Wave-packet dynamics in energy space of a chaotic trimeric Bose-Hubbard system

We study the energy redistribution of interacting bosons in a ring-shaped quantum trimer as the coupling strength between neighboring sites of the corresponding Bose-Hubbard Hamiltonian undergoes a sudden change $\delta k$. Our analysis is based on a threefold approach combining linear response theory calculations as well as semiclassical and random matrix theory considerations. The $\delta k$ borders of applicability of each of these methods are identified by direct comparison with the exact quantum-mechanical results. We find that while the variance of the evolving quantum distribution shows a remarkable quantum-classical correspondence (QCC) for all $\delta k$ values, other moments exhibit this QCC only in the nonperturbative $\delta k$ regime.

Figure 1

(Color online) Scheme of the wave-packet dynamics scenario: the perturbation is a rectangular pulse of duration $t_i$ at which the measurement is done. The function $f\left(t\right)$ represents the rescaled time dependence of the perturbation $\delta k\left(t\right)=\delta kf\left(t\right)$ (black line), while the dashed red line indicates its time derivative $\dot{f}\left(t\right)$.

Figure 2

Poincaré sections of the phase space belonging to the classical trimer for $N=1$ and different parameter values (a) $\lambda =0.005$, (b) $\lambda =0.05$, and (c) $\lambda =2$. On the $y$ axis we plot the action $I_3$, while on the $x$ axis the difference ${\phi }_2-{\phi }_3$ (in units of $\pi$) is plotted. The Poincaré section corresponds to the plane ${\phi }_1={\phi }_3$ and ${\dot{\phi }}_1>{\dot{\phi }}_2$ of the energy surface $\tilde{E}=0.2$.

Figure 3

(Color online) Parametric evolution of the eigenvalues ${\tilde{E}}_n^{\left(0\right)}$ as a function of the parameter $\lambda$. The number of bosons is $N=40$, and the effective interaction strength is $\tilde{U}=280$. In the main figure the entire spectrum is plotted, while the inset is a magnification of the small box. One observes a qualitative change in the spectrum as $\lambda$ is changed. See text for details.

Figure 4

(Color online) The level spacing distribution $\mathcal{P}\left(S\right)$ of the BHH trimer for three representative values of the dimensionless ratio $\lambda =k∕\tilde{U}$, which controls the underlying classical dynamics: (a) $\lambda =0.025(k=7)$, (b) $\lambda =0.05(k=14.5)$, and (c) $\lambda =0.35(k=100)$. The red dash-dotted line corresponds to the Poissonian distribution (14), which is expected for integrable systems: the dashed blue line corresponds to the Wigner surmise (13) (chaotic systems), while the solid green line represents the fitted Brody distribution (15). In Fig. 15 we report the fitted Brody parameter $q$ for various values of $\lambda$. The system corresponds to $N=230$ bosons and $\tilde{U}=280$. The histograms include the 400 relevant levels around $\tilde{E}=0.26$.

Figure 5

The Brody parameter $q$ for the BHH plotted against the dimensionless ratio $\lambda$, which controls the underlying classical dynamics. The values of $q$ are obtained from fits to $P_q\left(S\right)$ around $\tilde{E}=0.26$ as reported in Fig. 4. Error bars are of the size of the circles. The system corresponds to $N=230$ bosons and $\tilde{U}=280$. See text for details.

Figure 6

(Color online) The power spectrum of the classical trimer (2) at energy $\tilde{E}=0.26$, $\tilde{U}=280$, and ${\lambda }_0=0.053$. The classical cutoff frequency ${\omega }_{\mathrm{cl}}={\tilde{\omega }}_{\mathrm{cl}}\tilde{U}\approx 280$ is indicated by vertical dashed lines.

Figure 7

(Color online) The band profile $(2\pi \tilde{U}∕N^2\Delta ){\mid {\mathbf{B}}_{nm}\mid }^2$ versus $\omega =(E_n-E_m)∕\hslash$ is compared with the classical power spectrum $\tilde{C}\left(\omega \right)$. The number of particles is $N=230$ and ${\lambda }_0=0.053$. Inset: a snapshot of the perturbation matrix ${\mathbf{B}}_{nm}$.

