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quantum pumps, S-matrix, adiabatic, dissipation, entropy production
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            $\scriptscriptstyle |$}}\kern-.40em{\rm C}}
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\title{Optimal Quantum Pumps}
\begin{document}
\input epsf
%%%%%%%%%%%%%%%%%
\tightenlines
\author{ J.~E.~Avron ${}^{(a)}$, A. Elgart ${}^{(b)}$, G.M. Graf ${}^{(c)}$
and L. Sadun ${}^{(d)}$}
 \draft
\maketitle
\centerline{ ${}^{(a)}$ Department of Physics, Technion, 32000 Haifa, Israel} 
\centerline{${}^{(b)}$ Department of Physics, Jadwin Hall, Princeton 
University, Princeton, NJ 08544, USA} 
\centerline{${}^{(c)}$ Theoretische Physik, ETH-H\"onggerberg, 8093 Z\"urich,
Switzerland} \centerline{ ${}^{(d)}$ Department of Mathematics,
University of Texas, Austin Texas 78712, USA}
%%%%%%%%%%%
\begin{abstract}
Optimal pumps  saturate a lower bound on energy dissipation. We
give a characterization of optimal pumps in terms of the
\emph{energy shift} matrix, which is computed from
the on-shell, frozen, scattering matrix. The energy shift
determines the charge transport, the dissipation and entropy
production in adiabatic pumps. We  give a geometric
characterization of optimal pumps and discuss an example.
\end{abstract}
\pacs {PACS numbers: 72.10.Bg, 73.23.-b}
%\newpage
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\begin{multicols}{2}
%\bigskip
\narrowtext
{\bf Overview:} Quantum pumps transport charge, spin,
energy, entropy, etc. in and out of electron reservoirs which are
themselves at equilibrium and at identical chemical potentials,
$\mu$, and identical temperatures, $1/\beta$. The pumping is done
by scattering from a time dependent scatterer
\cite{bpt,marcus,b,aa,saa,aegs,levi,br}.



Let $\dot E$ denote the net energy flux (power) and $\dot Q$ the charge
flux (current) in a given channel, flowing out from the scatterer to a 
distant reservoir. Let $R_k= h/e^2\approx 25812.80\,\Omega$ be the (von
Klitzing) quantum unit of resistance. If the reservoir is held at 
zero temperature, we shall prove an elementary
lower bound on the power dissipated at each and every channel:
\begin{equation}\label{lb}
\dot E-  {\mu  \over e}\dot Q \ge \frac {R_k} 2\, \dot Q^2.
\end{equation}
If $R_{k}/2$ is interpreted as the
contact  resistance of each channel, the right hand side gives  
the dissipation at the contact. This bound does not depend on the adiabaticity
of the pump. It is a consequence of the fact that a channel is one
dimensional, that the charge carriers are non-interacting fermions
and the reservoirs are at zero temperature. Channels that saturate
the lower bounds Eq.~(\ref{lb}) will be called optimal. An pump 
is called optimal if all of its channels are optimal.

We henceforth use units with $e=\hbar=1$, so $R_k=2\pi$ and the
chemical potential can be interpreted as a voltage. 

Adiabatic pumps admit an explicit characterization of optimal
channels in terms of their scattering matrix. A pump is adiabatic
when the frequency $\omega$ of the pump is slow compared with the natural
time scale  $\tau$ of the problem, \cite{comment}, e.g., the Wigner time 
delay \cite{w}.  That is, $\varepsilon = \omega \tau \ll 1$.

Let ${\cal S}(t;\mu)$ be the (time independent) scattering matrix
at energy $\mu$, for a scatterer frozen  at time $t$. In the case
of $n$ channels, ${\cal S}(t;\mu)$  is an $n\times n$ unitary
matrix. $\dot{\cal S}(t;\mu)=O(\varepsilon)$ denotes the $t$
derivative of the matrix. The \emph{energy shift} matrix, ${\cal E}$, is
defined to be:
\begin{equation}\label{es}
  {\cal E}(t,\mu)= i\hbar\, \dot {\cal S}(t,\mu) {\cal S}^\dagger(t,\mu),
\end{equation}
and is of order $\varepsilon$. 
The matrix elements of the energy shift can be expressed as:
\begin{equation}\label{m}
{\cal E}_{jk} = i\hbar\, \langle \psi_k|\dot \psi_j\rangle
\end{equation}
where $|\psi_j\rangle$ is the $j$-th row (not column!) of the scattering 
matrix.
The energy shift is conjugate to the Wigner time delay ${\cal
T}(t,\mu)= -i\hbar\, \partial_\mu{\cal S}(t,\mu) {\cal
S}^\dagger(t,\mu)$, \cite{w}.

