Eective potential and superuidity of microwave-dressed polar molecules Fulin Deng1 2Xing-Yan Chen3 4Xin-Yu Luo3 4Wenxian Zhang1 5Su Yi2 6 7and Tao Shi2 6 7y 1School of Physics and Technology Wuhan University Wuhan Hubei 430072 China

2025-05-03 0 0 791.54KB 12 页 10玖币

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Eﬀective potential and superﬂuidity of microwave-dressed polar molecules

Fulin Deng,1, 2 Xing-Yan Chen,3, 4 Xin-Yu Luo,3, 4 Wenxian Zhang,1, 5 Su Yi,2, 6, 7, ∗and Tao Shi2, 6, 7, †

1School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China

2CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,

Chinese Academy of Sciences, Beijing 100190, China

3Max-Planck-Institut f¨ur Quantenoptik, 85748 Garching, Germany

4Munich Center for Quantum Science and Technology, 80799 M¨unchen, Germany

5Wuhan Institute of Quantum Technology, Wuhan, Hubei 430206, China

6CAS Center for Excellence in Topological Quantum Computation & School of Physical Sciences,

University of Chinese Academy of Sciences, Beijing 100049, China

7Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China

(Dated: October 25, 2022)

For microwave-dressed polar molecules, we analytically derive an intermolecular potential com-

posed of an anisotropic van der Waals shielding core and a long-range dipolar interaction. We

validate this eﬀective potential by comparing its scattering properties with those calculated using

the full multi-channel interaction potential. It is shown that scattering resonances can be induced by

a suﬃciently strong microwave ﬁeld. We also show the power of the eﬀective potential in the study

of many-body physics by calculating the critical temperature of the Bardeen-Cooper-Schrieﬀer pair-

ing in the microwave-dressed NaK gas. It turns out that the eﬀective potential is well-behaved and

extremely suitable for studying the many-body physics of the molecular gases. Our results pave the

way for the studies of the many-body physics of the ultracold microwave-dressed molecular gases.

Introduction.—Ultracold gases of polar molecules [1, 2]

provide a unique platform for the exploration of quantum

information [3], quantum computing [4, 5], quantum sim-

ulation [6, 7], quantum chemistry [8, 9], and precision

measurement [10–12]. From the condensed-matter per-

spective, the strong long-range and anisotropic dipole-

dipole interaction (DDI) make ultracold polar molecules

an ideal platform for investigating strongly correlated

many-body physics [13, 14]. Over the past decade, there

are tremendous experimental eﬀorts for the creation of

the ultracold molecular gases by both direct cooling [15]

and cold-atom assembly. Particularly, indirect produc-

tion of the high-phase-space-density molecular gases from

ultracold atomic gases via the Feshbach resonance and

stimulated Raman adiabatic passage has been success-

fully employed to create bialkali molecules of KRb [16],

RbCs [17, 18], NaK [19–21], NaRb [22], NaLi [23], and

NaCs [24, 25]. Recently, starting from the association of

double degenerate Bose-Fermi mixtures [26, 27] and evap-

orative cooling [28–30] enabled by the collisional shielding

with either a d.c. [28, 29, 31] or a microwave ﬁeld [30, 32],

degenerate Fermi gases of polar molecules have ﬁnally be-

come available in experiments.

Unlike the conventional DDI induced by a d.c. electric

ﬁeld, the long-range DDI between microwave-shielded

molecules in the highest dressed state is attractive in

the plane of the microwave ﬁeld [30], which may lead

to exotic p-wave superﬂuids [33–37]. Because DDI cou-

ples diﬀerent rotational states, a complete description of

the intermolecular interaction involves multiple dressed

∗Electronic address: syi@itp.ac.cn

†Electronic address: tshi@itp.ac.cn

rotational states of the molecules [38, 39], which is cum-

bersome for the studies of the many-body physics of a

single shielded dressed state. Therefore, a simple and

accurate eﬀective potential is an essential ingredient for

exploring the many-body physics of molecular gases.

In this Letter, we analytically derive an eﬀective poten-

tial between two microwave-dressed polar molecules. At

large inter-molecular distance, this potential is a negated

DDI such that it is attractive in the plane of the mi-

crowave ﬁeld and repulsive along the propagation direc-

tion of microwave. While at short range, the potential

is of the 1/r6type and is anisotropically repulsive. As

a result, the eﬀective potential has a shielding core in

all three dimensions. The validity of this eﬀective po-

tential is justiﬁed by comparing it with numerically ob-

tained adiabatic potential and by exploring the scattering

properties of two molecules. We show that the eﬀective

potential not only leads to the correct scattering cross

sections, but also accurately predicts the position of the

scattering resonance. Finally, as an application of the ef-

fective potential, we study the Bardeen-Cooper-Schrieﬀer

(BCS) superﬂuidity in the microwave-dressed NaK gas,

where the Rabi-frequency of the microwave ﬁeld plays

the role as a control knob to tune the superﬂuid critical

temperature. It turns out that the eﬀective potential is

well-behaved and suitable for studying the many-body

physics of molecular gases.

