# Transport through a lattice with local loss: from quantum dots to lattice
gases

J.R. Anglin  State Research Center OPTIMAS and Fachbereich Physik,Rheinland-
Pfälzische Technische Universität,D-67663 Kaiserslautern, Germany

###### Abstract

Recent work has studied fermion transport through a finite one-dimensional
lattice of quantum dots, with localized particle loss from the central lattice
site. The dots at each end of the lattice are connected to macroscopic leads,
represented as zero-temperature reservoirs of free fermions with a given
potential difference. Here we show how this model represents one limiting case
of a larger class of models that can be realized with cold quantum gases in
optical lattices. While quantum gas realizations allow many system parameters
to be varied, we note limitations from finite size effects, and conclude that
quantum dots and quantum gases offer complementary views on transport through
lossy finite lattices.

##  I Introduction

###  I.1 Motivation and background

The effects of localized loss on quantum transport are of high current
interest both as examples of fundamental phenomena and as practical tools for
controlling quantum states and processes [3, 4, 5, 6, 7, 8, 9, 10]. In
particular, recent theoretical work [1, 2] has investigated fermions traveling
from one reservoir to another through a finite chain of sites, with one of the
sites having a finite rate of particle loss. The finite number of sites means
that the current is dramatically affected by reflection and discrete
eigenfrequencies, and so losing particles from one site does not just drain
away some current classically, but also modifies quantum interference effects.

Reference [2] has solved this general problem with nonequilibrium Green’s
functions in the Keldysh formalism extended to open quantum systems. The main
result for the steady-state fermionic current through the finite chain is that
conductivity shows discrete steps, as a function of voltage between the
reservoirs across the chain of sites, and that the lossy site systematically
smooths out these steps, even with moderate loss rates. Higher loss rates also
significantly lower the steps, reducing overall conductivity.

Letting the single lossy site be the central site, with an odd number of sites
in the chain, elegantly isolates the effect of loss on the conductivity steps.
The conductivity steps are associated with energetic access to single-particle
eigenstates in the finite chain; with reflection symmetry, odd-parity
eigenstates have nodes at the loss site, and transport through these channels
is much less affected by the loss than transport through even-parity channels.
This shows up clearly in [2] in conductivity steps that alternate between
smooth and steep.

Although [2] explicitly raises the possibility of experimental realizations
with ultracold fermi gases in optical lattices, however, its concrete results
are obtained for parameter regimes that rather represent electron transport
through a chain of quantum dots, with the reservoir roles played by
macroscopic wire leads. In this paper we therefore look critically at a
quantum gas realization of this lossy transport problem. Instead of applying
an open-system formalism, we replace the macroscopic fermion reservoirs of [2]
with more lattice sites like the ones in the finite middle chain. The finite
middle chain is only separated from the “reservoir” parts of the lattice by
“weak links” of slower tunneling rates between sites. We will show, first of
all, that this particular class of Hamiltonian realizations of the particle
reservoirs can indeed reproduce exactly the results of [2]. We will then re-
tune our parameters to represent experiments that will likely be more
straightforward with quantum gases. We will investigate both finite-
temperature and finite-size effects, and conclude that experiments with
quantum gases are realistically feasible, and can provide complementary
insights to those offered by quantum dots.

###  I.2 Paper structure

In Section II immediately below we will introduce our lattice Fermi gas and
show how all single-particle observables can be found by solving a single-
particle Schrödinger equation. Loss from the central lattice site, which is
defined through a Lindblad master equation, is shown to be described exactly
by an imaginary potential term in the Schrödinger equation.

In Section III we will then explain the class of initial states which let this
infinite closed system exactly reproduce the observables of the open system in
which a finite lattice is coupled at its ends to particle reservoirs. In
particular we will present a case of our model which exactly reproduces the
results of [2] for finite chains of quantum dots coupled to macroscopic leads.

In Section IV we will then re-tune our infinite model to correspond more
closely to the most straightforward kind of quantum gas experiments, with
chemical potential differences between the left and right reservoirs, but no
on-site potential differences, and with densities of states in the middle and
outer sections of the lattice all equal, instead of having much greater
densities of states in the reservoirs. We predict steady-state currents that
are qualitatively similar to those found in [2] for the quantum dot system
with leads. There are abrupt steps in the conductivity at chemical potential
differences corresponding to eigenfrequencies of the isolated middle lattice;
the steps are smoothed by the loss at the central site, or remain quite sharp,
depending on the parity of the corresponding eigenfunctions. Although
qualitatively very similar, the straightforward quantum gas scenario is found
to differ in detail from the quantum dot version of the experiment.

In Section V we will consider the experimentally important constraint of
finite lattice size, which in the presence of loss means that the only true
steady state of the system is absence of particles (except in cases with
reflection symmetry, in which odd-parity modes may be occupied, but no
transport occurs). We will show, however, that if the outer ends of the
lattice contain sufficiently many sites, then there is a time interval during
which the system can can sustain a quasi-steady flow of particles through the
middle region. We will further show that the current during this quasi-steady
era corresponds closely to the steady-state current in the infinite system,
with residual finite-size effects that are small for lattices with total site
numbers of order 100.

We will conclude in Section VI with a summary of our results and an outlook
toward further theoretical and experimental work to address bosons, finite
temperatures, interactions, and multiple loss sites. A series of Appendices
supply technical details and derivations of formulas provided in the main
text.

##  II Reduction to a one-particle Schrödinger equation

###  II.1 Infinite lattices with weak links as finite lattices between
reservoirs

We consider an infinite one-dimensional lattice of sites that can be occupied
by non-interacting particles that can move between sites. In second-quantized
notation the Hamiltonian operator takes the form

$\displaystyle\hat{H}=\hbar\tau\sum_{m=-\infty}^{\infty}\left(-T_{m}(\hat{a}^{\dagger}_{m+1}\hat{a}_{m}+\hat{a}^{\dagger}_{m}\hat{a}_{m+1})+V_{m}\hat{a}^{\dagger}_{m}\hat{a}_{m}\right)\;,$
(1)

where $m$ labels each site, $\tau$ is a frequency scale, and $\hat{a}_{m}$ and
$\hat{a}^{\dagger}_{m}$ are canonical destruction and creation operators. We
will have effectively spinless fermions in mind throughout this paper, but the
reduction to a single-particle problem that we are about to describe proceeds
identically for bosons as well; the only difference between fermions and
bosons in this problem is whether Fermi-Dirac or Bose-Einstein distributions
are appropriate as initial states. Throughout this paper we will use $m$ and
$m^{\prime}$ to refer to sites or numbers of sites, and reserve $n$ for
numbers of particles.

The site-dependent parameters $T_{m}>0$ and $V_{m}$ do not depend arbitrarily
on $m$, but have a simple pattern that defines a symmetrical finite chain of
$2M+1$ sites ($|m|\leq M$) centered between semi-infinite reservoir zones on
the left and right:

$\displaystyle T_{m}$ $\displaystyle=\left\\{\begin{matrix}1&,&-(M+1)\leq
m\leq M\\\ \Lambda&,&m<-(M+2)\;\hbox{or}\;m>M+1\\\
\tau_{1}\sqrt{\Lambda}&,&m=-(M+2)\;\hbox{or}\;m=M+1\end{matrix}\right.$ (2)
$\displaystyle V_{m}$ $\displaystyle=\left\\{\begin{matrix}0&,&|m|\leq M\\\
+\frac{V}{2}&,&m<-M\\\ -\frac{V}{2}&,&m>M\end{matrix}\right.\;.$ (3)

Our overall frequency scale $\tau$ is thus the inter-site hopping rate in the
middle chain of sites, while the dimensionless parameters $\Lambda$ and
$\tau_{1}$ effectively define, respectively, the density of states in the
infinite outer ends of the lattice, and the links between the outer ends and
the finite middle section. We have included the factor of $\sqrt{\Lambda}$
with the factor $\tau_{1}$ because this will turn out to leave a non-trivial
model in the limit $\Lambda\to\infty$. In particular this limit will let the
infinite outer ends of the lattice reproduce the effects of macroscopic leads
that are treated as reservoirs, when $\tau_{1}<1$ implies a “weak link”
between the finite middle lattice and the outer “reservoir” ends.

###  II.2 Localized loss at site $m=0$

Following [2], we add particle loss at site $m=0$ to our model by assuming the
time evolution of the many-body density operator $\hat{\rho}(t)$ to be
governed by the Lindblad master equation

$\frac{d}{dt}\hat{\rho}=-\frac{i}{\hbar}[\hat{H},\hat{\rho}]+\gamma\tau\left(\hat{a}_{0}\hat{\rho}\hat{a}_{0}^{\dagger}-\frac{\hat{a}_{0}^{\dagger}\hat{a}_{0}\hat{\rho}+\hat{\rho}\hat{a}_{0}^{\dagger}\hat{a}_{0}}{2}\right)\;.$
(4)

With the middle-chain hopping rate $\tau$ as our basic time scale, the
dimensionless rate factor $\gamma$ defines the average rate $\gamma\tau$ at
which particles are lost, in a probabilistic Poisson process, from site $m=0$.
In quantum gas realizations of our model, this loss can be tuned by ejecting
particles from the central lattice site using lasers or electron beams.