The energy shift plays a role in adiabatic pumping. For example,
it is known that the diagonal matrix elements of ${\cal E}$
determine the net current entering the reservoir through the $j$-th
channel \cite{bpt,b,aegs}
\begin{equation}\label{q}
\dot Q_j= {\cal E}_{jj}/2 \pi + O(\varepsilon^2).
\end{equation}
As we shall see, the
energy-shift also provides information on  dissipation and noise.

For a channel connected to a reservoir at zero temperature we
shall show:
\begin{equation}\label{e}
\dot E_j -\mu\dot Q_j= \frac 1 {4 \pi} \sum_{k}\vert{\cal
E}_{jk}\vert^2 + O(\varepsilon^3).
\end{equation}
It follows that the
$j$-th channel is optimal if ${\cal E}_{jk}=0$ for $j\neq k$, and
the pump is optimal if and only if ${\cal E}$ is a diagonal matrix.

The scattering matrix of an optimal pump is 
${\cal S}(t)= U_d(t){\cal S}_0$
where $U_d(t)$ is a \emph{diagonal} unitary and ${\cal S}_0$ a
constant unitary matrix. This  coincides with Andreev and
Kamenev's characterization of noiseless pumps \cite{ak,ll}.

The notion of optimal pumps {is} geometric. This is seen from the
fact that the vanishing condition on matrix elements of the energy
shift, Eq.~(\ref{m}) is invariant under reparameterization of
time.

For $n$-channel scattering, the space of Hermitian matrices ${\cal
E}$ has $n^2$ real dimensions, while the space of diagonal
matrices is $n$ dimensional. Optimal pumps are therefore a set of
codimension $n^2-n$. In particular, pumps with a single channel
are automatically optimal.  Below we shall give an example of an
optimal pump with $n=2$ that models the quantum Hall effect.



Optimal channels are related to integral charge transport: If the
$j$-th channel is optimal, then the charge transport in a cycle of
the pump is an integer. This can be seen from Eq.~(\ref{q}):
\begin{equation}\label{psi}
Q_j=  -\frac i {2\pi}\, \int \langle \psi_j|\dot \psi_j\rangle
\,dt= -\frac i {2\pi}\, \oint \langle \psi_j|d \psi_j\rangle.
\end{equation}
For an optimal channel  $|\dot\psi_j\rangle $ is parallel to
$|\psi_j\rangle$,  since $\langle \psi_k|\dot \psi_j\rangle =0$,
hence  it can only accumulate a phase along the path. Since
$|\psi_j\rangle$ is single valued, the total phase accumulated on
a closed path must be an integer multiple of $2\pi$. It is worth remarking
that in this case Eq.~(\ref{psi}) not only expresses the expectation value of
the charge transport in a cycle, as it does by virtue of its derivation, but 
also the actual charge transport, since the stated condition implies absence of
noise, i.e., variance. The right
hand side of Eq.~(\ref{psi}) shows that charge transport is
geometric: It depends on  the  path but is independent of its
parameterization. In spite of all of this, the charge transport is
not quantized and has no topological content: A small deformation
of the scattering matrix will be a deformation away from
optimality and will deform the charge transport away from the
integers. In \cite{aa,saa} it is argued that non-dissipative
processes make quantum pumps similar to systems with energy gaps
where charge transport is related to topological invariants.
Although the conclusion about integrality of transport is correct
for optimal pumps, the argument is  misleading because the charge
transport in pumps is not associated with topological invariants.
Moreover, it is not possible on the basis of the order of
dissipation alone to distinguish between optimal and non-optimal
channels; in either case the dissipation is $O(\varepsilon^2)$. It
is however true that absence of \emph{excess} dissipation
 implies integrality.


The dissipation formulas admit a geometric interpretation.
$\ket{\psi_j}$, being normalized, can be viewed as a point on the
sphere $ S^{2n-1}\subset\C^n$. The Hopf map \cite{hopf} $\pi:
S^{2n-1}\to \C P^{n-1}$, which `forgets' the phase of
$|\psi\rangle$, turns $S^{2n-1}$ into a fiber (circle) bundle with
base space $\C P^{n-1}$.  The $j$-th row of ${\cal E}$ describes
the velocity of $\ket{\psi_j}$ in $S^{2n-1}$.  Of this, ${\cal
E}_{jj}$ is the projection of this velocity onto the fiber --- the
changing phase of $\ket{\psi_j}$ --- while the matrix elements 
${\cal E}_{jk}$, with $k \ne j$,
give the projection of this velocity onto $\C P^{n-1}$. The
current $\dot Q_j$, and the minimal dissipation $|{\cal
E}_{jj}|^2/4\pi$, are both functions of motion in the fiber, while
the excess dissipation is the ``energy'' (that is, squared
velocity) associated with motion in the base.


