Hard-core bosons on optical superlattices: Dynamics and relaxation in the superfluid and insulating regimes

Marcos Rigol, Alejandro Muramatsu, and Maxim Olshanii

Phys. Rev. A 74, 053616 – Published 16 November 2006

Abstract

We study the ground-state properties and nonequilibrium dynamics of hard-core bosons confined in one-dimensional lattices in the presence of an additional periodic potential (superlattice) and a harmonic trap. The dynamics is analyzed after a sudden switch-on or switch-off of the superlattice potential, which can bring the system into insulating or superfluid phases, respectively. A collapse and revival of the zero-momentum peak can be seen in the first case. We study in detail the relaxation of these integrable systems towards equilibrium. We show how after relaxation time averages of physical observables, like the momentum distribution function, can be predicted by means of a generalization of the Gibbs distribution.

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Received 14 July 2006

DOI:https://doi.org/10.1103/PhysRevA.74.053616

Authors & Affiliations

Marcos Rigol

Physics Department, University of California, Davis, California 95616, USA

Alejandro Muramatsu

Institut für Theoretische Physik III, Universität Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany

Maxim Olshanii

Department of Physics & Astronomy, University of Southern California, Los Angeles, California 90089, USA

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Vol. 74, Iss. 5 — November 2006

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Figure 1
(Color online) Decay of one-particle correlations (averaged per unit cell as described in the text) in the superfluid phase. These systems have up to 1900 lattice sites and $L = 2$ . The plot for $ρ = 0.25$ , $A = 5 t$ was displaced down for clarity. Straight solid lines correspond to power laws $1 ∕ \sqrt{x}$ . The inset shows $n_{k = 0}$ (top) and $n_{k = \pm π ∕ a}$ (bottom) vs $N_{b}$ for systems at quarter filling and $A = 0.5 t$ . The straight lines signal $\sqrt{N_{b}}$ behavior. Distances are normalized by the lattice constant $a$ .Reuse & Permissions
Figure 2
(Color online) Momentum profiles for periodic systems with 900 lattice sites in a superlattice potential with $L = 2$ . In (a) and (c) the system is in a superfluid state. The peaks at $k = 0$ and $k = \pm π ∕ a$ signal the presence of quasi-long-range one-particle correlations (Fig. 1). On the other hand, in (b) and (d) the system is in an insulating state where one-particle correlations decay exponentially (Fig. 3).Reuse & Permissions
Figure 3
(Color online) Decay of one-particle correlations (averaged per unit cell as described in the text) in the insulating (half-filled) case. The systems considered have up to 1500 lattice sites and $L = 2$ . The straight lines signal an exponential decay with a correlation length $ξ$ , which is a function of $A$ . In the inset we show $ξ$ vs $A$ . The straight lines depict the asymptotic behavior of $ξ$ . For very small values of $A$ one has that $ξ ∕ a \sim 1 ∕ (A ∕ t)$ while for large values of $A$ one obtains that $ξ ∕ a \sim 1 ∕ \sqrt{A ∕ t}$ (where $a$ is the lattice constant).Reuse & Permissions
Figure 4
(Color online) Normalized occupation of the $k = 0$ momentum state as a function of the density in periodic systems with $N = 1000$ . For $A \neq 0$ one can see that $n_{k = 0}$ is strongly suppressed around $n = 0.5$ , where the system is an insulator.Reuse & Permissions
Figure 5
(Color online) Evolution of the occupation of the zero-momentum state after $A$ is changed from $A = 0$ in the initial (superfluid) state to $A = 1.0 t$ (a), $A = 4.0 t$ (b), and $A = 8.0 t$ (c). The insets show in more detail the first three revivals of $n_{k = 0}$ . Dashed lines show $n_{k = 0}$ obtained from the constrained thermodynamics theory (see text). These systems have 300 lattice sites, 150 particles, and $L = 2$ .Reuse & Permissions
