Adiabatic spin cooling using high-spin Fermi gases
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] M a r Adiabatic spin cooling using high-spin Fermi gases
M. Colom´e-Tatch´e , C. Klempt , L. Santos , and T. Vekua Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, Appelstr. 2D-30167, Hannover, Germany Institut f¨ur Quantenoptik, Leibniz Universit¨at Hannover, 30167 Hannover,Germany
Abstract.
Spatial entropy redistribution plays a key role in the adiabaticcooling of ultra-cold lattice gases. We show that high-spin fermions with aspatially variable quadratic Zeeman coupling may allow for the creation of aninner spin-1 / / / diabatic spin cooling using high-spin Fermi gases
1. Introduction
Ultra-cold atoms in optical lattices offer an extraordinary controllable scenario forthe study of strongly-correlated systems [1, 2], exemplified by the observation of thesuperfluid to Mott-insulator (MI) transition in ultra-cold bosons [3]. Remarkableprogress has been achieved in lattice fermions as well, allowing for the preciseanalysis of the Fermi-Hubbard model, a key model in condensed-matter physics ofparticular relevance in the study of high-temperature superconductivity [4]. Excitingrecent experiments have reported the realization of the metal to MI transition intwo-component fermions [5, 6]. Due to super-exchange, the MI phase of spin-1 / s ,for N´eel ordering in a 3D cubic lattice ( s/k B = ln 2) [7]. However quantum correctionsreduce the critical s down to s N /k B ≃ .
35 [8, 9].Reaching such an extraordinary low entropy constitutes nowadays a majorchallenge, which demands novel types of cooling especially designed for many-bodysystems in optical lattices[10]. A number of cooling proposals have been recentlysuggested [11, 8, 12, 13, 14, 15, 16, 17], most of them based on the redistributionof entropy within the trap, where certain regions act as entropy absorbers from theregion of interest, i.e. a Mott insulator at the trap center.Interestingly, spin degrees of freedom may be employed for designing coolingtechniques resembling adiabatic demagnetization cooling in solid-state physics [18].In this method, a decrease in the strength of an externally applied magnetic fieldallows the magnetic domains of a given material to become disoriented. If thematerial is isolated, temperature drops as the disordered domains absorb thermalenergy in order to perform their reorientation. In cold atoms, this technique waspioneered in Chromium experiments, where the spin-flip mechanism was provided bydipole-dipole interactions [19]. Recently, a novel demagnetization cooling mechanismhas been proposed for two-component fermions based on time-varying magnetic fieldgradients [20]. In that method, scalar domains are cooled by transferring particle-holeentropy into magnetic entropy in overlapping regions between the two components.Note, however, that gradient cooling does not address cooling of the spin degrees offreedom, contrary to the method discussed below.In this paper, we study the spatial entropy distribution of multi-component spin- S fermions with an inhomogeneous quadratic Zeeman effect (QZE), and how thisspatially-dependent entropy profile may be employed for designing an adiabatic coolingmethod which specifically targets the reduction of spin entropy. We show in particular,that an inhomogeneous QZE may lead to an effective pseudo-spin-1 / S fermions at the wings. We show that, remarkably, the spin- S wings act asentropy absorbers all the way to vanishing temperatures in the presence of frustration.We illustrate the idea with the specific example of one-dimesnional (1D) spin-3 / / diabatic spin cooling using high-spin Fermi gases
2. Entropy and frustration
