Statistical mechanics of a Feshbach coupled Bose-Fermi gas in an optical lattice
aa r X i v : . [ c ond - m a t . o t h e r] J u l Statistical mechanics of a Feshbach coupled Bose-Fermi gas in an optical lattice
O. Søe Sørensen and N. Nygaard
Lundbeck Foundation Theoretical Center for Quantum System Research,Department of Physics and Astronomy, University of Aarhus, DK-8000 Århus C, Denmark
P. B. Blakie
Jack Dodd Centre for Quantum Technology, Department of Physics, University of Otago, New Zealand
We consider an atomic Fermi gas confined in a uniform optical lattice potential, where the atoms can pair intomolecules via a magnetic field controlled narrow Feshbach resonance. The phase diagram of the resulting atom-molecule mixture in chemical and thermal equilibrium is determined numerically in the absence of interactionsunder the constraint of particle conservation. In the limiting cases of vanishing or large lattice depth we derivesimple analytical results for important thermodynamic quantities. One such quantity is the dissociation energy,defined as the detuning of the molecular energy spectrum with respect to the atomic one for which half of theatoms have been converted into dimers. Importantly we find that the dissociation energy has a non-monotonicdependence on lattice depth.
I. INTRODUCTION
The use of Feshbach resonances with ultracold atoms in op-tical lattices provides a new avenue for creating molecules.This system has seen experimental realizations in a variety ofsystems with bosonic and fermionic species [1, 2, 3, 4, 5].For the case of fermionic atoms the Feshbach assisted conver-sion is to molecular bosons with strikingly different behaviorin the degenerate regime. In this article we study the statis-tical mechanics of such a degenerate gas of fermionic atomsconfined in an optical lattice and subject to a Feshbach reso-nance. Our interest is the chemical equilibrium between thefermionic atoms and their dimerized, bosonic counterparts.For an ultracold atomic gas a Feshbach resonance is ob-served when a closed channel bound state is coupled to thescattering continuum of an energetically open channel. Theposition of this resonance is tunable, since the different chan-nels correspond to different combinations of internal atomicstates and hence experience different Zeeman shifts in an ap-plied magnetic field. As the closed channel bound state istuned from above to below the open channel threshold, theresonance becomes a true molecular bound state of the two-body system [6]. The parameter controlling the position of theresonance as the magnetic field is varied is the resonance en-ergy E res , which is the detuning of the closed channel boundstate from the open channel threshold.Previously the thermodynamics of an atomic Fermi gaswith Feshbach resonant atom-molecule conversion has beenstudied both in free space [7] and in a harmonic confining po-tential [8, 9]. For E res < the dimers are stable against disso-ciative decay and hence are real molecules. For E res > thedimers have a finite lifetime, but we can still discuss chem-ical equilibration of the gas in the steady state limit, wherea detailed balance is established between molecule formationand decay. In this respect we also use the term molecules forthe unstable dimer states, which in an ensemble will have afinite occupation at any instant. Note that this implies a nar-row Feshbach resonance, for which the scattering resonanceis a consequence of a long-lived quasi-stationary state embed-ded in the continuum [11]. In the structured continuum of the optical lattice the molecular states may also be stable againstdissociative decay at energies above the continuum threshold,if their energies lie in the band gaps [12, 13, 14, 15].For a narrow Feshbach resonance it has been establishedthat the existence of a small parameter facilitates a quanta-tively correct description of the many-body physics based ona pertubative expansion [10, 11]. Building on this insight, weconsider for simplicity an ideal gas mixture, such that the onlyeffect of the Feshbach resonance is to maintain the chemicalequilibrium between the two species. Even this simple modelcaptures the essential physics of atom-molecule conversionsin experiments [16], and it is