Squeezing out the entropy of fermions in optical lattices

Abstract

At present, there is considerable interest in using atomic fermions in optical lattices to emulate the mathematical models that have been used to study strongly correlated electronic systems. Some of these models, such as the 2-dimensional fermion Hubbard model, are notoriously difficult to solve, and their key properties remain controversial despite decades of studies. It is hoped that the emulation experiments will shed light on some of these long-standing problems. A successful emulation, however, requires reaching temperatures as low as 10⁻¹² K and beyond, with entropy per particle far lower than what can be achieved today. Achieving such low-entropy states is an essential step and a grand challenge of the whole emulation enterprise. In this article, we point out a method to literally squeeze the entropy out from a Fermi gas into a surrounding Bose–Einstein condensed gas, which acts as a heat reservoir. This method allows one to reduce the entropy per particle of a lattice Fermi gas to a few percent of the lowest value obtainable today.

Currently, many laboratories are trying to realize the anti-ferromagnetic (AF) phase of the 3D Hubbard model by using ultra-cold fermions in optical lattices (1). Although the AF phase is well known in condensed matter, its realization will be a major step for the emulation program because it requires overcoming the serious challenge mentioned above that is common to all cold atom emulations (2). To understand the origin of the problem, recall that the strongly correlated states of lattice fermions emerge in the lowest Bloch band of the optical lattice. To put all the fermions in the lowest band, a sufficiently deep optical lattice is required. In such deep lattices, many known methods of cooling fail. For example, standard evaporative cooling does not work because the magnetic repulsive potential used in the evaporation process is not strong enough to overcome the deep lattice. As a result, all current experiments on lattice quantum gases resort to the conventional cooling scheme, first cooling the quantum gas in a harmonic trap (without optical lattice) to the lowest-entropy state possible, and then turning on the lattice adiabatically (3, 4). The hope is that one could reach the state of interest when the lattice depth is sufficiently high.

To realize these strongly correlated states with the current scheme, it is necessary that the entropy of the gas prior to switching on the lattice be less than that of the strongly correlated state one wishes to achieve. For 3D fermion Hubbard models, recent studies (5) show that it is possible to reach the AF phase slightly below the Neel temperature T_N with the conventional cooling method. A similar calculation by Tremblay et al. (6) for the 2D Hubbard model, however, showed that the conventional scheme cannot reach even the pseudo-gap regime, which exists at a higher temperature than the anticipated superconducting phase. Although these calculations are for homogenous systems, they apply to confining traps as long as the majority of the sample is a Mott insulator with one fermion per site. These studies show that, even under optimal conditions, one can at most reach the AF phase close to the magnetic ordering, but not low enough temperatures to study ground-state properties.

The problem of the conventional method is that it can only cool atoms before they are loaded onto an optical lattice. There is no way to reduce the entropy further once the lattice is switched on. Thus, the lowest entropy attainable today in an optical lattice plus harmonic trap is just the lowest entropy achievable in a harmonic trap alone. In the case of Fermi gases, it is S/N = π²(T/T_F) (7), where T_F is the Fermi temperature in the trap. For T/T_F = 0.05, which is very much the limit today, we have S/N ∼ 0.5. Any strongly correlated states with lower entropy are unreachable by this method. It is therefore important to find ways to reduce the entropy of the system significantly below this value.

Our Entropy Removal Scheme

The purpose of this article is to present a scheme to produce a lattice Fermi gas with about one fermion per site with entropy per particle much lower than what can be achieved today. Our method is based on the principle of entropy redistribution and the removal of entropy by isothermal compression. It consists of the following steps.

1. We immerse the lattice fermions in a Bose–Einstein condensate (BEC) that acts as a heat reservoir. The fermions are confined in a harmonic trap and a strong optical lattice. The bosons sees a much weaker trapping lattice than the one that traps for the fermions. They are confined in a loose trap and cover the entire fermion system. The traps for bosons and fermions are species specific (8) so that they can be varied separately. (See below for discussions of these potentials.)

2. We compress the fermion harmonic trap adiabatically to turn the fermions at the center into a band insulator, which has 2 fermions per site and essentially zero entropy. During this process, a substantial amount of the original fermion entropy is pushed into the bosons, while the entire system has little temperature change because of the large heat capacity of the BEC compared with the lattice fermions. Hence, even though the process is adiabatic for the entire Bose–Fermi system, it is essentially isothermal as far as the fermions are concerned.

