Emergence of Stationary Many-body Entanglement in Driven-dissipative Rydberg Lattice Gases

8/10/2019 Emergence of Stationary Many-body Entanglement in Driven-dissipative Rydberg Lattice Gases

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Emergence of stationary many-body entanglement

in driven-dissipative Rydberg lattice gases

Sun Kyung Lee,1 Jaeyoon Cho,2 and K. S. Choi1, 3

1Spin Convergence Research Center, Korea Institute of Science and Technology, Seoul 136-791, Korea2School of Computational Science, Korea Institute for Advanced Study, Seoul 130-722, Korea

3Institute for Quantum Computing and Department of Physics & Astronomy,

University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Non-equilibrium quantum dynamics represents an emerging paradigm for condensed matter physics, quantuminformation science, and statistical mechanics. Strongly interacting Rydberg atoms offer an attractive platformto study driven-dissipative dynamics of quantum spin models with long-range order. Here, we explore theconditions under which stationary many-body entanglement persists with near-unit fidelity and high scalability.In our approach, coherent many-body dynamics is driven by Rydberg-mediated laser transitions, while atomsat the lattice boundary reduce the entropy of the many-body state. Surprisingly, the many-body entanglementis established by continuously evolving a locally dissipative Rydberg system towards the steady-state, as withoptical pumping. We characterize the dynamics of multipartite entanglement in a 1D lattice by way of quantumuncertainty relations, and demonstrate the long-range behavior of the stationary entanglement with finite-sizescaling, reaching hectapartite entanglement under experimental conditions. Our work opens a route towardsdissipative preparation of many-body entanglement with unprecedented scaling behavior.

Quantum control ofopen many-body systems has become a

major theme in the quest to explore new physics at the inter-face between condensed matter physics, quantum informationscience, and statistical mechanics15. The ability to controlthe many-body interactions and their dissipative processes hasbeen identified as a powerful resource for the preparation ofsteady-state entanglement613 and the investigation of noise-driven quantum phase transitions2. Indeed, quantum reser-voir engineering provides the framework for dissipative quan-tum computation3,4 and communication14 with built-in fault-tolerance. Furthermore, open system dynamics offers newprospectives to the relationship between entanglement andquantum thermodynamics5.

Laser-driven Rydberg atoms offer unique possibilities for

creating and manipulating open quantum systems of dipo-lar interacting spin models1517. By exciting atoms to high-lying Rydberg states, strong and long-range interactions be-tween the Rydberg atoms can be exploited to induce spin-spininteractions, whereas atoms comprising the many-body statecan couple to their local radiative reservoirs by spontaneousemission18. The competition between the coherent and in-coherent dynamics can drive the system to bipartite entangledstates for two atoms19,20 and novel states of matter for a meso-scopic number of atoms, exhibiting topological order, glassi-ness, and crystallization dynamics2127. Remarkably, the basicprimitives behind such a principle have been demonstrated inthe laboratory by several groups2833.

Despite the tantalizing prospects of quantum-reservoir en-gineering, the main obstacle has been that local decoherence(e.g., spontaneous emission) generally destroys the global en-tanglement of the system. Most proposals reported to datethereby achieve the required non-local jump operator byway of collective system-bath coupling613 in order to sup-press the information loss by local dissipation. In practice,such a coupling is achieved in the highly challenging, strongcoupling regime for an array of qubits interacting with a com-mon reservoir (e.g., cavity mode). Furthermore, the inher-ently local nature of the driving fields hardly allows only a

single entangled state to be distinctively separated from the

coupling to the reservoir, which enforces the introduction ofauxiliary coherent manipulations and multiple time-steps ofquantum gates and dissipations to single out a particular en-tangled state from a broader subspace21, diluting the very na-ture of quantum-reservoir engineering.

Another challenge is the characterization of entanglementin the many-body state (t)under evolution1,34. For interact-ing spin systems, spin waves are the quasiparticle excitationsdescribing the collective state of the atomic mode. Entan-glement in such a system can be defined by the correlationsamong the collective excitations1. Hence, verification proto-cols for mode entanglement can be extended to extract themany-body entanglement of these quantum spin systems3538.

Uncertainty relations, as defined in Refs. 37, 38, can be ap-plied to access the genuine multipartite entanglement of(Ref.39).

Here, we explore such many-body entangled states persist-ing with high fidelity in the stationary limit for laser-drivenRydberg atoms in a lattice. As illustrated in Fig. 1, ourprotocol conceptually begins by globally pumping regularly-arranged Rydberg atoms(AB)with a driving field, wherethe lattice is separated into two partitions A,B . Rydberg ex-citation coherently delocalizes within the subspace defined bysystem atoms A, while reservoir atoms B at the latticeboundary serve as an entropy sink for A with local fieldsdthat enhance the spontaneous decay. By preparing a dark state

in the Markovian dynamics, the atomic sample evolves to-wards the entangled steady-state in the form of an eigenstate

|1 =|WA |g gB of a many-body Hamiltonian Hxyin the single-excitation subspace, where |WA(|g gB)isaW-like entangled state (ground state) forA(B).

We apply our method to generate bipartite entanglementfor N = 4 atoms, and extend our work for N = 6 to in-vestigate the driven-dissipative dynamics of the many-bodyentanglement with the uncertainty relations37,38. We findthat quadripartite W-state persists indefinitely with fidelity

arXiv:1401.00

28v3[quant-ph]2

9May2014


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FIG. 1: Production of stationary entanglement with Rydbergatoms in 1D lattice. (a)Schematic of optically-driven, dissipativeRydberg atoms in a 1D staggered triangular lattice. Distances a0anda1are defined between spins in neighbor and next-neighbor configu-rations. Inset(i)The decay rates1,N= rfor the edge atomsare enhanced by mixing the Rydberg states|r with short-lived|ewith fields d (Methods). Inset(ii) Atoms are pumped by a driv-ing fieldwith detuning. (b) Rydberg blockaded atomic structureshowing a rich family of anharmonic levels separated by subspacen. (Left) Two-photon process H2optically pumps the population tothen = 1manifold. (Right) XY Hamiltonian Hxy dictates the de-localization dynamics within then = 1subspace. Jij is driven by

Raman transitions|r(1)i |r(1)j , while (i)ls is induced by lightshifts (Methods).

