Numerical study of two-body correlation in a 1D lattice with perfect blockade

We compute the dynamics of excitation and two-body correlation for two-level"pseudoatoms"in a 1D lattice. We adopt a simplified model where pair excitation within a finite range is perfectly blocked. Each superatom is initially in the ground state, and then subjected to an external driving laser with Rabi frequency satisfying a Poissonian distribution, mimicking the scenario as in Rydberg gases. We find that two-body quantum correlation drops very fast with the distance between pseudoatoms. However, the total correlation decays slowly even at large distance. Our results may be useful to the understanding of Rydberg gases in the strong blockade regime.

, spurred largely by fast developments in laser cooling and trapping. In such systems, the dipolar interaction between two nearby atoms will shift the pair excitation out of resonance with the driving laser. Local excitation is then greatly suppressed, showing the well-known phenomenon of dipole blockade. It is an important mechanism responsible for the explanation of many-body effects such as spectral line broadening [1, 2] and the expected sub-Poissonian atom counting statistics [4,8,9]. It is also proposed to utilize dipole blockade to implement quantum gates useful in quantum computation and information [10,11]. To date, there have been many theoretical proposals and numerical simulations in exploring the excitation dynamics and coherence properties such as pair correlation and number correlation, which demonstrate the important role that dipole blockade played in these systems.
Previous investigations of Rydberg gas have focused on the mean number of excitations and how the excitations are correlated in a gas. However, there hasn't been a study of genuine quantum correlation. This paper will discuss other useful information that can be extracted from the wave function, especially the two-body quantum correlation and total correlation (including classical correlation). As we will see later, two-body correlation is another important quantity for characterizing such systems. To investigate correlation properties, we are mainly interested in the following two questions. First, in such systems, how does correlation behave as the excitation begins to build up? Second, which correlation is more important? It has been shown that a mean field approach is not appropriate in dealing with correlation and a full quantum calculation is necessary [12]. However, the direct simulation of correlation is still absent due to numerical complexities in real Rydberg gases.
We first consider the scenario as in current experiments where Rydberg atoms are confined in a volume V and driven by a laser. In the spirit of the dipole blockade model, we can divide V into equally sized regions V = k v k with each region approximated as a sphere of blockade radius R b . We follow the usual notation and call the ensemble of atoms in each region v k a "superatom" [13]. For van der Walls interaction and a narrow band laser, R b is determined from C 6 /R 6 b = Ω, where C 6 is the van der Waals coefficient in the atomic unit and Ω is the Rabi frequency of the driving laser. Rydberg atoms are randomly distributed in V so that the number of atoms in each v k is in general different. For example, in the homogeneous case, the atom numbers in each superatom follow a Poissonian distribution with the same mean atom number. Such a picture is useful in explaining certain experimental results, e.g., the population dynamics under a driving laser. In this case, each superatom evolves with different Rabi frequencies. This leads to a fast and almost linear increase in the initial stage and saturation in the long time. However, such a model is not so interesting in showing the two-body correlation properties because correlation between different superatoms is apparently 0. Therefore, a proper modification to the superatom model is necessary in order to give nontrivial information on two-body correlation.
In this paper, we discuss an elongated Rydberg gas whose transverse size is smaller than R b . This allows us to treat the Rydberg gas as a quasi 1D system. The advantage is two fold. On the one hand, the edge effect in numerical simulation is smaller and the results converge relatively faster than those in higher dimensions with the same atom number. On the other hand, it permits an easier readout of the quantum state for ensemble of atoms in any region along the longitudinal direction. Similar to the 3D case, we divide such a quasi 1D gas into a collection of superatoms aligned along the longitudinal direction. We investigate the two-body correlation by further dividing each superatom (v k ) into smaller subregions labelled as w k,α , i.e., v k = Nw α=1 w k,α . The center of w k,α is denoted as r k,α . The number of partitions in each v k , labelled as N w , is assumed the same, ∀k. For the interest of this paper, we term each subregion w k,α as a "pseudoatom" located at position r k,α . in Fig.   1, we give a schematic view of partitioning 3 superatoms into 9 pseudoatoms. The subscripts k and α can be further combined into a single one by relabelling them as in a spin chain.
Interaction between two pseudoatoms on site j and k are assumed to be a van der Waals interaction with the distance |r j − r k |. This approximation neglects the position variance of excitation in each pseudoatom. By partitioning each superatom into more pseudoatoms, the error can be reduced. Different from the case of nearest neighboring superatoms (with a distance 2R b ) where the interaction is usually ignored, we can see that nearest neighboring pseudoatoms (with a distance of a fraction of R b ) interact much more strongly so all of the pseudoatoms are correlated analogous to a spin chain.
