Topological charge pumping with subwavelength Raman lattices D. Burba1M. Rači unas1I. B. Spielman2 3and G. Juzeli unas1y 1Institute of Theoretical Physics and Astronomy

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Topological charge pumping with subwavelength Raman lattices

D. Burba,1M. Rači¯unas,1I. B. Spielman,2, 3, ∗and G. Juzeli¯unas1, †

1Institute of Theoretical Physics and Astronomy,

Vilnius University, A. Goštauto 12, Vilnius LT-01108, Lithuania

2Joint Quantum Institute, University of Maryland,

College Park, Maryland 20742-4111, 20742, USA

3National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

(Dated: October 12, 2022)

Recent experiments demonstrated deeply subwavelength lattices using atoms with Ninternal

states Raman-coupled with lasers of wavelength λ. The resulting unit cell was λ/2Nin extent, an

N-fold reduction compared to the usual λ/2periodicity of an optical lattice. For resonant Raman

coupling, this lattice consists of Nindependent sinusoidal potentials (with period λ/2) displaced

by λ/2Nfrom each other. We show that detuning from Raman resonance induces tunneling

between these potentials. Periodically modulating the detuning couples the s- and p-bands of the

potentials, creating a pair of coupled subwavelength Rice–Mele chains. This operates as a novel

topological charge pump that counter-intuitively can give half the displacement per pump cycle

of each individual Rice–Mele chain separately. We analytically describe this behavior in terms of

inﬁnite-system Chern numbers, and numerically identify the associated ﬁnite-system edge states.

I. INTRODUCTION

The behavior of one-dimensional (1D) systems is

frequently tractable by analytic and numerical methods,

often making them ideal prototypes for understanding

phenomena that are intractable in higher dimensions.

Even non-interacting systems such as those described

by the Rice–Mele (RM) model [1] can have non-

trivial topology manifesting as protected edge states and

quantized topological charge pumping [2]. Here we focus

on a recently developed 1D subwavelength lattice for

ultracold atoms built from NRaman-coupled internal

states [3, 4] and show that adding temporal modulation

to the detuning away from Raman resonance can drive

transitions between the s- and p-band Wannier states

in adjacent lattice sites. In the tight-binding limit,

this gives rise to a pair of coupled RM chains with

new regimes of topological charge pumping as well as

topologically protected edge states.

Conventional optical lattices for ultracold atoms rely

on the ac Stark shift to produce potentials proportional

to the local optical intensity. As a result, the lattice

period can never be be smaller than half the optical

wavelength λ. Recently two techniques have emerged

to create deeply sub-wavelength lattices [3–6], both can

be understood in terms of “dressed states” created by

coupling internal atomic states with one- or two-photon

optical ﬁelds [7–13]. Here we consider the scheme

depicted in Fig. 1(a) relying on sequentially coupling

Ninternal atomic states using two photon Raman

transitions. For resonant couplings of equal strengths,

this results in independent adiabatic potentials for each

of the Ndressed states, displaced by λ/2Nfrom each

other as shown by the dashed curves in Fig. 1(b).

∗spielman@nist.gov; http://ultracold.jqi.umd.edu

†gediminas.juzeliunas@tfai.vu.lt

This idealized situation is disturbed by imbalancing

the coupling strengths, as studied in Ref. [3], or by

detuning one or more of the transitions from resonance;

the latter situation is plotted in Fig. 1(b). The addition of

such perturbations makes evident the λ/(2N)periodicity

of the adiabatic potential, giving rise to nearest-neighbor

(NN) tunneling between sites spaced by a single reduced

unit cell. This induced tunneling is generally much

stronger than the natural N’th neighbor tunneling of the

undisturbed lattice.

Here we focus on the eﬀects of an additional time-

modulated detuning which gives rise to an eﬀective

tunneling matrix element between s- and p-band

Wannier states spaced by ±λ/(2N), leading to a novel

subwavelength optical lattice. In this lattice the

proximity between adjacent sites allows the modulation

induced matrix element to be comparable or larger

than that of the NN tunneling induced by static

detuning. Fig. 1(c) shows the resulting lattice geometry

arising from this description, and (d) unwraps this into

a pair of coupled Rice-Mele (RM) chains described

by a highly tunable two-leg ladder Hamiltonian with

novel topological properties that are the focus of this

manuscript.

