# Light scattering in Cooper-paired Fermi atoms

###### Abstract

We present a detailed theoretical study of light scattering off superfluid trapped Fermi gas of atoms at zero temperature. We apply Nambu-Gorkov formalism of superconductivity to calculate the response function of superfluid gas due to stimulated light scattering taking into account the final state interactions. The polarization of light has been shown to play a significant role in response of Cooper-pairs in the presence of a magnetic field. Particularly important is a scheme of polarization-selective light scattering by either spin-component of the Cooper-pairs leading to the single-particle excitations of one spin-component only. These excitations have a threshold of where is the superfluid gap energy. Furthermore, polarization-selective light scattering allows for unequal energy and momentum transfer to the two partner atoms of a Cooper-pair. In the regime of low energy () and low momentum (, being the Fermi velocity) transfer, a small difference in momentum transfers to the two spin-components may be useful in exciting Bogoliubov-Anderson phonon mode. We present detailed results on the dynamic structure factor (DSF) deduced from the response function making use of generalized fluctuation-dissipation theorem. Model calculations using local density approximation for trapped superfluid Fermi gas shows that when the energy transfer is less than , where refers to the gap at the trap center, DSF as a function of energy transfer has reduced gradient compared to that of normal Fermi gas.

###### pacs:

03.75.Ss,74.20.-z,32.80.Lg## 1 Introduction

Cold atoms are of enormous research interest in current physics. The tremendous advancement in technology of cooling, trapping and manipulation [1] of atomic gases during 80’s and 90’s has enabled researchers to achieve a low temperature down to a few hundredth of a microKelvin. This led to the first realizations of Bose-Einstein condensation (BEC) [2] in dilute gases of ultracold bosonic atoms about a decade ago. Predicted in 1924 by Einstein [3] based quantum statistics of indistinguishable particles discovered by Bose [4], BEC in gaseous systems had long been thought a subject of mere academic pursuit beyond experimental reach because of the requirement of ultralow temperature which was unimaginable even two decades ago. The success in BEC is a breakthrough prompting researchers to look for experimental realizations of many other theoretical predictions of quantum physics using cold atoms. The most remarkable property of such atoms is the tunability of the atom-atom interaction over a wide range by an externally applied magnetic field or other means. This provides an unique opportunity to explore physics of interacting many-particle systems in a new parameter regime. In this context, the focus of attention has been now shifted to cold atoms obeying Fermi-Dirac statistics. Since fermions are the basic constituents of matter, research with Fermi atoms under controlled physical conditions has important implications in the entire spectrum of physical and chemical sciences. In particular, it has significant relevance in the field of superconductivity [5, 6].

The quantum degeneracy in an atomic Fermi gas was first realized by Jin’s group [7] in 1999. Since then, cold Fermi atoms have been in focus of research interest in physics today. In a series of experiments, several groups [8, 9, 10, 11, 12, 13] have demonstrated many new aspects of degenerate atomic Fermi gases. In a remarkable recent experiment, Ketterle’s group [14] has realized quantized vortices as a signature of Fermi superfluidity in a trapped atomic gas. Two groups-Innsbruck [15] and JILA [16] have independently reported the measurement of pairing gap in Fermi atoms. Furthermore, Duke and Innsbruck groups [17, 18] have measured collective oscillations which indicate the occurrence of superfluidity [19]. One of the key issues in this field is the crossover [20, 21, 22] between BCS state of atoms and BEC of molecules formed from Fermi atoms. Several groups have demonstrated BEC [23] of molecules formed from degenerate Fermi gas. There have been many other experimental [24] and theoretical investigations [25] revealing many intriguing aspects of interacting Fermi atoms.

The analysis of response of Cooper-paired Fermi atoms due to external perturbation (such as photon or rf field) is important for understanding the nature of atomic Fermi superfluid. A method has been suggested to use resonant light [26] to excite one of the spin components into an excited electronic state and thereby making an interface between normal and superfluid atoms. This is analogous to superconductive tunnelling which has a threshold equal to the gap energy . This has been recently implemented (albeit using rf field) [15, 27] to estimate gap energy. There have been several other proposals [26, 28] for probing pairing gap.