Figure 8

(Color online) Distribution of rescaled matrix elements $w$ around $\tilde{E}=0.26$ rescaled with the averaged band profile. The dashed red line corresponds to the standard normal distribution, while the circles (○) correspond to a best fit from Eq. (24) with a fitting parameter $N=342$. The system corresponds $N=230$, $\tilde{U}=280$.

Figure 9

(Color online) Schematic representation of the two notions of the kernel $P(n\mid m)$. Left: projection of one perturbed eigenstate $\mid n(k_0+\delta k)⟩$ (bold blue level) on the basis $\mid m\left(k_0\right)⟩$ of the unperturbed Hamiltonian. Averaging over several $\mid n^{\prime }⟩$ states around energy $E_n$ yields the averaged shape of eigenfunctions (ASOE). Right: alternatively, if $P(n\mid m)$ is regarded as a projection of one unperturbed eigenstate $\mid m⟩$ (bold blue level) on the basis $\mid n⟩$ of the perturbed Hamiltonian and averaged over several states around $E_m$, it leads to the local density of states (LDOS).

Figure 10

(Color online) The kernel $P(n\mid m)$ of the BHH plotted as a function of the perturbed energies $E_n$ (LDOS representation) and for various perturbation strengths $\delta k>\delta k_{\mathrm{qm}}$. The averaged shape of eigenfunctions is given by the same kernel $P(n\mid m)$ and is obtained by just inverting the energy axis. Here, $N=70$, and ${\lambda }_0=0.053$.

Figure 11

(Color online) The quantal profile $P(n\mid m)$ (solid black line) as a function of $E_n-E_m^{\left(0\right)}$ for the BHH model is compared with $P_{\mathrm{prt}}$ (dashed red line) and with the corresponding $P_{\mathrm{IRMT}}$ (dash-dotted green line) of the IRMT model. The perturbation strength $\delta k$ is in (a) standard perturbative regime $\delta k=0.05$ and (b) extended perturbative regime $\delta k=0.3$. The system corresponds to $N=230$, $\tilde{U}=280$, and $k_0=15$. Here $\delta k_{\mathrm{qm}}=0.09$ and $\delta k_{\mathrm{prt}}=1.02$.

Figure 12

(Color online) Various measures of the spreading profile for the BHH and IRMT models: the quantal spreading $\delta E$ (black line with ○), the quantal spreading $\delta E_{\mathrm{prt}}$ of the perturbative profile given by Eq. (29) (red line with ◻), the spreading $\delta E_{\mathrm{IRMT}}$ (violet line with ◇) obtained from the IRMT modeling, the analytical spreading $\delta E_{\mathrm{ana}}$ (green line with △) obtained from (34), the core-width $\Gamma$ (orange line with ▽), and the classical spreading $\delta E_{\mathrm{cl}}$ (blue line). The dashed line has slope 1, while the dash-dotted line has slope 2 and are drawn to guide the eye. The systems correspond to $N=70$ bosons, $k_0=15$, and $\tilde{U}=280$. See text for details.

Figure 13

(Color online) The parameters (a) $\delta k_{\mathrm{qm}}$ and (b) $\delta k_{\mathrm{prt}}$ for various $\tilde{U}$, $N$, and for ${\lambda }_0=0.053$. A nice scaling in accordance with Eqs. (28, 31) is observed.

Figure 14

(Color online) Upper panel: the kernel $P(n\mid m)$ (LDOS representation) in the nonperturbative regime $\delta k=10$ for $N=230$ and $\lambda =0.053$. The results of the BHH model (solid black line) are compared with $P_{\mathrm{prt}}$ (red dashed line), $P_{\mathrm{IRMT}}$ of the IRMT model (dash-dotted green line), and the classical profile $P_{\mathrm{cl}}$ (blue line with ○). Lower panel: a time series $E\left(t\right)$ which leads to the classical profile $P_{\mathrm{cl}}\left(E\right)$ (see text for details).

Figure 15

(Color online) The classical energy spreading $\delta E_{\mathrm{cl}}\left(t\right)$ for the BHH (normalized with respect to the perturbation strength $\delta k$ and the boson number $N$) is plotted as a function of time. The dashed line has slope one and is drawn to guide the eye.

