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PHYSICAL REVIEW A 76, 023605 共2007兲 Probing n-spin correlations in optical lattices Chuanwei Zhang, V. W. Scarola, and S. Das Sarma Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA 共Received 27 November 2006; revised manuscript received 2 May 2007; published 13 August 2007兲 We propose a technique to measure multispin correlation functions of arbitrary range as determined by the ground states of spinful cold atoms in optical lattices. We show that an observation of the atomic version of the Stokes parameters, using focused lasers and microwave pulsing, can be related to n-spin correlators. We discuss the possibility of detecting not only ground state static spin correlations, but also time-dependent spin-wave dynamics as a demonstrative example using our proposed technique. DOI: 10.1103/PhysRevA.76.023605 PACS number共s兲: 03.75.Lm, 03.75.Mn, 39.25.⫹k, 75.10.Pq I. INTRODUCTION The advent of optical lattice confinement of ultracold atomic gases 关1–4兴 opens the possibility of observing a vast array of phenomena in quantum condensed systems 关5兴. In particular, optical lattice systems may turn out to be the ideal tools for the analog simulation of various strongly correlated interacting lattice models 共e.g., Hubbard model 关2,3兴, Kitaev model 关6兴兲 studied in solid state physics. The great advantage of optical lattices as analog simulators of strongly correlated Hamiltonians lies in the ability of optical lattices to accurately implement lattice models without impurities, defects, lattice phonons, and other complications which can obscure the observation of quantum degenerate phenomena in the solid state. In this context optical lattices can support a variety of interacting spin models which to date have been only approximately or indirectly observed in nature or remain as rather deep but unobserved mathematical constructs. Three exciting possibilities are currently the subject of active study 关5兴. The first 共and the most direct兲 envisions simulation of conventional condensed matter spin-lattice models in optical lattices. Quantum magnetism arising from strong correlation leads to many-body spin ground states that can be characterized by spin-order parameters. Spin order can, in some cases, show long-range behavior arising from spontaneous symmetry breaking, e.g., ferromagnetism and antiferromagnetism. Such long-range spin-ordering phenomena are reasonably well understood in most cases. Recent work also relates conventional spin-order parameters to entanglement measures which yield scaling behavior near quantum phase transitions 关7,8兴. The second possibility, simulation of topological spin states, arises from the surprising fact that optical lattices can also 共at least in principle兲 host more complicated spin models previously thought to be academic. The ground states of these models do not fall within the conventional Landau paradigm, i.e., there is no spontaneously broken symmetry, but show topological ordering and, as a result, display nontrivial short-range behavior in spin correlation functions. Examples include the chiral spin-liquid model 关9兴 and the Kitaev model 关6,10–12兴. Finally, optical lattices are also particularly well suited to realize coherent and collective spin dynamics because dissipation can be kept to suitably low levels 关13兴. While optical lattices offer the possibility of realizing all three of the above examples one glaring question remains. 