Kangkang Li and Zhaoxin Liang

Department of Physics,Zhejiang Normal University,Jinhua 321004,China

Abstract Dimensionality is a central concept in developing the theory of low-dimensional physics.However,previous research on dimensional crossover in the context of a Bose–Einstein condensate (BEC) has focused on the single-component BEC.To our best knowledge,further consideration of the two-component internal degrees of freedom on the effects of dimensional crossover is still lacking.In this work,we are motivated to investigate the dimensional crossover in a three-dimensional (3D) Rabi-coupled two-component BEC.The spin degrees of freedom consist of the Rabi-like and inter-and intra-interaction coupling constants.The dimensional crossovers from 3D to 2D or 1D are controlled by the continuous increase of 1D or 2D lattice depth respectively.Then we analyze how the dimensionality of the model system combined with spin degrees of freedom can affect quantum fluctuations.Accordingly,the analytical expressions of the ground-state energy and quantum depletion of the system are obtained.Our results show that the dimensional crossover induces a characteristic 3D to quasi-2D or 1D crossover in the behavior of quantum fluctuations,with an emphasis on the separated effects of Rabi-like and inter-and intra-interaction coupling constants on the quantum fluctuations.Conditions for possible experimental realization of our scenario are also discussed.

Keywords: two-component Bose gas,dimensional crossover,Bogoliubov theroy,quantum fluctuation,optical lattice

1.Introduction

Dimensionality is a fundamental aspect of quantum many-body physics.In particular,investigating quantum fluctuations along the crossovers from three dimensions(3D)to quasi-2D or 1D can set up the bridge between the theoretical low-D models and the actually 3D physics world.For instance,Tomonaga–Luttinger liquid [1] exists in 1D,while the high-Tc superconductivity [2]and magic-angle graphene [3–5] occur in 2D;these exotic phenomena at various low dimensions have stimulated ongoing interests and efforts to explore how the dimensionality affects quantum many-body systems using Bose–Einstein condensates(BEC).With state-of-the-art technology,quasi-1D[6]and quasi-2D [7,8] BECs can be realized by controlling the depth of optical lattices.Along this research line,numerous tight-confinement schemes [9–17] have been proposed where BECs can undergo dimensional crossovers directly from 3D to 2D or 1D(i.e.3D-2D or 3D-1D crossover).

These prior works have focused on single-component BECs,where the low-lying excitation is just the density excitation.To our best knowledge,however,a further account of the spin degrees of freedom (d.o.f.),a key ingredient playing out in modern physics,is still lacking in the study of dimensional crossovers.Compared to the single-component case,quantum gases with spin d.o.f can display novel phenomenology through the emergence of both the density and spin-density excitations.Therefore,we anticipate the spin d.o.f.greatly enriches dimensional-crossover physics.

Here,we theoretically investigate the dimensional crossovers in a Rabi-coupled two-component BEC.Such systems have attracted great attention recently [18,19],since they not only allow for extending the well-known Rabi problem of atom optics to the interacting systems,it turns out that the population transfer between the two levels can be described by Josephson dynamics,leading to internal Josephson effects.

In a Rabi-coupled two-component BEC,compared to the one-component case,the additional spin d.o.f introduces two new aspects.The first is associated with the existence of interand intra-spin interactions,whose interplay can significantly influence quantum fluctuations.In particular,[20]has predicted that a mixture of repulsive quantum gases in 3D with finelytuned mutual attraction may lead to the self-bound states stabilized by quantum fluctuations,known as the quantum droplet[20–22].In low dimensions,where quantum fluctuations are enhanced,the quantum droplet states exhibit properties distinct from the 3D case at equilibrium[23]and near equilibrium[24].The second aspect is associated with the Rabi coupling between the two internal levels,which makes the relative phase excitations gapped.Very recently,[19] has experimentally measured the beyond-mean-field equation of state in a coherently coupled two-component BEC.Thus,it is interesting and desired to study the dimensional crossovers based on the Rabi coupled two-component BEC.