Figure 16

(Color online) The profile $P_t\left(r\right)$ of the BHH plotted as a function of time for various perturbation strengths $\delta k<\delta k_{\mathrm{qm}}$ (a), $\delta k_{\mathrm{qm}}<\delta k<\delta k_{\mathrm{prt}}$ (b), and $\delta k>\delta k_{\mathrm{prt}}$ (c). Note the different scale in (c). Here, $N=70$, $\tilde{E}=0.26$ and ${\lambda }_0=0.053$.

Figure 17

(Color online) Left panel: the (normalized) energy spreading $\delta E\left(t\right)$ for the BHH and the IRMT model (inset) for three different perturbation strengths $\delta k=0.05<\delta k_{\mathrm{qm}}$ (solid black line), $\delta k_{\mathrm{qm}}<\delta k=0.3<\delta k_{\mathrm{prt}}$ (dashed red line), and $\delta k=5>\delta k_{\mathrm{prt}}$ (dash-dotted green line). The classical expectation $\delta E_{\mathrm{cl}}\left(t\right)$ is represented in all three plots by a dashed blue line for comparison. In the inset the black dash-dotted lines have slope 1 and $\frac{1}{2}$ respectively, and are drawn to guide the eye. While for the BHH model one observes restricted quantum-classical correspondence in all regimes, this is not the case for the IRMT model (inset): For perturbations $\delta k>\delta k_{\mathrm{prt}}$ the energy spreading $\delta E\left(t\right)$ exhibits a premature crossover to diffusive behavior. Right panel: the evolution of the corresponding core width $\delta E_{\mathrm{core}}\left(t\right)$ for the BHH model is plotted. In the perturbative regimes one observes a separation of scales $\delta E_{\mathrm{core}}\left(t\right)<\delta E\left(t\right)<{\Delta }_b$, which is lost for strong perturbations $\delta k>\delta k_{\mathrm{prt}}$, where $\delta E_{\mathrm{core}}\left(t\right)$ approaches more and more the classical expectation $\delta E_{\mathrm{cl}}\left(t\right)$. Here, $N=230$, $\tilde{U}=280$, $\tilde{E}=0.26$, and ${\lambda }_0=0.053$.

Figure 18

(Color online) The survival probability $P\left(t\right)$ for the BHH and three different perturbation strengths (a) $\delta k=0.05<\delta k_{\mathrm{qm}}$ (inset), $\delta k_{\mathrm{qm}}<\delta k=0.3<\delta k_{\mathrm{prt}}$ (main figure) and (b) $\delta k=5>\delta k_{\mathrm{prt}}$. The solid black line represents the exact numerical result, while the dash-dotted green line is the LRT result (39) calculated using the IRMT model. The inset of panel (a) represents the FOPT regime, while the main figure corresponds to the extended perturbative regime. Here the break time is $t_{\mathrm{prt}}\sim 0.1$ (see Sec. 6B2). In the nonperturbative regime (b), the LRT breaks down close to the calculated break time $t_{\mathrm{prt}}\sim 0.001$. In this panel we superimpose the Fourier transform of the LDOS as blue circles. The agreement with $\mathcal{P}\left(t\right)$ is excellent. Here, $N=230$, $\tilde{U}=280$, $\tilde{E}=0.26$, and ${\lambda }_0=0.053$.

Figure 19

(Color online) Snapshots of the evolving quantum profile $P_t\left(r\right)$ obtained from the BHH (black line) and the IRMT model (dash-dotted green line) as well as the classical profile $P_t^{\mathrm{cl}}\left(r\right)$ (blue line with ○) in the nonperturbative regime $\delta k=5>\delta k_{\mathrm{prt}}$ plotted against the energy difference $E-E_0$. After the quantal transition period $t\sim 0.002$ [see Fig. 17b], there is no scale separation between the core and the tail component and one observes overall detailed QCC. However, the initially excited component $\mid n_0⟩$ decays slower in the quantum case. Here, $N=230$, $\tilde{U}=280$, $\tilde{E}=0.26$, and ${\lambda }_0=0.053$.