Optimal control of Rydberg lattice gases
Jian Cui, Rick van Bijnen, Thomas Pohl, Simone Montangero, Tommaso Calarco
OOptimal control of Rydberg lattice gases
Jian Cui, Rick van Bijnen,
2, 3
Thomas Pohl,
2, 4
Simone Montangero,
1, 5 and Tommaso Calarco Institute for complex quantum systems & Center for Integrated Quantum Science and Technology (IQST),Universit¨at Ulm, Albert-Einstein-Allee 11, D-89075 Ulm, Germany Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Strasse 38, 01187 Dresden, Germany Institut f¨ur Quantenoptik und Quanteninformation, Technikerstr. 21a, 6020 Innsbruck, Austria Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK 8000 Aarhus C, Denmark Theoretische Physik, Universit¨at des Saarlandes, D-66123 Saarbr¨ucken, Germany (Dated: August 3, 2017)We present optimal control protocols to prepare different many-body quantum states of Rydberg atoms inoptical lattices. Specifically, we show how to prepare highly ordered many-body ground states, GHZ states aswell as some superposition of symmetric excitation number Fock states, that inherit the translational symmetryfrom the Hamiltonian, within sufficiently short excitation times minimising detrimental decoherence effects. Forthe GHZ states, we propose a two-step detection protocol to experimentally verify the optimized preparationof the target state based only on standard measurement techniques. Realistic experimental constraints andimperfections are taken into account by our optimisation procedure making it applicable to ongoing experiments.
I. INTRODUCTION
Quantum simulation and quantum information processingcrucially rely on the ability to create precisely controllablemultipartite quantum systems, with designed Hamiltoniansand low decoherence rates compared to experimental timescales. Ultracold atoms in optical lattices, laser-coupled tohigh-lying Rydberg states, provide an appealing platform forengineering such quantum systems. Optical potentials trap-ping the atoms provide highly flexible control over spatial ge-ometries [1, 2], with lattice sites that can be loaded with singleatoms with near-unit fidelity [3, 4]. Quantum gas microscopesrepresent an established technology for observing the quan-tum state of individual atoms within the lattices [5].Strong and tunable long-range interactions between atomsacross lattice sites can be established by laser-coupling themto Rydberg states, with interaction strengths that can befar in excess of all other energy scales in the system [6,7]. A striking consequence is the so-called Rydberg block-ade [8, 9], which was succesfully employed to entangle pairsof atoms [10–12], as well as ensembles of atoms [13–19].Rydberg-excited atoms in lattice geometries can be describedwith Ising spin models [20–23], which have recently seen im-pressive experimental confirmation [19, 24, 25]. Extendedspin models can be realised by adding exchange interactionsthrough coupling of multiple Rydberg levels [26–32], or byintroducing controlled dissipation [33–40]. Finally, even ageneral purpose Rydberg quantum simulator [41] and quan-tum annealer [42] have been proposed.Evidently, Rydberg atoms hold high promise for applicabil-ity in quantum information processing and quantum simula-tion. Yet, thus far most experimental investigations have beenlimited to studying dynamics of Rydberg-excited systems,while previously predicted interesting ground state physicsand associated quantum phase transitions [20–23, 43, 44] re-main largely unexplored. The primary limiting factor prevent-ing observation of many-body ground states is the finite life-time of the Rydberg states [7]. Although Rydberg atoms boastrelatively long lifetimes of up to tens of microseconds [45], it is still a very stringent requirement that the typically com-plex ground state preparation scheme is executed well beforea single decay event occurs. Preliminary experimental successhas been achieved in preparing ‘crystalline’ states of regu-larly spaced Rydberg excitations on a 1D chain of atoms [46].These experiments effectively probed the first few steps of afull Devil’s staircase, i.e. the stepwise increase of the Rydbergatom number in the many-body ground state with increasinglaser detuning or system size, that characterises the groundstate phase diagram of a lattice gas with power-law interac-tions [47]. The experiment in Ref. [46] employed a carefullydesigned adiabatic pulse scheme [22, 48–50], slowly evolvingthe initial ground state with no Rydberg excitations into thedesired crystalline state.An adiabatic state preparation scheme, however, has someinherent limitations. Firstly, it has to be executed slowly com-pared with the minimum energy gap by definition , which isdirectly at odds with the previously stated neccessity of per-forming the state preparation