Quantum Non-Demolition Photon Counting in a 2d Rydberg Atom Array Christopher Fechisin1 2Kunal Sharma2 3Przemyslaw Bienias1 2Steven L. Rolston1J. V. Porto1Michael J. Gullans2and Alexey V. Gorshkov1 2

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Quantum Non-Demolition Photon Counting in a 2d Rydberg Atom Array

Christopher Fechisin,1, 2, ∗Kunal Sharma,2, 3 Przemyslaw Bienias,1, 2 Steven

L. Rolston,1J. V. Porto,1Michael J. Gullans,2and Alexey V. Gorshkov1, 2

1Joint Quantum Institute, NIST/University of Maryland, College Park, MD, 20742, USA

2Joint Center for Quantum Information and Computer Science,

NIST/University of Maryland, College Park, MD, 20742, USA

3IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA

(Dated: October 21, 2022)

Rydberg arrays merge the collective behavior of ordered atomic arrays with the controllability

and optical nonlinearities of Rydberg systems, resulting in a powerful platform for realizing photonic

many-body physics. As an application of this platform, we propose a protocol for quantum non-

demolition (QND) photon counting. Our protocol involves photon storage in the Rydberg array,

an observation phase consisting of a series of Rabi ﬂops to a Rydberg state and measurements, and

retrieval of the stored photons. The Rabi frequency experiences a √ncollective enhancement, where

nis the number of photons stored in the array. Projectively measuring the presence or absence of a

Rydberg excitation after oscillating for some time is thus a weak measurement of photon number.

We demonstrate that the photon counting protocol can be used to distill Fock states from arbitrary

pure or mixed initial states and to perform photonic state discrimination. We conﬁrm that the

protocol still works in the presence of experimentally realistic noise.

Quantum non-demolition photon counting is a key

paradigm in quantum optics with direct applications in

quantum information processing and quantum network-

ing. Photon counting is most simply performed by cap-

turing photons with a detector which converts each pho-

ton to an electrical signal [1,2], but this process destroys

the quantum state of the photons. Methods of QND

photon counting circumvent this issue in various ways,

preserving the quantum state of the photons after the

measurement [3–14]. This is ideal for applications like

quantum networking and state preparation which make

further use of the photonic state after the measurement.

Rather than directly measuring the photon number, it

is sometimes preferable to make a series of weak mea-

surements via projective measurement of an alternative

observable which is more readily accessible experimen-

tally. Many weak measurements performed sequentially

can progressively collapse the system into an eigenstate

of the photon number with high ﬁdelity [4–7,15,16]. So

long as the series of weak measurements does not de-

stroy the quantum coherence of the photons, it serves as

an eﬀective QND measurement of the photon number.

Methods for QND photon counting have been stud-

ied in a wide variety of experimental platforms, includ-

ing cold atomic gases [3], microwave cavities [4–8], opti-

cal cavities [9,10], superconducting circuits [11,12,17],

waveguides [13,18], and nonlinear metamaterials [14].

These proposals each carry drawbacks and advantages

relative to one another, and can be ideal in diﬀerent sit-

uations. Protocols which encode photon number in a

phase, for instance, are well-suited for resolving small

photon numbers, but are only well-deﬁned within a sin-

gle period of the phase [5]. Some proposals approach

the task of non-destructively counting itinerant photons

[10,13,14,17–20], while others require conﬁnement to a

r’

Individual Atoms Collective State

Storage and Retrieval

Driven Oscillation

Rydberg Measurement

FIG. 1. The level structure of each atom in the array, as well

as the collective states of the array. The dashed line indicates

the shift in the Rydberg level |r0idue to the presence of a

Rydberg excitation |ri. This disrupts the EIT condition by

pushing the |eito |r0itransition far oﬀ resonance, making the

system fully reﬂective in the presence of an |riexcitation.

cavity [3–9,11,12]. The protocol which we study in this

work has no fundamental limitation in discerning large

or small photon numbers and is able to count itinerant

photons by storing them in a Rydberg array in free space.