We study the topological aspects of this lattice both

by considering adiabatic pumping and in terms of edge

states. In the former case we show that the added inter-

chain tunneling enables simple pumping trajectories

giving per-cycle displacements of 0, 1 or 2 unit cells; by

contrast only displacements in units of 2 sites are possible

for the uncoupled RM chains.

This manuscript is organized as following. In Sec. II we

formally derive the subwavelength Hamiltonian described

above. Section III focuses on the subwavelength

symmetry operations and solves the resulting band

structure problem. In Sec. IV we obtain a tight binding

description of this lattice in terms of localized s- and

p-band Wannier orbitals. The band-changing tunneling

arXiv:2210.05515v1 [cond-mat.quant-gas] 11 Oct 2022

(a) Conceptual geometry

BEC

(b) Adiabatic potentials for N = 3

0λ/6λ/3λ/2

Position, x

−4

−2

Energy, E/ER

|2

|1|0

s-band

p-band

(d) Unraveled tight binding model

(d) Unraveled Rice-Mele model

FIG. 1. Lattice concept. (a) Experimental geometry with

a single frequency Raman beam traveling along exand N

Raman laser beams sharing the same spatial mode traveling

along −ex. The level diagram for cyclic coupling is depicted

on the right. (b) Dressed state energies for N= 3 and Ω0=

Ω1= Ω2= 1ER. The dashed curves are computed for zero

detuning, whereas the solid ones are calculated for a detuning

described by Eq. (17) with l= 1 and δ= 0.5ER. All curves

are colored according to ternary plot on the right, marking

the occupation probabilities in the three dressed states (not

the bare internal atomic states) obtained by diagonalizing

Eq. (16). (c) Resonant driving gives nearest neighbor coupling

J±1between the s- and p-bands. Coupling within bands is

induced by a static detuning with matrix elements J0sand

J0p. (d) The same lattice unraveled into coupled RM chains.

induced by time-dependent detuning is derived in Sec. V.

Section VI discusses the novel regimes of of topological

pumping in the ladder. The regimes of topological edge

states are discussed in Sec. VII. Finally in Sec. VIII we

expound on the implications of this work and conclude.

II. HAMILTONIAN

A. Physical geometry

As illustrated in Fig. 1(a), we consider an ensemble

of ultracold atoms with Ninternal atomic ground or

metastable states |jiwith j= 0,1,...N−1. These states

have nominal energies ~ωj, giving frequency diﬀerences

δωj=ωj+1 −ωj, where here and below we adopt a

periodic labeling scheme for which the labels jand j+N

are equivalent; for example, this implies |ji=|j+Ni

and ωj=ωj+N. Notice that for the speciﬁc energies

depicted in (a), the state vector |N−1ihas the largest

energy and |0ihas the smallest energy, making their

frequency diﬀerence δωN−1=ω0−ωN−1negative.

The atoms are illuminated by the pair of

counterpropagating laser beams depicted in Fig. 1(a)

with wavelength λ, deﬁning the single photon recoil

momentum ~kR= 2π~/λ and energy ER=~2k2

R/2m

for atoms of mass m. The right going beam (green

arrow) has angular frequency ω+while the left

going beam (red/blue arrow) has angular frequencies

ω−

j=ω+−δωj. These lasers drive two-photon Raman

transitions that cyclically couple the internal atomic

states; each transition from |j+ 1ito |jiis characterized

by an independent coupling strength Ωj. The overall

transition amplitude −Ωje−i2kRxincludes a phase

factor accounting for the two-photon recoil momentum

2~kRimparted by the counter propagating lasers. The

resulting light-matter interaction is described by

V(x) = −

N−1

j=0

Ωje−i2kRx|jihj+ 1|+ H.c., (1)

where a hat signiﬁes an operator that acts on the internal

atomic states, and we leave implicit the operator nature

of spatial variables such as the atomic position x. Each

state can be detuned in energy by δjfrom Raman

resonance, giving the contribution to the Hamiltonian

N−1

j=0

δj|jihj|.(2)

Finally including the kinetic energy yields the full

Hamiltonian

H=p2

2m+ˆ

V(x) + ˆ

U , (3)

where p=−i∂xis the momentum operator, and in what

follows we take ~= 1.