Specifically,we study the quantum fluctuation of an optically trapped Rabi-coupled two-component BEC along the dimensional crossover.Using the Green function approach and the Bogoliubov approximation,we calculate the ground-state energy and the quantum depletion.Our results show that the lattice induces a characteristic 3D to quasi-2D or quasi-1D crossover in the behavior of ground-state energy.We analyze how the combined effects of spin d.o.f and dimensionality affect the quantum fluctuation.

The paper is structured as follows.In section 2,we introduce our general theoretical model,based on which we study the dimensional crossover from 3D to quasi-low-dimensional cases in the following two sections.In section 3,we first study a 3D BEC trapped in a 1D optical lattice and investigate the dimensional crossover behavior from 3D to quasi-2D by increasing the lattice depth.Then,we study 3D BEC trapped in a 2D optical lattice,and investigate the dimensional crossover from 3D to quasi-1D in the ground state energy and quantum depletion.We conclude with a summary in section 4.Hamiltonian [22,19,18,25]

Herex1=x(x2=y) denotes the space coordinate.The lattice strengthVoptis in units of the recoil energy ofwithqBthe Bragg momentum andlB=π/qBthe lattice period.

We assumeVoptare relatively large(Vopt≥5),so that the size of the interband gapEgapis larger than the chemical potential μ,i.e.Egap?μ.Meanwhile,we assume the overlap of the wave functions of two consecutive wells is still sufficient to ensure full coherence.By this assumption[9–12],we restrict ourselves to the lowest Bloch band,where the physics is governed by the ratio between the chemical potential μ and the bandwidth of 4dJ,withJthe tunneling rates between neighboring wells.In general,for 4dJ?μ,the system retains an anisotropic 3D behavior,whereas for 4dJ?μ,the system crossovers to the (3-d) dimension.In the limit of 4dJ?μ,the model system can be treated as (3-d)-dimensional.

2.Hamiltonian of an optically trapped twocomponent BEC

We consider a Rabi-coupled two-component BEC in 3D,with two internal states labelled by σ=a,b,which is trapped in ad-dimensional (d=1 or 2) optical lattice.At zero temperature,the system can be well described by theN-body

withVthe system volume.Compared to the Hamiltonian[19]of a Rabi-coupled two-component BEC without optical confinement,Hamiltonian(3) in the presence ofd-dimensional optical lattice acquires two important differences: (i) the interaction coupling constants have been renormalized by the optical lattice as=C dgandwithThis is because the tight optical lattice effectively increases the repulsive interactions.(ii) Instead of taking the quadratic form as in the free space,the kinetic energyεk0along the confinement direction becomes

We aim to derive the beyond-mean-field ground state energyEgand the quantum depletion (N-N0)/Nof an optically-trapped Bose gas.To this end,we exploit the approach developed by Hugenholtz and Pines[26].Using the single-particle Green function [27],we have

Equations (7) and (8) provide the central equations for our subsequent study of the dimensional crossover of the optically-trapped Rabi-coupled two-component Bose gas.

3.Dimensional crossovers of an optically-trapped Rabi-coupled two-component BEC

According to equations(7)and(8),the spin d.o.f.introduces two new ingredients compared to the scalar BEC.Namely,the beyond-mean-field ground state energyEgand the quantum depletion (N-N0)/Nnow depend on,firstly,theassociated with the spin-dependent interactions,and secondly,the Rabi-frequency ?Ω.In particular,the Rabi-coupling ∝?Ω between the two components gives rise to phase correlations between the two components,in contrast to the density-density correlations coming from the interspecies interaction ?g.Note that such coupling can be implemented via a two-photon (Raman)process or direct coupling between the two internal states.

To identify the respective roles of the spin-dependent interaction and the Rabi coupling on the dimensional crossover,we devise two scenarios.In scenario(i),we takeand investigate how the Rabi-coupling affects the dimensional crossover.In scenario(ii),we turn off the Rabi coupling,i.e.?Ω=0,and study how the dimensional crossover is affected by the spin-dependent interaction quantifeid bywith λ ≠1.

We shall first consider a 1D optical lattice along thexdirection and study the dimensional crossover from 3D to quasi-2D in section 3.1.Then,we consider a 2D optical lattice along bothx-andy-directions and study the dimensional crossover from 3D to quasi-1D in section 3.2.