F 0.99for N = 6, and that entanglement depth k showsfavorable scaling relative to its system size, reaching hec-tapartite (k = 100) entanglement for N = 128 atoms.Unlike previous methods with auxiliary unitary and time-sequential manipulations613,21, the many-body entanglementemerges purely out of the open system dynamics in a time-independent, continuous fashion with local decoherence, aswith the original spirit of optical pumping. Our methodthereby allows the scalable production of high-fidelity dissi-pative many-body entanglement with Rydberg atoms.

ResultsSchematics of optically-driven, dissipative Rydberg atoms.

We consider the many-body states ofNatoms configured in alattice [See Fig. 1(a)], irradiated by a uniform driving field that couples the atomic ground state |g to the highly excitedRydberg state|r with detuning . A pair of atoms i, j inthe Rydberg state at lattice sites xi, xj couple each other via

the potential(ij)p =Cp|xi xj |p with power-law scaling,

for which we take p = 6 for the van der Waals regime ofblockade shifts18. In a frame rotating with the laser frequency,

the Hamiltonian is given by

H=

Ni=1

(i)rr +

(i)x

Ni,j

(ij)p (i)rr

(j)rr , (1)

where (i) =|i| is the projection operator for states |

with {g, r}, and(i)k are the canonical Pauli operators foratomi withk

{x,y,z,

}.

i, j

denotes the sum over alli =j . In the following, we denote the ground state (n= 0) as|G =|g g, the singly-excited (n = 1) states as |r(1)i =|g1 ri gN, and the doubly-excited (n = 2) states as|r(2)ij = |g1 ri rj gN for the excitation subspacen=

i(i)rr .

The open many-body dynamics for the atomic state is

governed by a Markovian master equation = i[H,] +Lwith the Lindblad superoperatorsL =i i2 ((i)+ (i) {(i)+ (i) ,}) for the atomic coupling to their local radiativereservoirs. As discussed in the Methods, in order to allowthe jump n n 1, we can arbitrarily set the decay ratei |

d|2/

erelative to its free-space rate

rby coherently

mixing the Rydberg level |r and a rapidly decaying |e withfieldd, whereeis the decay rate of|e (inset of Fig.1).Dissipative production of many-body entanglement. Asshown by Fig. 1(b), our dissipative protocol starts by opti-cally pumping the population inton = 1subspace by driving

|G |r(2)i,i+1 with the field through two-photon tran-sition H2 with =

(i,i+1)p /2. Higher-order transitions

(n = 1 n = 3) are suppressed for moderate N due tothe long-range nature of

(ij)p (Methods). We thereby adi-

abatically eliminate|G and|r(2)ij in the off-resonant limit

|

(ij)p

| wd, and obtain an effective Hamiltonian

Hxy =i,j

Jij

(i)+

(j) +

(i)

(j)+

i=1

(i)ls

(i)rr (2)

in the single-excitation manifold, wherewd=

2r/4 + 22

is the power-broadened linewidth. The XY Hamiltonian Hxydelocalizes the Rydberg excitation between sites i, jat a hop-

ping rateJij = 2

2

(ij)p

, with each siteisubjected to a

magnetic field (i)ls =

2

j=i

2

(ij)p

(Methods).

The dissipative many-body entanglement for the steady-state limt = ss is prepared as follows. We first

identify the spectrum

{i,

|i

} ofHxy in the n = 1 sub-

space. Our goal is to set Jij , (i)ls such that one and only oneof the eigenstates, say|1, corresponds to a product ofWstate|WA =

iA |r(1)i for a subset A of atoms (sys-

tem atoms) and ground state|g gB for another subsetB (reservoir atoms), thereby leading to|1 = |WA|g gB . We control the relative hopping ratesJi,ix be-tween nearest neighbors (x = 1) and next-nearest neighbors(x= 2) in a lattice to obtain dark resonance for atoms B. Byenhancingi for atomsB , the atomic sample is dissipativelydriven to the entangled dark state |1.


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FIG. 2:Driven-dissipative dynamics of bipartite atomic entangle-

ment. (a)Contour of stationary entanglement fidelityF2with inter-action parameter and distancea0(in units of blockade radiusdB).(b)Dynamics of entanglement fidelityF2(t)as a function of pump-ing time (in units of). Inset. Temporal evolution of concurrence Cfrom unentangledC = 0to maximal entanglement C = 1for theparameters:= wd/2(a0/dB)

6, = 103, and1,4 = 104r for

atoms1, 4with {, a0/dB} = { 6

3, 0.26}.

Emergence of dark states for open many-body dynamics.

Qualitatively, the dark resonances Ji,i+1 = Ji,i+2 occurfor atoms{1, N} at the lattice boundary of a 1D staggeredtriangular lattice in Fig. 1(a) for

=

6

3 and

N = 4.

determines the relative strength between nearest (a0) andnext-nearest (a1) neighbor interactions by the relation =a1/a0. More generally, for N 4, quantum interferencebetween multiple pathways|r(1)i |r(1)j occurs so that|1 =|WA |g gB emerges as the unique dark state(Methods). This process is analogous to coherent popula-tion trapping (CPT) for levels consisting of radiative states

{|r(1)iB} with decay rateiB rcoupled to metastablestates {|r(1)iA}. We define atomsB as suitable atomic reser-voirs, whereby the atoms are continuously projected to theground state by spontaneous emission (Ref. 42). In orderto enable this process, we locally enhance the decoherence

|d|2/e for the reservoir atoms B by 104 relative tothe radiative ratesr of the system atoms A. Any Rydbergpopulation in atoms B will cause the overall atomic state tobecome bright and decay until it reaches the unique steady-state |1. Many-body entanglement is thereby established forthe stationary state ss= |11|.