Our purpose here is to calculate response function of superfluid Fermi gas due to stimulated light scattering that does not cause any electronic excitation in the atoms. We particularly emphasize the role of light polarization in single-particle excitations which have a threshold . We present a scheme by which it is possible to have single-particle excitation in only one partner atom (of a particular hyperfine spin state) of a Cooper-pair using proper light polarizations in the presence of a magnetic field. This may lead to better precision in spin-selective time-of-flight detection of scattered atoms. Furthermore, spin-selective light scattering allows for unequal energy and momentum transfer into the two partner atoms of a Cooper-pair. This may be useful in exciting Bogoliubov-Anderson (BA) phonon mode of symmetry breaking by making small difference in momentum transfers received by the two partner atoms from the photon fields. A number of authors [29, 30, 31] have theoretically investigated Bogoliubov-Anderson (BA) mode [32, 33, 34] in fermionic atoms as a signature of superfluidity. BA mode constitutes a distinctive feature of superfluidity in neutral Fermi systems since it is associated with long wave Cooper-pair density fluctuations. However, experimental detection of this mode is a challenging problem.

We present a detailed theoretical analysis of the response function of Cooper-paired atoms at zero temperature due to light scattering. The stimulated light scattering we discuss here is similar to Bragg spectroscopy used by Ketterle’s group for measuring structure factor of an atomic BEC [35]. The response function we derive is applicable for most general case of polarization-selective single-particle excitations for unequal (or equal) momentum as well as energy transfers to the two partner atoms of a Cooper-pair. We develop the theoretical framework for stimulated light scattering off Cooper-paired Fermi atoms following the method used for describing Raman scattering in superconductors [36, 37]. We use standard Nambu-Gorkov formalism of superconductivity [39, 38] to calculate the response function taking into account the vertex correction due to final state interactions. We deduce dynamic structure factor (DSF) from the response function applying generalized fluctuation-dissipation theorem. We present detailed analytical and numerical results of our calculation of DSF of trapped superfluid Fermi gas of atoms using local density approximation. The inhomogeneity of trapped gas has a role in distinguishing the DSF of superfluid gas from that of normal gas. When the energy transfer is smaller than where is the gap at the trap center, the DSF of superfluid gas as a function of energy transfer shows much reduced gradient in comparison to that of normal gas. This is because of the fact that the gap has an inhomogeneous distribution gradually vanishing at the edge of the trap.

The paper is organized in the following way. In the following two sections, we define bare vertex in light scattering and response function, respectively. In the fourth section, we discuss stimulated light scattering in two-component Li Fermi atoms in the presence of a magnetic field. We next describe in detail the method of vertex correction in light scattering off Cooper-paired Fermi atoms. In the sixth section, we discuss our analytical results followed by description on numerical results in the seventh section, and then we conclude.

## 2 Bare vertex in light scattering

To begin with, let us consider an elementary process of photon scattering by a neutral atom. Let the atom’s initial and scattered electronic state be denoted by and , respectively. The frequencies of the incident and scattered photon are represented by and , respectively. According to second order perturbation theory, the strength of scattering is given by Kramers-Heisenberg formula [40]

(1) |

where denotes all the intermediate atomic states that can be coupled to the initial and final atomic states and by the incident and scattered photon fields. Here and are the momentum and mass of the valence electron of atom, denotes the polarization state of the incident (scattered) photon, is the atomic frequency between the states and . The atomic transition () probability and the differential scattering cross section of photons is proportional to [40]. It should be mentioned that does not depend on the momentum transfer associated with the scattering, but it is sensitive to light polarization directions. Let us now consider the particular case: that is, before and after the scattering, the atom remains in the same electronic state. Then, making use of the completeness of the intermediate states , one can rewrite the term as [40]

(2) |

Further, let us assume , that is, the incident as well as scattered light fields are in near resonance with the atomic frequency. In such a case, the second term within the third bracket on the right hand side (RHS) of Eq. (1) is much smaller than the first term, because energy denominator of the second term is of the order of optical frequency while that of the first term can be chosen to be smaller by several orders of magnitude. Thus, neglecting the second term, the bare vertex can be written as [40]

(3) |

Next, using electric-dipole approximation and the fact , one can express

(4) |

where is the transition dipole moment between the states and , is the electric field and

(5) |

with being the number of incident photons.

