Driving alkali Rydberg transitions with a phase-modulated optical lattice R. Cardmanand G. Raithel Department of Physics University of Michigan Ann Arbor MI 48109 USA

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Driving alkali Rydberg transitions with a phase-modulated optical lattice

R. Cardman∗and G. Raithel

Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

(Dated: October 6, 2022)

We develop and demonstrate a spectroscopic method for Rydberg-Rydberg transitions using a

phase-controlled and -modulated, standing-wave laser ﬁeld focused on a cloud of cold 85Rb Rydberg

atoms. The method is based on the ponderomotive (A2) interaction of the Rydberg electron,

which has less-restrictive selection rules than electric-dipole couplings, allowing us to probe both

nS1/2→nP1/2and nS1/2→(n+ 1)S1/2transitions in ﬁrst-order. Without any need to increase

laser power, third and fourth-order sub-harmonic drives are employed to access Rydberg transitions

in the 40 to 70 GHz frequency range using widely-available optical phase modulators in the Ku-band

(12 to 18 GHz). Measurements agree well with simulations based on the model we develop. The

spectra have prominent Doppler-free, Fourier-limited components. The method paves the way for

optical Doppler-free high-precision spectroscopy of Rydberg-Rydberg transitions and for spatially-

selective qubit manipulation with µm-scale resolution in Rydberg-based simulators and quantum

computers.

Innovations in quantum technologies based on Ryd-

berg atoms rely on manipulation of their internal states.

Technologies include simulators exploring quantum phase

transitions and walks [1–5], quantum processors [6,7],

and Rydberg-atom-based sensors [8,9]. It is often beneﬁ-

cial to trap and arrange the Rydberg atoms using tightly

focused laser beams, optical-tweezer arrays, or optical

lattices to conﬁgure such systems. Coherent interactions

on Rydberg qubits can be performed with rf to sub-THz

radiation. The diﬀraction limit of &1 mm then poten-

tially disallows single-qubit operations or short-distance

gates. One method to achieve spatial selectivity of Ryd-

berg transitions on a µm-scale, required in many of these

applications, is through optical addressing of isolated-

core excitations (ICE) in alkaline-earth atoms [10–16],

but limitations of the ICE method include autoioniza-

tion of low-lRydberg states due to Rydberg-ICE interac-

tion. Also, ICE addressing is not practical in commonly-

used alkali atomic species. Here we explore direct optical

drives of Rydberg transitions as a more widely-applicable

method with µm-scale spatial selectivity.

Rydberg transitions can be directly optically driven

through ponderomotive interactions, e2A2/2me, where

Ais the vector potential of the driving laser [17]. Driv-

ing ponderomotive transitions entails generating an op-

tical intensity gradient that is spatially varying within

the Rydberg-electron’s wavefunction, and modulating

the intensity distribution at (a sub-harmonic of) the

atomic transition frequency. Suitable control of the

modulation frequency leads to transitions between Ryd-

berg states. Ponderomotive transitions in modulated op-

tical lattices typically have a Doppler-free component

with an interaction-time-limited linewidth [18]. Pon-

deromotive Rydberg atom spectroscopy has been pre-

viously performed by amplitude-modulating an optical

lattice, allowing transitions between states {|0i,|1i} of

equal parity, i. e. h0|ˆ

Π|0i=h1|ˆ

Π|1i[17]. Odd-

parity (h0|ˆ

Π|0i=− h1|ˆ

Π|1i) transitions are forbid-

den for this drive method, unless the modulation is de-

tuned from resonance by the lattice trap-oscillation fre-

quency [18,19] and the atom’s motional quantum state

νis changed, which is generally undesirable. On the

other hand, Rydberg quantum simulators operating on

electric-dipole interactions between atoms sometimes re-

quire the preparation of a mixed-parity system (e. g.,nS

and n0Patoms [3,5,20]). Harnessing optically-driven

ponderomotive transitions for Rydberg quantum simu-

lators therefore requires a generalization that will allow

local odd-parity transitions without motional excitation

of the atoms.

