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Josephson phase dynamics in 3He weak links
Physica B 284}288 (2000) 285}286
Josephson phase dynamics in He weak links Keiko Matsunaga *, Munehiro Nishida , Daichi Matsumoto , Susumu Kurihara , Noriyuki Hatakenaka , Hideaki Takayanagi Department of Physics, Waseda University, Shinjuku-ku, Tokyo 169-8555, Japan NTT Basic Research Laboratories, Atsugi, Kanagawa 243-0198, Japan
Abstract We propose a simple phenomenological model for dissipative dynamics of a super#uid. Qualitative agreement between the results of our numerical calculations and of the experiment on a super#uid helium three weak link system gives strong support for our model. 2000 Elsevier Science B.V. All rights reserved. Keywords: Ginzburg}Landau theory; He super#uid; Josephson e!ect; Nonlinear SchroK dinger equation; Time-dependent Ginzburg}Landau
1. Introduction Recent discovery [1] of AC Josephson e!ect in super#uid He has raised a renewed interest in the behavior of super#uid in the mesoscopic regime. Even at low temperatures, dissipation is known to be of crucial importance in such a system. However, we still do not have a reliable equation for super#uid with dissipation. Here we propose a phenomenological equation, which is a combination of the non-linear SchroK dinger equation with time-dependent Ginzburg}Landau-type damping. This model may be applicable to various kinds of super#uid, like that in Bose}Einstein condensation or in mesoscopic superconductor.
*t (t) (i!c) "[E #; N !k]t (t)#Kt (t), *t *t (t) (i!c) "[E #; N !k]t (t)#Kt (t), *t
2. Model The equation we propose for the macroscopic onebody wave function (the order parameter) W(x, t) is * (i!c) W(x, t) *t
where k is the chemical potential, g is the coupling constant and <(x) is the single-particle potential. In the limit of small amplitude, long wavelength and constant <, the above equation gives the dispersion relation u"(2k!ic. We can thus interpret c as a damping constant at least in this limit. Let us now suppose that <(x) represents a barrier separating the condensate into two parts coupled weakly by tunneling. The wave function can then be written as W"t (t)U (x)#t (t)U (x), where U is the ground state of each part slightly overlapping each other. We then obtain the equations for time-dependent coe$cients:
" ! !k#<(x)#g"W(x, t)" W(x, t), 2m * Corresponding author. Fax: #81-3-5286-3447. E-mail address: [email protected] (K. Matsunaga)
where E , ; and K correspond to kinetic energy, interaction energy and tunneling rate, respectively. Under the symmetrical condition E "E "0 and ; "; ";, we rewrite the equations in terms of phase di!erence h, normalized number di!erence z, and the total number of condensed particles N(t "(N e F , N"N #N , z"(N !N )/N, h"h !h ). Furthermore, we adopt normalized para meters q"(K/ )t and a"N ;/K, where N is N(q"0). AC Josephson e!ects are conveniently visualized as the
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 6 1 7 - 4
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K. Matsunaga et al. / Physica B 284}288 (2000) 285}286
modes of the pendulum, and the transition between the two would be induced by the e!ect of dissipation. From our model, since the frequency of the rotational mode is given by u +(a/2((az !4(cos h !1) #(az !4(cos h #1)), the dimensionless coupling constant a must be su$ciently large a*a +[4(cos h #1)]/z to have rota tional mode. Other features in the undamped (c"0) case has already been extensively studied [3,4]. In cO0 regime, the damping constant for h is roughly ca. If a<1, the equations for h reduces to the RSJ model and c corresponds to E (E R)\ (R is the resistivity, E the charg! ( ! ing energy, E the Josephson energy). The time evolution ( of total particle number N generally appears as follows: "rst, it decreases as e\AO, then starts increasing at q"(ac)\, and "nally is restored to the initial number. This process could o!er quasi-particle excitations and recondensation as losing their energy. The results of numerical calculation, z versus q, are shown in Fig. 1. The main features of the experimental results are qualitatively well reproduced, for example: the transition between the two kinds of oscillations, the change in frequency with time, damping processes, sharp waveforms near the transition point and so on. We must take the experimental situation, including mechanical forces, into account to "t our calculation quantitatively.
4. Conclusion
Fig. 1. Normalized number di!erence as a function of time. c"3;10\, (a) a"4800, (b) a"1200, (c) a"850.
motion of a pendulum by the Feynman two-state model [2]. In this case, z corresponds to the angular momentum of the pendulum and h to the angular position.
3. Damping and transition Two distinct kinds of oscillations observed in the experiment [1] are interpreted as rotational and oscillatory
We have found that the characteristic behavior observed in Ref. [1] can be successfully interpreted by our phenomenological model. Justi"cation of our model by a microscopic calculation is the subject of future work.
References [1] S. Backhaus, S.V. Pereverzev, A. Loshak, J.C. Davis, R.E. Packard, Science 278 (1997) 1435. [2] R. Feynman, Statistical Mechanics, Addison-Wesley, Redwood City, 1972. [3] A. Smerzi, S. Fantoni, S. Giovanazzi, S.R. Shenoy, Phys. Rev. Lett. 79 (1997) 4950. [4] N. Hatakenaka, J. Phys. Soc. Japan 67 (1998) 3672.