as fast as possible. Secondly,many-body states that are not adiabatically connected to a triv-ial initial state are out of reach of adiabatic preparation. Toovercome these limitations, we turn to the tools of OptimalControl (OC) [51–55]. Stimulated by earlier succeses of OCin quantum information processing [56–64], and the designof many-body quantum dynamics [65–67], as well as the suc-cessful applications in experiments [68–71], especially thosewith Rydberg atoms [63, 64, 72–74], we adopt the “choppedrandom basis” (CRAB) and dressed CRAB (dCRAB) opti-mal control method [66, 75, 76] for quantum state prepara-tion in Rydberg lattice gases. We will showcase three typi-cal examples: (i) crystalline states of regularly spaced excita-tions [46] as a prominent and experimentally relevant exampleof the Rydberg blockade effect, (ii) GHZ states with maximalmultipartite entanglement, relevant for quantum informationprocessing tasks [77–82], and (iii) an arbitrary superpositionstate, for which no other preparation method is known so far,demonstrating the generality of our method.The paper is organized as follows. In Sec. II we pro-vide a description of the Rydberg system under study, aswell as an outline of the relevant experimental considerations. a r X i v : . [ qu a n t - ph ] A ug Sec. III demonstrates the results for the Rydberg crystallinestate preparation and the obtained excitation staircase. InSec. IV we show the optimized dynamics for creating and de-tecting a GHZ state which encodes the qubits in groups ofatoms collectively sharing an excitation, complemented by anarbitrary quantum superposition state preparation scheme de-scribed in Sec. V. Finally, Sec. VI summarizes the paper andprovides an outlook on exploring the so-called quantum speedlimit of state preparation in Rydberg atoms.
II. BASIC DESCRIPTION
The system we consider is composed of a two-dimensionallattice with one atom per site, which can be realized exper-imentally either in an optical lattice [46, 83] or in an arrayof optical dipole traps [2], or even in dense disordered gasesby targeted laser excitation [84]. Given the short time scalesconsidered in this paper and other works in the literature [22–25, 32–37, 46, 83, 85, 86], only the internal electronic degreesof freedom are considered. Initially the system is prepared inthe Mott insulating phase in which every atom is in its elec-tronic ground state ∣ g ⟩ . Laser light couples the atomic groundstate ∣ g ⟩ to a high-lying Rydberg state ∣ e ⟩ with a Rabi fre-quency Ω and frequency detuning ∆ , as illustrated in Fig. 1(a).Experimentally, such Rydberg state transitions can either bedriven by a two-photon transition via a low-lying intermediatestate [7] or by a direct single-photon transition [12, 17, 25]. Inthe present calculations we focus on the specific situation ofprevious lattice experiments [46, 83] where Rubidium atomshave been excited to S / Rydberg states via a far detunedintermediate P / state with two laser beams. This essentialstate picture is well justified, as near-resonant state mixing[85, 87] can be neglected [22].If two atoms at different lattice sites with positions r i and r j are excited to the Rydberg level, they experience strong vander Waals interactions, V ij = C /∣ r i − r j ∣ . For the selected S state the corresponding C = . × − Jm [46, 88].The interaction between two ground-state atoms or betweenone ground- and one Rydberg atom is negligible [6, 89, 90].In the interaction picture, this system can be described by theHamiltonian [19, 46] H ( t ) = ̵ h ( t ) ∑ i ( ˆ σ ( i ) eg + ˆ σ ( i ) ge ) + ∑ i ≠ j V ij σ ( i ) ee ˆ σ ( j ) ee −̵ h ∆ ( t ) ∑ i ˆ σ ( i ) ee , (1)where the operators ˆ σ ( i ) αβ = ∣ α i ⟩⟨ β i ∣ denote the atomic transi-tion and projection operators for the i th atom at position r i .We investigate the Rydberg atom excitation dynamics by inte-grating the Schr¨odinger equation governed by H ( t ) , employ-ing a numerical approach described in [22, 49].In Fig. 1(c) we show the spectrum of eigen-energy levels ofthe system described by the Hamiltonian (1) in the classicallimit Ω = . In this case, all eigenstates are tensor productsof excitation number Fock states on each site, i.e., many-bodyFock states corresponding to a given spatial configuration ofsite-localized Rydberg excitations. FIG. 1. Rydberg atomic gas. (a) Level scheme of two Rb atomsin optical lattice sites. The atomic ground state ∣ g ⟩ is coupled, withRabi frequency strength Ω , to an excited Rydberg state ∣ e ⟩ . The laseris detuned by ∆ . The two atoms are separated by a distance r ; theblue curve represents their mutual energy shift due to van der Waalsinteraction. (b) The unit-filling optical lattice is tailored into a × N bar shape for the crystalline state preparation. The