Universal Relations for a Fermi Gas Close to a -wave Interaction Resonance
We investigate the properties of a spinless Fermi gas close to a -wave interaction resonance. We show that the effects of interaction near a -wave resonance are captured by two contacts, which are related to the variation of energy with the -wave scattering volume and with the effective range in two adiabatic theorems. Exact pressure and virial relations are derived. We show how the two contacts determine the leading and sub-leading asymptotic behavior of the momentum distribution ( and ) and how they can be measured experimentally by radio-frequency and photo-association spectroscopies. Finally, we evaluate the two contacts at high temperature with a virial expansion.
Introduction. In the past decade, degenerate Fermi gases close to scattering resonances have attracted both theoretical and experimental attention Zwerger2011 . In the unitary Fermi gas close to an -wave resonance, it is understood that thermodynamic properties are universal Ho2004 , depending only on a single function, called the “contact” Tan2008 ; Braaten2008 ; Zhang2009 ; Werner2009 . Its manifestations in physical properties have been extensively explored and confirmed in experiments Stewart2010 ; Sagi2012 ; Hoinka2013 . Extension to arbitrary dimensions has been considered Valiente2011 ; Valiente2012 . However, so far contact has only been considered for s-waves, even though p-wave and higher partial wave resonances have been explored experimentally Regal2003 ; Zhang2004 ; Gunter2005 ; Schunck2005 ; Gaebler2007 ; Fuchs2008 ; Inada2008 ; Nakasuji2013 ; Ticknor2004 ; Chevy2005 ; Levinsen2007 ; Levinsen2008 and theoretically Pricoupenko2006 ; Lasinio2008 ; Zhang2010 ; Braaten2012 ; Nishida2013 ; Zinner2014 ; Nishida2014 ; Gao2014 ; Peng2014 ; Gridnev2014 ; Jiang2015 ; Ohashi2005 ; Gurarie2007 ; Gubbels2007 ; Inotani2012 .
In this Letter, motivated by the recent radio-frequency (rf) spectroscopic data near a -wave Feshbach resonance in K Luciuk2015 , we generalize the concept of contact to -waves. In the case of an -wave resonance, a single contact, which depends on the -wave scattering length , is sufficient for the characterization of universal properties of the system. For example, the two-body binding energy is given by , where is the mass of the atoms, and the effective range correction is in general small Werner2012 . In the case of -wave scattering, however, the phase shift is given by for a short-range potential, and the effective range is of fundamental relevance, in addition to the scattering volume . This can be seen clearly in the binding energy of a shallow -wave bound state com1 , depending crucially on both and . As a result, to capture the universal properties of a spinless Fermi gas around a -wave resonance, it is necessary to introduce two contacts, related to the variation of and , separately. We show how two adiabatic theorems [see Eqs. (11) and (12)] can be established and how the two contacts relate to the leading () and the sub-leading () terms of the high-momentum distribution. We also show how the two contacts can be measured spectroscopically. Finally, we use a virial expansion to determine each contact as a function of , , and at high temperature.
General Formulation. To start, let us consider the two-body problem, where two identical fermions of mass interact via a short-range potential of range , tuned close to a -wave resonance. The relative wave function in the -wave channel can be written as , where labels the projection of angular momentum along -direction and is the relative wave vector. For low-energy -wave scattering, the radial wave function can be expanded in powers of , . In the asymptotic regime where , we fix the normalization such that the explicit form of [and hence and ] is
It is important to note that the above asymptotic forms for also hold for any shallow -wave bound state in the corresponding asymptotic regime. Once the asymptotes are determined through Eq. (1), the short-range form () of is completely fixed by two-body physics, due to competition between kinetic and potential energy and in particular, independent of the asymptotic wave vector Zhang2009 .
To proceed to the many-body case, we first need to derive two important identities, relating the change of and to that of the variation of the potential . Consider two slightly different potentials , each with scattering volume and effective range . The radial Schrödinger equation is
where and similarly for .
