Isothermal sweep theorems for ultra-cold quantum gases in a canonical ensemble
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Isothermal sweep theorems for ultra-cold quantum gases in a canonical ensemble
M. Iskin
Department of Physics, Koc¸ University, Rumelifeneri Yolu, 34450 Sariyer, Istanbul, Turkey. (Dated: October 15, 2018)After deriving the isothermal Hellmann-Feynman theorem (IHFT) that is suitable for mixed states in a canon-ical ensemble, we use this theorem to obtain the isothermal magnetic-field sweep theorems for the free, aver-age and trapping energies, and for the entropy, specific heat, pressure and atomic compressibility of strongly-correlated ultra-cold quantum gases. In particular, we apply the sweep theorems to two-component Fermi gasesin the weakly-interacting BCS and BEC limits, showing that the temperature dependence of the contact parame-ter can be determined by the variation of either the entropy or specific heat with respect to the scattering length.We also use the IHFT to obtain the Virial theorem in a canonical ensemble, and discuss its implications forquantum gases.
PACS numbers: 03.75.Ss, 03.75.Hh
I. INTRODUCTION
Dilute samples of ultra-cold quantum gases are well-described by zero range interactions, a consequence of whichis the existence of some universal relations that govern thebehavior of these systems [1]. The first examples of suchrelations were originally derived by solving the many-bodySchr¨odinger equation [2–4], and they relate energy and pres-sure of the system to a single parameter that comes from theshort-range physics. What makes these relations remarkableis that they hold for any finite-energy state of the system, itdoes not matter whether the system is few- or many-body, su-perfluid or normal, weakly or strongly interacting, in equilib-rium or nonequilibrium, at zero or finite temperature. Theseuniversal relations have more recently been rederived usingmany other approaches including the quantum field theoreticaltechniques [1, 5–9], and have also been verified via numericalfew-body calculations [10].There is one common element in these universal relations:they all involve a single parameter called the contact. In thecase of two-component Fermi gases, the contact parameter isgiven by the coefficient of the /k tail of the momentum dis-tribution, and therefore, it measures the number of fermionpairs with small separations. It has been found that the con-tact parameter also appears in many other short-range (high-momentum) or short-time (high-frequency) properties of thesystem [11–16]. This parameter has recently been measuredin an ultracold K gas via the measurements of the high-momentum tail of the momentum distribution and of the high-frequency tail of the radio-frequency signal [17], and alsomeasured in an ultracold Li gas via the measurements of thestatic structure factor [18]. The measured temperature andscattering length dependence of the contact parameter com-pare well with the theoretical predictions [19, 20].In particular, the so-called adibatic sweep theorem for theenergy [3], which has recently been experimentally veri-fied [17], relates the contact to the change in total energy of thesystem when the atom-atom scattering length ( a ) is changedadiabatically, i.e. with zero heat transfer. In atomic systems,the value of a can be tuned at will from very large and nega-tive to very large and positive values, thanks to the presenceof magnetically induced Feshbach resonances, and an adia- batic (constant entropy) sweep is accomplished by changingthe bias magnetic field over a time scale longer than the relax-ation time [17].Motivated by these earlier works, here we generalize anumber of these universal relations to finite temperatures, andthe rest of the manuscript is organized as follows. In Sec. II,we derive the isothermal Hellmann-Feynman theorem (IHFT)that is suitable for mixed states in a canonical ensemble. Then,we use this theorem in Sec. III to obtain the isothermal (con-stant temperature) sweep theorems (IST) for the free, averageand trapping energies, and for the entropy, specific heat, pres-sure and atomic compressibility of strongly-correlated ultra-cold Fermi gases. We also use the IHFT to derive the Virialtheorem in a canonical ensemble in Sec. IV. A brief summaryof our conclusions is given in Sec. V. II. HELLMANN-FEYNMAN THEOREMS
