Metastable quantum phase transitions in a periodic one-dimensional Bose gas: Mean-field and Bogoliubov analyses

We generalize the concept of quantum phase transitions, which is conventionally defined for a ground state and usually applied in the thermodynamic limit, to one for metastable states in finite-size systems. In particular, we treat the one-dimensional Bose gas on a ring in the presence of both interactions and rotation. To support our study, we bring to bear mean-field theory, i.e., the nonlinear Schrödinger equation, and linear perturbation or Bogoliubov–de Gennes theory. Both methods give a consistent result in the weakly interacting regime: there exist two topologically distinct quantum phases. The first is the typical picture of superfluidity in a Bose-Einstein condensate on a ring: average angular momentum is quantized and the superflow is uniform. The second is where one or more dark solitons appear as stationary states, breaking the symmetry, the average angular momentum becomes a continuous quantity. The phase of the condensate can therefore be continuously wound and unwound.

Figure 1

Amplitude $\left|{\psi }_{J,j}^{\left(\text{ST}\right)}\left(\theta \right)\right|$ (upper panels) and phase profiles ${\phi }_{J,j}\left(\theta \right)$ (lower panels) of solitonic condensate wave function (19) for various numbers of density notches $j$. Left: $\left(j,\gamma ,\Omega \right)=\left(1,0.7,0.45\right)$; middle: $\left(j,\gamma ,\Omega \right)=\left(2,0.7,0.55\right)$; right: $\left(j,\gamma ,\Omega \right)=\left(3,0.7,0.45\right)$. For the weakly interacting regime the phase-winding number $J$ is determined by a fixed set of parameters $\left(j,\gamma ,\Omega \right)$.

Figure 2

(Color online) Parameter space where soliton solutions can exist. Filled regions present the regimes where the stationary (a) $j=1$, (b) $j=2$, and (c) $j=3$ soliton solutions coexist with plane-wave solutions. Boundaries are given by the parabolas of Eq. (56) with a phase-winding number $J$, as indicated by an integer value next to the corresponding parabolic curve. The path $C$ in panel (c) denotes a typical path for observing the continuous change in the topology of the order parameter (see Secs. 3C, 3D). (d) Elliptic parameter $k^2$ for a single soliton on the ring $\left(j=1\right)$ as a function of rotational drive $\Omega ∊\left[0,0.5\right]$ and mean-field strength $\gamma ∊\left[-1.5,1.5\right]$. For $\Omega =0.5$ the soliton solution is given by the Jacobi sn and cn functions for repulsive and attractive interactions, respectively. For $\Omega =0$ the soliton solution is given by the Jacobi dn function for attractive interactions.

Figure 3

(a) Energy diagram of stationary states for a fixed strength of repulsive $\left(\gamma =1\right)$ interaction. The zero of energy is taken as $\gamma /2$. The parabolas correspond to the energies of the plane wave, $E_J^{\left(\text{PW}\right)}-\gamma /2$, for various phase-winding numbers. Other branches bifurcating from the parabolas denote the energies of soliton trains, $E_{J,j}^{\left(\text{ST}\right)}-\gamma /2$, for $j=1$ (located in the first excited-state regime), $j=2$ (second-excited-state regime), and $j=3$ (third-excited-state regime) dark solitons, respectively. Integers in the figures denote the phase-winding number $J$ of solitons, which is equivalent to that of the plane wave. The right panel enlarges the path $C$ which connects the plane-wave and soliton branches in $0.3\lesssim \Omega \lesssim 0.7$ (rough end points are indicated with the open circles), starting from the plane-wave branch, passing through the soliton branch between the phase boundaries (filled circles), and again goes back to the plane-wave branch. (b) Energy diagram of stationary states for attractive $\left(\gamma =-0.4\right)$ interaction. The $j=1$ bright soliton is located in the ground-state regime, and $j=2$ and $j=3$ bright-soliton trains are located in the first and the second excited regimes, respectively.

Figure 4

Evidence of a second-order metastable quantum phase transition: determinants of Hessian matrix for (a) energy and (b) chemical potential.

Figure 5

(Color online) Changes in the amplitude $\left|\psi \left(\theta \right)\right|$ and phase $\phi \left(\theta \right)$ in the third-excited-state regime for $\gamma =0.6$ for the higher-energy path $C$ shown in Fig. 3. For $\Omega \simeq 0.3$ the amplitude has a constant value of $\left|\psi \right|=1/\sqrt{2\pi }$ and the gradient of the phase is equal to 2. As $\Omega$ increases, a bifurcation in the energy occurs at $\Omega \simeq 0.39$ and both the amplitude and phase start to wind in the higher-energy soliton branch while keeping $\phi \left(2\pi \right)-\phi \left(0\right)=4\pi$. The phase-winding number changes from $J=2$ to $J=-1$ at $\Omega =0.5$ and the amplitude has three nodes where the phase jumps by $\pi$. For $\Omega >0.5$ both the amplitude and phase start to unwind with $\phi \left(2\pi \right)-\phi \left(0\right)=-2\pi$, and the winding disappears at $\Omega \simeq 0.61$. The sequence of unwinding through a single soliton was described in our previous work [24].

Figure 6

Change in the average angular momentum $L/N$ along path $C$ for a triple soliton train in the third-excited-state regime, where the phase-winding number is $J=2$ $\left(-1\right)$ for $0\le \Omega <0.5$ $\left(0.5\le \Omega <1\right)$. The average angular momentum takes the value equivalent to the phase-winding number $J$ in the plane-wave regime, and linearly depends on $\Omega$ in the soliton regime.

Figure 7

Eigenvalues of the BdG equations for a fixed strength of repulsive interaction $\left(\gamma =1\right)$. The input condensate mode (zero excitation energy shown in the solid line) is taken as the higher-energy metastable state in the third-excited-state regime of Fig. 3a. Dashed and thick curves plot excitation energies from the plane-wave and soliton-train states, respectively.