exact in the limit where the res-onance is infinitey narrow [11]. With a finite atom-moleculecoupling an effective atom-atom interaction arises upon theelimination of the molecular degrees of freedom. We empha-size that even though this effective interaction may divergeas the magnetic field is varied across the resonance, all ther-modynamic quantities such as the chemical potential and thecondensate fraction remain well behaved (see e.g. [17, 18]).In particular, for a narrow Feshbach resonance the ideal gasthermodynamics remains qualitatively correct when the atom-molecule coupling is small but finite [11].We believe that the optical lattice introduces considerablenew physics, and that the best way to understand the result-ing changes in the thermodynamics is to consider the idealcase first. Thus, for a narrow Feshbach resonance we an-ticipate that the inclusion of interactions will only impactour results quantitatively. Notwithstanding, for a broad Fes-hbach resonance interactions play a crucial role, and hencein that case the thermodynamics of the BCS-BEC crossoverrequires more advanced modelling than presented here, i.e.a full many-body theory for resonantly interacting Fermiatoms [19, 20, 21, 22, 23, 24]. For a deep lattice this hasbeen studied at zero temperature by Koetsier et al. [25]. How-ever, even for a broad resonance a thermodynamic descriptionbased only on the molecular binding energy, the temperatureand the total number of atoms can give a molecule fractionwhich agrees well with experimental data across the Feshbachresonance [26].The thermal properties of ultracold atoms in optical latticeshave received considerable recent attention. In the degenerateregime ideal Bose and Fermi gases have been investigated inthe uniform lattice [27, 28] and in the presence of addition-al external harmonic confinement [29, 30, 31]. Interactionsplay an important role in deep lattices, and more recent workhas examined the effect these interactions have on the thermalexcitation generated during the preparation of bosonic Mott-insulating states [32, 33], and on the feasibility of achievingthe fermionic Neel state [34]. Finite temperature mixtures ofatomic Bose and Fermi gases in lattices have been studied [35]in an attempt to explain recent experiments [36].Our system has several parameters which can be varied in-dependently in a numerical calculation. We only show resultsfor half filling which is of the greatest interest in relation tocurrent experiments. However, we have found that systemswith filling fractions less than or equal to unity have much thesame behavior. For filling fraction larger than unity the ex-cited bands play a larger role, and some of the conclusionspresented here have to be amended. For simplicity we re-strict our analysis to the case of a spin-balanced Fermi gas.There are many interesting effects associated with spin polar-ization, but these do not play a major role in the transitionbetween atoms and molecules which we intend to study here.Instead the essential quantities governing the phase diagramare the resonance energy, i.e. the energy offset between atomsand molecules, the lattice depth, and the temperature. Tun-ing the resonance energy at a fixed lattice depth shifts the en-ergy spectra of the atoms and the molecules with respect toeach other, thereby moving the point of chemical equilibrium.Since the atoms and molecules experience different lattice po-tentials, the depth of the periodic potential determines wherethe transition between the two species occurs. II. FORMALISM
We consider a simple cubic optical lattice with M sitescontaining N tot identical fermionic atoms of mass m a withtwo internal states labeled |↑i and |↓i , which we will referto as spin-up and spin-down respectively. In this paper werestrict our attention to the case of an equal number of atomsin the two internal states N ↑ tot = N ↓ tot = N tot , (1)and we define the filling fraction η to be the average numberof each kind of atom per site η ≡ N ↑ tot M = N ↓ tot M = N tot M . (2)With the Feshbach resonance a spin-up and a spin-downatom can couple to a bosonic dimer with mass m m = 2 m a .In the following we determine the phase diagram of the sys-tem under the assumption that the only interaction