Nonlinear transport in the presence of a local dissipation

We characterize the particle transport, particle loss, and nonequilibrium steady states in a dissipative one-dimensional lattice connected to reservoirs at both ends. The free-fermion reservoirs are fixed at different chemical potentials, giving rise to particle transport. The dissipation is due to a local particle loss acting on the center site. We compute the conserved current and loss current as functions of voltage in the nonlinear regime using a Keldysh description. The currents show steplike features that are affected differently by the local loss: the steps are either smoothened, nearly unaffected, or even enhanced, depending on the spatial symmetry of the single-particle eigenstate giving rise to the step. Additionally, we compute the particle density and momentum distributions in the chain. At a finite voltage, two Fermi momenta can occur, connected to different wavelengths of Friedel oscillations on either side of the lossy site. We find that the wavelengths are determined by the chemical potentials in the reservoirs rather than the average density in the lattice.

Figure 1

(a) A lattice of $M$ sites is connected to reservoirs at both ends, and a local particle loss with amplitude $\gamma$ acts on the center site. The coupling to the reservoirs ${\tau }_1$ is in general different from the tunneling amplitude $\tau$ within the lattice. (b) Representation of the different energy scales. In the reservoirs, states up to the chemical potentials ${\mu }_{L,R}$ are filled at zero temperature, as shown by the Fermi distributions $n_F\left(\omega \right)$ on either side. The eigenenergies of an isolated lattice are depicted by the horizontal lines within the box in the middle. The current through the lattice changes in steps when the chemical potentials coincide with eigenenergies of the lattice. Adapted from Ref. [36].

Figure 2

Conserved current $I$ through a quantum dot coupled to leads as a function of voltage for (a),(c) $\epsilon =0$ and (b),(d) $\epsilon =1$. The current is scaled by the coupling $\mathrm{\Gamma }=\pi {\rho }_0{\tau }_1^2$. The tunneling amplitude is ${\tau }_1=0.1$ in panels (a),(b) and ${\tau }_1=0.5$ in panels (c),(d). The vertical line in panels (b) and (d) marks the voltage at which the chemical potential of the left reservoir coincides with the quantum dot energy level. Here, $V, \gamma$, and $\epsilon$ are in units of $\tau , {\tau }_1$ is in units of $\tau \sqrt{V}$, and $\tau =1/\left(\pi {\rho }_0\mathcal{V}\right)$.

Figure 3

Integrand $g\left(\omega \right)$ of Eq. (25) for the three-site system. There are three maxima approximately at the eigenenergies of the isolated system, marked by dashed vertical lines. Here, $\epsilon =0$ in the left column and $\epsilon =1$ in the right, ${\tau }_1=0.1$ in panels (a),(b) and ${\tau }_1=0.5$ in panels (c),(d).

Figure 4

Conserved current $I$ as a function of voltage in a three-site system for (a),(c) $\epsilon =0$ and (b),(d) $\epsilon =1$. The current is scaled by $\tilde{\mathrm{\Gamma }}=\mathrm{\Gamma }{\tau }^2/({\mathrm{\Gamma }}^2+{\tau }^2)$. In panels (a),(b), ${\tau }_1=0.1$, and in (c),(d), ${\tau }_1=0.5$. The units of the parameters are the same as in Fig. 2. The current has a steplike structure with steps approximately at voltages $2|E_n|$, marked by dashed vertical lines, where $E_n$ are the single-particle eigenenergies of an isolated system. For large dissipation $\gamma$, the steps arising from symmetric eigenstates disappear, while a larger coupling ${\tau }_1$ leads to a smoothening of all steps.

Figure 5

Conserved current $I$ as a function of voltage in a system of (a) 5 and (b) 23 sites, with $\epsilon =0$ and ${\tau }_1=0.1$. The saturation current at $\gamma =0$ is close to $\mathrm{\Gamma }$ and is scaled by this value.

Figure 6

For three or more sites, the saturation value of the current at large voltage decays with $\gamma$. For $M=23$, the value of the current at $V=5$ is used as $I_{\text{sat}}$. We set here ${\tau }_1=0.1$.

Figure 7

The loss current as a function of $\gamma$ at $V=\epsilon =0$ for different system sizes and cutoffs $\mathrm{\Lambda }$. (a) For the quantum dot, the $\mathrm{\Lambda }\to \infty$ limit is given by Eq. (31). In this limit, the loss current saturates at $2\mathrm{\Gamma }$, while for finite $\mathrm{\Lambda }$ there is a nonmonotonic dependence. (b) For three or more sites, $I_{\text{loss}}$ is nonmonotonic independently of the value of the cutoff. The solid, dashed, and dotted lines mark the different values of $\mathrm{\Lambda }$ as in panel (a). The many-site system and the quantum dot with a finite cutoff have the same $\sim 1/\gamma$ dependence at large $\gamma$, where the curves for different lattice sizes and different values of $\mathrm{\Lambda }$ overlap. We set here ${\tau }_1=0.5$.