Rydberg atoms, due to their intrinsic controllability

and strong dipole-dipole and van der Waals interactions,

have become a prototypical system for facilitating inter-

actions between photons [21–25] and, empowered by the

development of Rydberg arrays, simulating many-body

physics [26–32]. The quantum optical properties of dis-

ordered ensembles of Rydberg atoms are well-known [33–

36], but the quantum optics of ordered Rydberg arrays is

arXiv:2210.10798v1 [quant-ph] 19 Oct 2022

less well-understood and has been the subject of much

recent theoretical [37–40] and experimental [41] work.

Ordered atomic arrays have been shown to exhibit emer-

gent behaviors arising from cooperative interactions, such

as acting as near-perfect mirrors [42–44], emitting light

in ﬁxed, geometrically determined directions [39], and—

most importantly for this work—storing light with sig-

niﬁcantly increased ﬁdelity [45]. Rydberg arrays combine

these collective phenomena with strong optical nonlinear-

ities, making for a powerful platform for the realization

of photonic many-body physics [46] and, as we study in

this work, QND photon counting.

Our protocol consists of three stages: (I) storing a

photonic pulse in the array, (II) measuring the num-

ber of photons ncontained therein, and (III) retrieving

the stored pulse. The photon number is progressively

pinned down through a series of observation cycles, each

of which consists of a partial Rabi ﬂop to a Rydberg spin

wave state followed by a direct measurement of the pres-

ence of a Rydberg excitation. The frequency of the Rabi

ﬂop is enhanced by a factor of √nfrom the single-atom

Rabi frequency, making it possible to discern arbitrary n.

The outcome of the Rydberg measurement along with the

known evolution time thus serves as a weak measurement

of photon number n.

Our main result is that, in the absence of dephasing

noise during the observation stage, we can perform QND

detection of nphotons with ﬁdelity limited only by stor-

age and retrieval error in time tγ=0 ∼√n/Ω, where Ω is

the single atom Rabi frequency and γis the dephasing

rate. In the presence of dephasing noise, our detection

is no longer completely non-demolition and takes time

tγ>0∼γn/Ω2. We discuss later precisely how destruc-

tive the protocol is in the presence of noise.

The physical system.—We consider an ordered two-

dimensional array of atoms with subwavelength spacing,

where each atom has the level structure shown in Fig. 1.

Lowercase letters label single-atom states, while upper-

case letters label collective states of the many-body sys-

tem. In particular, |giis a ground state, |eiis an excited

state, |siis a metastable shelving state, and |riand |r0i

are Rydberg states. In a later section, we will propose

particular levels in Yb to realize these states.

|Giis the many-body ground state in which all atoms

are in the state |gi.|Sniis the symmetrized collec-

tive state with nexcitations of individual atoms to |si:

|Sni=1

√NCnPN

ij=1 ˆσ(i1)

sg ··· ˆσ(in)

sg |Gi, where ˆσij := |iihj|.

Similarly, |Rniis the symmetrized spin wave state with

n−1 atoms excited to |siand one atom excited to the

Rydberg state |ri:|Rni=1

√nPN

i=1 ˆσ(i)

rs |Sni. We assume

that the entire array is within a blockade radius, so that

further excitations to |riare forbidden by the blockade.

The protocol.—In Stage I, an initial photonic state cou-

ples to the |gi − |eitransition and is stored in the array

using an auxiliary classical control ﬁeld acting on the

|si−|eitransition [45]. The initial photonic state is sent

through a beamsplitter so that it is normally incident

upon the array symmetrically from both sides, as is nec-

essary for optimal storage eﬃciency [45]. We denote the

photonic state as a superposition of number states |ni,

|ψphi=PN

i=1 cn|ni.Nis the number of atoms in the

array and therefore the upper bound on the number of

excitations which can be stored in the array. Photon

storage is performed on the |gi−|ei−|siΛ-subsystem

as described in [45]. Storage maps the state of the ar-

ray from |Gito Pncn|Sni, where the amplitudes cnare

inherited from |ψphi. Each photon is therefore stored as

an excitation from |gito |si.

