On-Demand Entanglement of Molecules in a Reconﬁgurable Optical Tweezer Array Connor M. Holland1Yukai Lu1 2and Lawrence W. Cheuk1

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On-Demand Entanglement of Molecules in a

Reconﬁgurable Optical Tweezer Array

Connor M. Holland,1, ∗Yukai Lu,1, 2, ∗and Lawrence W. Cheuk1, †

1Department of Physics, Princeton University, Princeton, New Jersey 08544 USA

2Department of Electrical and Computer Engineering,

Princeton University, Princeton, New Jersey 08544 USA

(Dated: October 13, 2022)

Entanglement is crucial to many quantum applications including quantum information process-

ing, simulation of quantum many-body systems, and quantum-enhanced sensing [1–5]. Molecules,

because of their rich internal structure and interactions, have been proposed as a promising plat-

form for quantum science [6–8]. Deterministic entanglement of individually controlled molecules

has nevertheless been a long-standing experimental challenge. Here we demonstrate, for the ﬁrst

time, on-demand entanglement of individually prepared molecules. Using the electric dipolar inter-

action between pairs of molecules prepared using a reconﬁgurable optical tweezer array, we realize an

entangling two-qubit gate, and use it to deterministically create Bell pairs [9]. Our results demon-

strate the key building blocks needed for quantum information processing, simulation of quantum

spin models, and quantum-enhanced sensing. They also open up new possibilities such as using

trapped molecules for quantum-enhanced fundamental physics tests [10] and exploring collisions

and chemical reactions with entangled matter.

INTRODUCTION

Entanglement lies at the heart of quantum mechan-

ics. It is both central to the practical advantage pro-

vided by quantum devices [1–3], and important to un-

derstanding the behavior of many-body quantum sys-

tems [4]. The ability to create entanglement control-

lably has been a long-standing experimental challenge.

Molecules have been proposed as a promising platform

for quantum simulation and quantum information pro-

cessing because of their rich internal structure and long-

lived interacting states [6–8]. In the past two decades,

much progress has been made in producing and control-

ling molecules at ultracold temperatures, both through

coherent assembly of ultracold alkali atoms [11] and di-

rect laser-cooling [12]. Rapid advances have been made in

recent years, including the creation of degenerate molec-

ular gases [13, 14], the creation of molecular magneto-

optical traps [12, 15–17], high-ﬁdelity detection of sin-

gle molecules [18–20] and laser-cooling of complex poly-

atomic molecules [21, 22]. In addition, coherent dipolar

interactions have also been observed in bialkali molecules

trapped in optical lattices [20, 23].

A major outstanding challenge to fully realizing the

potential of molecules has been achieving deterministic

entanglement with microscopic control. In this work, we

report the ﬁrst realization of a two-qubit entangling gate

between individual laser-cooled molecules trapped in a

reconﬁgurable optical tweezer array (Fig. 1a). The ap-

proach of molecular tweezer arrays [18, 19, 24, 25] com-

bines the microscopic controllability oﬀered by reconﬁg-

urable optical tweezer traps [26–30] with the ability to

generate entanglement through the electric dipolar in-

teraction between molecules. We speciﬁcally make use

of eﬀective spin-exchange interactions that arise between

FIG. 1. Laser-cooled Molecules in a Reconﬁgurable Op-

tical Tweezer Array. (a) Single CaF molecules trapped in

an optical tweezer array are prepared into closely separated

tweezer pairs. Molecules in each pair held by separate tweezer

traps interact via the long-range electric dipolar interaction

HSE. (b) The electric dipolar interaction leads to dipolar

spin-exchange of rotational excitations. (c) Molecules are

loaded stochastically, detected non-destructively, and rear-

ranged into the desired 1D conﬁguration. The molecules are

then initialized into a single internal state and the pair sepa-

rations are reduced in order to switch on interactions. After

speciﬁc interaction times, the pairs are then separated and

detected state-selectively.

rotational states to realize an iSWAP gate, which along

with single qubit rotations, is suﬃcient for universal

quantum computation [31] (Fig. 1b). We subsequently

use this to entangle pairs of molecules into Bell states,

which are prototypical entangled states [9] often used

to benchmark quantum platforms such as photons [32],

trapped ions [33], neutral atoms [34], superconducting

circuits [35], nitrogen-vacancy centers in diamond [36],

and semiconductor quantum dots [37].

arXiv:2210.06309v1 [cond-mat.quant-gas] 12 Oct 2022

FIG. 2. Tweezer Rearrangement and Internal State Initial-

ization. (a) Probability of creating defect-free molecular ar-

rays via rearrangement. A ﬁt to pn, where nis the array size,

gives a single particle rearrangement ﬁdelity of p= 0.974(1).

