Polaron residue and spatial structure in a Fermi gas Christian Trefzger, Yvan Castin
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Christian Trefzger, Yvan Castin. Polaron residue and spatial structure in a Fermi gas. EPL - Euro- physics Letters, European Physical Society/EDP Sciences/Società Italiana di Fisica/IOP Publishing, 2013, 101, pp.30006. hal-00747260v2
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Polaron residue and spatial structure in a Fermi gas
Christian Trefzger and Yvan Castin
Laboratoire Kastler Brossel, Ecole´ Normale Sup´erieure and CNRS, UPMC, 24 rue Lhomond, 75231 Paris, France
PACS 03.75.Ss – Degenerate Fermi gases
Abstract –We study the problem of a mobile impurity of mass M interacting via a s-wave broad or narrow Feshbach resonance with a Fermi sea of particles of mass m. Truncating the Hilbert space to at most one pair of particle-hole excitations of the Fermi sea, we determine ground state properties of the polaronic branch other than its energy, namely the polaron quasiparticle residue Z, and the impurity-to-fermion pair correlation function G(x). We show that G(x) deviates from 4 unity at large distances as −(A4 + B4 cos 2kFx)/(kFx) , where kF is the Fermi momentum; since A4 > 0 and B4 > 0, the polaron has a diverging rms radius and exhibits Friedel-like oscillations. In the weakly attractive limit, we obtain analytical results, that in particular detect the failure of the Hilbert space truncation for a diverging mass impurity, as expected from Anderson orthogonality 4 catastrophe; at distances between ∼ 1/kF and the asymptotic distance where the 1/x law applies, they reveal that G(x) exhibits an intriguing multiscale structure.
Introduction. – The physics of atomic Fermi gases Model. – We consider in three dimensions an ideal has recently experienced a fast development, thanks to the gas of N same-spin-state fermions of mass m, enclosed in a Feshbach resonance technique that allows to tune the s- cubic quantization volume V with periodic boundary con- wave scattering length a of the interaction, and to obtain ditions. The gas is perturbed by an impurity, that is a dis- highly degenerate strongly interacting Fermi gases [1, 2]. tinguishable particle, of mass M. The impurity interacts The first realizations of spin polarized configurations [3,4] with each fermion resonantly on a s-wave broad or narrow asked for theoretical interpretation of the experimental Feshbach resonance, as described by a two-channel model results, and shortly afterwards it was proposed that, in [17–19], with a Hamiltonian Hˆ written in [11]: The parti- the strongly polarized case, the minority atoms dressed cles exist either in the form of fermions or impurity in the by the Fermi sea of the majority atoms form a normal open channel, or in the form of a tightly bound fermion- gas of quasiparticles called polarons [5, 6], which agrees to-impurity molecule in the closed channel. These two with the experimental phase diagram [3,7]. Strictly speak- forms are coherently interconverted by the interchannel ing, such polarons should be called “Fermi polarons”, so coupling of amplitude Λ 1. We restrict to the zero-range as to distinguish from the traditional condensed-matter- (or infinite momentum cut-off) limit, so that the inter- physics polaron, i.e. an electron coupled to a bosonic bath action is characterized by the s-wave scattering length a of phonons [8]. and the non-negative Feshbach length R∗ [20]. In terms of The basic single-polaron properties, such as its binding the effective coupling constant g and of the interchannel energy to the Fermi sea and its effective mass, are now coupling Λ, one has well understood, both for broad [5, 6, 9, 10] and narrow 2π 2a π 4 [11–13] Feshbach resonances. Here we study the polaron ~ ~ g = and R∗ = 2 2 , (1) ground state properties going beyond the energy search, µ Λ µ for both types of resonances. We study the quasiparticle where µ = mM/(m + M) is the reduced mass. A broad residue Z, already investigated experimentally in [14, 15] Feshbach resonance corresponds to R∗ = 0. and theoretically in [12, 16], and the intriguing issue of the density distribution that surrounds the impurity, as Polaronic ansatz. – Whereas the ground state of characterized by the pair correlation function: This gives the system presents two branches, a polaronic branch and access to the spatial extension of the polaron, of potential a dimeronic branch [9,11–13,16,21,22], we restrict to the important consequences on the properties of the polaronic 1We can neglect the open-channel fermionic interaction if its gas. (background) scattering length abg is ≈ the van der Waals length.