For our purpose, we need to derive the Hellmann-Feynmantheorem (HFT) suitable for a canonical ensemble of a fixednumber N of identical particles that are in thermal equilibriumwith a heat reservoir, which is discussed next. A. HFT for pure states
For any stationary normalized eigenvector | ψ n ( λ ) i with thecorresponding eigenvalue E n ( λ ) of a Hamiltonian H ( λ ) , i.e. h ψ n | ψ n i = 1 and E n = h ψ n | H | ψ n i , the usual HFT statesthat ∂E n ∂λ = (cid:28) ψ n (cid:12)(cid:12)(cid:12)(cid:12) ∂H∂λ (cid:12)(cid:12)(cid:12)(cid:12) ψ n (cid:29) , (1)where λ is an arbitrary real parameter, explicitly appearing in H . This well-known theorem has proved to be very useful inmany fields ranging from quantum chemistry, quantum statis-tics and many-body physics to molecular physics, with a lotof applications.In particular, the HFT has recently been used to derive anumber of exact relations for strongly-correlated systems withshort-range interations, in the context of ultra-cold quantumgases [5, 6, 26]. Since this theorem applies only for purestates, these exact relations strictly hold at zero temperaturefor any change or equivalently at finite temperatures for anadiabatic change. In order to generalize these relations to fi-nite temperatures, here we derive the IHFT for a canonicalensemble of a fixed number of identical particles that are inthermal equilibrium with a heat reservoir. B. IHFT for mixed states in a canonical ensemble
For this purpose, we assume the temperature T is fixed,and the Hamiltonian H is a function of some arbitrary realparameter λ . There are two complimentary ways to obtain theisothermal HFT in a canonical ensemble.The simplest way follows from the definition of the statis-tical Helmholtz Free energy F , i.e. F = − T ln Z in a unit of k B = 1 here and throughout, where Z = Tr( e − H/T ) is thecanonical partition function of the system. Then, it is easy toshow that (see also [21, 22]) ∂F∂λ = (cid:28) ∂H∂λ (cid:29) , (2)where the ensemble average of any operator is defined by h A i = Tr( ρA ) and ρ = e − H/T /Z is the density operatorin the canonical ensemble. We recall that the derivatives withrespect to λ are evaluated at fixed T here and throughout. Inaddition, using the definitions of the thermodynamic Free en-ergy, average energy and entropy, i.e. F = E − T S , E = h H i and S = − Tr( ρ ln ρ ) = − ∂F/∂T , respectively, Eq. (2) canbe written as ∂E∂λ = (cid:18) − T ∂∂T (cid:19) (cid:28) ∂H∂λ (cid:29) . (3)This is the isothermal HFT for mixed states in a canonicalensemble, and it will play a central role in the remaining partsof this manuscript.An alternative way of deriving Eq. (3) is as follows. Westart directly from the definition of E , i.e. E = Tr( ρH ) , andevaluate the derivative to obtain ∂E∂λ = (cid:28)(cid:18) E − HT (cid:19) ∂H∂λ (cid:29) . (4)In the intermediate steps, we used Tr( ∂e − H/T /∂λ ) = − (1 /T )Tr( e − H/T ∂H/∂λ ) and Tr(
H∂e − H/T /∂λ ) = − (1 /T )Tr( He − H/T ∂H/∂λ ) , both of which can be derivedby writing Tr( A ) = P n h ψ n | A | ψ n i where | ψ n i is the normal-ized eigenvector and E n is the corresponding eigenvalue of H .In addition, we used ∂e − E n /T /∂λ = h ψ n | ∂e − H/T /∂λ | ψ n i , which can be obtained from the usual HFT given in Eq. (1).For fixed T , starting from the definition h A i = Tr( ρA ) , it iseasy to show for the observables that commute with H , i.e. [ H, A ] = 0 , that ∂ h A i /∂T = h ( H − E ) A i /T . Therefore,Eqs. (3) and (4) are equivalent.Having derived the IHFT for a canonical ensemble, nextwe use this theorem to obtain a number of exact relations for strongly-correlated ultra-cold quantum gases, which is themain purpose of this manuscript.
III. ISOTHERMAL SWEEP THEOREMS
In this section, we use the IHFT given in Eq. (3) to derivethe isothermal sweep theorems (IST) for the free, average andtrapping energies, and for the entropy, specific heat, pressureand atomic compressibility in a canonical ensemble.