The entanglement dynamics displays an intricate behavior,as the atomic sample is driven to the steady-state ss. At theearly stage of Liouvillian dynamics (0 t1 1/), atomsin |G =|g g are rapidly pumped to then = 1subspace.

The Rydberg excitation then delocalizes under Hxy with off-resonant Raman transitions Jij. At the final stage (t2 1/),the Rydberg lattice gas is dissipatively pumped to aW-likeentangled state|WA, which separates from|g gB. Theentanglement fidelityF is thereby determined by the branch-ing ratior/ 104 between the lifetimes of dissipativeand coherent dynamics. Because our procedure does not in-volve adiabatic evolutions, our dark-state pumping protocol

is in principle scalable to arbitrarily large N with extendedsamplesL dB only limited by F = 1 O(r/). Bycontinuously driving the system towards ss, the many-bodyentanglement is auto-stabilized in the presence of noise anddecoherence.

Open-system dynamics for bipartite atomic entanglement.

In the following, we perform a numerical analysis of the re-laxation behavior of the Rydberg gas to a stationary bipartiteentanglement for atom numberN= 4and enhanced radiativerates1,N = for the edge atoms. Fig. 2(a) displays thecontour map of entanglement fidelity F2 =2|TrB [ss]|2for the stationary state ss relative to |2 = 1/

2(|g2r3 +

|r2g3

) as a function of interaction parameter and distance

a0 (in units of blockade radius dB = 6

C6/wd). The pro-file of fidelity along a0 depicts the requirement of Rydberg-blockade regimea0 < dB to provide sufficient nonlinearity

inn[Fig.1(b)], selectively driving transitions |G |r(2)i,i+1and adiabatically eliminating subspacesn = 0, 2 (Methods).Atoms in the region 0.2 a0/dB 0.5 are thereby effi-ciently pumped to the single-excitation subspace. The interac-tion parameteris tuned to numerically maximize the steady-state entanglement fidelity up to F2 = 0.9982for 1 =

6

3and 2= 0.36at a0/dB = 0.26. To validate our entanglementpumping scheme, we further show the dissipative dynamics ofconcurrenceC(Ref. 34)for 1 in the inset of Fig. 2(b). Theatomic sample is driven to a maximally entangled state withF2 = 0.9965withint = 200.

Evolution of many-body entanglement and uncertainty-

based entanglement witness. Now, let us treat the case ofmany-body entanglement withN= 6atoms in the 1D lattice,as an example of multipartite system. With same parameterset and , we simulate the dissipative dynamics of entan-glement fidelity F4(t) =4|TrB [(t)]|4, with respect tothe ideal symmetric W state|4 = 12

5i=2 |r(1)i by way

of quantum-trajectory method [See Fig. 3(a)]. Here, we haveoptimized the steady-state fidelitymax(F4) = 0.9912for theparameters {, a0/dB} ={1.1996, 0.285}, thereby setting asymmetric quadripartiteW-state

|1

=

|4

|g1, g6

.

The dissipative transitions of genuine many-body entan-glement is detected by the uncertainty relations3739, whichserves as the collective entanglement witness{(t), yc(t)}(Ref. 1). The uncertainty =

i2i measures the to-

tal variance of projection operators i =|WiWi| to NA-dimensional W-state basis|Wi, while yc = 2NANA1

p2p0p21

detects the amount of higher-order spin-waves (e.g., p2 =i=j(i)rr (j)rr) and ground-state fraction p0 =

i(i)gg

relative to the singly-excited spin wavep1 =

i(i)rr , where


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FIG. 3:Driven-dissipative dynamics of many-body entanglementfor six atoms. (a) Dynamics of entanglement fidelity F4 at max-imum point{, a0/dB} ={1.1996, 0.285} simulated by way ofMonte-Carlo wavefunction. Inset. 3D map of steady-state entan-glement fidelity F4 for interaction parameter and distance a0 (inunits of blockade radius dB). F4 > 0.99 for0.25 a0 0.35(b) Dissipative preparation of genuine quadripartite entangled state.The entanglement parameters {(), yc} transit from fully separa-ble (black) to bipartite entanglement (purple, t2 2), to tripartiteentanglement (green,t3 65), and to stationary quadripartite en-tanglement (red,t4 100).

NA is the number of atoms in A. For an ideal W-state,

min{, yc} {0, 0}, while the boundary (k1)b repre-sents the minimum uncertainty for(k 1)-partite entangledstates for a given yc. Violation of the uncertainty bound

()< (k1)b then signals the presence of genuinek-partite

entanglement stored in (t), with the full NA-partite entan-

glement certified by 0 () < (NA1)b .{, yc} can bemeasured by the transverse collective spin variance and by theexcitation statistics (Methods).