1050-2947/2007/76共2兲/023605共7兲 Once a suitable spin Hamiltonian is realized, how do we observe the vast array of predicted phenomena in spinoptical lattices? To date, time-of-flight measurements have proven to yield detailed information related to two types of important correlation functions of many-body ground states of particles trapped in optical lattices. The first, a first-order correlation function 共the momentum distribution兲, indicates ordering in one-point correlation functions 关14兴. The second is a second-order correlation function 共the noise distribution兲 which indicates ordering in two-point density-density correlators 关15–18兴. The former can, for example, detect longrange phase coherence while, as we will see below, the latter is best suited to probe long-range order in two-point correlation functions, e.g., the lattice spin-spin correlation function. We note that recent proposals suggest that time-of-flight imaging can in principle be used to extract other correlation functions 关19,20兴. In this paper we propose a technique to observe equal time n-spin correlation functions characterizing both longand short-range spin ordering useful in studying all three classes of spin-lattice phenomena mentioned above. Our proposal utilizes realistic experimental techniques involving focused lasers, microwave pulsing, and fluorescence detection to effectively measure a general n-spin correlation function defined by n 兵␣ jk,k = 1, . . . ,n其 ⬅ 具⌿兩 兿 j jk兩⌿典, k=1 ␣ k 共1兲 where ⌿ is the many-body wave function of the atomic en␣ semble, 兵jk其 is a set of sites, and j jk 共␣ jk = 0 , 1 , 2 , 3兲 are k Pauli spin operators at sites jk with the notation 0 = I, 1 = x, 2 = y, and 3 = z. Examples of order detectable with one-, two-, and three-spin correlation functions are magnetization 共具zj 典 = 1兲, antiferromagnetic order 关具zj zj⬘典 = 共−1兲 j−j⬘兴, and chiral spin-liquid order 关具 j · 共 j⬘ ⫻ j⬙兲典 = 1兴, to name a few. In general our proposed technique can be used to experimentally characterize a broad class of spin-lattice models of the form 冉 M 冊 H共J;A兲 = J共t兲 兺 A兵jk其 兿 j jk , 兵jk其 k=1 ␣ k 共2兲 where J has dimensions of energy and can vary adiabatically as a function of time, t, while the dimensionless parameters 023605-1 ©2007 The American Physical Society PHYSICAL REVIEW A 76, 023605 共2007兲 ZHANG, SCAROLA, AND DAS SARMA A兵jk其 are kept fixed. For example, M = 2 represents the usual two-body Heisenberg model. Several proposals now exist for simulating two-body Heisenberg models 关5,10兴. In the following we, as an example, consider optical lattice implementations of the Heisenberg XXZ model, 冉兺 具j,j⬘典 x y z 冊 共xj j⬘ + yj j⬘兲 + ⌬zj j⬘ , 共3兲 where 具j , j⬘典 denotes nearest neighbors and ⌬ and J are model parameters that can be adjusted by, for example, varying the intensity of lattice laser beams 关10兴. The paper is organized as follows: in Sec. II we show that, in practice, short-range spin correlations are difficult to detect in noise correlation measurements. Section III lays out a general procedure for detecting n-spin correlations with local probes. An experimental scheme for realizing such a procedure is proposed and a quantitative feasibility analysis is presented. Section IV is devoted to an investigation of the time dependence of correlation functions using the technique. In particular, we show how the technique can be used to engineer and probe spin-wave dynamics. We conclude in Sec. V. II. LOCAL CORRELATIONS IN TIME OF FLIGHT We first discuss the measurement of spin-spin correlation functions by analyzing spatial noise correlations 共two-point density-density correlations兲, i.e., 关具n共rជ兲n共rជ⬘兲典 − 具n共rជ兲典 ⫻具n共rជ⬘兲典兴 / 具n共rជ兲典具n共rជ⬘兲典 in time-of-flight images from atoms confined to an optical lattice modeled by the XXZ Hamiltonian. The ground states of this and a variety of spin models can be characterized by the spin-spin correlation function between different sites. For instance, the spin-spin correlation function in a one-dimensional XXZ spin chain 共with J ⬎ 0兲, shows power-law decay 具zj j⬘典 ⬃ 共− 1兲 j−j⬘/兩j − j⬘兩 z 共4兲 in the critical regime 共−1 ⬍ ⌬ ⱕ 1兲, where = 1 / 共1 − 1 cos−1 ⌬兲. In principle this correlation function can be probed by spatial noise correlation in time of flight. We argue that, in practice, short-range correlations 共e.g., ⬎ 1 in the XXZ model兲 are difficult to detect in time-offlight noise correlation measurements. To see this note that the noise correlation signal is proportional to 关15兴 G关Q共r − r⬘兲兴 = 兺 eiQ共j⬘−j兲a具zj j⬘典, z Correlation Amplitude HXXZ共J;⌬兲 = J 1 (a) 共5兲 j,j⬘ where Q is the lattice wave