3.1.Dimensional crossover from 3D to quasi-2D

In this section,we study a Rabi-coupled two-component Bose gas in a 1D optical latticeVopt(x)=Vopt×ERsin2(qB x)in thex-direction;atoms are unconfined in they-zplane.In this case,the single-particle energy in equation (9)can be written as

We begin with scenario (i),whereand ?Ω ≠0.To explicitly construct an analytic solution,we exploit the fact that typical experimental mixtures of hyperfine states of bosonic alkali atoms are near the boundary of phase separation instability,namely the coupling constants satisfy the inequalityin order to avoid the phase separation.For the two hyperfine states of Na,for example,one hasthe Green function (9) can be simplified as

Plugging equation (12) into equations (7) and (8),we derive the quantum depletions in each component,i.e.withn1ex=n2ex,as well as the beyond-mean-field ground state energyEg/V.After straightforward but tedious calculation,we obtain

It turns out that the Rabi term ?Ω does not explicitly enter the expressions of quantum depletions and beyondmean-field ground state energy.Such surprising results can be understood as follows.Mathematically,the ?Ω only appears in the second term of the Green function(12).It can be shown that the corresponding integrals in equations (13) and (14)yield the zero value,i.e.whereCdenotes the contour of integration.As a result,?Ω disappears in equations (13) and (14).From the physical angle,the second term of equation(12)shows that the energy levelε2(k)=+?Ωacts only as the transition state such astoand there are no particles really occupying this spectrum.Therefore,the second term of equation (12) does not contribute to equations (13) and (14).

Next,we turn to scenario (ii) where ?Ω=0 and show how the spin-dependent interactions,i.e.with λ ≠1,can affect the dimensional crossover.In this case,the Green function (9) becomes

Obviously,both the two terms in equation (17) contribute to the quantum fluctuations,different from scenario (i) where only the first term of equation (12) contributes.The quantum depletionsn1ex=n2exand the beyond-mean-field energyEg/Vcan be calculated by directly plugging equation (17)into equations (7) and (8).We find

Figure 1.(a)Scaling function γ(s)(solid line)in equation(19)and its asymptotic behavior (dashed line) of in the limit of s ?1 with λ=0.95.(b) Scaling function δ(s) (solid line) in equation (18) and its asymptotic behavior (dashed line) of with λ=0.95.For vanishing s,δ(s)saturates to the value π,no matter what the value of λ is.

To visualize the behavior of quantum fluctuations in the entire crossover regimes,we focus on equations(18)and(19),as ?Ω does not explicitly play a role as stated previously.In figure 1(a),we show the scaling function γ(s) and its asymptotic behavior with λ=0.95.In the limits?1,the asymptotic law is.in figure 1(b) we show the scaling function δ(s) and its asymptotic behavior with λ=0.95.For vanishings,δ(s)saturates to π,no matter what the value of λ is.In the limits?1,the asymptotic law is

Note that we do not consider the effect of the confinement-induced resonance (CIR) [30,31] on the coupling constant ?g.This is because the basic physics of CIR can be understood in the language of Feshbach resonance [32],where the scattering open channel and closed channels are,respectively,represented by the ground-state transverse mode and the other transverse modes along the tight-confinement dimensions.Within the tight-binding approximation,however,the ultracold atoms are frozen in the states of the lowest Bloch band and can not be excited into the other transverse modes.Thus the effect of CIR oncan be safely ignored due to the absence of closed channels [30–32].

3.2.Dimensional crossover from 3D to quasi-1D

In this section,we study the dimensional crossover from 3D to quasi-1D of a Rabi-coupled two-component Bose gas trapped into a 2D optical lattice asVopt(r)=VoptER[s in2(qBx)+sin2(qB y)].In this case,the dispersion relation of the single particle becomes2J[2 -coskx-cosky].

