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\title{Phonon relaxation of subgap levels in superconducting quantum point
       contacts}

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\rtitle{Phonon relaxation of subgap levels\ldots}

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\sodtitle{Phonon relaxation of subgap levels in superconducting quantum
          point contacts}

%%% author(s) ( + e-mail)
\author{D.\,A.\,Ivanov$^{+*}$,
M.\,V.\,Feigel'man$^+$\/\thanks{e-mail: feigel@landau.ac.ru}}

%%% author(s) - for colontitle (at the top of the page)
\rauthor{D.\,A.\,Ivanov D.A., M.\,V.\,Feigel'man M.V.}

%%% author(s) - for table of contents
\sodauthor{Ivanov, Feigel'man}

%%% author's address(es)
\address{$^+$L.D.Landau Institute for Theoretical Physics RAS,
117940 Moscow, Russia\\~\\
$^*$12-127 M.I.T. Cambridge, MA, 02139 USA}

%%% dates of submition & resubmition (if submitted once, second argument is *)
\dates{12 November 1998}{12 January 1999}

%%% abstract
\abstract{Superconducting quantum point contacts are known to possess two
subgap states per each propagating mode. In this note we compute
the low\d temperature relaxation rate of the upper subgap state
into the lower one with the emission of an acoustic phonon. If
the reflection in the contact is small, the relaxation time may
become much longer than the characteristic lifetime of a bulk
quasiparticle.}

%%% PACS numbers
\PACS{74.50.+r, 74.80.Fp}

\begin{document}

\maketitle

In the present paper we address the question of phonon
relaxation of subgap levels in superconducting quantum point
contacts (SQPC) [1\ch 3]. We use this term for a class of
junctions between two superconductors (of the BCS type), were
only a small number of modes propagates and the scattering
center of the contact is much shorter than the superconducting
coherence length. Under the latter condition, the internal
structure of the scatterer is not important, but may be
described by a scattering matrix for normal electrons~[4,~5].
Thus, in the class of SQPC we include both SNS junctions with a
thin normal layer and with quantized propagating modes~[6] and
mechanically\d controllable breakjunctions~[7].

Theoretically, SQPCs in different setups are predicted to
exhibit various quantum phenomena. The most advanced predictions
involve quantum\d mechanical evolution of a superconducting
island connected to external leads by SQPCs~[8,~9]. Other works
discuss interference between time evolutions of localized states
and/or excitations of localized states by an external
electromagnetic field [10\ch 12]. For experimental observation
of these predictions, the decay time of excited localized states
must be sufficiently long. This decay time directly enters the
expressions for the thermal fluctuations of the Josephson
current in SQPC [13\ch 16]. Potential use of SQPC in quantum
computing devices also crucially depends on their ability to
preserve quantum coherence~[17].

At low temperatures ($T\hm\ll\Delta$, where $\Delta$~is the
superconducting gap), the dominant inelastic processes are the
transitions between localized subgap levels, without
participation of continuum spectrum. There are two major
channels of inelastic relaxation. One is the emission or
absorption of a phonon, the other is due to the electromagnetic
coupling to the external environment. It will depend on the
particular experimental setup, which of the two relaxation
channels dominates. In the present paper we consider only the
phonon relaxation in its simplest form, when the superconducting
phases on the contact terminal are assumed to be rigidly fixed.

In several works, the time of decay into phonons was estimated
as the characteristic time~$\tau_0$ of a bulk quasiparticle
[10\ch 12]. The latter lifetime is of order
$\tau_0\hm\sim\Theta_D^2/\Delta^3$\t the same as the relaxation
time of a normal electron with energy~$\Delta$ above the Fermi
level~[18] (we set $\hbar\hm=1$ throughout the paper). We want
to point out that this assumption is not justified for decay of
localized states. In refs.~[16,~19] the relaxation of subgap
states in SQPC is associated with bulk subgap states appearing
due to electron\d phonon interaction. This contribution is
exponentially small for temperatures much lower that the
superconducting gap.

We compute the \emph{direct} matrix element for decay of
localized states into phonons. This direct decay leads to a
non\d vanishing relaxation rate even at zero temperature.
Further, we simplify our discussion by setting the temperature
much lower than the energy of the subgap level. Then the only
allowed process is the transition from the upper to the lower
level with the emission of an acoustic phonon. The extension of
our result to include thermal phonons is obvious (see eq.~(16)
below).

The characteristic energy scale of the subgap levels is
$\Delta$, thus the wavelength of the phonon is of order
$s/\Delta\hm\sim\xi s/v_F\hm\ll\xi$, where $s$ and~$v_F$ are
sound and Fermi velocities respectively, $\xi$~is the
superconducting coherence length. The rate and the angular
distribution of the emitted phonon may depend on the particular
geometry of the contact (or, more precisely, on the geometry of
the wavefunction of the subgap states). In this paper we discuss
the simplest setup of a narrow one\d dimensional contact. By
this we mean that the whole subgap state is localized in a
narrow strip of width much smaller than the phonon wavelength.
Although very idealistic, this assumption is consistent with the
model of adiabatic constriction~[2] and gives an upper bound for
the actual decay rate.

