Shape effects of localized losses in quantum wires: Dissipative resonances and nonequilibrium universality

We study the effects of the spacial structure of localized single-particle losses in weakly interacting fermionic quantum wires. We show that multiple dissipative impurities give rise to resonant effects visible in the transport properties and the particles' momentum distribution. These resonances can enhance or suppress the effective particle losses in the wire. Moreover, we investigate the interplay between interactions and the impurity shape and find that, differently from the coherent scatterer case, the impurity shape modifies the scaling of the scattering probabilities close to the Fermi momentum. We show that, while the fluctuation-induced quantum Zeno effect is robust against the shape of the impurities, the fluctuation-induced transparency is lifted continuously. This is reflected in the emergence of a continuous line of fixed points in the renormalization group flow of the scattering probabilities.

Figure 1

Upper panel: Pictorial representation of a system of one-dimensional spinless fermions with a dissipative impurity of finite width. $\gamma \left(x\right)$ is the dissipative impurity profile, while $V\left(x\right)$ is the coherent impurity profile. Individual particles coming in from the left may be reflected, transmitted, or lost from the system with probabilities ${\mathcal{R}}^{+},\mathcal{T},{\eta }^{+}$, respectively. Lower panel: The three examples of extended dissipative impurities discussed in this article. Black lines represent coherent barriers while wiggled red arrows represent particle losses.

Figure 2

The regime of scattering parameters allowed by condition (8): the physical region is enclosed between the plot axes and the surface lying above. Coherent impurities correspond to the blue line, while delta-shaped purely dissipative impurities correspond to the red one. Delta-shaped impurities lie on the surface connecting these lines.

Figure 3

Loss (${\eta }_k$), transmission (${\mathcal{T}}_k$), and reflection (${\mathcal{R}}_k$) probabilities for a purely dissipative box-shaped impurity of width $s$ (blue lines), compared to a delta-shaped impurity (black line).

Figure 4

Scattering probabilities for double impurities, as functions of particle momentum $k$ and for different interimpurity distances $s$. Upper three panels: Loss (${\eta }_k$), transmission (${\mathcal{T}}_k$), and reflection (${\mathcal{R}}_k$) probabilities for a double delta-shaped dissipative impurity (blue). The envelope of the functions is given by the scattering parameters of a single delta-shaped impurity (black) and the curve of extremal resonance (dashed), found from maximizing $\eta$ w.r.t. $s$. Lower panel: Reflection probability for a double delta-shaped coherent impurity (blue), compared to a single one of equal strength (black).

Figure 5

Loss (${\eta }_k$), transmission (${\mathcal{T}}_k$), and reflection (${\mathcal{R}}_k$) probabilities for the impurity (12) with $s=\pi m\gamma$ as functions of the momentum $k$ and for different values of $V$.

Figure 6

Particle density profile (left panels) and momentum distribution (right panels) for different impurity types. For all the plots $\gamma =k_{\text{F}}/m$. Dashed lines denote the prequench distribution. Upper panels: Results for a delta-shaped impurity (black solid line), symmetric double delta-shaped (dark blue line), and asymmetric double delta-shaped impurity (lighter black line). For all curves $s=11\pi /\left(2m\gamma \right)$, and ${\gamma }_{\text{L}}={\gamma }_{\text{R}}/3$ with $\gamma ={\gamma }_{\text{L}}+{\gamma }_{\text{R}}$ for the asymmetric double impurity. Lower panels: Results for the box-shaped impurity for different values of the impurity width.

Figure 7

Symbolic representation of the first order perturbation theory for the scattering parameters $r_k^{+}$ and $t_k$. Due to interactions, an effective impurity potential $U_{\text{eff}}\left(x\right)$ is seen by the particles instead of the bare impurity potential $U\left(x\right)$. To first order in the interactions, only the presented processes, involving a single scattering from the Friedel oscillations, contribute to the correction of the scattering parameters. Such a scattering corresponds to a divergent integral $I_k^{\pm }$ in Eq. (21).

Figure 8

RG flow for attractive interactions ($\alpha <0$) and ${\eta }^{★}=0.4$ (arrows). Surfaces with constant $\mathcal{D}>0$ are shown for ${\eta }^{★}=(0,0.2,0.4,0.6,0.8,1)$. Blue, thick lines correspond to fixed points stable either for $\alpha >0$ or $\alpha <0$. The red, thick lines are unstable fixed points of the RG for both repulsive and attractive interactions.

Figure 9

Conserved quantity $\mathcal{D}$ for the box-shaped impurity. Both a finite width and a finite loss are required for $\mathcal{D}>0$ while coherent potentials weakly modify it additionally. For $s\to 0$ we find $\mathcal{D}\sim s^2{\gamma }^2$.

Figure 10

Momentum distribution for different interaction strengths and impurity shapes. For all panels ${\mathrm{\Delta }}_{\text{UV}}=k_{\text{F}}/2,k_{\text{F}}=m\gamma$.

Figure 11

Loss probability for a localized dissipation near the end of a wire. Depending on the distance $s$, the dissipation is strongly enhanced or perfectly removed. For $k=m\gamma$, the loss probability can reach 1 perfectly.

Figure 12

Renormalization of the effective strength of coherent and dissipative localized impurities for attractive interactions. The stable fixed point lies at $\gamma =V=0$ corresponding to perfect transmission. $V$ vanishes algebraically while $\gamma$ vanishes only logarithmically slow so that the dissipative part dominates close to the Fermi edge.