Quantum signatures of charge flipping vortices in the Bose-Hubbard trimer

In this work we study quantum signatures of charge flipping vortices, found in the classical discrete nonlinear Schrödinger trimer, by use of the Bose-Hubbard model. We are able to identify such signatures in the quantum energy eigenstates, for instance when comparing the site amplitudes of the classical charge flipping vortices with the probability distribution over different particle configurations. It is also discussed how to construct quantum states that correspond to the classical charge flipping vortices and which effects can lead to deviations between the classical and quantum dynamics. We also examine properties of certain coherent states: classical-like quantum states that can be used to derive the classical model. Several quantum signatures are identified when studying the dynamics of these coherent states, for example, when comparing the average number of particles on a site with the classical site amplitude, when comparing the quantum and classical currents and topological charge, and when studying the evolution of the quantum probability amplitudes. The flipping of the quantum currents are found to be an especially robust feature of these states.

Figure 1

First column: $|{\psi }_j{\left(t\right)|}^2$ for classical CFVs with $\beta =1.00$. From top to bottom: $\mathrm{\Delta }{\mathcal{H}}_{\text{rel}}$ = 0.024, 0.141, 0.670, and 0.996. Second column: time evolution of $\langle {\widehat{\nu }}_j\rangle =\langle C_0\left[\psi \right]\left(t\right)|{\widehat{\nu }}_j|C_0\left[\psi \right]\left(t\right)\rangle$ for CFV coherent states with $\psi =\psi \left(0\right)$ of the classical CFV on the same row, $N=100$ and $\alpha =\beta /(N-1)=1/99$. The insets show $\langle {\widehat{\nu }}_j\rangle$ over a longer time. Third column: average currents $\langle {\widehat{J}}_j\rangle$. Fourth column: quantum topological charge $\tau$. Both the currents and topological charge are calculated from the CFV coherent state on the same row. $\langle {\widehat{J}}_2\rangle$ follows $\langle {\widehat{J}}_3\rangle$ quite closely and is hard to distinguish in the plots.

Figure 2

(a) Trajectories of classical solutions for $\beta =1.00$. The black crosses mark dimerlike solutions while the black circles mark the SDW solutions. The orange (dark gray) and yellow (light gray) lines are the CFVs of Figs. 1(b1) and 1(d1), respectively. For (b)–(j) $N=100$ and $\alpha =\beta /(N-1)=1/99$. (b)–(g) $|\langle n_1,n_2,n_3|E_m\left(k\right)\rangle {|}^2/{max}_{n_1,n_2,n_3}\left(\right|\langle n_1,n_2,n_3|E_m\left(k\right)\rangle {|}^2)$, where (b)–(f) $k=0$ and $m=1,2,3,7,11$ [energy increasing from (b) to (f)], and (g) $k=2\pi /3$ and $m=11$. (h) Lower part of the energy spectrum for $k=0$ (crosses) and $k=2\pi /3$ (circles). $N{\mathcal{H}}_{\text{SDW}}$ and $N{\mathcal{H}}_{\text{dimerlike}}$ are include as the top and bottom dashed lines, respectively. (i) ${\mathrm{\Gamma }}_m$ of Eq. (15). (j) $2\pi /|E_m\left(0\right)-E_m(2\pi /3)|$.

Figure 3

Time evolution of the probability distribution $|\langle n_1,n_2,n_3|C_0\left[\psi \right]\left(t\right)\rangle {|}^2$ of the CFV coherent states corresponding to (a1)–(a5) $\leftrightarrow$ Fig. 1(b2); (b1)–(b10) $\leftrightarrow$ Fig. 1(c2); (c1)–(c10) $\leftrightarrow$ Fig. 1(d2). The time $t$ is shown in the top-right corner of each plot. The corresponding position of the classical associated CFVs are marked with blue (gray) crosses. In (c1), (c4), and (c8) the probability distribution is localized close to the triangle border and may be difficult to distinguish.

Figure 4

$\alpha =1.00/(N-1)$ and $N=100$. (a, b) $|\langle E_m\left(k\right)|C\left[\psi \right]\rangle {|}^2$ for the lowest eigenstates with (a) $k=0$ and (b) $k=2\pi /3$, with $\psi$ following the CFV subfamily with $\beta =1.00$ from the dimerlike to SDW solution. All $\psi$ are chosen so that site 1 has an amplitude minimum, similar to the initial conditions in the first column of Fig. 1. (c)–(f) show these projections in more detail for the four CFV coherent states in Figs. 1–1, respectively, with crosses for $k=0$ and circles for $k=2\pi /3$.

Figure 5

$\alpha =1.00/(N-1)$ and $N=100$. First row: $|\langle E_m\left(0\right)|C\left[\psi \right(t\left)\right]\rangle {|}^2$ for the four classical CFVs shown in Fig. 1 for different $t$, with $T$ being the period of ${\left|\psi \left(t\right)\right|}^2$. Going from left to right corresponds to going from top to bottom in Fig. 1. The time $t$ only needs to run over one quarter of a period, because of symmetries in the classical CFVs dynamics. (e, f) Time evolution of $\langle {\widehat{\nu }}_j\rangle$ for the CFV coherent states of (c) and (d), respectively, with $t=T/4$. The line coloring is the same as in Fig. 1. (g, h) Average currents, with (g) connected with (e), and (h) with (f).

Figure 6

$\alpha =1.00/(N-1)$. Top row: $N=10$. Bottom row: $N=30$. (a, e) Lower part of energy spectrum for $k=0$ (crosses) and $k=2\pi /3$ (circles). (b, f) ${\mathrm{\Gamma }}_m$ of Eq. (15). (c, d, g, h) Similar to Figs. 4 and 4, with (c, g) $k=0$, and (d, h) $k=2\pi /3$.

Figure 7

$N=30, \alpha =1.00/(N-1)$. Top row: Time evolution of $\langle {\widehat{\nu }}_j\rangle$ for CFV coherent states corresponding the those in Fig. 1 but with $N=30$. The line coloring is the same as in Fig. 1. Second row: Associated average currents. Bottom row: Projection of CFV coherent states onto energy eigenstates.

Figure 8

Similar to Fig 7, but for $N=10$.

Figure 9

(a) $2\pi /|E_m\left(0\right)-E_m(2\pi /3)|$ for $\alpha =1.00/(N-1)$. (b) $2\pi /|E_m\left(0\right)-E_m(2\pi /3)|$ for (from top to bottom), $m=1,2,3,4,5$, and $N=100$. (c) $2\pi /|E_1\left(0\right)-E_1(2\pi /3)|$. (d) $2\pi /|E_m\left(k\right)-2E_{m+1}\left(k\right)+E_{m+2}\left(k\right)|$ for $N=100$. The curves for $k=0$ and $k=2\pi /3$ are almost identical.