Photoemission spectrum and effect of inhomogeneous pairing

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Sep 30, 2010 - Using this technique, one can probe single- particle excitations that allow us to investigate many-body effects in the BCS-BEC crossover [3–14].
PHYSICAL REVIEW A 82, 033629 (2010)

Photoemission spectrum and effect of inhomogeneous pairing fluctuations in the BCS-BEC crossover regime of an ultracold Fermi gas Shunji Tsuchiya,1,2 Ryota Watanabe,1 and Yoji Ohashi1,2 1

Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan 2 CREST(JST), 4-1-8 Honcho, Saitama 332-0012, Japan (Received 2 March 2010; revised manuscript received 8 June 2010; published 30 September 2010) We investigate the photoemission-type spectrum in a cold Fermi gas which was recently measured by the JILA group [Stewart et al., Nature (London) 454, 744 (2008)]. This quantity gives us very useful information about single-particle properties in the BCS-BEC crossover. In this paper, including pairing fluctuations within a T -matrix theory, as well as effects of a harmonic trap within the local density approximation, we show that spatially inhomogeneous pairing fluctuations due to the trap potential are an important key to understanding the observed spectrum. In the crossover region, while strong pairing fluctuations lead to the so-called pseudogap phenomenon in the trap center, such strong-coupling effects are found to be weak around the edge of the gas. Our results including this effect are shown to agree well with the recent photoemission data of the JILA group. DOI: 10.1103/PhysRevA.82.033629

PACS number(s): 67.85.−d, 03.75.Ss, 05.30.Fk

I. INTRODUCTION

The recent photoemission-type experiment developed by the JILA group [1] provides a powerful method to study microscopic properties of cold Fermi gases. This experiment is an analog of angle-resolved photoemission spectroscopy (ARPES) [2], which has been extensively applied in condensed matter physics. Using this technique, one can probe singleparticle excitations that allow us to investigate many-body effects in the BCS-BEC crossover [3–14]. Indeed, the observed spectra exhibit a dramatic change in the crossover region [1]. One typical many-body effect on single-particle excitations expected in the BCS-BEC crossover regime of a cold Fermi gas is the pseudogap effect. In this phenomenon, preformed pairs cause a gaplike structure in the density of states (DOS) even above the superfluid phase transition temperature Tc . The pseudogap has been observed in the underdoped regime of high-Tc cuprates [15]. However, as the origin of this phenomenon in high-Tc cuprates, in addition to pairing fluctuations, various possibilities have been proposed, such as spin fluctuations and a hidden order. Since the BCS-BEC crossover in a cold Fermi gas is dominated by pairing fluctuations, the study of the pseudogap in this system would also be useful for clarifying the validity of the pseudogap mechanism based on preformed pairs in high-Tc cuprates. Since the pseudogap appears in single-particle excitations, a photoemission experiment would be useful for this purpose. In considering the pseudogap effect in cold Fermi gases, one should note that the presence of a trap naturally leads to spatially inhomogeneous pairing fluctuations. While the pseudogap structure in the DOS is expected to be remarkable in the trap center, such a many-body effect may be weak around the edge of the gas cloud. Since the pseudogap in high-Tc cuprates is a uniform phenomenon, this inhomogeneous pseudogap effect is unique to trapped Fermi gases. Indeed, the double-peak structure in the photoemission spectrum observed in the strong-coupling Bose-Einstein condensation (BEC) regime of the 40 K Fermi gas [1] seems difficult to explain if a simple uniform system is considered [16]. In this paper, we study pseudogap phenomena in a trapped Fermi gas above Tc , addressing the recent photoemission 1050-2947/2010/82(3)/033629(5)

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experiment by the JILA group [1]. Extending our previous paper for a uniform Fermi gas to include the effects of a harmonic trap within the local density approximation (LDA), we calculate the local DOS, as well as the local spectral weight (SW), over the entire BCS-BEC crossover region within the T -matrix approximation in terms of pairing fluctuations [16–20]. We clarify how the inhomogeneous pseudogap phenomenon appears in these quantities. Recently, Refs. [21,22] have studied photoemission spectra using phenomenological theories with the BCS ansatz. While they capture some features of experimental results in the unitarity limit, they cannot explain the observed spectrum in the BEC regime, which consists of upper sharp and lower broad peaks. In contrast, including the inhomogeneous strongcoupling effect, we show that our ab initio calculation of the photoemission spectra can naturally explain the experiments [1] from the weak-coupling BCS to the strong-coupling BEC regimes, without introducing any free parameters. II. FORMALISM

We consider a two-component Fermi gas in a harmonic trap. Assuming a broad Feshbach resonance, we employ the ordinary BCS model, described by the Hamiltonian  ξ p c†pσ c pσ H = p,σ

−U

 q, p, p





c p+q/2↑ c− p+q/2↓ c− p +q/2↓ c p +q/2↑ .

