Nonequilibrium time evolution and rephasing in the quantum sine-Gordon model

We discuss the nonequilibrium time evolution of the phase field in the sine-Gordon model using two very different approaches: the truncated Wigner approximation and the truncated conformal space approach. We demonstrate that the two approaches agree for a period covering the first few oscillations, thereby giving a solid theoretical prediction in the framework of the sine-Gordon model, which is thought to describe the dynamics of two bosonic condensates in quasi-one-dimensional traps coupled via a Josephson tunneling term. We conclude, however, that the recently observed phase-locking behavior cannot be explained in terms of homogeneous sine-Gordon dynamics, which hints at the role of other degrees of freedom or inhomogeneity in the experimental system.

Figure 1

Classical trajectories of the pendulum model (1), plotted in terms of phase ${\phi }_0$ and rescaled particle number difference $z_0=2n_0/N$. The separatrix (black) separates the self-trapped regime with trajectories constrained to $z_0>0$ or $z_0<0$ half-plane (orange) from the trajectories visiting both half-planes (blue). The initial states considered in Secs. 5a and 5b, corresponding to a well-defined particle number difference and a random phase, can be visualized as horizontal lines (green, dashed). In the self-trapped regime, the small frequency shift between the trajectories cut by this line (brown) amounts to beating phenomena. The experimentally relevant squeezed initial state, considered in Sec. 5c, is depicted as a green ellipse.

Figure 2

Time-dependent expectation value of $cos\phi$ and $\mathcal{N}:cos\beta \varphi {:}^{\text{pl}}$ (a) and the standard deviation of half of the particle number difference (b). Continuous blue curves correspond to extrapolated TCSA data, while dashed red lines correspond to TWA. For TWA, we have used $K=1.56$, $L=14.86\mu \mathrm{m}$, $N=400$, $c=2800\mu \text{m}/\text{s}$, $J/h=7\mathrm{Hz},$ and $N_s=60$, corresponding to a Josephson frequency $f_J=410.8\mathrm{Hz}$. The parameters of TCSA are $\beta =1.42$, ${\nu }_1=299.8\mathrm{Hz}$, and the dimensionless length $l=10$. Time evolution is measured in terms of the bare Josephson frequency $f_J$ (TWA) and the renormalized Josephson frequency, corresponding to the frequency associated with the first breather, i.e., ${\nu }_1$ (within TCSA). Here and in all subsequent figures, the upper index “pl” indicates the perturbed conformal field theory (PCFT) expectation value computed on the $(z,\overline{z})$ complex plane as specified in Sec. 3b.

Figure 3

Comparing results for $cos\phi$ and $\mathcal{N}:cos\beta \varphi {:}^{\text{pl}}$ (a) and $sin\phi$ and $\mathcal{N}:sin\beta \varphi {:}^{\text{pl}}$ (b), obtained from TWA and TCSA, respectively. Blue continuous curve corresponds to TCSA with $e_{\text{cut}}=18$, blue dotted curve to the extrapolated TCSA data, and red dashed curve to TWA, for an initial particle number imbalance $\left(N_R-N_L\right)/2=25$. The parameters of TWA are $K=1.56$, $L=14.86\mu \mathrm{m}$, $N=400$, $c=2800\mu \text{m}/\text{s}$, $J/h=7\mathrm{Hz}$ and $N_s=60$, corresponding to a Josephson frequency $f_J=410.8\mathrm{Hz}$. For TCSA, we used $\beta =1.42$, ${\nu }_1=299.8\mathrm{Hz}$, and the dimensionless length $l=10$. Time evolution is measured in terms of the bare (TWA) and renormalized (TCSA) Josephson frequency, $f_J$ and ${\nu }_1$, respectively.

Figure 4

Time-dependent expectation value of half of the particle number difference (a) and its standard deviation (b), for an initial particle number imbalance $\left(N_R-N_L\right)/2=25$, with blue continuous, blue dotted, and red dashed curves corresponding to TCSA with $e_{\text{cut}}=18$, to the extrapolated TCSA data and to TWA, respectively. The parameters of TWA are $K=1.56$, $L=14.86\mu \mathrm{m}$, $N=400$, $c=2800\mu \text{m}/\text{s}$, $J/h=7\mathrm{Hz},$ and $N_s=60$, corresponding to a Josephson frequency $f_J=410.8\mathrm{Hz}$. For TCSA, we used $\beta =1.42$, ${\nu }_1=299.8\mathrm{Hz}$, and the dimensionless length $l=10$. Time is measured in units of the bare (TWA) and renormalized (TCSA) Josephson frequency, $f_J$ and ${\nu }_1$, respectively.