## 3 The response function

To define response function of fermionic atoms due to an applied laser field, we use the second-quantized operator which describes the annihilation(creation) of an atom with hyperfine spin and center-of-mass momentum . These operators satisfy fermionic algebra. The effective atom-field hamiltonian is , where

(6) |

with being the frequency-difference between incident and scattered photons. We assume that, except the center-of-mass momentum, the spin or any other internal degrees of atom does not change due to light scattering. By treating light fields classically, the effective interaction hamiltonian can then be written as

(7) |

where is the momentum transferred to the atom due to photon scattering and represents the bare vertex corresponding to the ground hyperfine spin magnetic quantum number .

Now, one can define the density operators by and

(8) |

One can identify the operator as the Fourier transform of the density operator in real space. The scattering probability of incident particles (photons in the present context) is related to the response or susceptibility

(9) |

of the target system by which the incident particles are scattered. Here means thermal averaging and is the complex time ordering operator. The Fourier transform of this susceptibility is

(10) |

where is the temperature and is the Matsubara frequency with being an integer. The scattering cross section is proportional to the generalized dynamic structure factor which can be obtained by the generalized fluctuation-dissipation theorem through the analytic continuation of as

(11) |

We define the following polarization matrix element:

(12) |

where . The polarization bubble is nothing but the susceptibility of Eq. (9). The dynamic structure factor is thus related to this polarization term by fluctuation-dissipation relation as expressed in Eq. (11). The spectrum of density fluctuation is proportional to the dynamic structure factor which can also be defined as the Fourier transform of the two-time density-density correlation function.

## 4 stimulated light scattering in two-component Fermi atoms

We would like to study stimulated light scattering in two-component Fermi atoms. In particular, we consider trapped Li Fermi atoms in their two lowest hyperfine spin states and . For simplicity, the number of atoms in each spin component is assumed to be the same. However, a mismatch in number densities of the two spin components may lead to interior gap superfluidity [41, 42] in a Fermi gas of atoms. An applied magnetic field tuned near the Feshbach resonance ( Gauss) results in splitting between the two spin states by MHz [43], while the corresponding splitting between the excited states and is MHz [10].

Figure 1 shows the schematic level diagram for stimulated light scattering by two-component Li atoms. Two off-resonant laser beams with a small frequency difference are impinged on atoms, the scattering of one laser photon is stimulated by the other photon. In this process, one laser photon is annihilated and reappeared as a scattered photon propagating along the other laser beam. The magnitude of momentum transfer is , where is the angle between the two beams and is the momentum of a laser photon. Let both the laser beams be polarized and tuned near the transition . Then the transition between the states and would be forbidden while the transition will be suppressed due to the large detuning MHz. This leads to a situation where the scattered atoms remain in the same initial internal state . Similarly, atoms in state only suffer scattering when two polarized lasers are tuned near the transition . Thus, it is possible to scatter atoms selectively of either spin components using circularly polarized lasers in the presence of magnetic field. Under such conditions, considering a uniform gas of atoms, the effective laser-atom interaction Hamiltonian in electric-dipole approximation can be written as

(13) |

If refers to then

(14) |

where is the transition dipole matrix element between the ground and the excited states. Similarly, if is then the subscript “22” should be replaced by “11”. In writing the above vertex term, we have also assumed that both the laser beams are of almost equal intensity. For both the laser beams having polarization tuned near as in Fig. 1, one finds . On the other hand, in the absence of magnetic field (or in the presence of a weak magnetic field), the hyperfine magnetic sub-levels of the ground and excited state would be degenerate (or nearly degenerate). In such a case, irrespective of whether both the laser beams are unpolarized or equally polarized, we have .

