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\def \cntl {\centerline}
\def\schrod{Schr\"odinger }
\def\no{\noindent}
\def\vx{\vec x}
\def\PRL{ Phys.~Rev.~Lett.~}
\def\PR{ Phys.~Rev.~}
\def\WS{Wannier-Stark }
\def\qual{1}
\def\diric{2}
\def\rrremark{3}
\def\wsl{4}
\def\cfks{5}
\def\jh{6}
\def\tp{7}
\def\remark{8}
\def\rremark{9}
\def\zak{10}
\def\starkg{11}
\def\starkd{12}
\def\rings{13}
\def\gt{14}
\def\pa{15}
\def\gta{16}
\def\bel{17}
\def\dss{18}
\def\GLG{19}
\def\rem{20}
\def\vxc{\vec {\rm x}_c}
\def\L{$\Lambda$ }
\def\half{${\scriptstyle {1\over 2}}$}
\def\da{\vec\nabla\times \vec a}
\def\O{$\Omega$ }
\def\Op{$\Omega$. }
\def\Oc{$\Omega^c$ }
\def\Ocp{$\Omega^c$. }
\def\ket{\langle\psi|}
\def\U{{\bf U}}
\def\K{{\bf K}}
\def\bra{|\psi\rangle}
\def\today{\ifcase\month\or
January\or February\or March\or April\or May\or June\or
July\or August\or September\or October\or November\or
December\fi
\space\number\day, \number\year}
{\nopagenumbers
\headline={\ifnum \count0=1 { \ninerm\hfil
\today}\else\hfil\fi}
\footline={\ifnum \count0=1 {PACS: 72.10.Bg \hfil}}
{~}
\vskip 2. cm
\twelvermb
\cntl{Quasienergies, Stark Hamiltonians and Growth of Energy}
\cntl{for Driven Quantum Rings}
\tenrm
\bigskip \vskip 2.0cm \cntl{
Joseph E.
Avron and Jonathan~Nemirovsky} \cntl{Department of Physics}
\cntl{Technion - Israel Institute of Technology}
\cntl{Haifa, 32000, Israel}
\bigskip
\vskip 1.8cm
We study time-dependent
\schrod operators in Aharonov-Bohm geometries
where the flux threading the hole increases linearly with time.
We show
that the the quasienergy operator has, in these cases, the
same spectrum as the {\it time independent} Stark
Hamiltonian on the {\it universal covering space}.
Combining known results on Stark Hamiltonians
with a
theorem of Bellissard, we prove that the energy
of a particle on a
{\it finite} ring, with smooth background potential, increases
without bound as $t\to\infty$. \vfill\eject }
\schrod operators that describe quantum particles
acted on by external {\it
time-independent} electric and magnetic fields may or may not
be time independent. When the domain is simply connected,
for example, for the Euclidean space, there is
always a
choice of gauge so that the \schrod operator is
time-independent. We
shall call this the {\it Stark
form} of the operator. When the domain is not
simply connected, as for example in the case of Aharonov-Bohm
geometry, there is no
time-independent form for the operators
describing (interacting or noninteracting) particles driven by a
time
independent electromotive force due to a linearly increasing flux
threading the hole [\qual ].
The Hamiltonians of Classical Mechanics, in contrast, can always
be
brought into a time-independent form when the external fields
are time-independent. The price one
has to
pay for doing so is to replace the domain \O with a hole by its
(universal) covering space
$\Omega^c$, which is
simply connected and which is where the static potential lives.
Recall that the (universal) covering spaces of the circle and
the annulus are the infinite line and
the infinite helix, respectively.
The \schrod operators associated to
time-independent electric and
magnetic fields on the (universal) covering space
can always be brought into
a Stark form.
However, (unlike the case in Classical Mechanics)
it is not a-priori clear what these operators have to do
with
the original problem on the multiply connected domain \Op
Our purpose here is to show
that the
Stark operator on the (universal) covering space is the
quasienergy operator
for the time-dependent operator on \Op
(This will be explained in
some detail below). We shall then apply this observation to give
a
proof which is both elementary and rigorous of the fact that the
energy
of a particle on a driven thin ring increases without bound for a
large class of background potentials and {\it all} initial
conditions.
For the sake of simplicity let us focus on the one particle
\schrod operator without magnetic fields on \O [\diric ]: $$H(t)
={1\over 2m} \Big(-i\hbar \vec\nabla - t \vec a(\vx)\Big)^2
+V(\vec
x),\quad \vec\nabla\times\vec a = 0,\quad
\vx\in\Omega.\eqno(1)$$
The
electric field, $e \vec E (\vx ) = \vec a(\vx) -\vec\nabla V(\vx
)$,
and magnetic field, (which vanishes identically on $\Omega$),
are both time independent.