vector which gets mapped into coordinates r and r⬘ in time of flight on the detection screen, and a is the lattice spacing. Including normalization the noise correlation signal is proportional to N−1 for systems with long-range order 共e.g., antiferromagnetic order giving = 0 in the above XXZ model兲 but shows a much weaker scaling for short-range correlations. In fact the ratio between correlators in a ground state with ⬎ 1 共short-range power-law order兲 and = 0 共long-range antiferromagnetic order兲 scales as N−1 making the state with power-law correlations rela- N = 20 0.5 0 1 (b) N = 200 0.5 0 − 3π a − π π a a Q (r − r') 3π a FIG. 1. 共Color online兲 Noise correlation plotted as a function of wave vector of the one-dimensional XXZ model. The solid 共dashed兲 line corresponds to a ground state with = 0, long-range 共 = 2, short-range兲 spin correlator. The amplitudes are normalized by the maximum for antiferromagnetic order 共 = 0兲. The number of atoms in panels 共a兲 and 共b兲 are N = 20 and N = 200, respectively. tively difficult to detect in large systems. To illustrate this we compare the calculated noise correlation amplitude, G, in Fig. 1 for two cases = 0 共solid line兲 and = 2 共dashed line兲 with N = 20 关Fig. 1共a兲兴 and N = 200 关Fig. 1共b兲兴 for the 1D XXZ model. We see that the correlation amplitude for short-range 共power-law兲 order is extremely small in comparison to longrange antiferromagnetic order for large N. The small correlation signal originates from the fact that the noise correlation method is in practice a conditional probability measuring collective properties of the whole system, while short-range spin correlations describe local properties and are therefore best detected via local operations. In the following we propose a local probe technique to measure local correlations thus providing an experimental scheme which compliments the time-of-flight–noise correlation technique, best suited for detecting long-range order. III. DETECTING n-SPIN CORRELATION WITH LOCAL PROBES A. General procedure We find that general n-spin correlators, 兵␣ jk , k = 1 , . . . , n其, can be related to the Stokes parameters broadly defined in terms of the local reduced density matrix = Tr兵jk,k=1,. . .,n其 兩 ⌿典具⌿兩 on sites 兵jk , k = 1 , . . . , n其, where the trace is taken on all sites except the set 兵jk其. The Stokes parameters 关21兴 for the density matrix are S␣ j . . ..␣ j 1 ␣ n = Tr共兿k=1 j jk兲 leading to the decomposition k 3 = 2−n ␣j 1 兺 ,¯,␣ jn=0 冉 n S ␣ j . . .␣ j 1 n ␣ j 兿 k=1 jk k 冊 . n 共6兲 Using the theory of quantum state tomography 关21兴, we find the n-spin correlators 023605-2 PHYSICAL REVIEW A 76, 023605 共2007兲 PROBING n-SPIN CORRELATIONS IN OPTICAL LATTICES n 兵␣ jk,k = 1, . . . ,n其 = 兿 共P兵兩␣ j 典其 ± P兵兩␣⬜j 典其兲, k k=1 (a) 共7兲 k A B ↑ A B Microwave ↓ A B where the plus 共minus兲 sign indicates a 0 共nonzero兲 index and 兵兩␣ j 典 , 兩␣⬜ 典其 denote the measurement basis for the atom k Focused lasers B (b) F=2 jk ↑ at jk. We define the measurement basis to be 兩1典 = 共兩 ↓ 典 ⬜ 冑 冑 + 兩 ↑ 典兲 / 冑2, 兩⬜ 1 典 = 共兩 ↓ 典 − 兩 ↑ 典兲 / 2, 兩2典 = 共兩 ↓ 典 + i 兩 ↑ 典兲 / 2, 兩2 典 = 共兩 ↓ 典 − i 兩 ↑ 典兲 / 冑2, 兩3典 = 兩 ↓ 典, 兩⬜ 3 典 = 兩 ↑ 典. Finally, P兵兩␣ j 典其 is k the probability of finding an atom in the state 兩␣ j 典. k The expansion of the product defining then yields a quantity central to our proposal, -2 0 -1 1 (c) 2 ↑ Microwave F=1 ↓ -1 0 1 (d) 3 B A B A ↓ A B 2 B Fluorescence Signals Detection laser A B ↑ F =1 n 兵␣ jk,k = 1, . . . ,n其 = 兺 共− 1兲l Pl , 共8兲 l=1 where Pl is the probability of finding l sites in the states 兩⬜ jk 典 and n − l sites in 兩 jk典. Equation 共8兲 shows that the n-spin correlation function can be written in terms of experimental observables. We can now write a specific example of the two-spin correlation function 共discussed in the preceding section兲 in terms of observables: 兵3 , 3其 = P兩↓典 j 兩↓典 j + P兩↑典 j 兩↑典 j 1 2 1 2 − 共 P兩 ↓ 典 j1兩 ↑ 典 j2 + P兩↑典 j 兩↓典 j 兲 . In the following subsection we dis1 2 cuss a specific experimental procedure designed to extract precisely this quantity using local probes of cold atoms confined to optical lattices. B. Proposed experimental realization We now describe and critically analyze an experimental procedure designed to find the probabilities, Pl, from which we can determine the spin-correlation function 兵␣ jk , k = 1 , . . . , n其 through Eq. 