Ultracold quantum wires with localized losses: Many-body quantum Zeno effect

We study a one-dimensional system of interacting spinless fermions subject to a localized loss, where the interplay of gapless quantum fluctuations and particle interactions leads to an incarnation of the quantum Zeno effect of genuine many-body nature. This model constitutes a nonequilibrium counterpart of the paradigmatic Kane-Fisher potential barrier problem, and it exhibits strong interaction effects due to the gapless nature of the system. As a central result, we show that the loss probability is strongly renormalized near the Fermi momentum as a realization of the quantum Zeno effect, resulting in a suppression of the emission of particles at the Fermi level. This is reflected in the structure of the particle momentum distribution, exhibiting a peak close to the Fermi momentum. We substantiate these findings by three complementary approaches: a real-space renormalization group of a general microscopic continuum model, a dynamical Hartree-Fock numerical analysis of a microscopic model on a lattice, and a renormalization group analysis based on an effective Luttinger liquid description incorporating mode-coupling effects.

Figure 1

Sketch of a fermionic wire subject to a localized loss. Before emptying out completely, a quasistationary state occurs characterized by constant currents $j$ flowing toward the loss site, analogous to a wire connected to reservoirs at its far ends.

Figure 2

Particle loss rate for the lattice model as a function of time elapsed from the quench, for increasing system sizes, $L=61,101,141,181$ (leftmost to rightmost curve, respectively). The dashed line indicates the constant loss rate (18) obtained analytically for $L\to \infty$. For all curves, $\gamma =J,N\left(0\right)/L=0.25$.

Figure 3

Density profile for the noninteracting lattice model (3). Upper panel: Density profiles for different times elapsed from the quench ($tJ=0,50,100,150,200$) and analytical average density $n_{\text{bg}}$ (dashed line). Lower panel: Friedel oscillations around the impurity at time $tJ=200$ obtained numerically (solid line) and analytically (hollow circles). For all curves $L=601,\gamma =2J,N\left(0\right)/L=0.25$.

Figure 4

Speed $v$ of the light cone (dots) separating depleted and unperturbed regions as a function of the initial filling factor $\nu =N\left(0\right)/L$, obtained numerically in the lattice system, with $L=401,\gamma =3J$. The dashed line indicates the Fermi velocity $v_F$ as a function of the filling factor.

Figure 5

Transmission probability ${\mathcal{T}}_k$, reflection probability ${\mathcal{R}}_k$, and loss probability ${\eta }_k$ as a function of momentum $k$ for the continuum (left panel, $\gamma /v_F=1.5$) and lattice model (right panel, $\gamma =3J$).

Figure 6

Left panel: Background density $n_{\text{bg}}$ (solid) and density at the loss site $n_{\text{imp}}$ (dashed) in the second temporal regime as a function of the dissipation strength $\gamma$ in the continuum model. Right panel: Particle loss rate $|dN/dt|$ in the second regime as a function of the dissipation strength $\gamma$ in the continuum model.

Figure 7

Momentum distribution $n_k\left(t\right)$ for the lattice model after the quench with solid lines indicating numerical simulations and dashed lines the analytical results. Upper panel: Momentum distribution in the second temporal regime for different times elapsed from the quench, $tJ=0,25,50,75,100$. For all curves $L=401,\gamma =3J,N\left(0\right)/L=0.75,T/J=0$. Lower panel: Momentum distribution at the end of the second temporal regime for different system sizes, for a fixed aspect ratio $t_{\text{II}}=L/\left(2v\right)$. For all curves, $\gamma =2J,N\left(0\right)/L=0.75,T/J=0$. For $N\left(0\right)/L=0.75$, $v=2J$.

Figure 8

Upper panel: Momentum distribution $n_k\left(t\right)$ from the numerical simulations (solid) in the lattice model approaching a stationary value within the spatial segment $j\in [-100,100]$ of a system with $L=501$ sites, for different times elapsed from the quench in the second temporal regime, $tJ=0,30,60,90,120$. The dashed line, indicating the stationary distribution $n_{\text{ness}}\left(k\right)$, is approached asymptotically in time. For all curves $\gamma =0.5J,N\left(0\right)/L=0.75,T/J=0$. Lower panel: Momentum distribution $n_k\left(t\right)$ from the numerical simulations (solid) in the lattice model initialized with a finite temperature $T/J=0.2$ for different times after the quench in the second temporal regime, $tJ=0,25,50,75,100$. The dashed lines indicate analytical results. For all curves $L=401,\gamma =3J,N\left(0\right)/L=0.75$.

