# Universal relations and normal-state properties of a Fermi gas with laser-dressed mixed-partial-wave interactions

###### Abstract

In a recent experiment [P. Peng, , Phys. Rev. A 97, 012702 (2018)], it has been shown that the -wave Feshbach resonance can be shifted toward the -wave Feshbach resonance by a laser field. Based on this experiment, we study the universal relations and the normal-state properties in an ultracold Fermi gas with coexisting - and -wave interactions under optical control of a -wave magnetic Feshbach resonance. Within the operator-product expansion, we derive the high-momentum tail of various observable quantities in terms of contacts. We find that the high-momentum tail becomes anisotropic. Adopting the quantum virial expansion, we calculate the normal-state contacts with and without a laser field for K atoms using typical experimental parameters. We show that the contacts are dependent on the laser dressing. We also reveal the interplay of laser dressing and different partial-wave interactions on various contacts. In particular, we demonstrate that the impact of the laser dressing in the -wave channel can be probed by measuring the -wave contacts, which is a direct manifestation of few-body effects on the many-body level. Our results can be readily checked experimentally.

## I Introduction

The interplay of - and -wave interactions can introduce interesting many-body physics in ultracold Fermi gases Zhangexp2017 ; Zhoulihong2017 ; Yi2016 ; Yang2016 ; Jiang2016 ; Hu2016 . Such a scenario exists in the two-component K Fermi gases, where the -wave Feshbach resonances near G are close to the wide -wave Feshbach resonance near G exp2002s ; exp2004s ; exp2003p ; review2010 ; exp2004p . In the previous studies, it has been shown that mixed-partial-wave interactions in such a system can give rise to fermion superfluid with hybridized - and -wave pairing Zhoulihong2017 , as well as the interesting normal-state properties exhibiting the interplay of - and -wave interactions Yi2016 . For a low-dimensional two-component K Fermi gas, the overlap of - and -wave interactions can be tuned by using confinement-induced resonance, which would favor the elusive itinerant ferromagnetism in certain parameter regimes Yang2016 ; Jiang2016 ; Hu2016 .

In a recent experiment Zhangexp2017 , the -wave Feshbach resonance with the magnetic quantum number is shifted to overlap with the -wave resonance via laser dressing (see Fig. 1). As illustrated in Fig. 1, a laser field is applied to couple the bound-to-bound transition between the -wave closed-channel molecular states with and an excited state . While the energy shift is different for molecular states with different magnetic quantum numbers , the -wave Feshbach resonances associated with these closed-channel molecular states are also shifted. The experiment thus offers an additional control on mixed-partial-wave interactions of the system, which is bound to give rise to the interesting many-body physics. As a first attempt at clarifying the impact of few-body physics on many-body properties of the system, we study the universal relations and normal-state properties in an ultracold Fermi gas with coexisting - and -wave interactions near a laser-dressed -wave Feshbach resonance. We expect the interplay between laser dressing and mixed-partial-wave interactions to have interesting effects on the many-body level, such that contact in the -wave sector should be affected by the laser field as well. This is especially interesting as the - and -wave scattering channels are decoupled on the two-body level.

In dilute atomic gases with short-range interaction potentials, it has been shown that universal behaviors emerge in the large-momentum limit. Physically, this is because when two atoms get close, the short-distance many-body wave function reduces to the two-body solution, yielding universal relations, as first studied by Tan for a three-dimensional two-component Fermi gas near an -wave Feshbach resonance Tan20081 ; Tan20082 ; Tan20083 . As a result of the universality, observable thermodynamic quantities such as the high-momentum tail of the momentum distribution, the radio-frequency (rf) spectrum, the pressure, and the energy are connected by a set of key parameters called contacts Tan20081 ; Tan20082 ; Tan20083 ; Zhang2009 ; exp_contact1 ; exp_contact2 ; exp_contact3 . Recently, much effort has been devoted to the study of universal relations under a synthetic gauge field Peng2017 ; Jie2018 ; Zhang2018 , with Raman-dressed Feshbach resonance Yi2018 , in low-dimensional atomic gases Zwerger2011 ; Klumper2017 ; Castin20121 ; Castin20122 ; Valiente2011 ; Valiente2012 ; Zhou2017 ; Peng20162 ; Zhang2017 ; Yin2018 ; Cui20161 , in high-partial-wave quantum gases Cui20162 ; Yu2015 ; Yu2015exp ; Zhou20161 ; Yoshida2015 ; Ueda2016 ; Peng20161 , and in terms of contact matrices Zhou20162 and tensors Yoshida2016 . In particular, the universal relations for the -wave Fermi gases have already been experimentally verified Yu2015exp .