A. IST for the free, average and trapping energies
As a first application of the IHFT, we consider the Hamilto-nian that describes two-component Fermi gases in an externalpotential U σ ( r ) , i.e. H = X σ Z d r ψ † σ ( r ) (cid:20) − ~ ∇ m σ + U σ ( r ) (cid:21) ψ σ ( r ) − g Z d r ψ †↑ ( r ) ψ †↓ ( r ) ψ ↓ ( r ) ψ ↑ ( r ) , (5)where ψ † σ ( r ) creates a pseudospin- σ fermion with mass m σ ,and g ≥ is the strength of the short-range interaction. Inthe following discussions, the numbers of ↑ and ↓ fermionsneed not be equal. As usual the theoretical parameter g can bewritten in terms of the experimentally more relevant scatter-ing length a via /g = − M V / (4 π ~ a )+( M/ ~ ) P k (1 /k ) , where M = 2 m ↑ m ↓ / ( m ↑ + m ↓ ) is twice the reduced massof ↑ and ↓ fermions, and V is the volume. This equationgives g = − π ~ a/ ( M V π − M V ak c ) , where k c is themomentum-space cut-off used in order to evaluate the ultravi-olet divergent sum over k .Following the recent work [5], the contact parameter C canbe defined as (cid:28) ∂H∂a (cid:29) = M V g π ~ a Z d r h ψ †↑ ( r ) ψ †↓ ( r ) ψ ↓ ( r ) ψ ↑ ( r ) i , (6) = ~ C πM a , (7)where we used dg/da = − M V g / (4 π ~ a ) . Note that C isan extensive quantity that depends on both a and T . This def-inition guarantees [5] that the average energy E of the systemis of the desired form [2, 8], E − X σ h U σ i = X σ, k ~ k m σ (cid:20) n σ ( k ) − Ck (cid:21) + ~ C πM a , (8)where n σ ( k ) is the momentum distribution of σ fermions. Asemphasized in [2], this relation holds for any finite-energystate of the system, it does not matter whether the system isfew- or many-body, superfluid or normal, weakly or stronglyinteracting, in equilibrium or nonequilibrium, at zero or fi-nite temperature. Combining Eq. (7) with the IHFT given inEqs. (2) and (3), we obtain ∂F∂a = ~ C πM a , (9) ∂E∂a = ~ πM a (cid:18) C − T ∂C∂T (cid:19) . (10)Here, the derivatives with respect to a are evaluated at fixed T , and therefore, Eqs. (9) and (10) correspond to the IST forthe free and average energies, respectively. Compared to thezero temperature expression for any change or equivalentlythe finite temperature expression for an adiabatic change,i.e. the adiabatic sweep theorem for the energy [1, 3], themain difference in Eq. (10) is an extra T ∂C/∂T term. Sim-ilarly, we can also calculate ∂F/∂k c = ~ C/ (2 πM ) and ∂E/∂k c = [ ~ / (2 πM )]( C − T ∂C/∂T ) , where we used dg/dk c = − M V g / (2 π ~ ) .For homogenous systems (no external potential), the T de-pendence of C has been recently calculated in the low, inter-mediate and high T regimes as [19] C = C + α T , ( T ≪ T c < T F ) (11) C = C + α T , ( T c < T ≪ T F ) (12) C = α T , ( T ≫ { T F , T a } ) (13)where, up to the leading orders in a , C = 4 π n a , α = 9 √ π n a / (40 T F ) and α = − − π k F n a / (5 T F ) in the BCS limit; C = 2 πn/a and α = 2 π p πk F /a ( ∂a m /∂a ) n/T F in the BEC limit; and α = 8 π n /M for all couplings. Here, k F is the Fermi mo-mentum, T c is the critical temperature for superfluidity, T F = ~ k F / (2 M ) is the Fermi temperature, T a = ~ / ( M a ) isthe binding energy of two fermions, n = N/V = k F / (3 π ) is the density of fermions, a m is the dimer-dimer scatteringlength between Cooper molecules (e.g. a m = 0 . a when m ↑ = m ↓ ), and the weakly-interacting BCS and BEC lim-its are characterized by k F a → − and k F a → + , respec-tively. Note that our definition of C is larger by a factor of π compared to the definition of Ref. [19]. Note also thatthe expression for intermediate T regime, i.e. Eq (12), isvalid only in the BCS limit, and that C has a maximum ata particular