By applying the witness{, yc}, we observe that atomsinitially in ground state are dissipatively driven to the quadri-partite entangled

Wstate by sequentially crossing the bound-

aries (1)b ,

(2)b ,

(3)b in Fig. 3(b). The dissipative transi-

tions of many-body entanglement are indicated by black, pur-ple, green, and red lines of Fig. 3(b) for the average tra-

jectory (t). For pumping time t4 100/, the many-body system exhibits a full quadripartite entanglement witha moderate atom number N = 6, and reach{, yc}|ss{1.5 102, 2 104}, as the atoms are pumped to the de-sired eigenstate |1.Finite-size scaling of steady-state entanglement. Next, we

FIG. 4: Finite-size scaling behavior of many-body entanglementdepth. Multipartite entanglement behavior of the many-body sys-tem ss is probed with quantum uncertainty witness for yc 0by way of direct diagonalization ofHxy as a function of atom num-ber N. We obtain stationary eigenstates ss = |11|, exhibit-ing up to hectapartite entanglement for N = 128 atoms. The un-certainty boundaries for20-partite,40-partite,60-partite,80-partite,100-partite entanglement are shown as dashed lines.

move on to the question of finite-size scaling behavior of thestationary many-body entanglement. Although the full dy-namical simulation for large N is beyond our computationalcapability, the steady-state entanglement can be establishedby analyzing the unique eigenstate|1 that meets the darkresonance condition = 1, for which Ji,i+1 =Ji,i+2.Perturbations by higher-order interactions are negligible, as

x>2 |Ji,i+x|/|Ji,i+1| 102. We truncate our analysisup to next-nearest-neighbor interactions for the following dis-cussion. We define the entanglement depthk in accord withthe concept ofk-producibility for qubits1,34, thereby identify-

ing the minimal depth for genuine km-partite entanglement toproduce the purported state ss.

We directly diagonalize the many-body Hamiltonian Hxyfor1, and characterize the resulting entanglement depth k ofthe stationary eigenstate|1 up to N 128. Fig. 4cap-tures our result of{, yc 0} for the dark state|1 =|WkA |g gB, where|WkA is the k-partite symmet-ricWstate. Due to the nonlinear sensitivity of our witnessfor some regionk, we characterize the scaling of the minimalentanglement depthkm k(Methods). The shaded area rep-resents the physical region, wherebykm-partite entanglementcould be defined for a given N, and the dashed lines are theuncertainty bounds for 0 100-partite entanglement (with20-partite increments). Remarkably, we observe a favorablescaling up to genuine hectapartite (km = 100) entangle-ment forN= 128atoms.

Discussion

Our entanglement pumping scheme is experimentally fea-sible. By exciting 85Rb atoms to Rydberg state|r =|100S1/2, quadripartite entangled states could be preparedforF4 > 0.99withint4 = 10s in the region1(1.2)ma0(a1) 1.5(1.8)m. The limit for any driven-dissipativeapproach with Rydberg lattice gases will be the photoioniza-


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tion lifetimetph 4 ms (Methods). Since the pumping timeto reach ssdepends onN, our method can be applied to gen-erate stationary hectapartite entanglement withintp < tph forN= 128atoms witha0 1m (Methods).

ForN 128, one could explore emergent atom-field sys-tems embedded in photonic crystals. Dispersive optical in-teractions near band edges can induce dipole-dipole oscilla-

tions Hxy and Rydberg blockades H2 with tailored scaling

(ij)p cxij between low-lying excited atoms43,44. Decayrates i =

can be controlled by the density of states45.

More generally, the delocalization dynamics Hxy in the high-order subspace n (Methods) can be extended to examine local-ity estimates of many-body systems46,47 and bosonic samplingfor quantum algorithms48.

We have examined the conditions under which driven-dissipative dynamics displays a rich family of many-body en-tangled states, and have provided a criteria for the purportedentanglement. The stationary many-body entanglement showsa favorable long-range behavior up tokm= 100forN= 128atoms. Our work thereby opens the door towards an opensystem simulator with well-controlled coherent and dissipa-

tive many-body dynamics, monitored by information-basedquantities13.

This work is funded by the KIST Institutional Programs,and, in part, by the Ontario Ministry of Research & Innova-tion and Industry Canada. We acknowledge the support ofNVIDIA Corporation with equipment donations.

Methods

Control of spontaneous emission rates. As discussed in the main

text, for reservoir sitesi B, atoms initially in the Rydberg state |rwith decay rater radiatively couple to a highly decoherring state

|e

with decay rate e

r so that atoms in bipartition B can

behave as an effective reservoir channel for the system atoms inpartitionA. In this section, we discuss how we could manipulatethe spontaneous emission rate i of the Rydberg state|r for thereservoir atoms.

As illustrated in Supplementary Fig. 1(a), we consider a -typeenergy level diagram, where |r is dressed with |e by auxiliary fieldd. In the rotating-wave frame of the dressing laserd, the Hamil-tonian is given by

Hd = d(i)ee + d

(i)er +

(i)re

. (3)

The resulting optical Bloch equations are, then,

(i)ge = e(i)ge +id(i)ge +id(i)gr (4)

(i)gr = r(i)gr +id(i)ge , (5)where e,r= e,r/2 and dis the detuning for the dressing field drelative to the transition |e |r. In writing Eqs. 45,we have ne-glected the Langevin noise forces F and assumedc-number coun-

terparts for (i) (i) . Hence, we find that the atomic coherence

(i)gr(t)between |g, |r obeys the following equation of motion(i)gr (i +r) (i)gr e(ir)t + 2d(i)gr e(ir)t = 0, (6)

with(i)gr =(i)gr e

rt and =ie+ d.

Supplementary Fig. 1(b) shows the dynamics of Rydberg popu-

lation (i)rr(t)obtained by numerically solving Eqs. 45for the pa-

rameters of Figs. 24 withe = 104r. The black solid (dashed)

line is the atomic dynamics ford = 10r (d {102r, , 9 102r} with102r increments). The red line is the result of atomicdecay 103r withd = 103r. As we increased e, wefind that the effective decay rate for the reservoir atoms scales with |d|2/eup tod 0.1e.

In order to understand the dynamics, we formally integrate Eq.