p-1 Christian Trefzger and Yvan Castin
M/m = 0.1505 M/m = 1 M/m = 6.6439 polaronic branch. The ground-state of the N fermions is 1 1 1 the usual Fermi sea (“FS”), of energy EFS(N). We de- 0.8 (a1) 0.8 (b1) 0.8 (c1)
0.6 0.6 0.6 = 0 termine the ground state of a single impurity interacting * Z 0.4 0.4 0.4 R F with the N fermions using the unexpectedly high-accuracy k 0.2 0.2 0.2 approximation proposed in [6] (and generalized to encom- 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -6 -4 -2 0 2 4 pass the narrow resonance case in [11,13]), that truncates 1 1 1 the Hilbert space to at most one pair of particle-hole ex- 0.8 (a2) 0.8 (b2) 0.8 (c2) 0.6 0.6 0.6 2 = 1 * citations of the Fermi sea . For a zero total momentum, Z 0.4 0.4 0.4 R F this corresponds to the ansatz 0.2 0.2 0.2 k 0 0 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -6 -4 -2 0 2 4 ˆ† X ˆ† X ˆ† † 1 1 1 |ψpoli = φ d0 + φqbquˆq + φkqdq−kuˆkuˆq |FSi, 0.8 (a3) 0.8 (b3) 0.8 (c3) q 0.6 0.6 0.6
k,q = 10 * Z
0.4 0.4 0.4 R
(2) F ˆ† † ˆ† 0.2 0.2 0.2 k where dk,u ˆk and bk are the creation operators of an im- 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -14 -12 -10 -8 -6 -4 purity, a fermion and a closed-channel molecule of wave 1/(k a) 1/(k a) 1/(k a) vector k. The prime above the summation symbol means F F F that the sum is restricted to q belonging to the Fermi sea Fig. 1: (Color online) Quasiparticle residue (6) for various of N fermions, and to k not belonging to that Fermi sea. mass ratios M/m (columns a,b,c) and various Feshbach lengths The successive terms in (2) correspond in that order to (rows 1,2,3). Black line (solid for Z > 1/2, dashed for the ones generated by repeated action of the Hamiltonian Z < 1/2): Numerical solution. Red line: Second order weakly Hˆ on the Λ = 0 polaronic ground state. One then has to attractive expansion (22), that diverges for R∗ > 0 when s of minimize the expectation value of Hˆ within the ansatz (2), Eq. (19) tends to unity, and for R∗ = 0 when 1/(kFa) → 0. −1 Circles in the inset [(kFa) ∈ [−3, 0],Z ∈ [0.3, 1]] of (c2): with respect to the variational parameters φ, φq and φkq, with the constraint hψ |ψ i = 1. Expressing φ in Experimental data of [15] for kFR∗ ' 0.9474. Vertical dot- pol pol kq ted lines [11]: Polaron-to-dimeron crossing point; on the left terms of φ and φ in terms of φ, as in [11], one is left with q q (resp. right) of this line, the ground state is polaronic (resp. a scalar implicit equation for the polaron energy counted dimeronic) [for (a1), there is a very thin zone to the right of with respect to EFS(N); in the thermodynamic limit: this line where the ground state is trimeronic [25]]. Z 0 d3q 1 ∆Epol ≡ Epol − EFS(N) = 3 , (3) (2π) Dq function [9]. Within the polaronic ansatz (2), it was shown where the prime on the integral over q means that it is in [16] that simply Z = |φ|2. We first write the amplitude restricted to the Fermi sea q < k , with the Fermi mo- F φq in terms of the denominator (4) as [11]: mentum kF related as usual to the mean density of the Fermi sea ρ = N/V by k = (6π2ρ)1/3. The function of A F φq = (5) the energy in the denominator of the integrand is Dq where A is a normalization factor. Then using Eq. (5) and 1 µk µ2R µ F ∗ the coupled equations for φ, φ and φ [11], we get the Dq = − 2 2 + 4 ∆Epol + εq q kq g π ~ π~ m following expression for the residue: Z 0 d3k0 1 2µ + 3 − 2 02 , (4) Z 0 3 (2π) Eq−k0 + εk0 − εq − ∆Epol ~ k 2 1 d q 1 Z ≡ |φ| = 1 + 2 3 2 2 2 2 2 Λ (2π) Dq where ε = ~ k for the fermions, E = ~ k for the k 2m k 2M 0 3 3 2 −1 0 Z d kd q 1/D impurity, and the prime on the integral over k means + q . (6) 0 (2π)6 E + ε − ε − ∆E that it is restricted to k > kF. q−k k q pol Quasiparticle residue. – The polaron is a well- In the limit R∗ = 0, this exactly reproduces the result of defined quasiparticle if it has a non-zero quasiparticle [24] from the diagrammatic formalism in the ladder ap- residue Z, which is defined in the Green’s function for- proximation. For R∗ > 0, this can be shown to reproduce malism from the long imaginary-time decay of the Green’s also the results of [12] obtained in the ladder approxima- tion generalized to a two-channel model. In Fig. 1 we plot 2 Such a truncation in principle requires some control of its ac- Z as a function of 1/k a for various mass ratios M/m and curacy. In the weakly interacting regime, we shall validate it by F comparison to a perturbative expansion. In the strongly interacting reduced Feshbach lengths kFR∗. We find that Z tends to regime, one can add a second pair of particle-hole excitation in the 1 when a → 0−, as expected, and to 0 when a → 0+. The ansatz, as done in [10], but this is out of the scope of the present polaronic ansatz a priori makes sense when Z is close to work. One can also compare to diagrammatic Monte Carlo tech- unity, and its accuracy becomes questionable when Z → 0. niques [9], which show (up to now for R∗ = 0 and M = m) that the values of Z obtained from (2) are in excelleent agreement with the In Fig. 1, we thus have plotted Z in dashed line for a pre- exact Z even at unitarity [23]. dicted value below 1/2. In the weakly attractive limit, we
p-2 Polaron residue and spatial structure in a Fermi gas shall give below a systematic expansion of Z up to second 0.6 (a) order in kFa. 0.4
Pair correlation function. – The pair correlation 0.2 [G(x)-1] function G(xu − xd) is proportional to the probability 2 x)
F 0 density of finding a fermion at position xu knowing that (k the impurity is localized at xd. It is thus an observable -0.2 quantity, that can be extracted from a measurement of 0 5 10 15 20 25 30 the positions of all particles in a given realisation of the 0 (b) gas, which in turn has to be averaged over many real- 3 isations . In terms of the fermionic and impurity field -5 operators ψˆ (x ) and ψˆ (x ): [G(x)-1] u u d d 4
x) -10 ˆ† ˆ† ˆ ˆ F hψu(xu)ψd(xd)ψd(xd)ψu(xu)i (k (i) (ii) (iii) G(xu − xd) = . (7) -15 ρρd 1 10 100 kFx Here ρ = N/V is the unperturbed mean fermionic den- Fig. 2: (Color online) Deviation of the pair correlation func- sity, and ρ = hψˆ†(x )ψˆ (x )i is the mean density of d d d d d tion from unity (its uncorrelated value), for M/m = 6.6439, impurity for the interacting system. Due to the inter- kFR∗ = 1 and 1/kFa = −2. (a,b) Black thick line: Nu- channel coupling, the impurity has a non-zero probability merical solution (9). (b) Red solid line: Multiscale predic- πclosed (studied in [11]) to be tightly bound within a closed- tion (31). Dashed line: Non-oscillating bit of the asymptotic channel molecule, where it cannot contribute to G(x) and prediction (14) with A4 given by (26). Dotted line: Non- to ρd. In terms of the probability πopen = 1 − πclosed for oscillating bit of the prediction (30). (i), (ii) and (iii): Zones the impurity to be in the open channel, one finds that of the multiscale structure of G(x) defined below Eq. (31). ρd = πopen/V . In the thermodynamic limit, the open- channel probability is related to the quasiparticle residue its large distance limit and its non-interacting limit)4. as The presence of the impurity induces oscillations in the 2 fermionic density that are still significant at distances of π Z 0 d3kd3q 1/D open q several 1/k . = 1+ 6 F Z (2π) Eq−k + εk − εq − ∆Epol (8) Properties of the pair correlation function. – A first property of the G(x) function is the sum rule: while the pair correlation function is Z d3x[G(x) − 1] = 0, (13) Z 0 3 Z h d q 2 G(x) = 1 + − 2f(x) + 3 |fq(x)| ρ πopen (2π) where the thermodynamic limit was taken and the integral Z 0 d3k i is over the whole space. This sum rule follows directly from − |f˜ (x)|2 , (9) (2π)3 k the integral representation of the Dirac delta distribution R d3x exp(ik · x) = (2π)3δ(k). where we have introduced the functions A second property is that, in the limit where x → +∞,
0 3 ik·x Z d k e /D A4 + B4 cos(2kFx) f (x) = q (10) G(x) − 1 ∼ − . (14) q 3 x→+∞ 4 (2π) Eq−k + εk − εq − ∆Epol (kFx) 0 Z d3q eiq·x/D 4 ˜ q The prefactor of 1/(kFx) is thus a periodic function of fk(x) = 3 (11) (2π) Eq−k + εk − εq − ∆Epol x of period π/kF, with a mean value A4 and a cosine 0 Z d3kd3q ei(k−q)·x/D contribution (of amplitude B4) reminiscent of the Friedel f(x) = q . (12) 6 oscillations. The fact that the mean value A4 differs from (2π) Eq−k + εk − εq − ∆Epol zero has an important physical consequence: It shows that Interestingly, the contribution involving f(x) is an inter- the polaron is a spatial extended object, since even the first ference effect between the subspaces with zero and one moment hxi of G(x) − 1 diverges (logarithmically) in the pair of particle-hole excitations in the Fermi sea. In Fig. 2 thermodynamic limit. we plot the numerically obtained deviation G(x) − 1 of Eq. (14) results from an asymptotic expansion of the pair correlation function from unity (which is both (10,11,12) in powers of 1/x, obtained by repeated integra- tion by parts as in [29], always integrating the exponential 3Single shot spatial cold-atom distributions integrated over some direction z can now be accurately measured by absorption imaging, 4The function x2G(x) has a finite limit in x = 0 since the giving access to spatial noise and its correlations [26]. The unwanted impurity-to-fermion wavefunction diverges as the inverse relative dis- integration over z can be undone by an inverse Abel transform [27]. tance. For R∗ = 0 and m = M, this limit is called the contact [28].
p-3 Christian Trefzger and Yvan Castin
M/m = 0.1505 M/m = 1 M/m = 6.6439 2 3 3 In the weakly attractive limit. – Following 10 10 10 (a1) (b1) (c1) 1 2 2 4 10 10 10 Ref. [11], we define for a < 0 A 4 = 0
0 1 1 * 10 10 10 R & B µ F 4 1/2
-1 0 0 k A 10 10 B 10 s ≡ (−aR∗) kF, (19) 4 m -2 -1 -1 10 10 10 -10 -5 0 5 10 -10 -5 0 5 10 -6 -4 -2 0 2 4 2 3 3 10 10 10 − (a2) (b2) (c2) and we take the limit a → 0 for fixed s < 1 (which 1 2 2 4 10 10 10 − implies R∗ → ∞). Then ∆Epol tends to 0 , so that the
0 1 1 = 1 10 10 10 * & B R integral appearing in (4) is bounded [11] and 4 -1 0 0 F k
A 10 10 10
-2 -1 -1 10 10 10 2 -10 -5 0 5 10 -10 -5 0 5 10 -6 -4 -2 0 2 4 gDq → 1 − (sq/kF) . (20) 2 3 3 10 10 10 a→0− (a3) (c3) 1 2 (b3) 2 4 10 10 10