(1)

Here, c pσ is the annihilation operator of a Fermi atom with pseudospin σ =↑ , ↓, and the kinetic energy ξ p = ε p − µ is measured from the chemical potential µ, where ε p = p2 /2m and m is the atomic mass. The pairing interaction −U (< 0) is assumed to be tunable by a Feshbach resonance [23], which is related to thes-wave scattering length as as [24] 4π as /m = −U/[1 − U p 1/(2ε p )]. Within a LDA, the effects of a trap are conveniently incorporated into the theory by replacing the chemical potential µ by µ(r) ≡ µ − V (r) [25], where V (r) = mωtr2 r 2 /2 is a harmonic potential (with ωtr the trap frequency). The LDA ©2010 The American Physical Society

SHUNJI TSUCHIYA, RYOTA WATANABE, AND YOJI OHASHI 0.6

µ/εF

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-2

-1

0

-1

1

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FIG. 1. Calculated Tc as a function of the inverse scattering as−1 . The inset shows µ at Tc . kF is the Fermi momentum, and εF is the Fermi energy. We use these Tc and µ in Figs. 2–5.

single-particle Green’s function is given by G p (iωn ,r) = 1/[iωn − ξ p (r) −  p (iωn ,r)], where ξ p (r) = ε p − µ(r), and ωn is the fermion Matsubara frequency. The LDA self-energy  p (iωn ,r) involves effects of pairing fluctuations within the T -matrix approximation [16–20], which is given by   p (iωn ,r) = T q,νn q (iνn ,r)G0q− p (iνn − iωn ,r). Here, νn is the boson Matsubara frequency, and G0p (iωn ,r) = 1/[iωn − ξ p (r)] is the LDA free fermion propagator. q (iνn ,r) = −U/[1 − U q (iνn ,r)] is the particle-particle scattering matrixwithin the T -matrix approximation, where q (iνn ,r) = 0 0 T p,ωn G p+q/2 (iνn + iωn ,r)G− p+q/2 (−iωn ,r) is the pair propagator [10]. In this paper, we focus on the case at Tc to compare with the experiments in Ref. [1]. The region above Tc will be discussed elsewhere. In the LDA, the superfluid phase transition is determined by the Thouless criterion at the trap center [26], i.e., q=0 (iνn = 0,r = 0)−1 = 0 [10]. Strongcoupling effects on µ are included by solving this Tc equation together with  the  equation for the total number of Fermi atoms, N = 2T d r p,ωn G p (iωn ,r)eiωn δ . Figure 1 shows the calculated Tc and µ as functions of (kF as )−1 , where kF is the Fermi momentum defined from the Fermi energy εF = (3N )1/3 ωtr = kF2 /(2m). In Fig. 1, Tc in a trapped gas is found to be higher than that in a uniform gas in terms of εF [26,27]. Using these values, we calculate the local SW A( p,ω,r) and local DOS ρ(ω,r) from the analytically continued Green’s function as, respectively, 1 A( p,ω,r) = − Im[G p (iωn → ω + iδ,r)], π  ρ(ω,r) = A( p,ω,r).

(2)

PHYSICAL REVIEW A 82, 033629 (2010)

Here, b p describes the third pseudospin state |3, and ω3 is the energy difference between |↑ and |3. The chemical potential µ3 (r) = µ3 − V (r) involves trap effects within the LDA. The transition from |↑ to |3 is induced by the tunneling  † Hamiltonian [26,28,29] HT = tF k (e−iωL t bk+q L ck↑ + H.c.), where tF is a transfer matrix element, and q L and ωL are the momentum and frequency of the rf pulse, respectively. The photoemission spectrum is obtained from the rf tunneling current I ( ,r) from |↑ to |3, where ≡ ωL − ω3 is the rf detuning. Within the linear response theory, we obt  tain I ( ,r) = −i −∞ dt  [Jˆ(r,t),HT (r,t  )]eδt , where Jˆ(t) ≡  † −itF k ei(H +H3 )t (e−iωL t bk+q L ck↑ − H.c.)e−i(H +H3 )t is the tunneling current operator in the Heisenberg representa+H3 )t tion, and HT (t) ≡ ei(H +H3 )t HT e−i(H . We thus obtain  the rf tunneling current I ( ,r) = p I ( p, ,r), where the momentum-resolved photoemission spectrum I ( p, ,r) has the form I ( p, ,r) = 2π tF2 A( p,ξ p (r) − ,r)f (ξ p (r) − ).