Each propagating mode in a quasi\d one\d dimensional contact may
be described by the Hamiltonian
\begin{multline}
H=\int_{-\infty}^{+\infty}dx\Bigl[i\Psi^+_{L\beta}\partial_x\Psi_{L\beta}
-i\Psi^+_{R\beta}\partial_x\Psi_{R\beta}+{}\\
{}+\Delta(x)\bigl(\Psi^+_{R\uparrow}\Psi^+_{L\downarrow}
-\Psi^+_{R\downarrow}\Psi^+_{L\uparrow}\bigr)+{}\\
{}+\Delta^*(x)\bigl(\Psi_{R\downarrow}\Psi_{L\uparrow}
-\Psi_{R\uparrow}\Psi_{L\downarrow}\bigr)\Bigr]+H_{scatt},
\end{multline}
where $\Psi^+$ and $\Psi$ are electron operators ($L$ and $R$
subscripts denote left- and right\d movers,
$\beta\hm=\uparrow,\downarrow$ is the spin index),
$\Delta(x)$~is the superconducting gap with the following
$x$\D dependence:
\begin{equation}
\Delta(x)=
\begin{cases}
\Delta,&x<0,\\
\Delta e^{i\alpha},&x>0.
\end{cases}
\end{equation}
(It will be convenient for us to use the units with Fermi
velocity equal to one throughout the paper.) The scattering
term~$H_{scatt}$ expresses elastic scattering at $x\hm=0$ and
may be described by a scattering matrix~[5]. Diagonalizing the
Hamiltonian~(1) gives the subgap state operators
$\gamma^+_\uparrow$ and $\gamma^+_\downarrow$ raising energy by
\begin{equation}
E(\alpha)=\pm\Delta\sqrt{1-t\sin^2\frac{\alpha}{2}}
\end{equation}
each ($t$ is the normal transparency of the contact)~[4,~20].
The two levels below continuum are the ground state $|0\rangle$
and the first excited state
$|1\rangle\hm=\gamma^+_\uparrow\gamma^+_\downarrow|0\rangle$.
The decay of the state $|1\rangle$ to the state $|0\rangle$ with
the emission of a phonon depends on the density matrix element
\begin{equation}
\langle0|n(x)|1\rangle=\langle0|\Psi^+_\beta(x)\Psi_\beta(x)|1\rangle.
\end{equation}
Since $\gamma^+_\uparrow$ and $\gamma^+_\downarrow$ are linear
in electron operators, the matrix element~(4) may be computed by
commuting density operator with them:
\begin{equation}
\langle0|n(x)|1\rangle=\left\{\left[n(x),\gamma^+_\uparrow\right],
\gamma^+_\downarrow\right\}=i|b|\kappa e^{-2\kappa|x|}\sign(x),
\end{equation}
where $b$ is the backscattering amplitude ($|b|\hm=\sqrt{1-t}$),
and $\kappa\hm=\sqrt{t}\Delta|\sin(\alpha/2)|$ is the inverse
length of the subgap state. The matrix element is purely
imaginary if the relative phases of $|0\rangle$ and $|1\rangle$
are chosen according to~[5]
\begin{equation}
\langle0|\frac{\partial}{\partial\alpha}|0\rangle
=\langle1|\frac{\partial}{\partial\alpha}|1\rangle=0,\qquad
\langle0|\frac{\partial}{\partial\alpha}|1\rangle\text{\ \ is real}.
\end{equation}
This fact is of no importance for the present calculation, but
will be used elsewhere in the discussion of the phonon emission
in the presence of dynamics in~$\alpha$.

The electron\d phonon interaction is described by the deformation
potential:
\begin{align}
H_{e-ph}&=g\int d^3r\varphi(r)\Psi^+_\beta(r)\Psi_\beta(r),\\
\varphi(r)&=\frac1{\sqrt V}\sum_k\sqrt\frac{\omega_k}2
(b_ke^{i(kr-\omega_kt)}+b_k^+e^{-i(kr-\omega_k t)}),
\end{align}
where $b_k$ are phonon operators normalized by
$[b_{k_1},b^+_{k_2}]\hm=\delta_{k_1k_2}$,
\begin{equation}
g^2=\frac{\pi^2\zeta}{2\varepsilon_F},
\end{equation}
(in the units with Fermi velocity equal to one), $\zeta$~is the
coupling constant of order one.

