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Effect of resonant pumping on activated decay rates
Volume 116, number 9
PHYSICS LETTERS A
14 July 1986
E F F E C T OF R E S O N A N T P U M P I N G ON ACTIVATED DECAY RATES B.I. IVLEV and V.I. M E L ' N I K O V L.D. Landau Institute for Theoretical Physics, Academy of Sciences, ul. Kosygina 2, 117334, Moscow, USSR Received 12 December 1985; accepted for publication 8 May 1986
The problem of oscillations of a brownian particle at the bottom of a potential well under the action of a force d~cos ~2t is considered. The rate of escapes out of the well is found to exhibit a threshold behavior when the resonance is crossed.
We discuss the effects of resonant pumping on the activated decay of zero-voltage states of a Josephson junction. Recently, these effects were observed experimentally [1]. Our approach to the problem is based on the observation of the similarity to a problem concerning the escape of a brownian particle out of a potential well under action of a force d~cos I2t. The effect of this force on the distribution function f(p, x) is most significant when the resonance condition [ ~(0) - I2 I << I2 is met (o~(E) is the energy-dependent frequency of free oscillations). Then f(p, x) shall deviate from the equilibrium shape near the bottom of the well; it still retains this equilibrium shape in the major part of the well, where the sole effect of pumping is a certain reduction of the effective depth of the well. In a narrow range I E - V I - T ( T is the temperature, V is the barrier height, T<< V) near the barrier top, the function f ( p , x) shows nonequilibrium behavior due to the escapes of particles across the barrier [2]. Henceforth, the two perturbations of f(p, x), obtained by pumping and by escapes across the barrier, may be considered separately. Then, our program is as follows: we solve the Fokker-Planck equation [3]
Of
p Of
0-7 + m ~-x +
(
~ cos ~2t
0 Of = 3"-~pp ( rnT-~pp + pf )
OV(x) ) Of Ox
0p
at the bottom of the well, determine the search for the reduction of effective depth of the well from the equilibrium asymptotics of the solution and use it as an input in the known expressions for activated decay rates [2,4]. Eq. (1) may be greatly simplified in the underdamped regime 3' << c0(0), where the total energy E-p2/2m + V(x) is the slowest varying quantity. It is convenient to introduce the energy-phase representation
xm~o(O) =
2v/2mE-sin(f2t + ~b),
p = 2eTa-cos(hi
+ ~,).
Provided that I A I << I2 (A = o~(0) -- 12), the timedependent corrections to f(E, ep) are small, and f(E, O) obeys a stationary equation
(A - qE)
+g~/E/2mcos ¢p OE sin q~ 0f
= 3'-3--/~ E T
+f
+ 4--E-0th 5'
where q - -~0~(0) > 0, q - o~/V. The next-toleading order expansion in d~/y2 v ~ T may be written as
f(E, (1)
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
q~) = (1 + (o~/7 2v/2m-7) Re[z e x p ( - i ~ ) ] +S(E)}
exp(-E/T),
(2) 427
Volume 116, number 9
PHYSICS LETTERS A
where the normalization constraint on S ( E ) , a quadratic function of E, reads
f} S ( E )
14 July 1986
At J < 0 the equation o~(E) = 52 has no solution, and the system drives off resonance. In this case we have
exp(-E/T)dE=O, D/D o = l +d2/12rntA[qTe,
The calculation of the absolute decay rates [2,4] requires the full account of escapes across the barrier. However, the relative variation of the decay rate under p u m p i n g can readily be found from eq. (2) and is given by the ratio D / D o = 1 + S(~c), which amounts to reduction of the effective depth of the well by TS(oc). After simple calculations one arrives at
D / D o = 1 + (d~2/4mTy 2 )
~5 × Re f0 z( y )[1 - exp( - 1 ' )] y
12 d l', (3)
where y = - E / T and the function z ( y ) obeys the equation v
d2z d2 (~ + (1 - y ) + dy 2 ~
-
iyqT Y
-
1) 4T' . z
= -¢,,.