(4)

Here, f ( ) is the Fermi distribution function. In Eq. (4), we have assumed that the momentum q L of the rf photon is negligible and |3 is initially empty [f (ε p − µ3 (r)) = 0]. In a uniform Fermi gas, the photoemission spectrum is related to the SW of occupied states as I ( p, → ξ p − ω) = 2π tF2 A( p,ω)f (ω). In particular, it is equal to the SW below ω = 0 at T = 0. When T > 0, thermally excited quasiparticles also contribute to the spectrum, so that I ( p, → ξ p − ω) becomes finite even when ω > 0. In the photoemission experiment [1], since the rf pulse is applied to the whole gas cloud, the observed spectrum involves contributions from all spatial regions of the cloud. To include this situation, we should take the spatial average of Eq. (4) as  2π tF2 Iav ( p, ) = d r A( p,ξ p (r) − ,r)f (ξ p (r) − ). V (5)  Here, V = 4π RF3 /3, where RF = 2µ/(mωtr2 ) is the ThomasFermi radius [30]. We emphasize that this gives a proper definition for the spatially averaged photoemission spectrum. For later convenience, we define the averaged occupied SW and DOS by, respectively,   (6) A( p,ω)f (ω) ≡ Iav ( p, → ξ p − ω) 2π tF2 ,    (7) ρ(ω)f (ω) ≡ Iav ( p, → ξ p − ω) 2π tF2 . p

(3)

p

To see the basic character of Eqs. (6) and (7), it is helpful to consider a free Fermi gas at T = 0. In this case, Eqs. (6) and (7) reduce to, respectively,

III. PHOTOEMISSION SPECTROSCOPY

The photoemission spectrum [1] can be calculated in the same way as the rf tunneling current spectrum [26,28,29], where atoms in one of the two pseudospin states (≡ |↑) are outcoupled into an unoccupied state |3 (= |↑ , ↓) by an applied rf pulse. Since the final-state interaction can be safely neglected in the 40 K Fermi gas [1,21], the Hamiltonian  † for |3 is simply given by H3 = p [ε p + ω3 − µ3 (r)]b p b p .

A( p,ω)f (ω) = |ω/µ|3/2 δ(ω − ξ p )θ (−ω), (8) √ 2 3/2 3/2 √ ρ(ω)f (ω) = (m / 2π )|ω/µ| ω + µθ (ω + µ)θ (−ω). (9) In the former, the peak position gives the one-particle √ energy ξ p . In the latter, the DOS in a uniform gas (∝ ω + µ) is modified by the factor |ω|3/2 .

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FIG. 2. (Color online) (a)–(c) Intensity of local SW A( p,ω,r) (units of εF−1 ). (d) Local DOS ρ(ω,r). We set T = Tc and (kF as )−1 = 0. The solid line in (a) is the BCS-like quasiparticle spectrum with pg = 1.06. IV. RESULTS

We now show our numerical results. Figure 2 shows the local SW A( p,ω,r) and DOS ρ(ω,r) at Tc in the unitarity limit [(kF as )−1 = 0]. In the trap center [Fig. 2(a)], a clear pseudogap structure exists, i.e., two prominent peaks appear along the particle branch and hole  branch of the BCS-like quasiparticle spectrum ω = ± ξ 2p + 2pg , where the superfluid gap  is replaced by the pseudogap pg . The origin of pg is a particle-hole coupling by pairing fluctuations [16–20]. Since pairing fluctuations also induce a finite lifetime of quasiparticle excitations, the particle and hole branches in Fig. 2(a) have finite widths, which is in contrast to the mean-field BCS case, where both branches appear as δ-functional peaks. We note that the deviation of the lower peak from the BCS-like quasiparticle spectrum is considered to be due to the presence of excited pairs [16,20]. Pairing fluctuations are weak around the edge of the gas due to low particle density. Thus, the pseudogap in the local SW gradually disappears, as one moves away from the trap center. [See Figs. 2(b) and 2(c).] In Fig. 2(c), only a single sharp peak line exists near the one-particle energy of a free Fermi gas ξ p (RF ). That is, the gas is pseudogapped in the center of the trap, while it is nearly noninteracting on its edges. These inhomogeneous features can also be seen in local DOS. As shown in Fig. 2(d), while a large dip structure (which is a characteristic pseudogap effect in the DOS) appears around ω = 0 in the trap center, the local DOS equal to that of √ is almost √ a free Fermi gas, ρ(ω,r) = [m3/2 /( 2π 2 )] ω + µ(r), when r = RF . Since spatial inhomogeneity is unique to trapped Fermi gases, the observation of the local SW and DOS by using tomographic techniques [31,32] would be interesting. We note that the inhomogeneous pseudogap structure depends on the strength of the pairing interaction. When pairing fluctuations are weak in the BCS regime, the doublepeak structure in A( p,ω,r) soon disappears, as one moves

