Shu Yang,Yue Chen and Peng Zhang,2,3

1 Department of Physics,Renmin University of China,Beijing 100872,China

2 Beijing Computational Science Research Center,Beijing 100084,China

3 Beijing Key Laboratory of Opto-electronic Functional Materials &Micro-nano Devices,Renmin University of China,Beijing 100872,China

Abstract The idea of manipulating the interaction between ultracold fermionic alkaline-earth (like)atoms via a laser-induced periodical synthetic magnetic field was proposed in Kanász-Nagy et al (2018 Phys.Rev.B 97,155156).In that work,it was shown that in the presence of the shaking synthetic magnetic field,two atoms in 1S0 and 3P0 states experience a periodical interaction in a rotated frame,and the effective inter-atomic interaction was approximated as the time-averaged operator of this time-dependent interaction.This technique is supposed to be efficient for 173Yb atoms which have a large natural scattering length.Here we examine this time-averaging approximation and derive the rate of the two-body loss induced by the shaking of the synthetic magnetic field,by calculating the zero-energy inter-atomic scattering amplitude corresponding to the explicit periodical interaction.We find that for the typical cases with shaking angular frequency λ of the synthetic magnetic field being of the order of (2π)kHz,the time-averaging approximation is applicable only when the shaking amplitude is small enough.Moreover,the two-body loss rate increases with the shaking amplitude,and is of the order of 10-10 cm3·s-1 or even larger when the time-averaging approximation is not applicable.Our results are helpful for the quantum simulations with ultracold gases of fermionic alkaline-earth (like) atoms.

Keywords:ultracold atom,spin-exchange interaction,Floquet engineering

1.Introduction

In recent years,the ultracold gases of alkaline-earth (like)atoms attracted much interest from both theorists and experimentalists [1,2].These atoms can be prepared in not only the electronic-orbit ground state(the1S0state)but also some longlived electronic-orbit excited states (the3P0and3P2states).In addition,there is a spin-exchange interaction between two fermionic alkaline-earth (like) atoms in1S0and3P0states,respectively.Explicitly,the nuclear-spin states of these two atoms can be exchanged during collision (figure 1(a)) [3–18].Therefore,the ultracold alkaline-earth(like)atoms can be used in quantum simulations for many-body physics related to the spin-exchange processes [19–28],e.g.the Kondo effects.To implement these quantum simulations,it is important to control the spin-exchange interaction between two atoms in1S0and3P0states,respectively.People have proposed various approaches to control this interaction with confinement potential or laser beams and experimentally realized the former in a quasi-(1+0)dimensional system of ultracold173Yb atoms [13–21].

In 2018,an optical-control approach for the above spinexchange interaction is proposed by Kana′sz-Nagy et al for173Yb atoms[19],which has relatively large natural scattering length.In this scheme,a single laser beam is applied to far-off-resonantly couple the3P0states to the3D1states.This beam induces a nuclear-spin-dependent AC-Stark shifts for the3P0states(figure 1(b)),which are proportional to the laser intensity and can be regarded as the effective Zeeman energies (EZEs) of a synthetic magnetic field.These EZEs can couple the scattering channels with respect to nuclear-spin singlet and triplet states.In this method,the laser intensity is periodically modulated,and thus the laser-induced EZEs for3P0are ‘shaking’.By changing the shaking amplitude and frequency,one can tune the effective inter-atomic interaction,i.e.realize a Floquet engineering.

Figure 1.(a)A schematic of spin-exchange scattering process of two alkaline-earth (like) atoms in 1S0 and 3P0 states,respectively,with different nuclear-spin states.(b) A schematic of the laser-induced nuclear-spin dependent AC-Stark shifts(i.e.the EZEs)of 3P0 states of an 173Yb atom.A detailed discussion is given in appendix C of [19].

