Zehan Li(李澤漢) Jian-Song Pan and W Vincent Liu

1Department of Physics and Astronomy,University of Pittsburgh,Pittsburgh,PA 15260,USA

2Wilczek Quantum Center,School of Physics and Astronomy and T.D.Lee Institute,Shanghai Jiao Tong University,Shanghai 200240,China

Keywords: d-wave interaction,quantum gas,Bogoliubov spectrum,superfluid

1. Introduction

Orbital high-partial-wave interacting quantum gases[1]have steadily attracted research interest,due to their potential to show exotic superfluidity. For example, d-wave interacting Fermi gases may be compared with a d-wave superfluid.Recently, d-wave scattering resonance has been observed in more and more ultracold atomic gases.[2–6]In particular, the observation of degenerate d-wave-interacting Bose gases with d-wave shape resonance[5]makes the hidden d-wave manybody correlation more experimentally accessible.

Unlike s-wave interactions, the closed channels of highpartial-wave Feshbach resonance carry finite momentum. For example, the closed channels of the p-wave Feshbach resonance carry a total angular momentum of 1and the interaction term is proportional to the momentumk. It is predicted that a finite-momentum superfluid will emerge in a p-wave interacting Bose gas.[7–9]The closed channels of d-wave Feshbach resonance carry a total angular momentum of 2and hence the many-body form is proportional to the square of the momentumk2. Although d-wave electronic Fermi superconductors have been extensively studied in condensed-matter physics, to the best of our knowledge, determination of the possible many-body states that a d-wave interacting atomic Bose gas can exhibit is an open question.

Inspired by recent experimental progress,[3,5,6]in this paper, we analyze the zero-temperature mean-field ground state and Bogoliubov spectrum of a d-wave interacting Bose gas.A two-channel model is adopted for a mixture of two components interacting via the d-wave interaction. Similarly to the p-wave interacting Bose gas,[7–9]the mean-field ground state typically has three quantum phases: atomic superfluid(ASF),molecular superfluid(MSF),and atomic–molecular superfluid(AMSF).However,unlike the p-wave case,the atomic superfluid does not carry finite momentum. The phase boundaries are analytically obtained. Furthermore, the Bogoliubov excitation spectrum is analyzed both numerically and analytically above the superfluid ground state with d-orbital aspects.

2. Model

3. Mean-field theory

We will obtain the Landau free energy by applying meanfield theory to our model and minimize it to establish the phase diagram and analyze the phase transition. This method is equivalent to solving the Gross–Pitaevskii equation. Replacing the atomic and molecular field operators with their relative classical order parametersΨσ,Φm,we obtain the Landau freeenergy functionF[Ψσ,Φm]=〈H〉.

We decompose our mean-field parameters to characterize the states of the system.For the atomic condensatesΨ1andΨ2,let us use a Fourier transform and make these fields complex periodic functions characterized by momentaQn,

3.1. Atomic superfluid phase

For large positive detuningν ＞0, the atomic channels have lower energy and the ground state is a molecule vacuum.The free energy is minimized by spatially uniform atomic order parameters[16]and leads to a free energy density of the form

Fig.1. Mean-field phase diagram of a d-wave resonant two-component Bose gas with a large positive detuning and 4λ11λ22 ?(λ12+λ21)2 ＞0.The atomic channels have lower energy.ASF1 and ASF2 refer to singleatom species’ superfluid states, and ASF12 refers to a double-atom species’s superfluid state.

Table 1. Sub-phases of the ASF phase. (i)Whenμ1 andμ2 are negative,both atomic species are in the normal(N)phase. (ii)Whenμ1＞0,μ2 ＜,atom1 formsacondensate.(iii) When , μ2 ＞0, atom 2 a forms condensate. (iv) When bothatom speciesformcondensates.

Table 1. Sub-phases of the ASF phase. (i)Whenμ1 andμ2 are negative,both atomic species are in the normal(N)phase. (ii)Whenμ1＞0,μ2 ＜,atom1 formsacondensate.(iii) When , μ2 ＞0, atom 2 a forms condensate. (iv) When bothatom speciesformcondensates.

Phase Chemical potentials Ψ1 Ψ2μ1 ＜0, μ2 ＜0 0 0 ASF1 μ1 ＞0, μ2 ＜λ12+λ21 Nμ1 ASF2 μ1 ＜λ12+λ21 2λ11 images/BZ_540_1402_527_1436_560.png μ1 λ11 0 images/BZ_540_1929_608_1962_641.png μ2 λ22 ASF12 μ1 ＞λ12+λ21 2λ22μ2, μ2 ＞0 0 2λ22μ2, μ2 ＞λ12+λ21 2λ11μ1 images/BZ_540_1258_689_1291_722.png images/BZ_540_1784_689_1818_722.png 4λ22μ1 ?2(λ12+λ21)μ2 4λ11μ2 ?2(λ12+λ21)μ1 4λ11λ22 ?(λ12+λ21)2 4λ11λ22 ?(λ12+λ21)2

Fig.2. Mean-field phase diagram of a d-wave resonant two-component Bose gas with large positive detuning and 4λ11λ22 ?(λ12+λ21)2 ＜0.A valid phase of the significant condensate fraction in both atomic fields is not found by mean-field calculation. The phases ASF1 and ASF2 are separated by a first-order transition boundary.

3.2. Molecular superfluid phase

In the MSF phase,there is large negative detuningν ＜0,that is,?ν ?|μ1,2|. The molecular channels have lower energy and the ground state is an atomic vacuum. The free energy densityfMis given as

whereDis anSU(5)matrix satisfyingD·D?=1. The ground state implies a broken symmetry groupSU(5).

