Critical fluctuations in a soliton formation of attractive Bose-Einstein condensates

We employ mean-field, Bogoliubov and many-body theories to study critical fluctuations in the position and momentum of a Bose-Einstein condensate whose translation symmetry is spontaneously broken due to attractive interactions. In a homogeneous system, the many-body ground state of the symmetry-preserving Hamiltonian is very fragile against superposition of low-lying states, while the mean-field theory predicts a stable bright soliton which spontaneously breaks translation symmetry. We show that weak symmetry-breaking perturbations cause the translation-symmetric many-body ground state to cross over to a many-body bright soliton. We argue that the center-of-mass fluctuations in the soliton state arise primarily from the depletion of the condensate to translation modes. We develop an extended mean-field theory to analytically reproduce these results obtained by the exact diagonalization method.

Figure 1

Schematic illustration of a quasi-one-dimensional torus.

Figure 2

(Color online) Many-body excitation spectrum $E_{\sigma ,\mathcal{L}}-E_{0,0}$ of the Hamiltonian $\widehat{H}$ with $N=200$, where $\mathcal{L}$ is the angular-momentum index and $\sigma$ is the index of the bands that appears for $gN\gtrsim 1$. We plot only low-lying states with indices $\sigma =0$, 1, and 2. The state ${\mid \mathcal{L}⟩}_{\sigma }$ denotes the many-body eigenstate whose total angular momentum is $\mathcal{L}$ and band index is $\sigma$.

Figure 3

(Color online) Fano factor $F\left(\theta \right)$ in Eq. (8) of the many-body ground-state wave function as functions of the strength of the interaction $gN$ and azimuthal angle $\theta$ with $N=200$.

Figure 4

The amount of decrease in the ground-state energy given by Eq. (11) in the presence of a symmetry-breaking potential for $gN=1.5$, where ${\kappa }_1\equiv {\epsilon }_1{N}^2$. The superimposed dashed line shows the result of the mean-field calculation given in Eq. (37).

Figure 5

(Color online) (a) Density profile $n\left(\theta \right)$ of the many-body wave function and (b) the peak density as a function of ${\kappa }_1$.

Figure 6

(Color online) (a) Distribution ${\mid {\beta }_{\mathcal{L}}\mid }^2$ of the angular momentum $\mathcal{L}$ in the many-body ground state $\mid {\Psi }_{\theta }^{\left({\epsilon }_1\right)}⟩$ for $gN=1.5$. (b) Width of angular-momentum distribution in the many-body ground state $\mid {\Psi }^{\left({\epsilon }_1\right)}⟩$ (open circles) and that in the Bogoliubov ground state $\mid {\Psi }^{\left(\mathrm{B}\right)}⟩$ (solid line) given by Eq. (55).

Figure 7

Depletions of the condensate obtained by the diagonalization of the Hamiltonian (open circles), and by the Bogoliubov theory given in Eq. (57) (solid line).

Figure 8

(Color online) Eigenvalues of the reduced single-particle density matrix for the double-well potential (solid curves) as a function of ${\kappa }_2\equiv {\epsilon }_2{N}^2$. Dashed curves show the results obtained for a single-well potential as a function of ${\kappa }_1\equiv {\epsilon }_1{N}^2$. In the latter case only one large eigenvalue survives for ${\kappa }_1\gtrsim 1$.

Figure 9

Ground-state energy as a function of ${\kappa }_2$ (solid curve) in the presence of a double-well potential measured from $E_0=⟨0\mid \widehat{H}\mid 0⟩$. Dashed curve shows the results of a single-well potential.

Figure 10

(Color online) (a) Cumulative number of counts for the center-of-mass position obtained by 2000 runs of independent simulation of quantum measurements for $g{N}_{\mathrm{init}}=1.6$ and $N_{\mathrm{init}}=300$. (b) Center-of-mass fluctuations obtained by the numerical simulations (open circles). The solid straight line obeys ${\left(\Delta {\theta }_{\mathrm{c.m.}}\right)}^2\propto j^{-1}$ and is drawn as a guide to the eye.

Figure 11

(Color online) (a) Ensemble average of the angular-momentum distribution ${\mid {\beta }_{\mathcal{L}}\mid }^2$ obtained from 2000 runs of the numerical simulation, and (b) its width (open circles) for $g{N}_{\mathrm{init}}=1.6$ and $N_{\mathrm{init}}=300$. The solid line is given by Eq. (65).

Figure 12

Depletion of the condensate for $g{N}_{\mathrm{init}}=1.6$. Symbols (엯,◇,×) show the results of numerical simulations, and the solid curve shows a theoretical one in Eq. (69), where $j$ is the number of measurements.

Figure 13

Width of the angular-momentum fluctuations in the GP ground state $\mid {\Psi }^{\left(\mathrm{GP}\right)}⟩$ (dashed curve) and that in the state $e^{r∕2({\widehat{T}}^{†2}-{\widehat{T}}^2)}\mid \mathrm{vac}⟩$ (solid curve).