FIG. 3. (Color online) Intensity of local SW A( p,ω,r) (units of εF−1 ). (a)–(c) (kF as )−1 = −1. (d)–(f) (kF as )−1 = +1. We set T = Tc .

away from the trap center. [See Figs. 3(a)–3(c).] In contrast, Figs. 3(d)–3(f) show that the pseudogap features persist even near the edge of the gas in the BEC regime. Figures 4 and 5, respectively, show the averaged occupied SW and DOS at Tc . In the weak-coupling BCS regime, these quantities are expected to be close to the free Fermi gas results in Eqs. (8) and (9). Indeed, the position of the peak line in Fig. 4(a) is almost at the single-particle energy of a free fermion ω = ξ p ( 0), and the averaged occupied DOS in

FIG. 4. (Color online) Intensity of averaged occupied SW p 2 A( p,ω)f (ω) (units of 1/2m) at Tc . The values of the pairing interaction (kF as )−1 are (a) −1, (b) 0, (c) 0.4, and (d) 1. The dashed line indicates the spectrum of a free Fermi gas ξ p . The solid line is the lower peak position of the intensity. Symbols are corresponding experimental data [1].

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ρ(ω)f(ω)/[mkF/(2π )]

SHUNJI TSUCHIYA, RYOTA WATANABE, AND YOJI OHASHI

0.02 0.2

0.01 0 -6 -4 -2 0 2 (ω+µ)/εF

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FIG. 5. (Color online) Averaged occupied DOS ρ(ω)f (ω) at Tc . The inset shows a magnification of the curve for (kF as )−1 = 1. Circles and crosses show the experimental data for (kF as )−1 = 0 and 1, respectively [1]. The data are fitted so that the peak value coincides with that of the theory curves. The data agree well with the theory curves.

Fig. 5 is close to the DOS for a free Fermi gas multiplied by |ω|3/2 when (kF as )−1 = −1. However, we also find that the peak line in Fig. 4(a) is slightly below the curve of ω = ξ p , which is a signature of the pseudogap effect near the trap center. That is, the  pseudogap pg lowers the hole branch as −|ξ p (r)| → − ξ 2p (r) + 2pg , and this effect still remains even after spatial averaging. In the crossover region, pairing fluctuations are strong around the trap center. In the unitarity limit, while the region around the edge of the gas still gives a sharp peak line at ω ξ p in A( p,ω)f (ω), the pseudogap in the SW around r = 0 causes the broadening of the lower part of the peak line around p/kF 0.5 shown in Fig. 4(b). In addition, the short lifetime of quasiparticle excitations caused by strong pairing fluctuations around the trap center also causes the broadening of ρ(ω)f (ω), as shown in Fig. 5. We also note that one can see the back-bending of the lower peak position in Figs. 4(b)–4(d) originating from the lower branch shown in Fig. 2(a), which agrees well with the experimental data [1].

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In the BEC regime, the pseudogap features persist to the edge of the gas. Thus, the double-peak structure in the local SW [see Figs. 3(d)–3(f)] is not smeared out by spatial averaging, as shown in Figs. 4(c) and 4(d). In particular, the double-peak structure in Fig. 4(d), consisting of an upper sharp peak and a lower broad peak, agrees well with the photoemission experiment by the JILA group [1]. We emphasize that this spectral structure cannot be explained within the previous phenomenological theories [21,22]. Such a double-peak structure also appears in Fig. 5, which is also consistent with the experiment [1]. In the BEC limit, the spectral weight only has the upper sharp peak describing the dissociation of two-body bound molecules. On the other hand, as discussed in Ref. [16], the lower broad peak in the SW in the BEC regime is evidence of a many-body character of paired atoms. Thus, the double-peak structure in Figs. 4(c) and 4(d) may be understood as a result of the fact that, in the BCS-BEC crossover, the character of the fermion pair continuously changes from the many-body bound state associated with the Cooper instability to the two-body bound state where the Fermi surface is not necessary. V. CONCLUSIONS

To summarize, we have discussed the inhomogeneous pseudogap effect in the BCS-BEC crossover regime of a trapped Fermi gas. Including this effect, we showed that the calculated spatially averaged photoemission spectrum agrees well with the recent experiment by the JILA group [1]. Since spatial inhomogeneity caused by a trap is inevitable in a cold Fermi gas, our results would be useful for clarifying strong-coupling phenomena of this system over the entire BCS-BEC crossover region. ACKNOWLEDGMENTS

We wish to thank A. Griffin for useful discussions. We acknowledge J. P. Gaebler and D. S. Jin for providing us with their experimental data.

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