Figure 6
(Color online) Time average of the momentum distribution function after the damping of the oscillations seen in Fig. 5. We averaged measurements done in time intervals $Δ τ = 20 t$ between times $τ = 20 t$ and $τ = 5000 t$ . The time average is compared with the results obtained in the usual thermal ensemble and the constrained theory explained in the text. These systems have 300 lattice sites, 150 particles, and $L = 2$ . In all cases the initial $A = 0$ and the final one $A = 1.0 t$ (a), $A = 4.0 t$ (b), and $A = 8.0 t$ (c). The corresponding temperatures in the grand-canonical ensemble are $k_{B} T = 0.86 t$ (a), $k_{B} T = 6.86 t$ (b), and $k_{B} T = 25.75 t$ (c).Reuse & Permissions
Figure 7
(Color online) Evolution of the occupation of the zero-momentum state after $A$ is changed from $A = 1.0 t$ , $A = 4.0 t$ , and $A = 8.0 t$ (from top to bottom) in the initial (insulating) state to $A = 0$ . Dashed lines show $n_{k = 0}$ obtained from the constrained thermodynamics theory (see text). These systems have 300 lattice sites, 150 particles, and $L = 2$ .Reuse & Permissions
Figure 8
(Color online) Time average of the momentum distribution function measured in time intervals $Δ τ = 20 t$ between times $τ = 20 t$ and $τ = 5000 t$ . The time average is compared with the results obtained in the usual thermal ensemble and the constrained theory explained in the text. These systems have 300 lattice sites, 150 particles, and $L = 2$ . Initially $A = 1.0 t$ (a), $A = 4.0 t$ (b), and $A = 8.0 t$ (c), and in all cases the final value of $A$ is $A = 0$ . The corresponding temperatures in the grand-canonical ensemble are $k_{B} T = 0.63 t$ (a), $k_{B} T = 2.06 t$ (b), and $k_{B} T = 4.03 t$ (c).Reuse & Permissions
Figure 9
(Color online) Comparison between the time average of the momentum distribution function of two systems starting from different initial conditions but having the same energy during the evolution. (As in the previous figures, $n_{k}$ for the time average was measured in intervals $Δ τ = 20 t$ between times $τ = 20 t$ and $τ = 5000 t$ .) In both cases the evolution is done with a Hamiltonian in which $A = 1.0 t$ , but the initial state is in one case a superfluid state $(A = 0)$ , and in the other an insulating state with $A = 6.6 t$ . The time average is compared with the results obtained in the usual thermal ensemble (which depends only on the final energy and the number of particles—i.e., is the same in both cases) and the constrained theory explained in the text. These systems have 300 lattice sites, 150 particles, and $L = 2$ . In the legend “(TA)” means time average in the out-of-equilibrium system and “C” the result of the constrained thermodynamics.Reuse & Permissions
Figure 10
(Color online) Time average of the modulus and real part of the one-particle density matrix compared with the power-law decay present in the initial state and with the results obtained using the usual (thermal) and generalized (constrained) Gibbs distributions. We calculated $ρ_{x}$ from the center of the box, and for the time average we measured $ρ_{x}$ in time intervals $Δ τ = 20 t$ between $τ = 20 t$ and $τ = 5000 t$ . These systems have 300 lattice sites, 150 particles, and $L = 2$ . The initial value of $A$ is $A = 0$ and the final one $A = 1.0 t$ . Straight lines following the constrained and thermal results signal an exponential decay of $ρ_{x}$ .Reuse & Permissions
Figure 11
(Color online) Density (averaged per unit cell) (a)–(d) and normalized momentum distribution function (e)–(h) for trapped systems with 1000 lattice sites, $V_{2} a^{2} = 3 \times 10^{- 5} t$ , $L = 2$ , and $A = 0.5 t$ . In (a)–(d), the regions with constant density are local insulators.Reuse & Permissions