###  II.3 Reduction to a Schrödinger equation with a complex potential

####  II.3.1 Single-particle observables

Equations (1) and (4) describe a many-body system of arbitrarily many non-
interacting fermions or bosons. The most accessible experimental observables
for many-body systems, however, are usually operators of the form
$\hat{O}=\sum_{m^{\prime}m}O_{mm^{\prime}}\hat{a}^{\dagger}_{m^{\prime}}\hat{a}_{m}$,
which for appropriate matrix elements $O_{mm^{\prime}}$ include particle
density, momentum density, kinetic energy density, and kinetic energy current
density, as well as first-order quantum coherences between arbitrary lattice
sites.

Observables of this class are called single-particle observables because they
do not directly probe interparticle correlations. Their expectation values in
an arbitrary many-body mixed quantum state can be given in terms of the
(unnormalized)_single-particle density matrix_ $R_{m^{\prime}m}$:

$\displaystyle\langle\hat{O}\rangle=\mathrm{Tr}(\hat{O}\hat{\rho})$
$\displaystyle=\sum_{m^{\prime},m}O_{mm^{\prime}}\mathrm{Tr}(\hat{a}^{\dagger}_{m^{\prime}}\hat{a}_{m}\hat{\rho})$
$\displaystyle=:\sum_{m^{\prime},m}O_{mm^{\prime}}R_{m^{\prime}m}\;.$ (5)

$R_{m^{\prime}m}$ is referred to as a single-particle density matrix because,
as a Hermitian matrix with non-zero eigenvalues in the Hilbert space of
lattice sites $m$, it is indistinguishable from the mixed-state density matrix
of a single particle, apart from a normalization factor.

(This normalization factor is simply the average total particle number,

$\sum_{m}R_{m^{\prime}m}=\mathrm{Tr}\Big{(}\hat{\rho}\sum_{m}\hat{a}^{\dagger}_{m^{\prime}}\hat{a}_{m}\Big{)}\;.$
(6)

If the particles are conserved, it is often conventional to normalize
$R_{m^{\prime}m}$ by dividing it by its trace. Since particles can be lost in
our case at site $m=0$, however, the trace of $R_{m^{\prime}m}$ can in general
be time-dependent. The unnormalized $R_{m^{\prime}m}$ that we use is the more
directly relevant quantity, anyway, for the simple $\hat{O}$-type observables
in the many-body system.)

Since no information about correlations between particles is included in
$R_{m^{\prime}m}$, $R_{m^{\prime}m}$ is in general a very incomplete
description of a many-body system. It is not an approximation, however, but
rather an exact representation of a restricted class of observables—which
happens to include most observables of physical interest.

####  II.3.2 Single-particle master equation

In the presence of interparticle interactions or nonlinear Lindblad operators,
interparticle correlations can still affect the single-particle observables
indirectly, even strongly, because time evolution couples $R_{m^{\prime}m}$ to
expectation values of higher-order combinations of
$\hat{a}^{\dagger}_{m^{\prime}}$ and $\hat{a}_{m}$ operators, in the typically
infinite BBGKY hierarchy. Since our $\hat{H}$ from (1) includes no
interactions among the particles, however, and our Lindblad operator is simply
$\hat{a}_{0}$, in our case the exact time evolution of $R_{m^{\prime}m}(t)$ is
given by a single-particle master equation which closes, in the sense that it
involves only $R_{m^{\prime}m}$ itself. Inserting our many-body Lindblad
equation (4) into the definition of $R_{m^{\prime}m}$ and applying canonical
(anti-)commutation relations
$\hat{a}_{m}\hat{a}^{\dagger}_{m^{\prime}}\pm\hat{a}^{\dagger}_{m^{\prime}}\hat{a}_{m}=\delta_{m^{\prime}m}$
for either fermions or bosons reveals

$\displaystyle\frac{i}{\tau}\dot{R}_{m^{\prime}m}=$ $\displaystyle
T_{m^{\prime}}R_{m^{\prime}+1,m}+T_{m^{\prime}-1}R_{m^{\prime}-1,m}$
$\displaystyle-T_{m}R_{m^{\prime},m+1}-T_{m-1}R_{m^{\prime},m-1}$
$\displaystyle+(V_{m}-V_{m^{\prime}})R_{m^{\prime}m}$
$\displaystyle-i\frac{\gamma}{2}\big{(}\delta_{m^{\prime}0}R_{0m}+\delta_{0m}R_{m^{\prime}0}\big{)}\;.$
(7)

####  II.3.3 Single-particle Schrödinger equation

Equation (II.3.2) can then be greatly simplified even further, by noting that
it can be solved by a factorizing _Ansatz_ of the form

$\displaystyle R_{m^{\prime}m}(t)=$ $\displaystyle\sum_{j,j^{\prime}}\int
d\omega\,d\omega^{\prime}\,A_{j^{\prime}j}(\omega,\omega^{\prime})$
$\displaystyle\times
e^{i\tau(\omega^{\prime}-\omega)t}\Psi^{j^{\prime}*}_{m^{\prime}}(\omega^{\prime})\Psi_{m}^{j}(\omega)$
(8)

as long as the $\Psi_{m}^{j}(\omega)$ are solutions
$\psi_{m}\to\Psi_{m}^{j}(\omega)$ of time-independent single-particle
Schrödinger equation

$\displaystyle\omega\psi_{m}=-T_{m}\psi_{m+1}-T_{m-1}\psi_{m-1}+(V_{m}-i\frac{\gamma}{2}\delta_{m0})\psi_{m}$
(9)

with a complex potential at $m=0$. The superscript $j$ in
$\Psi^{j}_{m}(\omega)$ is used to distinguish independent solutions to (9)
having the same eigenfrequency $\omega$, if these exist.

###  II.4 Open or closed?

This concludes our definition of our non-interacting lattice gas with weak
links and loss. Although our system is open in the sense of having loss, the
loss can be represented exactly with an imaginary potential. We do not include
any external sources of particles or energy, or noise of any kind. We
nevertheless propose to consider the middle chain of $2M+1$ sites, between the
weak links, as an open system, for which the infinite outer ends of the
lattice constitute reservoirs that can send particles into or through the
middle chain of sites. In the next Section we will explain how this works.

##  III Non-equilibrium stationary states

###  III.1 Non-equilibrium many-body system states from single-particle wave
functions?

So how do we use solutions to our single-particle Schrödinger equation (9) to
describe non-equilibrium steady states as in [2], with particle transport
between the two reservoirs through the finite middle chain with its single
lossy site?

We will consider special cases of (II.3.3) of the form

$R_{m^{\prime}m}=\sum_{j=L,R}\int\\!d\omega\,D_{j}(\omega)f_{j}(\omega)\Psi^{j*}_{m^{\prime}}(\omega)\Psi_{m}^{j}(\omega)\;,$
(10)

with $j=L,R$ denoting wave functions that are incident from the left and
right, respectively. Here $D_{j}(\omega)$ are the corresponding densities of
states, and $f_{j}(\omega)$ are Fermi-Dirac distributions with independent
temperatures and chemical potentials for $j=L$ and $j=R$. (In future work it
will be straightforward to use Bose-Einstein distributions instead, and
describe analogous transport problems with quantum Bose gases. In our
concluding Section VI we suggest that the bosonic problem will be less
interesting, however, unless interactions are present.)

Since in general we allow a potential difference $V$ between the left and
right ends of our infinite lattice, as well as different equilibrium
distributions for $f_{L}$ and $f_{R}$, our stationary states will include
transport in general.

In Appendix A we will explicitly construct the $\Psi^{L,R}_{m}(\omega)$
solutions to (9); in the remainder of our main text we will use reflection and
transmission coefficients that are derived in that Appendix. Here we explain
why an $R_{m^{\prime}m}$ of the form (10) really does represent non-
equilibrium steady states of our system, and then outline how we will use
these $R_{m^{\prime}m}$ to compute observable quantities in these states.

###  III.2 Left- and right-incident normalized eigenfunctions

The solutions $\Psi_{m}^{L,R}(\omega)$ that we construct explicitly in
Appendix A have the following “scattering” forms for $|m|>M$:

$\displaystyle\sqrt{2\pi}\Psi_{n}^{L}(\omega)$
$\displaystyle=\left\\{\begin{array}[]{ccc}e^{ik_{L}m}+\mathcal{R}_{L}(\omega)e^{-ik_{L}m}&\;,&m<-M\\\
\sqrt{\frac{sin(k_{L})}{sin(k_{R})}}\mathcal{T}(\omega)e^{ik_{R}m}&\;,&m>M\end{array}\right.$
(13) $\displaystyle\sqrt{2\pi}\Psi_{n}^{R}(\omega)$
$\displaystyle=\left\\{\begin{array}[]{ccc}\sqrt{\frac{sin(k_{R})}{sin(k_{L})}}\mathcal{T}(\omega)e^{-ik_{L}m}&\;,&m<-M\\\
e^{-ik_{R}m}+\mathcal{R}_{R}(\omega)e^{ik_{R}m}&\;,&m>M\end{array}\right.\;,$
(16)

for transmission and reflection coefficients $\mathcal{T}$ and
$\mathcal{R}_{L,R}$ that we compute in Appendix A and will use below. The
behaviors of $\Psi^{L,R}_{m}$ for $|m|\leq M$ are shown in Appendix A, but
will not be needed for any results we will present in our main text.

