Stimulated adiabatic passage in a dissipative Rydberg superatom
aa r X i v : . [ qu a n t - ph ] F e b Stimulated adiabatic passage in a dissipative Rydberg superatom
David Petrosyan
Institute of Electronic Structure and Laser, FORTH, GR-71110 Heraklion, Crete, Greece
Klaus Mølmer
Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark (Dated: July 10, 2018)We study two-photon excitation of Rydberg states of atoms under stimulated adiabatic passagewith delayed laser pulses. We find that the combination of strong interaction between the atomsin Rydberg state and the spontaneous decay of the intermediate exited atomic state leads to theRydberg excitation of precisely one atom within the atomic ensemble. The quantum Zeno effectoffers a lucid interpretation of this result: the Rydberg blocked atoms repetitively scattering photonseffectively monitor a randomly excited atom which therefore remains in the Rydberg state. Thissystem can be used for deterministic creation and, possibly, extraction of Rydberg atoms or ions oneat a time. The sympathetic monitoring via decay of ancilla particles may find wider applicationsfor state preparation and probing of interactions in dissipative many-body systems.
PACS numbers: 32.80.Ee, 32.80.Rm, 32.80.Qk
I. INTRODUCTION
Rydberg atoms strongly interact with each other vialong-range dipole-dipole (DD) or van der Waals (vdW)potentials [1]. Within a certain interatomic distance, theinteraction-induced level shifts can suppress resonant op-tical excitation of multiple Rydberg atoms [2–9]. A col-lection of atoms in the corresponding blockade volumethen forms a “superatom” which can accommodate atmost one shared Rydberg excitation [10–13].The Rydberg blockade mechanism constitutes the ba-sis for a number of promising quantum informationschemes [2, 3, 14] and interesting multiatom effects [15–34]. Resonant two-photon excitation of Rydberg states isemployed in several schemes [34–46] utilizing the effectsof atomic coherence, such as electromagnetically inducedtransparency [47] and coherent population trapping andtransfer [48]. Stimulated adiabatic passage with delayedpulses in an ensemble of three-level atoms was previouslyconsidered in [46], where all the atomic states were as-sumed to be stable, while the lower atomic transitionwas driven either by a microwave field or by a pair ofoptical fields in the Raman configuration. It was shownthat, under the Rydberg blockade, the application of the“counterintuitive” pulse sequence results in a multiatomentangled state with strongly correlated population of thetwo lower states.The purpose of the present work is to investigate themore typical experimental situation [36–40] in which bothtransitions of three-level atoms are driven by opticalfields in a ladder (Ξ) configuration, while the interme-diate excited state of the atoms undergoes rapid spon-taneous decay. For non-interacting (distant) atoms, thesituation is analogous to what is usually referred to asstimulated Raman adiabatic passage (STIRAP) in a Λ-configuration [48]. Adding interatomic interactions leadsto highly non-trivial behavior of the Rydberg superatom.In the earlier part of the process, the system is in a com- pletely symmetric superposition of N atoms, each under-going adiabatic passage towards the Rydberg state with-out populating the intermediate excited state. But onceany one atom is excited to the Rydberg state, it blocksfurther Rydberg excitations and triggers the cycling ex-citation and decay of the intermediate excited state ofall the other N − N -atom master equation, we obtain atthe end of the process a mixed state of the system with asingle Rydberg excitation incoherently shared among all N atoms. We can also understand the underlying physi-cal mechanism in terms of the quantum Zeno effect [49],in which atoms emitting spontaneous photons throughthe decay of the intermediate excited state reveal thatinteractions block their adiabatic passage towards theRydberg state and thereby perform frequent projectivemeasurements of the presence of a Rydberg excitation inthe ensemble. Remarkably, the larger is the number ofatoms within the blockade volume the more robust is thetransfer process resulting in a single Rydberg excitationof the superatom. II. ADIABATIC PASSAGE IN A MULTIATOMSYSTEMA. Single-atom STIRAP
