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Macroscopic quantum tunneling and thermal activation in dc SQUID
Physica C 153-155 (1988) 1409-1410 North-HoUand, Amsterdam
MACROSCOPICQUANTUMTUNNELINGAND THERMALACTIVATIONIN A DC SQUID
F. Sharifi, J. L. Gavilano*, and D. J. Van Harlingen Department of Physics, University of I l l i n o i s , Urbana, IL 61801 We have studied the transition rate from the zero-voltage state in a dc SQUID as a function of applied f l u x and temperature. We observe a crossover from thermal activation to macroscopic quantum tunneling at a flux-dependent temperature. The thermal rates are s i g n i f i c a n t l y suppressed, suggesting that the potential barrier for escape is e f f e c t i v e l y enhanced by the interaction of the macroscopic degrees of freedom in the two-dimensional dc SQUID potential. I . INTRODUCTION Recent experiments tested the a p p l i c a b i l i t y of quantum mechanics to macroscopic physical systems by measuring transition rates from the zero-voltage state of Josephson tunnel junctions or between d i f f e r e n t f l u x states in an r f SQUID. In these systems, the condensate phase difference across a Josephson junction evolves in a one-dimensional "washboard" potential. In the zero-voltage state, the phase is trapped in a metastable potential well and can only escape by thermal activation (TA) over the barrier or by macroscopic quantum tunneling (MQT) through the barrier. The effect of dissipation and the interplay of these processes has been a subject of much theoretical concentration(1). We have extended this work to a multidimensional potential system by investigating the thermal and quantum tunneling transition rates in a dc SQUID. The two-dimensional SQUID potential has two degrees of freedom, enabling us to investigate the dynamics and interaction of macroscopic variables. In addition, the amplitude and shape of the potential barriers around a metastable minimum in the dc SQUID potential can be varied widely by applying external bias current and magnetic f l u x . The barrier form affects strongly the thermal activation and quantum tunneling rates and t h e i r dependence on dissipation. 2. DC SQUID POTENTIAL The system studied is a symmetric dc SQUID with loop inductance L and junction c r i t i c a l currents I O. The potential energy contour at current bias I and f l u x bias @ is a twodimensional washboard potential(2) given by
U(61,62) : _cos61 _ cos62 _ ½(61+62)i + ½~#j2 where i = I / I 0 is the normalized bias current and j = (61-62-2xf)/n# is the induced * Present address: I n s t i t u t de Physique, Universit~ de NeuchAtel, Switzerland. 0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
c i r c u l a t i n g current for f l u x bias f = @/@0. Here 61 and 62 are the gauge-invariant phases and # = 2LIo/@0 is the inductance parameter. For #<
] 410
F. Sharifi et aL / Macroscopic quantum tunnefing
transition is substantially rounded, increasing by t y p i c a l l y 0.03% below Tc. Basically, is proportional to the transition rate, while a is a measure of the variation of the rate with bias current. The temperature independence of and o as T approaches zero is the signature of macroscopic quantum tunneling transitions from the zero-voltage state. The temperature variation of the rates at f=I/2 to below 100mK puts an upper bound on the effect of external noise on the t r a n s i t i o n rates, while agreement of the rates at f=O and f=1 precludes f l u x noise as a s i g n i f i c a n t source of escape.
0.17 / / /
f=O /
< :L o.16
•/
b
• f
/ /0 •
,,t1,/0
~i---/
T
""
Tc
s
/
0.15
0.0
n
t
0.5
1.0
1.5
106.6
MOT ( R = 1 3 . 2 n )
.~L106.1
f=O
Tc
105.6
o.o
o15
1.5
T(K) FIGURE 1. D i s t r i b u t i o n peaks and widths f o r f=O showinq thermal a c t i v a t i o n and MQT f i t s .