between theatoms is the Feshbach resonance, which maintains chemicalequilibrium between unbound atoms and diatomic molecules.Since we are assuming thermodynamic equilibrium, it is thenmeaningful to define the number of molecules N m and the number of atoms N a as the average number of molecules andatoms, respectively, in a long time period where all the exter-nal parameters are kept constant. At all times particle conser-vation leads to N tot = N a + 2 N m = N ↑ a + N ↓ a + 2 N m , (3)where N ↑ a and N ↓ a are the numbers of unbound spin-up andspin-down atoms respectively.In thermal equilibrium the temperatures of the atoms andthe molecules are the same, T , while the condition of chemicalequilibrium can be expressed from the atomic and molecularchemical potentials as µ m = µ ↑ a + µ ↓ a . (4)Since, in the spin-balanced case, the spin-up and spin-downatoms have identical thermal behavior, we have µ ↑ a = µ ↓ a ≡ µ and therefore µ m = 2 µ .Our choice of energy convention is to measure all singleparticle states from the lattice potential zero of energy, andfurthermore, we subtract the magnetic field shift E res fromthe molecular states. Thus the distribution functions take theform f a ( E ) ≡ e ( E − µ ) /k B T + 1 , (5a) f m ( E ) ≡ e ( E + E res − µ ) /k B T − . (5b)As we discuss in further detail below E ra and E rm are the avail-able single particle energy states for atoms and molecules, re-spectively, where r = 0 , , , . . . , with r = 0 correspondingto the ground state. Importantly, the molecular chemical po-tential is bounded from above by the lowest possible single-molecule energy µ m ≤ E m + E res , (6)since the molecules are governed by Bose-Einstein statistics.The density of states for atoms and molecules are given by ρ a ( E ) = 2 ∞ X r =0 δ ( E − E ra ) , (7a) ρ m ( E ) = ∞ X r =0 δ ( E − E rm ) , (7b)where the factor of two in (7a) arises because there are twotypes of atoms with identical energy spectra. The total numberof atoms and molecules for given values of E res , µ and T canbe found by integrating the density of states weighted by theoccupation over all energies. Because the energy levels arediscrete, this becomes a sum N a = Z dE ρ a ( E ) f a ( E ) = 2 X r f a ( E ra ) , (8a) N m = Z dE ρ m ( E ) f m ( E ) = X r f m ( E rm ) . (8b)Given the energy levels and the total particle number thechemical potential is determined at any temperature and reso-nance energy by the constraint (3). PSfrag replacements
E/E
R,a
E/E
R,a ρρ V ≈ E R,a (b) V = 2 E R,a (c) V = 4 E R,a (d) V = 8 E R,a
AtomsMolecules × × × × Figure 1: (Color online) The density of states of a 3D optical lattice calculated by binning the energy levels into intervals of length ∆ E =0 . E R,a . The blue (dark gray) and red (light gray) curves show the density of the atomic and molecular states, respectively.
A. Energy levels in the lattice
We consider a simple cubic optical lattice created by theoverlap of three orthogonal pairs of counterpropagating laserswith wavelength λ L = 2 π/k L . This gives rise to the potential V σ ( x ) = V ,σ L ( x ) , (9)for the particle type σ (corresponding to atoms “ a ” ormolecules “ m ”), where L is the dimensionless shape of thepotential L ( x ) = (cid:2) sin ( k L x ) + sin ( k L y ) + sin ( k L z ) (cid:3) . (10)The molecules experience a lattice potential twice as deep asthat in which the atoms move, V ,m = 2 V ,a ≡ V , since theStark shift of a molecule is the sum of those for each atom.Now the problem is to understand how the difference in par-ticle mass and apparent lattice depth for atoms and moleculesaffects the behavior of their respective single particle states,obtained by solving the time-independent Schrödinger equa-tion (cid:26) − ℏ ∇ m σ + V ,σ L ( x ) (cid:27) ψ rσ ( x ) = E rσ ψ rσ ( x ) . (11)There is a simple relationship between the atomic and molec-ular spectra, which is revealed by transforming (11) foreach species into their respective recoil energies, E R,σ ≡ ℏ k L / m σ : (cid:26) − ∇ k L + ¯ V ,σ L ( x ) (cid:27) ψ rσ ( x ) = ¯ E rσ ψ rσ ( x ) , (12)where barred quantities are in recoil units. The advantage ofEq. (12) is that, since the left hand side operator only dependson σ via the quantity ¯ V ,σ , the spectrum is of the form ¯ E rσ = ¯ E r ( ¯ V ,σ ) . (13) The relationship between ¯ V ,a and ¯ V ,m is quite simple: themolecules are twice as heavy as the atoms and see a latticethat is twice as deep as the one experienced by the atoms. Asa result the atomic and molecular potentials are related as ¯ V ,m = 4 ¯ V ,a , (14)i.e. in the respective recoil units the molecules see a latticethat is four times deeper. In general, an optical lattice withdepth exceeding one recoil unit has a considerable effect onthe spectral properties of the confined particles. Therefore,the difference in lattice depth for the atoms and the moleculeshas a rather profound effect on the properties of our system.We note that our choice to define the atomic energy originas the lattice potential zero, rather than the atomic ground-state, is to emphasize the differential lattice confinement ef-fects on the atoms and the molecules. III. NUMERICS
Because we are considering a separable potential (10), the3D spectrum of the time-independent Schrödinger equationis most efficiently calculated via the 1D eigenvalues, whichare easily determined by numerical diagonalization. For thedata presented here we have used a lattice with × × sites; this is sufficiently large that this system can be regardedas being approximately in the thermodynamic limit. We onlyexemplify results for η = since these represent the genericbehaviour of the systems with a filling fraction less than unity.We numerically determine the chemical potential as a func-tion of T and E res under the condition of conservation of thetotal number of particles, (3). For a given point ( µ, E res , T ) in the phase diagram any thermodynamic quantity, such as theenergy or the entropy, may then be calculated as the sum of anatomic and a molecular contribution. A. Density of states
The exact 3D energy levels can be found by making all pos-sible combinations of three 1D energy levels. However, for thepurposes of calculation the number of individual states neededis unwieldy, and it is desirable to construct a binned density ofstates by gathering the energy levels in small energy intervalsand representing them by the centre of the interval. Doingthis we obtain the graphs shown in Figure 1. The reliability ofthe binning procedure is confirmed in Figure 1(a), where werecover the usual √ E dependence of the densities of statesin the limit of vanishing lattice depth. By comparing Figures1(b) and (d) one sees that the molecular density of states for V = 2 E R,a and the atomic density of states for V = 8 E R,a have the exact same shape, though the latter is scaled on bothaxes by a factor of two (see Eqs. (13) and (14)). To simpli-fy the numerical task without losing effects due to details inthe energy spectrum we have used the exact energy levels forthe lowest energy bands only and the binned density of statesfor the higher energy levels. The justification for this approx-imation is that the high energy domain is only relevant if thetemperature is high, in which case the distributions f σ ( E ) aresufficiently slowly varying that we can consider them constantover small energy intervals.As the lattice depth is increased, band gaps emerge in the3D density of states due to the gaps in the 1D energy spec-trum. With reference to results shown in Figure 1 we makethe following observations of the spectral properties:1. The molecular density of states has clear characteris-tics of a being in a much deeper lattice than the atom-ic system, i.e. for a given V the molecular spectrumhas smaller band widths and larger band gaps than theatomic system. Most importantly, even for shallow lat-tices (e.g. V = 2 E R,a ), the molecule ground bandis very narrow compared to the atomic ground band.These observations are consistent with the discussionbelow Eq. (14).2. There is a positive offset in the ground state energy ofthe molecules relative to the atom states. As shown inAppendix A, this shift arises from anharmonic effectsin the lattice, and for the deep lattice limit it is given by E m − E a ≈ p V ,a /E R,a ! E R,a . (15)Such shifts should be measurable in experiments as adisplacement of the magnetic field position where halfof the atoms have been converted to molecules in an adi-abatic sweep across the Feshbach resonance (see sec-tion V). IV. PHASE DIAGRAM