In the following we consider multi-component fermions loaded in an optical lattice.Increasing the number of spin components from 2 to N (effective spin S = ( N − / T by a factorof ln N/ ln 2. This guarantees that at high T , well over the N´eel ordering, spatialregions with a larger effective S act as entropy absorbers in the cooling process shownbelow. However, this simple argument does not apply at low T if the system acquiresconventional N´eel ordering [25]. The entropy of the Heisenberg antiferromagnet (HAF)in d dimensions scales for T ≪ T N (N´eel temperature [25]) as s ∼ S − d , and hence theentropy of a spin- S system decreases as compared to that of a spin-1 / T leading to a lower entropy.The situation is reversed in frustrated systems, which present a large degeneracyof classical ground states with many branches of soft excitations at low T . One arrivesat the simplest frustrated large- S model starting from the SU ( N ) symmetric Hubbardmodel H = − t X m, ( c † m,i c m,j + h.c. ) + U/ X i n i , (1)where m = ( − S, . . . , S ), c m,i annihilates fermions with spin m in the site i , n i = P m n m,i = P m c † m,i c m,i , and t and U are the hopping and interaction couplingconstants, respectively. In the strong-coupling limit, U ≫ | t | , and retaining onefermion per site, one can derive the effective permutation model in second order ofperturbation theory, H = J/ X P i,j , (2)with P i,j being the permutation operator and J = 4 t /U . This model is the SU ( N )generalization of the HAF. It is exactly solvable in 1D [27] and has N − v / /N , where v / = πJ/ T , the entropy SU ( N ) spin modelis larger than that of spin-1 / s S ( T ) = N ( N − πT / v / [27]. For equivalent 2D and 3Dlattice models an increase of entropy with unbinding number of degrees of freedom isalso expected at low T . There, the classical ground state shows extensive degeneracy,and it is believed [28] that the N´eel order does not get stabilized for SU ( N >
2) dueto high frustration [29].Hence, due to frustration, with increasing S , N´eel order gets suppressed leadingto a larger entropy storage capacity at low T . Thus, if the Mott edges get frustratedwhile preserving S = 1 / one can use the frustratededges as entropy absorbers all the way from high to extremely low T .
3. 1D Spin- / fermions in the Mott-phase In the following, we illustrate the possibilities provided by high-spin lattice fermionswith the specific case of a balanced mixture of spin-3 / N m with spin m satisfies N m = N − m . Interparticleinteractions are characterized by the s -wave scattering lengths for channels with diabatic spin cooling using high-spin Fermi gases Figure 1.
Ratio between the entropy per particle s at q = 0 (continuous line)or at q = ˜ q ( T ) (dashed line) (˜ q ( T ) is the QZE at which entropy is largest at agiven T ), and the entropy s c at high QZE ( q = q ), versus the temperature T inthe lattice. Observe the crucial role of frustration at low T . If we were dealingwith an unfrustrated S = 3 / T at ≃ . total spin 0 and 2, a , . For s -wave interacting fermions S = 3 / ± / ± / q , induces a finite chirality τ = L [( N / + N − / ) − ( N / + N − / )].For large-enough interactions and at quarter filling (one fermion per site) the 1Dsystem enters into the Mott insulator regime, for which the ground state propertiesunder QZE were studied in Ref. [31]. For large QZE the ground state is a pseudo spin-1 / q < q cr ( q cr = J ln 2 / a = a ) the system enterseither a spin liquid phase (for a < a ) or a dimerized phase (for a > a ). For a ≃ a (the typical situation unless a , are externally modified), the gap of the dimerizedphase is exceedingly small, and hence the system behaves in practice as a spin liquiddown to extremely low temperatures. For a = a , in the presence of a spatiallyvariable QZE, the model Hamiltonian becomes H = H + P i µ m,i n m,i , with H givenby Eq. (2) with N = 4. For homogeneous µ m,i , this model is exactly solvable andits thermodynamic properties may be calculated by means of thermodynamic BetheAnsatz. We follow the method of Ref. [32], based on the self-consistent solution of 14coupled integral equations, to obtain the corresponding free energy f . The chemicalpotentials for each component are µ m,i = µ + m q i , where µ is the global chemicalpotential, and q i denotes the QZE constant at site i . The entropy is then given by s = − ∂f /∂T and the chirality by τ = − ∂f /∂q .As mentioned above, for q > q cr the system becomes pseudo-spin-1 /
2. Hence, theratio between the entropy per spin for q = 0 ( s ) and that for q > q cr should followthe same dependence as s / ( T ) /s / ( T ). We illustrate this point in Fig. 1, where wecompare s at large q = q = 5 J ( s c ) to s . Note that at large T s /s c = 2, whereas atlow T frustration leads to s /s c = 6. diabatic spin cooling using high-spin Fermi gases Figure 2.