strong repulsivevan der Waals interactions result in the Rydberg blockade effect witha blockade radius approximately a such that each group of 3 atomson the y axis effectively forms a super-atom. Such a system canbe described as a one-dimensional chain along the x direction of N super-atoms with √ enhanced Rabi coupling. (c) Energy spectrumof an N = atom chain in the classical limit ( Ω = ), plotted as afunction of the detuning in units of V L = C / L , which is the inter-action energy between atoms located at opposite ends of the chain,with L = ( N − ) a the length of the chain. The dashed vertical linemarks the phase transition point ∆ = . Each eigenstate of the sys-tem has a well-defined total number of excitations N e , indicated by acolor code: blue ( N e = ), magenta ( N e = ), green ( N e = ), yellow( N e = ), black ( N e = to ), red ( N e = N ). For the ground states,these excitations are regularly spaced, minizing the interaction en-ergy and forming a crystalline state. The inset shows a zoom of thelow-lying spectrum near the quantum phase transition point and thefirst ground-state level-crossing point. Increasing the laser detuning lowers the energy of the ex-cited atomic state and, therefore, favours the excitation ofRydberg atoms as seen in Fig. 1(c). Therefore, the low-energy sector of the spectrum is composed of ordered Ryd-berg atom configurations which minimize the total interactionenergy [22].Accurate pulse shaping of the Rydberg excitation laser pro-vides precise experimental control of both Ω ( t ) and ∆ ( t ) .This permits to steer the many-body quantum dynamics ofthe atomic lattice and to prepare specific many-body statesstarting from the simple initial state ∣ gg . . . g ⟩ , with all atomsin their ground state. While the basic idea of this ap-proach [22, 48, 49] has been demonstrated in recent exper-iments [46, 83], preparation fidelities have remained limitedby lattice imperfections and unavoidable transitions betweenthe ground state and the low-lying excited many-body eigen-states of eq. (1). Here, we use optimal control techniques tomitigate such limitations.We apply the dCRAB method to the preparation of crys-talline states, GHZ states as well as an arbitrary superposi-tion state in Rydberg atom lattices. In general the dCRABmethod identifies the optimal temporal shapes of the con-trol parameters, which have been expanded on a randomizedtruncated Fourier basis, through iteratively updating the co-efficients of the basis functions using a numerical minimiza-tion (e.g. simplex) method, which enables to obtain bet-ter fidelities from iteration to iteration. In order to drawa close connection to ongoing experiments, we incorporatetypical parameter constraints, limiting the Rabi frequency to Ω / π ≤ kHz [46, 83], and imposing a truncation on thehighest Fourier frequency for synthesising Ω ( t ) and ∆ ( t ) at . MHz and . MHz, which translate into a minimum riseand fall time for Ω and ∆ of ns and ns, respectively.In this paper, we constrain the amplitude of ∆ to be within ± MHz as in the experiments [46]; however, this cutoff is nota fundamental limit. We will see later that even with this limi-tation we can prepare high-fidelity crystalline states and GHZstates, and if we allow for larger detunings ∆ in the optimiza-tion, the results can only improve.Finally, in order to account for lattice defects, we consideran ensemble of N r = realizations with a lattice filling frac-tion of . in the optimization. We use the average fidelity, F C ≡ ∣ ⟨ ψ C ∣ ψ ( τ )⟩∣ and F G ≡ ∣ ⟨ ψ G ∣ ψ ( τ )⟩∣ for crystallinestate ∣ ψ C ⟩ and the GHZ state ∣ ψ G ⟩ , respectively, as the figureof merit for the optimization. Here the bars represent the en-semble average over N r realizations, and ∣ ψ ( τ )⟩ is the finalstate at time τ . This choice for the figure of merit ensuresthat while the obtained control parameters do not just opti-mize certain individual configurations but yield an optimizedaverage dynamics with a high degree of robustness with re-gards to lattice defects. In this paper we neglect other sourcesof imperfections, such as dephasing due to instrumentation orstray fields, which were found to be of minor relevance undertypical experimental conditions [46, 83]. III. CRYSTALLINE STATE PREPARATION