The holonomy of $\vec a$
represents the magnetic flux threading the hole
which is linearly increasing with time. So, we may take the
flux to be our clock, and choose the unit of time so that after
$t=1$ the flux through the hole increased by the unit of quantum
flux.
This sets the normalization of the loop integral of the vector
potential:
$\oint\vec a(\vx)\cdot d\vx = 2\pi\hbar $.
Since $\da = 0$, $\vec a$ is locally the gradient of a function:
$$\Lambda (\vxc)
\equiv \int^{\vxc} \vec a(\vx ')\cdot
d\vx '.\eqno(2)$$
$\Lambda (\vxc)$ is (in general)
a function on the (universal) covering space $\Omega^c$, where
the Hamiltonian $H(t)$ is related by a gauge
transformation to a time independent Stark Hamiltonian.
Indeed, let the gauge transformation be:
$$G(t)\equiv \exp i\big(t
\Lambda(\vxc)/\hbar\big), \eqno(3)$$ where $\vxc$ denotes a
point in
\Ocp In the new gauge the Hamiltonian has the
Stark form [\rrremark ]:$$
H_{Stark} \equiv G^\dagger(t) H(t) G(t) -
i\hbar G^\dagger(t)\big(\partial_t G(t)\big)
=
{-\hbar^2\over 2m} \Delta +V(\vxc) +\Lambda(\vxc).\eqno(4)$$
$V(\vxc)+\Lambda(\vxc)$ is a periodic + linear (=washboard)
potential, on the covering space. Because
of the presence of a periodic potential, we dub such operators
Wannier-Stark Hamiltonians [\wsl]. In the presence of holonomy,
the gauge
transformation $G(t)$ is well defined only for {\it
integer} times and can not be substituted in the time dependent
\schrod
equation, as is evident from Eq.~(4), where
differentiation with respect to time is taken. It is then natural
to ask
what is the relation, if any, between the time-independent Stark
Hamiltonian, $H_{Stark}$, on the covering space and the
time-dependent one on the ring.
The evolution generated by {\it general} time-dependent
Hamiltonians, (including cases where
$\vec E$ and $\vec B$ {\it are} time dependent),
can be reduced to considering time-independent ones.
In classical
mechanics the price is that the time-independent
Hamiltonians are defined on a larger phase space, where $E$ and
$t$
are the additional conjugate coordinates, (see e.g. [\cfks ]). In
quantum
mechanics, the analogous procedure, due to J.~Howland [\jh ], is
to
enlarge the Hilbert space to
$L^2(\Omega\otimes R)$, with elements $\psi(\vx,s)$,
$\vx\in\Omega$
and $s\in R$. On this larger (=Grand) Hilbert space, with one extra
coordinate, one considers the (quasienergy) operator $\K$
$$\K\equiv
-i\hbar\partial_s +H(s). \eqno(5)$$ It has the property that
$$\big(\exp - i(\sigma\K/\hbar)\psi\big)(\vx,s) =
U(s,s-\sigma)\psi(\vx,s-\sigma),\eqno(6)$$ where $U(t,s)$ is the
unitary propagation operator from time s to time t for $H(t)$.
Because of Eq.~(6), the evolution $U(s,t)$ can be studied via the
evolution generated by $\K$ which is characterized by the
spectral analysis of $\K$.
In the case of Eq.~(1), $H(t)$ has the property that it is
periodic {\it up to unitary}, i.e. [\tp ]:$$H(t+1) = G^\dagger H(t)
G\eqno(7)$$
where $G\equiv G(1)$. This property is inherited by $\K$. The
{\it analog}
of the usual Bloch type analysis then says that the spectral
analysis
of $\K$ reduces to the study of the spectra of
$\K$ restricted to the spaces of
``Bloch waves" in the s variable. In particular the analog of the
periodic Bloch waves, normalized, as usual, in
$L^2\big(\Omega\otimes [0,1]\big)$, are those that satisfy the
condition [\remark ]:
$$\psi(\vx,s+1)= G^\dagger\psi(\vx,s).\eqno(8)$$ Combined with
Eq.~(6) this gives:
$$\eqalign{\big(\exp -i(\K/\hbar)\psi\big)(\vx,s)
&= U(s,s-1)G\psi(\vx,s)\cr = G
U(s+1,s)\psi(\vx,s) &\equiv \ M(s) \psi(\vx,s)}.\eqno(9)$$
We used: $U(t+1,s+1) = G^\dagger
U(t,s)G$, which is a direct consequence of Eq.~(7). It follows
from Eq.~(9) that the spectral type of $\K$ (defined as an
operator on
the Grand Hilbert space) and those of each of the
Monodromy (=Floquet)
operators, $M(s)$, for all $s$, (defined as operators on the
``small"
Hilbert space $L^2(\Omega )$), coincide [\rremark ].