Finally, we remark, without proof \cite{aegs2}, that the entropy
flow into a cold reservoir,  $\beta \gg \hbar/\tau$,  from a slowly
operated pump,  $\hbar\omega\ll\beta^{-1}$ \cite{comment-scales}, is proportional to the 
excess energy dissipation, and not to the total energy dissipation:
\begin{equation}\label{entropy}
\dot S_j= \frac \beta {4\pi} \sum_{k\neq j}\vert{\cal
E}_{jk}\vert^2.
\end{equation}

 {\bf A lower bound on dissipation:}
We first consider a single channel and let $\dot E_\pm$ ($\dot
Q_\pm$) be the energy (charge) current transported across
it by the outgoing ($+$) or incoming ($-$) carriers separately.
Then
\begin{equation}
\dot E_\pm\ge \pi\dot Q_\pm^2, \label{sl}
\end{equation}
with equality holding if and only if the carriers fill a Fermi sea
up to chemical potentials $\mu_\pm$. Here is why. Let $0\le n(k)\le
1$ be the filling of states at momentum $k$ and energy $\epsilon(k)$,
which move with velocity $\epsilon'(k)$. Then
\begin{eqnarray}
\dot Q_+&=&\frac{1}{2\pi}\int_{\epsilon'(k)>0}dk\, n(k)
\epsilon'(k),\nonumber \\
 \dot E_+&=&\frac{1}{2\pi}\int_{\epsilon'(k)>0}dk\, n(k)
\epsilon(k)\epsilon'(k). \end{eqnarray}
 The bathtub
principle (w.r.t. the measure $\epsilon'(k)dk$ on
$\{k\mid\epsilon'(k)>0,\}$; see e.g., \cite{ll}) states that for
fixed $\dot Q_+$ the quantity $\dot E_+$ is minimized by
$n(k)=\theta(\mu_+-\epsilon(k))$ for some $\mu_+$. Assuming 
$ k\epsilon'(k)\ge 0,\, \epsilon(0)=0,$   we find for the minimizer
\begin{equation}
 \dot Q_+=\frac{\mu_+}{2\pi},\qquad \dot
E_+=\frac{\mu_+^2}{4\pi}, \label{min} \end{equation} which
proves (\ref{sl}). 



Consider next a channel connected to a reservoir at chemical
potential $\mu_{-}$.  Let $\dot E=\dot E_+-\dot E_-$ 
($\dot Q = \dot Q_+ - \dot Q_-$) be the net
energy (charge) flowing into the reservoir (i.e., out of the system). 
Then $\dot E_- = \pi \dot Q_-^2 = \mu_-^2/4\pi$, while $\dot E_+$ is at 
least $\pi \dot Q_+^2$.  We compute
\begin{eqnarray}
\dot E\ge \pi(\dot Q_+^2-\dot Q_-^2) =
\pi\dot Q^2+\mu_-\dot Q.
\end{eqnarray}
This is Eq.~(\ref{lb}) in natural units, and  equality occurs if and only if the flow into the reservoir
also fills a Fermi sea with Fermi energy $\mu_+$. Note that (\ref{lb})
gives the dissipation in each lead\cite{comment2}.


{\bf Dissipation in adiabatic pumps:}
To derive Eq.~(\ref{e}), we note that the  left hand side of
Eq.~(\ref{e}) can be written as
\begin{equation}\label{in-out} \dot
E_j-\mu\dot Q_j=\frac{1}{2\pi}\int dE (E-\mu)\big(
n_{+j}(E)-n_-(E)\big).
\end{equation}
where  $n_{-}(E)= \theta (E-\mu)$ is the distribution of the
electrons that arrive from the (zero temperature) reservoirs and
$n_{+j}(E)$ is the distribution of the electrons that enter the
reservoirs. The pump scrambles the incoming distribution and
produces a (non-thermal) outgoing distribution $n_{+j}(E)$. To
calculate the outgoing distribution is a problem in adiabatic
scattering theory.