Since we have neither time-reversal nor (unless $V=0$) spatial reflection
symmetry, the fact that the transmission coefficient $\mathcal{T}(\omega)$ is
the same for both left- and right-incident waves is not obvious, but it turns
out to be so in all possible cases of our model. The reflection coefficients
$\mathcal{R}_{L,R}(\omega)$, in contrast, do not coincide in general. In the
special cases that we will consider explicitly within our main text, however,
$\mathcal{R}_{L,R}\to\mathcal{R}(\omega)$ _are_ in fact the same, because
$k_{L,R}$ are the same, because these cases will either have $V=0$ or
$\Lambda\to\infty$.

In the $|m|>M$ regions the wave numbers $k_{L,R}(\omega)$ can be identified
immediately, by recognizing that plane waves $e^{\pm ik_{L}m}$ are local
solutions to the single-particle Schrödinger equation (9) for $m<-M$, as are
$e^{\pm ik_{R}m}$ for $m>M$, if

$-2\Lambda\cos(k_{L,R})\pm\frac{V}{2}=\omega$ (17)

where the $+(-)$ branch of $\pm$ applies for $L(R)$. The ratios of
$\sin(k_{L,R})$ that appear as factors before $\mathcal{T}_{L,R}$ are the
usual ratios of group velocities $d\omega/dk$ that appear in one-dimensional
transmission and reflection problems when the left and right asymptotic group
velocities may be different, as here if $V\not=0$. We can also note from (17)
that the densities of states which appear in our stationary $R_{m^{\prime}m}$
integral (10) are

$D_{L,R}(\omega)=\frac{1}{\frac{d\omega}{dk_{L,R}}}=\frac{1}{2\Lambda\sin\big{(}k_{L,R}(\omega)\big{)}}\;,$
(18)

which as usual in one-dimensional systems are the inverse group velocities.

###  III.3 Why scattering wave functions?

The first reason for defining the left- and right-incident wave functions
$\Psi_{m}^{L,R}$ in this way (13) is that they allow us easily to create
arbitrary incident wave packets at infinity. For example, in a superposition
of the form

$\int\\!dk_{L}\,e^{-\frac{1}{2}\Delta
m^{2}(k_{L}-\bar{k})^{2}}e^{-ik_{L}\bar{m}}\Psi_{m}^{L}$ (19)

for arbitrary $\bar{k}\in[0,\pi]$ and large $\Delta m\gg 1$ but even larger
negative $\bar{m}\ll-\Delta m$, only the incoming wave term $\propto
e^{ik_{L}m}$ in $\Psi_{m}^{L}$ will contribute significantly—and it will
contribute essentially the same Gaussian wave packet, centered on $m=\bar{m}$
and wave number $k_{L}=\bar{k}$, that we would obtain if $\Psi_{m}^{L}$
consisted of nothing but that incoming wave because we were dealing with an
infinite uniform lattice with no weak or lossy links anywhere.

It is then the implication of these incoming wave packets from infinity that
is our main reason for defining the $\Psi_{m}^{L,R}$. An equilibrium ensemble
of incoming wave packets represents the particles which approach our finite
middle chain, with its lossy site, from an infinite reservoir of particles in
equilibrium. Consequently we must be able to represent the single particle
density matrix $R_{m^{\prime}m}$ of a steady state of our system, with
distinct equilibrium distributions at left and right infinity, in the form
(10) that we defined at the beginning of this Section.

###  III.4 Cases with imaginary $k_{L,R}$

In general it may happen that one or the other of $k_{L,R}(\omega)$ as given
by (17) is complex. No cases with complex $k_{L,R}$ will need to be considered
here in our main text, but for possible use in future work we discuss this
technicality further in Appendix B.

###  III.5 Observables in the steady state

From our steady-state single-particle density matrix (10), and the explicit
eigenfunctions we derive in Appendix A, it is straightforward to compute
several steady-state observables as functions of the Hamiltonian parameters
and the initial states of the left and right lattice ends. The most basic
observable is the local occupation number at each site $m$, which is simply a
diagonal element of the unnormalized density matrix:

$\bar{n}_{m}=\mathrm{Tr}(\hat{\rho}\hat{a}^{\dagger}_{m}\hat{a}_{m})=R_{mm}\;.$
(20)

Within the finite middle chain of sites, $\bar{n}_{m}$ for $|m|\leq M$ will
depend on the $\Psi_{m}^{L,R}(\omega)$ for $|m|\leq M$, and will therefore not
have any simple relation to the $\mathcal{T}$ and $\mathcal{R}_{L,R}$
coefficients that define the wave functions for $|m|>M$. We will focus instead
on observables that relate directly to transport.

The simplest of these is the particle current. From the single-particle master
equation (II.3.2) we obtain

$\displaystyle\frac{d}{dt}\bar{n}_{m}$
$\displaystyle=-\gamma\tau\delta_{m0}\bar{n}_{0}-(J_{m+1}-J_{m})$
$\displaystyle J_{m}$ $\displaystyle=i\tau T_{m-1}(R_{m,m-1}-R_{m-1,m})\;.$
(21)

Since the first line of (III.5) is an obvious discrete version of the
continuous continuity equation, we recognize $J_{m}$ as the particle number
current from left to right, which is also proportional to the average momentum
at site $m$.

In a stationary state, $\bar{n}_{m}$ is time-independent, and so we conclude
that the particle number current is piecewise constant as a function of $m$:

$J_{m}=\tau\left\\{\begin{matrix}I+\frac{\gamma\bar{n}_{0}}{2}&\;,\;&m\leq
0\\\ I-\frac{\gamma\bar{n}_{0}}{2}&\;,\;&m>0\;,\end{matrix}\right.$ (22)

for constant average current $I$. Since the difference between the piecewise
constant currents on each side of $m=0$ thus represents particle transport
_into_ the central lossy site from the two reservoirs, we can follow [2] and
take $I$ as a representation of net transport _through_ the finite middle
lattice from the left reservoir into the right reservoir. With our stationary
$R_{m^{\prime}m}$ represented in the form (10), we can evaluate $J_{L,R}$ by
evaluating $J_{m}$ for $|m|>M$, and then use our $|m|>M$ forms for
$\Psi_{m}^{L,R}$ in (13) to conclude

$I=\frac{1}{4\pi}\int\\!d\omega\,\Big{(}f_{L}(\omega)-f_{R}(\omega)\Big{)}\Big{(}1+|\mathcal{T}|^{2}-|\mathcal{R}|^{2}\Big{)}\;,$
(23)

where $f_{L,R}(\omega)$ are equilibrium occupation distributions that can be
different for the two ends of the lattice. In (23) we have assumed
$\mathcal{R}_{L,R}\to\mathcal{R}$ as they do in the cases $V=0$ or
$\Lambda\to\infty$; a more general formula for $I$ is given in Appendix C. The
density of states factors $D_{L,R}(\omega)$ from (10) have not been omitted
accidentally from (23): they are truly absent, being conveniently canceled by
group velocity factors that appear in $J_{m}$.

Additional single-particle observables that may be of interest, and that can
probably be measured in quantum gas experiments, include the local energy at
site $m$, which is also simply related to the momentum current, as well as the
energy current, which could offer insight into how the central lossy cite
absorbs heat as well as particles. We will only consider the particle current
$I$ here in our main text, but we derive the corresponding formulas for the
other observables in Appendix C.

###  III.6 Semi-infinite lattices as macroscopic leads: The $\Lambda\to\infty$
limit

We now confirm that our infinite lattice model, with weak links between the
finite middle chain and the semi-infinite ends, does indeed represent a finite
middle chain coupled to particle reservoirs.

####  III.6.1 Transmission and reflection coefficients $\mathcal{T}$ and
$\mathcal{R}$ for $\Lambda\to\infty$

As we show in Appendix A, the dimensionless parameter $\Lambda$ does not
appear explicitly anywhere in the expressions for $\mathcal{R}_{L,R}$ and
$\mathcal{T}$. $\Lambda$ decisively affects $k_{L,R}(\omega)$ according to
(17), however, and $k_{L,R}$ do both appear in $\mathcal{R}_{L,R}$ and
$\mathcal{T}$. In the limit $\Lambda\to\infty$, in particular, the complicated
general formulas of Appendix A for $\mathcal{R}_{L,R}$ and $\mathcal{T}$
simplify dramatically.