The operator for the quasienergy, $\K$, is the phase
of the Monodromy.
We shall now show that it is unitarily equivalent to the
Wannier-Stark Hamiltonian $H_{Stark}$ of Eq.~(4),
defined on the covering space, $L^2(\Omega^c )$.
The Zak transform [\zak] from the Bloch states in
the Grand
Hilbert space to the Hilbert space associated with the covering
space
$\Omega^c$ is:
$$\widetilde \psi (\vxc) \equiv \int_0^1 ds\ \psi(\vx,s)
\exp i\Big(s \Lambda (\vxc)/\hbar\Big) .\eqno(10)$$
$\vxc\in\Omega^c$ are the preimages of $\vx\in\Omega$.
The inverse transform is:
$$\psi(\vx,s) = {1\over 2\pi\hbar} \sum_{Preimages\ of \ \vx}
\widetilde\psi(\vxc) \exp
-i\Big(s\Lambda(\vxc)/\hbar\Big).\eqno(11)$$ Eq.~(10) and (11)
are
compatible with the boundary conditions, Eq.~(8), and
preserve the appropriate norms.
It is a simple exercise, using Eq.~(10,11), and the boundary
condition,
Eq.~(8), to show that for the operators appearing in
Eqs.~(1) and (5) one has:
$$\eqalign{ (\widetilde{V\psi})(\vx_c)&=
V(\vx_c)\widetilde\psi(x_c),\cr
\big(\widetilde{(-i\hbar\vec\nabla -s\vec a)\psi}\big)(\vxc)
&=
\big(-i\hbar\vec\nabla_{x_c}\tilde\psi\big)(\vxc),\cr
\big( \widetilde{-i\hbar\partial_s\psi}\big)(\vxc) &=
\Lambda(\vxc)\tilde \psi (\vxc).}\eqno(12)$$
It follows that $\widetilde\K= H_{Stark}$, which is our main
result.
There is a basic intuition from tunneling that
says that Stark operators do not have normalizable
eigenvectors. This
intuition has been established rigorously for a wide class of
potentials [\wsl ,\cfks ,\starkg ]. In particular, it is known that
in one
dimension, if e.g.~$V$ is twice differentiable, the spectrum
of Wannier-Stark Hamiltonians is
absolutely
continuous -- there are no normalizable eigenstates [\starkd].
The question whether the
energy of a particle in a driven ring is bounded or not bears on
the
question whether idealized rings (i.e.\ without inelastic
processes)
provide a model for dissipation. Partly because of this
it had been studied by many authors, using various
techniques, including: Simulations, approximate and analytic
methods. Some of these authors arrived at conflicting
conclusions [\rings,\gt ,\pa ].
Recently, Gefen and Thouless [\gt ] studied the
growth of energy in driven rings by considering Zener tunneling
between the energy bands of the (adiabatic) spectra of $H(t)$.
They
have shown that provided the gaps are random and uncorrelated, a
phenomenon related to Anderson localization takes place in
energy
space which keeps the energy of the particle bounded.
We shall prove that for a one dimensional ring, with any smooth
(twice differentiable will do)
background potential the energy will,
eventually, runaway.
That this is so can also be seen from localization theory
if one takes into account the asymptotic decrease of the
gaps [\pa ,\gta ].
The ultimate growth of the energy is therefore something
that can be seen in more than one way.
The point we want to make is partly that of
simplicity and rigor, partly to settle an issue that had been
somewhat controversial, and mostly to
illustrate the use of the relation between the time dependent
Hamiltonian on the ring and the Stark Hamiltonian on the line.
The strategy is closely related to the one in [\pa ]
where \WS Hamiltonians are studied. The basic tool
is due to Bellissard [\bel ] originally devised for
for time-periodic Hamiltonians. It
makes use of two facts: That the Monodromy has no
(normalizable) eigenvalues, and that $H(t=0)$ is bounded below
and has {\it only} discrete spectrum whose point of accumulation
is at infinity.