Due to the R −6 dependence of interaction, the transition from blockade to no blockade has a narrow width of R. This gives us the motivation to assume a perfect blockade with pseudoatoms within a certain distance and completely no blockade outside of it. From the many body point of view, the introduction of this approximation allows us to concentrate represents a superatom which is further divided into 3 pseudoatoms w k,α (α = 1, 2, 3).
on the much simpler short range interaction rather than the long range interaction. It not only lets us see the effect due to pure blockade, but also greatly reduces the needed basis which makes the numerical simulation in a much larger system possible. The validity of this approximation will be discussed in the later part of this paper. Because of this type of partition for a superatom, the maximal number of pseudoatoms that can be blocked is just the maximal distance between two pseudoatoms j and k which have an interaction V j,k much larger than the Rabi frequency Ω, i.e., max j,k {j − k|V j,k /Ω > η}. Besides the vagueness of this definition, in practice we choose η = 6. A little algebra shows that the maximal number of pseudoatoms that can be blocked is given by max k=0,1,2... {k|k < 0.37N w }. The first few values are given here: for N w = 2, 3, 4, 5, 6, ..., the maximal number of pseudoatoms that can be blocked is found to be 0, 1, 1, 1, 2, ..., respectively. So the simplest nontrivial situation starts from N w = 3, perfect blockade only between two nearest neighboring pseudoatoms.
In this paper, we will limit our discussion to the situation where perfect blockade only exists between two nearest neighboring pseudoatoms.
Because of our approximation, the Hamiltonian can be written asĤ = where N is the total number of pseudoatoms.ˆ σ (k) are Pauli matrices for the k-th pseudoatom. J k = X k /λ is the scaled Rabi frequency for the k-th pseudoatom. We are assuming a resonant laser field with a constant intensity. X k is a random variable satisfying a Poissonian distribution with mean value λ. In this case, λ denotes the average number of Rydberg atoms in a pseudoatom. More explicitly, λ = N r /N w with N r the number of Rydberg atoms in a superatom which can be determined from the Rydberg gas density n, the blockade radius R b and the area in the transverse direction A through the relation The underlying motivation for the assumed form of J k is the number fluctuation and the collective excitation in each pseudoatom [6]. A given set of {J 0 , ..., J N −1 } is said to define a configuration. Our numerical results are obtained by averaging over many configurations. We note that the Hamiltonian formally describes a nonphysical multi-body interaction, which is of course an effective Hamiltonian constrained by the perfect blockade requirement. All the pseudoatoms are initially in the ground state |g ⊗N , and then subjected to a resonant external laser field. The subsequent dynamics constrained by the perfect blockade are what we are after in this paper.
For nearest neighboring blockade, the maximally possible excitation for all pseudoatoms is found to be [N/2] where [·] denotes the integer part. As a consequence, the number of restricted basis, N b , is significantly reduced from that of the full basis set. To compute the wave function, we expand it in the restricted basis labelled as |µ p , i.e., the fraction of excitation is found to be In  The excitation fraction of a physical Rydberg gas can be qualitatively obtained by mean field approach [3], which is essentially a reduction to single-atom picture. However, the two-atom correlation, where the off-diagonal terms of the density matrix are important, cannot be simply accounted for from the mean field approach. In the following, we will focus on correlation properties of reduced two-pseudoatom subsystem from a full quantum mechanical calculation.
The first quantity we will look into is the pair correlation, P ee , at saturation time, which is defined as the ratio of the probability of both pseudoatoms excited divided by the square of probability of single-pseudoatom excitation [12]. In Fig. 3, we show P ee as function of distance for different λ. Compared with the results in Ref. [12] (see Fig. 2) which uses a continuous model, our results captures the main physics in a much simpler way. We can see that for both the discrete and continuous model, P ee exhibits similar behaviors: it is negligible within a distance and there is a sharp increase after that; at large distance, it saturates to 1, corresponding to no correlation. For all three λ in our calculation, the next to nearest neighbor always has the largest value. This can be interpreted as follows. Assuming the index of the first pseudoatom to be 0, the separation d is simply the label of the other pseudoatom. P ee for d = 1 is 0 because pseudoatom 1 cannot be excited once pseudoatom 0 is excited. For d = 2, P ee is well above 1. This is because if pseudoatom 0 is in the excited state, then pseudoatom 1 must be in the ground state. Thus pseudoatom 2 is more likely to be in the excited state. Similarly, pseudoatom 3 will be less likely in the excited state, resulting in a value lower than 1. The fast approach to 1 at large distance means that such pair correlation is only short ranged. We do not present the result for λ = ∞ due to the non-negligible oscillation even at long enough time. While the above pair correlation only relies on the diagonal elements of the reduced two-body density matrix, the genuine quantum correlation must be computed from the full reduced two-body density matrix. We use entanglement of formation (EOF) as our measure for quantum correlation [14]. It is related to the so-called concurrence, C = max(0, , where λ i are the eigenvalues of a Hermitian matrix, R ≡ √ ρ(σ y ⊗ σ y )ρ * (σ y ⊗ σ y ) √ ρ, in descending order. EOF (ε) is then given by ε = h((1 + √ . ε gives 0 for separable states and 1 for maximally entangled states. In Fig. 4, we show EOF for pseudoatom pairs (0, 1) for different λ. We find that ε (0,1) shows similar pattern as excitation fraction. During the same time scale, they approach a maximal value and then decay slowly. This reflects two simple facts that entanglement is originated from excitation and it is superposed coherently in the initial stage. The behaviors of entanglement for other pairs are shown in Fig. 5. We can see they again behave in a similar way, although the first peaks are offset in time.