(b) Exemplary images of defect-free arrays. (c) Optical pump-

ing (orange arrow) prepares molecules in |Di. Microwave

sweeps (dashed green arrows) transfer |Dimolecules to |↑i.

PREPARING PAIRS OF LASER-COOLED

MOLECULES AND INITIALIZING THEIR

INTERNAL STATES

Our work starts with single laser-cooled CaF molecules

trapped in a dynamically reconﬁgurable array of optical

tweezer traps [19, 25]. Through a series of steps involv-

ing laser-cooling, optical trapping and transport, single

molecules are transferred from a magneto-optical trap

into a 1D array of 37 identical optical tweezer traps with

a uniform spacing of 4.20(6) µm. Since our laser-cooling

scheme relies on a closed optical cycle present only for the

X2Σ(v= 0, N = 1) in CaF [38], the molecules initially

loaded into the tweezers occupy a single rovibrational

manifold.

To remove the randomness in tweezer occupation, we

use a rearrangement approach pioneered in neutral atom

experiments [26, 27]. We non-destructively detect the

tweezer occupations using a variant of Λ-imaging [39].

The empty tweezers are identiﬁed, switched oﬀ, and the

remaining occupied tweezers are then rearranged into the

desired 1D pattern. We characterize the rearrangement

procedure by measuring the probability of successfully

creating uniform arrays, and ﬁnd a single particle rear-

rangement ﬁdelity of 97.4(1)%. As shown in Fig. 2a, we

are able to create uniform arrays up to a size of 16 with a

probability >0.6. The rearrangement ﬁdelity is limited

by the non-destructive detection ﬁdelity, with minimal

loss (0.2(10)%) due to movement of the tweezer traps.

After rearrangement, we initialize the internal state

of the molecules, which are distributed among the 12

hyperﬁne states in the X2Σ(v= 0, N = 1) rovibra-

tional manifold. To prepare molecules into a single hy-

perﬁne state, we optically pump molecules into |Di=

X2Σ(v= 0, N = 1, J = 3/2, F = 2, mF= 2). Subse-

quent microwave sweeps along with an optical clean-out

pulse transfer the molecules into the target ﬁnal state

|↑i =X2Σ(v= 0, N = 1, J = 1/2, F = 0, mF= 0)

(Fig. 2c). The overall ﬁdelity of preparing molecules in

|↑i is 82.4(11)%. Our preparation sequence ensures that

the dominant preparation error is in the form of unoccu-

pied tweezers, with a small contribution (≈1%) coming

from molecules prepared in the incorrect internal state

|+i=X2Σ(v= 0, N = 0, J = 1/2, F = 1, mF= 1). The

state initialization errors come from imperfect microwave

transfer, polarization impurity of the optical pumping

light, and loss due to heating in the tweezer traps.

PROBING SINGLE-MOLECULE COHERENCE

In order to produce entanglement via the dipolar in-

teractions between molecules, we require long coher-

ence times compared to the typical interaction timescales

of h/J ∼10 ms at our tweezer separations. Achiev-

ing long coherence times for optically trapped molecules

has been an ongoing experimental challenge with steady

advances being made. For molecules, diﬀerent inter-

nal states can experience diﬀerent trapping potentials,

which, in combination with motion due to ﬁnite temper-

ature, can lead to decoherence. For 1Σ bialkali molecules,

long coherence times of diﬀerent nuclear spin states

have been reported [40, 41], and work using “magic”

trapping conditions have demonstrated extended coher-

ence times between rotational states in both 1Σ and 2Σ

molecules [20, 42–45].

Since the eﬀective spin-exchange interactions couple

diﬀerent rotational states, we desire long rotational co-

herence times between the two interacting states |↑i and

|↓i =X2Σ(v= 0, N = 0, J = 1/2, F = 1, mF= 0).

Building upon previous work in CaF [45], we identify a

pseudo-magic trapping condition where both spin states

experience approximately identical trapping potentials.

Our pseudo-magic condition takes into account vector

and tensor shifts and is achieved by applying a magnetic

ﬁeld orthogonal to the tweezer light polarization at a re-

duced tweezer depth compared to that used for initial

loading and imaging.

To measure the resulting coherence time, we prepare

tweezer pairs with only one molecule initialized in |↑i.

We next apply a Ramsey pulse sequence consisting of

two π/2 microwave pulses (ﬁrst pulse along ˆx, second

pulse along ˆn= cos θˆx+ sin θˆy) separated by a variable

free evolution time (Fig. 3a). The remaining fraction of

|↑i molecules, P↑, oscillates as a function of θ, with the

oscillation amplitude directly measuring the coherence.