The phase diagram is characterized by the molecule frac-tion, defined as χ ( E res , T, V , η ) ≡ N m N tot (16)ranging from zero (all atoms) to unity (all molecules). In thissection we first examine the molecule fraction as a functionof the position of the resonance and the temperature of thesystem. We then consider how the low temperature chemicalequilibrium is affected by changing the lattice depth.A convenient energy scale is the Fermi energy E F ( η, V ) ,which is taken to be the highest occupied energy level, whenall the atoms are unbound. We note that for our choice of en-ergy origin the relevant Fermi temperature for characterizingdegeneracy is given by T F = ( E F − E a ) /k B , with E a theatomic ground state energy. A. Properties at fixed lattice depth
The molecular fraction is shown in Figure 2 as a functionof E res and T for a fixed lattice depth. Those results showthat tuning the Feshbach resonance, i.e. E res , provides a di-rect way of varying the composition of the gas. For large andpositive values of E res , the molecular states are at much high-er energy than the atomic states and are not thermally accessi-ble, thus realizing a pure atomic gas. Conversely, for large andnegative values of E res , the system exists as a pure moleculargas. At low temperature the transition between these two lim-iting regimes occurs when E res is close to E F (see Figure 2).We remark that for a free system or a gas trapped in a har-monic potential the transition from atoms to molecules startsat E res ≈ E F [8, 11]. PSfrag replacements k B T / E F E res /E F
00 0 0.10.1 0.20.2 0.30.3 0.40.4 0.50.5 0.60.6 0.70.7 0.80.8 0.90.91.0 11 1 2 3 4-1-2-3-4105-5-10-15-20-25-30 χ Molecules Atoms
Figure 2: (Color online) Molecule fraction in a ( E res , T ) phase dia-gram for V = 5 E R,a and η = . PSfrag replacements
Energy N u m b e r o f s t a t e s N u m b e r o f s t a t e s N u m b e r o f s t a t e s E a µ ≈ E a µ . E a E m E m E m µ µ m µ m E m + E res E m + E res µ m ≈ E m + E res (a) E res ≫ E F (b) E res ≈ E F (c) E res ≪ E F N a N m f a f m ρ a ρ m Figure 3: (Color online) Schematic illustration of how µ changes when the resonance energy is varied across E F . For clarity we considera deep lattice where the density of states of the lowest atomic- and molecular band is a narrow peak as illustrated by the solid lines. Thedistributions f a ( E ) and f m ( E ) are indicated by the dashed lines, and the shading under the peaks indicates the population of the atomicand molecular levels. Note that the E res -dependence is included in the position of the peak in the molecular density of states and not in theBose-Einstein distribution. The characteristic asymmetric fan-shape of the χ -contoursoccurs because when the temperature increases, moleculeswill dissociate as two seperate atoms are entroptically favor-able over a single molecule. Thus if we follow vertical linesin the phase diagram in Figure 2 towards higher temperatures,the fraction of molecules must decrease eventually.The chemical potentials for degenerate Bose and Fermi gas-es exhibit markedly different behavior, the former being con-strained by the lowest available single particle level, while thelatter depends explicitly on the number of particles in the sys-tem. Furthermore, in our system chemical equilibrium con-strains the molecular chemical potential to be twice that of theatoms. Thus it is clear that the change in atomic and molecu-lar populations induced by varying the applied magnetic fieldmust be accompanied by a change in the behavior of the chem-ical potential. This can be described qualitatively by consid-ering how the atoms and molecules are distributed over theirrespective energy levels when E res is swept across the Fes-hbach resonance as sketched in Figure 3. We identify threeseparate regimes: (a) Pure Fermi gas limit, E res → ∞ : If E res is sufficientlyhigh, the lowest molecular energy level E m + E res isnearly unoccupied and all atoms are unbound. We thenhave µ ≈ E F for T ≈ . (17)This corresponds to Figure 3(a) where the narrow ener-gy bands are sketched as peaks. For filling fraction η ≤ the Fermi energy lies in theground band. In the limit of a deep lattice the lowestband becomes sufficiently narrow that we can