For $\Lambda\to\infty$ the continuous bandwidth $|\omega\mp V/2|\leq 2\Lambda$
of the semi-infinite outer parts of the lattice becomes much broader than the
bandwidth $|\omega|\leq 2$ of the finite middle chain of sites. Wave vectors
for $|\omega|\gg 2$ decay so sharply within $|m|<M$ that they contribute
negligibly to any observables within the middle chain, including the particle
current through the middle chain, so as far as any such observables are
concerned, we can restrict our attention to $|\omega|\ll\Lambda$. For all
these $\omega$, the limit $\Lambda\to\infty$ means that $k_{L,R}$ are both
negligibly different from $\pi/2$. This implies that the density of states in
the “reservoir” ends of the lattice is uniform over the entire frequency range
that matters for transport through the finite middle chain.

With the replacement $k_{L,R}\to\pi/2$, then, our expressions from Appendix A
simplify—in particular $\mathcal{R}_{L,R}\to\mathcal{R}$ because
$k_{L}=k_{R}$. Defining $k(\omega)$ without subscript to be the wave number in
the middle chain,

$k(\omega)=\cos^{-1}\Big{(}-\frac{\omega}{2}\Big{)}\;,$ (24)

we obtain (see Appendix A)

$\displaystyle 1+|\mathcal{T}|^{2}-|\mathcal{R}|^{2}=$
$\displaystyle\frac{2}{|Z(\omega)|^{2}}\left[\sin^{2}(k)+\frac{\gamma}{4}\left(\frac{\sin^{2}[k(M+1)]}{\tau_{1}^{2}}+\tau_{1}^{2}\sin^{2}(kM)\right)\right]$
$\displaystyle|Z(\omega)|^{2}=$
$\displaystyle\left(\frac{\sin[k(M+1)]\cos[k(M+1)]}{\tau_{1}^{2}}+\tau_{1}^{2}\sin(kM)\cos(kM)\right)^{2}$
$\displaystyle+\left[\sin(k)+\frac{\gamma}{4\sin(k)}\left(\frac{\sin^{2}[k(M+1)]}{\tau_{1}^{2}}+\tau_{1}^{2}\sin^{2}(kM)\right)\right]^{2}\;.$
(25)

Since we consider small $\tau_{1}$, to make weak links between the outer ends
and the middle chain, the important feature in (III.6.1) is that the
denominator $|Z(\omega)|^{2}$ is large (of order $\tau_{1}^{-4})$, making the
current contribution from a given $\omega$ small, unless either
$\sin[(M+1)k(\omega)]$ or $\cos[(M+1)k(\omega)]$ is small. These are precisely
the conditions $\sin[2k(\omega)(M+1)]=0$ that define the $2M+1$ discrete
eigenfrequencies

$\displaystyle\omega_{n}$ $\displaystyle=-2\cos(k_{n})$ $\displaystyle k_{n}$
$\displaystyle=\frac{n\pi}{2(M+1)}\quad,\quad n\in[1,2M+1]$ (26)

of the middle chain when it is isolated ($\tau_{1}\to 0$ so that the outer
ends of the lattice can be ignored). Transport through the weakly linked
middle chain will therefore be small except for particles with frequencies
close to one of these resonant frequencies $\omega_{n}$.

It can further be noted, however, that the numerator in
$1+|\mathcal{T}|^{2}-|\mathcal{R}|^{2}$ in (III.6.1) can also somewhat
large—of order $\gamma\tau_{1}^{-2}$—unless $\sin[(M+1)k(\omega)]$ is small.
For moderate $\gamma$ this broadens the transmission resonances at
$\cos[(M+1)k_{n}]=0$, although for large $\gamma$ the denominator term of
order $\gamma^{2}\tau_{1}^{-4}$ also lowers the contributions of these
resonances.

At the resonances with $\sin[(M+1)k_{n}]=0$, in contrast, all the terms with
negative powers of $\tau_{1}$ vanish, and small $\tau_{1}$ makes the
transmission factor $(1+|\mathcal{T}|^{2}-|\mathcal{R}|^{2})/2$ close to one.
This is easy to understand: the isolated-chain eigenfrequencies $\omega_{n}$
with $\sin[(M+1)k_{n}]=0$ are those for which the isolated-chain
eigenfunctions are odd functions of $m$ and so have nodes at the lossy site
$m=0$. Transport through these channels is protected against loss by parity
symmetry, as reported in [2]. Transport at their resonant frequencies is only
affected by loss insofar as the weak links provide non-resonant coupling to
other modes of the finite chain that do not have nodes at $m=0$ and so do have
loss.

####  III.6.2 Current $I$ through the chain for $\Lambda\to\infty$

To confirm that we are reproducing the results of [2], we follow it in
considering fermions at zero temperature, with a non-zero voltage difference
$V$ ranging through the finite chain bandwidth $-2<\omega<2$. Since the
density of conduction-band electrons in a metal lead is a nearly constant
property of the material regardless of electrostatic potential, we keep the
chemical potentials (i.e. Fermi energies) $\mu_{L,R}$ in the left and right
reservoir $f_{L,R}(\omega)$ equal to the voltages $\pm V/2$, so that the
reservoirs differ in potential but have equal fermion densities:

$f_{L,R}(\omega)=\theta\big{(}\pm\frac{V}{2}-\omega\big{)}\;.$ (27)

Figure 1: The dimensionless current $I$ through a finite middle chain of five
sites ($M=2$), divided by $\tau_{1}^{2}$ as in [2], as a function of potential
difference $V$ between the left and right semi-infinite ends of the infinite
lattice, for four different values of $\gamma$ (from top to bottom at the left
side of the plot: $\gamma=0$, $0.5$, $5$, $100$.) In all cases $\tau_{1}=0.1$.
The thin, dark curves that only extend to $V=4.5$ are computed by numerical
integration of our (23) with (III.6.1) and (27), while the broader, fainter
curves that extend to the right edge of the plot are taken from Figure 5a) of
Reference [2]. The thin, dark curves lie directly on top of the broad, faint
curves, confirming the equivalence of the present model for $\Lambda\to\infty$
and the open system treated in [2].

As we see in Fig. 1, our infinite lattice model exactly reproduces the results
from [2] for a finite lattice between two macroscopic leads, treated with
Kheldysh path integrals. Figure 1 differs only in plotting resolution from
Figure 5 of [2]. Scaling the lattice weak link as $\tau_{1}\tau\sqrt{\Lambda}$
while the outer lattice ends have hopping rate $\Lambda\tau$, and then taking
$\Lambda\to\infty$ to make the reservoir density of states effectively
constant, maps the infinite lattice gas model onto the macroscopic leads
problem of [2].

##  IV A quantum gas version

###  IV.1 The quantum gas set-up

Having confirmed that the infinite lattice model with weak links can indeed
describe the finite chain of quantum dots coupled to leads as studied in [2],
we now examine a version of the lattice problem that would be more
straightforward to realize with an ultracold Fermi gas in a one-dimensional
optical lattice (or a three-dimensional array of many one-dimensional
lattices, allowing single-shot measurements of ensembles of the one-
dimensional system). While the limit of large $\Lambda$ could be achieved by
making $\tau$ small, with high potential barriers between the sites in the
middle chain, this would slow the system’s approach to a steady state,
possibly to the point where steady-state behavior could not be seen within the
sample lifetime. We therefore focus on the experimentally more straightforward
case $\Lambda=1$.

The on-site potential in the lattice can also be made non-uniform in cold atom
experiments, but it may be challenging to make the potential piecewise uniform
with abrupt steps between the middle and the outer ends of the lattice. We
therefore set $V\to 0$, leaving the two weak links as the only nonuniformity
in the lattice. With no analogous constraint to the fixed fermion density of
metal leads, however, we are free to let the chemical potentials $\mu_{L,R}$
in the two lattice ends be arbitrary. Varying $\mu_{L}-\mu_{R}$ while keeping
$V=0$ will turn out to have similar effects to varying $V$ with Fermi energies
locked to $\mu_{L,R}=\pm V/2$; the mechanism of opening more transmission
channels as the Fermi level rises is exactly the same, after all.

With no need to maintain fixed Fermion densities in both reservoirs, moreover,
we are free in the quantum gas scenario to let $\mu_{R}$ lie below the ground
state energy, so that no particles at all are incident from the right, and the
entire current is from left reservoir to right reservoir, through the middle
chain with its central site loss, minus reflection from the weak links back
into the left reservoir. This means that instead of (27) we now take

$f_{L}(\omega)=\theta(\mu_{L}-\omega)$ (28)

and $f_{R}=0$.

This less symmetrical but simpler set-up also has the advantage of showing
more conductivity steps. In the $\pm V\not=0$ scenario with equal fermion
densities in leads, the currents from right and from left exactly cancel in
the frequency range $\omega<-V/2$, so that the integral (23) over $\omega$ is
effectively over the symmetrical middle range $|\omega|<V/2$. Since the
resonant frequencies of the transmission channels through the finite middle
chain are in symmetrical pairs $\pm\omega_{n}$, raising $V$ effectively only
opens new channels in pairs, producing only $M+1$ conductivity steps. In the
quantum gas scenario with particles coming only from the left reservoir, in
contrast, raising $\mu_{L}$ through the system’s frequency band sweeps through
each of the $2M+1$ transmission resonances separately.