Our basic tool is a formula which relates outgoing and incoming
distribution in the adiabatic limit. Let $\ket{e_k}$ denote the
eigenvectors of ${\cal E}$ with eigenvalue $e_k$. The outgoing
density at energy $E$, on the $j$-th channel, is
\begin{equation}\label{io}
  n_{+j}(E)\approx \sum_k\, n_{-}(E-e_k)|\braket{e_k}{j}|^2,
\end{equation}
where $\approx$ means that this is an equality for distributions, to
leading order in $\varepsilon$. Inserting this in
Eq.~(\ref{in-out}) we obtain
\begin{equation} \dot E_j-\mu\dot
Q_j=\frac{1}{4\pi}\sum_{k=1}^n\, e_k^2 |\braket{e_k}{j}|^2
=\frac{1}{4\pi}\sum_{k=1}^n|{\cal E}_{jk}|^2. \label{diss1}
\end{equation}

{\bf Adiabatic scattering:}
It remains to derive Eq.~(\ref{io}). We start
with the standard point of view \cite{krein} of time-dependent
scattering theory, which views the S-matrix as a comparison of a
``free'' dynamics, generated by $H_0$ and the interacting
dynamics, generated by $H(t)$. For example, we can pick for $H_0$
the Hamiltonian associated with disconnected channels.  Let
$U(t,s)$ and $U_0(t,s)$ denote the corresponding evolutions. Then,
the S-matrix is defined by the limit (which we assume exists)
\begin{equation}\label{S}
  {\cal S}_d(t)=\lim_{T\to\infty} U_0(t,T)U(T,-T)U_0(-T,t).
\end{equation}
In the absence of a scatterer, $U=U_0$ and  ${\cal S}$ is the
identity, as it should be. When the scatterer is time independent,
$H(t)=H$,  the existence of the $T\to \infty$ limit for the
factors  $U(0,-T)U_0(-T,t)$ and $U_0(t,T)U(T,0)$ implies that
${\cal S}_d(t)$ is independent of $t$, as it should.

${\cal S}_d(t)$ maps incoming wave packet to outgoing ones. When
applied to a wave packet near the scatterer, it describes the
mapping for an incoming wave, originating at the reservoirs, that
hits the scatterer at time $t$.

It follows  from  Eq.~(\ref{S}) that, provided the limit exists,
\begin{equation}\label{em}
 i\, \dot{\cal S}_d(t)= [H_0,{\cal S}_d(t)].
\end{equation}
The free dynamics controls when a wave packet hits the scatterer.
Eq.~(\ref{em}) determines $n_+(H_0)$ to be
\begin{equation}\label{dotS}
{\mathcal S}_d(t)n_-(H_0){\mathcal S}_d(t)^\dagger=
n_-\big(H_0-i\dot{\mathcal S}_d(t){\mathcal S}_d(t)^\dagger\big).
\label{out}
\end{equation}
In the  time independent case,  $\dot {\cal S}_d=0$, and
Eq.~(\ref{dotS}) is an expression of  conservation of energy. For
time dependent scattering, Eq.~(\ref{dotS}), describes the
scattering out of the energy shell. This is the motivation for
calling $i\dot{\mathcal S}_d(t){\mathcal S}_d(t)^\dagger$ the operator
of energy shift.

In the adiabatic limit, the (exact) time dependent scattering
matrix ${\cal S}_d(t)$ can be approximated by the time independent
scattering matrix for the scatterer frozen at the time of the
scattering $t$. Namely, ${\cal S}_d(t)\approx
\delta(\mu-\mu')\,{\cal S}(t;\mu)$, with ${\cal S}(t;\mu)$ the
on-shell scattering matrix of Eq.~(\ref{es}). For the on-shell
densities Eq.~(\ref{dotS}) takes the form
\begin{equation}\label{dotSf}
n_+(\mu,t)\approx n_-\big(\mu-{\mathcal E}),
\end{equation}
which is a matrix version of Eq.~(\ref{io}).


{\bf A 2-lead optimal pump:}
We  conclude with an example of a nontrivial optimal pump. Besides
demonstrating the possibility of optimal pumps with multiple leads,
this example serves as a schematic of the quantum Hall effect.