Josephson phase dynamics in He weak links Keiko Matsunaga *, Munehiro Nishida , Daichi Matsumoto , Susumu Kurihara , Noriyuki Hatakenaka , Hideaki Takayanagi Department of Physics, Waseda University, Shinjuku-ku, Tokyo 169-8555, Japan NTT Basic Research Laboratories, Atsugi, Kanagawa 243-0198, Japan
Abstract We propose a simple phenomenological model for dissipative dynamics of a super#uid. Qualitative agreement between the results of our numerical calculations and of the experiment on a super#uid helium three weak link system gives strong support for our model. 2000 Elsevier Science B.V. All rights reserved. Keywords: Ginzburg}Landau theory; He super#uid; Josephson e!ect; Nonlinear SchroK dinger equation; Time-dependent Ginzburg}Landau
1. Introduction Recent discovery [1] of AC Josephson e!ect in super#uid He has raised a renewed interest in the behavior of super#uid in the mesoscopic regime. Even at low temperatures, dissipation is known to be of crucial importance in such a system. However, we still do not have a reliable equation for super#uid with dissipation. Here we propose a phenomenological equation, which is a combination of the non-linear SchroK dinger equation with time-dependent Ginzburg}Landau-type damping. This model may be applicable to various kinds of super#uid, like that in Bose}Einstein condensation or in mesoscopic superconductor.
*t (t) (i!c) "[E #; N !k]t (t)#Kt (t), *t *t (t) (i!c) "[E #; N !k]t (t)#Kt (t), *t
2. Model The equation we propose for the macroscopic onebody wave function (the order parameter) W(x, t) is * (i!c) W(x, t) *t
where k is the chemical potential, g is the coupling constant and <(x) is the single-particle potential. In the limit of small amplitude, long wavelength and constant <, the above equation gives the dispersion relation u"(2k!ic. We can thus interpret c as a damping constant at least in this limit. Let us now suppose that <(x) represents a barrier separating the condensate into two parts coupled weakly by tunneling. The wave function can then be written as W"t (t)U (x)#t (t)U (x), where U is the ground state of each part slightly overlapping each other. We then obtain the equations for time-dependent coe$cients:
" ! !k#<(x)#g"W(x, t)" W(x, t), 2m * Corresponding author. Fax: #81-3-5286-3447. E-mail address: [email protected] (K. Matsunaga)
where E , ; and K correspond to kinetic energy, interaction energy and tunneling rate, respectively. Under the symmetrical condition E "E "0 and ; "; ";, we rewrite the equations in terms of phase di!erence h, normalized number di!erence z, and the total number of condensed particles N(t "(N e F , N"N #N , z"(N !N )/N, h"h !h ). Furthermore, we adopt normalized para meters q"(K/ )t and a"N ;/K, where N is N(q"0). AC Josephson e!ects are conveniently visualized as the
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 6 1 7 - 4
286
K. Matsunaga et al. / Physica B 284}288 (2000) 285}286
modes of the pendulum, and the transition between the two would be induced by the e!ect of dissipation. From our model, since the frequency of the rotational mode is given by u +(a/2((az !4(cos h !1) #(az !4(cos h #1)), the dimensionless coupling constant a must be su$ciently large a*a +[4(cos h #1)]/z to have rota tional mode. Other features in the undamped (c"0) case has already been extensively studied [3,4]. In cO0 regime, the damping constant for h is roughly ca. If a<1, the equations for h reduces to the RSJ model and c corresponds to E (E R)\ (R is the resistivity, E the charg! ( ! ing energy, E the Josephson energy). The time evolution ( of total particle number N generally appears as follows: "rst, it decreases as e\AO, then starts increasing at q"(ac)\, and "nally is restored to the initial number. This process could o!er quasi-particle excitations and recondensation as losing their energy. The results of numerical calculation, z versus q, are shown in Fig. 1. The main features of the experimental results are qualitatively well reproduced, for example: the transition between the two kinds of oscillations, the change in frequency with time, damping processes, sharp waveforms near the transition point and so on. We must take the experimental situation, including mechanical forces, into account to "t our calculation quantitatively.
4. Conclusion
Fig. 1. Normalized number di!erence as a function of time. c"3;10\, (a) a"4800, (b) a"1200, (c) a"850.
motion of a pendulum by the Feynman two-state model [2]. In this case, z corresponds to the angular momentum of the pendulum and h to the angular position.
3. Damping and transition Two distinct kinds of oscillations observed in the experiment [1] are interpreted as rotational and oscillatory
We have found that the characteristic behavior observed in Ref. [1] can be successfully interpreted by our phenomenological model. Justi"cation of our model by a microscopic calculation is the subject of future work.
References [1] S. Backhaus, S.V. Pereverzev, A. Loshak, J.C. Davis, R.E. Packard, Science 278 (1997) 1435. [2] R. Feynman, Statistical Mechanics, Addison-Wesley, Redwood City, 1972. [3] A. Smerzi, S. Fantoni, S. Giovanazzi, S.R. Shenoy, Phys. Rev. Lett. 79 (1997) 4950. [4] N. Hatakenaka, J. Phys. Soc. Japan 67 (1998) 3672.