The transition rate is then given by
\begin{multline}
\tau^{-1}=2\pi\sum_k\left|\langle0,k| H_{e-ph}|1\rangle\right|^2
\delta(\omega_k-2E)={}\\
{}=\pi^3\zeta\frac{E}{\varepsilon_F^2}\int\frac{d^3k}{(2\pi)^3}
\delta(\omega_k-2E)\left|\langle0|n_k|1\rangle\right|^2.
\end{multline}
Here $E$ is the energy of the subgap states given by~(3) (so
that the energy of the emitted phonon is~$2E$),
$\varepsilon_F$~is the Fermi energy, $\omega_k$~is the phonon
dispersion relation, and $n_k$~is the three\d dimensional
density operator. Assume the linear isotropic phonon spectrum:
\begin{equation}
\omega_k=s|k|
\end{equation}
and the narrow contact limit, where the matrix element of~$n_k$
depends only on the component of~$k$ parallel to the
constriction and is given by the Fourier transform of the one\d
dimensional matrix element~(5). Finally, using $k\hm\gg\kappa$,
we arrive to the answer for~$\tau^{-1}$:
\begin{multline}
\tau^{-1}=\pi^2\zeta\frac{E^2}{(c\varepsilon_F)^2}
\int_{-\infty}^\infty dx\left|\langle0|n(x)|1\rangle\right|^2={}\\
{}=\frac{\pi^2\zeta}{2}(1-t)\frac{E^2\kappa}{(s\varepsilon_F)^2}.
\end{multline}
Returning to the physical units, we find up to a constant factor
of order one
\begin{equation}
\tau^{-1}=\sqrt t(1-t)\left|\sin\frac\alpha2\right|
\left(1-t\sin^2\frac\alpha2\right)\frac{\Delta^3}{\Theta_D^2},
\end{equation}
where $\Theta_D$ is the Debye temperature.

If compared to the characteristic bulk quasiparticle inverse
lifetime $\tau_0^{-1}\hm\sim\Delta^3/\Theta_D^2$~[18], the
result~(13) is smaller by a factor depending on the
backscattering probability~$1-t$\,\footnote{The actual lifetime
of a bulk quasiparticle diverges at the bottom of the
quasiparticle band. The ``characteristic'' lifetime~$\tau_0$
corresponds to energies of order~$\Delta$ above the bottom of
the band. There is no \emph{a priori} reason for the relaxation
rate $\tau^{-1}$ of the localized states to be of order
$\tau_0^{-1}$.}. In the case of weak backscattering, this factor
may contribute up to orders of magnitude to the decay time. This
effect is easy to understand: in ideally conducting contact
($t\hm=1$) the two Andreev states carry opposite momenta equal
to the Fermi momentum, and the matrix element~(4) contains only
a rapidly oscillating with momentum~$2k_F$ part\footnote{We
neglected the possibility of umklapp scattering.}:
\begin{equation}
\langle0|n(x)|1\rangle=\kappa e^{-2\kappa|x|}e^{2ik_F x}.
\end{equation}
This oscillating part of the matrix element gives the lower
bound for the relaxation rate~(13) as $t\hm\to1$:
\begin{equation}
\tau^{-1}_{t=1}
\sim\frac{E^3\kappa^4}{ s^3\varepsilon_F^6}
\sim\left|\cos^3\left(\frac\alpha2\right)\right|
\sin^4\left(\frac\alpha2\right)\frac{\Delta^7}{\varepsilon_F^3\Theta_D^3}.
\end{equation}
For realistic values of $\Delta$, $\Theta_D$, and
$\varepsilon_F$, this relaxation time is unphysically large. The
actual relaxation time at $t\hm=1$ will be bounded by other
factors such as finite thickness of the interface and non\d
one\d dimensionality of the contact. These effects go beyond the
simple model of the present paper\footnote{Our results (13),
(15) disagree with ref.~[15] where the relaxation rate is
computed in the same approximation. The relaxation rate of
ref.~[15] is not suppressed in the ballistic limit $t\hm\to1$.}.

Another important feature of the relaxation rate~(13) is that it
vanishes as $\alpha\hm\to0$, since in this limit the subgap
states become delocalized.

Our assumption of one\d dimensionality of the contact also
results in overestimating the relaxation rate. If the ``tails''
of the localized states are smeared in the terminals, they give
weaker contribution to phonon emission. Unfortunately, this
effect is highly geometry\d dependent, and should be considered
separately in each experimental realization.

As a direct application of the above result, the decay rate
entering the fluctuations of the Josephson current in a single
SQPC~[13,~14,~16] is given by
\begin{equation}
\gamma=\tau^{-1}(1+2n_B(2E(\alpha))=\tau^{-1}\coth\frac{E(\alpha)}T,
\end{equation}
where $n_B(2E(\alpha))$ is the Bose occupation number for the
phonons involved in the transition~[15], $\tau^{-1}$ is the
zero\d temperature rate~(13). The low\d frequency current noise
is~[13,~16]
\begin{equation}
S(\omega)=S_0\frac{2\gamma}{\omega^2+\gamma^2},
\end{equation}
where $S_0$ is the integral low\d frequency noise.

To summarize, we calculated the rate of direct relaxation of
subgap states in SQPC into acoustic phonons at low temperature
in the simplest one\d dimensional geometry. The relaxation rate
does not vanish in the $T\hm\to0$ limit, but it is strongly
suppressed in the case of a nearly ballistic contact.

We thank Gordey Lesovik for helpful discussions. The research of
M.V.F. was supported by INTAS-RFBR grant 95\ch 0302, Swiss
National Science Foundation collaboration grant 7SUP J048531,
DGA grant 94\ch 1189, and the Program ``Statistical Physics''
from the Russian Ministry of Science.

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\end{document}