Let us first recall the essence of adiabatic transferof population in an isolated three-level atom using apair of delayed laser pulses (STIRAP) [48]. A coher-ent optical field with Rabi frequency Ω ge resonantly cou-ples the stable ground state | g i to an unstable (decay-ing) excited state | e i , which in turn is resonantly cou-pled to another stable state | r i by the second coherentfield of Rabi frequency Ω er [Fig. 1(a)]. The eigenstatesof the corresponding Hamiltonian V af = ¯ h (Ω ge | e ih g | +Ω er | r ih e | + H . c . ) are given by | ψ i = (cos θ | g i− sin θ | r i )and | ψ ± i = √ (sin θ | g i ± | e i + cos θ | r i ), where the mix-ing angle θ is defined via tan θ = Ω ge / Ω er . The “dark”state | ψ i with energy λ = 0 does not have any contribu-tion from the fast decaying state | e i , while the “bright”states | ψ ± i having energies λ ± = ¯ h q Ω ge + Ω er do con-tain | e i and thus are unstable against spontaneous decay.The aim of the STIRAP process is to completely trans-fer the population between the two stable states | g i and | r i without populating the unstable state | e i , which isachieved by adiabatically changing the dark state super-position. With the system initially in state | g i , one firstapplies the Ω er field, resulting in h g | ψ i = 1 (Ω ge ≪ Ω er and therefore θ = 0). This is then followed by switch-ing on Ω ge and switching off Ω er [Fig. 1(b)], resulting in |h r | ψ i| = 1 (Ω ge ≫ Ω er and therefore θ = π/ θ ≪ h | λ ± − λ | ,the system adiabatically follows the dark state | ψ i , andthe bright states | ψ ± i , and thereby | e i , are never pop-ulated. Hence, the decay of | e i is neutralized and thepopulation of the system is completely transferred from | g i to | r i . For what follows, it is useful to remember thatthe dark state | ψ i does not contain | e i because the reso-nant coupling of | g i to | e i by Ω ge interferes destructivelywith the resonant coupling of | r i to | e i by Ω er . B. The N -atom master equation Consider now an ensemble of N three-level atomsconfined in a small volume with linear dimension L of several µ m. All the atoms uniformly interact withtwo optical fields of Rabi frequencies Ω ge and Ω er asshown in Fig. 1(a). The atom-field interaction Hamil-tonian reads V j af = ¯ h (Ω ge ˆ σ jeg + Ω er ˆ σ jre + H . c . ), whereˆ σ jµν ≡ | µ i jj h ν | are the transition operators for atom j .The intermediate excited state | e i decays to the groundstate | g i with the rate Γ eg ; the corresponding Liouvil-lian acting on the density matrix ˆ ρ of the system isgiven by L jeg ˆ ρ = Γ eg [2ˆ σ jge ˆ ρ ˆ σ jeg − ˆ σ jee ˆ ρ − ˆ ρ ˆ σ jee ]. Thedecay rate Γ re of the highly excited Rydberg state | r i is typically much smaller (and can be neglected whenΓ re ≪ Ω er / Γ eg ), but for completeness we include it via L jre ˆ ρ = Γ re [2ˆ σ jer ˆ ρ ˆ σ jre − ˆ σ jrr ˆ ρ − ˆ ρ ˆ σ jrr ]. Note that bothtransitions of the three-level atoms are assumed closed.We next include the interatomic interactions. Thelong-range potential between pairs of atoms i, j in theRydberg state | r i induces level shifts ∆ ij = C p /d pij ofstates | r i r j i , where d ij is the interatomic distance and C p is the DD ( p = 3) or vdW ( p = 6) coefficient. The atom-atom interaction Hamiltonian reads V ij aa = ¯ h ˆ σ irr ∆ ij ˆ σ jrr .We assume that all the atoms are within a blockade dis-tance from each other, ∆ ij ≫ max[ w ] ∀ i, j ∈ [1 , N ],where w = Ω ge +Ω er √ ge +Γ eg / is the Rydberg-state excita-tion linewidth of a single three-level atom. We define V aa Time ( µ s) P r ob a b iliti e s P r ( n ) o f n R ydb e r g e x c it a ti on s Time ( µ s) P r (0) P r (1) P r (2) Time ( µ s) R a b i fr e qu e n c y ( π M H z ) Ω er Ω ge re Γ Ω ereg Γ ge Ω gre (a) (b)(c6)(c4)(c2)(c5)(c3)(c1) FIG. 1. (a) Level scheme of atoms interacting with the fieldsΩ ge and Ω er on the transitions | g i → | e i and | e i → | r i , whileΓ eg and Γ re are (population) decay rates of states | e i and | r i . V aa denotes the interaction between atoms in Rydbergstate | r i . (b) Time dependence of the Ω ge and Ω er fields.