Quantum stochastic processes: boson and fermion Brownian motion

Dynamics of quantum systems which are stochastically perturbed by linear coupling to the reservoir can be studied in terms of quantum stochastic differential equations (for example, quantum stochastic Liouville equation and quantum Langevin equation). In order to work it out one needs to define t...

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Дата:	2003
Автори:	Kobryn, A.E., Hayashi, T., Arimitsu, T.
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Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2003
Назва видання:	Condensed Matter Physics
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Цитувати:	Quantum stochastic processes: boson and fermion Brownian motion / A.E. Kobryn , T. Hayashi, T. Arimitsu // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 637-646. — Бібліогр.: 40 назв. — англ.

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spelling	irk-123456789-1207652017-06-13T03:03:57Z Quantum stochastic processes: boson and fermion Brownian motion Kobryn, A.E. Hayashi, T. Arimitsu, T. Dynamics of quantum systems which are stochastically perturbed by linear coupling to the reservoir can be studied in terms of quantum stochastic differential equations (for example, quantum stochastic Liouville equation and quantum Langevin equation). In order to work it out one needs to define the quantum Brownian motion. As far as only its boson version has been known until recently, in the present paper we present the definition which makes it possible to consider the fermion Brownian motion as well. Динаміка квантових систем, що зазнають збурення через лінійну взаємодію із термостатом стохастично, може бути описана за допомогою квантових стохастичних диференціальних рівнянь (наприклад, квантового стохастичного рівняння Ліувіля, або квантового рівняння Ланжевена). Для цього необхідно дати означення квантового броунівського руху. Оскільки донедавна воно було наведене лише для бозонних систем, у даній роботі показується як таке означення можна ввести у випадку ферміонних систем. 2003 Article Quantum stochastic processes: boson and fermion Brownian motion / A.E. Kobryn , T. Hayashi, T. Arimitsu // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 637-646. — Бібліогр.: 40 назв. — англ. 1607-324X PACS: 02.50.Ey, 05.30.-d DOI:10.5488/CMP.6.4.637 http://dspace.nbuv.gov.ua/handle/123456789/120765 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Dynamics of quantum systems which are stochastically perturbed by linear coupling to the reservoir can be studied in terms of quantum stochastic differential equations (for example, quantum stochastic Liouville equation and quantum Langevin equation). In order to work it out one needs to define the quantum Brownian motion. As far as only its boson version has been known until recently, in the present paper we present the definition which makes it possible to consider the fermion Brownian motion as well.
format	Article
author	Kobryn, A.E. Hayashi, T. Arimitsu, T.
spellingShingle	Kobryn, A.E. Hayashi, T. Arimitsu, T. Quantum stochastic processes: boson and fermion Brownian motion Condensed Matter Physics
author_facet	Kobryn, A.E. Hayashi, T. Arimitsu, T.
author_sort	Kobryn, A.E.