make theapproximation E F ≈ E a (see Appendix A). (b) Intermediate region, E res ≈ E F : When E m + E res ap-proaches µ m from above, molecules will start to formwhile the number of atoms decreases as shown in Fig-ure 3(b). When E res decreases further, µ m also has todecrease due to the condition (6). Since µ m = 2 µ ,the atomic chemical potential therefore also starts to de-crease. (c) Pure Bose gas limit, E res → −∞ : Finally, when theresonance energy becomes low enough, µ has decreasedso much that almost no atomic states are occupied, andthe result is a nearly pure molecular gas correspondingto Figure 3(c). In this limit we have µ m ≈ E m + E res at low temperatures, i.e., µ = E m + E res for T ≈ , (18)From this description it is clear that the transition from atomsto molecules is more abrupt the narrower the lowest energybands are.The low temperature behaviour of µ at half filling is shownin Figure 4 for a range of different potential depths and theeffects discussed above are evident. Linear fits to µ in themolecular regime turns out to be in excellent agreement withthe pure Bose gas limit estimate (18) and the µ -plateaus in PSfrag replacements E res /E F µ / E R -15 -10-10-5 -4-4 -3 -2-2 -1 0 00 0 1 22 3 4 45 10101520 V = 20 E R,a V = 15 E R,a V = 10 E R,a V = 5 E R,a V ≈ E R,a
Figure 4: Chemical potential as a function of E res for η = anddifferent potential depths in the limit T ≈ . This corresponds tofollowing a horizontal line in the bottom of phase diagrams like inFigure 2. The linear behaviour in the left side is characteristic for thepure Bose gas regime, and the difference in the slopes arises becausethe Fermi energy varies with the potential depth. Inset: chemicalpotential curves for V = 5 , , , and E R,a (from bottom totop) for a finite temperature of . E R,a /k B . the pure Fermi gas limit reveal the dependence of E F on thelattice depth. Note that the arguments above do not tell us any-thing about the behaviour in the transition zone between theconstant and the linear regime, but merely that at sufficientlyhigh E res the chemical potential must be constant, while for E res tuned sufficiently below the resonance the chemical po-tential is linear. The fact that the transitions between these tworegimes is so sharp as in Figure 4 is a consequence of the verylow temperature. However, the general behaviour of plateauson the atomic side of the resonance and a linear variation of µ on the molecular side remain also at higher temperatures inthe limits E res → ±∞ as illustrated in the inset of Figure 4. V. DISSOCIATION ENERGY
A noticeable feature in Figure 4 is that the transition pointwhere the atoms start to form molecules varies with the poten-tial depth. In the absence of the optical lattice potential it liesat E res ≈ E F , as is well-known from BCS-BEC cross-overtheories for a narrow Feshbach resonance [11]. As the latticedepth is increased the resonance energy corresponding to theonset of the transition first decreases and then increases slight-ly towards a limiting value of E res ≈ E F in the deep latticelimit. We will elaborate on these effects in the following andexplain the behavior for deep lattice potentials.To quantify the location of the chemical transition we de-fine the dissociation energy E dis as the value of the reso-nance energy where half of the atoms have been convertedto molecules, χ ( E dis ) = [8]. In addition, we introduce thequantities E ± dis , delineating the zero temperature conversion zone, such that no unbound atoms are found for E res < E − dis and no molecules exist for E res > E +dis . The lower limit isdefined by the condition E m + E − dis = 2 E a , (19)which specifies where the first molecules start to break up asthe resonance energy is increased from the molecular side ofthe resonance. The other end of the transition zone, wherethe first molecules are formed as the resonance energy is de-creased starting with a pure atomic gas, is defined by the con-dition that the lowest molecular level passes twice the energyof the highest occupied atomic state, i.e. E m + E +dis = 2 E F . (20) A. Free space limit
