It is predicted that a nonequilibrium first-order phase transition analogous to optical bistability can occur when external coherent radiation is applied to a suitably prepared Josephson junction with an external resistance across it. The size of the hysteresis region can be tuned by varying the resistance

Agarwal, G. S.

Shenoy, Subodh R.

English

Publications of the IAS Fellows

VOLUME 44, +UMBER 23 PHYSICAL REVIEW LETTERS 9 JuNE 1980
Radiation-Induced Bistability in Josephson Junctions
Subodh R. Shenoy and Q. 8. Agarwal
School of I'hysics, University of Hyderabad, Hyderabad 500 00l, India
(Received 91 December 1979)
It is predicted that a nonequilibrium first-order phase transition analogous to optical
bistability can occur when external coherent radiation is applied to a suitab1y prepared
Josephson junction with an external resistance across it. The size of the hysteresis re-
gion can be tuned by varying the resistance.
PACS numbers: 74.50.+r, 74.30.Gn, 42.65.Gv
Recently there has been an upsurge of activ-
ity~ ' in nonequilibrium steady states and non-
equilibrium phase transitions in externally driven
systems. Remarkable new physical effects are
seen in such systems. For example, a supercon-
ductor irradiated by photons of less than pair-
breaking energies is driven into a nonequilibrium
steady state in which the order parameter is en-
hanced. ' Optical bistability' is an example of non-
equilibrium first-order phase transition that oc-
curs when an absorbing medium is subjected to
resonant coherent irradiation. These systems
exhibit interesting switching and hysteresis ef-
fects as a function of external parameters.
In this Letter we predict that such effects con-
trolled by external coherent radiation can occur
in suitably prepared Josephson junctions. The
coherent radiation produces a dc current in a
circuit consisting of the junction and a resistance.
This dc current, or equivalently, the dc voltage
Vd, developed across the resistance, is shown to
exhibit bistable behavior and hysteresis as a
function of the external radiation intensity. The
frequency of the associated ac Josephson current
will also exhibit bistability. For a given junc-
tion, the size of the hysteresis region in the vol-
tage-intensity plot can be tuned by varying the
resistance. This example of nonequilibrium su-
perconductivity differs from previous' ones in
that the external drive acts directly on the pairs
rather than directly on the quasiparticles. The
absence of an applied voltage in the steady state
distinguishes this effect from the Fiske and Sha-
piro steps in the dc voltage.
The system considered is depicted schematical-
ly in Fig. 1. An external source of frequency 0
irradiates the junction with coherent photons.
The oxide region'7 functions as a resonance cav-
ity of frequency cu, = c'nw/l„, with transverse
junction dimensions l„, l» the oxide having a di-
electric constant e and a thickness l, . Here c'/c
=[l,/e(2X + l, )]~~. The (similar) superconduct-
ing electrodes have a penetration depth & and gap
& and are at temperature &«7, so that the quasi-
particle tunneling current is negligible. For
& 2&/I, the external radiation will not create
qua siparticles.
The Hamiltonian of the junction can be writ-
ten, ' ' in terms of pseudospin operators and ne-
glecting contributions from the bulk electrodes,
as
H = — ' — ' (S + S') + k&u, a a+ iAT(k, ~)(a+ a )(S —S ) + 2i hT,„,N' 'cosQt(S —S+).2sS, s RI,
The first and second terms describe the energy of the charged-junction capacitance C =el„l, /4&l, ,
and the pair-tunneling term leading to the Josephson current. The third and fourth terms represent
the cavity photon energy and the absorption and/or emission of cavity photons by the Josephson cur-