Figure 2: Dimensionless fermion current $I$ at zero temperature, as a function
of left-side chemical potential $\mu_{L}$, for two different central-site loss
rates $\gamma$. Both plots show $\tau_{1}=0.1$ as in Fig. 1, but with $V=0$,
$\Lambda=1$, and no fermions incident from the right. Although the middle
chains here have seven sites ($M=3$), the leftmost and rightmost current steps
are small and gentle in a), and completely flattened in b), leaving only five
current steps clearly visible. In a) we have $\gamma=0.5$, while b) shows
$\gamma=5$. As well as suppressing the outermost of the seven current steps,
the moderate loss in a) has noticeably smoothed out two of the five more
visible steps in the current, while the stronger loss in b) has smoothed them
completely into gradual slopes. The other three visible current steps are not
significantly affected by loss in either case, since they are due to transport
through odd-parity channels having nodes at the lossy site $m=0$. Although
$I(\mu_{L})$ differs in detail from analogous plots of $I(V)$ for quantum dots
between leads, the same phenomena of conductivity steps with parity-dependent
smoothing by loss are revealed.

Figure 2 shows two quantum gas analogs of Fig. 1, plotting the dimensionless
steady-state current $I$ as a function of left-side chemical potential
$\mu_{L}$, with no fermions incoming from the right. The current plotted in
Fig. 2 is the same integral (23) as in Fig. 1, but with the different
$f_{L,R}(\omega)$ of (28) as well as the more complicated $\mathcal{R}$ and
$\mathcal{T}$ coefficients that are given by inserting $V\to 0$ and
$\Lambda\to 1$ in the general formulas of Appendix A.

Five steplike rises in conductivity through the finite middle chain of seven
sites ($M=3$) are clearly visible in Fig. 2; looking closely at a), one can
also discern two more small, smooth steps at the left and right edges of the
plot. Also apparent is the effect of loss in making some steps noticeably
steeper than others. In Fig. 2a) the tiny first and seventh steps, as well as
the more visible third and fifth steps, are all due to even-parity channels
that are affected by the loss at $m=0$, and so more gently sloped than the
three other steeper steps, which are due to odd-parity channels with nodes at
$m=0$. The ten-times-higher loss rate in Fig. 2b) is enough to suppress the
first and seventh steps entirely, and eliminate the shoulders of what were the
third and fifth steps in a), leaving them as gradual slopes, while still
leaving the three parity-protected transport resonances as sharp steps.

Many different parameter regimes of this kind of lossy transport can be
investigated in quantum gas experiments, which are highly tunable. While
quantum gases in optical lattices may in this sense be a more flexible
platform than quantum dots, quantum gases do have their own constraints. While
room temperature can easily be effectively zero temperature for conduction
electrons in metals, even nanokelvin temperatures may not be negligible for
ultracold Fermi gases. And while macroscopic leads are effectively infinite in
size, optical lattices in experiments are bound to be finite. We will examine
finite temperature effects next, and treat the conceptually greater difference
of finite system size in the following Section.

###  IV.2 Finite temperature effects

Finite temperature effects are incorporated straightforwardly by replacing the
step function form (27) of the zero-temperature Fermi-Dirac function with the
general case

$f_{L}(\omega)=\frac{1}{e^{\frac{\omega+\mu_{L}}{T}}+1}$ (29)

for temperature $T_{L}\to T$, leaving $f_{R}(\omega)=0$. The effect of finite
temperature is likewise simple: it smears out sharp features of frequency.

Since the main phenomena to be studied in this kind of transport are the
conductivity steps and their parity- and $\gamma$-dependent steepness, thermal
smearing of the conductivity curve is a serious issue for experiments. Since
the frequency spacing of our transmission resonances is of order $2/(2M+1)$,
temperatures higher than this spacing (in units of $\hbar\tau/k_{B}$) will
wash out the conductivity steps. Even if $T/\tau$ is below the $2/(2M+1)$
threshold, moreover, thermal smoothing of the conductivity steps can reduce
their dramatic signature of discrete transmission channels to only a slight
waviness in $I(\mu_{L})$, as seen in Fig. 3a); in this regime, moreover, the
additional smoothing of even-parity channels due to the $m=0$ loss can be hard
to discern. Both the steps and the effect of parity can still be seen at
sufficiently low temperatures, as in Fig. 3b), but such low temperatures may
be experimentally challenging. If the basic $T<2\tau/(2M+1)$ limit can be
beaten, the problem of losing visibility of the parity effect for
$T\lesssim\gamma$ can be solved by increasing $\gamma$, as in Fig. 3c).

Figure 3: Ultracold Fermi gas current $I$ through the lossy impurity, as a
function of left-side chemical potential $\mu_{L}$, at finite left-side
temperature (thin black curves), with zero-temperature results for comparison
(thick gray curves). All three plots show $V=0$, $\tau_{1}=0.1$, and $M=3$ as
in Fig. 2 (seven sites in the middle chain, although at most five current
steps are large enough to see clearly), with no fermions incoming from the
right. In a) we have $\gamma=0.5$ and $k_{B}T_{L}=0.1\hbar\tau$ (black curve);
the comparatively high temperature smooths out all the abrupt zero-temperature
steps into slopes that are similarly gradual. In b) we lower the temperature
to $k_{B}T=0.01\hbar\tau$; the steepest zero-temperature steps are only
slightly less steep at this low temperature, and the effect of loss in
smoothing out the second and fourth visible steps is still noticeable. In c)
we raise the loss rate $\gamma$ to 5; with stronger loss, the same temperature
$k_{B}T=0.1\hbar\tau$ as in a) no longer buries the smoothing effect of the
loss on the second and fourth visible steps.

In condensed matter systems with Fermi temperatures in thousands of Kelvin,
even room temperature is negligibly different from zero temperature. In
quantum gas experiments, in contrast, the ratio $k_{B}T/(\hbar\tau)$ of
temperature to hopping rate cannot be made arbitrarily low. While it is easy
to make $\tau$ small by increasing laser power in the optical lattice to
reduce tunnelling rates exponentially, reducing the lattice strength in order
to accelerate tunnelling will also weaken the tight binding approximation and
turn on excitations out of the lowest band, so that our single-band lattice
model (1) breaks down. A realistic upper bound on $\tau$ is probably in the
range of $10^{4}$ s${}^{-}1$, corresponding to a temperature range in hundreds
of nanoKelvin[11]. With the currently achievable temperatures for lattice
fermions in tens of nK, reaching $k_{B}T/(\hbar\tau)<0.1$ will likely require
future technical progress. As Fig. 3 shows, a lower ratio like
$k_{B}T/(\hbar\tau)=0.01$ may be needed to compete with the current step
sharpness of quantum dot experiments, but a ratio of $0.1$ can still be
sufficient to show the steps as well as the smoothing effect of loss in
dependence on parity. As in many potentially interesting cold atom
experiments, reaching significantly lower temperatures will be an important
challenge.

We now turn to the issue of finite size effects in quantum gas realizations,
which requires some new concepts but probably poses less serious problems than
finite temperature.

##  V Finite size effects

###  V.1 Finite size and finite time

Cold atom experiments can flexibly tune many parameters, but the number of
lattice sites cannot really be tuned to infinity. Although the finite sizes of
cold atom “reservoirs” can limit the effects that can be seen in cold atom
experiments, we will now show how measurements of finite time evolution can in
fact reproduce the steady-state behavior of infinite systems.

In the finite system with a lossy impurity there can be no steady state
current: currents can only be transient, strictly speaking, because in the
limit of infinite time all initial particles will have simply been lost.
Technically transient currents do not have to be short-lived phenomena,
however. They can easily be quasi-steady over intermediate time scales.

###  V.2 Quasi-stationary currents

For example, a natural initial state to prepare experimentally is a finite-
system analog of the current-from-the-left scenarios shown for infinite
lattices in Fig. 2. To prepare the initial state we set $\tau_{1}\to 0$, fully
breaking the weak link and isolating the middle chain of our lattice from its
outer ends. We then populate the isolated left end of the lattice with
fermions in equilibrium, leaving the middle and right-end lattices empty. At
$t=0$ we then restore non-zero $\tau_{1}$ and let the left-end population of
fermions flow through the weakly linked middle chain, with its loss at $m=0$,
and into the right lattice end. At different times $t>0$ we measure the
decreasing population difference between left and right sides,

$\frac{\Delta\mathcal{N}}{2}=\frac{1}{2}\sum_{m=M+1}^{M+N}(n_{-m}-n_{m})\;,$
(30)

which is due to the same left-to-right current that we have examined in
previous Sections, albeit no longer in a steady state. For clarity we return
to the zero-temperature scenarios; the frequency smoothing effect of finite
temperature can be added straightforwardly to the finite-size and finite-time
effects that we now examine.