4to obtain (i)ge eit = id

(i)gr eitdt d (i)gr eit. As-suming slowly-varying amplitude (i)gr ford e, we obtain thefollowing equation of motion

(i)gr =

r+i2d

(i)gr , (7)

where the effective decay rate is given by eff= r+ e|d|

2

||2+2ewith

= 2eff. As further discussed below, |r =|100S1/2 and |e =|5P1/2 have decay rates withe/r 104. Hence, decay rates forreservoir sites could be enhanced up to 4 order of magnitude with/r 104.Optical pumping to arbitrary n-subspace in an anharmonic Ryd-

berg ladder.Now, let us discuss the possibility of optically pumpingthe system ofNatoms to an arbitrary target nt-excitation subspacewith nt< N2, for which nt = 1in the main text. This is achievedby a set ofnt lasers resonantly driving the two-photon transitionsn n + 2 (n {0, , nt 1}) with effective Rabi frequencies(n)2 [See Supplementary Fig. 2(a)] and the three-photon transition

nt 2 nt+ 1with effective Rabi frequency (n)3 [see Supple-mentary Fig. 2(b)]. BecauseL dissipates the levels n n 1,the atomic population is pumped to the target subspacent[See Sup-

plementary Fig. 2(b)]. For the case ofnt = 1,(0)2 is provided by

a single global field for the entire atoms [See Supplementary Fig.2(c)].

The efficacy of this procedure to address only a particular transi-tionn

n depends on the anharmonicity in the Rydberg spectrum

Vn =n|Vp|n, where Vp = Ni,j(ij)p (i)rr(j)rr and|n repre-sents the most shifted state of the n subspace. TheVn is obtainedby degenerate Rydberg configurations with n-nearest neighbor ex-citations (e.g.,|n =|r1, , rn, gn+1, , gN). The Rydbergspectrum is then given by

Vn =n1i=1

nj=i+1

(ij)p . (8)

The transition energy forn n+ 2is then

Vn+2 Vn= 2n

i=1

(i,n+1)p + (1,n+2)p , (9)

so that the anharmonicity is given by

Vn+2,n= (1,n+1)p +

(1,n+2)p . (10)

As shown in Supplementary Fig. 2(b), for a given target subspace

nt, we terminate the two-photon excitations to nt 1 nt+ 1.All subspaces with n {0, , nt 1, nt + 1} are resonantlyconnected by two-photon transitions

(n)2 with detunings

(2)n =

(Vn+2 Vn)/2 and by three-photon 3 coupling with detuning(3)nt2 = (Vnt+1 Vnt2)/3, except for thentsubspace [See Sup-plementary Fig. 2 (b)]. The Rydberg blockade condition for the


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two-photon transitionn n+ 2is then given by

Vn+2,n> w(2)d , (11)

wherew(2)d =

2r+ 2|(n)2 |2 is the power-broadened width of the

two-photon transition n n+ 2 and (n)2 = 22/(2)n is theeffective Rabi frequency.

Optical pumping to the single-excitation subspace. Fornt

= 1,by driving the two-photon transition n = 0 n = 2 with(2)0 = (V2 V0)/2 = (1,2)p /2, the atoms are pumped to adecoherence-free subspace (DFS) for atoms A of the nt = 1sub-space [see Supplementary Fig. 2(d)]. As discussed in the maintext, the DFS is defined by the space spanned by superpositions of

{|r(1)iA}, and the subspace (DS) is set for the reservoir atoms B .In this case, high pumping efficiency to nt = 1 is assured if thehigher-order transitionn = 1 n = 3is blockaded for the leastshifted state |r1, r2, g3, , gN1, rN ofn = 3subspace, thereby(1,N1)p +

(2,N1)p > w

(2)d . For the 1D lattice in Fig. 1(a),

our dissipative pumping scheme works in the region a0/dB 0.13even for N 100, where we take = 102, /r = 104 and = a1/a0 =

6

3. For |r =|100S1/2, the blockade distance isdB = 5.8m, so that(a0, a1)

(750nm, 900nm).

Derivation of effective spin Hamiltonian. In the off-resonant limit

| (ij)p | wd, we obtain the effective Hamiltonian Heff(Eq.2)by truncating the perturbative expansion to the second order and bytime-averaging highly oscillating terms49,

Heff =m,n

[hm, hm]

mnei(mn)t

+m,n

hnhme

i(m+n)t

mn+h.c

, (12)

with the interaction Hamiltonian given by

HI =N

n=1

hneint +hne

int, (13)

whereHI = eiH0tH1e

iH0t, mn = [(1/2)(1/m+ 1/n)]1,

and mn= [(1/2)(1/m 1/n)]1. In particular, we use

H = H0+ H1 (14)

H0 =Ni=1

(i)rrNi 2 due tothe 1/r6 vdW scaling. In the following discussion, we therebytruncate our analysis up to next-nearest neighbor interactions withthe sparse-arrayHxy as

Hxy = Ni=1

(i)ls (i)rr +

Ni


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with

(i)ls =

J2 (4 + 2

26 N) fori = 1, N

J2 (6 + 2

26 N) fori = 2, N 1

J2 (6 + 4

26 N) for2 < i < N 2

(20)

Jij =

J for |i j| = 1J ( 1

26) for |i j| = 2

0 for |i j| >2(21)

with overall factorJ= 42/(i,i+1)p .Eigenstates | withi,= 0can be obtained by controlling the

ratio between nearest and next-nearest terms forJij with . As dis-cussed in Supplementary Fig. 3 (a), this process is analogous to the

behavior of CPT, where destructive quantum interference occurs for

the excitation pathways that connects the bright state |r(1)i (decayrate 104r) to metastable states |r(1)j=i (decay rater). Theemergence of dark state for such a toy model provides an insighton our choice of interaction parameter 1 = 6

3 for sym-

metric (antisymmetric) eigenstates, wherebyJi,i+1 =Ji,i+2. Forinstance, in the case ofN= 4 with

Hxy =

J J J 0J 0 J

J

J J 0 J0 J J J

, (22)destructive interference in the form J1,2(J3,4) =J1,3(J2,3) oc-curs for 1 = 4 = 0. For N 4, the eigenstate|1 withiB = 0cannot be obtained by locally considering the atoms nearthe boundaries (i.e., atoms 1, 2, 3 and N 2, N 1, N). Instead,the uniqueness of the dark state |1 is a manifestation of the many-body interferences forJij,

(i)ls , leading toiB = 0. Nonetheless,

Ji,i+1 =Ji,i+2provides a reasonable guiding principle for us toguess the dark resonance conditions for atoms near the edges for acertain value ofN, due to symmetric sparse characteristics ofHxy.