T , since C , α , α and α are all positive con-stants with respect to T . Using Eqs. (11)-(13) in Eq. (10),we find ∂E/∂a = ~ ( C − α T ) / (4 πM a ) , ∂E/∂a = ~ ( C − α T ) / (4 πM a ) and ∂E/∂a = ~ C/ (2 πM a ) forthe low, intermediate and high T regimes, respectively.Note also that taking the derivative of the Virial theoremgiven in Eq. (29) with respect to a , and using the IST for theenergy given in Eq. (10), we obtain ∂E tr ∂a = ~ πM a (cid:18) C + a ∂C∂a − T ∂C∂T (cid:19) , (14)which corresponds to the IST for the trapping energy. Here, E tr = 12 * U + 12 N X i =1 r i · ∇ r i U + (15)is the effective trapping energy, which reduces to the trappingenergy h U i in the case of harmonic trapping potentials. B. IST for the entropy and specific heat
The IHFT can also be used to find other thermodynamic re-lations. For instance, we obtain an entropy relation by takingthe derivative of S = ( E + F ) /T with respect to λ , and usingEqs. (2) and (3), leading to ∂S∂λ = − ∂∂T (cid:28) ∂H∂λ (cid:29) . (16)In addition, we obtain a specific heat relation (at constant vol-ume) from C V = ∂E/∂T = T ∂S/∂T, leading to ∂C V ∂λ = − T ∂ ∂T (cid:28) ∂H∂λ (cid:29) . (17)In Eqs. (16) and (17) the derivatives with respect to λ areevaluated at fixed T . We hope that these relations couldbe tested with thermodynamic measurements in strongly-interacting Fermi gases [27].For ultra-cold quantum gases described by the Hamiltoniangiven in Eq. (5), we can use Eq. (7) in Eqs. (16) and (17),leading to ∂S∂a = − ~ πM a ∂C∂T , (18) ∂C V ∂a = − ~ T πM a ∂ C∂T . (19)Here, the derivatives with respect to a are evaluated at fixed T , and therefore, Eqs. (18) and (19) correspond to the IST forthe entropy and specific heat, respectively. Equation (18) wasfirst derived in Ref. [19]. Using Eqs. (11)-(13) in Eqs. (18)and (19), we find ∂S/∂a = − α T / ( πM a ) and ∂C V /∂a = − α T / ( πM a ) for the low T , ∂S/∂a = − α T / (2 πM a ) and ∂C V /∂a = − α T / (2 πM a ) for the intermediate T , and ∂S/∂a = C/ (4 πM a T ) and ∂C V /∂a = − C/ (2 πM a T ) for the high T regimes. Therefore, since α and α are posi-tive constants, while ∂S/∂a is negative for low T , it becomespositive at high T , indicating that ∂S/∂a vanishes at a partic-ular temperature above T F . Note also that ∂S/∂a ∝ ∂C V /∂a in these regimes, and that the T dependence of C could be de-termined by the variation of either S or C V with respect to a . C. IST for the pressure
For the Hamiltonian given in Eq. (5), but in the absence ofthe potential term, i.e. a homogenous system where U ( r ) = 0 ,the IHFT can be used to derive the IST for the pressure.In general, via a dimensional analysis, the Freeenergy F can be written as F ( η , . . . , η r ) =( ~ λ /M ) f ( λη , . . . , λη r ) where λ has the dimensionof the inverse of a length, f is a dimensionless functionof its parameters, and η j labels r parameters (all with thedimension of a length) that F may depend on for a given H . For homogenous ultra-cold quantum gases, since F is afunction of T , V , N σ and a , dimensional analysis [6] requiresthat F must satisfy F = − r X q =1 η q ∂F∂η q = − a ∂F∂a + 2 T ∂F∂T − V ∂F∂V . (20)Using F = E − T S , S = − ∂F/∂T , P = − ∂F/∂V , and theIST for the free energy given in Eq. (9), we obtain P = 2 E V + ~ C πM aV . (21)Therefore, the universal pressure relation in a canonical en-semble is of the same form as the zero temperature one [1, 3].Taking the derivative of Eq. (21) with respect to a , and us-ing the IST for the energy given in Eq. (10), we obtain ∂P∂a = ~ πM a V (cid:18) C + a ∂C∂a − T ∂C∂T (cid:19) , (22)which corresponds to the IST for the pressure. We re-call