Stage II consists of two operations on the collective

state: (1) Rabi ﬂops between the states |siand |riand

(2) projective measurement of the presence of a Ryd-

berg excitation (see Fig. 2). To drive the collective os-

cillation, we couple |siito the Rydberg state |riivia

the rotating frame Hamiltonian ˆ

H= Ω PN

i=1 ˆσ(i)

rs + h.c.,

where Ω is the Rabi frequency of the transition and the

sum is over all atoms in the array. This induces a cou-

pling between the collective states |Sniand |Rniwhich

depends explicitly on the number of stored photons n:

hSn|ˆ

H|Rni=√nΩ =: Ωn. The objective of this stage is

to indirectly and progressively measure the photon num-

ber nby directly and repeatedly measuring the pres-

ence of a Rydberg excitation after evolution under ˆ

for some known time. Each observation cycle consists

of a driven oscillation for time τiand a projective mea-

surement of the collective state yielding the measure-

ment outcome mi∈ {Rydberg,No Rydberg}, building

over time a measurement record through cycle Tdenoted

MT={(τi, mi)}i=1,...,T . In Appendix C, we discuss how

to choose times {τi}to expedite convergence.

Our measurements project the array into the subspace

with or without a Rydberg excitation in |ri, conditioned

on the outcome of the measurement. We are able to ef-

ﬁciently perform such a measurement by tuning to EIT

and applying a weak classical probe light. EIT is ap-

plied to the |gi − |ei − |r0isubsystem, with the |si − |ri

subsystem acting as a ‘switch’. Thus in the absence of

a Rydberg excitation, the EIT condition is satisﬁed and

the array is transmissive, but in the presence of a Ryd-

berg excitation the EIT condition is disrupted and the

array is once again near-perfectly reﬂective [39]. This

measurement takes ﬁnite time which is limited by the

width of the EIT transparency window. We note that

a similar mechanism was used in [47,48] for performing

photon subtraction with atomic clouds.

Stage III consists of retrieval of the stored photons.

Assuming that the measured photon number was n, the

array must be in the state |Snito retrieve the photons. If

the preceding measurement instead projected the state to

|Rni, we can simply drive for time τ=π/2Ωnand arrive

at |Sni, so long as the measurement has converged to

Driven

Oscillation

Rydberg

Measurement

Measurement Record

Stage III:

Photon Retrieval

For time

Outcome

State Inference

stored in record

informs choice of

Stage I:

Photon Storage

Stage II:

Observation

FIG. 2. a) The three stages of the protocol. Stage II is the focus of this work. b) Driven oscillation between the collective

states |Sniand |Rni, where the red atom is in the Rydberg state and the dashed line depicts the blockade radius. The

traﬃc light reﬂects the fact that the array permits the probe light to pass only when there is no Rydberg excitation present.

This is because the array is tuned to EIT, which is disrupted by a Rydberg excitation. c)-d) Projective measurement in the

{Rydberg,No Rydberg}basis, the results of which are stored in the measurement record Mi. The two sides of the array indicate

the possible measurement outcomes. The measurement is performed by shining weak classical light on the array, indicated

by the yellow arrow. e) The measurement record is used to infer the state of the system. Here we schematically depict an

intermediate inference based on a small measurement record Miwhich has not yet converged to a Fock state.

n. Retrieval can then be performed as in [45]. The ﬁnal

photonic state comes out symmetrically from both sides

of the array and can be combined onto a single path using

a beamsplitter.

Measurement dynamics.—The focus of our analysis is

Stage II of the protocol. The measurement dynamics

of the system during this stage determine the protocol’s

capabilities and eﬀectiveness. This stage also raises the

question of how to best choose driving times τi.