(c) The corresponding time-dependent probabilities P r ( n ) of n = 0 , , N =1 − eg , Γ re given in the text), while thinner lines to the fullycoherent dynamics of non-decaying atoms (Γ eg , Γ re ≃ the probabilities P r ( n ) = h ˆΣ ( n ) r i of n Rydberg excita-tions of superatom through the corresponding projec-tors ˆΣ (0) r ≡ Q Ni =1 (ˆ σ igg + ˆ σ iee ) = Q Ni =1 ( − ˆ σ irr ), ˆΣ (1) r ≡ P Nj =1 ˆ σ jrr Q Ni = j ( − ˆ σ irr ), etc. Note that ˆ σ igg + ˆ σ iee + ˆ σ irr = ∀ i ∈ [1 , N ].The density operator ˆ ρ of the N -atom system obeysthe master equation [50] ∂ t ˆ ρ = − i ¯ h [ H , ˆ ρ ] + L ˆ ρ, (1)with the Hamiltonian H = P j V j af + P i We solve the master equation (1) numerically assumingthe atoms are irradiated by the two pulsed fields havingGaussian temporal shapesΩ ge,er ( t ) = Ω exp (cid:20) − ( t − t end ∓ σ t ) σ t (cid:21) , where Ω = 2 π × σ t = t end is the temporal width and relative delay of thepulses, and t end = 30 µ s is the process duration [seeFig. 1(b)]. We take cold Rb atoms [36–40], with theground state | g i ≡ S / | F = 2 , m F = 2 i , the interme-diate excited state | e i ≡ P / | F = 3 , m F = 3 i withΓ eg = 38 MHz, and the highly excited Rydberg state | r i ≡ nS / with the principal quantum number n ∼ re = 1 KHz. Within the trapping volume of lin-ear dimension L ∼ µ m we then have large interatomic(vdW) interactions [51] ∆ ij ≥ w ∀ d ij ≤ L , where w = √ +Γ eg / ≃ π × . N = 1 , . . . , N , even or odd, the“counterintuitive” sequence of pulses Ω ge,er ( t ) leads, withlarge probability P r (1) ≥ . 98, to a single Rydberg exci-tation of the superatom, while the probabilities of multi-ple excitations P r ( n > 1) are negligible, due to the strongblockade. Once a Rydberg excitation is produced, thesmall decay Γ re of state | r i leads to a slow decrease of P r (1).The response of the Rydberg superatom to the “coun-terintuitive” sequence of pulses may look analogous tothe coherent adiabatic passage of a single three-levelatom, but this similarity is superficial and the physicsbehind it is more involved. This is perhaps best illus-trated in Fig. 1(c) by the strikingly different behavior ofsuperatom in the absence of dissipation, Γ eg , Γ re = 0,which was studied in [46]. Without dissipation, in thetransition region Ω ge ( t ) ∼ Ω er ( t ) the probabilities of zero P r (0) and one P r (1) Rydberg excitation do not changemonotonically but alternate N − N the final state ofthe system does not contain a Rydberg excitation. [For N ≥ 4, the fast oscillations of probabilities P r (0 , 1) andtheir final values noticeably different from 0 and 1 aredue to the violation of adiabaticity with the increasedsystem size and the corresponding decrease in the sepa-ration between its eigenstates [46]]. A. Analysis For a dissipationless system, it is convenient to use thefully symmetrized states | n g , n e , n r i denoting n g atomsin state | g i , n e atoms in | e i , and n r atoms in | r i . Dueto the Rydberg blockade, only n r = 0 , n g + n e + n r = N . The field Ω ge cou-ples the ground state of the superatom | N g , e , r i suc- cessively to the collective single | ( N − g , e , r i , dou-ble | ( N − g , e , r i etc. excitation states, which arein turn coupled to the single Rydberg excitation states | ( N − g , e , r i , | ( N − g , e , r i etc. by the field Ω er [43]. The corresponding