In this work, we demonstrate optically-driven alkali

Rydberg transitions using an optical lattice that is

phased-modulated at a sub-harmonic ωm=ω0/q of the

atomic transition frequency ω0, with an integer sub-

harmonic order q. The transitions occur in ﬁrst-order

perturbation theory even at large q, no intermediate

atomic states are involved, and the required optical-

ﬁeld strengths do not increase with q. Optical setup, se-

lection rules, and transition Rabi frequencies in phase-

modulated lattices fundamentally diﬀer from the case of

amplitude modulation. Phase modulation of the laser

allows both odd- and even-parity transitions without

change of the motional number ν. Here we perform pon-

deromotive optical spectroscopy of 85Rb transitions by

scanning the lattice phase-modulation frequency ωmover

a sub-harmonic of the atomic resonance, ωm≈ω0/q. A

one-dimensional lattice with counter-propagating laser

beams is used (wavelength λ= 2πc/ωL= 2π/kL=

1064 nm), and, in a single setup, both odd- nS1/2→

nP1/2(h0|ˆ

Π|0i=− h1|ˆ

Π|1i) and even-parity nS1/2→

(n+ 1)S1/2(h0|ˆ

Π|0i=h1|ˆ

Π|1i) spectra are studied.

The lattice is constituted of three co-linear beams, a

pair of left- and right-propagating unmodulated beams

with optical ﬁelds E(i)

uand E(r)

u, plus a beam with ﬁeld

E(i)

mthat is phase-modulated at ωmand co-aligned with

E(i)

u. These ﬁelds are, in that order,

arXiv:2210.01874v1 [physics.atom-ph] 4 Oct 2022

E(i)

u(R0+re, t) = ˆ(i)E(i)

u(R0) cos [kL(Z0+ze)−ωLt+η2(t)],

E(r)

u(R0+re, t) = ˆ(r)E(r)

u(R0) cos [kL(Z0+ze) + ωLt],

E(i)

m(R0+re, t) = ˆ(i)E(i)

m(R0) cos [kL(Z0+ze)−ωLt+η0+η1cos (ωm∆s/c −ωmt) + η2(t)],(1)

where R0is the atom’s center-of-mass (CM) vector, re

is the Rydberg-electron vector operator, ˆ(i),ˆ(r)are po-

larization vectors of the right- (i) and left-propagating

(r) beams, η0accounts for spurious phase oﬀsets between

modulated and unmodulated beams, η1is the modulation

amplitude, and ∆sis the path length of the modulated

beam from the phase modulator to the atoms. The step-

function phase jump η2(t) is optionally applied to both

i-beams immediately after laser-excitation of the Ryd-

berg atoms in the lattice. The phase jump, if applied,

eﬀects a sudden translation of the lattice relative to the

atoms before the spectroscopic sequence. In our experi-

ments, E(i)

mis about one-tenth the magnitudes of E(i)

uand

E(r)

u. The three beams are overlapped and focused down

to a waist of ∼15 µm in the center of a 85Rb optical

molasses (see Supplement for a detailed schematic of the

optics).

2112

( a )

( b )

( c )

FIG. 1. (a) Qualitative sketch of the trapping potential,

U0cos (2kLZ0)+Uofs, created by E(i)

uand E(r)

u, vs. CM posi-

tion Z0, with two Rydberg atoms roughly to scale. (b) Qual-

itative magnitude of the atom-ﬁeld drive, |UAF |(Z0), formed

by E(i)

mand E(r)

u. (c) Phase of the atom-ﬁeld drive, ξ(Z0) (red

solid), in comparison with phase functions that would apply

to Raman transitions (blue dashed). Several trapped (1) and

un-trapped (2) atom trajectories vs. time, Z0(t), are plotted

on top (details, see text).