Entropy per particle (in k B units) and chirality profiles for L =120 particles for a Gaussian QZE profile q ( x ) = q exp (cid:0) − x /L (cid:1) , with q = 5 J . Dashed lines indicate the initial entropy prior to the switching of theinhomogeneous QZE profile. Before the lattice loading, T i /T F = 0 . .
016 (right). Note that for q > q cr , τ ≃ / T .The entropy bump for small T at q = q cr reflects the Van-Hove singularity at thebottom of the depleted Hubbard band.
4. Spatially-dependent QZE
We consider at this point the entropy and chirality profiles for the case of anon-homogeneous QZE, which may be achieved by means of microwave or opticaltechniques [22, 23, 24]. We perform local QZE approximation (similar to the localdensity approximation standard in trapped gases), i.e. we solve for the free energyat different positions x by varying q ( x ). The local QZE approximation demands asufficiently slow variation of the QZE at the scale of the inter-site spacing. In this waywe can evaluate the entropy profile inside the Mott insulator region. Note finally thatalthough the calculation is done for the case a = a , the conclusions may be extendedto the actual case in which a and a are slightly different, and spin redistribution viaspin-changing collisions occurs.Figure 2 shows chirality and entropy profiles. Note that the entropy per site (i.e.per particle) is significantly larger at the Mott wings than at the center. Hence, ifthe total entropy is conserved in a process in which an initially spatially-independententropy (deshed line on Fig. 2) is brought to the profile of Fig. 2, the outer regionswill remove entropy from the central core. This process is even more effective at low T . One may indeed estimate the entropy reduction (for T →
0) at the Mott centerfor model (2) in 1D, when considering an initial uniform spin-1 / / S = ( N − / γ ≡ s i /s c = [(1 − L /L ) N ( N − / L /L ] , (3)where L /L is the ratio of the number of sites in the spin-1 / s i ( s c ) is the initial (final) entropy per particle at the center. The gain ishence much bigger than the one at T ≫ T N , which is obtained after one substitutes N ( N − / N/ ln 2 in Eq. (3). diabatic spin cooling using high-spin Fermi gases S k B Figure 3.
Sliding down the isentropic curves (solid curves) with reducinguniformly QZE induces the reduction of the temperature in the lattice at thesecond stage of the proposed adiabatic cooling scheme.
5. Adiabatic cooling
The spatially-dependent entropy distribution opens interesting perspectives foradiabatic spin cooling. A possible scheme would consist on three steps. On a firststage a two-component balanced mixture is created at the lowest possible temperature T i , being stabilized against spin-changing collisions [21] by means of a sufficientlylarge homogeneous QZE. On a second stage a lattice is adiabatically grown andthe homogeneous magnetic field is adiabatically decreased allowing for spin-changingcollisions throughout the sample. The drop of the temperature in the lattice withthe adiabatic decrease of the homogeneous magnetic field can be estimated fromthe isentopic curves on Figure 3. However, to enter the spin coherent regime localentropy should be reduced. This is achieved in the final step which consists onslowly changing the QZE into a non-uniform profile by means of microwave oroptical techniques [22, 23, 24], leading to the coexistence of a spin-1 / S spin liquid at the wings (Fig. 2).Figure 4 shows, as a function of the initial T i /T F ( T F is the Fermi temperatureof the original spin-1 / s/k B = 0 .
35 is achieved for the homogeneouscase at T i /T F ≃ .
035 whereas for the inhomogeneous QZE profile it is reached at T i /T F ≃ .