As we can see in Fig. 4a), there is a brief initial period in which the
fermions first begin to move through the middle chain of sites, via the weak
links. Within a time of order $M/\tau$, however, this initial onset phase ends
and we can see a remarkably steady fall of $\Delta\mathcal{N}$. The initial
equilibrium state in the left end of the lattice can be viewed as an ensemble
of wave packets moving in both directions; these packets move to and through
the middle chain from further back in the left reservoir, just as they would
if the reservoir were infinite. Even the packets which begin moving to the
left, and then approach the middle chain after reflecting from the left end of
the lattice, simply reproduce the effect of incoming packets from beyond the
lattice end. The first effect which does not look just like an infinite
system, in fact, is the return to the weak link of packets which have been
reflected from it, and then reflected again from the left end of the lattice.
This occurs for packets with group velocity $v_{G}$ at time $t=2N/v_{G}$. The
maximum group velocity in our system is $2\tau$, occurring at $\omega=0$, and
so the earliest finite size effects are those that appear in Fig. 4 at
$t=N/\tau$. Between $t\lesssim M/\tau$ and $t<N/\tau$ there can be plenty of
time to observe a quasi-stationary current, as long as $N$ is sufficiently
larger than $M$. And as Fig. 4 shows, the rate at which $\Delta\mathcal{N}$
falls, during this quasi-stationary epoch between the initial transient and
the onset of finite-size effects, can closely match the steady-state current
of the infinite system.

Figure 4: Finite-time evolution in finite systems. The initial state is
prepared in a finite lattice with $N=20$, 40, or 80 sites in the outer lattice
on each side. With the weak links between reservoir ends and middle chain
initially turned off completely ($\tau_{1}\to 0$), the $N$-site left end
lattice is initially half-filled at zero temperature, with the right end and
central chain of seven sites ($M=3$) both initially empty. At $t=0$ the weak
links are turned on $\tau_{1}\to 0.1$ and the system’s single-particle density
matrix is evolved in time under (II.3.2) numerically, for a duration
$2.5N/\tau$. The difference in total left- and right-side populations $\Delta
N/2=(N_{L}-N_{R})/2$ is plotted in each case of $N$. The dashed line has the
slope $-I$, where $I$ is the steady-state current for the $N\to\infty$ limit
as shown in Fig. 2. The early transient epoch is so brief that it can only be
seen clearly with shorter time axis of the $N=20$ plot. The onset of finite-
size effects at $t=N/\tau$ is clear in all cases: with group velocity $2\tau$
for half filling, this is how long it takes for particles that initially
reflect from the weak link to return to the weak link after reflecting again
from the end of the lattice. For times after the short initial transient and
before $t=N/\tau$, the finite system’s particle difference closely reproduces
the infinite system’s steady-state current—in the case $\mu_{L}=0$ of initial
half-filling. See Fig. 5 for other cases of $\mu_{L}$.

###  V.3 Extended transient effects near transmission resonances

In fact Fig. 4 provides an overly optimistic view of how easy it is to see
steady-state current with finite-time measurements: although $\mu_{L}=0$ is
the worst case for finite-size effects, since $v_{G}$ is highest there, it
turns out to be the best case for short duration of the initial transient
period. While $\mu_{L}=0$ is not so much better than typical $\mu_{L}$ that
are not close to resonances, the slope of $\Delta\mathcal{N}$ can take longer
to settle down steadily for $\mu_{L}$ near transmission resonances other than
the one at $\mu_{L}=0$; see Fig. 5. The curves of $\Delta\mathcal{N}(t)$
appear quite similar for different outer-lattice sizes $N$, when both $t$ and
$\mu_{L}$ are rescaled with $N$ appropriately, indicating that the extended
transient epochs near transmission resonances are probably not really finite-
size effects, but rather finite-time effects. We cannot avoid these problems
by letting the system evolve for longer times, however, because finite-size
effects will appear if we wait too long.

Figure 5: Zero-temperature fermion flow over time in finite systems with
different $N$ as in Fig. 4, for $\mu_{L}$ near the first conductivity step at
$\mu_{L}=-\sqrt{2}$, and times less than $t=2.5N$. Top row: $N=20$ sites in
the lattice ends, with initial Fermion numbers 3,4,5,6 from left plot to
right. Middle row: $N=40$, initial numbers 8,9,10,11. Bottom row: $N=40$,
initial numbers $19,20,21,22$. The different initial particle numbers all
correspond to $\mu_{L}$ near $-\sqrt{2}$; in particular the second column of
plots have $\mu_{L}$ just below $-\sqrt{2}$ and the third column have
$\mu_{L}$ just above it. Since the group velocities at frequencies near these
$\mu$ are near $\sqrt{2}$ rather than 2 as at $\omega=0$, finite-size effects
are not yet apparent. Finite-time effects, however, are clear: the initial
transient epoch lasts much longer near the conductivity resonance.

###  V.4 Quasi-stationary current versus $\mu_{L}$

While finite lattices may thus force finite-time problems upon us, though,
these problems may not actually be too severe. Fig. 6 shows finite-system
analogs of Fig. 2a), with three different cases of $N$ (20, 40, and 80 in
Figs. 6a), 6b), and 6c), respectively). Each of these three plots further
shows three different simple attempts to extract the quasi-steady current from
the growth of $\Delta\mathcal{N}(t)$ before the onset of finite-size effects
at $t=N/\tau$. These attempts represent $I$ as local slopes
$[\Delta\mathcal{N}(t_{m}+1/\tau)-\Delta\mathcal{N}(t_{m}-1/\tau)]/(2\tau)$,
for three different choices of the measurement time $t_{m}$: the early choice
$t_{m}=N/(4\tau)$, the late choice $t_{m}=3N/(4\tau)$, and the intermediate
choice $t_{m}=N/(2\tau)$. If $\Delta\mathcal{N}(t)$ were as straight as it is
in Fig. 4, at $\mu_{L}=0$, these three choices of $t_{m}$ would all yield very
similar $I$ estimates. Discrepancies between the three attempts to find $I$
show the non-stationary nature of the fermion flow at other $\mu_{L}$.

Figure 6: Zero-temperature current $I$ versus initial left-side chemical
potential $\mu_{L}=\mu$ computed at three different times before the onset of
finite-size effects, for each of three different finite lattice sizes. All
cases have $\tau_{1}=0.1$, $\gamma=0.5$, $M=3$, and initial $\mu_{R}<-2$ (no
particles in the right reservoir section). All three plots a), b), c) show the
same thick, gray curve, which is $I(\mu)$ for the steady state of the infinite
lattice with $\Lambda=1$ and $V=0$. In plots a), b), and c) we show cases with
$N=$ 20, 40, 80. Each plot shows $I$ computed as
$\big{(}[n_{R}(t+1)-n_{L}(t+1)]-[n_{R}(t-1)-n_{L}(t-1)]/2$ for $t=0.25N$ (o
markers), $t=0.5N$ (+ markers), and $t=0.75N$ (x markers). The markers are
discrete points rather than continuous lines because zero-temperature fermions
in a finite lattice simply occupy the $n$ lowest modes, for integer $n$
between 1 and $N$.

As Fig. 6 shows, it is possible to resolve the steady-state current
$I(\mu_{L})$ from Fig. 2a) fairly well with finite lattices as small as
$N=20$, and lattices as large as $N=80$ allow essentially duplicating the
steady-state current curve of the infinite system, with clear conductivity
steps that are noticeably smoother for the even-parity channels that are
affected by loss. Even though extended transient effects near the steps still
exist even for $N=80$, as shown in Fig. 5, this issue only really affects the
two data points on either side of the step, and for $N=80$ there are many
other points to map the curve out accurately.

Ultimately the main limitation from finite lattices turns out to be the
fundamentally limited frequency resolution from only having an $N$-dimensional
single-particle Hilbert space in an $N$-site lattice. As long as $N$ is larger
than $M$ by enough both to delay finite-size effects until the initial
transient epoch is mostly over and to resolve the $2M+1$ conductivity steps in
the frequency band $-2<\omega<2$, experiments with ultracold Fermi gases in
finite optical lattices can indeed reveal the same interesting transport
phenomena with a lossy site that have been predicted for finite chains of
quantum dots coupled to leads.

##  VI Discussion and Outlook

In conclusion, we have constructed a class of fermionic lattice gas models
which can reproduce the results of [2] in the limit $\Lambda\to\infty$, and
then represent quantum gas experiments in the limit $\Lambda\to 1$. Different
but similar effects are predicted for quantum gases in optical lattice, in
comparison with the parity-dependent conductivity steps that are predicted to
be seen in quantum dot chains between leads. Finite-temperature effects seem
likely to be a practical challenge in near-future experiments with quantum
gases. While these effects will not prevent observation of the most
interesting conductivity features, they will likely prevent quantum gas
experiments from matching the sharpness of conductivity steps that can be seen
in condensed matter systems.

One new possibility that can appear with quantum gases in lattices is
transport of bosons instead of fermions. In the absence of interactions,
however, bosons do not readily display the dramatic step-like conductivity of
zero-temperature fermions, because Bose-Einstein distributions cannot
replicate the moveable sharp edge of a low-temperature Fermi-Dirac
distribution with variable Fermi energy. For non-interacting bosons, the
chemical potential can rise no higher than the bottom of the single-particle
energy band; the only way to have varying exposure to the different discrete
transport channels through the finite middle chain is to vary the reservoir
temperatures.