These estimates simplify in the limit V = 0 where thelowest atomic and molecular energy levels are only separatedby E res , i.e. E a = E m = 0 . We then have E − dis = 0 E +dis = 2 E F for V = 0 . (21)To find E dis for a vanishing potential we note that in this casethe density of states has a squareroot dependence on the ener-gy ρ a ( E ) = C √ E (22)with C = πM / E / R,a .The following argument applies atzero temperature: the number of unbound fermionic atoms,found by integrating the atomic density of states from zero tothe chemical potential, is N a = Cµ / . If µ = E F , thisequals the total number of atoms in the system, and in generalwe obtain the expression χ = 1 − N a N tot = 1 − (cid:18) µE F (cid:19) / , (23)for the molecule fraction (at T = 0 ). In the transition zone E res = 2 µ (since E m = 0 ), such that χ = 1 − (cid:18) E res E F (cid:19) / , (24)and the dissociation energy in this limit then follows from thecondition χ = : E dis = 2 / E F ≈ . E F for V = 0 . (25)This is confirmed by the numerical calculations, see Figure 5.Similar arguments can be used to show that E dis = 2 / E F for a gas trapped in a harmonic potential [8]. For a harmonictrap one also finds E − dis = 0 and E +dis = 2 E F since E m = E a ≪ E F . B. Deep lattice limit
In the opposite limit of a deep optical lattice the 3D ener-gy bands become increasingly narrow and E − dis and E +dis ap-proach each other. It is therefore reasonable to estimate E dis to lie halfway between them, and inserting (19) and (20) yields E dis → E F − ( E m − E a ) for V → ∞ . (26)In the deep lattice limit the narrow ground band is well-described by a tight-binding approach (see Appendix A),which gives an analytic expression for E m − E a (A13). Wethen have that E dis ≈ E F − p V ,a /E R,a ! E R,a . (27)This estimate, valid for a deep lattice, is plotted in Figure 5,where we also show the numerical halfway mark for the atom-molecule conversion, E dis , and there is good agreement be-tween the numerical results and (27) in the deep lattice limit.Note that since E F ≈ E a for a deep lattice, (A12) indicatesthat in this limit E F /E R,a ≈ p V /E R,a and hence that E dis eventually approaches E F as the lattice depth increases E dis → E F for V → ∞ , (28)since the first term in (27) is then the dominant one. The de-viation from this limit in a deep lattice is given by the anhar-monic corrections in (27). C. Lattice induced resonance shift
These results demonstrate that the lattice induced energyshift between atomic and molecular degrees of freedom altersthe dissociation condition for the Feshbach resonance. In anadiabatic sweep of the magnetic field across the resonance theeffect of the lattice is to modify the molecule formation curve( χ vs. B ). In particular, the magnetic field corresponding toconversion of half of the atoms to molecules, B V dis , is shiftedwith respect to its value in the absence of the optical lattice, B , by an amount δB V dis = B V dis − B , which in the deeplattice limit may be estimated from δB V dis ≈ E V dis − E ∆ µ ≈ E V F − . E ∆ µ . (29)Here we have used that E res varies linearly with B with aslope ∆ µ given by the magnetic moment difference betweenthe open and the closed channels. The Fermi energy with nolattice is E = (cid:0) ηπ (cid:1) / E R,a and for deep lattices we canapproximate E V F ≈ p V E R,a . Inserting this we get δB V dis ≈ q V E R,a − . (cid:0) ηπ (cid:1) / (cid:16) ∆ µµ B (cid:17)(cid:0) m a amu (cid:1)(cid:0) λ L nm (cid:1) · . × − G . (30) PSfrag replacements V /E R,a E r e s / E F χ E dis E dis Approximation
AtomsMolecules
Figure 5: (Color online) Dissociation energy vs. optical lattice depth.For deep lattice potentials E dis can be approximated by Eq. (27)(white line), while the limiting behavior for a vanishing lattice depthis given by Eq. (25) (dashed white line). Note that since the Fer-mi energy increases with increasing lattice depth, the scaling on thevertical axis depends on V . For Li in a lattice with wavelength λ L = 1032 nm and tak-ing ∆ µ = 2 µ B we find that B V dis − B ≈ .
11 G for alattice depth of V = 15 E R,a assuming a filling fraction of η = . This is comparable to the magnetic field width of theresonance ∆ B = 0 .
23 G at resonance at .
26 G [37]. Asimilar expression has been found experimentally for the lat-tice induced shift of the magnetic position where the Feshbachmolecule enters the continuum [15].