Explicitly,in the presence of the shaking EZEs,there is a rotated frame where the free Hamiltonian of each atom is time-independent and the two-body interaction potential is a periodical function of time,which depends on the shaking of the EZEs.Moreover,in [19]the effective inter-atomic interaction is approximated to be the time-averaged value of this explicit periodical interaction potential.

In this work we investigate the system of this scheme,and try to answer the following two questions:

(i) How good is the above‘time-averaging approximation’for the effective inter-atomic interaction?

(ii) How serious is the two-body loss (i.e.the heating effect) induced by the shaking of the laser intensity?

To this end,we calculate the inter-atomic zero-energy scattering amplitudes corresponding to the explicit timedependent interaction potential for the173Yb atoms,via the Floquet scattering approach.We answer question (i) by comparing the results with the scattering amplitudes given by the time-averaged interaction potential and answer question(ii)by deriving the two-body loss rate from the imaginary part of the zero-energy scattering amplitudes.

We study the typical cases with the shaking angular frequency λ ～kHz.We find that for these cases the time-averaging approximation is applicable only when the shaking amplitude of the EZE is small enough.In addition,the two-body loss rate K2increases with δ0.when the time-averaging approximation is not applicable we have K2?10-10cm3·s-1,which yields that for ultracold gases with typical density(1013–1014)/cm3the lifetime can be decreased to the order of (0.1–1) ms.

Our results yield that,in the quantum simulations for a closed system (e.g.the quantum simulations for Kondo physics),the shaking amplitude δ0should be smaller enough so that the two-body loss rate is low enough.In this parameter region,the time-averaging approximation is usually applicable.On the other hand,since the two-body loss rate can be controlled by the amplitude and frequency of the shaking field,this system may be used for the studies of open quantum many-body systems with atom-number dissipation.

The remainder of this paper is organized as follows.In section 2,we introduce the principle of the control for the interaction between alkaline-earth(like)atoms in1S0and3P0,respectively,with shaking EZEs.In section 3 we show the Floquet scattering approach for the calculation of inter-atomic zero-energy scattering amplitude.Our results are discussed in section 4 and a brief summary is given in section 5.

2.Control of inter-atomic interaction with a shaking laser beam

In this section,we introduce the interaction-manipulation scheme proposed in [19].

2.1.System and inter-atomic interaction

As shown in figure 1(a),we consider two ultracold173Yb atoms in the1S0-(g-)and3P0(e-)states,respectively.In the two-body problem,these two atoms can be considered as two distinguishable atoms,i.e.the g-atom and e-atom.We further assume that the nuclear-spin state of each atom can be either ↑or ↓,with magnetic quantum numbersmF(↑)ormF(↓),respectively,while the nuclear-spin states of the g-and e-atoms are different.Therefore,for our system,the two-atom internal state could be either

or

Furthermore,in the presence of a natural magnetic field B and a laser beam which can induce nuclear-spin dependent ACStark shifts (i.e.the EZEs) for the3P0states (figure 1(b)).Therefore,the total Zeeman energies Eα(β)of the state|α(β)〉is the summation of the one given by the natural magnetic field and the laser-induced EZE,and thus the Zeeman-energy difference between these two states is

where δBis the difference of the Zeeman energies induced by the real magnetic field for states|β〉and|α〉,and δLis the EZE difference between these two states.Explicitly,we have

where μe(g)is the nuclear magnetic moment of the e-(g-)atom andis the laser-induced AC-Stark shift of the3P0state with a magnetic quantum numbermI(↑)(mI(↓)).