Fitting to an exponential decay curve yields a bare coher-

ence time T∗

2of 2.5(3) ms. Adding a spin-echo improves

the coherence time to T2= 29(2) ms. Following previous

work exploring dipolar interactions of KRb molecules in

FIG. 3. Single Particle Coherence and Spin-Exchange Oscillations. (a) Ramsey pulse sequence used to measure rotational

coherence. The bottom Bloch sphere diagrams show the action of the various pulses for a molecule initialized in |↓i. (b) The

XY8 dynamical decoupling sequence. (c) Ramsey contrast of non-interacting molecules versus free evolution time t. Green

triangles, red squares, and blue circles show the cases when no spin-echo, one spin-echo, and the XY8 sequence is applied,

respectively. Exponential ﬁts give coherence times (1/e) of 2.5(3) ms, 29(2) ms, and 215(30) ms, respectively. Insets show

exemplary Ramsey fringes with corresponding sinusoidal ﬁts shown by the dashed lines. (d) Spin-exchange oscillations at

a tweezer separation of 1.93(3) µm. Shown are the |↑↑i populations measured after the Ramsey pulse sequence, P↑↑ , as a

function of interaction time t, for molecular pairs initialized in |↑↑i. The solid curve is a ﬁt to a phenomenological model.

Insets show exemplary ﬂuorescence images at the indicated times. (e) Spin-exchange oscillations at separations of 1.26(2) µm

(red pentagons), 1.43(2) µm (orange hexagons), 1.60(2) µm (yellow diamonds), 1.68(2) µm (green squares), 1.93(3) µm (blue

circles), and 2.35(3) µm (purple triangles). Curves are oﬀset vertically by 0.3 for clarity. (f) The extracted spin-exchange

strength Jversus pair separation r. The light red band indicates the theoretical prediction taking into account the ﬁnite

temperature of the molecules and the uncertainty in the electric dipole moment of CaF. The dashed blue curve shows the

prediction without taking into account ﬁnite temperature. Inset shows the single particle loss rate γDversus pair separation r.

an optical lattice [23, 46], we implement the XY8 dynami-

cal decoupling sequence depicted in Fig. 3b, and ﬁnd that

the 1/e coherence time is further extended to 215(30) ms

(Fig. 3c). This is consistent with our understanding that

the bare coherence times are primarily limited by slow

(ms-timescale) ﬂuctuations of ambient magnetic ﬁelds.

OBSERVING COHERENT DIPOLAR

SPIN-EXCHANGE INTERACTIONS

Having achieved suﬃciently long rotational coherence

times, we next set out to observe coherent spin-exchange

interactions. The long-range electric dipolar interaction

between the molecules gives rise to resonant exchange of

rotational excitations between |↑i and |↓i. The resulting

spin-exchange interaction is described by the Hamilto-

nian

HSE =J

2ˆ

S+

1ˆ

S−

2+ˆ

S−

1ˆ

S+

2=Jˆ

1ˆ

2+ˆ

1ˆ

2,

where ˆ

S+

i,ˆ

S−

i,ˆ

iare spin-1/2 operators for molecule

iand

J=d2

4πε0

r3(1 −3 cos2θ0),

with d=h↑|ˆ

d|↓i being the transition dipole moment,

r=|~r|being the intermolecular separation, θ0being the

angle between ~r and the quantization axis, and ε0being

the free space permittivity. Starting with two molecules

in a product state, time evolution under ˆ

HSE can lead

to entanglement. For example, two molecules initially

prepared in the product state |↑i⊗|↓i become maximally

entangled after interacting for a time t=π~/(2J).

To observe the eﬀect of spin-exchange interactions, we

ﬁrst create pairs of |↑i molecules at an initial separation

of 4.20(6) µm, over which interactions are negligible. We

next reduce the pair separation to 1.93(3) µm over 3 ms,

where the interaction strength J(J=h×43 Hz) becomes

appreciable on the coherence timescale. Subsequently,

we apply the Ramsey pulse sequence used above with

θ= 0. To retain long coherence times, the XY8 decou-

pling pulses are kept on during the free evolution time.

Because the π-pulses in the XY8 sequence leave ˆ

HSE un-

changed, spin-exchange interactions are preserved [47].

For a molecular pair initialized in |↑↑i, the resulting

state following the Ramsey sequence is given by

|ψi=ie−iJ t

4~sin Jt

4~|↑↑i +icos Jt

4~|↓↓i,(1)

and P↑↑ oscillates at an angular frequency of J/(2~). As

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On-DemandEntanglementofMoleculesinaRecongurableOpticalTweezerArrayConnorM.Holland,1,YukaiLu,1,2,andLawrenceW.Cheuk1,y1DepartmentofPhysics,PrincetonUniversity,Princeton,NewJersey08544USA2DepartmentofElectricalandComputerEngineering,PrincetonUniversity,Princeton,NewJersey08544USA(Dated:October13,20...

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On-Demand Entanglement of Molecules in a Reconﬁgurable Optical Tweezer Array Connor M. Holland1Yukai Lu1 2and Lawrence W. Cheuk1

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