To investigate how entanglement varies with the distance between pseudoatoms, we use the first peak of EOF for different pairs to characterize its dependence on the distance. Our results are shown in Fig. 6. We note that EOF drops almost exponentially as the distance increases for all λ.  We can see that quantum correlation is significant only for nearest neighboring pseudoatoms. However, this is not the case for the total correlation (including classical correlation). To compute the total correlation, we adopt the measure as suggested by D. L. Zhou and L. You [15]. For density matrix ρ (12) of pseudoatoms 1 and 2, it is given by where ρ (k) is the reduced density matrix of the k-th pseudoatom. Different from Ref. [15], here the prefactor 2/3 is chosen so that this measure gives 1 for a maximally entangled state.
In Fig. 7, we show the total correlation as function of time for different λ. We again see that they show similar pattern as EOF. To investigate the distance dependence of total correlation, we show the first peak value of total correlation in Fig. 8. We can see that total correlation does not drop like a single exponential function with distance as that in EOF. Rather interestingly, they even do not decrease monotonically as increasing distance.
A simulation with N = 20 atoms shows similar behaviors with distance. We find that they give almost identical results for several initial distances. However, close to the middle point of the lattice (distance N/2), those values of N = 20 have correspondingly lower values than N = 16. Therefore, such non-monotonicity will become negligible for N → ∞, i.e., a finite size effect. We conclude that although quantum correlation is not important in most cases, the total correlation cannot be simply ignored, i.e., there is non-negligible classical correlation in such systems. It is necessary to include not just nearest neighboring pseudoatoms when considering correlation-related properties.
The extension of our results to higher dimensions seems difficult due to the computational limitation. For 2D configurations and nearest neighboring blockade, we can only simulate up to 5 × 5 square lattice. Without presenting more figures, we briefly discuss our results.
Our major results of 1D still hold in 2D configurations. We find that EOF shows similar patterns but drops even faster as distance increases. This is possibly due to the fact that in  2D, as more neighboring pseudoatoms are involved, the fluctuation is more intensive than that in 1D. In addition, the corresponding pair entanglement is found to be smaller than that in 1D. Thus we conjecture that the pair entanglement is even smaller in a 3D cubic lattice.
Before concluding, we want to justify the validity of our simplified Hamiltonian. We will show that the fast decay of entanglement on distance is not an artifact of the assumed short range interaction. To confirm this point, we carry out simulations using the long range where J k takes the same form as in the HamiltonianĤ. The dimensionless parameter D quantifies the interaction strength between two nearest neighboring pseudoatoms. Because we always use the distance between two nearest neighboring pseudoatoms as the length scale, D also depends on N w , i.e., the decrease in the distance is equivalent to the increase in the interaction strength. We carry out similar calculations by using the full basis ofĤ ′ .
Selected results for N w = 5 with nearest neighboring blockade are shown in Figs. 9 and 10, for ε max and M c,max as function of distance, respectively. We find that the main features by usingĤ ′ are qualitatively reproduced by the simplified HamiltonianĤ, which leads us to conclude that the behavior of entanglement on distance is not an artifact of short range interaction in the simplified HamiltonianĤ. And for the total correlation, the overall shape of the curves remains the same. More importantly, the total correlation does not drop as fast as entanglement with distance, which provides another evidence that our simplified model captures the main physics. We emphasize that this calculation not only justifies our simplified model, but also means that those effects in Figs. 6,8,9, and 10 are mostly due to blockade, which shows that the long range interaction is actually not so important to correlation properties. In conclusion, we have performed full quantum calculations on correlation properties for pseudoatoms in a 1D lattice structure with perfect blockade. By comparing the results from a reduced basis and a full basis calculation, we justify the validity of the simplified Hamiltonian for perfect blockade among pseudoatoms. This agreement also means that single-and twoparticle correlation properties (e.g., average excitation, correlation) are determined mostly by the pure blockade effect. From numerical simulation, we find that there are both quantum and classical correlations accompanying the building up of atomic excitation. Our results show that two-body entanglement is only important for nearest neighboring pseudoatoms and it drops exponentially fast with the distance between them even when there is no fluctuation in the system. However, the total correlation decays much more slowly with distance, showing the system in this paper is mostly classically correlated. As a simple extension to higher dimension, we compute our model system in a two dimensional 5 × 5 square lattice. We find 2D results agree qualitatively with those of 1D. From the theoretical point of view, our findings imply that a better description of Rydberg gas beyond mean field or superatom picture should also at least take classical correlation into consideration.
We hope that our study can be helpful to the understanding of Rydberg gases in the strong blockade regime.
We would like to thank J. V. Hernández for some helpful discussions. This work is supported by the NSF under grant no. 0355039.