共8兲. To illustrate our technique we consider a specific experimental system: 87Rb atoms confined on a single two-dimensional 共xy plane兲 optical lattice with two hyperfine ground states 兩 ↓ 典 ⬅ 兩F = 1 , mF = −1典 and 兩 ↑ 典 ⬅ 兩F = 2 , mF = −2典 chosen as the spin of each atom. Here the atomic dynamics in the z direction are frozen out by high frequency optical traps 关22兴. However, the scheme can be directly applied to three-dimensional optical lattices by using one additional focused laser which propagates along the xy plane and plays the same role as the focused laser 共propagating along the z axis兲 discussed in the following step 共II兲. In the Mott insulator regime with one atom per lattice site spin Hamiltonians, defined in Eq. 共2兲, may be implemented using spin-dependent lattice potentials in the superexchange limit 关10兴. The spin coupling J共t兲, i.e., the overall prefactor in Eq. 共2兲, can be controlled by varying the lattice depth. Our proposed experimental procedure will build on such spin systems, although it can be generalized to other implementations where H is generated by other means. In the spin systems, the ground states are strongly correlated many-body spin states and their properties can be characterized by the spin correlations between atoms at different lattice sites. Step 共I兲. We start with a many-body spin state ⌽0 and turn off the spin-spin interactions generated by superexchange between lattice sites. We achieve this by ramping up the spindependent lattice depth to ⬃50ER adiabatically with respect FIG. 2. 共Color online兲 Schematic plot of the experimental procedure used to measure n-spin correlation functions. A indicates one of the target atoms. B indicates all other atoms. 共a兲 Target atoms A are transferred to a suitable measurement basis using a combination of focused lasers and microwave pulses 关see step 共II兲兴. 共b兲 Atoms at state 兩 ↓ 典 are transferred to state 兩2典 by two microwave pulses, then atoms at state 兩 ↑ 典 are transferred to state 兩 ↓ 典 by another microwave pulse 关see step 共III兲兴. 共c兲 Target atoms A are transferred back to state 兩 ↑ 典 from 兩 ↓ 典 关see step 共IV兲兴. 共d兲 A detection laser is applied to detect the probability of finding target atoms at 兩 ↑ 典 关see step 共IV兲兴. to the band splitting, where ER = h2 / 2m2 is the photon recoil energy and is the wavelength of the optical lattice. The ramp up time for each lattice is chosen properly so that only the overall energy scale J共t兲 in the spin Hamiltonian 共2兲 is modified, which preserves the highly correlated spin state ⌽0. In the deep lattice, the time scale for the spin-spin interactions 共⬃ប / J ⬎ 10s兲 becomes much longer than the time 共⬃1 ms兲 taken to perform the spin-correlation measurement. The spin-spin interactions play no role in the measurement process and the following detection steps are quickly performed on this “frozen” many-body spin state ⌽0. Step 共II兲. In this step, we selectively transfer target atoms A at site共s兲 jk 共chosen a priori兲 to a suitable measurement basis 兵兩 jk典 , 兩⬜ jk 典其 from initial states 兵兩 ↓ 典 jk , 兩 ↑ 典 jk其, without affecting nontarget atoms B at other sites 关Fig. 2共a兲兴. Selective manipulation of quantum states of single atoms in optical lattices is currently an outstanding challenge for investigating physics in an optical lattice, because spatial periods of typical optical lattices are shorter than the optical resolution. In Ref. 