The ponderomotive interaction is given by the mean-

square of the ﬁeld in Eq. 1, averaged in time over many

optical cycles and a small fraction of 2π/ωm. One ﬁnds

a time-independent component that constitutes a posi-

tive optical-lattice atom-trapping potential with an oﬀ-

set, U0cos (2kLZ0) + Uofs, and a time-dependent atom-

ﬁeld interaction potential, UAF (Z0, t), that couples states

|0iand |1i. For the latter we ﬁnd

UAF (Z0, t) = ~

2Ωq,0|cos (2kLZ0)|ei(ξ−qωmt)|1i h0|+ h.c.,

(2)

where Ωq,0|cos (2kLZ0)|is the Z0-dependent Rabi-

frequency magnitude for the q-th sub-harmonic drive,

and ξis the Z0-dependent phase of the atom-ﬁeld cou-

pling. The trapping function, |UAF |, and the phase func-

tion ξ(Z0) are plotted in Fig. 1. The staircase shape

of ξ(Z0) is unique to ponderomotive lattice-modulation

drives and gives rise to a novel paradigm of Doppler-free

spectroscopy. We will develop this insight after presen-

tation of the experimental data.

In our ﬁrst demonstration of lattice phase-modulation

drive, we prepare 85Rb atoms in |0i=

46S1/2from a

sample of ground-state atoms laser-cooled and localized

near local maxima of the 1064-nm lattice intensity using

oﬀ-resonant (∆ = +140 MHz), two-photon laser excita-

tion with 780- and 480-nm light. It is η0=η2= 0, while

η1is pulsed on from zero to 1.3(3)πfor the duration of the

drive, τ= 6 µs. We measure the |0i → |1i=

46P1/2

transition, which has a lattice-free transition frequency

ω0/2π= 39.121294 GHz. We use a sub-harmonic or-

der q= 3, and the modulation frequency ωm/2πis

scanned from 13.040211 GHz to 13.040633 GHz in steps

of 3 kHz. Notably, the q= 3 sub-harmonic drive allows us

to project the Rydberg transition (frequency ∼39 GHz)

into the Ku-band (12-18 GHz), for which eﬃcient op-

tical ﬁber modulators exist. Internal-state populations

of |0iand |1iare counted with state-selective ﬁeld ion-

ization (SSFI) [21]. Fig. 2shows the spectrum with an

overlapped numerical simulation (for details of the sim-

ulation, see Supplement).

The peak Rabi frequency Ωq=3,0in Eq. 2for the case

of Fig. 2has the following dependence on experimental

parameters,

Ωq=3,0=−αe(ωL)

~E(i)

mE(r)

u(ˆ(i)·ˆ(r))

× h1|sin (2kLze)|0iJ3(η1),(3)

where αe(ωL) is the free-electron polarizability for light

at angular frequency ωL. This quantity is −545 atomic

units for 1064 nm. For the spectrum in Fig. 2, we es-

timate Ω3,0∼2π×70 kHz and U0∼h×2.5 MHz by

comparing simulated and experimental signals. When

E(r)

uis extinguished, the Rabi frequency vanishes, as the

intensity gradient in the laser ﬁeld no longer varies within

the Rydberg-electron wavefunction. This test proves

that there is no population transfer into |1iby means

of higher-order A·pinteractions originating from mi-

crowave leakage, and that the observed transfer into |1i

is entirely due to the A2interaction in the modulated

optical lattice.

The sub-harmonic order qdoes not appear in the

atomic matrix element h1|sin (2kLze)|0iin Eq. 3. For

this reason, sub-harmonic ponderomotive lattice phase-

modulation spectroscopy does not require higher in-

tensity laser beams with increasing order q. Hence,

the frequency reduction aﬀorded by the sub-harmonic

drive does not come at the expense of larger ac shifts

and lattice-induced photo-ionization rates. This bene-

ﬁt stands in contrast with multi-photon microwave spec-

troscopy, where both ﬁeld intensity required and ac shifts

increase drastically with q. From the Bessel function

Jq(η1) in Eq. 3, it is apparent that sub-harmonic pon-

deromotive lattice phase-modulation spectroscopy comes

with two minor penalties: (1) while no increase in optical

power is required, a moderately higher microwave power

is needed to drive the phase modulator, and (2) the maxi-

mum achievable atom-ﬁeld coupling exhibits a mild drop-

oﬀ as a function of sub-harmonic order q. These small

penalties are heavily outweighed by the massive expan-

sion of accessible Rydberg-transition frequency ranges af-

forded by high-order sub-harmonic drives.