0 1 1 = 10 After integration over q in Eq. (3), we get as in [11]: 10 10 10 * & B R 4 -1 0 0 F k A 10 10 10 2k2 k a 1 arctanh s -2 -1 -1 ~ F F 10 10 10 ∆Epol ∼ − 1 . (21) -10 -5 0 5 10 -10 -5 0 5 10 -14 -12 -10 -8 -6 -4 − 2 1/(k a) a→0 µ π s s 1/(kFa) 1/(kFa) F Also, the interchannel coupling amplitude Λ scales as Fig. 3: (Color online) Coefficients A and B in the asymp- 4 4 (−k a)1/2, see Eq. (1), and a systematic expansion of ob- totic expansion (14) of the pair correlation function, for var- F ious mass ratios M/m (columns a,b,c) and various Feshbach servables may be performed in powers of kFa, treating ˆ† ˆ lengths (rows 1,2,3). Upper and lower black lines (solid for the interchannel coupling Λb uˆd + h.c. within perturba- Z > 1/2, dashed for Z < 1/2): Numerical evaluation of A4 tion theory. The terms neglected in the ansatz (2) have 3 6 and B4, respectively. Red lines: Leading order analytical re- an amplitude O(Λ ), that is a probability O(Λ ). One can 4 sults (26) and (27), that diverge for R∗ > 0 when s of Eq. (19) thus extract from Eq. (6) the exact value of Z up to Λ : tends to unity, and for R∗ = 0 when 1/(kFa) → 0. Vertical dotted lines: Polaron-to-dimeron crossing point. 2 " 2 # 1 m kFa kFa 3 = 1 + c1 + c2 + O(kFa) . (22) Z a→0− µ2 π π function eik·x or e±iq·x to pull out a 1/x factor 5: The first coefficient of the expansion (22) is given by
2 2 1 arctanh s µ/(2π ~ ) X 0 ik xu c = − − , (23) f (x) ∼ F (k , u; q, u )e F (15) 1 2 q x2k D F 1 − s s F q u=±1 2 2 −µkF/(2π ) X 0 it originates from the second term in Eq. (6), a closed- f˜ (x) ∼ ~ F (k, u; k , u0)eikFxu , (16) k 2 2 F channel contribution. This is why c1 = 0 on a broad x k Dk e F z u0=±1 Feshbach resonance where s = 0. The second coefficient is 4 2 −µ/(8π ) 0 ~ X 0 ikFx(u−u ) f(x) ∼ 4 F (kF, u; kF, u )e .(17) 2 x Dk e 1 (1 + α) (s arctanh s − α arctanh α) F z u,u0=±1 c = 2 2 1 − s2 s2 − α2 1 2m2 arctanh s 1 + s2 arctanh s Here ez is the unit vector along z, u is the cosine of the an- − 1 + − 1 − gle between x and k, u0 is the cosine of the angle between 2 µ2 s (1 − s2)2 s x and q, and the function F is defined as follows: 1 s2(1 − α)(1 + α)2 arctanh α + + 2(1 − s2) (1 − s2)2(s2 − α2) 4m2 q2 F (k, u; q, u0) = − (1 − u2)(1 − u02) [s4 + (1 + α)2s2 − α2] arctanh s (m + M)2 k2 − , (24) 2s(1 − s2)(s2 − α2) 2 2 −1/2 m − M q 2µ q 0 2µ + 1 + 2 − uu − ∆Epol 2 2 . where the mass contrast α = (M −m)/(M +m) also obeys m + M k M k ~ k 6 (18) 2 arctanh α = ln(M/m) . The first term in Eq. (24) orig- inates from the last term in (6), an open channel contri- 2 bution. For a broad Feshbach resonance (s = 0), it is non- Therefore, in Eq. (9) the integrals containing |fq(x)| and ˜ 2 4 zero, whereas the sum of the other terms of (24), originat- |fk(x)| provide a 1/x contribution with an oscillating ln(M/m) ing from the closed channel, vanishes, and c2 = 2 2 . prefactor, as f(x) does. In Fig. 3 we plot A4 and B4 1−m /M as functions of 1/kFa for various values of the mass ratio It is instructive to analyze the perturbative expansion M/m and reduced Feshbach length kFR∗. in the exactly solvable limit of M/m → +∞: The impu- rity can then be considered as a pointlike scatterer of fixed 5One also uses the fact that, uniformly in the integration do- position, for convenience at the center of a spherical cavity + main: ∀n1, n2, n3, n4 ∈ N, there exists Cn1,n2,n3,n4 ∈ R such that n1 n2 n3 n4 0 n2+n4 6 |∂k ∂u ∂q ∂u0 F (k, u; q, u )| ≤ Cn1,n2,n3,n4 q . Contrarily to a first impression, c2 has a finite limit when s → α.
