the additional dim-ple potential [62–66]. This stems from the stronger | S (cid:105)−| P (cid:105) transition with the decay rate of ≈ Srand ≈
95 mHz for
Yb. This in turn allows for a reduc-tion of the dimple laser intensities and consequently ofthe additional dimple potential by a factor of 95 / ≈ V. CONCLUSIONS
We have studied the enhancement of cooling of aSU( N ) Fermi gas exploiting state selective trapping ofa subset of N A atomic levels for which the trappingpotential forms a dimple. We could demonstrate suchenhancement and quantify the cooling using the high-temperature expansion of the Hubbard model by explicitevaluation of the entropy densities and final tempera-tures leading to a SU( N A ) Mott-insulator in the dimple.We could also demonstrate that optimal cooling occurs when the chemical potentials for both families are equalin the reservoir, leading to the symmetry restoration ofthe SU( N ) Hubbard model therein. While these resultsare encouraging for the current experiments with coldfermionic gases featuring N sub-levels, such as Yb or Sr, the high temperature expansion used here is notsuitable to describe the regime of sufficiently small tem-peratures where exotic magnetic phases driven by the su-perexchange interaction could be achieved. To this end,further work using for instance dynamical mean-field the-ory [44, 67] and taking into account various experimentalconsiderations such as the precise profile of the trap isnecessary to faithfully quantify the cooling.
VI. ACKNOWLEDGMENTS
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Here we discuss the particle densities in the dimple andthe reservoir in the atomic and zero temperature limit.The particle densities are given by Eq. (8), which in theatomic limit and using LDA reduces to¯ n F,j = − ∂ µ F Ω ,j = N A (cid:80) n A =0 N B (cid:80) n B =0 (cid:0) N A n A (cid:1)(cid:0) N B n B (cid:1) e − β(cid:15) j ( n A ,n B ) n F z ,j , (A1)where we have used the expression Ω ,j = − /β log z ,j for the atomic limit grand potential, cf. Eq. (14), and F = A, B . We note that in the infinite temperature limit β → n F,j = N F / β → ∞ , theEq. (A1) is dominated by a single term with the lowest energy (cid:15) j , cf. Eq. (12), which we rewrite as (droppingthe site index j for simplicity and setting ˜ V B = 0, cf. theEq. (7))2˜ (cid:15) = n A + n B +2 n A n B + n A (2 ˜ V A − µ A − − n B (2˜ µ B +1) . (A2)Here we have denoted by tilde the quantities rescaledby the interaction energy, ˜ (cid:15) = (cid:15)/U , ˜ V F = V F /U , ˜ µ F = µ F /U . It should be noted that the fact that the sum inEq. (A1) is dominated by a single term of given n A , n B implies that the particle densities correspond to these,¯ n F = n F . In order to determine the particle numbers¯ n F as a function of ˜ µ F it thus suffices to identify thecombination ( n A , n B ) which minimizes the energy (A2)for a given set of parameters ˜ µ F , ˜ V A .To demonstrate this, let us first consider a limit ˜ µ A →−∞ , such that the lowest energy corresponds to n A = 0and Eq. (A2) becomes2˜ (cid:15) ( n A = 0 , n B ) = n B ( n B − − µ B ) . (A3)Similarly, the minimum of (A3) implies n B = 0 for ˜ µ B →−∞ . Increasing ˜ µ B then leads to a series of transitions,in steps of 1, in the particle number ¯ n B and the thresholdvalues of ˜ µ B can be obtained from the relation˜ (cid:15) (0 , n B ) = ˜ (cid:15) (0 , n B + 1) (A4)which leads to ˜ µ ( n B ↔ n B +1) B = n B . (A5)This allows us to analyze the situation of Fig. 2 and toidentify the particle numbers as ˜ µ A is varied. For ˜ µ B = − . n B = 0. As we increase ˜ µ A from −∞ , more A-particles will populate the dimple andthe reservoir and thus n B remains zero. The energy (A2)simplifies to2˜ (cid:15) ( n A , n B = 0) = n A (cid:16) n A + 2 ˜ V A − µ A − (cid:17) . (A6)From the condition ˜ (cid:15) ( n A ,
0) = ˜ (cid:15) ( n A + 1 ,
0) we get thethreshold values for ˜ µ A ˜ µ ( n A ↔ n A +1) A = ˜ V A + n A . (A7)for which the number of A-particles changes from n A to n A +1 until the saturation n A = N A for ˜ µ A > ˜ V A + N A − Finite temperature.