关23兴, a scheme for single atom manipulation using microwave pulses and focused lasers is proposed and analyzed in detail. Here we apply the scheme to selectively transfer atoms between different measurement bases. To manipulate a target atom A, we adiabatically turn on a focused laser that propagates along the ẑ axis having the maximal intensity located at A. The spatially varying laser intensity I共r兲 induces position-dependent energy shifts ⌬Ei共r兲 = iI共r兲 共9兲 for two spin states 兩 ↓ 典 and 兩 ↑ 典, where the parameter i for state 兩i典 共i = ↓ or ↑兲 is determined by the focused laser parameters. Different polarizations and detunings of the focused laser lead to different i and thus yield different shifts of the 023605-3 PHYSICAL REVIEW A 76, 023605 共2007兲 ZHANG, SCAROLA, AND DAS SARMA ⍀共t兲 = ⍀0 exp共− 20t2兲 共− t f ⱕ t ⱕ t f 兲 and parameters 0 = 14.8ER / ប, ⍀0 = 13.1ER / ប, and t f = 5 / 0. The pulse transfers the measurement basis of the target atoms A in 16.9 s, while the change in the quantum state of nontarget atoms is found to be below 3 ⫻ 10−4 by numerically integrating the Rabi equation that describes the coupling between two spin states by the microwave pulse. The focused lasers are adiabatically turned off after the microwave pulse. During the step 共II兲, the probability for spontaneous scattering of one photon from target atoms inside the ⌫ focused laser can be roughly estimated as P ⬇ 兰 f ប V共t兲dt i −4 ⬃ 2 ⫻ 10 , where i and f represent the times when the focused laser is turned on and off, ⌫ is the decay rate of 6P state, is the detuning, and V共t兲 is the potential depth of the focused laser. The distance between any two target atoms can be as short as a lattice spacing. This is because the basis transfer processes for different target atoms are preformed sequentially in time, i.e., the process for an atom at site j2 starts after the process for an atom at site j1 is accomplished. In the special case that sites j1 and j2 are spatially well separated and the final basis 兵兩 jk典 , 兩⬜ jk 典其 at two sites are the same, the transfer process can be done simultaneously for two sites with one microwave pulse. Step 共III兲. In this step, we transfer all atoms to the 兩F = 1典 hyperfine level 关Fig. 2共b兲兴 to avoid stray signal in the detection step 共IV兲. We apply two microwave pulses to 20 (a) (d) n pA 0.2 n pA ρ 33A 0.4 10 0 20 (b) 0 10 1×10 −4 (e) n Bp 0 1×10 −5 (c) 5 ×10 −5 n Bp hyperfine splittings 兩⌬E共r兲 兩 = 兩⌬E↑共r兲 − ⌬E↓共r兲兩 between two spin states. We choose a +-polarized laser that drives the 5S → 6P transition to obtain a small diffraction limit. The wavelength f ⬇ 421 nm 共corresponding to a detuning ⌬0 = −2 ⫻ 1209 GHz from the 6 2 P3/22 state兲 is optimized to obtain the maximal ratio between energy splittings of two spin states and the spontaneous scattering rate 关23兴. Because of the inhomogeneity of the focused laser intensity I共r兲, 兩⌬E共r兲兩 reaches a maximum at the target atom A and decreases dramatically at neighboring sites. Therefore the degeneracy of hyperfine splittings between different atoms is lifted. By adjusting the focused laser intensity, the differences of the energy shifts ␦ = 兩⌬E共0兲 兩 −兩⌬E共 / 2兲兩 between target atom A and nontarget neighboring atom B can be varied and calculated through Eq. 共9兲. Here we choose ␦ = 74ER because it balances the speed and fidelity of single spin manipulation, leading to less spontaneously scattered photons from the target atoms in the spin-dependent focused lasers. To avoid excitations of atoms to higher bands of the optical lattice, the rise speed of the focused laser intensity should satisfy the adiabatic condition. We use the adiabatic condition to estimate the ramp up time of the focused laser to be 35 s to give a 10−4 probability for excitation to higher bands. We then change the measurement basis by applying a microwave / 2 pulse that drives a suitable rotation to target atoms A 关Fig. 2共a兲兴. The microwave is resonant with the hyperfine splitting between two spin states of the target atoms A, but has a detuning larger than ␦ for nontarget atoms B. Consider a pulse with Rabi frequency 5 ×10−6 0 0 10 20 30 40 50 Γt 0 0 1 2 Ω/Γ 3 FIG. 3. 共a兲 Time evolution of the probability for the target atoms A to be in the excited state 兩3典. 共b兲 The number of scattering photons nAp versus time for atoms A at state 兩 ↑ 典. 共c兲 The number of scattering photons nBp versus time for atoms 共A or B兲 at states 兩 ↓ 典 and 兩2典. 