3. After pushing out the fermion entropy into the BEC, we remove it by evaporating away the bosons all at once, leaving the remaining fermions to equilibrate. Because the band insulator is incompressible, only its density and entropy near the surface are affected during this process. This in turn severely limits entropy regeneration during the equilibration process. As a result, the entropy of the rethermalized band insulator has a similar ultralow value as before boson evaporation.

4. We open up the the fermion harmonic trap adiabatically to lower the density of the lattice fermions. In this way, one can produce a Mott insulator or other states with fractional filling with the same ultralow entropy.

Before proceeding, we return to discuss the construction of the trapping potentials mentioned in Step 1. To have an optical lattice that confines the fermions tightly and the bosons loosely, the energy difference between the ground state (S state) and the excited state (P state) of fermions must be smaller than that of the bosons, ΔE_f < ΔE_b. In this way, one can choose a laser with frequency (ω) red detuned with respect to both excitation energies (ℏω < ΔE_f,ΔE_b), such that the detuning for the fermions is smaller than that for the bosons, ΔE_b − ℏω > ΔE_f − ℏω. For ⁴⁰K fermions, its 4S to 4P transition has a wavelength 740 nm, whereas the wavelength of the 2S to 2P transition of ⁷Li boson is 671 nm, which satisfies the aforementioned condition. Moreover, the difference between these excitation energies large enough so that the detuning ΔE_b − ℏω and ΔE_f − ℏω sufficiently large to suppress heating due to spontaneous emission. Condition (1) can therefore be satisfied.

Another scheme that makes use of the hyperfine structure of the P state of ⁸⁷Rb was pointed out in ref. 8. By tuning the laser frequency to 790.01 nm, which is between the 2 hyperfine states P_3/2 and P_1/2 of ⁸⁷Rb, Rb bosons see no lattice because the red detuned lattice due to P_3/2 is canceled by the blue detuned lattice due to P_1/2. However, such laser will generate an attractive potential for ⁶Li and ⁴⁰K, because its wavelength is longer than those of the S – P transitions of ⁶Li and ⁴⁰K, respectively.

To illustrate our scheme, we will first discuss the basic properties of lattice fermions and the important process of entropy redistribution.

Number Density and Entropy Distributions of Lattice Fermions

A Fermi gas in the lowest band of a (3D) optical lattice is described by the Hubbard model

where $\widehat{J}$ = −J∑_{〈R,R′〉,σ} a_R,σ^†a_R′,σ describes hopping of fermions with spin σ, (σ = ↑,↓) between neighboring sites R and R′, J is the tunneling integral, $\widehat{W}$ = $U{\sum }_R{n}_{R,\uparrow }{n}_{R,\downarrow }$ describes the on-site repulsion (U) between spin-up and spin-down fermions; a_R,σ^† and n_R,σ = a_R,σ^†a_R,σ are the creation and number operators of a fermion with spin σ at site R; $\widehat{N}$ = ∑_R,σ n_R,σ is the total fermion number, and μ is the chemical potential.

The possible states of the fermions are: band insulator (BI), Mott insulator (MI), and “conducting” state (C), corresponding to two, one, and a noninteger number of fermions per site, respectively. The Mott insulator will develop AF order at the Neel temperature T_N ∼ J²/U, whereas the conducting state is expected to have a superfluid ground state. The strongly correlated regime emerges when

with J²/U being the smallest energy scale. At present, experiments operate in the temperature range

For simplicity, we set Boltzmann's constant k_B = 1. Our goal is to perform operations in this high-temperature regime so that the fermions will lose a substantial amount of entropy.

Within the temperature range described by Eq. 3, $\widehat{J}$ can be treated as a perturbation and $\widehat{K}$ is site-diagonal to zeroth order in J. The number occupation at site R is then

The entropy per site is s_R = ∂(TlnZ_R)/∂T, where Z_R(T,μ) = Tre^{−$\widehat{K}$_R/T} = 1 + 2e^μ/T + e^(2μ−U)/T is the partition function at site R. Explicitly, we have

and E_R(T,μ) = U〈$\widehat{n}$_R,→$\widehat{n}$_R↓〉 = e^{(2μ − U)/T} / (Z_R(T,μ).