Figure 7: Current $I$ through the finite chain of seven sites ($M=3$) as in
Fig. 2, but for non-interacting bosons at maximal chemical potential
$\mu_{L}=-2$, as a function of temperature. No conductivity steps are
apparent: while the sharp edge of the zero-temperature Fermi-Dirac
distribution rises with increasing $\mu$, increasing temperature in the Bose-
Einstein distribution merely spreads the distribution out flatter, steadily
overlapping more and more with all seven conductivity resonances.

As Fig. 7 disappointingly shows, increasing temperature simply mixes in
higher-frequency channels continuously. No conductivity steps can be seen—at
least not in scenarios like those we have considered in this paper, and not
without interactions among the bosons. Perhaps different kinds of experiments
will be able to show more abrupt unlocking of discrete transport channels for
bosons. Further studies may also reveal more interesting transport behavior in
interacting bosons, which can, among other complications, have higher chemical
potentials.

###### Acknowledgements.

The author thanks Corinna Kollath and A.-M. Visuri for very useful
discussions, and acknowledges support from the Deutsche Forschungsgemeinschaft
(DFG) through SFB/TR185 (OSCAR), Project No. 277625399. In particular the
essential ideas of representing lead contacts as weak links, and addressing
finite-size effects, are due to C. Kollath.

## References

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##  VII Appendix A: The single-particle wave functions
$\Psi^{L,R}_{m}(\omega)$

###  VII.1 Unnormalized eigenfunctions of definite parity for $|m|\leq M$

For $V\not=0$ our entire system lacks reflection symmetry, but there is always
reflection symmetry within the middle chain $|m|\leq M$. This makes it
straightforward to construct $\psi_{m}$ eigenvectors that are either odd or
even in $m$, for $|m|\leq M$. Once we have these solutions, we will be able to
combine them into left-incident and right-incident $\Psi^{L,R}_{m}$, and
finally normalize them.

The odd solutions are the simplest, since they vanish at $m=0$ and are
therefore unaffected by the imaginary potential term $\gamma$. They can simply
be written as

$\displaystyle\psi^{-}_{m}(\omega)$ $\displaystyle\underset{|m|\leq
M}{=}\sin\big{(}k(\omega)m\big{)}$ $\displaystyle\omega$
$\displaystyle=-2\cos(k)$ $\displaystyle k(\omega)$
$\displaystyle=\cos^{-1}\Big{(}-\frac{\omega}{2}\Big{)}\;.$ (31)

Note that we may need to consider $|\omega|>2$, for which $k(\omega)$ is
complex, because for $\Lambda>1$ the outer parts of the lattice support
eigenmodes in a wider bandwidth than that of the middle chain.

The modes that are even in the middle chain are a bit less trivial, because of
the complex potential at $n=0$:

$\displaystyle\psi^{+}_{m}(\omega)$ $\displaystyle\underset{|m|\leq
M}{=}\cos\big{(}k(\omega)m\big{)}-\frac{i\gamma}{4\sin\big{(}k(\omega)\big{)}}\sin\big{(}k(\omega)|m|\big{)}\;.$
(32)

How these eigenvectors behave for $|m|>M$ is then determined firstly by
solving (9) for $|m|>(M+1)$, which implies

$\displaystyle\psi_{n}^{\pm}(\omega)$
$\displaystyle\underset{|m|>M}{=}\alpha^{\pm}_{L,R}e^{ik_{L,R}(\omega)|m|}+\beta^{\pm}_{L,R}e^{-ik_{L,R}(\omega)|m|}\;,$
(33)

for $|m|>M$, with the subscript $L$ applying for $m<-M$ and $R$ for $m>M$, and
with $k_{L,R}(\omega)$ defined in (17) above.

To fix the separate left and right coefficients $\alpha^{\pm}_{L,R}$ and
$\beta^{\pm}_{L,R}$, we finally solve (9) for $|m|=M$ and $|m|=M+1$, which
provide two conditions at each end of the finite middle chain that are
analogous to the continuity conditions on the wave function and its first
derivative in continuous Schrödinger problems. After some algebra we obtain

$\displaystyle\psi_{m}^{-}\underset{|m|>M}{=}$
$\displaystyle\mathrm{sgn}(m)\Big{[}\frac{A_{L,R}}{2}e^{ik_{L,R}|m|}+\frac{A^{*}_{L,R}}{2}e^{-ik_{L,R}|m|}\Big{]}$
$\displaystyle\psi_{m}^{+}\underset{|m|>M}{=}$
$\displaystyle\Big{(}B_{L,R}-\frac{i\gamma}{4\sin(k)}A_{L,R}\Big{)}\frac{e^{ik_{L,R}|m|}}{2}$
$\displaystyle+\Big{(}B^{*}_{L,R}-\frac{i\gamma}{24\sin(k)}A^{*}_{L,R}\Big{)}\frac{e^{-ik_{L,R}|m|}}{2}$
(34)

where the $L$ subscripts apply for $m<-M$ and the $R$ subscripts for $m>M$.
The coefficients here are

$\displaystyle A_{L,R}(\omega)=$
$\displaystyle\frac{e^{-ik_{L,R}M}}{i\sin(k_{L,R})}$
$\displaystyle\times\left(\frac{1}{\tau_{1}}\sin[k(M+1)]-\tau_{1}\sin(kM)e^{-ik_{L,R}}\right)$
$\displaystyle B_{L,R}(\omega)=$
$\displaystyle\frac{e^{-ik_{L,R}M}}{i\sin(k_{L,R})}$
$\displaystyle\times\left(\frac{1}{\tau_{1}}\cos[k(M+1)]-\tau_{1}\cos(kM)e^{-ik_{L,R}}\right),$
(35)

where $k$ without $L,R$ subscripts is the middle-chain $k(\omega)$ from (24),
which is reproduced in (VII.1). Although we have the same $\Lambda$ for both
$m>M$ and $m<-M$, if $V\not=0$ then $k_{L}\not=k_{R}$ and so $\psi_{m}^{\pm}$
do not actually have definite parity for $|m|>M$.

###  VII.2 Left- and right-incident wave functions

We then simply construct linear combinations of $\psi^{\pm}_{m}$ which contain
no incoming wave on one side, and in which the amplitude of the incoming wave
on the other side is $1/\sqrt{2\pi}$. The result is

$\displaystyle\Psi^{L}_{m}(\omega)$
$\displaystyle=\frac{A^{*}_{R}\psi_{m}^{+}(\omega)-\Big{(}B_{R}^{*}-\frac{i\gamma}{4\sin(k)}A^{*}_{R}\Big{)}\psi_{m}^{-}(\omega)}{\sqrt{2\pi}Z}$
$\displaystyle\Psi^{R}_{m}(\omega)$
$\displaystyle=\frac{A^{*}_{L}\psi_{m}^{+}(\omega)+\Big{(}B^{*}_{L}-\frac{i\gamma}{4\sin(k)}A^{*}_{L}\Big{)}\psi_{m}^{-}(\omega)}{\sqrt{2\pi}Z}$
$\displaystyle Z(\omega)$
$\displaystyle=\frac{A^{*}_{R}B^{*}_{L}+A^{*}_{L}B_{R}^{*}}{2}-\frac{i\gamma}{4\sin(k)}A^{*}_{L}A^{*}_{R}$
(36)

which further implies

$\displaystyle\mathcal{T}$
$\displaystyle=\frac{i}{Z}\frac{\sin(k)}{\sqrt{\sin(k_{L})\sin(k_{R})}}$
$\displaystyle\mathcal{R}_{L}$
$\displaystyle=\frac{1}{2Z}\left(A^{*}_{R}B_{L}+B^{*}_{R}A_{L}-\frac{i\gamma}{2\sin(k)}A^{*}_{R}A_{L}\right)$
$\displaystyle\mathcal{R}_{R}$
$\displaystyle=\frac{1}{2Z}\left(A^{*}_{L}B_{R}+B^{*}_{L}A_{R}-\frac{i\gamma}{2\sin(k)}A^{*}_{L}A_{R}\right)\;.$
(37)

As we note in Section III of our main text, the dimensionless parameter
$\Lambda$ does not appear explicitly in the results for $\mathcal{R}_{L,R}$
and $\mathcal{T}$. $\Lambda$ does appear in Equation (17), however, which
defines $k_{L,R}(\omega)$, and so $\Lambda$ actually does affect
$\mathcal{R}_{L,R}$ and $\mathcal{T}$ substantially. In particular both for
$\Lambda=1$ and for $\Lambda\to\infty$ we obtain $k_{L}=k_{R}$ and the
distinctions between $A_{L,R}$, $B_{L,R}$ disappears, so that we can take
$\mathcal{R}_{L,R}\to\mathcal{R}$ as in our main text.

