Many-Body Quantum Spin Dynamics with Monte Carlo Trajectories on a Discrete Phase Space

Interacting spin systems are of fundamental relevance in different areas of physics, as well as in quantum information science and biology. These spin models represent the simplest, yet not fully understood, manifestation of quantum many-body systems. An important outstanding problem is the efficient numerical computation of dynamics in large spin systems. Here, we propose a new semiclassical method to study many-body spin dynamics in generic spin lattice models. The method is based on a discrete Monte Carlo sampling in phase space in the framework of the so-called truncated Wigner approximation. Comparisons with analytical and numerically exact calculations demonstrate the power of the technique. They show that it correctly reproduces the dynamics of one- and two-point correlations and spin squeezing at short times, thus capturing entanglement. Our results open the possibility to study the quantum dynamics accessible to recent experiments in regimes where other numerical methods are inapplicable.

Popular Summary

Understanding the quantum dynamics exhibited by large-spin systems is fundamental for various fields of physics, quantum information science, and biology. However, quantum many-body systems are complex; computing their exact time evolution is, in general, impossible. Numerical approximation methods have been developed but are mostly limited to dealing with low-dimensional systems or weakly interacting regimes. We develop and test a new semiclassical approach for dealing with the time evolution of generic spin-lattice models. Our method substantially improves the accuracy of existing semiclassical methods by accounting for the inherently discrete nature of quantum-mechanical spin.

Our technique, which relies on a Monte Carlo sampling, is able to accurately capture the dynamics of the $x$, $y$, and $z$ moments of a single spin and also entanglement and correlations between pairs of spins, regardless of the dimension of the model. We test our method using coupled two-level systems with interactions that decay with increasing distance in different ways and find that our method yields results that are in agreement with analytical and numerically exact calculations.

This method opens up the possibility of studying quantum dynamics in long-range spin models accessible to recent state-of-the-art experiments in regimes where other numerical techniques cannot be applied. We expect that our method will be useful for analyzing the quantum dynamics of polar molecules, Rydberg atoms, and photonic crystals, for instance.

Figure 1

Discrete phase space and the DTWA. (a) Our method considers the quantum uncertainties of $N$ spin-$1/2$ particles individually, rather than the noise of the collective spin $S=N/2$. (b) The quantum physics of a spin-$1/2$ particle can be fully described by a discrete four-point Wigner quasiprobability distribution, $w_{(p,q)}$. The probability for a spin to point along the $\pm x$, $\pm y$, and $\pm z$ directions ($p_{\pm 1}^{x,y,z}$) is given by the sum over the vertical, diagonal, and horizontal lines in phase space, respectively [20]. (c) Discrete Wigner function of a spin pointing along $z$. (d) The idea behind the DTWA is to (i) randomly sample the phase points for each spin according to $w_{(p,q)}$, (ii) calculate the time evolution according to classical equations, and (iii) evaluate observables in phase space from the statistical mixture of $n_t$ trajectories.

Figure 2

Dynamics for Ising interactions. Circles denote the exact solution, dashed lines are traditional TWA results, solid lines denote DTWA results, for 1D, $N=100$ spins. (a),(b) Evolution of $⟨S_x⟩$, for all-to-all (decay exponent $\alpha =0$) and dipolar ($\alpha =3$) interactions, respectively. Traditional TWA captures only the initial decay and no oscillations or revivals. In contrast, DTWA becomes exact (on top of the black symbols). (c),(d) The evolution of the correlation functions $\mathrm{\Delta }{S}_x=⟨S_x^2⟩-⟨S_x{⟩}^2$ and $\mathrm{Re}⟨S_y{S}_z⟩$ for dipolar interactions. While $\mathrm{Re}⟨S_y{S}_z⟩$ is exactly captured in DTWA, $\mathrm{\Delta }{S}_x$ shows deviations. DTWA improves traditional TWA predictions in all panels.

Figure 3

Dynamics for $XY$ interactions. Circles denote exact, solid lines DTWA, and dotted lines traditional TWA results. (a) Time evolution of $⟨S_x⟩$ for the $XY$ model in 1D, for $N=100$ spins and with long-range interactions with decay exponent $\alpha =3$. In contrast to the Ising case, DTWA is exact for short times only but can capture longer times than traditional TWA. (b) Time evolution of the spin-squeezing parameter $\xi$. The DTWA gives exact results for short times and good estimates for the achievable $\xi$. (b) Achievable $\xi$ as a function of the filling fraction (averaged over 1000 random configurations). Exact diagonalization for $N<20$, tDMRG otherwise. Inset: Rapid increase of the entanglement entropy for noninteger filling $\overline{n}<1$ (single configuration), rendering tDMRG calculations inefficient in this regime.

Figure 4

Spreading of correlations. Time evolution of ${\mathcal{C}}_j^{yy}$ in a system with $N=100$ spins on $M=100$ sites for long-range interactions with decay exponent $\alpha =3$. We consider three models [Eq. (4)]: (a) Ising interactions, (b),(c) Ising interactions plus a transverse field term $\mathrm{\Omega }\sum _i{\widehat{\sigma }}_i^x$, and (d) $XY$ interactions. We compare exact or tDMRG results (left-hand side) with results obtained from the DTWA (right-hand side).

Figure 5

Range of interactions and higher dimensions ($XY$ model). Circles denote exact diagonalization, solid lines DTWA results. The time evolution of the spin-squeezing parameter $\xi$ is shown (a) in 1D ($N=20$ spins) for different decay exponents $\alpha$ and (b) with fixed $\alpha =3$ and for different dimensions (1D, $20\times 1\times 1$ lattice; 2D, $5\times 4\times 1$; 3D, $3\times 3\times 2$). With increasing range of interactions, the DTWA improves and spin squeezing increases. (c) DTWA prediction for increasingly large 2D systems ($N\le 6400$) with $\alpha =1$. The exact diagonalization result is shown for the $4\times 4$ lattice.

Figure 6

Statistical convergence. (a),(c) Time evolution of the squeezing parameter in DTWA for different number of trajectories: $n_t=500$, 1000, 2000, 4000, 8000, 16 000, 32 000, 64 000 from light to dark. (a) Small $4\times 4$ system with $XY$ interactions ($\alpha =1$). (c) $20\times 20$ system. In both cases, for $n_t>4000$, curves are essentially indistinguishable. Panels (b),(d) show the maximum relative difference of the calculations from the $n_t=64000$ result. The upper line is for the spin-squeezing parameter, the lower line for $⟨S_x⟩$; there are smaller relative differences for this simple observable. Both differences decrease as $1/\sqrt{n_t}$ (dashed line) as expected. Panels (b) and (d) are for the small and large system, respectively.