We confirmed this prediction by solving the full spectrum of thesparse Hamiltonian matrix Hxy withJ|ij|>2

0and by numer-

ically simulating the stationary state of the master equation. Weobtain two sets of eigenstates| =

ii,|r(1)i with 1, =

N, = 0for arbitraryNthat meetsJi,i+1= Ji,i+2at = 1asbelow

set 1 : N= 4 + 6m(m= 0, 1, ) (23){i,} = {0, 1, 1, 0, 1, 1, 0, 1, 1, 0, , 1, 1, 0, 1, 1, 0}

set 2 : N= 6 + 10m(m= 0, 1, ) (24){i,} = {0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0,

1, 1, 1, 1, 0, 1, 1, 1, 1, 0}.Therefore, our method could produce stationary k-partite entangle-ment in the form of an eigenstate |1 =|WkA |g gB withk = 2 + 4m(set 1) andk = 4 + 8m(set 2), plotted as blue dotsin Supplementary Fig. 4 (b). In the hindsight, we can attribute theexistence of symmetric entangled steady-states in Eqs. 2324to thespecial structure ofHxy. As Hxyis sparse, highly symmetric and re-dundant, a kind of commensurability requirement is imposed to theeigenstate under the restriction that the coefficients at the edges arezero.

Even if we were to consider for J|ij|>2, our result would nothave changed for = 1. The truncation would slightly mod-ify the exact eigenstate as|i |i + |i withi|i = 0up to a normalization constant 1. Roughly speaking,i|iscales linear to the perturbation in the energy scale 102 up to

a leading order. Since the energy perturbation to (ij)p 1/r6ijis at most p =

(i,i+3)p /

(i,i+1)p 102 [see Supplementary

Fig. 3(b)], the perturbation to the entanglement fidelity is at mostF 102, which is well within the numerical uncertainty of thequantum trajectory method [see Fig. 3]. In terms of dark resonanceJi,i+1= Ji,i+2, the higher-order interactionsJ|ij|>2for = 1(blue dots) are suppressed by at least 102 relative toJi,i+1, Ji,i+2.By takingN , the higher-order contributions

x>2 |Ji,i+x|

would still be far too negligible to have any impact on the final statewith

x>2 |Ji,i+x| 102 min(|Ji,i+1|, |Ji,i+2|), leading to

F 0.99.The infinitesimally reduced fidelity can then be recovered to

F 1 by displacing to an optimal value by the more gen-eral conditionJ1,2 =

x2x,1J1,1+x for|1 at the expense

of having a slightly modified steady-states, i.e., a new eigenstate|1 = |WA |g gB. Thanks to the inherent symmetry of thesystem, this modified steady-state |W would only marginally dif-fer from the original one. Furthermore, the original steady-state |1could be recovered by re-adjusting the arrangement of the atoms. Inany case, the only sensitive parameter that determines the optimal fi-

delity is the branching ratior/ 104, which sets the balancebetween the lifetimes for dissipative and coherent evolutions, therebythe final fidelityF 1 O(r/).

On the other hand, in the region of 1 (J|ij|>2 J|ij|=1, J|ij|=2), the optimal value cannot be predicted by thedark resonance conditions of the sparse-array matrix Hxy. In thiscase, Jij displays zigzag oscillatory decay as shown in Supplemen-tary Fig. 3(b), and higher-order terms J|ij|>2 must be included inthe analysis.

N-partite uncertainty witness. In this section, we describe ourmethod of constructing the N-partite uncertainty witness fromRef. 37. Our entanglement witness{, yc} consists of identify-ing the boundaries (k1)b for all possible states

(k1)b produced

by convex combinations of pure (k 1)-partite entangled states|(k1)b as well as their mixed siblings with less k. As shownin Refs. 37, 50, the lower bound of (k

1)b is attained by tak-

ing a convex set of{b((k1)

b ), yc(

(k1)

b )} for all pure states(k1)b = |(k1)b (k1)b |. In Fig. 3, we depict the bound-aries

(1)b ,

(2)b ,

(3)b for all possible realizations of fully separable

states, bipartite entangled and tripartite entangled states, respectively,by following the procedures of Refs. 38, 39.

Generally, we can determine the projection operators i =|WiWi| with i {1, , 2m} for arbitrary number of systemsNm= 2

m with the recursive relationship,

|W(m)i = 1

2

|W(m1)i , G(m1) |G(m1), W(m1)i

,

from the initial condition|W(1)1,2 = 1/

2(|gr |rg). Here,|W(m)i = (1/

2m)

2m

i |r(1)i and|G(m) =|g g for Nmatoms. As discussed in Ref. 37, we then construct the uncertaintywitness =

i2i to identify thebounds{(k1)b } for (k1)-

partite entanglement up tok Nm. For convenience, we set themaximalNm NA to be larger than the number NA of atoms inA, so that we could distinguish the entanglement depth k for anyk NA.