Now we consider the inter-atomic interaction of these two atoms.The bare interaction of these two atoms is diagonal on the basis of nuclear-spin singlet and triplet states,i.e.

and thus can be expressed as

where r ≡|r|with r is the relative position of these two atoms,and V±(r) is the interaction potential corresponding to the states |±〉.In this work,we consider the low-energy cases where both the relative kinetic energy of the two atoms and the amplitude and shaking frequency of the Zeeman energy gap δ are much smaller than the characteristic energy(the van der Waals energy) of the inter-atomic interaction potential5This low-energy condition is satisfied by the current experiments.For instance,for ultracold gases of 173Yb atoms EvdW is about ?(2π)3 MHz,while in the experiments the two-body relative kinetic energy is on the order of ?(2π)104 Hz,and the energy gap δ is also of this order when the magnetic field B is below 100 G..Therefore,we model the interaction potential(r) with the energy-independent Huang-Yang pseudopotential (HYP),i.e.we have (?=m=1,with m being the single-atom mass)

where a±are the scattering lengths of the two atoms in states|±〉.In our calculation we take a+=1900a0and a-=200a0,which are approximations of the theoretical computations or the experimentally measured values [4–6].

According to the above discussions,the total Hamiltonian for the relative motion of these two atoms is given by

2.2.Shaking of δ and rotated frame

In this scheme,the intensity of the laser beam is periodically modulated,and thus the EZE difference δLof equation(5)can be expressed as

with δ0＞0 and λ＞0 being the shaking amplitude and frequency,respectively.In addition,we further assume that the real magnetic field is tuned so that the difference δBof the real Zeeman energies of states |α〉 and |β〉 satisfies

Thus,the total energy gap δ between these two states is

Therefore,in the Schr?dinger picture,the time-dependent Hamiltonian H of our two-body problem is given by equation (9),with δ andV? (r) being given by equation (12)and equations(7),(8),respectively.Nevertheless,as[19],we do our calculation in the rotated frame induced by the unitary transformation

Explicitly,the state |ψ(t)〉(R)in the rotated frame is related to the state |ψ〉(S)in the Schro¨dinger picture via

Direct calculation shows that in this frame the evolution of|ψ(t)〉(R)satisfies

where the Hamiltonianin the rotated frame is given by

with

Here the operatoris defined as

with

and Jn(z)being the nth order first-type Bessel function.Notice that in the rotated frame the two-body spin states before and after spin-exchanging (e.g.|α〉 and |β〉) are degenerate when the two atoms are far away with each other.

In the rotated frame the inter-atomic interaction is described by the time-dependent two-component pseudopotential,which depends on the amplitude δ0and the shaking frequency λ of the laser beam.Therefore,one can control the inter-atomic interaction via this laser beam.

2.3.Time-averaging approximation

Nevertheless,it is not easy to directly use the pseudopotentialin the many-body calculations.Therefore,it would be useful ifcan be approximated by an effective interaction in the rotated frame.In [19]the authors use the time average ofas the effective potential,which can be denoted asand is given by

with

3.Floquet scattering

In this section,we show our approach for the calculation of the inter-atomic zero-energy scattering amplitude and the two-body loss rate.To perform these calculations,we require to solve the scattering problem with respect to the Hamiltonianof equation (16).Since the interaction potentialof this Hamiltonian is a periodic function of time,we use Floquet scattering theory with the formalism of Sykes,Landa,and Petrov in [29].

3.1.Scattering with incident channel |α〉

We first consider the scattering of the g-atom with nuclear spin ↓and the e-atom with nuclear-spin ↑,i.e.the scattering process incident from spin channel |α〉.This scattering is described by the Floquet scattering wave function |Ψ(α)(r,t)〉which satisfies

and the out-going boundary condition in the limit r→∞,and thus can be expressed as:

where r=|r| and

Substituting the wave function |Ψ(α)(r,t)〉 into the Schro¨dinger equation (15),and using the relation=-4πδ(r),we can find that Ψ(α)(r,t)〉 should also satisfy the Bethe-Peierls boundary condition (BPC)

Hereis the inverse operator of ‘scattering length matrix’defined in equation (18).Substituting equation (24) into (26),we derive the linear algebraic equations for the amplitudes fα(β)←αand:

Numerically solving equations (27)–(29),we can derive elastic scattering amplitude fα←αand the spin-exchanging amplitude fβ←α.