Previously,we have stated that the Rabi-coupling constant of ?Ω does not enter the analytical expressions of quantum depletion and beyond-mean-field ground state energy.We found that this statement still hold in the case of the dimensional crossover from 3D to 1D.As such,we only consider scenario (ii) whereand Ω=0.We find

Now,we study the dimensional crossover from 3D toward 1D based on equations (20) and (21).In the limit ofs?1,we find equations (20) and (21) recover the wellknown 3D results [29],namely the system retains an anisotropic 3D behavior.Specifically,fors?1,the asymptotic law isin equation (23)and in equation (22).It follows from equations (20) and (21) that we asymptotically obtain

In figure 2,we plot the functionsh(s)in equation(20)andf(s) in equation (21).Their asymptotic behaviors are also shown.Thus we conclude that equations (20) and (21) provide the analytical expressions of quantum depletion and beyondmean-field ground state along the 3D-1D dimensional crossover.

Figure 2.(a) Scaling function (solid line) in equation (21) and its asymptotic behavior (dashed line) of with λ=0.95.(b) Scaling function (solid line) in equation (20) and its asymptotic behavior (dashed line) of H with λ=0.95.

4.Discussion and conclusion

The calculations of this work are based on the mean-field Bogoliubov theory,which can be justified as aposterioriby estimating the quantum depletion [9].Note that there is no experimental study of quantum depletion of Rabi-coupled two-component BEC.In what follows,we plan to adopt the experimental parameters of one-component BEC to estimate the quantum depletion in equations (13) and (20).In more detail,the experimental work [33,34] has shown that the Bogoliubov theory provides a semiquantitative description for an optically-trapped one-component BEC even in the case of the quantum depletions being in excess of 50%.It is supposed that such a statement is still valid for an optically-trapped Rabi-coupled two-component.We limit ourselves into the case ofin order to simplifying the estimation.For a uniform BEC,the quantum depletionand the Bogoliubov approximation is valid providedis small.For an opticallytrapped Rabi-coupled two-component BEC in the regimethe above quantum depletion is modified qualitatively as (8m* /3mwithm*the effective mass.Considering typical experimental parameters as in [9],this modification remains small.For an optically-trapped Rabicoupled two-component Bose gas along the dimensional crossovers,we take the parameters in the experiment in[35]:n=3×1013cm-3,lB=430 nm,a3D=5.4 nm,andlB/ξ ～1.The corresponding quantum depletion is evaluated as(N-N0)/N～0.0036×δ(s) orh(s) with δ(s) andh(s) shown in figures 1(b) and 2(b) respectively.It is clear that the quantum depletion (N-N0)/N<20%,and therefore,the Bogoliubov approximation is valid in the sprit of [33,34].

Summarizing,we have investigated a 3D Rabi-coupled two-component BEC trapped in a 1D and 2D optical lattice,respectively.We have analytically derived the ground-state energy and the quantum depletion.Our results show the 3D to quasi-2D or quasi-1D crossovers in the behavior of quantum fluctuations.The underlying physics involves the interplay of three quantities: the strength of the optical lattice,the interaction between bosonic atoms,and the spin d.o.f.All these quantities are experimentally controllable at present.Notably,the state-of-the-art technology allows the depth of an optical lattice to be arbitrarily tuned by changing the laser intensities,enabling realizations of quasi-1D [6] and quasi-2D [7,8]BECs.[36] has demonstrated fast control of the interatomic interactions by coherently coupling two atomic states with intra-and interstate scattering lengths almost at will.Furthermore,the beyond-mean-field equation of the state of a Rabicoupled two-component has been experimentally measured by[19].Therefore,the phenomena discussed in this paper are supposed to be observable within the current experimental capabilities.Directly observing such dimensional effects on the novel quantum phases in a Rabi-coupled two-component BEC,e.g.quantum droplet[20,23],would present an important step in revealing the interplay between dimensionality,quantum fluctuations and spin d.o.f in quasi-low dimensions.

Acknowledgments

We thank Chao Gao for stimulating discussions.This work was supported by the Zhejiang Provincial Natural Science Foundation (Grant Nos.LZ21A040001 and LQ20A040004),the National Natural Science Foundation of China (Nos.12074344,and 12104407)and the key projects of the Natural Science Foundation of China (Grant No.11835011).