Fulde–Ferrell–Larkin–Ovchinnikov states in a superfluid Fermi gas

Journal of Physics and Chemistry of Solids 66 (2005) 1359–1361 www.elsevier.com/locate/jpcs

Fulde–Ferrell–Larkin–Ovchinnikov states in a superfluid Fermi gas Takeshi Mizushima*, Masanori Ichioka, Kazushige Machida Department of Physics, Okayama University, Okayama 700-8530, Japan

Abstract Recently, both the experimental and theoretical studies to realize superfluid states in atomic gases have been performed by using a 50–50 mixture of atoms in two hyperfine spin states. We discuss favorable superfluid states in an unequal mixture. Such situation is analogous to a superconducting state in the presence of a magnetic field acting on the electron spins and then the natural candidate for the ground state is known as the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state which is a spatially inhomogeneous superfluid state. We examine the possibility of the FFLO state in two species gas of Fermions confined in a realistic three-dimensional harmonic trap. We propose a clear experimental way to create and directly detect the spatially modulated FFLO state from the macroscopic signature. q 2005 Elsevier Ltd. All rights reserved.

1. Introduction After the realization of Bose–Einstein condensation in an atomic Bose gas, the further challenging subject for the theory and the experiment has been to succeed in the achievement of the superfluid phase transition in an atomic Fermion systems. The adiabatic sweeping of the magnetic field by the Feshbach resonance technique has opened up a breakthrough for cooling degenerated Fermi gases below the superfluid transition temperature [1]. Recently, the experimental observation of a Cooper-pairing in the BECBCS crossover region has been reported by two groups [2,3]. It should be emphasized that all the experiments have been done by using a 50–50 mixture of atoms in two hyperfine spin states. In the present paper, we discuss on the possibility of the spatially modulated superfluid phase, called the Fulde– Ferrell–Larkin–Ovchinnikov (FFLO) state [4,5], in an atomic Fermion system with unequal population in two hyperfine spin states. After introducing the Bogolinbov-de Gennes formalism in Section 2, we briefly provide the analytic solution for the FFLO state of an ideal quasi-one-dimensional system. Then, we go on showing the numerical calculation for the realistic three-dimensional system. We also propose an

most suitable experimental situation to directly observe the FFLO state by using currently available techniques. 2. Bogoliubov-de Gennes equation Throughout this paper, since we consider a Fermi gas with the unequal population in two species sZ[, Y, the following chemical potential is used: ms Z mC ðs^ 3 Þs;s dm with the Pauli matrix s^ 3 . Following the standard mean-field formalism, Bogoliubov-de Gennes equation is given as " #    K[ K m[ DðrÞ uq uq Z 3 ; (1) q   vq D ðrÞ KKY C mY vq where Ks ZKðZ2 =2mÞV2 CVðrÞCgrs ðrÞ with gZ4pZ2 a=m being the attractive interaction (a the s-wave scattering length) and an axisymmetric trap potential VðrÞZ ð1=2Þmðu2r r 2 C u2z z2 Þ. The self-consistent condition is taken P as the paring field DðrÞZ geff q uq ðrÞvq ðrÞf ð3 Þ and the P q 2 particle density of each component, r ðrÞZ ju ðrÞj f ð3 [ q q qÞ P and rY Z q jvq ðrÞj2 ½1K f ð3q Þ. When calculating D(r), we have used a regularized coupling constant geff to avoid the ultraviolet divergence rather than the bare g. 3. Quasi-one-dimensional system

* Corresponding author. E-mail address: [email protected] (T. Mizushima).

0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.05.067

In the one spatial dimension without a trap, which is approximate for a long cigar-shaped gas uz/ur/0, one admits an exact analytical solution for the FFLO state [6]: D(z)ZD1k1sn(D1z, k1) where the order parameter D1 and the modulus k1 are self-consistently determined. sn(z, k) is the

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T. Mizushima et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1359–1361