Let $|e_j\rangle$, $j=1,\dots,\infty$ be the normalized
eigenvectors
of $H(t=0)$ with (ordered) eigenvalues $E_j$. Let $c(k,n)\equiv
|\langle
e_k|M^n|\psi\rangle|^2$, with $\bra$ an initial normalized state
of finite energy, $M\equiv M(0)$.
By the completeness of $|e_j\rangle$ and the
unitarity of
$M$, $$\sum\limits_{k=0}^\infty c(k,n)=1\eqno(13)$$
for all
integer n's. On the other hand, $$\lim_{N\to\infty}
{1\over N}\sum\limits_{n=0}^{N} c(k,n) = |\langle
e_k|P_{pp}|\psi\rangle|^2 =0,\eqno(14)$$ where $P_{pp}$ is a
projection
on the pure point part of the spectrum of $M$. The
first
equality is Wiener (=RAGE) theorem (see [\cfks ]). The second
equality is
the statement that $M$ has no pure point part. Now use the
fact that
$U(n,0)= (G^\dagger)^n M^n$ together with Eq.~(7),
(13) and (14) to get
$$\eqalign{
{1\over N}\sum\limits_{n=0}^N \langle
\psi(n)|H(n)|\psi(n)\rangle &=
{1\over N}\sum\limits_{n=0}^N \langle
\psi|\big(M^\dagger\big)^n
H(0)M^n|\psi\rangle \cr
={1\over N}\sum\limits_{n=0}^N \sum\limits_{k=0}^\infty
E_k c(k,n) &>
\epsilon E_0 + (1-\epsilon) E_{j} },\eqno(15)$$
where $\epsilon\equiv
{1\over N}\sum\limits_{n=0}^N\sum\limits_{k=0}^{j-1}c(k,n)$
can be chosen as small as one wants, and $j$,
and
thus $E_j$, as large as one wants, provided N is large enough.
This
completes the proof that the energy is unbounded.
We close this paper with a sequence of remarks, mostly about
open problems. \hfil\break
1. The proof does not provide information
on the rate of growth of energy: Wiener theorem is ``soft" and
gives
no a-priori information on the rate of convergence to zero.
\hfil\break
2. It is instructive that the cases with infinitely large domains
remain
open. The proof fails because $H(t=0)$
may have essential spectrum in which case $E_j$ can get
stuck at finite energies even as $j\to\infty$. From a na\"\i ve
physical intuition it appears surprising that
the energy growth can be hindered when the
(adiabatic) spectrum becomes more dense
(and ultimately is essential). An example where something
related
happens is
a one dimensional ring or radius $R$,
with ``random potential" $V$.
>From Landauer formula for the conductance one sees that (since
the driving potential is independent of $R$)
energy grows at the rate which {\it decreases} with $R$.
\hfil\break
3. Ping Ao [\pa ] gave a fascinating argument suggesting that the
Dirac Comb (Kronig-Penney model) is critical, that is, the
spectrum is made of localized states for weak electric fields,
and
has no localized states if the field is large [\dss ].
This would mean that for
rings with Dirac Comb potential
the energy will remain bounded (in fact,
it will be an almost periodic function of time) for weak
driving.\hfil
\break
4. By adapting standard methods from scattering theory
of Strak Hamiltonians [\cfks ,\starkg ] in the multidimensional,
and
also multiparticle case, one would have that the spectrum has
an absolutely continuous component.
This would mean that for lots of
finite-energy initial states, the energy would grow. In fact,
the basic intuition that ``reasonable" Stark Hamiltonians
have no eigenvalues suggests that this will hold for {\it all}
initial
states [\GLG].\hfil\break
\bigskip
\centerline{ACKNOWLEDGEMENTS} We are
grateful to
S.~Fishman and D.~Iliescu for useful discussions. This research
is
supported by BSF,
the Fund for the Promotion of Research at the Technion and
Elron-Elbit
research grant.
\bigskip \cntl{REFERENCES}
\bigskip
\item {\qual .} For the sake of simplicity we
assume throughout that there is a a single hole.
\item {\diric .} We impose
Dirichlet boundary conditions on the boundary
$\partial\Omega$.
\item {\rrremark .} When \O is simply
connected \O and \Oc coincide. Eq.~(4) gives for
$\vec a = e \vec E$, the textbook form of a dc
Stark effect $\Lambda(\vxc ) =
e\vec E\cdot\vxc$.
\item{\wsl .} There is tremendous literature on \WS
Hamiltonians. The experimental situation
is reviewed in G.~Bastard, J.~A.~Brum
and R.~Ferreira, Solid State Physics {\bf 44},
F.~Seitz and
D.~Turnbull, Editors, Academic Press, (1991).