For Fig. 4,we assumed the stationary limit, so that yc 0. Inorder to verify the minimum bounds

(k1)b , we only need to op-

timize the overlap of pure (k 1)-partite entangled states of theform|(k1)b = |G(Nk+1)

k1i i|r(1)i with one of the

projectors|Wi. This is achieved when the test state is a balanced


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8

(k 1)-partiteW-state (i.e., |i| = 1/

k 1). Supplemtary Fig.4 (a) depicts the uncertainty bounds(k

1)b withk {1, , Nm}

calculated for yc = 0 and Nm = 27 = 128. The shaded regions

represent the parameter spaces for which ambiguity exists for the

tiered structure (k)b (k+1)b . This is caused by the nonlinearstructure of(|(k)b (k)b |)to POVM values i. For such regions,we conservatively quote the minimum value ofkm and certify thepresence of genuine entanglement depthkm+ 1 stored in the pur-

ported state with() < (km)

b [See Supplementary Fig. 4(b)].Hence, the entanglement depth km (red dot) is a conservative es-timate, which can be detected in an experiment, as opposed to themodel-dependent analysis of k (blue dot) for the pure state form|1 =|WkA |g gB (i.e., by counting the number of non-zero probability amplitudes in |WkA).

Experimentally, the witness {(), yc()} can be determined bydetecting the fluctuation2 St in the collective transverse spin com-ponent St =

i{cos d(i)x , sin d(i)y } and the excitation statistics{p0, p1, p2}, wheredis the detection angle in the transverse plane

x y. As discussed in Ref.39,() (), (25)() = N 1

N

1

N2d2 , (26)

where d= 2N(N1)

ij |dij | is the average off-diagonal coherencedij = |gir| |rjg| for the reduced density matrix 1 in thesingle-excitation subspace. Sincemin2Std = 2

ij |dij|, we

find the following upper bound of the measured variance

() = N

N 1

1

min2StdN 1

2. (27)

The quantum statisticsyc = 2NN1

p2p0

p21can be detected by the

total excitation statistics {p0, p1, p2} with MCP ionization signals.Hence, our entanglement witness can be readily implemented evenforlow-resolutionRydberg experiments without the capability to lo-

cally detect the state of single atoms in the lattice.

Experimental parameters with alkali atoms. Let us consider85Rb atoms interacting with optical field near the transition be-tween |g =|5S1/2 and |r =|nS1/2 with two-photon Rabi fre-quency = 12/

and detuning that globally addresses theatomic sample. As shown by Supplementary Fig. 5, this could beachieved by a two-photon transition with Rabi frequencies 1, 2via the intermediate state|e = |5P1/2 with one-photon detun-ing . The Rydberg excitation spectrum displays a highly non-linear excitation spectrum n due to the dipole-dipole interaction

(ij)p = Cp|ri rj |p, with the most shifted level given by con-figuration states consisting of nearest-neighbor excitations|r(2)i,i+1with(i,i+1)p .

In order to achieve the parameter sets of Figs. 24,we take theprincipal quantum numbernp = 100so that |r = |100S1/2, mj =1/2. The radiative lifetime is given by = 0(np), wherenp =np nl is the effective principal number and nl is the quantumdefect. With0 = 1.43ns and = 2.94for |100S1/2 (Ref. 18),we find that the Rydberg lifetime is r = 1ms (i.e.,r 1kHz).On the other hand,e 36MHz for |e. Since ein the limitof strong dressing fields 1 for reservoir atoms, 104r canbe achieved in an experiment.

By setting 1,2 = 100 GHz and = 101,2 (i.e., = 10

GHz), the photo-ionization rate can be determined by

= I

w =2Isat

w

1,2e

2, (28)

where e = 36 MHz is the spontaneous decay rate for |e,Isat = 4.5 mW/cm

2 is the saturation intensity for

|g

|e

and

2 107A2 is the photo-ionization cross-section that cou-ples the Rydberg state|100S1/2 to the continuum free-electronwavefunctions18. Hence, we find that the photo-ionization lifetimeis limited to = 1/ 4ms 1/r (Ref.51).

The blockade shift (ij)p is determined by Rydberg coefficient

Cp, for which we take C6 = 56 THzm6 for the vdW interac-tion between two Rydberg atoms in|r =|100S1/2, mj = 1/2(Refs. 5254). For = 10 GHz and N = 4, 6, the blockade

shift for nearest-neighbors is (i,i+1)6 = 20 THz (a0/dB 0.3),while the power-broadened linewidth for the transition |g |r iswd

2. The resulting blockade radius is rB 6

C6/wd =

4m and 1(1.2)m a0(a1) 1.5(1.8)m for F4 > 0.99.Even for N = 128 atoms, in which a0/dB 0.13, we can set = 102, /r = 10

4 and = a1/a0 = 6

3, thereby leading

to the blockade distance dB = 6

Cp/wd = 5.8m and(a0, a1)(750nm, 900nm). In terms of spatial localizations, the variance ofthe lattice constants would need to be less than a0, a1 0.99 in Figs. 24. This could be readilyachieved in deep optical lattice experiments with zero-point motionx10nm. Hence, Rydberg atoms interacting in the strong block-ade regime with the lattice constants a0, a1 1m> 0/2(Figs.24)can be spatially resolved, so that d can be locally addressedto the reservoir sites without the requirement for diffraction-limitedimaging resolutions0/2(Ref.55).

Therefore, the pumping time for F > 0.95for Figs. 23is thenp 102/ = 10s, which is not limited by the photo-ionizationtime 4 ms. In addition, since the quantum jumps in then = 1 subspace occur on a time-scale oftj O(N2) due to therandom walk for|r

(1)i until it reaches the reservoir sites with

1,N = r, we expect that the pumping time to reach sta-tionarity also scales astp O(N2). On the other hand, if we wereto address every zeros in Eqs. 2324withd, the pumping timetp O(N)will scale linear to the number Nof eigenstates {|}spanningn = 1. By extrapolation, we estimate that our methodcould be extended to generate 100-partite entangled steady-stateswith the parameters{1,2, , , |g, |e, |r}. Further improve-ment in the entanglement depth k may be possible by optimizing

the driving field under the constraint 22

r for a given |r,

which reduces the ionization time t (Ref. 51). Alternative strate-gies, including the use of photonic crystals with atoms in low-lyingelectronic states, will be discussed elsewhere.