Jacobi elliptic function. That has the asymptotic form D(z)xD1k1 sin (D1z) in the limit of k1/0 where the spatially modulated phase changes into the paramagnetic phase as the second-order phase transition. In the opposite limit k1/1, the pairing field exhibits a rectangular shape which is quite different from the sinusoidal modulation. Here, kZ1 means uniform BCS states. Comparing with the free energies of two quantum phases, the FFLO and BCS states, the phase boundary can be determined as follows dncri h

jn[ K nYj 1 D0 Z ; p 3F n

(2)

where we use the experimentally controllable variables, the population difference dn and the amplitude of the Cooperpair at Ð the zero-temperature D0, with each particle P number ns Z drrs ðrÞ and the total particle number nZ s ns $3F is the Fermi energy. Beyond the above critical population difference dnRdncri, the uniform BCS state changes into the modulated FFLO state. Since D0/3FZ0.2–0.4 in the present experiments, the critical population difference should be of an order of 10–20% difference. This modulated phase accompanies the spin variation with half of the fundamental modulation periodicity. In the limit of k1/1, since the paring field has a rectangular form, the excess density of the up-spin Fermions periodically accumulates at the zeros of D(z). As the population difference approaches to dncri, the accumulation becomes sharp in space: the asymptotic form of the local magnetization mðzÞZ r[ðzÞK rYðzÞf 1K tanh2 ðD0 zÞ. 4. Three-dimensional system We consider the axisymmetric trap potential with ur/2pZ1.67uz/2pZ1000 Hz and the 6Li atoms with the chemical potential mZ 12:5Zur , corresponding to nw1100 atoms. Throughout this paper, D 0/3FZ0.35 and the temperature kB T=Zur Z 0:05 are fixed. Fig. 1(a) shows the spatial variation of the pairing field for dnZ0.15. It is seen that the pairing field changes its sign at the plane zZ0 and also near the edge. It should be noted that the total density exhibits a smooth Thomas-Fermi profile for the normal Fermions (see Fig. 1(b) in Ref. [7]). The cross-section D(rZ0, z) at the rZ0 axis is displayed in Fig. 1(b) for various population differences dnZ0.07, 0.15, 0.32. The critical population difference for the present system is estimated from Eq. (2) as dnZ0.11. As dn increases, the modulation period tends to become short, thus the extra zeros appears near the edge. For dnZ 0.32 where the pairing field has five zeros, the pairing field near the center of the trap then remains the large amplitude against increasing dn, while the pairing field near the edge immediately becomes small. That originates from the presence of a confinement potential, namely the difference of the local transition temperature Tc(r). It is also noted that

Fig. 1. (a) The spatial profile of the pairing field D(r, z) for dnZ0.15 and D0/3FZ0.35. (b) The cross-section of D(r, z) at rZ0 is shown for dnZ0.07, 0.15, 0.32.

for dnZ0.32 the modulation period becomes shorter towards the edges where the local density tends to be dilute. In Fig. 2, we display our most important results. Ð Fig. 2(a) shows the columnar density profile rs ðzÞ h2p dr rrs ðr; zÞ and the local magnetization defined as m(z)Zr[(z)KrY(z). It is seen from this that m(z) exhibits the sharp peak at zZ0 and also small peaks appear near the edge of the pairing field. The cross-sections of m(r, z) at rZ0, 3, 6 mm are illustrated in Fig. 2(b). At rZ0, the local magnetization are strongly bounded at the zeros of D(r) and as r becomes large, that profile tends to become broad and starts overlapping between the neighboring peaks. Near the edge of the cloud rZ6 mm, which corresponds to the effectively weak coupling regime, there is only single peak. The characteristic features of such the density profile are the macroscopic signature for the FFLO state, which can be directly imagined either by optical absorption or by Stern–Gerlach experiments [8]. The origin will be discussed from the Fermionic excitation spectrum in the following part. Here, we should mention the dn dependence of this macroscopic feature for a fixed D0/3F, which is shown in Fig. 2(c). It is seen that the local magnetization has sharp peaks near the critical population difference dncriZ0.11, while as dn increases the clear peak structure spreads all over the cloud. This tendency qualitatively agrees with the analytic result for an uniform system. However, it should be emphasized that even in the situation far from the phase boundary between BCS and FFLO states, e.g. dnZ0.32 in Fig. 2(c), m(r) remains the clear peaks at the plane zZ0. Fig. 3 shows the gap structures in the BCS and FFLO states where the Fermionic excitation spectrum is distinctly altered form the uniform BCS state. P The local density of states (LDOS) is defined by N[ðr;EÞZ q juq ðrÞj2 dðEK3q Þ;