Hamiltonian can be expressedas H = ¯ h (Ω ge ˆ e † ˆ g + Ω er ˆ r † ˆ e + H . c . ), where operators ˆ g (ˆ g † ), ˆ e (ˆ e † ) and ˆ r (ˆ r † ) annihilate (create) an atom in thecorresponding state | g i , | e i and | r i ; ˆ g and ˆ e are stan-dard bosonic operators, while ˆ r describes a hard-core bo-son (ˆ r † ) = 0. As was shown in [46] for non-decayingatoms, an ideal adiabatic passage leads to the final stateof the system | J x = 0 i for N even, and | J x = 0 i ⊗ | r i for N odd, where | J x = 0 i is the eigenstate of opera-tor ˆ J x ≡ (ˆ e † ˆ g + ˆ g † ˆ e ) with zero eigenvalue. The state | J x = 0 i involves an equal number of atoms [ N/ N − / 2] in ( | g i ± | e i ) / √ eg > ∼ Ω ge of state | e i , the dynamic of the system is completely different.If we had non-interacting atoms, the adiabatic passagewould yield a product state (cos θ | g i − sin θ | r i ) ⊗ N con-taining multiple Rydberg excitations but no atoms instate | e i . The strong interatomic interactions, how-ever, shift the energies of multiply excited Rydberg statesout of resonance with the Ω er field. Starting from theground state | N g , e , r i = Q Ni =1 | g i i the populationtransfer beyond the symmetric single Rydberg excitationstate | ( N − g , e , r i = √ N P Nj =1 | r i j Q Ni = j | g i i is thenblocked. In this superposition, the atoms in state | g i can be now excited to state | e i by the strong resonantfield Ω ge since the coupling of | e i to | r i by Ω er and theresulting destructive interference are suppressed. Atomsexcited to | e i rapidly decay back to | g i with randomphases. This leads to continuous dephasing of the super-position | ( N − g , e , r i , turning it into an incoherentmixture of single Rydberg excitation, which is decoupledfrom states | ( N − n ) g , n e , r i and | N g , e , r i . A relatedeffect is described in [12], where the dephasing of col-lective Rydberg excitation was brought about by an in-homogeneous light field with short-range space and timecorrelations. Here, instead, the superposition containinga Rydberg atom is dephased by the field Ω ge and decayΓ eg through the cycling transition | g i ↔ | e i of all theatoms that do not populate the Rydberg state.The field Ω ge ( t ) driving the N − t , the average populations of state | g i for allthe atoms is therefore approximately given by the steady-state expression for an independent two-level atom [50], h ˆ σ gg i ≈ Ω ge +Γ eg / ge +Γ eg / ≡ κ . The probability that all but theRydberg excited atom are in the ground state | g i is then h ˆ σ jrr Q Ni = j ˆ σ igg i ≈ [ κ ( t )] N − . At large times κ ( t end ) ≃ ρ = N P Nj =1 ˆ σ jrr Q Ni = j ˆ σ igg . Yet,the total probability of finding one Rydberg excitationis P r (1) = h ˆΣ (1) r i ≃ ge > Ω er [Fig. 1].These results are fully reproduced by the exact numericalsolution of the density matrix equations (1). B. Quantum Zeno effect An alternative and perhaps more elegant explanationas to why the superatom attains near unity Rydberg ex-citation h ˆΣ (1) r i ≃ | e i and sponta-neous decay back to the ground state | g i , the Rydbergblocked atoms perform continuous projective measure-ments of the Rydberg excitation in the ensemble.To verify this physical picture, we have performedquantum Monte Carlo simulations [52] of the dissipa-tive dynamics of a few-atom system. In such simula-tion, the state of the system | Ψ i evolves according to theSchr¨odinger equation ∂ t | Ψ i = − i ¯ h ˜ H | Ψ i with an effectivenon-Hermitian Hamiltonian ˜ H = H − i ¯ h P j (Γ eg ˆ σ jee +Γ re ˆ σ jrr ) which does not preserve the norm of | Ψ i . Theevolution is interrupted by random quantum jumps | Ψ i → ˆ σ jge ( er ) | Ψ i with probabilities determined by thedecay rates Γ eg ( re ) .In the early part of evolution, the states of all theatoms share the same overlap with the current dark andbright states, and the state of the system | Ψ i is sym-metric under permutation of the