5. ANALYSIS
We have performed computer simulations to determine d i s t r i b u t i o n peaks and widths as a function of temperature and f l u x bias using a two-dimensional thermal activation model(3). The t r a n s i t i o n rate is given by the usual form r : ~ exp (-AU/kBT), where AU is the barrier height of the saddle and the prefactor ~ depends on the curvatures(2) of the well and saddle both parallel and transverse to the tunneling direction. The only parameter not independently determined from the data is the thermodynamic c r i t i c a l current. We are unable to f i t our thermal results with any choice of I c or any reasonable change of other parameters (see dashed lines labeled TA). Instead, we find that the thermal activation rate is s i g n i f i c a n t l y suppressed in this system. In fact, we can only model our results by assuming that the barrier height is e f f e c t i v e l y increased by a factor of about 2.5; this f i t is denoted MTA. We have no direct physical j u s t i f i c a t i o n for such an enhancement. However, Suhl(4) has recently shown that under some conditions the thermal activation rate in a multidimensional potential system can be s i g n i f i c a n t l y reduced from the usual Kramers prediction. This occurs because of coupling between the degrees of freedom in the system via the thermal bath, diverting thermal energy away from the mode capable of activating from the well. With the enhanced barrier height, we are able to f i t our thermal results accurately well above Tc for all bias fluxes and extract a value for I c. We have not attempted a WKB calculation to determine MQT rates for the dc SQUID. However, for small # we expect the single junction result to be a good approximation(5). Using t h i s model, we find good agreement with the asymptotic peak value for an effective damping resistance of 15~, which corresponds closely to the normal state resistance of our SQUID. This resistance also couples thermal and quantum processes(6) and may contribute to the rounding of the crossover region peaks; we have included quantum corrections to the thermal rate above Tc in our simulations.
9.8 ACKNOWLEDGEMENTS \
\\ ~
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We thank R. Wakai, G. Hilton, K. Springer, Y. Chert, and A. Leggett for valuable discussions and technical advice. This work supported by NSF grants DMR-84-11631 and DMR-86-12860 (MRL).
MOT(R=15.6O)
:L
._v 9.3
\\
A
REFERENCES
v
f=1/2 8.8 0.o
I
\ \ i
n
0.5
1.0 T(K)
FIGURE 2.
Distribution peaks for f=I/2.
.5
( I ) A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983). (2) C. Tesche, J. Low T. Phys. 44, 119 (1981). (3) E. Ben-Jacob, et a l . , J. Appl. Phys. 54, 6533 (1983). (4) H. Suhl, Phys. Rev. Lett. 60, 473 (1988). (5) Y-C. Chen, J. Low T. Phys. 65, 133 (1986). (6) H. Grabert, P. Olschowski, and U. Weiss, Phys. Rev. B 36, 1931 (1987).
MACROSCOPICQUANTUMTUNNELINGAND THERMALACTIVATIONIN A DC SQUID
F. Sharifi, J. L. Gavilano*, and D. J. Van Harlingen Department of Physics, University of I l l i n o i s , Urbana, IL 61801 We have studied the transition rate from the zero-voltage state in a dc SQUID as a function of applied f l u x and temperature. We observe a crossover from thermal activation to macroscopic quantum tunneling at a flux-dependent temperature. The thermal rates are s i g n i f i c a n t l y suppressed, suggesting that the potential barrier for escape is e f f e c t i v e l y enhanced by the interaction of the macroscopic degrees of freedom in the two-dimensional dc SQUID potential. I . INTRODUCTION Recent experiments tested the a p p l i c a b i l i t y of quantum mechanics to macroscopic physical systems by measuring transition rates from the zero-voltage state of Josephson tunnel junctions or between d i f f e r e n t f l u x states in an r f SQUID. In these systems, the condensate phase difference across a Josephson junction evolves in a one-dimensional "washboard" potential. In the zero-voltage state, the phase is trapped in a metastable potential well and can only escape by thermal activation (TA) over the barrier or by macroscopic quantum tunneling (MQT) through the barrier. The effect of dissipation and the interplay of these processes has been a subject of much theoretical concentration(1). We have extended this work to a multidimensional potential system by investigating the thermal and quantum tunneling transition rates in a dc SQUID. The two-dimensional SQUID potential has two degrees of freedom, enabling us to investigate the dynamics and interaction of macroscopic variables. In addition, the amplitude and shape of the potential barriers around a metastable minimum in the dc SQUID potential can be varied widely by applying external bias current and magnetic f l u x . The barrier form affects strongly the thermal activation and quantum tunneling rates and t h e i r dependence on dissipation. 2. DC SQUID POTENTIAL The system studied is a symmetric dc SQUID with loop inductance L and junction c r i t i c a l currents I O. The potential energy contour at current bias I and f l u x bias @ is a twodimensional washboard potential(2) given by
U(61,62) : _cos61 _ cos62 _ ½(61+62)i + ½~#j2 where i = I / I 0 is the normalized bias current and j = (61-62-2xf)/n# is the induced * Present address: I n s t i t u t de Physique, Universit~ de NeuchAtel, Switzerland. 0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
c i r c u l a t i n g current for f l u x bias f = @/@0. Here 61 and 62 are the gauge-invariant phases and # = 2LIo/@0 is the inductance parameter. For #<
] 410
F. Sharifi et aL / Macroscopic quantum tunnefing
transition is substantially rounded, increasing by t y p i c a l l y 0.03% below Tc. Basically, is proportional to the transition rate, while a is a measure of the variation of the rate with bias current. The temperature independence of and o as T approaches zero is the signature of macroscopic quantum tunneling transitions from the zero-voltage state. The temperature variation of the rates at f=I/2 to below 100mK puts an upper bound on the effect of external noise on the t r a n s i t i o n rates, while agreement of the rates at f=O and f=1 precludes f l u x noise as a s i g n i f i c a n t source of escape.