In this work, adopting the operator-product expansion (OPE) approach Cui20161 ; Cui20162 ; Zhang2018 ; Yi2018 ; Wilson ; Kadanoff ; Braaten20081 ; Braaten20082 ; Braaten20083 ; Platter2016 ; Yu20171 ; Yu20172 ; Yu20173 ; Qi2016 , we derive universal relations of the system with laser-dressed hybrid interactions. We show that the leading-order terms in high-momentum tails of the momentum distribution can be expressed by five contacts, with one laser-field-dependent open-channel contact and four laser-field-dependent closed-channel contacts. Interestingly, one of the contacts is anisotropic, and the high-momentum tail in the momentum distribution shows anisotropic features. Notice that the anisotropy here is not due to the laser dressing, which does not induce a momentum transfer. Rather, it comes from the anisotropy of the many-body system, due to either an anisotropic environment or spontaneous symmetry breaking Cui20162 .

The comparisons between our results and those in previous studies are as follows:

First, for one-dimensional pure -wave Fermi gases in a two-channel model of the previous work Cui20162 , there are four contacts which are similar to our results. We will point out this in the end of the first paragraph below Eq. (41). In Fermi gases with -wave interactions, the leading-order term in the high-momentum tail should also feature a contact Zwerger2011 . Therefore, when one considers a system with mixed-- and -wave interactions, there should be five contacts in the lead-order terms. In a previous study Yi2016 , we also considered contacts in a system with mixed-- and -wave interactions, but in the absence of laser dressing. The discrepancy lies in the fact that we have previously neglected the anisotropy associated with a finite center-of-mass momentum.

Second, anisotropy in contacts can be induced either by the anisotropic interactions, such as the -wave interaction or by finite center-of-mass momentum. The former has been discussed in previous works Yu2015 ; Yu2015exp ; Zhou20161 ; Yoshida2015 ; Ueda2016 ; Peng20161 , under which contacts associated with different magnetic quantum numbers behave differently. The latter has been discussed, for example, in Ref. Cui20162 , where the contact in the tail ( is the relative momentum) is anisotropic, and the other contacts are isotropic. Specifically, what we mainly focus on here is the anisotropic feature induced by the center-of-mass momentum. For three-dimensional pure -wave Fermi gases in Refs. Yu2015 ; Yu2015exp , they assumed that the distribution of center-of-mass momentum is isotropic, so that the contacts which they obtained do not show an anisotropic feature induced by the center-of-mass momentum and they did not have the tail. Reference Zhou20161 just considered the zero center-of-mass-momentum case, which does not show an anisotropic feature induced by the center-of-mass momentum. Reference Yoshida2015 calculated only the leading order term ( tail) of the high-momentum distribution in three-dimensional pure -wave Fermi gases, and it does not show an anisotropic feature induced by the center-of-mass momentum.

We then calculate the normal-state contacts and spectral function using the quantum virial expansion, both with and without the laser field for K atoms using typical experimental parameters. We find that, with the addition of the laser dressing in the closed channel of the -wave interaction, the -wave contact significantly decreases around the -wave Feshbach resonance. Such a behavior is a direct manifestation of few-body effects on the many-body level and is useful for detecting the impact of dressing lasers on the system. Furthermore, the interplay of laser dressing and -wave interaction leads to a much larger -wave contact than the one without a laser. Additionally, we show the -wave contacts decrease very rapidly in the Bose-Einstein condensation (BEC) limit under the influence of -wave interaction, which is due to the interplay of - and -wave interactions on the many-body level as discussed in Ref. Yi2016 . Our results can be readily checked under current experimental conditions.

The paper is organized as follows: In Sec. II, we give the model Lagrangian density to describe the two-component ultracold Fermi gas with laser coupling. In Sec. III, we present a brief derivation of the renormalization of bare interactions. In Sec. IV, we calculate the high-momentum distribution of this system within the OPE quantum field method. In Sec. V, we derive the corresponding universal relations such as high-frequency rf spectroscopy, adiabatic relations, pressure relations, and virial theorem for this system. In Sec. VI, we present the formalism of the quantum virial expansion and express the contacts and the spectral function in the normal state up to the second order. In Sec. VII, we numerically evaluate the high-temperature contacts and spectral functions. We summarize in Sec. VIII.