In a harmonic trap V_ω(R) = Mω²R²/2, the density and entropy distributions in the temperature range U > T > J can be calculated from Eqs. 4 and 5 using local-density approximation (LDA) by replacing μ with μ(R) ≡ μ−V_ω(R). A typical distribution is shown in Fig. 1. The chemical potential μ and temperature T in Eqs. 4 and 5 are determined from the number and entropy constraints,

where N is the total number of fermions and S is the total entropy produced in the convention cooling scheme before the lattice is turned on. Fig. 1 shows the following phases: [(BI) : n_R → 2, for U < μ(R)]; [(MI) : n_R → 1 for 0 < μ(R) < U]; [(Vacuum) : n_R → 0 for μ(R) < 0]. There are also two conducting phases (C1) and (C2) with noninteger number of fermions per site. They are (C1) : 1 < n_R < 2, with μ(R) ∼ U, and (C2) : 0 < n_R < 1 with μ(R) ∼ 0. We will denote the centers of the conducting regions (C1) and (C2) as R₁ and R₂. They are determined by

Fig. 1.

Density distribution n_R (red) and entropy distribution s_R (blue) of a Fermi gas in the lowest band of an optical lattice and in the presence of an harmonic trap. BI and MI denote band insulator and Mott insulator, respectively. The regions labeled C1 and C2 are the conducting regions where the fermions have a large number of fluctuations and are mobile.

The (BI) region has essentially zero entropy per particle, s_R ∼ 0, since the pair (↑↓) is the only possible configuration at each lattice site. At temperatures U > T > T_N, (MI) has spin entropy s_R ∼ ln 2, since both ↑ and ↓ are equally probable. At the same temperature range, (C1) has s_R ∼ ln 3, since the doublet (↑↓) and the single-spin states ↑ and ↓ are all equally probable. At even higher temperatures, s_R of both (MI) and (C1) can rise as high as ln 4, as the probability of having an empty site increases from zero. Similar situation occurs in (C2).

While the total entropy and other properties of the system can be calculated in the temperature range U > T > J by using Eqs. 4, 5, and 6, it is useful to understand them by using simple estimates. The total entropy of the system is

where d is the lattice spacing, and ΔR₁ and ΔR₂ are the widths of the (C1) and (C2) regions. They are given by Δμ(R₁) ∼ T and Δμ(R₂) ∼ T, or

The quantities ${\bar{s}}_C$ and ${\bar{s}}_M$ are the average entropy per site in (C) and (MI) regions: ln 3 < ${\bar{s}}_C$ < ln 4, and ln 2 < ${\bar{s}}_M$ < ln 4. From Eq. 7, we also have

showing that the (MI) region shrinks as ω increases.

Entropy Localization and Reduction

From Fig. 1, we see that if we compress the trap adiabatically (by increasing ω), the band insulator will grow at the expense of other phases. As a result, the Mott regions and the conducting regions, and hence all the entropy, are pushed to the surface, as shown in Fig. 2A and B. Since all the entropy in the bulk is squeezed into the surface layer, the entropy density at the surface must rise above its former value to keep the total entropy constant, which means the temperature the system will increase, as shown in Fig. 2. The fact that adiabatic compression causes heating is a consequence of thermodynamics (2).

Fig. 2.

Density and entropy distributions in adiabatic (A and B) and isothermal (C and D) processes. (A) Shown is the number density n_R of N = 2 × 10^{6 40}K fermions in an optical lattice with lattice height V_o = 15E_R and U = 0.5E_R at harmonic trap frequencies ν_F = ω_F/2π = 100 Hz (blue), 150 Hz (purple), and 600 Hz (red), calculated from Eqs. 4–6. The entropy per particle is fixed at S/N = 0.5. The corresponding entropy distributions s_R are shown in B, with temperatures 27 nK (blue), 88nK (purple), and 1,880 nK (red), respectively. As the trap frequency is increased from 100 Hz (blue) to 150 Hz (purple), A and B show that the Mott phase melts away and the entropy density at the center of the conducting layer increases, leading to a rise in temperature. This rise is very rapid. It scales as ω² as seen in Eq. 11. C and D show the number density n_R and entropy density of s_R for the same parameters as A and B, but now with the temperature fixed (T = 50 nK) instead of entropy per particle. As the harmonic trap is compressed from 100 Hz (blue) to 150 Hz (purple) and then 600 Hz (red), S/N decreases from 0.732 (blue) to 0.333 (purple) and finally, 0.021 (red), reaching a value that is only 4% of the lowest value attainable today, and has not yet reached the limit shown in Eq. 12.