title	Quantum stochastic processes: boson and fermion Brownian motion
title_short	Quantum stochastic processes: boson and fermion Brownian motion
title_full	Quantum stochastic processes: boson and fermion Brownian motion
title_fullStr	Quantum stochastic processes: boson and fermion Brownian motion
title_full_unstemmed	Quantum stochastic processes: boson and fermion Brownian motion
title_sort	quantum stochastic processes: boson and fermion brownian motion
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2003
url	http://dspace.nbuv.gov.ua/handle/123456789/120765
citation_txt	Quantum stochastic processes: boson and fermion Brownian motion / A.E. Kobryn , T. Hayashi, T. Arimitsu // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 637-646. — Бібліогр.: 40 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT kobrynae quantumstochasticprocessesbosonandfermionbrownianmotion AT hayashit quantumstochasticprocessesbosonandfermionbrownianmotion AT arimitsut quantumstochasticprocessesbosonandfermionbrownianmotion
first_indexed	2025-07-08T18:32:38Z
last_indexed	2025-07-08T18:32:38Z
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fulltext	Condensed Matter Physics, 2003, Vol. 6, No. 4(36), pp. 637–646 Quantum stochastic processes: boson and fermion Brownian motion∗ A.E.Kobryn 1,2 , T.Hayashi 1 , T.Arimitsu 1 1 Institute of Physics, University of Tsukuba, Ibaraki 305-8571, Japan 2 Institute for Molecular Science, Myodaiji, Okazaki, Aichi 444-8585, Japan Received September 5, 2003 Dynamics of quantum systems which are stochastically perturbed by lin- ear coupling to the reservoir can be studied in terms of quantum stochastic differential equations (for example, quantum stochastic Liouville equation and quantum Langevin equation). In order to work it out one needs to de- fine the quantum Brownian motion. As far as only its boson version has been known until recently, in the present paper we present the definition which makes it possible to consider the fermion Brownian motion as well. Key words: stochastic processes, boson Brownian motion, fermion Brownian motion PACS: 02.50.Ey, 05.30.-d The description of time-dependent behavior of non-equilibrium systems involv- ing stochastic forces can be given with the help of the Langevin equation. It is the stochastic differential equation for dynamic variables and is of fundamental impor- tance in the theory of Brownian motion [1,2]. Random forces in Langevin equation are usually described by Gaussian white stochastic processes [3] since such a descrip- tion is a convenient model for the processes with short correlation times. Similarly, from the mathematical point of view, white noise can be considered to be a limit of some well specified stochastic processes. Stochastic integral with respect to such processes is defined as a kind of a Riemann-Stieltjes one [4] where multiplication between the stochastic increment and integrand is commonly considered in the form of Itô [5] or Stratonovich [6]. The Langevin equation can be used to calculate various time correlation func- tions. Now it is radically extended to solve numerous problems arising in different areas [7–11]. In particular, the theory of Brownian motion itself has been extended to the situations where the “Brownian particle” is not a real particle anymore, but instead it possesses some collective properties of a macroscopic system. A corre- sponding equation in the phase space or the Liouville space of quantum mechanical ∗Invited paper. c© A.E.Kobryn, T.Hayashi, T.Arimitsu 637 A.E.Kobryn, T.Hayashi, T.Arimitsu