09, showing that an inhomogeneous QZE may allow for a large entropyreduction at the center as a result of the entropy excess at the storage wings. Notethat for T i /T F ≃ . / d = 2 ,
3) systems. A simpleestimate may be obtained from a spin-wave analysis when a > a , where at least inthe limit a ≫ a N´eel order sets in on bipartite lattices [33], however due to frustration diabatic spin cooling using high-spin Fermi gases Figure 4.
Entropy at the center s c (in k B units) as a function of the temperaturebefore lattice loading, T i /T F , for the inhomogeneous q ( x ) profile of Fig. 1 (solid)and for a homogeneous one q = q (dashed), for which the whole system retainsa spin-1/2 character. In order to associate a central s c with a given T i /T F ,we calculated for different T inside the lattice the entropy profiles and the totalentropy S tot /L = π k B T i /T F . the modulus of the N´eel order parameter decreases with increasing spin. There is justone spin-wave mode at q → ∞ and three spin-wave modes at q = 0. Taking the valuesof spin-wave velocities from [34], we obtain that at T → γ is given byEq. (3), albeit with N ( N − / a + a ) / ( a − a )] d , implying that the cooling should be even more efficient for higher dimensions . Note that the spin-wave analysis predicts γ to increase indefinitely when approaching a = a , althoughspin-wave analysis becomes less reliable in the vicinity of that point [34].
6. Experimental feasibility
A possible experimental sequence may be devised for e.g. K. Prior to the latticeloading, a balanced mixture of e.g. F = 9 / , M F = − / F = 9 / , M F = − / T i /T F using standard techniques. At thisstage a sufficiently large homogeneous magnetic field guarantees that the initialtwo-component mixture is stable against spin-changing collisions. The gas can beapproximated with a good accuracy by a free Fermi gas and therefore the initialentropy per particle is given by s = k B π T i /T F [30]. State of the art experiments mayreach at this stage T i/T F ≈ .
1, corresponding to an entropy per particle of s ≈ k B .Once the gas is cooled down, a 3D lattice is grown and under proper conditions a Mottinsulator with one particle per site develops at the trap center [5, 6]. The next stepconsists in slowly lowering the magnetic field to allow for quasi-resonant spin-changingcollisions [35] throughout the sample leading to a redistributed population (between F = 9 / M F = − / − / − /
2, and − / T inthe lattice. Temperature drop after this stage for 1D case, (assuming adiabaticity)is depicted on Fig. 3. Note that fermions with S > / S > / F = 9 / , M F = − / F = 9 / , M F = − / F = 9 / M F = − / F = 7 / M F = − / F = 9 / M F = − / diabatic spin cooling using high-spin Fermi gases F = 7 / M F = − /
2. This ∆ M F = − M F = − / − / q cr . Furthermore, the Raman beams shift the resonance condition for spin-changing collisions in the center of the trap such that no more atoms in M F = − / M F = − / s N is reached). In this way the problem of approaching the ground statedespite the presence of many metastable low energy states may be circumvented.Let us mention some final remarks. For simplicity of our calculations we haveconsidered cooling only within a Mott insulator. In the presence of particle-holeexcitations we expect an even larger cooling efficiency [6]. Note that the S = 1 / S > / S > /
7. Conclusions
In summary, we studied a possible route for adiabatic spin cooling using high-spinlattice fermions. The process resembles demagnetization cooling, but the role of themagnetic field is played by a spatially dependent QZE, and spin flip is substituted byspin-changing collisions. The spatially dependent QZE leads to two distinct regions ofdifferent effective spin ( S = 1 / S > / T the outer spin- S region actsas an entropy absorber simply due to the larger spin, whereas the same remains true ateven very low T due to frustration. As a result we showed that a significant reductionof the entropy (more pronounced at lower T and higher dimensions) of the spin-1 / / Acknowledgements.
We thank A. Seel, J. Grelik and U. Schneider for discussions. We acknowledgesupport from the Center of Excellence QUEST, the ESF (EuroQUASAR), and theSCOPES Grant IZ73Z0-128058.
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