Eﬀective molecule-molecule interaction.—We consider

a gas of the NaK molecules in the 1Σ(v= 0) state which

exhibits a molecular-frame dipole moment d= 2.72 De-

bye. Under ultracold temperature, only the rotational

degree of freedom is relevant such that the Hamiltonian

of a single molecule is ˆ

hrot =BrotJ2, where Brot/~=

2π×2.822 GHz is the rotational constant and Jis the an-

gular momentum operator. Since the rotation spectrum,

arXiv:2210.13253v1 [physics.atom-ph] 24 Oct 2022

BrotJ(J+ 1), is anharmonic, we focus on the two lowest

rotational manifolds (J= 0 and 1) which are split by an

energy ~ωe= 2Brot. Correspondingly, the Hilbert space

for the internal states of a molecule is deﬁned by four

states: |J, MJi=|0,0i,|1,0i, and |1,±1i. To achieve the

microwave shielding, molecules with electric dipole mo-

ment d0ˆ

dare illuminated by a position-independent σ+-

polarized microwave propagating along the z axis, where

dis the unit vector along the internuclear axis of the

molecule and the microwave ﬁeld rotates circularly in the

xy plane with frequency ω0. Within the internal-state

Hilbert space, the coupling between the microwave and

the molecular rotational states gives rise to the Hamilto-

nian ˆ

hmw =~Ω

2e−iω0t|1,1ih0,0|+h.c., where Ω is the Rabi

frequency. Then, in the interaction picture, the eigen-

states of the internal-state Hamiltonian, ˆ

hin =ˆ

hrot+ˆ

hmw,

are |0i≡|1,0i,| − 1i≡|1,−1i,|+i ≡ u|0,0i+v|1,1i,

and |−i ≡ u|1,1i − v|0,0i, where u=p(1 −δ/Ωeﬀ )/2

and v=p(1 + δ/Ωeﬀ )/2 with δ=ωe−ω0being the de-

tuning and Ωeﬀ =√δ2+ Ω2the eﬀective Rabi frequency.

The corresponding eigenenergies are E0=E−1=δand

E±= (δ±Ωeﬀ )/2. Figure 1(a) schematically shows the

level structure of a molecule.

For two molecules with dipole moments dˆ

d1and dˆ

d2,

the dipole-dipole interaction (DDI) between them is

V(r) = d2

4π0r3hˆ

d1·ˆ

d2−3(ˆ

d1·ˆ

r)(ˆ

d2·ˆ

r)i,(1)

where 0is the electric permittivity of vacuum, r=|r|,

and ˆ

r=r/r. To express DDI in the two-molecule internal

Hilbert space, we note that the two-particle Hamiltonian

H2=Pj=1,2ˆ

hj+V(r1−r2) possesses a parity sym-

metry, where ˆ

hj=−~2∇2

j/(2M) + ˆ

hin(j) with Mbeing

the mass of the molecule. This suggests that the sym-

metric and antisymmetric two-particle internal states are

decoupled in the Hamiltonian ˆ

H2. Here we focus on the

ten-dimensional symmetric subspace in which the shield-

ing states of the molecules lie. It turns out that, un-

der the rotating-wave approximation, V(r) in the seven-

dimensional (7D) symmetric subspace, S7≡ {|νi}7

ν=1,

is decoupled from the remaining three-dimensional sym-

metric subspace, where |1i=|+,+i,|2i=|+,0is,|3i=

|+,−1is,|4i=|+,−is,|5i=|−,0is,|6i=|−,−1is, and

ingly, with respect to the asymptotical state |ν= 1i, the

energies of these states are Eν={0,1

2(δ−Ωeﬀ ),1

2(δ−

Ωeﬀ ),−Ωeﬀ ,1

2(δ−3Ωeﬀ ),1

2(δ−3Ωeﬀ ),−2Ωeﬀ }. In below,

we shall consider the two-molecule problem only in the

subspace S7.