3.2.Scattering with incident channel |β〉

Similarly,we can also consider the zero-energy scattering of the g-atom with nuclear spin ↑and the e-atom with nuclearspin ↓,i.e.the scattering process incident from spin channel|β〉.This process is described by the Floquet scattering wave function |Ψ(β)(r,t)〉 which satisfies

and the out-going boundary condition in the limit r→∞.Thus,this wave function can be expressed as:

Figure 2.The scattering amplitudes of 173Yb atoms,as a function of δ0 for λ=(2π)3 kHz,λ=(2π)5 kHz,and λ=(2π)10 kHz.(a1)–(a3):Re[fα←α](black solid line)and (blue dotted line).(b1)–(b3):Re[fβ←α](black solid line)and(blue dotted line).The insets show the behavior of each quantity for δ0 ≤5 kHz.

Numerically solving equations (33)–(36),we can derive elastic scattering amplitude fβ←βand the spin-exchanging amplitude fβ←α.

3.3.Two-atom loss rate

Our above calculation shows that due to the shaking of the interactionthe atoms have some probability to be scattered to the Floquet bands which differs from the incoming one.This is described by the terms with n＞0 in equations (24),(32).When the shaking frequency λ is high,the out-going momentum knfor these bands are also very large.As a result,the atoms scattered to these bands can escape from the trap.This is the two-body loss induced by the shaking of the control beam of our system.

According to the optical theorem,for our system,the two-body loss rate of atoms in state |α〉 and in state |β〉 are 8πIm[fα←α]and8πIm[fβ←β],respectively.Furthermore,the direct calculation for the above equations (36) and (29)show that fα←α=fβ←β.Thus,the two-body loss rate of our system can be expressed as

4.Results

Figure 3.Im[fβ←α]/|fβ←α| of 173Yb atoms as a function of δ0,for λ=(2π) 3 kHz (red solid line),λ=(2π) 5 kHz (blue dashed line),and λ=(2π) 10 kHz (green dotted line).Here we show the results for (a):δ0 ≤(2π)100 kHz and (b):δ0 ≤(2π)4 kHz.

In this section,we use the results given by the calculations of section 3 to address the questions (i) and (ii) of section 1.We first notice that,according to direct calculations,the zero-energy scattering amplitudesfrom channel|l〉to|j〉(l,j=α,β) with respect tocan be expressed as

On the other hand,the zero-energy scattering amplitudes fl←j(l,j=α,β)derived by the Floquet scattering approach satisfy fα←α=fβ←βand fβ←α=fα←β.

The above facts yield that to answer the question (i),i.e.examine the applicability of the time-averaging approximation,we just require to compare fα←αand fβ←αwithand,respectively.In figures 2 and 3 we perform this comparison for cases with typical shaking angular frequency λ=(2π) 3 kHz,λ=(2π) 5 kHz,and λ=(2π) 10 kHz.Our results show that the time-averaging approximation is applicable only when the shaking amplitude δ0is small enough,or roughly speaking,under the condition δ0?λ.As shown in figure 2,when this condition is violated,either Re[fα←α]or Re[fβ←α]would be quite different fromor,respectively.Furthermore,as shown in figure 3,when δ0is too largeIm[fβ←α]becomes comparable with the norm of fβ←α,while the time-averaging approximation yields=0.

Figure 4.The two-body loss rate K2 of 173Yb atoms as a function of δ0.Here we show the results for (a):δ0 ≤(2π)100 kHz and (b):δ0 ≤(2π)5 kHz.

In figure 4 we show the two-body loss rate K2for these three cases,which is calculated by the Floquet scattering approach.It is shown that K2increases with the shaking amplitude δ0,and can be enhanced to the order of 10-10cm3·s-1or even larger when δ0is so large that the timeaveraging approximation is not applicable.