共d兲 The number of scattering photons nAp versus the Rabi frequency of the resonant laser for atoms A at state 兩 ↑ 典. 共e兲 The number of scattering photons nBp versus the Rabi frequency of the resonant laser for atoms 共A or B兲 at states 兩 ↓ 典 and 兩2典. The time period for 共d兲 and 共e兲 is 50/ ⌫. transfer all atoms at 兩 ↓ 典 first to 兩F = 2 , mF = 0典 then to 兩2典 ⬅ 兩F = 1 , mF = 1典. Another microwave pulse is then applied to transfer all atoms at 兩 ↑ 典 to 兩 ↓ 典. The microwave pulse can be implemented within 12.5 s for a microwave Rabi frequency ⍀ = 2 ⫻ 40 kHz. Step 共IV兲. We transfer target atoms A at jk from 兩 ↓ 典 back to 兩 ↑ 典 with the assistance of the focused lasers and microwave pulses 关Fig. 2共c兲兴, using the same atom manipulation procedure as that described in step 共II兲. We then apply a detection laser resonant with 兩↑典 → 兩3典 ⬅ 兩5 2 P3/2:F = 3,m = − 3典 to detect the probability of finding all target atoms at 兩 ↑ 典 关corresponding to the basis state 兩⬜ jk 典 because we transferred atoms to the measurement basis in step 共II兲兴 关Fig. 2共d兲兴. The beam size of the detection laser should be large enough so that target atoms at different sites 兵jk其 experience the same laser intensity, and scatter the same number of photons if they are in the 兩 ↑ 典 state. The fluorescence signal 共the number of scattered photons兲 is from one of the n + 1 quantized levels, where the lth level 共l = 0 , . . . , n兲 corresponds to states with l sites of target atoms on state 兩 ↑ 典 共兩⬜ jk 典兲. By repeating the whole process many times, we obtain the probability distribution Pl, and thus the spin correlation function 兵␣ jk , k = 1 , . . . , n其 via Eq. 共8兲. The scattering photons of the fluorescence signals come mostly from the target atoms A at state 兩 ↑ 典 共Fig. 3兲. Signal from atoms at any 兩F = 1典 state is suppressed because of the large hyperfine splitting 共 ⬇ 2 ⫻ 6.8 GHz兲 between 兩F = 1典 and 兩F = 2典 states. Atoms B do not contribute to the fluorescence signal because they are already transferred to the 兩F = 1典 state in step 共III兲. The dynamics of photon scattering is described by the optical Bloch equation 关24兴, 023605-4 d i 33 = − ⌫33 + ⍀共13 − 31兲, dt 2 PHYSICAL REVIEW A 76, 023605 共2007兲 PROBING n-SPIN CORRELATIONS IN OPTICAL LATTICES 冉 冊 冉 冊 d ⌫ i 13 = − + i0 13 + ⍀共233 − 1兲, dt 2 2 ⌫ d i 31 = − − i0 31 − ⍀共233 − 1兲, dt 2 2 共10兲 where 33共t兲 = 兩c3共t兲兩2 is the probability for the atom to be in the excited state 兩3典, 兩1典 represents state 兩 ↑ 典 for target atom A at state 兩 ↑ 典 and 兩F = 1典 for other cases, 13共t兲 = c1共t兲c*3共t兲ei0t, * 31共t兲 = 13 共t兲. The detuning of the laser 0 is zero for target atoms A at state 兩 ↑ 典 and ⬃ for all nontarget atoms B and part of the target atoms A at hyperfine state 兩2典. ⌫ = 2 ⫻ 6.07 MHz is the decay rate of the excited state 兩3典, ⍀ is the Rabi frequency of the resonant laser that is related to the on-resonance saturation parameter by s0 = 2 兩 ⍀兩2 / ⌫2. We numerically integrate the optical Bloch equation and calculate the number of scattering photons n p共t兲 = ⌫ 冕 t 33共t⬘兲dt⬘ 共11兲 0 for both target and nontarget atoms. In Fig. 3共a兲, we plot the A for target atoms A at state 兩3典 with respect to probability 33 A increases initially and reaches the saturation time. We see 33 value s0 / 2共1 + s0兲. The number of scattering photons reaches 20 in a short period, 1.3 s for atoms A at 兩 ↑ 典 关Fig. 3共b兲兴, but it is only 10−5 for nontarget atoms B and target atoms A at state 兩2典 关Fig. 3共c兲兴. Therefore the impact of the resonant laser on the nontarget atoms B can be neglected. In Figs. 3共d兲 and 3共e兲, we see that for a wide range of Rabi frequencies 共laser intensities兲, the scattering photon number for the nontarget atoms B is suppressed to undetectable levels, below 10−4. Unlike the noise correlation method, the accuracy of our detection scheme does not scale with the number of total atoms N, but is determined only by manipulation errors in the above steps. We roughly estimate that n-spin correlations can be probed with a cumulative error ⬍n ⫻ 10−2, which is sufficient to measure both long- and short-range spincorrelation functions. Our estimate takes into account possible experimental errors in all four steps, and the fact that the same experimental procedure is repeated many times to determine the probability Pl. The total failure probability of the four-step detection process is p共n兲 ⬃ n ⫻ 10−3 for n target atoms to give an incorrect measurement of the target atom quantum state for determining the quantity Pl. Assuming that the experimental procedure is repeated times, the total probability for obtaining one incorrect measurement result is about p共n兲. Since one incorrect measurement result leads to an error ⬃1 / in Pl, the expectation for the error is p共n兲 ⫻ 1 / = p共n兲, which should be chosen to be 1 / to minimize the total error. In addition, the uncertainty in measurements of probability Pl itself in repeated experiments is also about 1 / . Taking into account all of these errors, we find that n-spin correlations can be probed with an error that scales as Cp共n兲, where C ⬃ 3 in our rough estimate. As a conservative estimate, we take C = 10 to give an overall error in measuring n-spin correlation function 兵␣ jk , k = 1 , . . . , n其 to be less than 10p共n兲 ⬃ n ⫻ 10−2. We note that the scheme requires repeated production of nearly the same condensate and repeated measurements, two standard techniques which have been realized in many experiments. We have proposed a powerful technique for investigating strongly correlated spin models in optical lattices and now consider one of several possible applications. IV. SPIN-WAVE DYNAMICS Our technique can be used to investigate time dependence of correlation functions. In the following, we show how our scheme can be used to engineer and probe spin-wave dynamics in a straightforward example, the Heisenberg XX model realized in optical lattices with a slightly different implementation scheme than the one discussed in the preceding section. Consider a Mott insulator state with one boson per lattice site prepared in the state 兩0典 ⬅ 兩F = 1 , mF = −1典 in a threedimensional optical lattice. By varying the trap parameters or with a Feshbach resonance, the interaction between atoms can be tuned to the hard-core limit. With large optical lattice depths in the y and z directions, the system becomes a series of one-dimensional tubes with dynamics described by the Bose-Hubbard Hamiltonian, Hs = − 共t兲 兺 共a†j a j+1 + a†j+1a j兲. 共12兲 j This Bose-Hubbard model can be solved exactly. It offers a testbed for spin-wave dynamics by mapping the Hamiltonian 共12兲 onto the XX spin model, HXXZ共−2 ; ⌬ = 0兲, with the Holstein-Primakoff transformation 关25兴. We now study the time-dependent behavior of the XX model using our proposed scheme. In the Heisenberg picture, the time evolution of the annihilation operator can be written as a j共t兲 = 兺 j⬘a j⬘共0兲i j⬘−jJ j⬘−j共␣兲, where J j⬘−j共␣兲 is the Bessel function of the interaction parameter ␣共t兲 = 2兰t0共t⬘兲dt⬘. To observe spin-wave dynamics, we first flip the spin at one site from ↑ to ↓, which, in the bosonic degrees of freedom, corresponds to removing an atom at that site. Because of the spin-spin interactions, initial ferromagnetic order gives way to a reorientation of spins at neighboring sites which propagates along the spin chain in the form of spin waves. This corresponds to a time-dependent oscillation of atom number at each site. Therefore, spin-wave dynamics can be studied in one- and two-point spin-correlation functions by detecting the oscillation of the occupation probability at certain sites and the density-density correlator between different sites, respectively. Single atom removal at specific sites can be accomplished with the assistance of focused lasers. With a combination of microwave radiation and two focused lasers 共propagating along y and z directions, respectively兲, we can selectively transfer an atom at a certain site from the state 兩0典 to the state 兩4典 ⬅ 兩F = 2 , mF = −2典. A laser resonant with the transition 兩4典 → 兩3典 ⬅ 兩5 2 P3/2 : F = 3 , m = −3典 is then applied to remove an atom at that site. Following an analysis similar to the one 023605-5 PHYSICAL REVIEW A 76, 023605 共2007兲 V共t兲 = V0/共1 + 4冑2PexeV0/ERrt兲, 共13兲 where Pexe is the probability of making an excitation to higher bands and R = ER / ប. For Pexe = 4 ⫻ 10−4, we find the interaction parameter to be ␣共thold兲 = 0.0146+ 0.0228Rthold, with the tunneling parameter 共t兲 = 共4/冑兲ER1/4V3/4共t兲exp关− 2冑V共t兲/ER兴. 