Next, consider a spinless Fermi system of total number confined in volume with density where is the Fermi wave vector. The two-body density matrix , where creates a fermion at position , is Hermitian and can be diagonalized
The eigenvalues satisfy the condition , and the associated pair wave functions form an orthonormal set. In a rotationally invariant system, they can be further written as
where is the center of mass and is the relative coordinate and . can be regarded as the center-of-mass momentum of a pair and label the quantum numbers of the relative radial direction, the angular momentum, and its projection respectively. Here the index is a shorthand for all the quantum numbers that label the pair wave function. In a single-component Fermi gas, must be odd. In the region , ; the -wave channel has the strongest penetration inside the interaction potential . As a result, we shall concentrate only on the -wave component, since it gives the dominant contribution to the interaction energy of the system.
The pair wave function , and hence is not an eigenfunction of the two-body Schrödinger equation, but can be expanded in terms of the -wave functions (setting and neglecting the subscript from thereafter)
where are the real expansion coefficients, the integration is over all scattering states, and we have also taken into account the possibility of a shallow bound state with radial wave function and binding energy , when and . Extension to multiple bound states is straightforward. An important consequence of such considerations is that, in the asymptotic region where , the form of , and hence , when expanded in power of , are identical to that of and . Furthermore, for , both are uniquely fixed by the two-body physics. Thus, when evaluating the expectation value of any short-range function such as potential , can be taken out of the integration over .
where we have defined two -wave contacts for each ,
Here the contribution from possible bound states is implicitly included in the integration over . We note that has dimension of length, while has dimension of inverse length. Just as in the -wave case, encapsulate all the short-range correlations of the many-body system. As a byproduct, the two-body density matrix for in the asymptotic regime can be written as
Now suppose that the potential can be controlled via an auxiliary parameter , such that a small change in can be written as . We can use the Hellmann-Feynman theorem and write , where is the total many-body Hamiltonian with denoting the kinetic energy, independent of . Using Eqs. (3,4,8), we find
In the simplest case of a shallow -wave two-body bound state, the wave function in the asymptotic region is given by with . It is then easy to obtain for the two-body bound state com1 . One can extract directly that and , consistent with the adiabatic theorems Eq. (12). The derivation also applies to thermal equilibrium, in which case, one should replace the energy by the free energy of the system and keep temperature constant Zhang2009 .
As in the -wave case, a pressure relation and virial theorem can be found. In a uniform system, the universal hypothesis is that the free energy can be written as close to a -wave resonance, where is the Fermi temperature. Using dimensional analysis Tan2008 ; Braaten2008 ; Zhang2009 ,
where is the energy density and is the total energy. In an external harmonic trap , the free energy can be written as near resonance, and we find
where denote the total potential energy due to the harmonic confinement.
Momentum Distribution. The correlations encapsulated by the two contacts determine the tail of momentum distribution, which can be measured using time-of-flight imaging Stewart2010 . Theoretically, the momentum distribution can be obtained by Fourier transforming the single-particle density matrix . While the leading term of momentum distribution is entirely determined by the internal structure of the two-body density matrix in the asymptotic regime [cf. Eq. (11)], the sub-leading term depends on the distribution of center of mass momentum . Assuming that the distribution of is isotropic, namely, etc., we find that, in fact for com1
which shows that while determines the strength of the leading Inotani2012 , the sub-leading term is not determined solely by the contact . In fact, the angular dependences of the momentum distribution is also not purely -wave, but has an -wave component. We also note that a sub-leading term in momentum distribution relating to the -wave effective range is found in Ref. Werner2012 .
Radio-frequency Spectroscopy. The rf coupling transfers fermions into an initially empty spin state , where is the rf Rabi frequency. For a perturbative , the transfer rate can be written as , where label the initial and final states, and denotes the initial state distribution. In the region , one finds Punk2007 ; Schneider2010 ; Braaten2010
Static Structure Factor. By definition, the static structure factor and can be measured by Bragg spectroscopy Vale2013 . Here is the density operator and other notations are the same as before. can be obtained directly by Fourier transforming , Eq. (11), and diverges linearly in the limit . It is cut off by the short-range potential and will be limited by .