In order to prepare a crystalline state with a given number N e of Rydberg excitations, one can drive the system througha sequence of level crossings by chirping the frequency de-tuning from negative to positive values as shown in Fig. 1(c).Such a near-adiabatic modification of the low-energy many-body states [22, 49, 91] has been demonstrated experimen- tally in [46]. However, a strictly adiabatic preparation of theabsolute ground state is hampered by the finite lifetime of theexcited Rydberg atoms, which limits the available evolutiontimes. Consequently, slight crystal defects emerge from un-avoidable transitions between the ground state and the low-lying excited many-body Fock states. In [46] the employedexcitation pulses allowed to prepare an ordered quantum stateof slightly delocalized Rydberg excitations, rather than the ac-tual ground-state crystal consisting of a single Fock state com-ponent. µ s) Ω , ∆ ( π M H z ) Ω∆ x position (a) n e ( x ) N e optimal controlGS N e GSquasi adiabatic ( a ) ( b )( c ) ( d ) FIG. 2. Rydberg crystalline state preparation. (a) Optimized controldriving parameters. (b) Excitation density n e ( x ) ≡ ⟨ e x ∣ ρ ∣ e x ⟩ of the3-excitation crystalline state (blue) and the average final state fromdCRAB optimal control (red). The average fidelity F C is larger than . . (c,d) Total excitation number staircase as a function of the lat-tice length for the classical states (blue curves with square markers),i.e., the ground states (GS) for Ω → and at fixed final detuning ∆ ,and the prepared states (purple curves with circle markers), obtainedby applying the pulses from optimized control (c) as well as from thequasi-adiabatic method of Ref. [46] (d) to systems with different size N . Below we demonstrate theoretically high-fidelity groundstate preparation within experimentally relevant preparationtimes using optimal control. Following the experimental sce-nario of [46], we consider a quasi-one-dimensional geometryin the form of a × N lattice as illustrated in Fig. 1(b), wherethe lattice spacing a = nm. Since the transverse extent isconsiderably smaller than the Rydberg blockade radius, thisgeometry behaves as a one-dimensional chain N super-atomsand of length L = ( N − ) a with a collectively enhanced Rabifrequency √ [46]. As described in the previous section,our method accounts for possible lattice defects and thereforeincludes resulting fluctuations of the effective Rabi frequencythe fluctuating number of atoms per super-atom.In Fig. 2(a) we show the pulse shape optimised via thedCRAB optimal control method [76] for the generation of a -excitation crystal in a chain of N = qubits for an excitationpulse duration of µ s. The resulting Rydberg excitation den-sity is nearly identical to that of a perfect three-atom crystal, asshown in Fig. 2(b) where only very weak fluctuations aroundthe optimal Rydberg atom positions occur. The quality of aprepared Rydberg crystalline state has been quantified throughthe total population of Fock states with given excitation num-ber n , i.e., P n ≡ ⟨∑ i ,...,i n ∣ e i ⟩ ⟨ e i ∣ ⊗ ⋯ ⊗ ∣ e i n ⟩ ⟨ e i n ∣⟩ [46].A more stringent evaluation than P n is the state fidelity F G .Our optimal control scheme reaches a high ground state av-erage fidelity over N r imperfect realizations of F G > . and a high final population P = . of -excitation Fockstates. Notice that for this Rydberg lattice gas system thequasi-adiabatic scheme employed in Ref. [46] tends to ob-tain states with low fidelity but relatively high P n , because ofunavoidable transitions to the low-lying excited many-bodyFock states. While these states have the correct number ofexcitations n , the excitations can be slightly displaced withrespect to positions of the actual ground state. Even thoughour protocol is run in an imperfectly prepared lattice with de-fects, the achieved fidelity yields a significant improvementover previous work for n = , where P = . could beachieved for an ideal lattice [46]. Note that these numberscan be further increased for higher Rabi frequencies, whichare now available for single-photon Rydberg excitation as re-cently demonstrated in [25].The enabled high preparation fidelity shows up most promi-nently in the so-called Rydberg blockade staircase [22]. Asshown in Fig. 2(c), this staircase appears as a stepwise in-crease of the Rydberg atom number of the many-body groundstate, when increasing the system length while keeping allother parameters fixed. In order to obtain the staircase, weapply the optimized control fields for the case of N = tosystems of different length N . As detailed in [22, 46], vary-ing the chain length is practically equivalent to a rescaling ofthe applied detuning, ∆ , via a change of V L ( see Fig. 1c).Hence, one can effectively target many-body ground stateswith different excitation numbers upon changing the chainlength for fixed parameters of the excitation pulse. As shownin Fig. 2(c), our optimised preparation pulse yields sharp tran-sitions between the different excitation numbers N e and en-ables the high-fidelity preparation of ordered Fock states with N e = . Both features represent significant improvementswith respect to the excitation pulses employed in both theoryand the experiment of Ref. [46]. For comparison, Fig. 2(d)shows