p-4 Polaron residue and spatial structure in a Fermi gas of arbitrarily large radius R, imposing contact conditions To confirm this expectation, restricting for simplicity to a of scattering length a and effective range −2R∗ on the broad Feshbach resonance (R∗ = 0), we have calculated fermionic wavefunction [20]. In the thermodynamic limit, the exact mean fermionic density in presence of a fixed one can then construct the Fermi sea of exactly calcula- pointlike scatterer, obtaining an expression equivalent to ble single-particle eigenstates in this scatterer-plus-cavity the one of §2.2.2 of [32] and leading at large distances to problem. As shown in [10] on a broad Feshbach reso- 3 kFa nance, the truncated ansatz (2) provides a good estimate 2ikFx G(x) − 1 ∼ 3 Re e . (28) x→+∞ 2(kFx) 1 + ikFa of ∆Epol for M/m → ∞. On the contrary, we find that it is qualitatively wrong for the quasiparticle residue: From As this 1/x3 law has a zero-mean oscillating prefactor, Eq. (6), it predicts a non-zero value of Z, whereas the G(x)−1 has a non-diverging integral over the whole space. exact Z vanishes for M/m → ∞, which proves the disap- Complement in augmented version: the mean fermionic pearance of the polaronic character. For an infinite mass density in the presence of the fixed scatterer of scattering impurity, indeed, Z is the modulus squared of the overlap length a placed at the origin of coordinates is ρ + δρ(r); between the ground state of the free Fermi gas and the for a < 0 we obtain the expression ground state of the Fermi gas interacting with the scat- " # Z kF 2 2 2 terer. This overlap was studied in [30], and vanishes in the dk k a sin kr 2 δρ(r) = 2 2 2 2 cos kr− − sin kr thermodynamic limit, a phenomenon called the Anderson 0 2π r 1 + k a ka orthogonality catastrophe. Satisfactorily, the perturbative This gives G(x) − 1 = δρ(x)/ρ. Interestingly the sum rule expansion (22) is able to detect this catastrophe: (13) is no longer obeyed. As the integral in (13) is no ln(M/m) longer absolutely convergent, some care must be taken. c2 ∼ . (25) M/m→∞ (1 − s2)2 We find that δρ(r) contains a mean number of fermions within the ball of radius R Such a logarithmic divergence with the mass ratio was al- Z 3 ready encountered in the context of the sudden coupling of N(R) ≡ d r δρ(r) r
p-5 Christian Trefzger and Yvan Castin
with the positive quantity ≡ −∆Epol/EF 1 and G(x) − 1 suggests that the resulting interaction may be γ ' 0.577 215 is Euler’s constant. For that order of tak- also long range. ing limits, the oscillating bit still obeys a 1/x4 law, with the same coefficient B4 as in (27); on the contrary, the ∗ ∗ ∗ non-oscillating bit obeys a different ln x/x4 asymptotic law (dotted line in Fig. 2b), which shows that the validity We acknowledge financial support from the ERC range of the 1/x4 law is pushed to infinity when a → 0−. Project FERLODIM N.228177. C. Trefzger acknowledges support from a Marie Curie Intra European grant IN- Remarkably, by keeping ∆Epol in the denominator of (12), one can obtain, see the appendix, an analytical ex- TERPOL N.298449 within the 7th European Community pression for G(x) − 1 that contains both the ln x/x4 and Framework Programme. the 1/x4 laws as limiting cases, and that describes the Appendix. – The integral over k and q in f(x) re- crossover region with cosine- and sine-integral functions: duces to a triple integral over k, q and the angle θ between k and q. Taking λ = |k − q| rather than θ as the vari- (0) able, and changing the integration order, we get f(x) = A4 n m h 1/2 1 i +∞ (0) G(x) − 1 = 4 Ci(kFx ) − Ci(kFx) R 2 kFx>1 (kFx) M 2 (−F/x) 0 dλ sin(λx)ϕ(λ) with F = (1 − s )A4 ρ/2, 1 hπ 2 2 Z 1 Z λ+q kq/(1 − s2q2) − kFx − Si(kFx/2) sin(kFx/2) 4 2 ϕ(λ) = dq dk m 2 2 2 i o max(1−λ,0) max(λ−q,1) M λ + k − q + − Ci(k x/2) cos(k x/2) + O(1) (31) F F 2 and 1/kF is the unit of length. ϕ(λ) is a C function over [0, 2] and [2, +∞[, with ϕ(0) = 0, ϕ00(0) = 1/[(1 − s2)], where the remainder O(1) is a uniformly bounded func- but with a jump J = ϕ00(2+)−ϕ00(2−) = M/[4m(1−s2)]. tion of kFx > 1 and 1. This formula satisfacto- →0 rily reproduces the numerical results, see Fig. 2b, where Triple integration by parts