In terms of the initialization of the atoms in the 1D lattice, the

atoms would need to be confined in each well with unit filling factor.In practice, this could be achieved by the superfluid-Mott insulatortransition or by the manipulation of laser-induced atomic collisionswith blue-detuned potentials56. The 1D staggered triangular latticecan be easily realized in a free-space superlattice configuration57.Since the general principle of our protocol is not necessarily confinedto a particular lattice configuration, one could explore other configu-

rations in 1D and 2D with arbitrary trap potential landscapes createdby spatial light modulators (SLM).


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Supplemental Figures for

Emergence of stationary many-body entanglement in driven-dissipative Rydberg lattice gases

Sun Kyung Lee,1 Jaeyoon Cho,2 and K. S. Choi1, 3

1Spin Convergence Research Center, Korea Institute of Science and Technology, Seoul 136-791, Korea2School of Computational Science, Korea Institute for Advanced Study, Seoul 130-722, Korea

3Institute for Quantum Computing and Department of Physics & Astronomy,

University of Waterloo, Waterloo, Ontario N2L 3G1, Canada


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12

Fig. S1: (a) Control of spontaneous emission rate via dressing field. Inset (i)The decay rates for the edge atoms are enhanced by dressing theRydberg states |r with auxiliary excited state |e via the dressing fieldsd. Inset(ii)Atoms are pumped by the driving field with detuning. (b) Enhancement in the decay rateeff |d|2/eas a function of the strength of dressing field d. The parameters aree = 104r, andresonant dressingd = 0.


13/15

13

A

|Gi

(a) (b)

(c) (d)

|1iA

3

(2)0 =

(i,i+1)p

2

(i,i+1)p

(i,i+1)p

|r(2)i,i+1i

2

n + 3

n + 2

n+

n

n 1

n =

n =

n = 2

nt + 1

t

nt 1

nt2

nt 3

{(n)2 }

(nt1)2

(0)2

Vn+3

Vn+2

Vn+1

Vn

n1

(n)2

Fig. S2: (a)Anharmonic Rydberg spectrum for two-photon transition(n)2 . Dotted (solid) line represents the energy level for non-interactingatoms (the most shifted energy level with Rydberg interaction).(b) A set ofnt1 two-photon transitions n n+2 (with n {0, , nt1})are resonantly driven at Rabi frequencies {(n)2 }, in tandem with a three-photon laser 3for nt 2 nt+ 1are required to pump atoms tothe target subspacent. Non-coupled subspaceA ofn excitations remains dark throughout the entire driving processes. The target eigenstate

is |1 =

{pi}A{pi}|r(n){pi}A |g gB . (c)A single laser is required to drive n = 0 n = 2for target subspace nt = 1,(d) For

= (2)0 , atoms are pumped to the target eigenstate |1 =

iAi,1|r(1)i A |g gB, whereby decoherence is enhanced for atoms B.


14/15

14

|r(1)N2i

|r(1)N1i

|r(1)N

i

! " # $ % &! &" &$ &% "!&!

'(

&!'$

&!')

&!'#

&!'*

&!'"

&!'&

&!!

&!&

(a)

(b)

|r(1)1 i |r

(1)2 i

|r(1)3 i

J12 J13

|r(1)N2i

|r(1)N1i

|r(1)N

i

JN,N1JN,N2

|r(1)1 i

|r(1)2 i

|r(1)3 i

J12J13

|Gi

|r(1)1 i |r

(1)2 i |r

(1)3 i

|r(2)1,2i

|r(2)1,3i

(1,2)p

2

(1,2)p

2 (1,3)p

(1,2)p

2 |Gi

|r(1)i

i

|r(1)j i

|r(2)i,ji

(1,2)p

2

(1,2)p

2

(i,j)p

J23

JN,N1JN,N2

JN1,N2

J23 JN1,N2

Fig. S3: (a)Interaction strengthJij and light shift (i)ls of the effective HamiltonianHxy are given by Raman transition |r(1)i |r(1)j via

two path mediated by|G and|r(2)i,j . (b) The power law scaling behavior of spin-spin coupling strength Jij with 1D staggered triangularlattices for 1 =

6

3 (blue) and 2 0.36 (red). For1, the spatial range ofJij depicts a monotonic power-law decay, whereas jig-jagoscillatory pattern exists for2.


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15

Fig. S4: (a) k-partite uncertainty bounds(k)b fork {1, , 128} andNm= 128. Shaded regions indicate the parameter spaces for whichambiguity exists for

(k)b (k+1)b , due to the nonlinear sensitivity of. For such regions, we conservatively quote the minimum value km

for the genuinek-partite entanglement with() < (km)b . (b)The minimum entanglement depth km certified by {, yc} (red dots), and

the entanglement depthk in the purported eigenstate |1 = |WkA |g gB (blue dots), with fully balancedk-partiteWstate |WkA.

(a) (b)

Fig. S5: (a) N-atom Rydberg blockade. Effective Rabi frequency between|g and|r is given by . (b) Level diagram for 85Rb atom.The effective transition between |g and |r is formed by a two-photon transition via the intermediate excited state |e, with |g =|5S1/2,|e = |5P1/2, and |r = |nS1/2. is the one-photon detuning respect to |e by field1(1 474nm) andis the two-photon detuningby the field2(2 795nm).

Documents

Emergence of Stationary Many-body Entanglement in Driven-dissipative Rydberg Lattice Gases