rent. The last term is the coupling of the Josephson current to an external coherent field of frequency
~. We will assume that the external source has finite bandwidth, which is quite important as seen
later. The coupling constant T is' [K = (c'/c)k]
T(k ~) I~ ks sin[(ks -k)l„] ~ T —T(K g)(@~ C)l/2 [k 2 k2] 1 1 Gxt P (2)
where k„=(2e/Ic)(PA+l, )H and H is an applied magnetic field that enables the current to couple to the
standing-wave junction-cavity photons and selects one mode for &~ k.
The pseudospin operators S, S, S, are defined in terms of those of the left and right electrodes. '
S (S+) transfers pairs from left (right) to right (left), and S, counts the number of excess pairs on
the left. For the usual dc Josephson effect; (S,) =0 and there is no pair imbalance of dc voltage across
1524 1980 The American Physical Society
VOLUME 44, NUMBER 23 PHYSlGAL REVIEW LETTERS 9 JUNE 1980
the junction. The commutator of S and 8 is neglected. '
The equations for the expectation values of coupled spin-photon system follow from (1) and (2). We
make the simplest' mean-field" type of approximation, replacing operators by their factored averages.
This yields
&
~ ) d;e(, & ~ 2eV(t) 4e' & )df ' 0 +C
&S,& = ——sin8 —2T(a+ a ) cos8 —4T,„,N cos Qt cos8,&S,& 1;
(3a)
(3b)
&a& =-~&a& —1'&, &a& —21T sin8$ (3c)
(4)
where decay terms have been added to represent resistance losses [in (3b)] and photon leakage (3c)
from the cavity, of quality S,ctory Q, at the rate & =~, /2Q.
We assume that a pair charge imbalance with a particular sign &S,& =sgn8 & 0, for example, has al-
ready been set up and, as shown below, that it can be maintained by the radiation power. We substi-
tute &a) =&a&e ' into (3) and take 8(f) =8(0)+ ~t (t&0). Keeping in mind the finite bandwidth of inci-
dent radiation, we time average Eq. (3). Further assuming that the photon mode is a fast mode, ' i.e.,RC»&, we put &a) =0, which yields &a&d, = 7'/[K+1(~, —&u)], where the subscript dc specifies dc
quantities. Following this, a physically transparent result for the rate of change of the pair charge
imbalance is obtained, i.e.,
2e&s „&= —4e7',~N' 'cos8(0) —(&d /R) 4e11'l&a&d I',
1/2 2Q ~
N, 1+ 4Q2(f —1)2
where we have set cos8(0) = —1, as this can be
(5)
a- gg--
FIG. 1. Schematic diagram of the system, with the
arrows denoting the direction of pair flow.
where the phase difference 8 (0) between the ap-
plied radiation and the resultant current can thus
be absorbed into the (adjustable) phase difference
between the two superconducting electrodes. N
is the number of external photons inside the junc-
tions. The escape of photons, otherwise fed back
and forth between cavity and current, contributes
to a pair imbalance decay 2e(2&&a a)d, ) 4«
x I& & ,I'.
When drive and dissipation balance, the scaled
steady-state voltage f -=&/~, =2eV~, /k~, is ob-
tained from
shown to minimize the Ginzburg-Landau-like
potential defined below. The dimensionless con-
stants are
Q'n& x WS,
Ã2)¹N~&N
(7a)
(7b)
where N» are the roots of dN/df =0. Equation
(7a) is equivalent to requiring N to exceed a
critical value. A plot, in the bistable regime of
the roots &u/~, vs (N/N, )'", is given in Fig. 2.
The hysteresis, as in other nonequilibrium sys-
tems, '~ is reminiscent of superheating and su-
percooling in equilibrium first-order transitions.
In the limit Q'a =(2/3v 3)(1+5), 0(6«1, we
find f„=f + —6' '/Q and N, = N "' + ' 6' '/Q
where f,= 1+-2'&3Q, (N, /N, )'"= 1+W3(1+-,' 5)/
2Q, and 5 = (R/R, —1), R, being defined implicitly
above. For - w, , the finite bandwidth can cover
the small hysteresis region. " In general, for a
given junction, the hysteresis size can be varied
by changing R.
The relative stability of these three states can