over each interval gives ' 0.160. It reveals that the pair correlation func- F h Z +∞ i f(x) = J cos 2x + ϕ00(0) + dλ ϕ(3)(λ) cos λx . tion has a multiscale structure for a weakly attractive x4 interaction, with three spatial ranges 8: (i) the logarith- 0 −1/2 The contribution of J reproduces Eq. (27). We find that mic range, 1 < kFx < , (ii) the crossover range, −1/2 −1 ϕ(λ) varies at three scales, , 1/2 and 0. For 0 < λ < 3/4, < kFx < 16 , and (iii) the asymptotic range, −1 we use the scaling λ = t and expand ϕ(3)(λ) in powers 16 < kFx. The logarithmic range is immediately re- 3/4 1/4 covered from Ci(u) = ln u + O(1). The −1/2 scaling of of at fixed t. For < λ < , we use the scaling 1/2 (3) 1/4 its upper limit is intuitively recovered if one assumes that λ = u and expand ϕ (λ) at fixed u. For < λ < 1, the relevant wave vectors in (12) obey |k − q| ≈ 1/x: Ne- we directly expand at fixed λ. With η = m/M, this gives glecting ∆E with respect to E in the denominator 2 −3 pol k−q (3) 4/(1 − s ) ηY (t − 1)/t + O(2t + 1) −1/2 ϕ (λ) = − − + ... of (12) then indeed requires kFx . (M/m) . 2(1 + 2t)3 2(1 − s2) (1 − ηu2)(1 + 4ηu2 + η2u4) 3 + O(u4) Conclusion. – The Fermi polaron, composed of an ϕ(3)(λ) = − + + ... impurity particle dressed by the particle-hole excitations 21/2u3(1 + ηu2)2(1 − s2) 4u4(1 − s2) of a Fermi sea close to a broad or narrow Feshbach reso- h η i [1 + O(λ)] ϕ(3)(λ) = + O(λ0) + + ... nance with zero-range interaction, is a spatially extended 2λ(1 − s2) 2λ3(1 − s2) object: The density perturbation induced by the impu- rity in the Fermi gas asymptotically decays as the inverse where Y is the Heaviside function, and the O( ) apply + + quartic distance, with a spatially modulated component for t ∈ R , u ∈ R and λ ∈ [0, 1], respectively. If one (3) reminiscent of the Friedel oscillations. In the weakly at- needs the integral of ϕ (λ) cos(λx) up to an error O(1) tractive limit, k a → 0− with |a|R fixed, where system- uniformly bounded in x and , one can apply several sim- F ∗ 1/4 atic analytical results are obtained, this density perturba- plifications over each interval. E.g., for < u < 1, (3) tion reaches its asymptotic regime over distances diverg- one can approximate ϕ (λ) by an expansion in pow- −1/4 2 ers of u, and for 1 < u < , by an expansion in ing as 1/(kF|a|) and exhibits at intermediate distances a rich multiscale structure. This may have important con- powers of 1/u. Adding contributions of all intervals, we concatenate them by pairs, and further noting that sequences on the interaction between polarons [33]: A po- −1/2 R dt −3 3 −4 −3 laron should indeed be sensitive to the deformation of the J = O(1), −1/4 |t − 2 t − (t + 1/2) | = O(1), underlying fermionic density profile induced by another R +∞ dt −3 R +∞ (3) −1/2 t = O(1), 1 dλ|ϕ (λ)| = O(1), we get polaron, since the impurity forming the polaron is cou- pled to that fermionic density. The long-range nature of (0) Z +∞ A4 n 1 dt cos(xt) G(x) − 1 = 4 − + 3 8 4 x 0 2 (t + 1/2) The first relative deviation of (kFx) [G(x) − 1] from its x → ∞ limit is −24/(k x)2 + m sin(k x). For m/M < 6, this is < 10% 1/2 1 F 2Mx F h Z dλ Z dλ i o for kFx > 16. From a similar first-deviation analysis, being in the + η cos(λx) − cos(λx) + O(1) . 1/2 1/2 1/2 logarithmic range actually requires kFx < (µ/M) . 2λ 2λ
p-6 Polaron residue and spatial structure in a Fermi gas
Explicitly evaluating the integrals gives Eq. (31) 9. [29] M. Holzmann, Y. Castin, Eur. Phys. J. D 7, 425 (1999). Finally, to obtain (30), one omits in the denominator [30] P. W. Anderson, Phys. Rev. Lett. 18, 1049 (1967). of ϕ(λ), so that it is no longer C2 at the origin: ϕ00(λ) = [31] E. M¨uller-Hartmann,T. V. Ramakrishnan, G. Toulouse, {η[ln(ηλ/2)+3/2]−s2/(1−s2)}/[2(1−s2)]+O(λ). We thus Phys. Rev. B 3, 1102 (1971). locally split ϕ00(λ) as the sum of a singular part ∝ ln λ and [32] S. Giraud, Ph.D. thesis, Universit´eParis VI, 2010, http: a C∞ function. The only trick is then to take, in the last //tel.archives-ouvertes.fr/tel-00492339 [33] S. Giraud, R. Combescot, Phys. Rev. A 85, 013605 triple integration by parts over λ and in the bit involving (2012). the singular part, (1 − cos λx)/x as a primitive of sin λx.