Figure 12
(Color online) Decay of one-particle correlations $ρ_{x}$ $(x = ∣ x_{i} - x_{j} ∣)$ (averaged per unit cell as described in Sec. 2) in the superfluid domains. We have chosen $x_{i}$ to be the center of the trap [see the corresponding density profiles in Figs. 11a, 11c]. Straight lines signal a power-law decay $ρ_{x} \sim 1 ∕ \sqrt{x}$ , which disappears when $n_{j} \to 0, 0.5$ , or 1. In the inset we show the scaling of the lowest natural orbital occupation with an increasing number of particles keeping constant the characteristic density of the system, which in this case is $\tilde{ρ} = 0.66$ . The straight line shows that $λ_{0} \sim \sqrt{N_{b}}$ .Reuse & Permissions
Figure 13
(Color online) Natural orbital occupations (a)–(d) and normalized wave function of the lower natural orbital $φ^{0} = {(N_{b} ζ ∕ a)}^{1 ∕ 4} ϕ^{0}$ (averaged per unit cell) (e)–(h), for trapped systems with 1000 lattice sites, $V_{2} a^{2} = 3 \times 10^{- 5} t$ , $L = 2$ , and $A = 0.5 t$ . In the insets of (a)–(d) we show the occupation of the lowest 11 natural orbitals, where degeneracy can be seen in the presence of the insulating domains. Comparing (e)–(h) with Figs. 11a, 11b, 11c, 11d one can see that the lowest natural orbitals only have a finite weight in the superfluid domains.Reuse & Permissions
Figure 14
(Color online) Occupations of the lowest 4 natural orbitals as a function of the characteristic density. These systems have 1000 lattice sites, $V_{2} a^{2} = 3 \times 10^{- 5} t$ , $L = 2$ , and $A = 0.5 t$ . Degeneracy sets in when insulating domains (with density $n = 0.5$ first and $n = 1.0 second$ ) appear in the middle of the trap.Reuse & Permissions
Figure 15
(Color online) Evolution of the momentum distribution function after a superlattice potential with $L = 8$ and $A = 2.0 t$ is applied for a short period of time $(Δ τ = 0.5 t)$ to the ground state of trapped system with 10 HCB’s, $V_{2} a^{2} = 4 \times 10^{- 6} t$ , and 2000 lattice sites. Undamped oscillations can be seen in $n_{k}$ during our simulation time.Reuse & Permissions
Figure 16
(Color online) Evolution of the occupation of the zero-momentum state (top plots) and the occupation of the lowest natural orbital (bottom plots) after a superlattice potential is turned off in trapped systems with 900 lattice sites and $V_{2} a^{2} = 3 \times 10^{- 5} t$ . The evolution starts from the ground state of a system with $L = 2$ and $A = 0.5 t$ . The number of particles is $N_{b} = 200$ (a) and $N_{b} = 299$ (b). The corresponding initial density profiles can be seen in Figs. 17a, 17d. The dashed lines are the results obtained with the generalized Gibbs distribution explained in the text.Reuse & Permissions
Figure 17
(Color online) Time average of the density profiles (a),(d), momentum distribution function (b),(e), and the occupation of the lowest 400 natural orbitals. The average is performed between $τ = 5000 t$ and $τ = 10000 t$ (see Fig. 16) with measurements done in time intervals $Δ τ = 40 t$ . The evolution is performed in trapped systems with 900 lattice sites, $V_{2} a^{2} = 3 \times 10^{- 5} t$ , starting from the ground state in the presence of a superlattice with $L = 2$ and $A = 0.5 t$ , after turning off the superlattice. The results of the time average are compared with the ones obtained in the usual thermal ensemble and the constrained theory explained in the text. The number of particles is $N_{b} = 200$ (a)–(c) and $N_{b} = 299$ (d)–(f). In (a) and (d) we included the averaged density per unit cell in the initial state. Flat regions correspond to insulating domains.Reuse & Permissions

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