As compression proceeds, it reaches the point where the width of the (MI) region becomes so thin that it is of the order of a few lattice spacings (see Fig. 2A and B). This occurs at R₂ − R₁ ≪ R₁ ∼ R₂ ∼ R, where R is the radius of the (BI) region related to the total fermion number N by N = 2(4π/3)(R/d)³. In this case, we have

Eq. 11 shows that T rises as ω² during an adiabatic compression.

If, however, we are able to keep the temperature constant during the compression, the entropy density at the surface will remain at its initial value (see Fig. 2C and D). As a result, the total entropy will be reduced by the same amount as the reduction of entropic volume (i.e., the region that contains entropy), which is substantial as the latter changes from a 3D volume to a surface layer. This can also be seen in Eq. 11, where S/N drops as ω⁻² in an isothermal compression.

As isothermal compression continues, the widths of the (C) and (MI) regions will shrink down to a lattice spacing d, and the entropic surface region will eventually be reduced to a surface of thickness d. At this point, LDA breaks down. Due to thermal fluctuation, the thickness of this surface layer will remain approximately one lattice spacing, regardless of further increase of the trapping potential. As long as T is above the spin-ordering temperature T_N, the entropy per site is still ln 2. This limits the lowest entropy attainable in isothermal compression for T > J²/U to $S^{*}=(4\pi R^{2}d)(\bar{s}/d^3)$, or

where $\bar{s}$ ∼ ln 2. For N = 5 × 10⁶ (3 × 10⁷), S*/N is 0.012 (0.007) which is 2% (1%) of the lowest value attainable today.

Entropy Transfer Between Fermions and BEC

To keep the fermions at roughly constant temperature, they must be in contact with a heat reservoir. A BEC is an ideal medium for this purpose, for it has a much higher heat capacity than the fermions and can therefore keep the temperature roughly constant. In this section, we will provide explicit calculations to demonstrate this fact. Our results show that the transfer of entropy between fermions and bosons is accurately described by the simple isothermal compression model in Entropy Localization and Reduction.

We will consider a mixture of BEC and fermions. The BEC is contained in a harmonic trap $V_B(R)=\frac{1}{2}{M}_B{\omega }_B^2{R}^2$ with frequency ω_B, where M_B is the boson mass, and another trap $V_F(R)=\frac{1}{2}{M}_F{\omega }_F^2{R}^2$ is used for the fermions. The fermions are also confined in an optical lattice. We will assume that these potentials are species specific as described in Step 1 in Our Entropy Removal Scheme, so that V_F can be varied independently of V_B. This is important because the compression of fermions should not lead to a substantial compression of the BEC. Otherwise the temperature of the BEC will rise, making it less efficient in absorbing the entropy of the fermions.

To study the entropy transfer between fermions and bosons, we first consider a homogenous Bose–Fermi system. In the grand canonical ensemble, the hamiltonian is

where $\widehat{K}$_F is the Hubbard hamiltonian (Eq. 1) with μ now denoted as μ_F. $\widehat{K}$_B is the hamiltonian for bosons with chemical potential μ_B,

where $\widehat{\varphi }$^† is the creation operator for bosons and g_BB = 4πℏ²a_BB/M_B. $\widehat{H}$_BF is the boson–fermion interaction,

with coupling constant g_BF, d is the lattice spacing, and $\widehat{n}$_F,R ($\widehat{n}$_F,R) is the number of fermions (bosons) in a unit cell centered at site R.

