Prog. Theor. Phys. Vol. 71 No. 6 (1984) pp. 1142-1151

[ Full Text PDF : FREE ACCESS (640K) ]

The Quantum Mechanical Tunneling under the Influence of Environment

— Using the Method of Density Matrix —

Hiroshi Kawamoto

Department of Physics, Kyoto University, Kyoto 606

(Received December 8, 1983)

Abstract:

The purpose of this paper is to study an effect of an environment on a quantum tunneling. Here, the system we consider is of a symmetric two-level type with “bare” tunneling frequency Δω₀, and the correlation time of actions due to the environment, τ_c, is assumed to be much shorter than Δω₀^-1. In the investigations, the method of density matrix is used, and a coupling between the system and the environment is included up to lowest nontrivial order. After calculations, we reach the following conclusions: First, the possibility of the quantum tunneling is, in a high temperature range β\hbarΔω₀ ≪1, washed out by the thermal actions from the environment. In the inverse case β\hbarΔω₀ ≫1, where the quantum fluctuation of the environment plays a dominant role, the possibility of the quantum tunneling remains. Secondaly, in the case β\hbarΔω₀ ≫1, the dissipation effect depresses the tunneling probability, while the potential barrier is effectively lowered by the actions due to the environment.

URL : http://ptp.ipap.jp/link?PTP/71/1142/
DOI : 10.1143/PTP.71.1142

References:

W. den Boer and R. de Bruyn Ouboter, Physica 98B (1980), 185.
R. F. Voss and R. A. Webb, Phys. Rev. Lett. 47 (1981), 265[APS].
I. M. Dmitrenko, G. M. Tsoi and V. I. Shnyrkov, Sov. J. Low Temp. Phys. 8(6) (1982), 330.
D. W. Bol. R. van Weelderen and R. de Bruyn Ouboter, Physica 122B (1983), 1.
A. J. Leggett, Prog. Theor. Phys. Suppl. No. 69 (1980), 80[PTP].
A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46 (1981), 211[APS].
A. Widom and T. D. Clark, Phys. Rev. Lett. 48 (1982), 63[APS].
V. Ambegaokar, U. Eckern and G. Schön, Phys. Rev. Lett. 48 (1982), 1745[APS].
S. Chakravarty, Phys. Rev. Lett. 49 (1982), 681[APS].
A. J. Bray and M. A. Moore. Phys. Rev. Lett. 49 (1982), 1545[APS].
A. Schmid, J. Low Temp. Phys. 49 (1982), 609[CrossRef].
A. A. Golub and V. P. Iordatii, JETP Lett. 36 (1982), 227.
A. I. Larkin and Yu. N. Ovchnnikov, JETP Lett. 37 (1983), 383.
J. Kurkijärvi, Phys. Rev. B6 (1972), 832[APS].
D. E. McCumber, J. Appl. Phys. 39 (1968), 3113[CrossRef].
W. C. Stewart. Appl. Phys. Lett. 12 (1968), 277[CrossRef].
Yu. M. Ivanchenko and L. A. Zil'bermann, JETP Lett. 8 (1968), 113.
V. Ambegaokar and B. I. Halperin, Phys. Rev. Lett. 22 (1969), 1364[APS].
K. K. Likharev and V. K. Semenov. JETP Lett. 15 (1972), 442.
R. H. Koch, D. J. Van Harlingen and J. Clarke, Phys. Rev. Lett. 45 (1980), 2132[APS]; ibid. 47 (1981), 1216[APS].
U. Fano, Rev. Mod. Phys. 29 (1957), 74[APS].
M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17 (1945), 323[APS].
R. Zwanzig, J. Chem. Phys. 33 (1960), 1338[CrossRef].
R. Kubo, in Lectures at Sitges International School of Statistical Mechanics, 1974, L. Garrid ed. (Springer Verlag, Berlin, 1975).
A. J. Dahm, A. Denenstein, D. N. Langenberg, W. H. Parker, D. Rogovin and D. J. Scalapino, Phys. Rev. Lett. 22 (1969), 1416[APS].
K. K. Likharev, Rev. Mod. Phys. 51 (1979), 101[APS].
H. B. Callen and T. A. Welton, Phys. Rev. 83 (1951), 34[APS].
H. Mori, Prog. Theor. Phys. 33 (1965), 127[PTP].
M. Lax, Phys. Rev. 145 (1966), 110[APS].
M. Simonius, Phys. Rev. Lett. 40 (1978), 980[APS].
A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 48 (1982), 1571[APS].
A. Widom and T. D. Clark, Phys. Rev. Lett. 48 (1982), 1572[APS].
N. R. Werthamer, Phys. Rev. 147 (1966), 225[APS].
M. Tinkham, Introduction to Superconductivity (McGraw-Hill, New York, 1975), chap 4.

Citing Article(s) :

Progress of Theoretical Physics Vol. 80 No. 6 (1988) pp. 973-985 :
Theoretical Study of Macroscopic Quantum Coherence in SQUIDS

Hiroshi Kawamoto