The effect of finite temperature is to “smear” out the staircase structure of ¯ n A as is apparentfrom the Fig. 2. Similarly, we note that for the parame-ters of Fig. 2 the non-zero value of ¯ n B in the ˜ µ A → −∞ limit is the consequence of non-zero temperature, whichinterpolates between ¯ n B = 0 for β → ∞ and ¯ n B = N B / β = 0. Appendix B: Derivation of the Eq. (15)
In this section we provide the details of the deriva-tion of the Eq. (15) following closely the treatment in[69] and [55, chap. 1,7,8] (cf. also [70–72] for related de-velopments). It is obtained using the high-temperatureexpansion of the Hubbard model Eq. (6) in the stronglyinteracting limit with t (cid:28) T (cid:28) U [55]. Splitting explic-itly the potential term for the two families and includingthe chemical potentials µ A,B as in Eq. (12), we first writethe Hamiltonian (6) as H ( t ) = U (cid:88) j ˆ n j (ˆ n j −
1) + (cid:88) j,α ∈ A ( V A,j − µ A )ˆ n α,j + (cid:88) j,α ∈ B ( V B,j − µ B )ˆ n α,j − t (cid:88) (cid:104) jk (cid:105) ,α c † α,j c α,k = H − t T (B1)Having denoted the hopping operator as T = (cid:80) (cid:104) jk (cid:105) ,α c † α,j c α,k , the lowest non-trivial term contribut-ing to the grand potential Ω is second order in the smallexpansion parameter t and is given by − β Ω = t (cid:90) β d τ (cid:90) τ d τ (cid:104) ˜ T ( τ ) ˜ T ( τ ) (cid:105) L , (B2)where ˜ T ( τ ) = e τH T e − τH , (cid:104) ˆ O (cid:105) =Tr [ e − βH ˆ O ] / Tr [ e − βH ] is the expectation value ofoperator ˆ O with respect to the atomic limit Hamiltonian H and (cid:104) O (cid:105) L stands for the term in (cid:104) O (cid:105) proportional tothe number of sites L , see [69] and chapter 8 of [55] fordetails.In the atomic limit H = H ( t = 0) = (cid:80) Lj =1 h j isa sum of Hamiltonians acting only on a single site j ofthe system. Similarly, T connects only nearest-neighborsites which differ by a single particle of color α . In thiscase, two such nearest-neighbor sites (denoted by 1 and2 hereafter) are spanned by eigenvectors of H | m (cid:105) = | m (cid:105) | m (cid:105) with the corresponding eigenenergy E m = (cid:104) m | H | m (cid:105) = (cid:15) m + (cid:15) m , where the single-site energies (cid:15) m j are given by Eq. (12). Using this and the LDA, theEq. (B2) can be written as − β Ω = − β (cid:80) j Ω ,j , where − β Ω ,j = t c (cid:96) z − (cid:88) m ,p e − β ( (cid:15) m + (cid:15) m ) |(cid:104) p |T | m (cid:105)| ×× I ( (cid:15) m + (cid:15) m − (cid:15) p − (cid:15) p ) , (B3) c (cid:96) is the coordination number of the lattice, z the single-site partition function Eq. (11) and I (∆) = (cid:90) β d τ (cid:90) τ d τ e τ ∆ e τ ∆ = (cid:40) β , ∆ = 0 ( e β ∆ − β ∆ − , ∆ (cid:54) = 0 . (B4)with the result stated in Eq. (16).The sum in (B3) can be evaluated as follows. Let usdenote the number of particles of family F and its com-plement ¯ F on site 1 and 2 as n F , n F , n F , n F respec-tively. Next, we consider a hopping of a particle of thefamily F from site 1 to site 2. The only non-vanishingcontribution to the sum (B3) comes from a configurationwhere there is exactly one particle of color α ∈ F on site 1and zero such particles on site 2. We can choose the color α on site 1 from N F possibilities. The remaining n F − F on site 1 can be chosen in (cid:0) N F − n F − (cid:1) ways. Similarly, there are (cid:0) N F − n F (cid:1) possible configurationsof particles of family F on site 2. The number of con-figurations of particles belonging to the complementaryfamily ¯ F is not constrained by the configurations of thefamily F and is given by (cid:0) N ¯ F n F (cid:1) , (cid:0) N ¯ F n F (cid:1) on site 1 and 2respectively. The overall combinatorial factor is thus theproduct of all these factors, namely N F (cid:18) N F − n F − (cid:19)(cid:18) N F − n F (cid:19)(cid:18) N ¯ F n F (cid:19)(cid:18) N ¯ F n F (cid:19) , (B5)which appears in the Eq. (15). We also note that toconvert the sum over m , p in the Eq. (B3) to a sumover n F , n F , n F , n F , we have exploited the fact thatthe single-site energies (cid:15) m j = (cid:15) m j ( n jA , n jB ), Eq. (12),are only functions of n jF , n j ¯ F . Appendix C: Extrema of the entropy density