Figure 8

The quantum dot occupation (a),(c) and the loss current (b),(d) as functions of voltage for different values of $\gamma$. In panel (a) the lines for $\gamma =5$ and 100 are very close to zero, and in panels (b) and (d) the loss current is zero for $\gamma =0$. Here, $\epsilon =1$ and the coupling to the reservoirs is ${\tau }_1=0.1$ (a),(b) and ${\tau }_1=0.5$ (c),(d). The vertical lines mark the voltage at which ${\mu }_L$ coincides with $\epsilon$.

Figure 9

The occupation of the lossy site and the loss current in a three-site system with $\epsilon =1$, (a),(b) ${\tau }_1=0.1$, and (c),(d) ${\tau }_1=0.5$. The line colors are as in Fig. 8. The vertical lines mark the voltages $V/2=|E_n|$, where $E_n$ are the single-particle eigenstates in the ${\tau }_1\to 0$ limit.

Figure 10

The quasimomentum distribution $\langle n_k\rangle$ in the lattice at different dissipation rates. The leftmost column [panels (a),(d),(g)] shows $\langle n_k\rangle$ in the full lattice, as given by Eq. (9). The middle and rightmost columns correspond to the left and right halves of the lattice, with momentum distribution given by Eqs. (32) and (33), respectively. On the first row (a)–(c), the voltage is $V=0$, on the second row (d)–(f), $V=1$, and on the third row (g)–(i), $V=5$. The Fermi momenta given by Eq. (34) are marked by vertical gray lines. The lattice chemical potential is set to $\epsilon =1$, so that at zero voltage and in the absence of dissipation, the lattice filling is $1/3$.

Figure 11

Particle density $\langle n_j\rangle$ for a lattice of 51 sites with $\epsilon =1$ and ${\tau }_1=0.5$ as in Fig. 10. The average filling for $\gamma =0$ is $1/3$. The density imbalance between the left and right sides develops as a combined effect of the finite voltage and dissipation.

Figure 12

The quasimomentum distribution $n_k$ in the lattice, as in Fig. 10, for $\epsilon =0$. The first row (a)–(c) corresponds to $V=0$, the second row (d)–(f) to $V=1$, and the third row (g)–(i) to $V=5$. At zero voltage and in the absence of dissipation, the lattice is half-filled.

Figure 13

Particle density $\langle n_j\rangle$ for a lattice of 51 sites as a function of the position $j$ for different voltages $V$ and losses $\gamma$ with $\epsilon =0$ and ${\tau }_1=0.5$.

Figure 14

The average imbalance $\delta n$ as a function of voltage in a lattice of seven sites, with (a) ${\tau }_1=0.1$ and (b) ${\tau }_1=0.5$. A larger coupling ${\tau }_1$ leads to the broadening of the steps. The vertical lines mark the values $V/2=E_n$, where $E_n$ are the single-particle eigenstates in the ${\tau }_1\to 0$ limit.

Figure 15

(a) The average imbalance as a function of voltage for a 51-site lattice, calculated as in Eq. (35), for ${\tau }_1=0.5$. The voltages where $V/2$ coincides with the eigenvalues of antisymmetric eigenstates are marked with vertical lines. The overall slope matches the estimate $(k_{F,L}-k_{F,R})/\pi$ with $k_{F,i}$ given by Eq. (34) (see the text). (b) The saturation imbalance at $V=5$ as a function of the dissipation rate coincides for 51 and 3 sites, and it is reproduced by the simple formula (36).

Figure 16

The energy diagram of a quantum dot with energy level $\epsilon$ coupled to a single reservoir. At zero temperature, states up to $\mu$ in the reservoir are filled and the rest are empty.

Figure 17

(a) The two contributions to the quantum dot occupation, given by Eqs. (37) and (38), indicated by the dotted and dashed lines. The different colors correspond to different values of ${\tau }_1$. The contribution of the bound states $n_P$ decays for increasing $\mathrm{\Lambda }$ while the contribution of the reservoir energy continuum $n_{BC}$ increases. When $\mu =\epsilon =0$, the occupation is 0.5 independent of the cutoff $\mathrm{\Lambda }$. (b),(c) The occupation $n_d$ as a function of the reservoir chemical potential $\mu$ with $\epsilon =0$. The steplike change in the occupation is smoothened out for larger tunneling amplitudes, similar to the average density imbalance in Fig. 14. (b) For a small cutoff $\mathrm{\Lambda }=1$, the bound-state contribution is finite when the coupling is large (${\tau }_1=1$). (c) For a larger cutoff $\mathrm{\Lambda }=3$, the bound-state contribution is negligible for all values of ${\tau }_1$ shown here.

Figure 18

(a) The quantum dot occupation and (b) the loss current as functions of voltage for a finite cutoff $\mathrm{\Lambda }=10$. When $\mathrm{\Lambda }$ is finite, the loss current has a nonmonotonic dependence on $\gamma$. The coupling to the reservoirs is ${\tau }_1=0.1$ and $\epsilon =1$.

Figure 19

The bound-state energies are found at the intersections marked by circles, as indicated by Eq. (D8). The $y$ axis is in units of $1/\left(\pi {\rho }_0\right)$.