The pump, shown in Fig.~({\ref{qhe}}), has two channels, each
connected to a reservoir on one end and to a loop of circumference
$\ell$ on the other.  The loop is threaded by magnetic flux $\Phi$
which is the engine of the pump. The scattering at the vertices is a
permutation matrix corresponding to the arrows in the
figure. Particles that enter from the left go clockwise around the
loop and exit to the left, while particles that enter from the right
go counterclockwise and exit to the right.  The scattering matrix of
the pump is therefore diagonal, with the phases of the two
reflection coefficients determined by $\Phi$:
\begin{equation}
 r= e^{i(k\ell+\Phi)}, \quad 
r'=e^{i(k\ell-\Phi)}. 
\end{equation}

\begin{figure}[h]
\centerline{\epsfxsize=3truein 
\epsfbox{tandem1.ps}}
\caption{A graph associated with the quantum Hall effect.}
\label{qhe}
\end{figure}

{From}  Eq.~(\ref{psi}), we see that increasing $\Phi$ by a unit of quantum
flux draws one particle in from the right and expels one particle
to the left.  This 
is independent of the chemical potential $\mu$ of the bath, 
and independent of the circumference $\ell$ of the loop.


At first glance this seems impossible, since there is no transmission
across the scatterer.  However, there is a simple semiclassical
explanation.  Increasing the flux creates an EMF around the loop that
accelerates the clockwise-moving particles and decelerates the
counterclockwise-moving particles.  Thus the outgoing states on
the left lead are filled to a higher energy than the outgoing states
on the right.  Furthermore, some low-energy counterclockwise-movers
that entered from the right are actually turned around by the EMF and
become clockwise-movers, after which they emerge to the left. This
accounts for the net transfer of charge from right to left.  Since the
scattering at each vertex is deterministic, the outgoing channels 
have no entropy, the inequalities (9) become equalities,
and the pump is optimal.

The quantum Hall effect can be described by a scatterer with four
leads (north, south, east and west) with a north-south voltage and an
east-west current.  However, if the the north and south leads are
connected by a wire, and if the resulting loop is threaded by a
time-varying magnetic flux to generate the north-south voltage, then
one obtains a geometry shown in Fig.~({\ref{qhall}}).
\begin{figure}[hall]
 \centerline{\epsfxsize=3truein
 \epsfbox{hall-bw.ps}
 }
\caption{The Hall effect as a pump driven by the emf $\dot \Phi$.}
\label{qhall}
\end{figure}


Indeed, the pump of Figure 1  models the essential features of the integer
QHE, \cite{dolgo}.  The Hall crystal, and the magnetic field applied to the Hall
crystal, are modeled by the vertices, which scatter particles in a
time-asymmetric manner.  The ``clockwise movers'' of Figure 1
correspond to electrons that enter the Hall crystal from the west,
move along the edge of the crystal until they reach the south lead, go
along the loop from south to north, move along the edge from north to
west, and emerge to the west.  The ``counterclockwise movers''
correspond to electrons that go from east to north (along the Hall
crystal) to south (along the loop) and then to east and out the east
lead.   By standard arguments \cite{fgw}, the edge states  reflect the  existence of localized bulk states 
in the crystal.



{\bf Concluding remarks:}
Adiabatic theory divides into two  chapters: One deals  with
systems with \emph{discrete spectra} and the other with systems
with \emph{continuous spectra}.  This division carries
over to the theory of adiabatic transport.  For systems with discrete 
spectra interesting transport properties often require that the 
system is multiply connected, see e.g., \cite{as};  
%The Thouless pump \cite{thouless-pump} is an example of a system that 
%has band spectrum, but can, nevertheless, be analyzed by methods that 
%normally apply to systems with discrete spectra\cite{footnote}.
charge transport admits a geometric interpretation, and 
is related to Chern numbers. Here 
we have analyzed quantum pumps as a problem in adiabatic scattering 
theory. It is known that charge transport in a closed cycle of a pump 
is,
again, geometric\cite{b,aegs}. But, unlike the discrete situation, 
charge transport is not quantized in the sense that it is not related 
to interesting topological invariants. Here we introduced the notion 
of optimal quantum pumps, which is again a geometric notion. 
Optimal pumps saturate a 
bound on dissipation, transport integer charges in closed cycles, 
are noiseless and minimize entropy production. We gave an explicit 
characterization of optimal pumps in terms of the matrix of
\emph{energy shift}.  


 {\bf
Acknowledgment:} We thank B. Altshuler, A. Kamenev, M. Reznikov, and U. Sivan
for discussion. The work was partially supported by 
the Israel Science Foundation and the Texas Advanced Research 
Program and the fund for the promotion of Research at the Technion.




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scatterers the two time scales coincide. If the scatterer is 
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To avoid this difficulty one  imposes the adiabaticity condition 
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%\bibitem{footnote} The Hamiltonian associated with Thouless
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%spectra.

\end{thebibliography}

\end{multicols}


\end {document}


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