(4)
The singularity-free function z ( y ) is found provided that z(0) = z(ov) = 0. Solutions of eq. (4) in limiting cases give some analytical results. If qT<< 7 << w(0) the distribution f(E, O) deviates from equilibrium in a region 8E - 7 / q , and
v/Tq T << 1,1 I << qT, D/D4}
and
(7)
I + d2/24mA2T,
qT<< 1,1 I << ~0(0).
(8)
We have neglected here all the q u a n t u m effects. This is justified, if T>>ko~(0) and besides, the perturbed part of f ( E , ~) comprises many energy levels. The last condition gives a lower bound for the d a m p i n g 7 >> q T ( k o ~ / T ) : . From eqs. (5) (8) it follows that the decay rate exhibits a steep threshold rise as ~2 decreases and crosses o~(0). The subsequent slow decrease of D, as ~ goes further down to ~ - o~(0), is beyond the scope of our approach. I n experiments [1] the potential V(x) was nearly cubic, and q = (5/36)o~(0)/V. The quality factor Q = o~(0)/7 was about 13. Estimation of the width of the nonequilibrium region 8 E - y / q - V / 2 shows, that substantial part of the well is perturbed by pumping. The observed enhancement of the decay rate [1] is in a broad agreement with our expectations, but the threshold behavior is smeared out because of the low Q factor. Our results are only applicable if 8E << V, which will be the case at somewhat larger values of Q.
D/D o = 1 + ( ~2/4yrnqT 2) ×[1-exp(-2v~l/y)]
'
(5)
If "{< > ~ , f ( E , q~) deviates from equilibrium in a narrow region 6E << T a r o u n d E 0 (o~ ( E 0 ) = ~2), and
D / D o = 1 + ( ' ~ 2 / 4 7 r n q T 2 )[1 - e x p ( - J / q T ) ] .
(6)
428
References
[1] M.H. Devoret, J.M. Martinis. D. Esteve and J. Clarke. Phys. Rev. Len. 53 (1984) 1260. [2] tt.A. Kramers. Physica 7 (1940) 284. [3] H. Risken, Springer series in sinergetics. Vol. 18. The Fokker-Ptanck equation (Springer, Berlin. 1984). [4] V.I. Mel'nikov, Physica A 130 (1985) 606.
PHYSICS LETTERS A
14 July 1986
E F F E C T OF R E S O N A N T P U M P I N G ON ACTIVATED DECAY RATES B.I. IVLEV and V.I. M E L ' N I K O V L.D. Landau Institute for Theoretical Physics, Academy of Sciences, ul. Kosygina 2, 117334, Moscow, USSR Received 12 December 1985; accepted for publication 8 May 1986
The problem of oscillations of a brownian particle at the bottom of a potential well under the action of a force d~cos ~2t is considered. The rate of escapes out of the well is found to exhibit a threshold behavior when the resonance is crossed.
We discuss the effects of resonant pumping on the activated decay of zero-voltage states of a Josephson junction. Recently, these effects were observed experimentally [1]. Our approach to the problem is based on the observation of the similarity to a problem concerning the escape of a brownian particle out of a potential well under action of a force d~cos I2t. The effect of this force on the distribution function f(p, x) is most significant when the resonance condition [ ~(0) - I2 I << I2 is met (o~(E) is the energy-dependent frequency of free oscillations). Then f(p, x) shall deviate from the equilibrium shape near the bottom of the well; it still retains this equilibrium shape in the major part of the well, where the sole effect of pumping is a certain reduction of the effective depth of the well. In a narrow range I E - V I - T ( T is the temperature, V is the barrier height, T<< V) near the barrier top, the function f ( p , x) shows nonequilibrium behavior due to the escapes of particles across the barrier [2]. Henceforth, the two perturbations of f(p, x), obtained by pumping and by escapes across the barrier, may be considered separately. Then, our program is as follows: we solve the Fokker-Planck equation [3]
Of
p Of
0-7 + m ~-x +
(
~ cos ~2t
0 Of = 3"-~pp ( rnT-~pp + pf )
OV(x) ) Of Ox
0p
at the bottom of the well, determine the search for the reduction of effective depth of the well from the equilibrium asymptotics of the solution and use it as an input in the known expressions for activated decay rates [2,4]. Eq. (1) may be greatly simplified in the underdamped regime 3' << c0(0), where the total energy E-p2/2m + V(x) is the slowest varying quantity. It is convenient to introduce the energy-phase representation
xm~o(O) =
2v/2mE-sin(f2t + ~b),
p = 2eTa-cos(hi
+ ~,).