statistical operators can be also considered as a sort of stochastic differential equa- tion. In order to investigate classical stochastic systems, the stochastic Liouville equation was first introduced by Anderson [12] and Kubo [13–15]. There were several attempts to extend the classical theory (both Langevin and stochastic Liouville equations) to the quantum cases. Study of the Langevin equa- tion for quantum systems has its origin in papers by Senitzky [16], Schwinger [17], Haken [18–20] and Lax [21], where they investigated a quantum mechanical damped harmonic oscillator in connection with laser systems. In particular, it was shown that the quantum noise, i.e. the spontaneous emission, can be treated in a way similar to the thermal fluctuations, and that the noise source has non-zero second moments proportional to a quantity which can be associated with a quantum analogue of a diffusion coefficient. As it was noticed by Kubo [22] in his discussion with van Kam- pen, the random force must be an operator defined in its own Hilbert space, which does not happen in the classical case since there is no consideration of space for the random force. Mathematical study of the quantum stochastic processes was initiated by Davies [23,24]. Essential progress in this subject has been made by mathematicians [25–31]. For instance, quantum mechanical analogues of Wiener processes [25] and quantum Itô formula for boson systems [26,27] were defined first by Hudson and co-authors. The classical Brownian motion is replaced here by the pair of one-parameter unitary group automorphisms, namely by the boson random force annihilation and creation operators with time indices in the boson Fock space, named quantum Brownian motion. Fermion stochastic calculus was first suggested by Applebaum, Hudson and Parthasarathy [32–35]. In these papers, they developed the fermion analog of the corresponding boson theory [27] in which the annihilation and creation processes are fermion field operators in the fermion Fock space. Within the framework of this formalism, the Itô-Clifford integral [36] – fermion analog of the classical Brownian motion – is contained as a special case. In this paper we give the definition of boson and fermion Brownian motion following the approach suggested by Hudson and Parthasarathy [27,34]. We do not pay much attention to purely mathematical aspects of the subject. Instead of that we try to represent the material in the terms which are easy for physicists to understand. We also consider quantum Brownian motion with allowance for thermal degree of freedom. Let Γ0 s denote the boson Fock space (the symmetric Fock space) over the Hilbert space H = L2(R+) of square integrable functions, and bt and b † t denote, respectively, boson annihilation and creation operators at time t ∈ [0,∞) satisfying the canonical commutation relations [bt, b † s] = δ(t− s), [bt, bs] = [b†t , b † s] = 0. (1) The bra- and ket-vacuums (\| and \|), respectively, are defined by (\|b†t = 0 and bt\|) = 0. The space Γ0 s is equipped with a total family of exponential vectors (e(f)\| = (\| exp { ∫ ∞ 0 dt f ∗(t)bt } , \|e(g)) = exp { ∫ ∞ 0 dt g(t)b†t } \|), (2) 638 Boson and fermion Brownian motion whose overlapping is (e(f)\|e(g)) = exp {∫ ∞ 0 dt f ∗(t)g(t) } . Here f, g ∈ H. The dense span of exponential vectors we denote by E. Operators bt, b † t and exponential vectors are characterized by the relations (e(f)\|b†t = (e(f)\|f ∗(t) and bt\|e(g)) = g(t)\|e(g)). Let us introduce operator Ut defined as Ut = σ<P[0,t] + σ>P(t,∞), (3) where σ< and