Decay and fragmentation in an open Bose-Hubbard chain

We analyze the decay of ultracold atoms from an optical lattice with loss from a single lattice site. If the initial state is dynamically stable, a suitable amount of dissipation can stabilize a Bose-Einstein condensate, such that it remains coherent even in the presence of strong interactions. A transition between two dynamical phases is observed if the initial state is dynamically unstable. This transition is analyzed here in detail. For strong interactions, the system relaxes to an entangled quantum state with remarkable statistical properties: The atoms bunch in a few “breathers” forming at random positions. Breathers at different positions are coherent, such that they can be used in precision quantum interferometry and other applications.

Figure 1

Model systems studied in the present paper: (a) an open Bose-Hubbard trimer with loss from site 2 and periodic boundary conditions; (b) an extended one-dimensional optical lattice with localized loss from a single lattice site.

Figure 2

Dynamics of the atom number and correlation functions in an open Bose-Hubbard trimer with loss from site 2 for weak interactions [$U=0.01J$; dashed (blue) line] and strong interactions [$U=0.1J$; solid (red) line]. Plotted are the total particle number $n_{\mathrm{tot}}$ (top row), the phase coherence between the site 1 and the site 3 $g_{1,3}^{\left(1\right)}$ (second row), the number fluctuations $g_{1,1}^{\left(2\right)}$ (third row), and the number correlations between the site 1 and the site 3 $g_{1,3}^{\left(2\right)}$ (bottom row). The dynamics has been simulated for three initial states: a BEC with a symmetric wave function (left), a BEC with an antisymmetric wave function (middle), and a Fock state (right). The loss rate is ${\gamma }_2=0.2J$ and the initial populations are $n_1\left(0\right)=n_3\left(0\right)=30$ and $n_2\left(0\right)=0$ in all cases. Time is given as $Jt$ in units of tunneling time.

Figure 3

Onset of breather formation in a triple-well trap for strong atomic interactions. (a) The total particle number $\langle {\widehat{n}}_{\mathrm{tot}}\rangle$, (b) the phase coherence $g_{1,3}^{\left(1\right)}$, and (c), (d) the density correlation functions $g_{1,1}^{\left(2\right)}$ and $g_{1,3}^{\left(2\right)}$ as a function of the interaction strength $U$ after a fixed propagation time $t_{\mathrm{final}}=50J^{-1}$ for ${\gamma }_2=0.2J$. The initial state is a pure BEC in state $|{\Psi }_{-}\rangle$ with $N=60$ atoms.

Figure 4

(a), (b) Full counting statistics of the breather state at time $t=10J^{-1}$ in the first and second wells. (c), (d) Density matrix $\rho (n_1,n_2,n_3;n_1^{\prime },n_2^{\prime },n_3^{\prime })$ of a breather state at time $t=10J^{-1}$ for a fixed particle number $n=50$ and (c) $n_2=n_2^{\prime }=0$ and (d) $n_2=n_2^{\prime }=1$. Parameters are $U=0.1J$ and ${\gamma }_2=0.2J$. The initial state is a pure BEC with $N=60$ atoms and an antisymmetric wave function, (23).

Figure 5

Interferometry of the NOON state according to the time evolution $\widehat{U}\left(t\right)=\exp [-J({\widehat{a}}_1^{†}{\widehat{a}}_3-{\widehat{a}}_3^{†}{\widehat{a}}_1)t]$. (a) Probability of detecting an even (solid line) and an odd (dashed line) number of atoms at site 1 as a function of time. (b), (c) Full counting statistics at site 1 at (b) the beginning of the interferometer stage $t=10J^{-1}$ and (c) during the interferometer stage at $t=13.92J^{-1}$, where $P_{\mathrm{even}}=1$. The breather state is generated starting from a BEC with an antisymmetric wave function as shown in Fig. 2, with $U=0.1J$ and ${\gamma }_2=0.2J$. During the interferometer stage we assume that $U={\gamma }_2=0$.

Figure 6

Evolution of the entanglement parameter, (18), for three initial states: (a) a BEC with symmetric wave function $|{\Psi }_{+}\rangle$, (b) a BEC with an antisymmetric wave function $|{\Psi }_{-}\rangle$, and (c) a Fock state $|{\Psi }_F\rangle$. Parameters are ${\gamma }_2=0.2J$, with $U=0.01J$ [dashed (blue) line] and $U=0.1J$ [solid (red) line].

Figure 7

Entanglement and decoherence of a breather state in the presence of phase noise. (a) Evolution of the entanglement parameter, (18), for the antisymmetric initial state $|{\Psi }_{-}\rangle$ and $\kappa =0$ (solid line), $\kappa ={10}^{-4}J$ (dashed line), and $\kappa ={10}^{-3}J$ (dash-dotted line), with $N\left(0\right)=60$. (b) Temporal maximum of the entanglement parameter ${\max }_t{E}_{1,3}\left(t\right)$ as a function of the phase noise rate $\kappa$ for the antisymmetric initial state $|{\Psi }_{-}\rangle$ and different particle numbers. Parameters are $U=0.1J$ and ${\gamma }_2=0.2J$.