Rigorous mathematical results,
can be traced from: G.~Nenciu, Rev.~Mod.~Phys., {\bf 63},
91, (1991);
V.~S.~Buslaev and L.~A.~Dmitrieva, Lenningrad Math J. {\bf 1}
287, (1990);
F.~ Bentosela {\it et. al.} J.~Phys.~A{\bf 21}, 3321, (1988);
J. Howland and I.~W.~Herbst, Comm. Math. Phys. {\bf
80}, 23, (1981).
\item {\cfks .} H.~L.~ Cycon, R.~G.~Froese, W.~Kirsch and
B.~Simon, {\it \schrod Operators}, Springer, (1987).
\item {\jh .} J.~Howland, Math. Ann. {\bf 207}, 315, (1974).
\item {\tp .} The analysis given below is patterned on the
analysis of
time-periodic operators where $G=1$. There is huge
literature on
the quasienergy spectrum of time periodic operators.
A recent review is S.~Fishman, Physica Scripta
{\bf 40}, 416, (1989).
A good part of the mathematical literature can be trace from
Ref.~[\cfks ]; Some of the original research papers are:
K.~Yajima H.~Kitada, Ann.~Inst.~Henri Poincare,
A vol.~{\bf XXXIX}, 145, (1983);
V.~En\ss \ and K.~Veselic, Ann.~Inst.~Henri Poincare,
A vol.~{\bf XXXIX}, 159, (1983).
Resonances for the Monodromy operators are
discussed e.g.~in S.~I.~Chu and \break W.~P.~Reinhardt,
\PRL {\bf 39}, 1195, (1977);
N.~Moiseyev and H.~J\"urgen~Korsch, \PR {\bf A41}, 498,
(1990) and references therein. Operators
periodic up to unitary have not been as popular, and the only
reference we are aware of is a poster in IAMP Conference (1991),
by
J.~Lebowitz and collaborators.
\item {\remark .} One could also study the general Bloch solution
with an
arbitrary phase. This, however, does not lead to anything
beyond what
one gets from the periodic solution.
\item {\rremark .} This shows that the
spectra
of $M(s)$ and $M(s')$ coincide, a fact that also follows directly
from
$M(s) = GU(1+s,s)= U(s,0)M(0)U(0,s)$.
\item {\zak .} J.~Zak, Solid State Phys.~{\bf 27}, F.~Seitz and
D.~Turnbull, Editors, Academic Press (1974).
\item {\starkg .} E.~l.~Korotyaev,
Math.\ U.S.S.R.\ Sbornik, {\bf 60}, 177, (1988)
M.~Ben-Artzi and A.~Devinatz, Mem.~A.M.S., {\bf 364}, (1987);
I.~W.~Herbst, Math.~Z.~{\bf 155}, 55, (1977); and
references therein.
\item {\starkd .} F.~Bentosela, R.~Carmona, P.~Duclos, B.~Simon,
B.~Souillard and R.~Weder, \break
Comm.~Math.~Phys.~{\bf 88}, 387, (1983).
\item{\rings .} D.~Lenstra and W.~J.~Van Herringen,
\PRL {\bf 57} 1623, (1986);
R.~Landauer, \PRL {\bf 58}, 2150, (1987);
G.~Blatter and D.~A.~Browne,
\PR {\bf B37}, 3856, (1988); N.~Triveldi and D.~A.~Browne,
\PR {\bf B38}, 9581, (1988).
\item {\gt .}Y.~Gefen and D.~J.~Thouless, \PRL {\bf 59}, 1752,
(1987).
\item {\pa .} Ping Ao, Phys.\ Rev.\ {\bf B41}, 3998, (1990).
\item {\gta .} The work of Gefen
and Thouless can be
understood to mean that there is an intermediate range of times,
much longer than Zener times,
so that the energy does not grow if the initial conditions
involve low energies.
\item {\bel .} J.~Bellissard,in {\it
Trends
and Developments in the Eighties}
S.~Albeverio and Ph.~Blanchard, Ed., World Scientific,
Singapore, (1985).
\item {\dss .} This is known to be the case
for the {\it Random} Dirac Comb with
electric field. See F.~Delyon,
B.~Simon and B.~Souillard, \PRL {\bf 52}, 2187, (1989).
\item {\GLG .} For a numerical study see
Y.~Gefen, D.~Lubin and I.~Goldhirsch,
WIS preprint, (1991).
Proving the absence of embedded eigenvalues is in general a
hard problem.
The known general results involve conditions on the decay
of the $V$. See ref [\cfks ,\starkg ].
\vfil\eject
\end