To calculate the properties the system, we make the following approximations:

• (i) a mean field decomposition of H_BF, replacing it by H_BF^M = g_BF ∑_R($\widehat{n}$_F,Rn_B + n_F$\widehat{n}$_B,R − n_Fn_B), where n_F = 〈$\widehat{n}$_F,R〉; n_B = 〈$\widehat{n}$_B,R〉;

• (ii) treating the hopping term J as a perturbation of $\widehat{K}$_F as in Number Density and Entropy Distributions of Lattice Fermions. This is justified in the temperature regime U > T > J;

• (iii) applying the Hartree–Fock approximation for thermodynamics of the bosons (9). With approximation i, $\widehat{K}$ becomes

  

The pressure P(T,μ_F,μ_B) = Ω⁻¹T ln Tre^−K′/T is then

where P_F,B(T,μ_F,B) = Ω⁻¹T ln e^{−K_F,B(μ_F,B)/T}, and Ω is the volume of the system. The fermion and boson densities are given by

Within approximation ii, the fermion density is readily given by Eq. 4 as

To apply approximation iii, we follow the procedure in ref. 9 to calculate the boson density for both the normal and superfluid parts of the boson cloud, which is of the form

according to Eq. 16. (The presence of the superfluid will depend on the value μ_B − g_BFn_F.)

Eqs. 19 to 20 form a complete set of equations that determine (n_B, n_F) self-consistently as a function of (T,μ_B,μ_F). All these have to be evaluated numerically. These solutions then allow one to evaluate the entropy density, which is $s(T,{\mu }_F,{\mu }_B)=\frac{\partial P}{\partial T}$, or

where s_F, s_B, and s_BF are defined as fermion, boson, and “interaction” entropy density, respectively,

which will also be functions (T,μ_B,μ_F). To find the density and entropy distribution in the presence of the boson and fermion traps, we can use LDA to replace μ_F,B by μ_B,F(R) = μ_B,F − V_B,F.

In our numerical calculations, we consider 5 × 10⁶ of ⁴⁰K fermions in a lattice immersed in a BEC of ⁸⁷Rb with N_B = 5 × 10⁷ bosons. The frequency of the fermion trap is initially set at ν_F = ω/(2π) = 100 Hz. The bosons are confined in a loose trap with frequency ν_B = ω_B/(2π) = 10 Hz, so that it covers the entire fermion system. We take a_BB = 5.45 nm, and g_BF = g_BB/2. The lattice height is V_o = 15E_R and the fermions have a Hubbard interaction U = 1.5E_R, where E_R is the recoil energy. To demonstrate the transfer of entropy from the fermions to the BEC, we consider an initial state with initial temperature 60 nK, corresponding to the entropy per particle S_F/N_F = 0.691 for the fermions. For these parameters, most of the bulk is already a band insulator, similar to that in Fig. 2A and the entropy density is accumulated at the surface. (See Fig. 3A.)

Fig. 3.

Density distribution n_R and entropy distribution s_R of bosons (blue) and fermions (red) in a mixture of BEC and lattice Fermi gas. The total number of fermions N_F (Bosons N_B) is 5 × 10⁶ (5 × 10⁷). The trapping frequency of bosons is fixed at ν_B = 10 Hz. The number density of bosons and fermions at fermion harmonic trap frequencies 100 Hz, 200 Hz, and 600 Hz are shown in A – C. The corresponding entropy densities are shown in D – F. As the frequencies ν_F of the fermion harmonic trap increases, the edge of the band insulator sharpens and the entropy distribution becomes more and more narrow, yet the temperature of the system changes very little as the fermion trap is tightened, from 60 nK to 64.2 nK, and to 65.6 nK. As a result, the height of the entropy distribution remains roughly constant. The narrowing of the entropy distribution, however, leads to a rapid drop in the entropy per particle, from 0.691 to 0.184 to 0.016, as shown in D – F.

When the fermion trap frequency ν_F increases from 100 Hz to 600 Hz, we see from Fig. 3A – C that the edge of the band insulator becomes sharper, while the entropy density becomes concentrated at the surface (Fig. 3D – F). Our calculation also shows that the fermion entropy S_F ≡ ∫ s_f decreases, whereas the boson entropy S_B ≡ ∫ s_B increases, while S_BF ≡ ∫ s_BF remains much smaller than S_F and S_B during the compression, and the entropy of the entire Bose-Fermi mixture is a constant.