The dynamics under driving are described by the fol-

lowing Hamiltonian written in the basis of collective exci-

tations: ˆ

Hcoll =PnΩn(|RnihSn|+|SnihRn|).We begin

in the state |ψ(0)i=Pncn|Sniwith amplitudes cnin-

herited from the stored light pulse. Driving under ˆ

Hcoll

for time τyields |ψ(τ)i=Pncn(cos(Ωnτ)|Sni −

isin(Ωnτ)|Rni).Performing the Rydberg measurement

as described in the previous section results in two

possible outcomes: |ψSi=1

pSPncncos (Ωnτ)|Snior

|ψRi=1

pRPncnsin (Ωnτ)|Rni, with probabilities pS=

Pn|cncos (Ωnτ)|2and pR=Pn|cnsin (Ωnτ)|2. The

amplitudes cnare therefore updated by a factor propor-

tional to cos (Ωnτ) or sin (Ωnτ) as a result of the measure-

ment, so that c(1)

n∼cnsin (Ωnτ) or c(1)

n∼cncos (Ωnτ),

depending on the measurement outcome. This reﬂects

the partial information learned about the photon num-

ber from a single measurement of the collective state.

After each measurement, the system is once again in a

state of the form |ψi=Pnc(i)

n|Xni, where X∈ {S, R}.

Further iterations of unitary evolution and measurement

will continue to update the amplitudes, so that after the

ith observational cycle the amplitudes are given by {c(i)

n}.

For an arbitrary pure or mixed initial state (see next

section), the state converges to a single photon number

state as the protocol iterates, analogous to the progres-

sive state collapse observed in [5], with the outcome al-

ways the distillation of a single photon number state.

Generalization to mixed states.—The analysis of the

preceding sections readily generalizes to mixed intial pho-

tonic states. The dynamics in this case are governed

by the master equation ˙ρ(t) = −i[ˆ

Hcoll, ρ], under which

the populations of ρevolve independently of the coher-

ences in the Fock basis. This implies that any two den-

sity matrices ρand ρ0which satisfy ρii(0) = ρ0

ii(0) will

also satisfy ρii(t) = ρ0

ii(t). In particular, for any mixed

state ρ(0) there exists some pure state ρ0(0) such that

ρii(0) = ρ0

ii(0) and therefore ρii(t) = ρ0

ii(t). This has

several important implications for the performance of the

protocol which are discussed in Appendix B.

Initial state inference.—An important application of

QND photon counting is state inference—gleaning in-

formation about the initial photonic state from the

measurement record MT={(τi, mi)}i=1,...,T . Con-

sider the set of distributions of photon number states

with maximum photon number N, which we denote

{Pα}N. Each distribution Pα= (p0, . . . , pN)∈ {Pα}N

describes a class of potential initial photonic states

which share the same populations. Note that we can-

not discern within these classes, as the protocol is in-

sensitive to coherences between number states. Bayes’

Theorem yields the probability that the initial state

was described by Pα, conditioned on the measurement

record MT: Pr(Pα|MT) = Pr(MT|Pα)Pr(Pα)

Pα0Pr(MT|Pα0)Pr(Pα0), where

Pr(Pα) is a prior over the set of initial distributions.

Pr(MT|Pα) the quantity which we must compute to

ﬁnd Pr(Pα|MT). Because diﬀerent number states are

not mixed through the protocol, this expression can be

written Pr(MT|Pα) = PnpnPr(MT|n),where Pr(MT|n)

is the probability of observing the measurement record

MTgiven initial state |Sni. This is readily given by

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QuantumNon-DemolitionPhotonCountingina2dRydbergAtomArrayChristopherFechisin,1,2,KunalSharma,2,3PrzemyslawBienias,1,2StevenL.Rolston,1J.V.Porto,1MichaelJ.Gullans,2andAlexeyV.Gorshkov1,21JointQuantumInstitute,NIST/UniversityofMaryland,CollegePark,MD,20742,USA2JointCenterforQuantumInformationandComput...

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Quantum Non-Demolition Photon Counting in a 2d Rydberg Atom Array Christopher Fechisin1 2Kunal Sharma2 3Przemyslaw Bienias1 2Steven L. Rolston1J. V. Porto1Michael J. Gullans2and Alexey V. Gorshkov1 2.pdf

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Quantum Non-Demolition Photon Counting in a 2d Rydberg Atom Array Christopher Fechisin1 2Kunal Sharma2 3Przemyslaw Bienias1 2Steven L. Rolston1J. V. Porto1Michael J. Gullans2and Alexey V. Gorshkov1 2

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