0.17 / / /
f=O /
< :L o.16
•/
b
• f
/ /0 •
,,t1,/0
~i---/
T
""
Tc
s
/
0.15
0.0
n
t
0.5
1.0
1.5
106.6
MOT ( R = 1 3 . 2 n )
.~L106.1
f=O
Tc
105.6
o.o
o15
1.5
T(K) FIGURE 1. D i s t r i b u t i o n peaks and widths f o r f=O showinq thermal a c t i v a t i o n and MQT f i t s .
5. ANALYSIS
We have performed computer simulations to determine d i s t r i b u t i o n peaks and widths as a function of temperature and f l u x bias using a two-dimensional thermal activation model(3). The t r a n s i t i o n rate is given by the usual form r : ~ exp (-AU/kBT), where AU is the barrier height of the saddle and the prefactor ~ depends on the curvatures(2) of the well and saddle both parallel and transverse to the tunneling direction. The only parameter not independently determined from the data is the thermodynamic c r i t i c a l current. We are unable to f i t our thermal results with any choice of I c or any reasonable change of other parameters (see dashed lines labeled TA). Instead, we find that the thermal activation rate is s i g n i f i c a n t l y suppressed in this system. In fact, we can only model our results by assuming that the barrier height is e f f e c t i v e l y increased by a factor of about 2.5; this f i t is denoted MTA. We have no direct physical j u s t i f i c a t i o n for such an enhancement. However, Suhl(4) has recently shown that under some conditions the thermal activation rate in a multidimensional potential system can be s i g n i f i c a n t l y reduced from the usual Kramers prediction. This occurs because of coupling between the degrees of freedom in the system via the thermal bath, diverting thermal energy away from the mode capable of activating from the well. With the enhanced barrier height, we are able to f i t our thermal results accurately well above Tc for all bias fluxes and extract a value for I c. We have not attempted a WKB calculation to determine MQT rates for the dc SQUID. However, for small # we expect the single junction result to be a good approximation(5). Using t h i s model, we find good agreement with the asymptotic peak value for an effective damping resistance of 15~, which corresponds closely to the normal state resistance of our SQUID. This resistance also couples thermal and quantum processes(6) and may contribute to the rounding of the crossover region peaks; we have included quantum corrections to the thermal rate above Tc in our simulations.
9.8 ACKNOWLEDGEMENTS \
\\ ~
<
We thank R. Wakai, G. Hilton, K. Springer, Y. Chert, and A. Leggett for valuable discussions and technical advice. This work supported by NSF grants DMR-84-11631 and DMR-86-12860 (MRL).
MOT(R=15.6O)
:L
._v 9.3
\\
A
REFERENCES
v
f=1/2 8.8 0.o
I
\ \ i
n
0.5
1.0 T(K)
FIGURE 2.
Distribution peaks for f=I/2.
.5
( I ) A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983). (2) C. Tesche, J. Low T. Phys. 44, 119 (1981). (3) E. Ben-Jacob, et a l . , J. Appl. Phys. 54, 6533 (1983). (4) H. Suhl, Phys. Rev. Lett. 60, 473 (1988). (5) Y-C. Chen, J. Low T. Phys. 65, 133 (1986). (6) H. Grabert, P. Olschowski, and U. Weiss, Phys. Rev. B 36, 1931 (1987).