the numerical excitation staircase obtained by using theadiabatic pulse employed in Ref. [46]).Fig. 3 illustrates the Rydberg excitation dynamics inducedby our optimised laser pulse. As demonstrated by the timeevolution of the energy [Fig. 3(a)], energy gap [Fig. 3(b)],the overlap between the instantaneous state and time-localground state [Fig. 3(c)] as well as the excitation number dis-tribution and the instantaneous state fidelity [Fig. 3(d)], theoptimized system dynamics indeed remains near adiabaticand closely follows the instantaneous many-body ground stateduring the first µ s. This suggests that adiabatic prepara-tion methods [22, 48, 49] indeed provide a useful strategy forpreparing low-energy many-body states [46]. However, thefinal stage of the optimised system dynamics significantly de-viates from adiabaticity, which ultimately yields the enhancedground state fidelity described above. Notice that the opti-mized control pulses presented here are robust against the lat-tice imperfections arising from non-unity filling of atoms. De-coherence process, e.g. Rydberg state radiative decay, onlyplays a minor role on a time scale of µs , as can be seen from the total decay probability P d ( t ) ≡ ∫ t Γ N e ( t ′ ) dt ′ , where N e ( t ′ ) = ⟨∑ i ∣ e i ⟩ ⟨ e i ∣⟩ is the total excitation number of thestate at time t ′ , and Γ = . kHz is the single atom radiativedecay rate for the S state of Rb [92]. For the optimizedevolution the total decay probability at the final time is only P d ( τ ) = . . µ s) E ( V L L / ) µ s) δ E ( V L L / ) µ s) O v e r l a p µ s) P r ob a b ilit y ( a ) ( b )( c ) ( d ) FIG. 3. Dynamics of the Rydberg crystallization. (a) Low-lying en-ergy spectrum of the laser-dressed system (blue dashed curve) andthe energy of the instantaneous state from the optimized dynamics(red solid curve). We zoom in at the curves between . and . µs in the inset, during which the energy of the optimized dynamicsgoes into the excited spectrum. (b) The energy gap between the firstexcited state and ground state (magenta dashed curves) as well asthe energy difference between the instantaneous state and the groundstate (green solid curve). In the time window from . to . µs the red curve is above the lowest blue curves, which means the en-ergy of the instantaneous state is higher than the first excited state.(c) The overlap between the instantaneous state ∣ ψ ( t )⟩ and the time-local ground state ∣ ψ G ( t )⟩ . This overlap measures how close theoptimized dynamics is to the adiabatic evolution. (d) Fidelity ( or-ange curve with circle marks) and the probability of excitations withgiven numbers, P n , for the instantaneous state (solid curves) and thetime-local ground state (dotted curves) by color code: black, red,green, blue, magenta correspond to n from to . Recent numerical work [50] pointed out that the prepara-tion scheme employed in [46] would yield a rather low groundstate fidelity F C ≲ . for the short pulse duration of µ s usedin the experiment [46]. It was, hence, concluded that adiabaticcrystal state preparation requires substantially longer excita-tions times at which dissipative processes would inevitablystart to play a significant role [50]. The above results (seeFig. 2 and Fig. 3), however, demonstrate that optimal controlallows to alleviate this problem by facilitating high-fidelityground state preparation for time scales for which the exci-tation dynamics remains highly coherent. IV. GHZ STATE PREPARATION AND DETECTION
Having demonstrated the power of optimal control tech-niques for preparing ordered low-energy states of Rydbergexcitations, we now consider the high-energy region of themany-body energy spectrum. One area of particular interestlies around ∆ c = N − ∑ i < j V ij /̵ h , as marked in Fig. 4(b),where the N -atom ground state, ∣ G ⟩ ≡ ∣ g , g , ..., g N ⟩ ,becomes degenerate with the fully excited state ∣ E ⟩ ≡∣ e , e , ..., e N ⟩ , which allows to generate maximally entangledGHZ states, ∣ ψ G ⟩ = (∣ G ⟩ + e iθ ∣ E ⟩)/√ [86]. FIG. 4. Lattice geometry and spectrum for the GHZ state. (a) Thelattice is tailored in such a way that four nearest neighbouring sitesare filled with atoms to form a super-atom at each of the corners en-coding a quantum bit, with the centers of the super-atoms separatedby a . (b) Energy spectrum of a × atomic array in the classicallimit, as a function of detuning ∆ , with V L = C / L the interactionenergy along one edge of the lattice of length L = a . Color cod-ing as in Fig. 1(c). The states ∣ ⟩ and ∣ ⟩ become resonantat a detuning ∆ c (defined in the main text). The second gray line isguideline for the crossing detuning point ∆ c . Due to the strong Rydberg-Rydberg atom interaction thepreparation of such high energy states requires a different lat-tice geometry than that of the previous section. Specifically,we consider an optical lattice with the aforementioned param-eters but filled in such a way [93] as to obtain qubits, eachof which located at one corner of a × square lattice, seeFig. 