T. Mizushima et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1359–1361

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ρσ(z) [µm–1]

12 10

ρ ρ m

(a)

8 6 4 2

m(z) [µm–1]

m(r,z) [µm–3]

0.5 0.4

r=0 µm r=3 µm r=6 µm

(b)

0.3 0.2 0.1 0 7 6 5 4 3 2 1 0

δn=0.07 δn=0.15 δn=0.32

(c)

–10

–5

0

5

10

z [µm] Fig. 2. (a) The columnar density profiles along the z-axis for dnZ0.15 and (b) the cross-sections of m(r, z) at rZ0, 3, 6 mm. (c) The columnar magnetization profiles m(z) for various dnZ0.07, 0.15, 0.32. The arrows indicate the positions of the zeros of D(rZ0, z), which is shown in Fig. 1. The number of the zeros is 1 (dnZ0.07), 3 (dnZ0.15), and 5 (dnZ0.32).

P NYðr;EÞZ q jvq ðrÞj2 dðEC3 q Þ. In Fig. 3(a), the DOS of Ð up-spins N[ðrZ 0; EÞZ dz N[ðrZ 0; z; EÞ for the BCS state is shown where the maximum value of the gap is D0 =Zur Z 4:7. It is seen that there is no precise energygap, because the presence of a confinement potential. The wavefunctions of the eigenmodes up to jEjZD0 are localized near the surface of the cloud. In the case of the FFLO state whose DOS is displayed in Fig. 3(b), the additional peaks appear at the mid energy-gap in addition to the above in-gap excitations. This mid-gap state has the strongly bounded wavefunction at the zeros of the pairing field, which is shown in Fig. 3(c). The mid-gap state for tip-spins is situated below the Fermi level, while the eigenenergy of that for down-spins is over the Fermi energy. The difference of the occupation results in the macroscopic peaks of the magnetization shown in Fig. 2. Such macroscopic signature for the FFLO state can be directly observed by using currently available techniques, e.g. Stern–Gerlach separation. On the other hand, the microscopic signature, such as the sharp peaks in N[ and NY which split by 2dm, may be observed by using stimulated Raman transition as an extra satellite. This technique is employed by Chin et al. [2], who identify the energy gap in Fermi condensates. The study on competing effect in between the mid-gap and surface excitations remains as the future problem. 5. Summary We have proposed ail experimental way oil resonance Fermion superfluid systems with unequal mixtures of two

Fig. 3. (a) The density of states (DOS) profiles N[ (rZ0, E) at rZ0 for D0/3FZ0.35 and the equal mixture dnZ0.0 at kB T=Zur Z 0:05 (the solid line) and 0.2 (the dashed line). (b) The DOS profiles for D0/3FZ0.35 and dnZ0.15 at kB T=Zur Z 0:05. The solid (dashed) line corresponds to the up-spin (down-spin) Fermions. (c) The local DOS of the up-spin N[ (rZ0, z, E).

species to achieve the FFLO state. We have briefly provided the analytical results of the BdG equation for an ideal onedimensional system. Then, we have presented the numerical results for the three-dimensional system. It has been found that the local magnetization exhibits sharp peaks at the zeros of the pairing field near the phase boundary between the uniform BCS and FFLO states. Such the macroscopic signature for the FFLO state can be observed by species-selective absorption, or by Stern–Gerlach separation and also the microscopic signature directly coming from the excitation spectrum may be observable by using stimulated Raman transition as an extra satellite.

References [1] [2] [3] [4] [5] [6] [7] [8]

M. Greiner, et al., Nature (London) 426 (2003) 537. C. Chin, et al., Science 305 (2004) 1128. M. Greiner et al., cond-mat/0407381. P. Fulde, A. Ferrell, Phys. Rev. 135 (1964) A550. A.I. Larkin, Y.N. Ovchinnikov, Sov. Phys. JETP 20 (1965) 762. K. Machicla, H. Nakanishi, Phys. Rev. B 30 (1984) 122. T. Machida, et al., Phys. Rev. Lett. 94 (2005) 060404. J. Stenger, et al., Nature (London) 396 (2003) 345.