atoms. But already thefirst quantum jump breaks this symmetry by transferringone randomly selected atom from | e i to | g i (or, with amuch smaller probability ∼ Γ re / Γ eg , from | r i to | e i ),while the dark state overlap increases for the atoms thatdid not jump. During the subsequent evolution underthe Rydberg blockade, the bright state overlap grows forall the atoms, while the following jump again suddenlyincreases the dark state contribution to the states of theatoms which did not jump. This proceeds untill eventu-ally only one atom has experienced no quantum jump.This atom closely follows the dark state superposition,and while small excursions away from the dark state oc-cur, they are reduced by each jump of the other atoms.Hence, atoms undergoing quantum jumps stabilize thealmost deterministic evolution of a non-decaying atomtowards the Rydberg state. C. Coherence relaxation We finally demonstrate the robustness of adiabatic pas-sage in a Rydberg superatom. In a single three-levelatom, the STIRAP—while immune to the decay andmoderate detuning of the intermediate state | e i —is verysensitive to coherence relaxation between the long-livedstates | g i and | r i [48]. The physical origins of relaxationof the atomic coherence include non-radiative collisions,Doppler shifts, laser phase fluctuations and electromag-netic field noise. In addition to the small decay Γ re ofthe Rydberg state | r i , we now include its dephasing γ r via the Liouvillian L jr ˆ ρ = γ r [(ˆ σ jrr − ˆ σ jee − ˆ σ jgg )ˆ ρ (ˆ σ jrr − Time ( µ s) R ydb e r g e x c it a ti on p r ob a b ilit y P r ( ) N P r ( ) N =1,2,3,4,5,6 FIG. 2. Probabilities P r (1) of single Rydberg excitation ofsuperatom composed of N = 1 − γ r and decaysΓ eg , Γ re . Main panel shows the time-dependence of P r (1),and inset shows P r (1) for different N at time t end = 30 µ s. ˆ σ jee − ˆ σ jgg ) − ˆ ρ ] [50]. In Fig. 2 we show the probability ofRydberg excitation of the superatom obtained from thesolution of the master equation (1) with the dephasingrate γ r = 2 π × . h ˆ σ rr i ≃ . 9, sincethe decoherence of the dark state superposition leads tothe population of the bright states [47]. With increas-ing the number of atoms N , however, the Rydberg exci-tation probability of the superatom grows according to h ˆΣ (1) r i ≈ N h ˆ σ rr i ( N − h ˆ σ rr i +1 [33], approaching P r (1) > ∼ . 97 for N = 6. The spontaneous decay, perhaps surprisingly,counteracts the detrimental effect of decoherence of thedark state superposition and facilitates the efficient pro-duction of a single Rydberg excitation. A similar resulthas been obtained for Rydberg superatoms composed ofincoherently driven two-level atoms [33]. IV. DISCUSSION To conclude, we have examined the excitation of a Ry-dberg superatom using adiabatic passage with delayedlaser pulses. We have found that spontaneous decay ofatoms from the intermediate excited state facilitates asingle Rydberg excitation of the superatom, with nearlyunit probability. An ensemble of N > N = 1 atoms, and the final count of theextracted Rydberg atoms would correspond to the initialnumber of ground state atoms in the trap.Experiments with the atomic ensembles much largerthan the blockade length have revealed significant sup-pression of the number of Rydberg excitations [4–7],which is consistent with a regular spatial arrangementof Rydberg atoms [25, 32]. We envisage that employingstimulated adiabatic passage to produce with unit prob-ability singe Rydberg atoms per blockade volume can re-sult in long-range order and tighter crystallization of Ry-dberg excitations in extended systems [33, 34] and mayshine more light onto the spatial correlation patterns ofRydberg excitations.Finally, our studies provide a new element to the veryactive field of dissipative generation of states and pro-cesses in open many-body systems [53, 54]. 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