In addition,figure 2 also shows that a scattering resonance occurs for λ=(2π) 3 kHz and δ0≈(2π) 6.58 kHz,where both the scattering amplitude and the two-body loss rate are significantly enhanced.To understand this resonance more clearly,in figure 5 we illustrate the real parts of the scattering amplitudes fα←αand fβ←αas a function of λ for fixed shaking amplitude δ0=(2π) 6.58 kHz.Five significant resonances A,B,C,D and E are clearly shown in this figure,which occur at λ ≈(2π)6.16 kHz,λ ≈(2π)3.00 kHz,λ ≈(2π)1.99 kHz,λ ≈(2π)1.49 kHz and λ ≈(2π)1.19 kHz,respectively.The resonance B is just the one in figure 2.These resonances are due to the coupling between the incident channel and other Floquet channels corresponding to the terms proportional to e-iωnt(n ≠0) of equation (32).

In the following,we provide more discussion on these resonances.We first notice that,for our Floquet scattering problem the total Hilbert space can be expressed as H=Hr?HS?HF,with Hrand HSbeing the Hilbert space for the two-atom relative motion and internal states,respectively,andFH being the space of periodical functions of time,i.e.the space spanned by the functions e-iωnt(n=0,±1,±2,...).Here we formally denote the states inSH,FH,and HS? HFwith |〉,|〉Fand |〉SF,respectively,and denote the functions e-iωnt(n=0,±1,±2,...)as|n).With these notations,the Schro¨dinger equation (23) can be re-expressed as

Figure 5.The scattering amplitudes of 173Yb atoms,as a function of λ for δ0=(2π) 6.58 kHz.(a):Re[fα←α](black solid line) and (blue dotted line).(b):Re[fβ←α](black solid line) and (blue dotted line).The insets show the behavior of each quantity for λ ≤0.8 kHz.

where |ψ(r)〉SFis the scattering wave function,

and

Hereandare defined as:

and

In our problem,the atoms are incident from the channel |0〉Fwith zero kinetic energy,and the channels |n〉Fwith n≥0 and n ＜0 are open and closed channels,respectively.Furthermore,direct calculations also show that for each given n ＜0,the self Hamiltonianof the closed channel can support two bound states with energies

In the weak-inter-channel-coupling cases with small shaking amplitude δ0,the scattering resonances can occur when these closed-channel bound states are resonant with the zero-energy incident state,i.e.when any one of the following equations

is approximately satisfied by δ0and λ.Here we say ‘a(chǎn)pproximately’ because the resonance points can actually be slightly shifted from the solutions of equation(46)by the inter-channel couplings,i.e.the inter-channel coupling can contribute to Lamb shifts for the self-energies of the closed-channel bound states.This physical picture is justified by our results of figure 5.Using direct calculations we find that the values of (δ0,λ)corresponding to the resonances A,B,C,D and E of this figure approximately satisfy equation (46) with ζ=+and n=1.21,5.29,18.33,24.12 and 15.82,respectively.In the parameter region of figure 5 there may be other resonances that are too narrow to be clearly resolved.

5.Summary

We calculate the scattering amplitude and two-atom loss rate for the ultracold gases of173Yb atoms with interaction being modulated with the Floquet-engineering approach proposed in [19].We use the energy-independent two-channel Huang-Yang pseudopotential to model the bare interaction between these two atoms.A more quantitatively accurate may be derived via the calculations with quantum defect theory.

Our results show that for the cases with typical shaking angular frequency λ ～(2π)kHz,the time-averaging approximation is applicable and the two-body loss rate K2is lowenough only when the shaking amplitude δ0is low enough.When δ0is too large,the inter-atomic scattering amplitude would be quite different from the one given by the simple time-averaged potential,and K2can be enhanced to the order of 10-10cm3·s-1,which is quite large for typical ultracold gases.

According to these results,when our system is used for the simulations of closed quantum systems,the shaking amplitude δ0should be small enough so that the two-body loss effect is weak enough.On the other hand,since one can increase the two-body loss rate by increasing δ0,this system may be used for the studies of open systems with particlenumber dissipations.