## Ii Model

We consider a two-component Femi gas close to an -wave Feshbach resonance. One of the spin components is also close to a laser-dressed -wave Feshbach resonance, as illustrated in Fig. 1. Physically, the closed-channel molecular states with different should feel a different laser-induced energy shift, which would lead to a state-dependent shift in the corresponding self-energies of the system Lagrangian. The local Lagrangian density (at coordinate ) is given by , where Zhangexp2017

(1) | |||||

(2) | |||||

(3) |

Here the self-energy in coordinate space is HePRL2018

(4) |

() denotes the open-channel fermionic atom-field operator, denotes the field operator for the closed-channel molecule in ground state with the magnetic quantum number , and denotes the direction of spin polarization. are the coefficients when transforming to the -wave spherical harmonics , which satisfies . Therefore, , ; , , . is the center-of-mass coordinate, is the time, is the atom mass, is the -wave bare coupling between two fermionic atoms, is the -wave bare coupling between two fermionic atoms and a bosonic molecule, and is the bare magnetic detuning. The difference in the energy levels of atoms and excited molecules is denoted by . is the strength of the effective laser-induced coupling between the molecular ground state and excited state . is the detuning of the laser light with respect to the energy difference between the ground and excited states of molecules. is the frequency of the laser light, and is the energy difference between the ground and excited states of molecules. The spontaneous decay of the excited molecular state is treated phenomenologically by a decay rate . The natural units will be used throughout the paper.

Accordingly, we can write the Hamiltonian in momentum space from the Lagrangian by the Legendre and Fourier transformations

(6) | |||

(7) | |||

(8) | |||

(9) |

where the self-energy in momentum space is

(10) |

() is the annihilation (creation) field operator of Fermi atom in momentum space, () is the annihilation (creation) field operator of ground-state bosonic molecule in momentum space, is the center-of-mass momentum, is the total incoming energy, is the volume of the system, is the fermionic chemical potential with spin , and the particle numbers are given by and .

## Iii Interaction renormalization

In this section, we will renormalize the bare interactions in the - and -wave channels, respectively. On the two-body level, different partial-wave scattering channels are decoupled. Therefore, the renormalization can be performed independent for these two cases.

### iii.1 wave

In -wave case, we consider zero total momentum for each pairing state, so that an incoming state can be set as with two fermions of different species having momentum and to an outgoing state with two fermions having momentum and .

As shown in Fig. 2, the two-body matrix for the -wave interaction is given by renormalization2007

(11) |

where the polarization bubble for wave is

(12) | |||||

The -wave scattering length is given by

(13) |

where is an ultraviolet momentum cutoff.

Further, we get the renormalization relation

(14) |

### iii.2 wave

We consider an incoming state with two fermions of different species having momentum and to an outgoing state with two fermions having momentum and .

As shown in Fig. 3, the two-body matrix for -wave interaction is given by renormalization2007 ; renormalization2012 ; amplitude2010pwave ; Yao20181 ; Yao20182

(15) | |||||

where the factor 2 in front of comes from the scattering of two identical fermions Yu2015exp ; renormalization2007 , the bare molecule propagator is

(16) |

the polarization bubble is

(17) | |||||

and the full molecule propagator satisfies

(18) |

In the absence of optical field, i.e., , the -wave scattering amplitude is given by

(19) | |||||

where is the -wave scattering volume and is the -wave effective range. Further, we have the renormalization relations review2010 ; exp2004p ; Yu2015exp

(20) | |||||

(21) |

where and are renormalized in the form of

(22) | |||

(23) |

In the presence of optical field, the -wave scattering volume is

and the -wave effective range is

Notice that, in the section of numerical calculations, we use a large detuning, i.e., Zhangexp2017 .

## Iv Momentum distribution

In this section, we study the tail of the momentum distribution for fermions with coexisting - and -wave interactions near a laser-dressed -wave Feshbach resonance using the quantum field method of OPE Cui20161 ; Cui20162 ; Zhang2018 ; Yi2018 ; Wilson ; Kadanoff ; Braaten20081 ; Braaten20082 ; Braaten20083 ; Platter2016 ; Yu20171 ; Yu20172 ; Yu20173 ; Qi2016 .

OPE is an ideal tool to explore short-range physics. Furthermore, OPE is an operator relation that the product of two operators at small separation can be expanded in terms of the separation distance and operators, which can be interpreted as a Taylor expansion for the matrix elements of an operator. Therefore, one can expand the product of two operators as

(26) |

where are the local operators and are the short-distance coefficients. can be determined by calculating the matrix elements of the operators on both sides of Eq. (26) in the two-body state for -wave interaction and for -wave interaction.