that Eq. (22) is derived for a homogenous system, andtherefore, it is incorrect to compare it with Eq. (14), andconclude that V ∂P/∂a = 4 ∂E tr /∂a for a trapped sys-tem. Using Eqs. (11)-(13) in Eq. (22), we find ∂P/∂a = ~ (3 C − α T ) / (12 πM a V ) in the BCS and ∂P/∂a = − ~ α T / (8 πM a V ) in the BEC limit for the low T regime, ∂P/∂a = ~ C / (4 πM a V ) in the BCS limit for theintermediate T regime, and ∂P/∂a = ~ C/ (4 πM a V ) forthe high T regime. Therefore, since α is a positive constant,while ∂P/∂a is positive in the BCS limit, it becomes negativein the BEC limit in the low T regime, indicating that ∂P/∂a vanishes at a particular scattering length around unitarity. D. IST for the atomic compressibility
The isothermal atomic compressibility is defined as κ T = − V − ∂V /∂P, and it can be obtained by taking the derivativeof Eq. (21) with respect to V at constant T . Using Eq. (9)to relate ∂C/∂V to P , P = − ∂F/∂V , and E = F + T S together with the Maxwell relation ∂S/∂V | T = ∂P/∂T | V , we obtain κ T = 5 P + a ∂P∂a − T ∂P∂T . (23)This equation also follows from a dimensional analysis of thepressure P which is a function of T , V , N σ and a , similar tothe analysis that lead to Eq. (20). The ∂P/∂a term is given byEq. (22), and the ∂P/∂T term can be obtained by taking thederivative of Eq. (21) with respect to T , leading to C V = 3 V ∂P∂T − ~ πM a ∂C∂T , (24)where C V = ∂E/∂T is the specific heat at constant V . Notethat this equation relates the specific heat to the pressure and contact parameter. Using Eqs. (22) and (19) in Eq. (23), weobtain κ T = 5 P − T V C V + ~ πM aV (cid:18) C + a ∂C∂a − T ∂C∂T (cid:19) , (25)which relates the isothermal atomic compressibility to thepressure, specific heat and contact parameter. When C = 0 ,Eq. (25) is satisfied for the ideal Fermi gases. The IST for theatomic compressibility can be easily obtained by taking thederivative of Eq. (25) with respect to a , and using Eqs. (22)and (19).Note that since the compressibility is related to the den-sity fluctuations via the fluctuation-dissipation theorem, κ T = V T − ( h ˆ N i − h ˆ N i ) / h ˆ N i , where ˆ N is the density operator,it can be used to extract some thermodynamic information inatomic systems. Although this was proposed as early as in2005 [23], it has recently been possible to extract this informa-tion for two-component Fermi gases, by measuring the den-sity fluctuations and atomic compressibility [24, 25]. Com-bining Eq. (25) with the fluctuation-dissipation theorem, pro-vides yet another universal relation that can be verified withatomic systems. Note also that the isoentropic (or adiabatic)compressibility κ S is related to the κ T via κ S /κ T = C V /C P , where C P is the specific heat at constant P . Since the spe-cific heats are also related to each other via C P = C V + T V κ T ( ∂P/∂T ) , using Eq. (24), we obtain κ S = 1 κ T + T VC V (cid:18) C V V + ~ πM aV ∂C∂T (cid:19) , (26)which relates κ S to κ T , C V and C . In atomic systems, itis easier to measure κ S than κ T , and Eq. (26) can be used toextract the temperarure dependence of C , given that κ T can beextracted from the fluctuation-dissipation theorem [24, 25].Having derived the IST for the free, average and trap-ping energies, and for the entropy, specific heat, pressure andatomic compressibility in a canonical ensemble, next we de-rive the Virial theorem. IV. VIRIAL THEOREM FOR TRAPPED SYSTEMS
In this section, we use the IHFT given in Eq. (3) to derivethe Virial theorem [4, 21, 26, 27] in a canonical ensemble.This can be most easily achieved following the recent workon the zero-temperature case [26].