be studied by constructing a Ginzburg-Landau-
like function 4(f ) whose minima are the multiple
steady states and whose lowest minimum occurs
at the most probable state." The probability of
a-=R ' ' ' N = R — "', (6)
(d
Equation (5) admits three real roots provided
1525
VOLUME 44, NUMBER 23 PHYSICAL REVIEW LETTERS 9 JuNE 1980
g, (N2/N, )'
FIG. 2. Schematic plot of the scaled dc voltage f
=2eVdc/Ru, vs the scaled (square root of) externally
supplied photon number in the cavity (N jN, ) . The
hysteresis region is bounded by the critical points
(ft
~
2 &t
~
2
j/2
a given f is P(f) ~exp(-2~f 4(f)/RC), where 7~ '
is the strength of a delta-correlated random force
in a stochastic equation for f, and the dimension-
less function C (f ) is just the integral of the left-
hand side of (5) and can be related to the dissipa-
tion in the system. " The function 4 (Fig. 3)
clearly shows the exchange of relative stabilities
and hysteresis.
The potential is symmetric, 4(f) =4(-f )
(with a cusp at f =0), reflecting the basic sym-
metry of the circuit. The zero-voltage state at
the cusp is unstable, and a spontaneous fluctua-
tion may perhaps be sufficient to break the sym-
metry. The symmetry can also be externally
broken if need be; e.g. , as follows. Apply a dc
voltage to the irradiated junction such that 2eV,~/
h =0 =w, . This will result, through the Fiske
step effect, ' in a dc current in a direction fixed
by V,~, corresponding to a point A in Fig, 2.
Now switch off V,~, leaving the microwave power
on. The voltage will fall (dotted line) till it reach-
es the steady-state curve.
The dc voltage variation of Fig. 2 also corre-
sponds to a variation of the frequency ~ =2e V/h
of an ac Josephson current component and of the
emitted radiation. For a given circuit, the fre-
quency can be intensity tuned and switched.
We now examine the feasibility of the bistability.
Keeping in view that lossy cavities make the pho-
ton leakage dissipation significant and large re-
sistances make the power requirements low for
FIG. 3. Schematic plot of the Ginzburg-Landau-like
potential 4 (f) vs the scaled voltage f. The dot denotes
the initial steady state.
a given Vd, -S&u, /2e, we take the typical values
of the junction parameters as E„=l,=0.1 cm (Q
=10"Hz), l, = 20 A, e =5, I, =30 ALA, R = 100 Q,
Q = 5, y = 4 x 10 ' cm; we then find N, = 5x 10' pho-
tons, and a= 0.02. The external photon density
in the cavity must be at least N/(21+i, ) l„l,
&& 10" cm '. This corresponds to an available
power of N, hQc'/l„4x 10 ' W-, and is less than
the power required by resistance and cavity loss-
es. The lifetimes jt: and Rc justify the original
S,st-mode assumption for photons. For a 10%
transmission from outside, this means an ex-
ternal power of at least 0.05 W/cm' is required.
Voltages and currents are typically around 0.1
mV and 1 p A. The fractional pair imbalance es-
tablished by the radiation is one pair in 10'. The
power requirement and the bistability condition of
(7) are very sensitive to the junction dimensions
and also depend on Q, R, and I, This could ex-
plain" why the predicted phenomena have not
previously been seen, apart from that fact that
here the junction must be "free running" with a
self-consistently developed, rather than an ap-
1526
VOLUME 44, NUMBER 2$ PHYSICAL REVIEW LETTERS 9 JUNs 1980
plied, ' voltage. In order for the bistability to be
observable, the lifetimes of the states must be
large. Estimating the photon fluctuation rate) K 10"sec ', we find vz '- 0.1 sec '. The
scaled lifetime in the probability exponent is vt /
AC-10', and the distribution is thus, indeed,
sharply peaked.
We thank I'. Chandrasekhar, P. A. Narayana,
and V. S. S. Sastry for useful discussions.