B. Dressed state basis

Because the internal states |jican be interpreted as

sites in a synthetic dimension [14, 15], it is convenient to

adopt a synthetic “momentum” representation, giving a

new basis of (position independent) dressed states

|εni=1

√N

N−1

j=0 |jiei2πnj/N ,with n= 0,1, . . . , N−1.

(4)

As above we periodically label states implying |εn+Ni=

|εni.

The light-matter coupling operator [Eq. (1)] can be

represented in the basis of dressed states as

V(x) = X

Vl(x),(5)

with terms

Vl(x) = −˜

Ωl

N−1

n=0

ei[2π(n+l)/N −qx]|εnihεn+l|+ H.c.(6)

resulting from the l-th Fourier component of the

transition amplitudes

Ωl=1

N−1

j=0

Ωjei2πlj/N .(7)

C. Dressed state potential V0(x)

We now consider the situation where the l= 0 Fourier

component is dominant, so

Ω0˜

Ωlwith l6= 0 .(8)

This component

Ω≡˜

Ω0=1

N−1

j=0

Ωj(9)

is the average of the Rabi frequencies Ωj. The

corresponding contribution to ˆ

V(x)is diagonal in the

basis of dressed states |εnibasis, giving

V0(x) =

N−1

n=0

εn(x)|εnihεn|,(10)

where each

εn(x) = −2Ω cos (2kRx−2πn/N)(11)

is a sinusoidal potential for atoms moving in |εni. The

potentials εn±1(x)for the neighboring dressed states

|εn+1iand |εn−1iare each spatially shifted from εn(x)

by a distance a=a0/N , giving a new unit cell that is N

times smaller than the a0=λ/2period of a conventional

optical lattice. The dashed curves in Fig. 1(b) illustrate

the lattice potentials εn(x)for the case of three internal

states (N= 3).

D. Coupling between dressed states

1. Coupling between dressed states via laser coupling

The Fourier components ˜

Ωlwith l6= 0 induce tunable

couplings Vl(x)[Eq. (6)] between atoms in dressed states

|εn+liand |εni; the corresponding potential minima are

separated by a distance l/N. The total contribution of

these components is

V0(x) = X

l6=0

Vl(x).(12)

Since each ˜

Ωlis a discrete Fourier transform of the

coupling matrix element Ωj, changing its j-dependence

can generate a range of tunneling amplitudes ˜

Ωlthat

can vary from short to long ranged. In the following

we consider a uniform atom-light coupling, Ωj= Ω and

thus ˆ

V(x) = ˆ

V0(x), and concentrate on the eﬀects of the

detunings to be considered next.

2. Coupling between dressed states via detuning

The dressed states are also coupled via inhomogeneous

(j-dependent) detunings δj. In the dressed state basis the

detuning operator (2) is

N−1

Ul|εnihεn+l|,(13)

where

Ul=1

N−1

j=0

δjexp i2πlj

N(14)

describes coupling between dressed states |εniand |εn+li

separated by l. The l= 0 term provides a uniform energy

oﬀset and will be omitted.

Similar to the case of inhomogeneous Rabi frequencies,

the coupling matrix element Ulbetween dressed states

|εniand |εn+liis a discrete Fourier transform of the

detunings δj. Therefore Ulcan achieve a desired long-

range structure on demand by properly choosing the j-

dependence of δj.

It is useful to represent Eq. (3) for the full Hamiltonian

H=ˆ

H0+ˆ

U , (15)

where the zero order Hamiltonian

H0=p2

2m+ˆ

V0(x)(16)

consists of the kinetic energy operator and the dressed

state potential ˆ

V0(x)deﬁned by Eq. (10). In what follows

we treat the detuning operator ˆ

Uas a perturbation which

couples the dressed states.

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TopologicalchargepumpingwithsubwavelengthRamanlatticesD.Burba,1M.Ra£iunas,1I.B.Spielman,2,3,andG.Juzeliunas1,y1InstituteofTheoreticalPhysicsandAstronomy,VilniusUniversity,A.Go²tauto12,VilniusLT-01108,Lithuania2JointQuantumInstitute,UniversityofMaryland,CollegePark,Maryland20742-4111,20742,USA3Natio...

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Topological charge pumping with subwavelength Raman lattices D. Burba1M. Rači unas1I. B. Spielman2 3and G. Juzeli unas1y 1Institute of Theoretical Physics and Astronomy

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