A. Virial theorem in a canonical ensemble
For this purpose, consider a general Hamiltonian H = K + I + U, that describes N particles with arbitrary statis-tics in arbitrary dimensions, where K is the kinetic energy, I is the interaction, and U is an arbitrary external potential.For ultra-cold quantum gases, the external potential is simply U = P Ni =1 U i ( r i ) , where U i ( r i ) has approximately harmonicdependence on the position r i of the particles.In general, via a dimensional analysis, U can be writ-ten as U ( r , . . . , r N ) = ( ~ λ /M ) u ( λ r , . . . , λ r N ) , where λ has the dimension of the inverse of a length, and u isa dimensionless function of its parameters. Therefore, wecan use the IHFT given in Eq. (3) to obtain λ∂E/∂λ =4(1 − T ∂/∂T ) E tr , where E tr is the effective trapping en-ergy defined in Eq. (15). In addition, we can write the en-ergy E , via again a dimensional analysis, as E ( ℓ , . . . , ℓ p ) =( ~ λ /M ) e ( λℓ , . . . , λℓ p ) , where ℓ q labels p parameters (allwith the dimension of a length) that E may depend on for agiven H , and e is a dimensionless function of its parameters.Evaluating the derivative with respect to λ for fixed values of ℓ q , we obtain λ∂E/∂λ = 2 E + P pq =1 ℓ q ∂E/∂ℓ q . Then, theVirial theorem is obtained by combining these two analysis,leading to E = 2 (cid:18) − T ∂∂T (cid:19) E tr − p X q =1 ℓ q ∂E∂ℓ q . (27)Compared to the zero temperature expression [26], the maindifferences here are an extra T ∂/∂T term in front of the po-tential terms, and an extra ℓ q term associated with the temper-ature. B. Trapped quantum gases
In particular, for the ultra-cold quantum gases, which arewell described by the s-wave scattering length a , Eq. (27) re-duces to (1 − T ∂/∂T )( E − E tr ) = − ( a/ ∂E/∂a. Notethat for finite k c , i.e. nonzero interaction range, there wouldbe an additional − ( k c / ∂E/∂k c = − ~ Ck c / (4 π M ) termon the right hand side of this equation. Furthermore, usingthe IST for the average energy given in Eq. (10) for the lastterm, solution of the resultant differential equation [28] canbe written as E = 2 E tr − ~ C πM a + κT, (28)where κ is a real constant independent of T . This is the mostgeneral form of the Virial theorem in a canonical ensemble.Compared to the zero temperature expression [4, 5, 26], themain difference here is an extra κT term. In the unitarity a →±∞ limit, it was shown via a dimensional analysis that theVirial theorem in a canonical ensemble does not have the lastterm [27], and hence, we know that κ vanishes in this limit,i.e. κ a →±∞ = 0 . We suspect κ = 0 for all a , however, sincea nonzero κ is allowed in general, this possibility needs to beclarified via other means.Similar to the dimensional analysis that lead to Eq. (20), itcan be shown that Eq. (28) follows from a dimensional analy-sis of the average energy E supplied with Eq. (10). Note that E is a function of T , N σ , a and the trapping frequency ω fora trapped system. However, applying a dimensional analysis to the free energy F of trapped systems, which is also a func-tion of T , N σ , a and ω , and using Eq. (9), we obtain Eq. (28)with κ = 0 [29]. Therefore, we conclude that κ = 0 for allparameter space, i.e. E = 2 E tr − ~ C πM a , (29)and that the Virial theorem in a canonical ensemble is of thesame form as the zero temperature one [4, 5, 26]. We recallthat taking the derivative of Eq. (29) with respect to a , andusing the IST for the energy given in Eq. (10), we obtainedthe IST for the trapping energy given in Eq. (14). V. CONCLUSIONS
To conclude, first we derived the isothermal Hellmann-Feynman theorem that is suitable for mixed states in a canon-ical ensemble. Then, we obtained the isothermal magnetic-field sweep theorems for the free, average and trappingenergies, and for the entropy, specific heat, pressure andatomic compressibility of strongly-correlated ultra-cold quan-tum gases. We applied the sweep theorems to two-componentFermi gases in the weakly interacting BCS and BEC limits,and showed that the temperature dependence of the contactparameter could be determined by the variation of either theentropy or specific heat with respect to the scattering length.We also obtained the Virial theorem in a canonical ensemble,and discussed its implications for quantum gases.One of the major challenges for the experiments with ultra-cold quantum gases is the lack of a precise thermometry,and even the measure of the temperature itself for strongly-interacting Fermi gases is a challenging problem [30]. Onone hand, this makes it more difficult to perform an isother-mal (constant temperature) magnetic-field sweep at ultra-coldtemperatures, compared to an adiabatic (constant entropy)sweep that is routinely performed in atomic systems, whiletuning the scattering length. On the other hand, it is theo-retically more easier to calculate thermodynamic quantititesat constant temperature, e.g. the calculation of isothermalatomic compressibility is much easier than the isoentropicatomic compressibility in the BCS-BEC crossover. Therefore,the isothermal Hellmann-Feynman and sweep theorems thatwe discussed in this paper are probably most useful for othertheoretical or numerical studies, and possibly for some specialexperiments where the thermometry is not an issue.