## 5 light scattering in Cooper-paired Fermi atoms: Vertex correction

To study light scattering in Cooper-paired Fermi atoms, we apply Nambu-Gorkov formalism that uses the four Pauli matrices

(15) |

The vertex equation is [44]

(16) |

where and is the energy-momentum 4-vector whose components are and . In pairing approximation, the Green function can be expressed in a matrix form as

(17) |

where and with . The bare vertex

(18) |

Using Pauli matrices and , this can be rewritten as

(19) |

where and . The susceptibility is given by

(20) |

### 5.1 vertex equation and its solution

To solve the vertex equation, let us expand the vertex function in terms of Pauli matrices as

(21) |

Using Eqs. (17) and (21) in Eq. (16), we can write

(22) |

where and

In writing the above equations, we have assumed . Further, we can write , where , and

Before performing the integration of Eq. (22), we note that the dominant contribution to the integral comes from -values near , that is, . Hence we can approximate

(23) |

where denotes the third component of energy-momentum 4-vector . If the potential is separable in two variables and , then Eq. (16) is analytically solvable. Let us, for simplicity, replace by the well-known mean field potential (where ) which is expressed in terms of s-wave scattering length . By doing so, we are basically considering the weak-coupling case. However, within mean-field approximation the strong-coupling limit may be accessed by first renormalizing the BCS mean-filed interaction and then taking the limit as will be discussed later.

With the assumption of a -independent gap , the double integrations on and then resemble to those appearing in relativistic equations in QED and so can be carried out analytically by Feynman’s method [45]. The angular integration is left to the last. There are basically two types of integrals:

(24) |

(25) |

These integrals are explicitly calculated in Ref. [46] using Feynman’s method of parametrization. For completeness, we here reproduce the method of calculation. The terms which are odd in will not contribute to the integration and so those terms can be omitted. Substituting where is a parameter varying between 0 to 1, the integral of Eq. (24) can be reexpressed as

(26) |

The -integration can be carried out by residue method of complex integration. The pole is , where . Since has infinitesimally negative imaginary part, the pole lies in the lower half of the real axis. The residue is . After performing - and -integration , one obtains the result , where

(27) | |||

(28) |

The -integration in Eq. (26) is divergent, therefore a cut-off frequency is required as the upper limit of integration. After having performed the integration, the vertex terms can be expressed as

(29) | |||||

(30) |

(31) | |||||

(32) | |||||

Since is decoupled from all other vertex terms including the bare ones (), we can set . Using the expansion of Eq. (21), the susceptibility can be written as

(33) | |||||

We note that the dressed part of is proportional to the momentum transfer , therefore we have in the low momentum transfer regime, that is, for , where is the BCS coherence length. Introducing the variable , where is the angle between and , we can drop all the terms odd in in the above equations, since upon integration over those terms vanish. Thus also becomes decoupled while and form only two coupled equations which can be analytically solved.

### 5.2 gap equation

The gap equation can be obtained from Eq. (31) by setting and equal to zero and replacing by the gap parameter . The resulting equation reads

(34) |

The cut-off frequency has been introduced ad-hoc to tackle the divergence problem for the time being. This needs to be eliminated by the method of regularization. To this end, we here recall that in carrying out the various momentum integration, we made an approximation: the integration was restricted near the chemical potential (which is nearly equal to Fermi energy in the weak coupling regime). To restore the actual gap equation, we here remove this approximation and let and thus obtain

(35) |

The gap defined by this equation is however, divergent. To remove this divergence, we define regularized mean-field coupling by subtracting from the right hand side of Eq. (35) the zero field contribution (i.e., and ). The resulting gap equation is

(36) |

which yields convergent results. In the weak-coupling regime (), . The strong-coupling regime () may be accessed by simultaneously solving for the interacting chemical potential from the single-spin BCS number-density equation

(37) |

This approach of solving the regularized gap plus the number equation to access strong-coupling regime within the simple mean-field framework fails to account for pairing fluctuation effects which are particularly significant near in the strong-coupling regime. However, far below , the correction due to the pairing fluctuation is very small as shown in Ref. [21]. The two coupled Eqs. (36) and (37) admit analytical solutions which are obtained by Marini et al. [47] for the entire range of the parameter starting from weak interaction () to the unitarity limit (). In the unitarity limit, the solutions provide and . For convenience in solving the two coupled equations numerically, we rewrite the equations in terms of the two dimensionless scaled variables and as

(38) |

(39) |

where . We have set . Calling the right hand side of Eqs. (38) and (39) as and , respectively; eliminating from both the equations, we obtain

(40) |

For given values of the parameters and , the Eq. (40) can be solved for . Then substituting this solution into Eq. (38), one evaluates and so also the gap