4(a). In every corner, only × lattice sites are filledwith one atom each, in which only one Rydberg excitationcan exist and be shared coherently by the × sites becauseof the blockade effect thus encoding the ∣ ⟩ state for the qubit.The ∣ ⟩ state of one qubit corresponds to all its constituentatoms in the ground state. A collection of N bl atoms ( N bl = in our example) in a blockade sphere is also called a “super-atom”, featuring in addition a collective enhancement of theeffective Rabi frequency with a factor of √ N bl [18, 94–96].The large qubit spacing ensures a moderate interaction energyof ̵ h − C /( a ) = . × π MHz for the S / Rydberg state used in [46, 83], while the use of multiple adjacent atomsreduces the detrimental effects of lattice defects as describedabove.Because of their highly entangled nature, the preparation ofGHZ states is much more sensitive to decoherence processesthan that of the classical crystalline states discussed in theprevious section. In particular, a single Rydberg state decaywould completely decohere a prepared GHZ state and projectthe system onto a separable state. Avoiding such undesiredeffects once more requires very short operation times, i.e. itcalls for optimised preparation pulses.Fig. 5(a) shows such an optimised pulse for a targeted GHZstate with θ = π / and a chosen pulse duration of µ s, andrequiring a vanishing initial and final Rabi frequency as wellas a detuning of ∆ ( τ ) = ∆ c at the end of the pulse. Thetime evolution of the corresponding fidelity is depicted inFig. 5(b) (cyan solid curve) and yields a final average value of F G = . . Note that such high fidelities are indeed obtaineddespite a significant fraction of lattice defects around .Remarkably, the fidelity that can be obtained for a defect-free atomic lattice is virtually perfect with infidelity × − .Such conditions and geometries can, for example, be realizedwith optical dipole-trap arrays as demonstrated in a numberof recent experiments [2–4, 19]. As can be seen from the P + P curve in the panel (b), the optimized quantum dy-namics differs significantly from the preparation protocol pro-posed in Ref.[86], where the accessible many-body states areconstrained to ∣ G ⟩ and ∣ E ⟩ , and GHZ states are generated byinducing Landau-Zener transitions between them. As shownin Fig. 5(b), the optimised preparation pulses presented here,on the contrary, exploit a significantly larger fraction of theunderlying Hilbert space for high-fidelity generation of GHZstates within a short preparation time. Indeed the chosen µ spreparation time of Fig. 5 is sufficiently short to ensure a to-tal decay probability P d ( T ) of less than . [see Fig. 5(c)the orange shaded area plotting the times amplified P d ( t ) ].The final value P d ( τ ) provides an upper bound on infidelitycaused by Rydberg state decay, assuming that any decay pre-vents the target state preparation. The overall preparation fi-delity can thus be estimated as F G × ( − P d ( T )) = . . Thereal part and the imaginary part of the final density matrix forthe prepared state (brown) and the targeted GHZ state (green)are shown in Fig.5(d) and (e), respectively.The experimental detection method for this system is lim-ited to the excitation probability on each site, which is suffi-cient to probe the crystalline state [46, 83] but not enough todemonstrate the presence of the GHZ state directly. Here wepropose to apply a sequence of measurements to probe GHZstates, exploiting the fact that information on the coherencepresent in the state can be extracted from the free time evolu-tion of the system [97].We start from the natural assumption that many copiesof identical final states can be obtained simply by repeatingthe experiment, as is routinely done to improve measurementstatistics. The first step is then to perform a standard excitationmeasurement [46] on many copies of the final states ∣ ψ ( τ )⟩ .If the system is in the GHZ state, 50% of the measurementoutcomes will result in no excitations while the other 50% ( a )( b )( c )( d )( e ) FIG. 5. GHZ state preparation. (a) Optimized control parame-ters. The detuning ∆ at the final time has been fixed to be ∆ C in Fig. 1, see the right vertical axis. (b) Fidelity for the GHZstate with θ = . π (cyan solid curve), and population in the sub-space spanned by ∣ ⟩ and ∣ ⟩ (green dotted curve), when ap-plying the optimized control in panel (a) to one realization with-out lattice defects. The fidelity between the final state of this re-alization and the GHZ state is approximately . . (c) Probabil-ity P n ≡ ⟨∑ i ,...,i n ∣ i ⟩ ⟨ i ∣ ⊗ ⋯ ⊗ ∣ i n ⟩ ⟨ i n ∣⟩ of excitations withgiven numbers n and the times amplified decay probability P d forthe instantaneous state ρ ( t ) . (d) Real part and (e) imaginary part ofthe average final states (brown) and the GHZ state (green). will result in 4 excitations. No other configuration should ap-pear for any individual measurement. That shows that the fi-nal state (not necessary pure) lives in the subspace spannedby the states ∣ ⟩ and ∣ ⟩ as ρ sub ψ = ( / γe iα γe − iα / ) with ≤ γ ≤ / . Clearly, the GHZ states ∣ ψ G ⟩ are described by ρ sub ψ for γ = / and α = θ .In the second step, we still need to distinguish between ∣ ψ G ⟩ and the other states in ρ sub ψ . One intuitive way to dis-tinguish between them is of course to measure the purity of the final state. Recently, the Greiner group has shownan experimental method to probe the purity of the state forcold atoms in an optical lattice through measuring the aver-age parity of the atomic interference between identical two-copy states [98]. However, this parity scheme is not particu-larly suitable for many-body Rydberg systems, since the long-range interactions between Rydberg atoms from the samecopy are difficult to switch off in the interference. Hence,we propose a free-evolution scheme in which one can distin-guish them by simply evolving the systems with a detectionHamiltonian H d = ̵ h Ω M ∑ i ( ˆ σ ( i ) eg + ˆ σ ( i ) ge ) + ∑ i ≠ j V ij ˆ σ ( i ) ee ˆ σ ( j ) ee ,where Ω M is the maximal Rabi coupling generated by thecontrol lasers. The coherence γ as well as the phase factor α for each individual initial state in ρ sub ψ will result in uniquedynamics. The difference between the targeted GHZ stateand any others states in ρ sub ψ is thus detectable from the dif-fering dynamics of the excitation probabilities for one qubit, E i ( ρ ( t )) ≡ Tr [ ρ ( t )∣ i ⟩ ⟨ i ∣] .As an example, Fig. 6(a) shows that the excitation dynam-ics of the targeted GHZ initial state (the γ = . , α = . π state in ρ sub ψ ) differs from that of a fully mixture state, la-belled as ρ mix , in ρ sub ψ with γ = . The excitation difference D t ≡ E i ( ρ sub ψ ( t )) − E i ( ρ GHZ ( t )) is a function of γ and α for a general initial state in ρ sub ψ , where the parameter t in thebrackets represents the evolution of the corresponding statefrom time to time t . We use a notation without t to denotethe time-maximal deviation within the experimental time as ∣ D ∣ = max t ∣ D t ∣ . Fig. 6(b) depicts ∣ D t ∣ for ρ mix . In this exam-ple ∣ D ∣ occurs at about 6 µs . This time only varies slightly bychanging parameters. Fig. 6(c) depicts ∣ D ∣ for different γ and α . In the small γ limit, ρ sub ψ is close to the fully mixed state,so that ∣ D ∣ is insensitive to the phase factors α . For γ = . , ρ sub ψ consists of the GHZ states with different phase factor,and therefore ∣ D ∣ significantly depends on α . In general, ev-ery state differs from each other in terms of D t , and ∣ D ∣ is agood measure of the difference.Thus, the detection scheme we propose is firstly measuringthe excitation profile of the prepared state and then evolvingthe prepared state under the detection Hamiltonian to comparethe dynamics of a single qubit excitation D t with respect tothat of the targeted GHZ state. The total experimental time,which is composed of the preparation time ( t p = µs ) andthe free evolution time in the second step ( t d = µs ) plus theexcitation detection time ( t e = µs [83]), is shorter than thelifetime of the Rydberg state. V. ARBITRARY STATE PREPARATION
Let us finally demonstrate the general applicability ofthe method by studying the preparation of arbitrary many-body states in a Rydberg lattice. As a specific exam-ple we choose the same lattice geometry as in section IVand consider symmetric target states, ∣ ψ s ⟩ = ∑ n a n ∣ s n ⟩ ,spanned by the number states ∣ s ⟩ = ∣ ⟩ , ∣ s ⟩ =(∣ ⟩ + ∣ ⟩ + ∣ ⟩ + ∣ ⟩) / , etc.. Again we imple-ment realistic experimental constraints for the excitation pulse E i time ( µ s) D t α ( π )o γ ( a )( b )( c ) FIG. 6. GHZ state detection. Evolution of the single superatomexcitation (a) under the detection Hamiltonian with different initialstate: GHZ state (∣ ⟩+ i ∣ ⟩)/√ (solid brown curve), equallymixed state ρ mix = (∣ ⟩ ⟨ ∣ + ∣ ⟩ ⟨ ∣)/ (dashed bluecurve), and (b) the deviation D t = E i ( ρ mix ) − E i ( ρ GHZ ( t )) (reddot-dash curve). Due to the symmetry of the lattice geometry all thesuperatoms behave the same. (c) The time-maximal deviation ∣ D ∣ asa function of γ and α . The target state is highlighted with purplecircle. and account for random lattice defects by performing an en-semble average over random spatial configurations.In Fig.7(a) we show the optimised excitation pulse forpreparing the state ∣ ψ a ⟩ with randomly generated coefficients, a = . , a = . , a = . , a = . and a = . . Even for a short preparation time of µ s theoptimized pulse allows to generate the target state with a highfidelity of . . This is illustrated in Fig.7(b) where we showthe difference ρ ( τ ) − ∣ ψ a ⟩ ⟨ ψ a ∣ between the target state andthe generated state. It’s elements are very small throughoutdemonstrating the high quality of the optimised state prepara-tion approach. VI. DISCUSSION AND SUMMARY
