Numerical computation of dynamically important excited states of many-body systems

We present an extension of the time-dependent density matrix renormalization group, also known as the time evolving block decimation algorithm, allowing for the computation of dynamically important excited states of one-dimensional many-body systems. We show its practical use for analyzing the dynamical properties and excitations of the Bose-Hubbard model describing ultracold atoms loaded in an optical lattice from a Bose-Einstein condensate. This allows for a deeper understanding of nonadiabaticity in experimental realizations of insulating phases.

Figure 1

Absorption spectrum obtained for an 8$\%$ modulation (i.e., $\delta =0.08$), after integration time $t=100$ [solid (red) line] and $t=300$ (filled circles connected by the black line). Data for $t=100$ are multiplied by 3. The higher resolution obtained for longer times allows partial separation of various components due to individual excited states. Circles correspond to numerical data; the black line connects them to guide the eye. Inset: Average occupation numbers of the initial ground state of the system.

Figure 2

Average occupation numbers per site, $\langle n_l\rangle$, for some selected excited states of the system analyzed in Fig. 1. The (black) line with open circles is the ground state, while (red) lines with filled squares show selected excited states. (a) One of the two states corresponding to the absorption peak at $\omega /U_0=0.24$ (the other state is symmetric with regard to the trap center). (b) The density for one of the states associated with the very small peak around $\omega /U_0=0.37$. (c, d) Two states associated with the “high” -frequency peak at $\omega /U_0=1.75$.

Figure 3

Average occupation numbers for another set of excited states of the system analyzed in Fig. 1. The (black line) with open circles represents the ground state, while (red) lines with filled squares show selected excited states.

Figure 4

Absorption spectrum for a system of 65 particles on 40 sites, in an harmonic trap with curvature $c=0.006,$ centered around $r_0=20.62345$. The mean lattice depth is $s_0=12$ (corresponding to $U_0=0.6752$ recoil energy). It is modulated sinusoidally with 1.67$\%$ relative amplitude. The solid (black) line with open circles corresponds to a modulation duration $t=50/E_R$; for a longer modulation time, $t=200/E_R,$ sharper peaks become resolved [thicker solid (red) line]. The absorbed energy is divided by 4 in the latter case, to compensate for a longer perturbation. The thick dashed (green) line shows the absorption curve for a much stronger, 20$\%$, modulation for $t=50/E_R$, rescaled by the intensity factor ${12}^2$.

Figure 5

Open circles: average occupation number for the ground state of 65 particles in a lattice with depth $s=12$ (40 sites) and an harmonic trap with curvature $c=0.006,$ centered around $r_0=20.62345$ (to separate left and right excited states with regard to the trap center). Filled (red) squares connected by the dashed line and (green) crosses connected by the dotted line show occupation numbers for two extracted excited states showing particle-hole excitation in the SF region. Excess energies (energies measured with respect to the ground-state energy) of the states are $\Delta E/U_0=2.2502$ (resp. $\Delta E/U_0=2.2434$) for the filled (red) squares [resp. (green) crosses], in excellent agreement with the positions of the two closely spaced peaks in the FT of the autocorrelation function shown in the inset. The lower panel shows the standard deviation of the occupation number at each site; for the ground state (open circles) and the excited states, the latter shows reduced fluctuations.

Figure 6

Long time evolution of a wave packet prepared by a modulation at frequency $\omega /U_0=2$ and then evolved with the TEBD algorithm. (a) The excess energy over the ground state. It is not conserved, but after the initial drop, it stabilizes. (b) Similar behavior occurs for the entanglement entropy, with additional oscillations because the wave packet is the sum of a relatively few localized states. (c) The FT of the autocorrelation function does not display any clear peaks if performed in the [0, 5000] time interval [thin (black) line] but definitely shows several well-defined excited states over the time interval [15 000,20 000] [thick (red) line].

Figure 7

Average occupation numbers of exemplary excited states showing a significant superfluid-type excitation, compared to the ground state [open (black) circles]. Filled (red) squares connected by dashed lines correspond to the excited state with excess energy $\Delta E/U_0=2.2418$ (lower energy peak of the doublet in the inset in Fig. 5). The (green) crosses connected by dotted lines correspond to a state (excess energy $\Delta E/U_0=1.9504$) with significant excitation around the trap center. The lower panel shows the standard deviation of the occupation of sites for the ground state (open circles) and two excited states. The latter show regions of increased superfluid-like fluctuations.

Figure 8

Average occupation numbers of exemplary excited states showing localized particle-hole excitations, compared to the ground state [open (black) circles]. Filled (red) squares connected by dashed lines correspond to excited states. (a) Excitation similar to those discussed for the smaller system, i.e., particle-hole excitation in the transition region between two Mott plateaus (excess energy $\Delta E/U_0=1.8417$). (b, c) Two particles being transferred between sites; the excess energies are $\Delta E/U_0=1.9988$ and $\Delta E/U_0=2.0539$. (d) An example of a higher lying excited state $\Delta E/U_0=7.5280$ with a partially melted, right $\langle n_l\rangle =2$ Mott plateau.

Figure 9

Standard deviations of the occupation numbers, for the states presented in Fig. 8. Local excitations lead to an increase in particle number fluctuations.

Figure 10

Autocorrelation spectra, Eq. (1), obtained dynamically for $s=14$ after switching on the optical lattice, for disorder strengths (a) $s_2=0$, (b) $s_2=0.4375$, and (c) $s_2=2.1875$. All parameters are taken to approximate the experimental situation [39]. The energy levels of the system appear as peaks (origin at the zero-point energy), with height $|c_i{|}^2$ from eigenbasis expansion. Peak labels are for further reference.

Figure 11

Properties of states of the trapped BEC in a deep optical lattice ($s=14$), without disorder ($s_2=0$), $r_0=0.12345$ in Eq. (10), to break the parity symmetry w.r.t. the trap center. The thick (black) line refers to the ground state; the thin (red) line, to the dynamically prepared wave packet; and the (green) line with crosses and (blue) line with squares, to the two excited states with the largest populations, denoted 1 and 2 in Fig. 10a. (a) Average occupation number $\langle n_l\rangle$ at each site. (b) Difference $D_l=\langle n_l\rangle -{\langle n_l\rangle }_{\mathrm{Gnd}\mathrm{state}}.$ (c) $\Delta n_l=\sqrt{\langle n_l^2\rangle -{\langle n_l\rangle }^2}.$ (d) Entanglement entropy, Eq. (4): almost zero in the Mott plateaus for eigenstates—implying their approximate separability; large for the wave packet. (e) Variance of the number of atoms to the left of site $l$. It is akin to the entanglement entropy, showing a marked difference between the eigenstates and the wave packet.

Figure 12

Same as Fig. 11, but for small disorder: $s_2=0.4375$, $r_0=0$, and $\phi =0.5432$ in Eq. (10). The (green) crosses and (blue) squares connected by lines correspond, respectively, to excitations A and B indicated in Fig. 10b. Mott insulating regions are clearly visible for all eigenstates, although more vague for the wave packet.

Figure 13

Properties of states of bosons trapped in a deep optical lattice ($s=14$) in the presence of strong disorder, $s_2=2.1875$. Here $r_0=0.1075$ and $\phi =0.0304$ in Eq. (10). The (green) crosses and (blue) squares connected by lines correspond, respectively, to excitations X and Y in Fig. 10c. The entanglement entropy is much larger for the dynamically created wave packet than for stationary states, and $\Delta n_l$ has many more peaks, indicating a significant melting of the Bose glass.