Figure 8

Semiclassical interpretation of breather-state formation. (a) Classical trajectories starting in the vicinity of the symmetric states $({\alpha }_1,{\alpha }_2,{\alpha }_3)=(1,0,1)/\sqrt{2}$ (red) and the antisymmetric states $(1,0,-1)/\sqrt{2}$ (blue). (b)–(g) The quantum dynamics of the Husimi $Q$ function follows the classical phase-space trajectories. (b), (e) A BEC with a symmetric wave function $|{\Psi }_{+}\rangle$ remains approximately pure. (c), (f) A BEC with an antisymmetric wave function$|{\Psi }_{-}\rangle$ is coherently split into two parts forming the breather state. (d), (g) A number state $|{\Psi }_F\rangle$ is also split into two parts, but number fluctuations and correlations are less pronounced. The Husimi function $Q({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is plotted as a function of the population imbalance $z=\left(\right|{\alpha }_3{|}^2-|{\alpha }_1{|}^2)/n_{\mathrm{tot}}$ and the relative phase ${\varphi }_3-{\varphi }_1$ for ${\alpha }_2=0$ and $|{\alpha }_1{|}^2+{\left|{\alpha }_3\right|}^2=n_{\mathrm{tot}}$ at times (b)–(d) $t=0$ and (e)–(g) $t=10J^{-1}$. Here, $n_{\mathrm{tot}}$ denotes the total atom number at the respective time. Parameters are $U=0.1J$, ${\gamma }_2=0.2J$, and $N\left(0\right)=60$.

Figure 9

Properties of the metastable solutions of the non-Hermitian DGPE, (22), for $J=1$ and ${\gamma }_2=0.2$ as a function of the interaction strength $g=U{n}_{\mathrm{tot}}$. (a) Decay rate per atom $\Gamma$ and (b) relative occupation of the first well $|{\alpha }_1{|}^2$. The icons at the right indicate the density distribution in the three wells and the dynamical stability for large $g$.

Figure 10

Analysis of the dynamical stability of the metastable solutions of DGPE (22) for $J=1$, $U{n}_{\mathrm{tot}}=3$, and ${\gamma }_2=0.2$. The dynamics was simulated starting from a metastable state (black dashed line) and the state plus a small perturbation of order $0.01$ (red line). Plotted is the relative occupation of the first well, $|{\alpha }_1{\left(t\right)|}^2/{\parallel \vec{\alpha }\left(t\right)\parallel }^2$. The density distributions of the initial states are illustrated by the icons in the corners: (a) the antisymmetric state, (b) a breather in the first well, (c) a breather in the leaky second well, and (d) the balanced state.

Figure 11

Dynamics of a leaky Bose-Hubbard chain with 50 wells. We have plotted (a) the atomic density $\langle {\widehat{n}}_j\left(t\right)\rangle$, (b) the density fluctuations $g_{j,j}^{\left(2\right)}\left(t\right)$ at each lattice site, as well as (c) the phase coherence $g_{j,j+k}^{\left(1\right)}\left(t\right)$ and (d) the density-density correlations $g_{j,j+k}^{\left(2\right)}\left(t\right)$ between site $j=25$ and the neighboring sites $k=1$ [solid (red) line] and $k=2$ [dashed (blue) line]. Parameters are $UN\left(0\right)=25J$, ${\gamma }_1=2J$, $M=50$, and $\rho (t=0)=N/M=1000$.

Figure 12

Transition to the breather regime in an open optical lattice. (a) The phase coherence $g_{j,j+k}^{\left(1\right)}$ and (b) the number correlation function $g_{j,j+k}^{\left(2\right)}$ between site $j=9$ and the neighboring sites $k=1$ [solid (red) line] and $k=2$ [dashed (blue) line] after a fixed propagation time $t_{\mathrm{final}}=50J^{-1}$. One observes a sharp transition when the interaction strength $UN\left(0\right)$ exceeds a critical value of $\sim 2.5J$. Parameters are ${\gamma }_1=2J$, $M=50$, and the atomic density is $\rho (t=0)=N/M=1000$.

Figure 13

The critical nonlinearity $\rho U_{\mathrm{crit}}$, at which breathers start to form, as a function of the lattice size $M$. Numerical results using the truncated Wigner method [filled (blue) circles] are compared to a fit using Eq. (27) [solid (red) line]. The fitting parameters we found are $L=0.075$ with bounds $(0.054,0.096)$ and $P_{\mathrm{th}}=1-2.4\times {10}^{-7}$ with bounds $(1+4.54\times {10}^{-7},1-9.3\times {10}^{-7})$, while the summed square of residuals is $\mathrm{SSE}=1.2\times {10}^{-3}$. The other parameters are ${\gamma }_1=2J$ and $\rho (t=0)=N/M=1000$.

Figure 14

Comparison of a truncated Wigner simulation [dashed (green) lines] and a quantum jump simulation [solid (red) lines] shows very good agreement. Shown are the time evolution of the total atom number $\langle {\widehat{n}}_{\mathrm{tot}}\rangle$, the phase coherence $g_{1,3}^{\left(1\right)}$, and the density-density correlations $g_{1,3}^{\left(2\right)}$. Parameters and initial states are the same as in Fig. 2, with $U=0.1J$.