VI. ACKNOWLEDGMENTS
The author thanks Eric Braaten and Felix Werner for com-ments. This work is financially supported by the MarieCurie International Reintegration (FP7-PEOPLE-IRG-2010-268239) and Scientific and Technological Research Councilof Turkey’s Career (T ¨UB ˙I TAK-3501) Grants. [1] See the recent reviews by F. Werner and Y. Castin,arXiv:1001.0774; and also by E. Braaten, arXiv:1008.2922(2010).[2] S. Tan, Annals of Physics , 2952 (2008).[3] S. Tan, Annals of Physics , 2971 (2008).[4] S. Tan, Annals of Physics , 2987 (2008).[5] E. Braaten and L. Platter, Phys. Rev. Lett. , 205301 (2008).[6] E. Braaten, D. Kang, and L. Platter, Phys. Rev. A , 053606(2008).[7] S. Zhang and A.J. Leggett, Phys. Rev. A , 023601 (2009).[8] R. Combescot, F. Alzetto, and X. Leyronas, Phys. Rev. A ,053640 (2009).[9] F. Werner, L. Tarruell, and Y. Castin, Eur. Phys. J. B , 401(2009).[10] D. Blume and K. M. Daily, Phys. Rev. A , 053626 (2009).[11] G. Baym, C. J. Pethick, Z. Yu, and M.W. Zwierlein, Phys. Rev.Lett. , 190407 (2007).[12] R. Haussmann, M. Punk, and W. Zwerger, Phys. Rev. A ,063612 (2009).[13] H. Hu, X.-J. Liu, and P. D. Drummond, Europhys. Lett. ,20005 (2010).[14] D. T. Son and E. G. Thompson, Phys. Rev. A , 063634(2010).[15] E. Taylor and M. Randeria, Phys. Rev. A , 053610 (2010).[16] W. Schneider and M. Randeria, Phys. Rev. A , 021601(R)(2010).[17] J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin, Phys.Rev. Lett. , 235301 (2010).[18] H. Hu, E.D. Kuhnle, X.-J. Liu, P. Dyke, M. Mark, P.D. Drum-mond, P. Hannaford, and C.J. Vale, Phys. Rev. Lett. ,070402 (2010). [19] Z. Yu, G. M. Bruun, and G. Baym, Phys. Rev. A , 023615(2009).[20] F. Palestini, A. Perali, P. Pieri, and G. C. Strinati, Phys. Rev. A , 021605(R) (2010).[21] D. T. Son, arXiv:0707.1851 (unpublished).[22] J. E. Thomas, Phys. Rev. A , 013630 (2008).[23] M. Iskin and C. A. R. S´a de Melo, Phys. Rev. B , 224513(2005); and Phys. Rev. A , 013608 (2006).[24] C. Sanner, E. J. Su, A. Keshet, R. Gommers, Y. Shin, W. Huang,and W. Ketterle, Phys. Rev. Lett. , 040402 (2010).[25] T. M¨uller, B. Zimmermann, J. Meineke, J.-P. Brantut, T.Esslinger, and H. Moritz, Phys. Rev. Lett. , 040401 (2010).[26] F. Werner, Phys. Rev. A , 025601 (2008).[27] J. E. Thomas, J. Kinast, and A. Turlapov, Phys. Rev. Lett. ,120402 (2005).[28] After the substitution, the resultant equation is of the form (1 − T ∂/∂T ) y = 0 , for which the general solution can be writtenas y = κT where κ is a real constant independent of T .[29] This was pointed out to us by Eric Braaten (private communi-cation).[30] So far there are two model-independent thermometry methodsemployed in the experiments. The first one is used in M. Zwier-lein et al., Nature , 54 (2006) for imbalanced Fermi gases,and it is based on fitting the noninteracting edge of the major-ity component to the Fermi function. The second one is usedin L. Luo, B. Clancy, J. Joseph, J. Kinast, and J. E. Thomas,Phys. Rev. Lett. , 080402 (2007), where the temperature isobtained from the thermodynamic relation T = ∂E/∂S , af-ter measuring the energy and entropy to obtain a smooth E ( S ))