Because of the large heat capacity of the bosons, the overall temperature T rises only moderately, from 60 nK to 65.6 nK. This shows that the simple isothermal compression model for the lattice fermions in Entropy Localization and Reduction is a reasonable approximation of the entropy transfer process between the bosons and the fermions. At ν_F = 600 Hz, the fermion entropy per particle is S_F/N_F ∼ 0.016, which is ≈3% of the best estimate achievable with conventional methods.

Equilibration of the Lattice Fermions After Boson Evaporation

After transferring the entropy of the fermions to the BEC as shown in Fig. 3C at the temperature 65.5 nK (≡ T⁽ⁱ⁾) and at trap frequency ν_F = 600 Hz, we evaporate all the bosons suddenly so as to obtain a pure fermion system. The lattice Fermi gas will then relax to its equilibrium state corresponding to the new trap frequency ω_F = 2πν_F. In this process, entropy will be generated and temperature will rise. However, because the band insulator is incompressible, and because the trap has a larger value of ω_F, entropy generation is limited. We find (in the calculation below) that the increases in entropy and temperature are about a few percent of their values prior to evaporation. In other words, the entropy of the band insulator after it reaches equilibrium is essentially the same as that before boson evaporation, i.e., S_F/N_F ∼ 0.02. This will be the entropy inherited by the Mott insulator that emerges from the band insulator as one decompresses the trap.

The calculation for the above processes is as follows. Immediately after evaporation, the lattice Fermi gas has total energy

where ɛ(T,μ) ≡ U〈$\widehat{n}$_R,→$\widehat{n}$_R↓〉 is the internal energy per site of a homogeneous lattice fermion with hamiltonian $\widehat{K}$_F at the temperature range U > T > J, and

n_B,F(R) are boson and fermion densities prior to boson evaporation, which were calculated from Eqs. 19 and 20. When the system finally relaxes to equilibrium, it will have a different temperature T^(f) and chemical potential μ_F^(f). The energy of the final system is

Since energy is conserved in this process, we have E^(f) = E⁽ⁱ⁾. In addition, the total number of particles is given by $N_F={\sum }_R{n}_F(T^{(f)},{\mu }_F^{(f)}-V_{\omega }(R))$. These two relations uniquely determine the final temperature T^(f) and chemical potential μ_F of the band insulator, from which one can calculate the final entropy. With the parameters we mentioned before, we find that the entropy increase in this process is negligible, only about a few percent of the value before evaporation; hence, our conclusions summarized in Our Entropy Removal Scheme.

Final Remarks

We have introduced a method that allows extraction of a substantial fraction of the entropy of a Fermi gas in an optical lattice after a strong lattice is switched on. Moreover, the extraction process is conducted at the temperature regime T > J, much higher than the Neel temperature T_N ∼ J²/U. Although our method makes explicit use of the band insulator, it is applicable to any system that has an equilibrium phase with a large gap. The idea is to use the gapful phase to push away all the entropy in the bulk into a surrounding Bose–Einstein condensed gas. It should also be stressed that, although the final stage of our method involves evaporating away the Bose–Einstein condensed gas, it is different from the usual sympathetic cooling both in purpose and in function. First, our evaporation must be preceded by the localization of fermion entropy to be effective. Second, the purpose of removing the bosons is not to decrease the energy of the system (as in the usual sympathetic cooling), but to remove the entropy it absorbed from the fermions. A natural question is whether one can efficiently reduce the entropy of a Mott phase which is in direct contact with a BEC by evaporating on the latter. The answer is negative, as shown by our calculations. The reason is that the interactions between bosons and fermions typically have very weak spin dependence. The removal of bosons therefore has little effect on the spin entropy of the Mott phase, which will remain close to its value before boson evaporation, ln 2 per particle.

Finally, we point out that our scheme assumes that it is possible to expand the trap adiabatically to turn a band insulator into a Mott insulator. This implies that the relaxation time for particle redistribution is sufficiently fast. Since mass transport occurs at the interface between different phases (i.e., regions (C1) and (C2)), it will take place within the timescale ℏ/J. It is therefore helpful to use lattices with sufficiently large tunneling, (say, ≈10–15 E_R), and to reach the large U limit by increasing the interaction by using a Feshbach resonance. (In the case of bosons, Cheng Chin's group at Chicago has recently succeeded in producing a Mott phase of cesium bosons by using a Feshbach resonance in a lattice with relatively large tunneling, which has shortened considerably the equilibration time.)