In this section we show by explicit computation in theatomic limit and in the regime of small particle densityin the reservoir, ¯ n AR + ¯ n BR <
1, that the symmetricchoice of chemical potentials µ A = µ B for the two familiescorresponds to the extremum of the entropy density perparticle¯ s = ¯ s i = L R s R + L D s D L R (¯ n AR + ¯ n BR ) + L D (¯ n AD + ¯ n BD )= s R + rs D n + r n D =: YW . (C1)investigated in Fig. 3a. Here n = (cid:80) F = A,B ¯ n F R , n D = (cid:80) F = A,B ¯ n F D and r = L D /L R is the ratio of the dimpleand the reservoir sizes. The functions Y, W in (C1) standfor the nominator and the denominator respectively andare defined for future convenience, cf. below. In the limit of zero tunneling (atomic limit), large in-teractions, βU (cid:29) µ F < U , the dominant contribu-tion to the single-site partition function in the reservoircomes from the configurations containing at most oneparticle such that the Eq. (11) can be approximated as z ≈ (cid:88) F N F e βµ F , (C2)where we have used the fact that V A,j = V B,j = 0 (wedrop the site index hereafter for simplicity as we will beconcerned solely with the quantities in the reservoir andthe atomic limit; we also use F = A, B and for a given F we denote its complement as ¯ F throughout this section).The corresponding particle and entropy densities (8),(9)read ¯ n F = 1 z N F e βµ F (C3) s = log( z ) − βz (cid:88) F µ F N F e βµ F . (C4)From (C3) we find e βµ F N F = ¯ n F z which allows to ex-press the partition function (C2) as z = 11 − n (C5)and consequently the entropy density (C4) as s = − log(1 − n ) − (cid:88) F βµ F ¯ n F . (C6)It is interesting to verify that combining (C3) and (C5)we also get n − n + N e βµ = 0 (C7)which has real solutions only on the interval 0 ≤ n ≤ N = N A + N B is the total number of colors).Next, we assume that the entropy and particle densi-ties in the dimple s D , ¯ n F D do not vary with the chemicalpotentials µ F , which is well satisfied when the dimple isin the Mott regime (we further comment on this assump-tion below). In what follows, we investigate the extremaof the reservoir density Eq. (C1) with respect to µ F . De-noting ∂ ≡ ∂ µ F , ¯ ∂ ≡ ∂ µ ¯ F to simplify the notation, theextremum has to satisfy ∂ ¯ s = ¯ ∂ ¯ s = 0. Applying thiscondition to the Eq. (C1) we find ∂ ¯ s = 0 ⇔ W ∂Y − Y ∂W = 0 (C8)which yields the constraint for the values of µ A , µ B ex-tremizing ¯ s . Using ∂ ¯ n F = β (1 − ¯ n F )¯ n F (C9a)¯ ∂ ¯ n F = − β ¯ n F ¯ n ¯ F (C9b) ∂z = βz ¯ n F . (C9c)0we have ∂Y = β ¯ n F [ βµ F (¯ n F −
1) + βµ ¯ F ¯ n ¯ F ] (C10a) ∂W = β ¯ n F (1 − n ) . (C10b)To proceed, rather than investigating the properties ofthe constraint (C8) for the general variables µ A , µ B , weask whether it can be satisfied for µ A = µ B = µ . In thiscase n = NN F ¯ n F (C11a)¯ n F = N F N ¯ F ¯ n ¯ F (C11b) βµ = log (cid:18) N n − n (cid:19) . (C11c)Substituting these expressions to (C8) we find β ¯ n F [ r ( n D βµ + s D ) + n βµ + s ] = 0 . (C12)The first solution is, with the help of (C11a), the triviallimit n = 0, i.e. vanishing particle density in the reser-voir. The second solution can be cast in the form PQ = r, (C13)where P = − ( n βµ + s ) = log(1 − n ) (C14a) Q = n D βµ + s D = n D log (cid:18) η D N n − n (cid:19) . (C14b) Here log η D = s D / n D and we have used the expression(C11c) for βµ . For a given dimple to reservoir size ratio r the Eq. (C13) thus represents the condition for n , andthrough (C11a) for ¯ n F and ¯ n ¯ F , which maximizes ¯ s . Forthe physically meaningful scenario n D > n ∈ (0 , P ∈ ( −∞ ,
0) and Q ∈ ( −∞ , ∞ ) with the limit lim n → + P = 0. This impliesthat the condition (C13) can be satisfied for arbitrary r for 0 < n <
1, proving that µ A = µ B corresponds to theextremum of ¯ s in the atomic limit as claimed.To demonstrate this, we consider the case studied inFig. 3a, where r = 1 /