Figure 5

Time-dependent expectation value of $cos\phi$ (a) and the standard deviation of half of the particle number difference (b), for an experimentally accessible parameter set, calculated within TWA. We used the particle number $N=1000$, system size $L=25\mu \text{m}$, tunnel coupling $J/h=30\mathrm{Hz}$, and Luttinger parameter $K=27$. For a condensate of ${}^{87}\mathrm{Rb}$ atoms, with atomic mass $m_{\mathrm{Rb}}$, this corresponds to $c=830\mu \text{m}/\text{s}$, and typical short distance cutoff ${\xi }_h\equiv \hslash /\left(m_{\mathrm{Rb}}c\right)=0.62\mu \text{m}$, consistent with the number of lattice sites $N_s=L/{\xi }_h=28$. Time is measured in terms of the bare Josephson frequency, $f_J=189.7\mathrm{Hz}$.

Figure 6

Time-dependent expectation value of $cos\phi$ (a) and $sin\phi$ (b) for an experimentally accessible parameter set, using a squeezed initial state, calculated within TWA. These squeezed initial states are also accessible by experiments: The initial particle number imbalance is zero and the initial expectation value of ${\phi }_j$ is ${\phi }_0=\pi /6$ (red dashed curve) and ${\phi }_0=\pi /3$ (orange dotted curve). The initial variance of $n_j$ and ${\phi }_j$ is given by Eq. (60). We used the particle number $N=1000$, system size $L=25\mu \text{m}$, tunnel coupling $J/h=30\mathrm{Hz}$, and Luttinger parameter $K=27$. For a condensate of ${}^{87}\mathrm{Rb}$ atoms, this results in $c=830\mu \text{m}/\text{s}$, and typical short-distance cutoff ${\xi }_h\equiv \hslash /\left(m_{\mathrm{Rb}}c\right)=0.62\mu \text{m}$, consistent with the number of lattice sites $N_s=L/{\xi }_h=28$. Time is measured in terms of the bare Josephson frequency, $f_J=189.7\mathrm{Hz}$.

Figure 7

Time-dependent expectation value of half of the particle number difference (a) and its standard deviation (b) for a squeezed initial state, using an experimentally accessible parameter set, calculated within TWA. The two initial states are also realizable by experiments: The initial particle number imbalance is zero and the initial expectation value of ${\phi }_j$ is ${\phi }_0=\pi /6$ (red dashed curve) and ${\phi }_0=\pi /3$ (orange dotted curve). The initial variances of $n_j$ and ${\phi }_j$ are given by Eq. (60). We used the particle number $N=1000$, system size $L=25\mu \text{m}$, tunnel coupling $J/h=30\mathrm{Hz}$, and Luttinger parameter $K=27$, leading to $c=830\mu \text{m}/\text{s}$, and typical short-distance cutoff ${\xi }_h\equiv \hslash /\left(m_{\mathrm{Rb}}c\right)=0.62\mu \text{m}$ for a condensate of ${}^{87}\mathrm{Rb}$ atoms, consistent with the number of lattice sites $N_s=L/{\xi }_h=28$. Time is measured in terms of the bare Josephson frequency, $f_J=189.7\mathrm{Hz}$.

Figure 8

Keldysh contour. The contour of the path integral in Eq. (D1), running from $t_0=0$ to $t$, consisting of forward (${\mathcal{C}}_{+}$) and backward (${\mathcal{C}}_{-}$) branches. All operators are ordered along the Keldysh contour. The path integral can be evaluated by introducing a discrete time step $\mathrm{\Delta }t=t/M$ and inserting the completeness relation (30) at each time step according to Eq. (D2).

Figure 9

The drawing shows time-reversed pairs of forward and backward paths, utilizing the analogy $\phi \leftrightarrow x$ and $n\leftrightarrow p/\hslash$. Such pairs only contribute to the classical fields in Eq. (D6), while they yield vanishing quantum fields.

Figure 10

First quantum correction to the TWA result. The correction term, Eq. (D19), for the operators $cos\widehat{\phi }$ (a) and ${({\widehat{N}}_R-{\widehat{N}}_L)}^2/4$ (b), plotted as a function of dimensionless time $f_{J}t$, for a quench recoupling two independent, identical condensates prepared in their ground state. Here, we use the parameters of Fig. 2: $K=1.56$, $L=14.86\mu \text{m}$, $N=400$, $c=2800\mu \text{m}/\text{s},$ and $J/h=7\text{Hz}$, with the number of lattice sites $N_s=60$. The leading-order results for the time evolution of this quench were analyzed in Sec. 5a.

Figure 11

Time evolution of the variance of half of the particle number difference, for the initial state and parameters of Fig. 2, with continuous blue and dashed red curves corresponding to TCSA and TWA, respectively. The parameters are $K=1.56$, $L=14.86\mu \mathrm{m}$, $N=400$, $c=2800\mu \text{m}/\text{s}$, $J/h=7\mathrm{Hz},$ and $N_s=60$. Time is measured in terms of the bare and renormalized Josephson frequencies $f_J$ and ${\nu }_1$ in TWA and TCSA, respectively.