VI. CONCLUSION
We have presented results for the thermodynamics of anatom-molecule mixture in an optical lattice potential under thecondition of chemical equilibrium adjusted by a Feshbach res-onance which controls the energy offset between the atomicand molecular levels.The phase diagram has been determined and we have ana-lyzed the behavior of the chemical potential and the fraction ofmolecules with emphasis on the low-temperature regime. Inparticular we have identified the dissociation energy definedas the energy offset where of the atoms have been dimer-ized into molecules. The chemical conversion takes place in atransistion zone of the magnetic field controlled resonance en-ergy, which narrows and shifts as the lattice depth is increasedand the Bloch bands approach discrete energy levels. Further-more, the dissociation energy shifts with the lattice depth, asthe molecules and the free atoms experience different latticepotentials, and hence their energy levels are displaced withrespect to one another as the lattice depth is changed. In thedeep lattice limit an analytic, precise expression for the centerof the transistion zone was obtained by treating anharmoniccorrections to the energy levels in the lattice wells perturba-tively.Our results thus show that in an optical lattice potential theposition and width of the Feshbach resonance, as indicated bythe conversion of atoms into dimers and vice versa, dependson the lattice depth. We have shown that the lattice inducedshift should be measurable for an atom with a narrow reso-nance such as Li.By focussing on the ideal gas case we have been able toclarify the effect of the uniform optical lattice potential on theatom-molecule equilibrium. The inclusion of interactions andan inhomogenous trap potential will be areas of future devel-opment to completely describe this system and make quan-tatative comparison with experiments possible.
Acknowledgments
N. N. acknowledges financial support by the Danish Natu-ral Science Research Council.
Appendix A: THE DEEP LATTICE LIMIT
When the optical lattice is very deep, we can make a tight-binding approximation and regard the potential as a collectionof M independent wells. The lowest energy levels are thenthose for one well with a degeneracy equal to the number ofsites. A Taylor expansion of the 1D potential ( V ,σ sin ( k L x ) )at x = 0 to 6th order gives V σ ( x ) ≈ V ,σ (cid:18) ( k L x ) −
13 ( k L x ) + 245 ( k L x ) (cid:19) , (A1)so each lattice well can be described as a harmonic oscillatorwith the frequency ω σ = s k L V ,σ m σ , (A2)with two anharmonic terms ˆ H = − V ,σ k L ˆ x ) , (A3) ˆ H = 2 V ,σ
45 ( k L ˆ x ) , (A4)that we will treat with perturbation theory. The unpertubedenergy levels for the harmonic oscillator are E n (0) σ = (cid:18) n + 12 (cid:19) ℏ ω σ = (2 n + 1) p V ,σ E R,σ , (A5)and the first order corrections ∆ E n (1) σ,i = h n | ˆ H i | n i , i = 4 , (A6) can be calculated by expressing the factor ( k L ˆ x ) i in terms ofthe ladder operators for the harmonic oscillator ( k L ˆ x ) i = ((cid:18) E R,σ V ,σ (cid:19) / (cid:0) ˆ a † + ˆ a (cid:1)) i . (A7)We thus obtain the anharmonic first order energy shifts ∆ E n (1) σ, = − E R,σ (cid:0) n + 2 n + 1 (cid:1) , (A8) ∆ E n (1) σ, = 136 E R,σ p V ,σ /E R,σ (cid:0) n + 6 n + 8 n + 3 (cid:1) , (A9)but we also need to consider the second order pertubation con-tributions from ˆ H to the n ’th energy level ∆ E n (2) σ, = X m = n | h m | ˆ H | n i | E nσ, − E mσ, = − E R,σ p V ,σ /E R,σ X m = n | h m | (cid:0) ˆ a † + ˆ a (cid:1) | n i | m − n , (A10)since this contribution is of the same order as the first orderpertubation contribution from ˆ H . In the general case the sumonly gets contributions from m = n ± and m = n ± ,and in the simplest case n = 0 , where the contributions from m = n − and m = n − vanish, we get ∆ E σ, = − E R,σ p V ,σ /E R,σ , (A11)resulting in the approximation E σ ≈ s V ,σ E R,σ − −
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