Three peaks appear in the spectrum in Fig. 2. The

central peak is free of Doppler shifts, conserves the mo-

tional state ν, and has a linewidth of ∼200 kHz, near

the Fourier limit. This peak corresponds with atoms ex-

periencing a constant phase ξthroughout the interaction

time; i.e., the atoms remain within a distance of .0.125λ

from a given lattice-intensity minimum (Trajectories 1 in

Fig. 1). The broad sidebands at about ±300 kHz corre-

spond to Doppler shifts of atoms traveling over many

lattice wells (trajectories 2 in Fig. 1). Those atoms move

across multiple steps of ξ(Z0) at a roughly constant ve-

locity valong z, and exhibit a Doppler shift according to

the time average

h˙

ξi ' dξ

dZ0

hvi ' 4π

λhvi= 2kLhvi.(4)

There, we use the fact that the average slope of the step

functions in Fig. 1is dξ/dZ0'4π/λ.

The strength of the Doppler-shifted features in Fig. 2

relative to the Doppler-free line follows from the Ryd-

berg excitation scheme. Rb 

5S1/2atoms, which have a

positive ac polarizability, are laser-cooled and trapped at

the lattice intensity maxima. Rydberg excitation lasers

are tuned to the ground-Rydberg resonance at the lattice

intensity maxima. Because the ponderomotive force gen-

erally repels Rydberg atoms from regions of high laser

- 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6

0 . 0 0

0 . 0 5

0 . 1 0

0 . 1 5

0 . 2 0

0 . 2 5

t r a n s i t i o n p r o b a b i l i t y

Eu(r) = 0

Eu(r) = 0 ; s i m u l a t i o n

E u(r) = 0

antenna spectrum

η2 = 0

( 3 ωm - ω0) / 2 π ( M H z )

q = 3

FIG. 2. Population in |1ias ωmis scanned over ω0/3 in

3 kHz steps. Here |0i=

46S1/2and |1i=

46P1/2. The

blue signal is an average of 10 individual ωm-scans with 400

measurements each and E(r)

uunblocked. The gold line is a

corresponding numerical simulation. The pink signal shows

the population in |1iwhen E(r)

uis blocked for an average of

4 individual scans with 400 measurements each. The shaded

spectrum shows a single-photon microwave drive using a horn

antenna without any 1064-nm light. In all spectra, τ= 6 µs.

intensity, most Rydberg atoms prepared in this way are

not trapped along zand traverse over multiple lattice

periods during the atom-ﬁeld interaction, giving rise to

the strong Doppler-shifted sidebands in Fig. 2. A minor-

ity of the prepared Rydberg atoms is barely trapped in

the Rydberg-atom lattice, which suﬃces to produce the

observed Doppler-free peak. The red-shift of this peak

relative to the ﬁeld-free atomic resonance reﬂects a small

Rydberg-state-dependent diﬀerential light shift.

To enhance the visibility of the Doppler-free peak rela-

tive to the Doppler-shifted side peaks, we suddenly shift

the optical lattice in zby λ/4 immediately after Rydberg-

atom preparation. The shift, implemented by a phase

step function η2(t) with step size πin Eq. 1, places the

atoms near a lattice intensity minimum during atom-ﬁeld

interaction [22]. Most atoms are then trapped while be-

ing probed, and the Doppler-free peak becomes larger

than the side peaks. In Fig. 3we show a spectrum for

the same transition as in Fig. 2, with the sudden λ/4

lattice translation applied.

While the Doppler-shifted sidebands in Fig. 2are sup-

pressed, as expected, they are stronger in the measure-

ment than in the simulation result. We attribute this

disagreement to the fact that the lattice translation is

not perfectly instantaneous. Classically, the Rydberg

atoms, which are initially excited near a maximum of the

Rydberg-atom trapping potential, begin rolling down the

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DrivingalkaliRydbergtransitionswithaphase-modulatedopticallatticeR.CardmanandG.RaithelDepartmentofPhysics,UniversityofMichigan,AnnArbor,MI48109,USA(Dated:October6,2022)WedevelopanddemonstrateaspectroscopicmethodforRydberg-Rydbergtransitionsusingaphase-controlledand-modulated,standing-wavelasereldf...

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Driving alkali Rydberg transitions with a phase-modulated optical lattice R. Cardmanand G. Raithel Department of Physics University of Michigan Ann Arbor MI 48109 USA

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