Pinning quantum phase transition in a Tonks-Girardeau gas: Diagnostics by ground-state fidelity and the Loschmidt echo

We study the pinning quantum phase transition in a Tonks-Girardeau gas, both in equilibrium and out of equilibrium, using the ground-state fidelity and the Loschmidt echo as diagnostic tools. The ground-state fidelity will have a dramatic decrease when the atomic density approaches the commensurate density of one particle per lattice well. This decrease is a signature of the pinning transition from the Tonks to the Mott insulating phase. We study the applicability of the fidelity for diagnosing the pinning transition in experimentally realistic scenarios. We find that fidelity can predict the particle number(s) at which the pinning occurs. In addition, we explore the out-of-equilibrium dynamics of the gas following a sudden quench with a lattice potential. We find that all properties of the ground-state fidelity are reflected in the Loschmidt echo dynamics, i.e., in the nonequilibrium dynamics of the Tonks-Girardeau gas initiated by a sudden quench of the lattice potential.

Figure 1

Ground-state fidelity (a) as a function of the number of particles $N$ for constant box size $L=30x_0$ and lattice wave vector $k=4\pi x_0^{-1}/3$ with different lattice amplitudes $V_l$. The black crosses are analytical results (for $N⩽M+1$) obtained with first-order perturbation theory. Ground-state fidelity (b) as a function of $N$ for constant lattice amplitude $V_l=0.1E_R$ and wave vector $k=4\pi x_0^{-1}/3$ with different box sizes $L$. See text for details.

Figure 2

Ground-state fidelity as a function of $N$ for different phases $\phi$ of the lattice $V_l\left(x\right)$. Outside the presented interval of $N$, GSF is approximately identical for all phases. See text for details.

Figure 3

Single-particle energy spectrum for cosine-squared $\phi =0$ (black crosses) and sine-squared $\phi =\pi /2$ (red circles) lattice; other parameters are $L=30x_0$ and $V_l=0.85E_R$. See text for details.

Figure 4

Single-particle density $\rho (N,x)$ obtained from the many-body ground states of $N$ atoms in $V_0\left(x\right)+V_l\left(x\right)$ potential (a) $\rho (40,x)$ (blue dashed line) in cosine-squared lattice (red solid line) and (b) $\rho (40,x)$ (blue dashed line) in sine-squared lattice (red solid line). For reference, we plot the corresponding densities ${\rho }_0(N,x)$ of $N$ atoms in the ground state of $V_0\left(x\right)$ trap (black dotted-dashed line). See text for details.

Figure 5

Single-particle density. (a) $\rho (40,x)$ (red cross) and two nearby densities $\rho (39,x)$ (blue circles) and $\rho (41,x)$ (green squares) in cosine-squared lattice (black dotted-dashed line). (b) $\rho (39,x)$ (red crosses) and two nearby densities $\rho (38,x)$ (blue circles) and $\rho (40,x)$ (green triangles) in sine-squared lattice (black dashed line). A smaller range of axis is chosen to provide good visibility. See text for details.

Figure 6

Ground-state fidelity for the pinning transition in a harmonic-oscillator potential. (a) GSF as a function of number of particles $N$ for different lattice amplitudes $V_l$ with constant frequency ${\omega }_0/2\pi =25$ Hz. (b) GSF as a function of $N$ for different ${\omega }_0$ with constant $V_l=0.45E_R$. See text for details.

Figure 7

The properties of SP eigenstates for an optical lattice superimposed with a harmonic potential $V_0\left(x\right)+V_l\left(x\right)$. (a) Single-particle energy spectrum for two values of amplitude $V_l=1.5E_R$ (blue circles) and $V_l=0.45E_R$ (red triangles). (b) The differences between nearest SP energies indicating the local “gaps.” (c) The density of (SP) states $g\left(E\right)$ for three values of $V_l$. Each plot starts from the ground-state value of the energy for a given potential. From these densities, one obtains the value of the global gap in the system as sketched; $h$ is the “height” between the global minimum and the first local maximum to the right of it as indicated in the figure. (d) The values of the global gaps as a function of $V_l$. See text for details.

Figure 8

Loschmidt echo $L\left(t\right)$ for different particle numbers $N$ in the infinitely deep box potential. Optical lattice $V_l\left(x\right)$ has 40 minima, i.e, $M=40$ lattice wells. LE reflects properties of GSF, i.e., decay is strongest for $N=(M-1,M)$, and the Loschmidt echo values come in pairs. The frequency of revivals decreases as we approach criticality. See text for details.

Figure 9

Dominant revival frequency of Loschmidt echo obtained with Eqs. (30) and (32) ${\omega }_R$ (blue circles) and with the Fourier transform of Loschmidt echo ${\omega }_{\text{FFT}}$ (red crosses). Parameters used for calculation of LE are the same as in Fig. 8. See text for details.

Figure 10

The Loschmidt echo $L\left(t\right)$ for different particle numbers $N$. (a) LE for particle numbers up to $N<N_{\mathrm{pinn}}$. (b) LE for $N>N_{\mathrm{pinn}}$. See text for details.

Figure 11

Dominant revival frequency of Loschmidt echo obtained with Eq. (33) ${\omega }_R$ (red circles) and with the Fourier transform of the Loschmidt echo ${\omega }_{\text{FFT}}$ (blue crosses). The parameters used to calculate the LE are the same as in Fig. 10. See text for details.