### 5.3 solutions

Now, to write down the solutions of the various vertex terms and the susceptibility is straightforward. Let , where represents the single particle density of states near the chemical potential. The various vertex terms can be expressed as

(41) |

(42) |

and

(43) |

Here

(44) |

(45) |

and

(46) |

The symbol implies average of a function over the chemical potential surface: , since is an even function of , we have . Making use of these vertex terms, the susceptibility can be written as

(47) |

We drop the second term inside the third bracket which leads to small corrections due to Landau-liquid-like behavior without adding any significant qualitative effect. Further, for , we have

(48) |

## 6 dynamic structure factor

The dynamic structure factor is obtained from the response function via analytic continuation of energy . By means of generalized fluctuation-dissipation theorem as embodied in Eq. (11), in the zero temperature limit the dynamic structure factor is related to the imaginary part of the density response function via analytic continuation of energy as

(49) |

The key function here is of Eq. (27), where is given in Eq. (28). As , . We have the following analytic properties of :

(50) |

where

(51) |

## 7 Analytical results and discussions

Equation (52) gives an expression for dynamic structure factor of a homogeneous Fermi superfluid when the excitations are of single-particle type for the parameters satisfying . Different amount of energy transfers (or excitations) to the two constituent partners of a broken Cooper-pair can be made by appropriately selecting the polarization states of the exciting two laser beams and tuning their frequency from the excited atomic state in the presence of a magnetic field. This fact is taken into account in the expression of (52), because any nonzero value of the term means unequal excitation of the two partners. For instance, two extreme cases can be mentioned: Case-I: For unpolarized light in the absence of magnetic field, equal amount of energy transfer occurs to the two partners resulting in ; Case-II: On the other hand, for circularly polarized light in the presence of strong magnetic field, we have meaning only either partner can be excited. We will present our numerical results for these two extreme cases. To compare our results with the known results for normal Fermi system in the limit , we will use in Case-I the limit meaning and . In Case-II, we will use the limit implying that . Intuitively, one may understand that the Case-II would be significantly different from Case-I both qualitatively and quantitatively. In the Case-II, upon receiving an energy () from an incident photon, one partner of a Cooper-pair moves out of the Fermi sphere, while the other partner remains within the Fermi sphere. Let us consider an elementary process of single photon scattering by a Cooper-pair. Suppose, the Cooper-pair consists of an atom A having spin and momentum and another atom B with spin and momentum . When this Cooper-pair is broken due to stimulated scattering of polarized photon in a situation like Case-II, atom A will move out of the Fermi surface as an excited quasi-particle with momentum with certain probability given by BCS correlation and atom B will have certain probability of remaining within the Fermi sphere moving as a quasi-particle with momentum . Thus, only one partner of the Cooper-pair will contribute to the intensity of scattered atoms reducing the strength of the density fluctuation spectrum compared to that of Case-I. However, there could be some advantage in detecting the scattered atoms in Case-II by spin-selective time-of-flight measurement technique as we will discuss later in the concluding section.

### 7.1 Case-I: Leading approximations

In this case, we have . In the limit ,

(55) |

which is given by Eq. (54). For , in the leading approximation in terms of , this reduces to the form

(56) |

which is devoid of any vertex correction. The same expression can be derived by taking , and meaning that we use bare vertex only. This is also obtainable from the static BCS- Bogoliubov mean-filed treatment as shown in the appendix. Because of the absence of vertex correction, it violates the Ward identities [48] that guarantee the conservation of total particle number and the obeyance of the continuity equation.

To perform the integration over in Eq. (56), it is convenient to change the variable into

(57) |

The condition implies . Then the Eq. (56) can be expressed as

where and

(59) |

For , we have and the result is

(60) |

where is the complete elliptic integral. Note that in the limit reduces to the form which is same as that of a normal quantum fluid of noninteracting quasi-particles within the energy range [49]. The dynamic structure factor reaches a maximum at . As increases above , decreases below unity and hence the integral in Eq. (LABEL:leadint) decreases.

In view of the forgoing analysis, we now verify how far f-sum rule is fulfilled by the dynamic structure factor as given by Eq. (56). To this end, we separate the integral over energy in the sum rule