To derive an eﬀective potential between two molecules,

we make use of the Born-Oppenheimer approximation

which holds when the kinetic energy of the molecules is

much smaller than the energy level spacings between in-

ternal states (∼Ωeﬀ ). After diagonalizing V(r) in S7, we

ﬁnd seven adiabatic potentials corresponding to diﬀerent

dressed-state channels [see, e.g., Fig. 1(b) for the typi-

cal adiabatic potential curves]. Particularly, the eﬀective

0 1000 2000

-30

-20

-10

10 (b)

500 1000 1500 2000

-1.2

-0.6

0.6

1.2

(c)

(a)

FIG. 1: (a) Schematic of the level structure of a microwave-

dressed molecule. (b) Typical adiabatic potential curves of

two colliding molecules for seven dressed state channels. (c)

Eﬀective potentials along θ=π/2 obtained by numerical di-

agonalization (solid lines), numerical ﬁtting (dashed line), and

analytical expressions (dash-dotted lines) for δr= 0.1 and

Ω/(2π) = 20, 50, and 80 MHz (for three sets of curves in

descending order).

potential for two molecules in the dressed state |+iis

the highest adiabatic curve. Remarkably, as shown in

the Supplemental Material (SM), there exists an approx-

imate expression for the eﬀective potential, i.e.,

Veﬀ (r) = C3

r3P2(cos θ) + C6

r6A(θ),(2)

where Pl(cos θ) is the Legendre polynomial with θbe-

ing the polar angle of rand A(θ) = 7 −5P2(cos θ)−

2P4(cos θ). Moreover, C3=d2/24π0(1 + δ2

r),C6=

d4/1120π22

0Ω(1 + δ2

r)3/2with δr=δ/Ω. The ﬁrst

term of Veﬀ represents DDI which, diﬀerent from the con-

ventional one, is attractive in the xy plane and repulsive

along the zaxis. Because A(θ)>0 when θ6= 0 or π, the

second term is repulsive and provides a shielding core

away from the zaxis. Interestingly, even along the z

axis on which A(θ) vanishes, DDI itself is repulsive and

prevents two molecules from getting close to each other.

In Fig. 1(c), the eﬀective potential (2) is benchmarked

by the highest adiabatic curve obtained from diagonal-

izing V(r). Generally speaking, the expression for C3

is accurate in the sense that it gives rise to the correct

long-range behavior, while the analytical expression for

C6is a good approximation only when Ω > d2

0/(4π0r3).

In any case, one can alternatively determine the values

of C3and C6by ﬁtting the adiabatic potential curve,

which, as shown in Fig. 1(c) and also in SM, yields satis-

factory results in the energy range of interest to us. The

advantage of Eq. (2) is that it establishes an intuitive

connection between the potential and the physical pa-

rameters of the microwave ﬁeld. In addition, as shall be

shown below, this eﬀective potential is well-behaved and

can be used for studying the many-body problems.

Two-body scatterings.—To further justify the eﬀective

potential, we investigate the low-energy scattering of two

shielding molecules interacting via Veﬀ (r). Since this

study only involves a single scattering channel (ν= 1

in S7), its results should be checked by the scattering

calculations involving all seven channels. To this end, let

us brieﬂy outline the theoretical treatment for the multi-

channel scattering [38–41]. The Schr¨odinger equations

governing the relative motion of two colliding molecules

are

ν0=1 −~2∇2

Mδνν0+Vνν0ψν0(r) = ~2k2

Mψν(r),(3)

where ψν(r) is the wave function of the νth scattering

channel, Vνν0=hν|V|ν0i, and kν=pk2

1−MEν/~2is

the incident momentum of the νth scattering channel.

To solve Eq. (3), we ﬁrst expand the wave functions in

the partial-wave basis as ψν(r) = Plm Ylm(ˆ

r)φνlm(r)/r,

where lis odd for identical fermions. The equations for

φνlm can be numerically evolved from r= 0 to a suf-

ﬁciently large value r∞using Johnson’s log-derivative

propagator method [42]. Then, by comparing φνlm with

the asymptotical boundary condition, we obtain the scat-

tering amplitude fν0l0m0

νlm and cross section σν0l0m0

νlm =

4πfν0l0m0

νlm 

2for the (νlm) to (ν0l0m0) scattering. It

should be noted that σν0l0m0

νlm is nonzero only when mand

m0satisfy m=mνand m0=mν0where, for ν= 1 to 7,

mν=m1, m1+ 1, m1+ 2, m1, m1+ 1, m1+ 2, and m1,

respectively, with m1being an integer. Numerically, to

ensure the convergence of the scattering cross sections,

we normally choose k0r∞>32 and lc>11, where lcis

the truncation imposed on the orbital angular momen-

tum.

As a special case of the multi-channel scattering, the

Schr¨odinger equation for single-channel scattering can be

obtained by projecting Eq. (3) onto the ν= 1 channel

with V11 being replaced by Veﬀ . In addition, we denote

the single-channel scattering cross section as σl0m0

lm .