By using the Fourier transformation on both sides of Eq. (26), we have the expression of momentum distribution Braaten20082

(27) |

where is the relative momentum.

In the following subsections, we will show the calculations for the momentum distribution , for instance.

### iv.1 -wave channel

As shown in Figs. 4(a)-4(d), there are four types of diagrams which can be used to denote the operators on the left-hand side of OPE equation (26). However, the only nonanalyticity comes from the diagram as shown in Fig. 4(d). Therefore, we can evaluate the diagram in Fig. 4(d) as

(28) | |||||

### iv.2 -wave channel

Similar to the case of -wave interaction, we can evaluate the diagram in Fig. 6(d) as

(31) |

where we average over the direction of as an approximation, is the total incoming energy, are the spherical Bessel functions, and are the Legendre polynomials.

To match the nonanalytic terms in Eq. (IV.2), we calculate the expectation values of the molecule operator as shown in Fig. 7:

(32) | |||||

Therefore, we get

(33) |

(34) |

(35) |

### iv.3 Coexistence of - and -wave channels

Matching Eq. (28) and Eq. (IV.2) with Eq. (30) and Eqs. (32)-(35), we get the momentum distribution in the large -limit ( with the total number density and the interaction range)

(36) |

where the corresponding contacts are defined as

(37) | |||

(38) | |||

(39) | |||

(40) | |||

(41) |

Notice that the distribution of here is anisotropic. Therefore, we find that is anisotropic and the tail and part of the tail of the momentum distribution Eq. (IV.3) show anisotropic behaviors of center-of-mass momentum . Especially in the previous studies, it has been shown that the contacts of a similar nature to in tail and in tail also exist for one-dimensional -wave Fermi gases Cui20162 .

Note that, if we consider finite total momentum in the -wave case, the center-of-mass-momentum-related contacts will appear in the and tails of the high-momentum distribution Yi2018 , where is the relative momentum. However, we calculate only the high momentum tails up to , so that the and tails are not included here. Therefore, we consider only zero total momentum in the -wave case.

We emphasize that, in our system, the center-of-mass momentum induces the anisotropic behavior of the contact. Here, only in tail and in tail are both center-of-mass-momentum and laser dependent, but only is anisotropic. The other three contacts are laser dependent but not center-of-mass-momentum dependent.

As the adiabatic relations shown in the next section, , and are associated to the inverse of -wave scattering length, the inverse of -wave scattering volume, and the inverse of -wave effective range. The last two contacts and are related to the velocity and the kinetic energy of the closed-channel molecules, respectively.

A simple physical picture to describe the anisotropic behavior of the contact is as follows. The contact is a many-body physical quantity which bridges the few- and many-body physics. Therefore, the anisotropic behavior of the contact is dependent both on the center-of-mass momentum and the anisotropy of the many-body wave function of the system. For example, a finite anisotropic contact can be probed in the Fulde-Ferrell state which supports a finite-momentum two-body bound state and pairing superfluidity Yi2018 . Specifically, the anisotropic contact induced by the center-of-mass momentum appears only in the subleading tails of the pure - and pure -wave high-momentum distributions, respectively.

We should clarify that, in this system, the center-of-mass momentum is a good quantum number and the Galilean invariance is not broken. Therefore, the high-momentum tail in the presence of the finite center-of-mass momentum can be obtained by doing a frame transformation from the center-of-mass frame. For concreteness, we explicitly assume that the system has a distribution of the center-of-mass momentum, which accounts for the anisotropy in the high-momentum tail.

For simplicity, in our model, we assume that the -wave interaction exists only between two spin-up fermions, and there is no interaction between two spin-down fermions. Therefore, the momentum distribution for the spin-down fermions has only the -wave contact.

(42) |

## V Universal relations

In this section, we derive the corresponding universal relations.

### v.1 High-frequency radio-frequency spectroscopy

The rf spectroscopy can be used as an important experimental tool to detect the contacts Yu2015 ; rf2010s-wave1 ; rf2010s-wave2 ; Hofmann2011 ; Thompson2010 . The high-frequency tails of the rf spectroscopy are governed by contacts. The rf with frequency is applied to transfers fermions from the internal spin state () into a third spin state . The resultant number of the atoms transferred to state is proportional to the transition rate, which is given by rf2010s-wave2 ; Hofmann2011

(43) |

where is the rf Rabi frequency determined by the strength of the rf signal, , and is the time ordering operator.