Figure 9

Particle loss rate $|dN/dt|$ (upper panel) and rescaled particle loss rate $|dN/dt|/\nu$ (lower panel) of the lattice model in the second temporal regime as a function of the temperature $T$ ($k_{\text{B}}=1$), for different initial filling factors $\nu =N\left(0\right)/L$. Dots indicate the results of numerical simulations, lines the analytical prediction. For all curves $\gamma =3J,L=101$.

Figure 10

RG flow of the loss probability $\eta$ near the Fermi level for different bare values of $\gamma$. In the upper panel interactions act repulsively, i.e., $\alpha >0$, in the lower panel attractively, i.e., $\alpha <0$.

Figure 11

Momentum distribution $n_{\text{ness}}\left(k\right)$ (solid red line) in the NESS, reconstructed from the RG flow of $\eta$. The momentum distribution of the noninteracting system (dashed black line) is shown for comparison. The right panels show the same distribution as in the left panels but with a cutoff scale ${\ell }_{T/L}=4$ (see Sec. 6g). Upper panels: $\alpha =0.5$ and $\gamma /v_F=2$. Lower panels: $\alpha =-0.5$ and $\gamma /v_F=4$.

Figure 12

Density profile from the Hartree-Fock approximation for different interaction strengths at time $tJ=65$ elapsed from the quench. For all curves $\gamma =3J,L=401,N\left(0\right)/L=0.25$.

Figure 13

Total particle loss rate in the second temporal regime as a function of $\gamma$ for different interactions strengths $U$, for $L=151,N\left(0\right)/L=0.4$ from the Hartree-Fock approximation (dots), in comparison to the analytical prediction (lines).

Figure 14

Momentum distribution $n_k$ for repulsive interactions $\nu U=3/5J$ (upper panels) and the noninteracting system $\nu U=0J$ (lower panels) for increasing system sizes $L$. The distribution is obtained at $tJ=L/7$ close to the end of second temporal regime, in a segment with sites $j\in [0,2L/5]$ right to the impurity corresponding to the depleted region at that time. The smaller panels show a magnified view of the distribution in the respective panel to the left around $k_F=\nu \pi$. For all curves $\gamma =3J,\nu =0.4$.

Figure 15

Schematic RG flow of the dissipation strength $\gamma$, respectively $D$, in dependence on the Luttinger parameter $g$ at first order.

Figure 16

Phase diagram for the dissipative impurity from the second-order RG equations, for different values of the microscopic dissipation strength ${\overline{\gamma }}_0$ (${\overline{D}}_0$ in the dual regime) and of the Luttinger parameter $g$. The blue-shaded area in the lower-right corner corresponds to values where the fluctuation-induced transparency is visible, while the upper-left one corresponds to values where the fluctuation-induced QZE is visible. The gray-shaded areas correspond to values where the effective temperature hinders these effects. The solid lines are obtained using Eqs. (52) and (54) with $c=ln10$. The striped regions correspond to values where the perturbative RG breaks down.

Figure 17

RG flow of $\overline{\gamma }$ and $\overline{T}$ from Eq. (49) with initial conditions $\overline{\gamma }(\ell =0)={\overline{\gamma }}_0$ and $T(\ell =0)=0$. The dashed lines correspond to the solution of Eq. (50). Upper (lower) panel: Values of ${\overline{\gamma }}_0$ and $g$ as indicated by the $+$ ($\times$) symbol in Fig. 16.

Figure 18

$\eta \left({\ell }_T\right)$ as a function of temperature $T$ for attractive interactions $\alpha =-0.5$ and different values of $\gamma /v_F$.

Figure 19

Real (upper panel, black lines) and imaginary part (lower panel, black lines) of the eigenvalues ${\lambda }_l$ of the non-Hermitian effective Hamiltonian $\tilde{H}$ as a function of $\gamma$, for $L=18$ and periodic boundary conditions. The red dashed lines indicate the analytical solutions (E7), respectively (E9).

Figure 20

Upper panel: Detail of the delocalized solution (E6) for $\gamma =1.5J$. Lower panel: Detail of the localized-state solution (E8), for $\gamma =3J$.

Figure 21

Integrals $I_T$ and $I_{\kappa }$ as a function of the Luttinger parameter $g$ for $T=0$. The gray-shaded areas indicate insufficient numerical convergence.