###  VII.3 Infinite $\Lambda$ limit

As we note in Section IV of our main text, the $\Lambda\to\infty$ limit which
makes the reservoir density of states constant over the middle chain bandwidth
lets us replace $k_{L,R}\to\pi/2$. Our expressions for $Z$, $\mathcal{T}$, and
$\mathcal{R}_{L,R}$ then simplify greatly. We obtain

$\displaystyle A_{L,R}\underset{\Lambda\to\infty}{\longrightarrow}$
$\displaystyle(-i)^{M+1}\left(\frac{\sin[k(M+1)]}{\tau_{1}}+i\tau_{1}\sin(kM)\right)$
$\displaystyle B_{L,R}\underset{\Lambda\to\infty}{\longrightarrow}$
$\displaystyle(-i)^{M+1}\left(\frac{\cos[k(M+1)]}{\tau_{1}}+i\tau_{1}\cos(kM)\right)$
$\displaystyle|Z|^{2}\underset{\Lambda\to\infty}{\longrightarrow}$
$\displaystyle\left[\frac{\sin[k(M+1)]\cos[k(M+1)]}{\tau_{1}^{2}}+\tau_{1}^{2}\sin(kM)\cos(kM)\right]^{2}$
$\displaystyle+\left[\sin(k)+\frac{\gamma}{4\sin(k)}\left(\frac{\sin^{2}[k(M+1)]}{\tau_{1}^{2}}+\tau_{1}^{2}\sin^{2}(kM)\right)\right]^{2}$
$\displaystyle|\mathcal{T}|^{2}\underset{\Lambda\to\infty}{\longrightarrow}$
$\displaystyle\frac{\sin^{2}(k)}{|Z|^{2}}$
$\displaystyle|\mathcal{R}_{L,R}|^{2}\underset{\Lambda\to\infty}{\longrightarrow}$
$\displaystyle\frac{1}{|Z|^{2}}\left[\left(\frac{\sin[k(M+1)]\cos[k(M+1)]}{\tau_{1}^{2}}+\tau_{1}^{2}\sin(kM)\cos(kM)\right)^{2}\right.$
$\displaystyle+\left.\frac{\gamma^{2}}{16\sin^{2}(k)}\left(\frac{\sin^{2}[k(M+1)]}{\tau_{1}^{2}}+\tau_{1}^{2}\sin^{2}(kM)\right)^{2}\right]\;,$
(38)

as used in our main text.

##  VIII Appendix B

As we noted in Section III of our main text, for the class of models that we
have defined it can in general happen that one of $k_{L,R}$ is complex—or even
that both of them are. This can happen for one of $k_{L,R}$, for $V>0$,
because the bandwidth $4\Lambda\tau$ of the outer lattice ends is finite, and
so a relative shift of $V$ between the bands of the outer ends means that they
do not fully overlap. Wherever an $\omega$ is in the continuous part of the
infinite model’s spectrum, because it is in the band of one lattice end,
$k_{L,R}$ will be purely imaginary for $\omega<-2\Lambda\pm V/2$, below the
corresponding lattice end’s frequency band, and it will be $\pi$ plus an
imaginary term for $\omega>2\Lambda\pm V/2$, above the band.

It can also happen that both $k_{L,R}$ are complex, for a discrete set of
complex $\omega$ that are in the spectrum of our non-Hermitian Hamiltonian
which includes the imaginary potential at $m=0$, if $\Lambda<1$ so that the
middle chain possesses eigenstates outside the bands of both the outer lattice
ends.

We have only considered $\Lambda\geq 1$ in this paper, leaving other cases for
future work. In fact we have only explicitly considered cases either with
$V>0$ but $\Lambda\to\infty$, so that $k_{L,R}$ are both real for all $\omega$
that are involved in transport through the middle chain, or else with
$\Lambda=1$ but $V=0$, so that all $k_{L,R}$ are real. For the benefit of
future work, however, we note that our formulas extend straightforwardly to
include complex $k_{L,R}$, when they are taken in the general forms shown in
Appendix A above, and when the observables include the $X_{L,R}$ factors as in
Appendix C below.

If for example $k_{R}(\omega)$ has an imaginary part for some $\omega$ but
$k_{L}(\omega)$ is real, we can keep our $\Psi_{n}^{L}(\omega)$ as described
above, with the stipulation that we take the positive imaginary branch of
$k_{R}$, but there will simply be no $\Psi_{n}^{R}$ for this $\omega$, because
the “incoming” wave would be exponentially growing. It will also be important
to note that the evanescent transmitted waves in these cases of
$\Psi_{n}^{L}(\omega)$ carry no particle or energy currents to $n\to+\infty$
even though $|\mathcal{T}|>0$.

##  IX Appendix C

####  IX.0.1 Particle current with complex $k_{L,R}$

In the general case where $k_{L,R}$ are not equal, and may be complex, it is
necessary to modify our main text’s formula (23) for the particle current
through the middle chain, by distinguishing $\mathcal{R}_{L}$ and
$\mathcal{R}_{R}$, and by including additional factors $X_{L,R}(\omega)$:

$\displaystyle I=$
$\displaystyle\frac{1}{4\pi\tau_{1}^{2}}\int\\!d\omega\,\left[f_{L}(\omega)\Big{(}1+X_{L}|\mathcal{T}|^{2}-|\mathcal{R}_{L}|^{2}\Big{)}\right.$
$\displaystyle\quad\quad\quad\quad-\left.f_{R}(\omega)\Big{(}1+X_{R}|\mathcal{T}|^{2}-|\mathcal{R}_{R}|^{2}\Big{)}\right]\;.$
(39)

The factor $X_{L}$ here is equal to one if $k_{R}$ is real, and is zero if
$k_{R}$ has a non-zero imaginary part; $X_{R}$ depends in the same way on
whether $k_{L}$ is purely real. These factors do not have to be inserted by
hand, but come out simply in evaluating $J_{m}$ in the cases where $k_{L}$ or
$k_{R}$ is either imaginary or imaginary plus $\pi$.

The subscripts are not accidentally switched: the factor $X_{L}$ depends on
$k_{R}$, and _vice versa_ , because the issue is whether or not the wave which
is transmitted through to the right from the left, or _vice versa_ , is
evanescent or propagating.

####  IX.0.2 Local energy at site $m$

The Hamiltonian (1) is a sum over sites $m$, and apart from site $m=0$, where
there is an imaginary potential, each term in the $m$-sum in (1) is an
observable that can be interpreted as the local energy associated with site
$m$; we can treat each site’s two neighbors symmetrically by averaging their
contributions. The expectation value of this local energy is then

$\displaystyle E_{m}=$
$\displaystyle\hbar\tau\Big{(}-\frac{T_{m}}{2}\big{(}R_{m+1,m}+R_{m,m+1}\big{)}$
$\displaystyle-\frac{T_{m-1}}{2}\big{(}R_{m,m-1}+R_{m-1,m}\big{)}+V_{m}R_{mm}\Big{)}$
(40)

according to (1).

####  IX.0.3 Energy current

Just as the particle number current $J_{m}$ could be inferred from the time
derivative of the average local number $\bar{n}_{m}$, we can also compute from
(II.3.2) that

$\displaystyle\frac{d}{dt}E_{m}$
$\displaystyle=-(Q_{m+1}-Q_{m})-\frac{\gamma\tau}{2}E_{0}(\delta_{m,-1}+\delta_{m0})$
$\displaystyle Q_{m}$
$\displaystyle=-i\frac{\hbar\tau^{2}}{2}T_{m-1}\Big{[}T_{m}(R_{m+1,m-1}-R_{m-1,m+1})$
$\displaystyle\qquad+T_{m-2}(R_{m,m-2}-R_{m-2,m})$
$\displaystyle\qquad-(V_{m}+V_{m-1})(R_{m,m-1}-R_{m-1,m})\Big{]}\;,$ (41)

and recognize $Q_{m}$ as the local energy current. In the steady state
$\dot{E}_{m}=0$ then implies that $Q_{m}$ is piecewise constant,

$Q_{m}=\frac{\tau}{4\pi}\Big{(}Q-2\pi\gamma\,\mathrm{sgn}(m)E_{0}\Big{)}\;,$
(42)

with $Q_{0}=Q$ also being the average energy current through the finite middle
chain from left to right.

Evaluating $Q=\lim_{m\to\infty}(Q_{-m}+Q_{m})/2$ in (10) and using (9) then
reveals a result which we might well have guessed without even thinking much
about what it meant: the average energy current in the stationary state is a
similar integral to the integral (23) for the average particle current $I$,
with the integrand simply multiplied by $\hbar\tau\omega$.

$\displaystyle Q=$
$\displaystyle\frac{\hbar\tau^{2}}{4\pi}\int\\!d\omega\,\omega\left[f_{L}(\hbar\tau\omega)\Big{(}1+X_{L}|\mathcal{T}|^{2}-|\mathcal{R}_{L}|^{2}\Big{)}\right.$
$\displaystyle\qquad\left.-f_{R}(\hbar\tau\omega)\Big{(}1+X_{R}|\mathcal{T}|^{2}-|\mathcal{R}_{R}|^{2}\Big{)}\right]\;.$
(43)