REFERENCES
[1] K.M. O’Hara et al., Science 298, 2179 (2002); T. Bourdel et al., Phys. Rev. Lett. 91, 020402 (2003). [2] The BCS-BEC Crossover and the Unitary Fermi Gas, LNIP 836, W. Zwerger editor (Springer, Berlin, 2012). [3] M. W. Zwierlein, C. H. Schunck, A. Schirotzek, W. Ket- terle, Nature (London) 442, 54 (2006). [4] G. B. Partridge et al., Science 311, 503 (2006). [5] C. Lobo, A. Recati, S. Giorgini, S. Stringari, Phys. Rev. Lett. 97, 200403 (2006). [6] F. Chevy, Phys. Rev. A 74, 063628 (2006); R. Combescot, A. Recati, C. Lobo, F. Chevy, Phys. Rev. Lett. 98, 180402 (2007). [7] N. Navon, S. Nascimb`ene,F. Chevy, C. Salomon, Science 328, 729 (2010). [8] T.K Mitra, Ashok Chatterjee, Sukla Mukhopadhyay, Phys. Rep. 153, 91 (1987). [9] N. Prokof’ev, B. Svistunov, Phys. Rev. B 77, 020408(R) (2008); ibid., Phys. Rev. B 77, 125101 (2008). [10] R. Combescot, S. Giraud, Phys. Rev. Lett. 101, 050404 (2008). [11] C. Trefzger, Y. Castin, Phys. Rev. A 85, 053612 (2012). [12] P. Massignan, Europhys. Lett. 98, 10012 (2012). [13] R. Qi, H. Zhai, Phys. Rev. A 85, 041603(R) (2012). [14] A. Schirotzek, Cheng-Hsun Wu, A. Sommer, M. W. Zwierlein, Phys. Rev. Lett. 102, 230402 (2009). [15] C. Kohstall et al., Nature 485, 615 (2012). [16] M. Punk, P.T. Dumitrescu, W. Zwerger, Phys. Rev. A 80, 053605 (2009). [17] M. Holland, S.J.J.M.F. Kokkelmans, M.L. Chiofalo, R. Walser, Phys. Rev. Lett. 87, 120406 (2001). [18] V. Gurarie, L. Radzihovsky, Annals of Physics 322, 2 (2007). [19] C. Mora, Y. Castin, L. Pricoupenko, C. R. Physique 12, 71 (2011). [20] D.S. Petrov, Phys. Rev. Lett. 93, 143201 (2004). [21] C. Mora, F. Chevy, Phys. Rev. A 80, 033607 (2009). [22] R. Combescot, S. Giraud, X. Leyronas, Europhys. Lett. 88, 60007 (2009). [23] K. Van Houcke, private communication. [24] P. Massignan, G.M. Bruun, Eur. Phys. J. D 65, 83 (2011). [25] C.J.M. Mathy, Meera M. Parish, D.A. Huse, Phys. Rev. Lett. 106, 166404 (2011). [26] S. F¨olling et al., Nature 434, 491 (2005). [27] Mark J.H. Ku, A.T. Sommer, L.W. Cheuk, M.W. Zwier- lein, Science 335, 563 (2012). [28] Shina Tan, Annals of Physics 323, 2952 (2008).
9We have checked that the leading correction to (20) gives, as expected, a contribution also O(1) to the expression in between curly brackets in (31).
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