Photo-association Spectroscopy. Photo-association has been used to measure the fraction of closed channel molecules in two-component Fermi gases Partridge2005 , which is related to the -wave contact Zhang2009 ; Werner2009 . In the case of a -wave resonance, if the internal wave function of the relevant excited molecule has a specific projection along the direction, namely , the transition rate is given by , with the Rabi frequency and the molecule creation operator.
Since usually the final molecular state has a finite decay rate , in the expression of should be replaced by a Lorentzian . Typically MHz Partridge2005 , much larger than the energy scales associated with the spatial motion of the Fermi gas. As a result, when the of the photo-association laser is tuned to resonance, dominates over typical values of , and we can approximate the Lorentzian by ,
The Franck-Condon factor can be computed once is known. What is important here is that it depends only on two-body physics, so the many-body dependence is encapsulated in . The contribution from is smaller by a factor if the excited molecular state is of extension . In the case when the photo-association process does not distinguish between final molecular states of different , the total transition rate will be the sum of the individual .
Virial Expansion for -wave Contacts. At high temperatures, the effects of the interaction can be taken into account by the second virial expansion HM2004 . The change of the free energy of the spinless fermions is given by , where is the free energy without interactions and . The second virial coefficient is given by
Let , then by the adiabatic theorems,
When , with
Figure 1 shows the dependences of and as a function of for and , appropriate for the case of K with a typical density of m at the -wave resonances near G Regal2003 . Here we note that while decreases monotonically as increases, shows non-monotonic behavior and reaches a maximum when , where it is comparable to , if non-dimensionalized by (see Fig. 1). The temperature dependence of the contacts and at is shown in the inset, for which is much smaller than ; the magnitude of both grows with increasing . In the temperature regime , Eqs. (20) and (21) give and , which should be contrasted with -dependence for -wave contact Yu2009 .
Away from -wave resonances where the scattering volume is small, Eq. (18) gives and when contribution from the deeply bound state is excluded. In this limit, the scaling and is also expected from perturbation calculations for small Pricoupenko2006 , which indicates the irrelevance of and in Fermi gases close to an -wave resonance.
Discussion. Our derivation of the -wave contacts is based on the single-channel model which does not take into account explicitly the presence of closed-channel molecules, as in the case of a Feshbach resonance. The same results shall be obtained for a two-channel model, provided that the closed-channel molecule is small (comparable to ), which is typically the case. This is because all the arguments so far depend only on the properties of the two-body wave function or two-body density matrix in the asymptotic regime, which, in our derivation, depend only on the scattering volume and effective range , irrespective of whether they arise from a shape resonance or a Feshbach resonance. For actual atomic systems, the van der Waals potential modifies the -wave scattering phase shift by introducing a term in the effective range expansion Bo1998 ; Zhang2010 . However, close to a Feshbach resonance, it was shown that , whose effects are thus negligible Zhang2010 ; Bo2011 . As a result, we expect that our main results Eqs. (11) to (17) to remain true close to a -wave Feshbach resonance.
Resonances for different can be split due to magnetic dipole-dipole couplings Ticknor2004 . For K, the and resonances around G are split by about G Ticknor2004 . To take this into account, we can introduce phase shifts for different , . Likewise, we can establish the relation and while Eqs. (11) and (15) to (17) stay intact.
Acknowledgements. We are grateful to Shina Tan, Chris Luciuk and Stefan Trotzky for useful discussions; to Pengfei Zhang for correcting an error in Eq. (16); and to W. Zwerger for drawing our attention to Ref. Werner2012 . ZY is supported by NSFC under Grant No. 11474179 and 11204152, and the Tsinghua University Initiative Scientific Research Program. JT is supported by AFOSR, ARO, and NSERC. SZ is supported by GRF HKU 17306414 and CRF HKUST3/CRF/13G, and the Croucher Foundation under the Croucher Innovation Award.
Note added. During the final preparation of this manuscript, closely related work by Yoshida and Ueda appeared Yoshida2015 , in which they discuss one of the contacts, , using a two-channel model.
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