I. M. Eliashberg, Pis'ma Zh. Eksp. Teor. Fiz. 11,
186 (1970) [JETP Lett. 11, 114 (1970)l; T. M. Kapwijk,J. ¹ Van den Berghand, and J. E. Mooij, J. Low.
Temp. Phys. 26, 385 (1977).
2A. Schmid, Phys. Rev. Lett. 38, 922 (1977).
3H. M. Qibbs, S. L. MeCall, and T. N. C. Venkatesan,
Phys. Rev. Lett. 36, 1135 (1976); R. Bonifacio and
L. A. Lugiato, Phys. Rev. A 18, 1129 (1978); Q. S.
Agarwal, L. M. Narducci, R. Gilmore, and D. H.
Feng, Phys. Rev. A 18, 620 (1978), and the references
therein.
4A. ¹itzan and J. Ross, J. Chem. Phys. 59, 241
(1973); G. Nicolis and I. Prigogine, Self-organization
in Nonequilibrium Systems (Wiley-lnterscience, New
York, 1977).
Cf. H. Haken, Syne~getics (Springer-Verlag, New
York, 1977).
6Cf, L. Solymar, S'Npercondlctive Vhnnezing and
Applicatioris (Chapman and Hall, London, 1972), Chaps.
9, 10.
YP. A. Lee and M. O. Soully, Phys. Rev. 8 3, 769
(1971).
P. W. Anderson, Phys. Rev. 112, 1900 (1958); P. R.
Wallace and M. J. Stavn, Can. J. Phys. 43, 411 (1965).
~M. J. Stephen, Phys. Rev. Lett. 21, 1629 (1968);
D. McCumber, Phys. Rev. Lett. 23, 1228 (1969).
' For the case of hysteresis region larger than the
bandwidth, one could consider two coherent sources
with two frequencies, close to the higher and lower
values at the transition, and with nonoverlapping band=
widths Eq.uation (6) would hold approximately for
either source. After the intensity variation of one
beam drives the system to the other frequency, the
second beam takes over control and the first can be
switched off.
"Cf. S. R. Shenoy, to be published.
'Kith systems parameters in the above range, one
finds the neglected I~ sin8 term yields corrections of
order 1%. However, see J. T. Chen, R. J. Todd, and
Y. W. Kim, Phys. Rev. 8 5, 1843 (1972).
Theoretical Studies of a Light-Induced Change in Multiplet Satellites
of 3p-Photoelectron Spectra of Transition-Metal Compounds
Satoru Sugano
Institute for Solid State Physics, University of Tokyo, Itoppongi, MinatoWu, Tokyo 106, Japan
Yoshihiro Miwa and Tsuyoshi Yamaguchi
Paculty of Engineering, Shieuoka University, Hamamatsu 432, Japan
(Received 31 December 1979)
It is shown that optical excitations of the d shell may induce changes in the multiplet
structure of the 3P -photoelectron emission spectra of transition-metal compounds. The
photoelectron spectra of Ni + and Cr'+ compounds in some excited states are calculated
and compared with those in the ground states.
PACS numbers: 79.60.Eq
It is well known' that the initial d-electron state
of a transition-metal compound is clearly reflect-
ed in multiplet satellites of photoelectron emis-
sion spectra (PES). In the present paper, we
shall show that an optical excitation of the d shell,
whose lifetime could be assumed extremely short
(as little as 10 "s), may induce a change in the
multiplet structure of, mainly, Sp PES of transi-
tion-metal compounds. We should like to empha-
size the point that observation of this change pro-
vides us with a unique method of detecting the ex-
cited state with a very short lifetime. As dis-
cussed later, a physically important point in this
method is that the satellite lines associated with
an excited state are still expected to be observed
even if the state is extremely shortlived.
Let us first examine how short a change of the
d-electron state could be and still be observed
as a change in the multiplet structure. As well-
known' shakeup satellites or multiplet satellites
1980 The American Physical Society 1527


Radiation-induced bistability in Josephson junctions

http://repository.ias.ac.in/46034/1/46034.pdf

Radiation-induced bistability in Josephson junctions

Abstract

Similar works

Full text

Available Versions

Publications of the IAS Fellows