3.3. Atomic–molecular superfluid

For intermediate detuning,both the atomic and molecular modes are gapless. To understand the phase boundaries and the behavior of the order parameters, it is convenient to approach the AMSF phase by starting from the MSF phase.[7]For simplicity, we specialize in a balanced mixture, whereμ1=μ2=μ. Applying the mean-field assumption, we obtain the free energy densityfAM=F[Ψσ,Φm]/V=fQ+fM,wherefQdescribes theQ-dependent portion of the free energy densityfAM,

whereDis anSU(5) rotation matrix. Similarly to the analysis in the MSF context,the broken symmetry group isSU(5).It is worth noting that a zero-momentum solution is needed to minimize the free energy,which is different from the finite momentum case that applies to p-wave interaction gases.[7]The condensate densities are

By settingnA=0 andnM=0,respectively,we obtain the two phase boundaries that separate the three phases, the molecular superfluid (MSF) phase, the atomic–molecular superfluid (AMSF) phase, and the atomic superfluid (ASF) phase.The relation between the condensate densities and the detuning is depicted in Fig.3,

Fig.3.Atomic and molecular condensate density versus FR detuning ν.The red curves denote the molecular condensate density,the blue curves denote the atomic condensate density. (i)MSF for ν ＜νd1; (ii)AMSF for νd1 ＜ν ＜νd2;(iii)ASF for ν ＞νd2.

4. Low energy excitations

In this section,we will focus on low-energy excitation of the d-wave FR to double-check the consistency of the meanfield results. To begin with, we expand the field operators in the ASF,MSF,and AMSF phases around their mean-field condensate values,[7,8]σ=Ψσ+δσandm=Φm+δm.With these perturbation field representations,the Hamiltonian(1)is expanded up to the second order in the momentum space with creation and annihilation operatorsσ,kandm,k,

4.1. Atomic superfluid

In the ASF phase,the previous section confirmed that the molecular modes are gapped. The relative mean fieldΦm=0,and the atoms are condensed at zero momentumQ=0. To discover the atomic modes, we need to integrate the molecular modes out(see Supplementary). In the low-energy regime,whenk →0,we calculate the dispersion up to thek2-th order.The atomic and molecular modes are given by

wherenAis the atomic condensate density at the mean-field level,nA=|Ψ1|2+|Ψ2|2. The atomic modes are gapless excitations in the superfluid states. The molecular modes have an energy gap given byν ?2λnA+gAMnA. When it vanishes,we have a transition from the ASF phase to the AMSF phase at the detuning value which is consistent with Eq.(17)(nA=μ/λ). Figure 4 shows the theoretical and numerical results. They are a good fit for the small-kregion.

Fig. 4. ASF phase excitation spectrum. Here, we use the parameters{m=1,μ =1,ν =3.2,λ11 =λ22 =3,λ12 =λ21 =1}. The units are arbitrary. All the molecular modes are gapped, but the atomic modes are gapless. The five molecular modes are degenerate. The numerical and theoretical results are a good fit for the small-k regime.

4.2. Molecular superfluid

Fig. 5. MSF phase-excitation spectrum. The parameters used for the MSF phase are{m=1,μ =0,ν =?1.44,g0=1}. The units are arbitrary. The atomic modes are gapped and degenerate. All the molecular modes are gapless;m=±1,±2 are degenerate and denoted by the lower green line,and m=0 is denoted by the upper green line.

which is consistent with Eq.(16). Figure 5 shows the consistency between the theoretical and numerical results.

4.3. Atomic–molecular superfluid

For the intermediate phase, both atomic and molecular condensates exist. Hence, they define a complicated coupled Hamiltonian(see supplementary). The molecular and atomic condensate mean-field solutions are given by

Similarly to what we achieved in the MSF phase, we choose the simplest case to compute the spectra,D=1.Diagonalizing this Hamiltonian yields the spectra up to thek-th order

Figure 6 shows the consistency between the theoretical results and numerical results.

Fig. 6. AMSF phase-excitation spectrum. The parameters here are:{m=1,μ=0,ν=?1,λ11=λ22=1.5,λ12=λ21=0.5,g0=2,gAM=?1,g=0.01}. The units are arbitrary. The atomic modes are gapless for the two blue lines. The molecular modes are also gapless:m=±1,±2 are degenerate and denoted by the lower green line;m=0 is denoted by the upper green line.

5. Final remarks and conclusions

In general,atomic loss is inevitable near a Feshbach resonance. In this case, the free energy becomes complex and the ground states are no longer stable. For simplicity, let us qualitatively estimate the effect of atomic loss by introducing imaginary parts into the chemical potentialsμ1,2,M. However,as a criterion, different quantum phases may be straightforwardly obtained from Table 1 by replacing the chemical potentials with their real parts,if we determine the ground states according to the real parts of the free energies in Eqs.(6),(8),(9), and (10). The imaginary parts of the free energies determine the damping rates of the corresponding ground states.When the relaxation times (to the equilibrium states) are far shorter than the lifetime of the atomic gas due to atomic loss,the ground states predicted here are still observable. The qualitative properties of low-energy excitation spectra(such as the numbers of gapless modes)are also expected to be unchanged due to the introduction of atomic loss when the continuous symmetries are not broken. It is hard to quantitatively estimate the effect of atomic loss at this stage, since the experiment lacks the necessary data.

In this paper,we studied the mean-field ground state of a d-wave interacting Bose gas,and found that there are three superfluid phases: the atomic,molecular and atomic–molecular superfluid phases. What was most surprising was that,unlike the p-wave case,[7–9]we found that the atomic superfluid does not carry finite momentum. Furthermore,we studied the lowenergy excitation spectrum above the superfluid phases. Our work provides a basic reference for experiments on degenerate d-wave interacting Bose gases.

Acknowledgment

The authors are indebted to Bing Zhu and Chao Gao for helpful discussion.