σ> are two independent parameters taking values ±1, and P[a,b] (a6b) is an operator on H of multiplication by the indicator function whose action reads P[a,b] ∫ ∞ 0 dt g(t) = ∫ ∞ 0 dt θ(t− a)θ(b− t)g(t). (4) Here, θ(t) is the step function specified as θ(t) = { 1, t > 0, 0, t < 0. (5) Operator P[a,b] has the following properties: P 2 [a,b] = P[a,b], P † [a,b] = P[a,b], P[a,b]P[c,d] = P[c,d]P[a,b], (6) which are easily verified using definition (4). Then we see that operator Ut is unitary, and satisfies U2 t = I, U † t = Ut, UtUs = UsUt, (7) where I is the identity operator. The so-called reflection process Jt ≡ Jt(Ut), t ∈ R+, whose action on E is given by [34] Jt\|e(g)) = \|e(Utg)) = exp { Ut ∫ ∞ 0 dt′ g(t′)b†t′ } \|), (8) inherits properties of the operator Ut (7), i.e. J2 t = I, J † t = Jt, JtJs = JsJt, (9) and does not change the vacuum: (\|Jt = (\|, Jt\|) = \|). Let us now consider new operators bt = Jtbt, b † t = b † tJt. (10) Apparently, they annihilate vacuums: (\|b† t = 0, bt\|) = 0. The following matrix elements (e(f)\|[Jt, bs]−σ\|e(g)) = (e(f)\|{1− σ[σ> + (σ< − σ>)θ(t− s)]}g(s)Jt\|e(g)), (11a) (e(f)\|[bt, b † s]−σ\|e(g)) = (e(f)\|{δ(t− s) + JtJs(σ>σ< − σ)f ∗(s)g(t)}\|e(g)), (11b) 639 A.E.Kobryn, T.Hayashi, T.Arimitsu are valid for f, g ∈ H, where for arbitrary operators A and B the (anti-)commutator is defined as [A, B]−σ = AB − σBA, σ = ±1. (12) Then the requirement of equal-time (anti-)commutativity between Jt and bt, i.e. [Jt, bt]−σ = 0, gives 1− σσ< = 0, while the requirement of canonical (anti-)commu- tation relation [bt, b † s]−σ = δ(t− s) (13) leads to σ>σ<−σ = 0. All those conditions are satisfied when σ< = σ and σ> = +1. Then the operator Ut turns out to be Ut = σP[0,t] + P(t,∞). (14) Note that for a boson system, i.e. σ = 1, Ut = I and the operators bt and b † t reduce, respectively, to bt and b † t . We see that the generalized quantum Brownian motion, defined by Bt = ∫ t 0 dt′ bt′ , B † t = ∫ t 0 dt′ b † t′ , (15) with B0 = 0, B † 0 = 0, satisfies [Bt, B † s]−σ = min(t, s). The case σ = 1 represents the boson Brownian motion [27,30], whereas the case σ = −1 represents the fermion Brownian motion [34]. Their increments annihilate the vacuum, i.e. dBt = Bt+dt −Bt = btdt, dBt\|) = 0, (16a) dB † t = B † t+dt −B † t = b † tdt, (\|dB † t = 0, (16b) and their matrix elements read (e(f)\|dBt\|e(g)) = (e(f)\|Jtg(t)dt\|e(g)), (e(f)\|dBtdBt\|e(g)) = 0, (17a) (e(f)\|dB † t \|e(g)) = (e(f)\|f ∗(t)dtJt\|e(g)), (e(f)\|dBtdB † t \|e(g)) = dt(e(f)\|e(g)). (17b) Here we neglected the terms of the order higher than dt. The latter equations are summarized in the following table of multiplication rules for increments dBt and dB † t : dBt dB † t dt dBt 0 dt 0 dB † t 0 0 0 dt 0 0 0 (18) Quantum stochastic calculus with consideration for thermal degree of freedom can be derived within the framework of Non-Equilibrium Thermo Field Dynamics (NETFD). It is a unified formalism, which enables us to treat dissipative quantum 640 Boson and fermion Brownian motion systems by the method similar to the usual quantum mechanics and quantum field theory, which accommodates the concept of the dual structure in the interpretation of nature, i.e. in terms of the operator algebra and the representation space. Infor- mation about the general structure of NETFD can be found in many papers and we first of all refer to the original source [37] and to the review article [38]. With that in mind, we now consider a tensor product space Γ̂ = Γ0 s⊗ Γ̃0 s. Its vacuum states \|)) and exponential vectors \|e(f, g))) are defined through