Quantum many-body dynamics of dark solitons in optical lattices

We present a fully quantum many-body treatment of dark solitons formed by ultracold bosonic atoms in one-dimensional optical lattices. Using time-evolving block decimation to simulate the single-band Bose-Hubbard Hamiltonian, we consider the quantum dynamics of density and phase engineered dark solitons as well as the quantum evolution of mean-field dark solitons injected into the quantum model. The former approach directly models how one may create quantum entangled dark solitons in experiment. While we have already presented results regarding the latter approach elsewhere [R. V. Mishmash and L. D. Carr, Phys. Rev. Lett. 103, 140403 (2009)], we expand upon those results in this work. In both cases, quantum fluctuations cause the dark soliton to fill in and may induce an inelasticity in soliton-soliton collisions. Comparisons are made to the Bogoliubov theory which predicts depletion into an anomalous mode that fills in the soliton. Our many-body treatment allows us to go beyond the Bogoliubov approximation and calculate explicitly the dynamics of the system’s natural orbitals.

Figure 1

(Color online) Density and phase engineering a standing soliton. To engineer a standing soliton in a lattice, we (a) take a free lattice, (b) apply a blue-detuned Gaussian beam to dig a density notch, (c) turn off the Gaussian beam and raise half the lattice to imprint a phase, and (d) lower the lattice to its original state. The lattice heights are the same in each case, the number of atoms indicates the density, and red indicates atoms lower in phase by amount $\pi$.

Figure 2

(Color online) Engineered standing soliton profile and DNLS dynamics. (a) The density and phase of the engineered standing DNLS soliton solution is plotted versus spatial position. Because we select an even number of sites, the two points with lowest density straddle the lattice center at $x=0$. (b) Real time dynamics in the DNLS of the initial condition in (a) show a stably propagating density notch with a spray of phonons. A plot of the phase (not shown) reveals a $\pi$ phase drop across the notch for all times. Note that all surface plots in this paper have space on the horizontal axis and time on the vertical axis.

Figure 3

(Color online) Average number and number fluctuations for engineered standing quantum soliton. (a) The average particle density during quantum evolution indicates that the dark soliton fills in over time; $⟨{\widehat{n}}_k⟩$ can be thought of as the sum of all natural orbitals weighted by their respective occupation numbers, cf. Fig. 4. (b) The normalized number variance measuring the relative on-site number fluctuations is greatest as the soliton fills in.

Figure 4

(Color online) Natural orbital dynamics for engineered standing quantum soliton. The densities of the four most highly occupied natural orbitals are plotted in (a)–(d) in order of decreasing occupation $N_j$. Population of the mode ${\varphi }_k^{\left(1\right)}$ is most responsible for the soliton decay shown in Fig. 3a. Here, we show the natural orbital densities ${\left|{\varphi }_k^{\left(j\right)}\right|}^2$ multiplied by their respective average occupation numbers $N_j$.

Figure 5

(Color online) Quantum measures for engineered standing quantum soliton. (a) The pair correlation function with respect to the center lattice site tells us that the soliton is in fact filling in during a single experiment. The initial condition has substantial spatial entanglement as indicated by (b) the local von Neumann entropy and (c) the average local impurity. At short times, before the soliton has filled in, the entropy in the region of the soliton is lower than at other points in the lattice, but this notch in entropy fills in over time. In (c) and (d), we see that both quantum depletion and particle entanglement grow in time.

Figure 6

(Color online) Engineered colliding soliton profile and DNLS dynamics. (a) As in Fig. 2, we show the DNLS initial condition, but this time we plot the state that results in a symmetric soliton collision during time evolution. (b) The resulting dynamics according to the DNLS reveal an elastic collision between the two dark solitons.

Figure 7

(Color online) Quantum-induced inelasticity for quantum engineered dark solitons. A collision between two engineered dark quantum solitons is shown for fixed $\nu U/J=0.35$ at filling factors (a) 1, (b) 0.7, (c) 0.4, and (d) 0.2. The average particle number is plotted in each case. The collision elasticity decreases with decreased filling due to an increase in the growth rate of quantum fluctuations for lower particle densities.

Figure 8

(Color online) Natural orbital dynamics during quantum evolution of a standing mean-field soliton. (a) The condensate wave function and the (b) second, (c) third, and (d) fourth most highly occupied natural orbitals during quantum evolution of a mean-field dark soliton. Note the different color bars in each case. The discontinuities are due to a swapping of natural orbital occupation numbers at specific times during the evolution as we indicate with dashed-dotted horizontal lines.

Figure 9

(Color online) Phase measures during quantum evolution of a standing mean-field soliton. (a) The average of the Pegg-Barnett phase operator and (b) the phase of the order parameter during quantum evolution of a mean-field dark soliton. The phase drop across the notch according to the Pegg-Barnett definition decays as the soliton depletes, whereas the order parameter phase maintains a $\pi$ phase drop across $x=0$ for all times.

Figure 10

(Color online) Anomalous Bogoliubov mode: Density profile and comparison to first depleted mode. (a) The density of the anomalous (negative frequency and positive norm) mode of the Bogoliubov excitation spectrum is plotted versus space for effective interactions $\nu U/J=0,0.05,0.20,0.35$ at $M=31$ sites. (b) For the latter case of $\nu U/J$, we show the time evolution at time steps $tJ/\hslash \approx 25,30,35,40$ of the depleted natural orbital most responsible for the decay of the dark soliton during the initial stages of evolution. We see that the Bogoliubov mode in (a) for $\nu U/J=0.35$ does not resemble the evolving symmetric mode shown in (b).

Figure 11

(Color online) Frequency of Bogoliubov mode versus interaction strength: Appearance of energetic and dynamical instability. For $M=31$ and $M=30$ sites, we show the real and imaginary parts of the calculated negative or complex frequencies appearing in the Bogoliubov excitation spectrum. For an odd number of sites, e.g., $M=31$, the frequency is always negative and real, while for an even number of sites, e.g., $M=30$, the frequency becomes purely imaginary past a critical value of $\nu U/J$. This behavior is general for any odd or even number of sites and is not specific to the particular choice of lattice sizes made above.