Provided that I A I << I2 (A = o~(0) -- 12), the timedependent corrections to f(E, ep) are small, and f(E, O) obeys a stationary equation
(A - qE)
+g~/E/2mcos ¢p OE sin q~ 0f
= 3'-3--/~ E T
+f
+ 4--E-0th 5'
where q - -~0~(0) > 0, q - o~/V. The next-toleading order expansion in d~/y2 v ~ T may be written as
f(E, (1)
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
q~) = (1 + (o~/7 2v/2m-7) Re[z e x p ( - i ~ ) ] +S(E)}
exp(-E/T),
(2) 427
Volume 116, number 9
PHYSICS LETTERS A
where the normalization constraint on S ( E ) , a quadratic function of E, reads
f} S ( E )
14 July 1986
At J < 0 the equation o~(E) = 52 has no solution, and the system drives off resonance. In this case we have
exp(-E/T)dE=O, D/D o = l +d2/12rntA[qTe,
The calculation of the absolute decay rates [2,4] requires the full account of escapes across the barrier. However, the relative variation of the decay rate under p u m p i n g can readily be found from eq. (2) and is given by the ratio D / D o = 1 + S(~c), which amounts to reduction of the effective depth of the well by TS(oc). After simple calculations one arrives at
D / D o = 1 + (d~2/4mTy 2 )
~5 × Re f0 z( y )[1 - exp( - 1 ' )] y
12 d l', (3)
where y = - E / T and the function z ( y ) obeys the equation v
d2z d2 (~ + (1 - y ) + dy 2 ~
-
iyqT Y
-
1) 4T' . z
= -¢,,.
(4)
The singularity-free function z ( y ) is found provided that z(0) = z(ov) = 0. Solutions of eq. (4) in limiting cases give some analytical results. If qT<< 7 << w(0) the distribution f(E, O) deviates from equilibrium in a region 8E - 7 / q , and
v/Tq T << 1,1 I << qT, D/D4}
and
(7)
I + d2/24mA2T,
qT<< 1,1 I << ~0(0).
(8)
We have neglected here all the q u a n t u m effects. This is justified, if T>>ko~(0) and besides, the perturbed part of f ( E , ~) comprises many energy levels. The last condition gives a lower bound for the d a m p i n g 7 >> q T ( k o ~ / T ) : . From eqs. (5) (8) it follows that the decay rate exhibits a steep threshold rise as ~2 decreases and crosses o~(0). The subsequent slow decrease of D, as ~ goes further down to ~ - o~(0), is beyond the scope of our approach. I n experiments [1] the potential V(x) was nearly cubic, and q = (5/36)o~(0)/V. The quality factor Q = o~(0)/7 was about 13. Estimation of the width of the nonequilibrium region 8 E - y / q - V / 2 shows, that substantial part of the well is perturbed by pumping. The observed enhancement of the decay rate [1] is in a broad agreement with our expectations, but the threshold behavior is smeared out because of the low Q factor. Our results are only applicable if 8E << V, which will be the case at somewhat larger values of Q.
D/D o = 1 + ( ~2/4yrnqT 2) ×[1-exp(-2v~l/y)]
'
(5)
If "{<
D / D o = 1 + ( ' ~ 2 / 4 7 r n q T 2 )[1 - e x p ( - J / q T ) ] .
(6)
428
References
[1] M.H. Devoret, J.M. Martinis. D. Esteve and J. Clarke. Phys. Rev. Len. 53 (1984) 1260. [2] tt.A. Kramers. Physica 7 (1940) 284. [3] H. Risken, Springer series in sinergetics. Vol. 18. The Fokker-Ptanck equation (Springer, Berlin. 1984). [4] V.I. Mel'nikov, Physica A 130 (1985) 606.