In this work, we have investigated the applicability of op-timal control approaches for the dynamical preparation ofmany-body states in a lattice of interacting Rydberg atoms.We have demonstrated that this opens up the fidelity prepa- µ s) Ω , ∆ ( π M H z ) Ω , ∆ ( ∆ c ) Ω∆ ( a )( b ) FIG. 7. Arbitrary state preparation. (a) Optimized control parame-ters. Panel (b) shows the elements of the difference ρ ( τ ) − ∣ ψ a ⟩ ⟨ ψ a ∣ between the generated and the target state. Here ρ ( τ ) is the aver-age final state over realizations of the system with lattice defects.Since the elements of the target state are real by construction we onlyshow the real part in panel (b). ration of ordered ground states, highly entangled GHZ statesand even arbitrary, randomly chosen, many-body states un-der realistic conditions and typical experimental constraintson excitation pulse shaping. In particular, for the latter opti-mal control techniques as demonstrated in this work presentlyprovide the only suitable approach to generate complex many-body states in an efficient and experimentally viable fashion.We have placed particular focus on limitations and imper-fections, such as lattice defects, that are practically unavoid-able in experiments. Our optimized control pulses are ro-bust against lattice defects, in the sense that they yield highpreparation fidelities for nearly every randomly sampled spa-tial configuration and a high average fidelity with a small sta-tistical spread. For example, comparing the average fidelitiesfrom two sets of random samples we found a difference ofless than − for all three studied target states.Table VI summarises the overall performance of thedCRAB optimal control method for the three example states.As one can see the major limitation on achievable fidelitiesin all three cases stems from the finite fraction of lattice de-fects and spontaneous decay of the Rydberg state. Whilewe have considered here a filling fraction of . [46], re-cent experiments have already reached considerably highervalues in optical lattices [25, 99] and optical dipole trap ar-rays [3, 4]. Equally important, spontaneous Rydberg state de- F F s F s − F P d ( τ ) ⟨ N e ⟩ τ / µs Crystallinestate 0.85 0.923 0.073 0.1 3 4GHZ state 0.921 0.99991 0.079 0.07 2 3 ∣ ψ a ⟩ defective re-alizations F and the single-shot fidelity for perfect lattice F s . Thefourth column measures how much infidelity is caused by the lat-tice defects. The fifth column, P d ( τ ) , lists the decay probability atthe final excitation time. The last two columns show the expectationvalue of the number of excitations, as well as the excitation times,respectively, for the three examples. cay ultimately limits achievable preparation fidelities, whichis why we have chosen relatively short pulse durations of afew µ s (see table VI).Within our optimisation approach, it should be possible tofurther reduce the total evolution time without a significantdegradation of the preparation fidelity until reaching the quan-tum speed limit. Since the preparation time eventually deter- mines the extend of undesired decoherence effects, the de-tailed exploration of the quantum speed limit in Rydberg lat-tices presents both a fundamentally interesting and practicallyimportant problem for future studies. In view of the recentadvances in optically controlling the many-body dynamics ofRydberg atom lattices, the control techniques demonstrated inthis work will enhance the capabilities of such systems forquantum simulations as well as the collective preparation ofcomplex nonclassical many-body states for quantum infor-mation applications. We hope that the first theoretical stepsin this direction, as presented in this article, will initiate fur-ther experimental and theoretical work to tap the full potentialof optimal control techniques for Rydberg-atom many-bodyphysics.Acknowledgement.— We thank Tommaso Macr`ı, VictorMukherjee, Johannes Zeiher, Christian Gross and ImmanuelBloch for valuable discussions. TP is supported by the DNRFthrough a Niels Bohr Professorship. SM gratefully acknowl-edges the support of the DFG via a Heisenberg fellowship.This work made use of the High Performance ComputingResource BwUniCluster and JUSUTS cluster. 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