共14兲 Two physical quantities that can be measured in experiments are the single atom occupation probability D j共␣兲 = 具兩a†j a j兩典 = J2l−j共␣兲 兺 l⫽ 共15兲 at the site j, and the density-density correlator † G jj⬘共␣兲 = 具兩a†j a ja j⬘a j⬘兩典 = − 2 J2l−j共␣兲J␥−j⬘共␣兲 兺 l⫽,␥⫽ 2 关J2l−j共␣兲Jl−j⬘共␣兲 兺 l⫽ + Jl−j共␣兲J−j共␣兲J−j⬘共␣兲Jl−j⬘共␣兲兴 共16兲 between sites j and j⬘, where is the initial wave function with one removed atom at site . The former is related to the local transverse magnetization through 具兩szj 共␣兲兩典 = D j共␣兲 − 1/2, 共17兲 and the latter is related to the spin-spin correlator via z G jj⬘共␣兲 = 具兩szj 共␣兲s j⬘共␣兲兩典 + 关D j共␣兲 + D j⬘共␣兲兴/2 + 1/4. 共18兲 In Fig. 4, we plot D j共␣兲 and G jj⬘共␣兲 with respect to the interaction parameters ␣ 共which scales linearly with holding time兲. Here the initial empty site at j = 0 is located at the center of the spin lattice. We see different oscillation behavior at different sites. Initially, all sites are occupied except j = 0, i.e., D j⫽0共0兲 = 1, D j=0共0兲 = 0 关Fig. 4共a兲兴. The initial spinspin correlation G jj⬘共0兲 between different sites is zero if either j or j⬘ is zero, and one if both of them are nonzero 关Fig. 4共b兲兴. As ␣ increases, atoms start to tunnel between neighboring sites and the spin wave propagates along the onedimensional optical lattice, which is clearly indicated by the 1 0.8 (a) 0.6 0.4 0.2 0 0 1 Dj (α) above, we see that the impact on other atoms can be neglected. To observe fast dynamics of spin-wave propagation, we may adiabatically ramp down the optical lattice depth 共and therefore increase 兲 from the initial depth V0 = 50ER to a final depth 13ER, with a hold time, thold, to let the spin wave propagate. Finally, the lattice depth is adiabatically ramped back up to V0 for measurement. The time dependence of the lattice depth in the ramping down process is chosen to be 1 0.8 (b) 0.6 0.4 0.2 0 0 1 6 Gjj ' (α) ZHANG, SCAROLA, AND DAS SARMA 2 3 α4 5 2 3 α 4 5 6 FIG. 4. Site occupation probability 共a兲 and density-density correlation 共b兲 with respect to scaled spin interaction parameter ␣⬀ time for N = 30 and j = 0 关共a兲 solid line兴; j = 1 关共a兲 dashed line兴; j = 0, j⬘ = 3 关共b兲 dashed line兴; and j = 2, j⬘ = 3 关共b兲 solid line兴. The site index j ranges from −N to N. increase 共decrease兲 of the site occupation at j = 0 共j ⫽ 0兲. In Fig. 4, the oscillation of D j共␣兲 and G jj⬘共␣兲 in a long time period and the decay of the oscillation amplitudes originate from the finite size of the spin lattice, which yields the reflection of the spin waves at the boundaries. To probe the single site occupation probability D j共␣兲, we use two focused lasers and one microwave pulse to transfer the atom at site j to 兩4典. A laser resonant with the transition 兩4典 → 兩3典 is again applied to detect the probability to have an atom at 兩4典, which is exactly the occupation probability D j共␣兲. To detect G jj⬘共␣兲, we transfer atoms at both sites j and j⬘ to the state 兩4典 and use the same resonant laser to detect the joint probability for atoms at 兩4典. The fluorescence signal has three levels, which correspond to both atoms G jj⬘共␣兲, one atom D j共t兲 + D j⬘共t兲, and no atoms at state 兩4典. A combination of these measurement results gives the spin-spin correlator 具 兩 szj 共␣兲szj⬘共␣兲 兩 典. V. CONCLUSION We find that a relation between general spin-correlation functions and observable state occupation probabilities in optical lattices allows for quantitative measurements of a variety of spin correlators with the help of local probes, specifically focused lasers and microwave pulsing. 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