Since the scattering cross section of the pwave is

dominant over all other partial waves [49], we compare,

in Fig. 2(a), σ11

11 and σ111

111 for δr= 0.1 and k1/kF=

0.04, 0.45, and 1, where kF= (6π2n0)1/3is the Fermi

wave vector with n0= 1012 cm−3being the density of

the experimentally realized molecular gas [30]. As can

be seen, away from scattering resonances, quantitative

agreements have been achieved for p-wave cross section

under diﬀerent incident momenta. Moreover, the single-

channel calculations can even predict the position of scat-

tering resonance with high accuracy. These results to-

gether with other comparisons in SM validate the usage

of the eﬀective potential, which, as shown below, sig-

niﬁcantly simpliﬁes the calculations in the many-body

problems.

As to the Ω dependence of the p-wave cross section, it

can be seen that, for small k1,σ111

111 barely changes as Ω

varies over a wide range. Then at Ω/(2π)≈87.7 MHz

a narrow scattering resonance appears, signaling the for-

mation of a quasi-bound state. Furthermore, as k1in-

creases to kF, the resonance peak shifts to Ω/(2π)≈

73.6 MHz and the width of the resonance is signiﬁcantly

broadened. To understand these features, let us recall

that 1) there is a centrifugal barrier for the p-wave poten-

tial; 2) a scattering resonance implies that a quasi-bound

state with energy in resonance with the incident energy

forms inside the barrier. Now, as k1increases, the res-

onant quasi-bound state energy also increases and gets

closer to the top of the potential barrier. Consequently,

the lifetime of the quasi-bound state is shortened due to

large decay rate, which leads to a broader resonance. In

addition, the increasing quasi-bound state energy implies

a weakened attractive interaction via reducing Ω (see the

relation between C6and Ω). Therefore, the resonance

peak shifts towards the lower Ω direction as k1increases.

To reveal more details about the scattering resonance,

we map out, in Fig. 2(b), σ111

111 on the Ω-δrparameter

plane for k1= 0.45kF. As can be seen, a resonant peak

appears in the parameter region δr>0.3 and Ω/(2π)?

80 MHz. To give an intuitive explanation to the relation

between the resonance and control parameters, a p-wave

bound state at threshold appears when the WKB phase

ϕp=Zvp(r)≤0q−Mvp(r)/~2dr ∝Ω2

(1 + δ2

r)51/12

(4)

is suﬃciently large, where vp(r) = Rdˆ

r|Y1m(ˆ

r)|2Veﬀ (r).

Clearly, both increasing Ω and decreasing δrfavor the

appearance of a shape resonance and the formation of a

bound state.

Superﬂuid phase transitions.—As an application of the

eﬀective potential, we now turn to explore the BCS su-

perﬂuid phase transition in a homogeneous gas of the

dressed-state molecules with density n0. In particular,

we focus on the transition temperature Tcand the pair-

ing wave functions. Previously, the superﬂuidity of the

fermionic dipolar gases have been extensively studied us-

ing the pseudopotential containing a contact part and a

bare DDI [33–37, 43–48]. To illustrate the eﬀect of mi-

crowave ﬁeld to the superﬂuidity, we apply the eﬀective

potential to the molecules in the dressed state |+i, and

write down the many-body Hamiltonian

H=Zd3rˆ

ψ†(r)−~2∇2

2M−µˆ

ψ(r)

2Zdrdr0ˆ

ψ†(r)ˆ

ψ†(r0)Veﬀ (r−r0)ˆ

ψ(r0)ˆ

ψ(r),(5)

where ˆ

ψ(r) is the ﬁeld operator of the molecules in the

dressed state |+iand µis the chemical potential.

In the superconducting phase, the order parameter in

the momentum space takes the form

∆(k) = Zdp

(2π)3e

Veﬀ (k−p)hc−pcpi,(6)

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Eectivepotentialandsuperuidityofmicrowave-dressedpolarmoleculesFulinDeng,1,2Xing-YanChen,3,4Xin-YuLuo,3,4WenxianZhang,1,5SuYi,2,6,7,andTaoShi2,6,7,y1SchoolofPhysicsandTechnology,WuhanUniversity,Wuhan,Hubei430072,China2CASKeyLaboratoryofTheoreticalPhysics,InstituteofTheoreticalPhysics,ChineseAcadem...

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Eective potential and superuidity of microwave-dressed polar molecules Fulin Deng1 2Xing-Yan Chen3 4Xin-Yu Luo3 4Wenxian Zhang1 5Su Yi2 6 7and Tao Shi2 6 7y 1School of Physics and Technology Wuhan University Wuhan Hubei 430072 China.pdf

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Eective potential and superuidity of microwave-dressed polar molecules Fulin Deng1 2Xing-Yan Chen3 4Xin-Yu Luo3 4Wenxian Zhang1 5Su Yi2 6 7and Tao Shi2 6 7y 1School of Physics and Technology Wuhan University Wuhan Hubei 430072 China

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