the “principle of correspondence” [37] \|)) ←→ \|)(\|, (19a) \|e(f, g))) ←→ \|e(f))(e(g)\|. (19b) Annihilation and creation operators acting on Γ̂ are defined through bt\|e(f, g)))←→ bt\|e(f))(e(g)\|, b̃t\|e(f, g)))←→ \|e(f))(e(g)\|b†t , (20a) b † t \|e(f, g)))←→ b † t \|e(f))(e(g)\|, b̃ † t \|e(f, g)))←→ \|e(f))(e(g)\|bt, (20b) and similarly for Jt and J̃t, i.e. Jt\|e(f, g)))←→ Jt\|e(f))(e(g)\|, J̃t\|e(f, g)))←→ \|e(f))(e(g)\|Jt. (21) Algebra of commutation relations between these operators reads [bt, b † s] = [b̃t, b̃ † s] = δ(t− s), [bt, b̃s] = [bt, b̃ † s] = 0, (22a) [Jt, b̃s] = [J̃t, bs] = 0, [Jt, bt]−σ = [J̃t, b̃t]−σ = 0. (22b) Let us now consider new operators defined by bt = Jtbt, b † t = b † tJt, (23a) b̃t = τ̂ J̃tb̃t, b̃ † t = τ̂ b̃ † t J̃t, (23b) where τ̂ is an operator whose characteristic is determined to ensure the (anti-)com- mutation relations [bt, b̃s]−σ = [bt, b̃ † s]−σ = 0, (24a) [bt, b † s]−σ = [b̃t, b̃ † s]−σ = δ(t− s). (24b) First of all, we require that it should commute with Jt and J̃t, and (anti-)commute with bt and b̃t, i.e. [τ̂ , Jt] = [τ̂ , J̃t] = 0 and [τ̂ , bt]−σ = [τ̂ , b̃t]−σ = 0. Other properties of the operator τ̂ can be derived using definitions (23). For example, from (23b) one has [b̃t, b̃ † s]−σ = στ̂ 2[J̃tb̃t, b̃ † sJ̃s]−σ = στ̂ 2δ(t− s), (25) which gives τ̂ 2 = σ. On the other hand, commutativity of tilde conjugation and hermitian conjugation, i.e. (b̃t) † = (b† t) ∼, means τ̂ † = στ̂ . From the requirement that (b̃t) ∼ = bt and noting that (b̃t) ∼ = τ̂ (τ̂)∼Jtbt = τ̂ (τ̂)∼bt, one obtains (τ̂ )∼ = τ̂ †. 641 A.E.Kobryn, T.Hayashi, T.Arimitsu Thermal degree of freedom can be introduced by the Bogolubov transformation in Γ̂. For this purpose we require that the expectation value of b † tbs should be 〈b† tbs〉 = n̄δ(t− s) (26) with n̄ ∈ R+, where 〈. . .〉 = 〈\| . . . \|〉 indicates the expectation with respect to tilde invariant thermal vacuums 〈\| and \|〉. The requirement (26) is consistent with the thermal state conditions for states 〈\| and \|〉 such that 〈\|b̃† t = τ ∗〈\|bt, b̃t\|〉 = τ n̄ 1 + σn̄ b † t \|〉, τ = √ σ. (27) Let us introduce annihilation and creation operators ct = [1 + σn̄]bt − στn̄b̃ † t , (28a) c̃ + ◦ t = b̃ † t − στbt, (28b) and their tilde conjugates. From (27) one has 〈\|c+ ◦ t = 〈\|̃c+ ◦ t = 0 and ct\|〉 = c̃t\|〉 = 0. With the thermal doublet notations b̄ µ t = (b† t ,−τ b̃t), b ν t = collon(bt, τ b̃ † t), (29) c̄ µ t = (c+ ◦ t ,−τ c̃t), c ν t = collon(ct, τ c̃ + ◦ t ), (30) (28) and their tilde conjugates can be written in the form of the Bogolubov trans- formation c µ t = Bµνbν t , c̄ ν t = b̄ µ t [B −1]µν , Bµν = ( 1 + σn̄ −σn̄ −1 1 ) . (31) This transformation is canonical since new operators satisfy canonical (anti-)com- mutation relations [ct, c + ◦ s ]−σ = δ(t− s). (32) In the following, we will use the representation space constructed on vacuums 〈\| and \|〉. Note that 〈\| 6= \|〉†, i.e., it is not a unitary representation. Let Γ̂β denote the Fock space spanned by the basic bra- and ket-vectors introduced by a cyclic operations of ct, c̃t on the thermal bra-vacuum 〈\|, and of c + ◦ t , c̃ + ◦ t on the thermal ket- vacuum \|〉. Quantum Brownian motion at finite temperature is defined in the Fock space Γ̂β by operators B ] t = ∫ t 0 ds b ] s, B̃ ] t = ∫ t 0 ds b̃ ] s, (33) with B ] 0 = 0 and B̃ ] 0 = 0, where ] stands for null or dagger. The explicit rep- resentation of processes B ] t and B̃ ] t can be performed in terms of the Bogolubov transformation. The couple Bt and B † t , for example, is calculated as Bt = ∫ t 0 ds [ cs + στn̄c̃ + ◦ s ] = Ct + στn̄C̃ + ◦ t , (34a) B † t = ∫ t 0 ds [ [1 + σn̄]c+ ◦ s + τ c̃s ] = [1 + σn̄]C+ ◦ t + τ C̃t, (34b) 642 Boson and fermion Brownian motion where we defined new operators C ] t = ∫ t 0 ds c ] s, C̃ ] t = ∫ t 0 ds c̃ ] s, (35) with C ] 0 = 0 and C̃ ] 0 = 0, and ] standing for null or the Venus-mark. Since matrix elements of dC ] t and dC̃ ] t in thermal space Γ̂β read 〈dCt〉 = 〈dC̃t〉 = 0, 〈dC + ◦ t dCt〉 = 〈dC̃tdC̃ + ◦ t 〉 = 0, (36a) 〈dC + ◦ t 〉 = 〈dC̃ + ◦ t 〉 = 0, 〈dCtdC + ◦ t 〉 = 〈dC̃ + ◦ t dC̃t〉 = dt, (36b) the calculation of moments of quantum Brownian motion in the thermal space Γ̂β can be performed, for instance, as 〈dBtdB † t〉 = 〈( dCt + στn̄dC̃ + ◦ t )( [1 + σn̄]dC + ◦ t + τdC̃t )〉 = [1 + σn̄]〈dCtdC + ◦ t 〉 = [1 + σn̄]dt. (37) Repeating this for other pair products of dB ] t, dB̃ ] t and dt, multiplication rules for these increments can be summarized in the following table: dBt dB † t dB̃t dB̃ † t dt dBt 0 [1 + σn̄]dt τ n̄dt 0 0 dB † t n̄dt 0 0 τ [1 + σn̄]dt 0 dB̃t στn̄dt 0 0 [1 + σn̄]dt 0 dB̃ † t 0 στ [1 + σn̄]dt n̄dt 0 0 dt 0 0 0 0 0 (38) As far as we know, the result (38) is the first report about quantum boson and fermion stochastic calculus valid for the stationary non-equilibrium case. Similar result for a purely boson system was obtained in [39]. In [40] we derive stochastic calculus valid for both stationary and non-stationary cases and use them for the analysis of quantum stochastic differential equations. The obtained equations include quantum Langevin equation and quantum stochastic Liouville equation together with the corresponding master equation. The Fokker-Planck equation is derived by taking the random average of the corresponding stochastic Liouville equation. The relation between the Langevin equation and the stochastic Liouville equation, as well as between the Heisenberg equation for the operators of gross variables and the Fokker-Planck equation obtained there, is similar to the one between the Heisenberg equation and the Schrödinger equation in quantum mechanics and field theory. The application of quantum stochastic differential equations in particular problems will be presented elsewhere. 643 A.E.Kobryn, T.Hayashi, T.Arimitsu References 1. Chandrasekhar S. // Rev. Mod. Phys., 1943, vol. 15, p. 1. 2. 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Phys., 1994, vol. 4, p. 26. 39. Saito T., Arimitsu T. // J. Phys. A, 1997, vol. 30, p. 7573. 40. Kobryn A.E., Hayashi T., Arimitsu T. // Ann. Phys., 2003, vol. 308, p. 395. 645 A.E.Kobryn, T.Hayashi, T.Arimitsu Квантове стохастичне числення: бозонний та ферміоний броунівський рух О.Є.Кобрин 1,2 , Ц.Хаяші 1 , Т.Аріміцу 1 1 Інститут фізики Університету м. Цукуба, Японія, префектура Ібаракі 305-8571 2 Інститут молекулярних досліджень, Міодаіджі, Японія, м. Оказакі, префектура Аічі 444-8585 Отримано 5 вересня 2003 р. Динаміка квантових систем, що зазнають збурення через лінійну взаємодію із термостатом стохастично, може бути описана за до- помогою квантових стохастичних диференціальних рівнянь (напри- клад, квантового стохастичного рівняння Ліувіля, або квантового рівняння Ланжевена). Для цього необхідно дати означення кванто- вого броунівського руху. Оскільки донедавна воно було наведене лише для бозонних систем, у даній роботі показується як таке озна- чення можна ввести у випадку ферміонних систем. Ключові слова: стохастичні процеси, бозонний броунівський рух, ферміонний броунівський рух PACS: 02.50.Ey, 05.30.-d 646

Quantum stochastic processes: boson and fermion Brownian motion

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