Exact density matrix of an oscillator-bath system: Alternative derivation

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Physics Letters A ••• (••••) •••–•••

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Physics Letters A

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Exact density matrix of an oscillator-bath system: Alternative derivation

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Fardin Kheirandish Department of Physics, Faculty of Science, University of Kurdistan, P.O. Box 66177-15175, Sanandaj, Iran

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a r t i c l e

i n f o

a b s t r a c t

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Starting from a total Lagrangian describing an oscillator-bath system, an alternative derivation of exact quantum propagator is presented. Having the quantum propagator, the exact density matrix, reduced density matrix of the main oscillator and thermal equilibrium fixed point are obtained. The modified quantum propagator is obtained in the generalised case where the main oscillator is under the influence of a classical external force. By introducing auxiliary classical external fields, the generalised quantum propagator or generating functional of position correlation functions is obtained. © 2018 Published by Elsevier B.V.

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Article history: Received 29 June 2018 Received in revised form 22 September 2018 Accepted 24 September 2018 Available online xxxx Communicated by M.G.A. Paris Keywords: Density matrix Oscillator-bath Propagator Generating function

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1. Introduction

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40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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The quantum propagator is the most important function in quantum theories [1,2]. Knowing the quantum propagator, we can obtain all measurable quantities related to the physical system exactly, that is, we have a complete physical description of the underline system in any time. Unfortunately, except for some simple physical systems, obtaining the exact form of quantum propagator is usually a difficult task and we have to invoke perturbative methods. Among different approaches to find quantum propagator, we can refer to two main approaches. In the first method, quantum propagator is written as a bilinear function in eigenvectors of the Schrödinger equation. The main task in this method is to find the eigenfunctions of the Hamiltonian which are usually difficult to find and even having these eigenfunctions, extracting a closed form quantum propagator from them may be cumbersome. The second approach is based on the Feynman path integral technique [3–5]. One of the most efficient features of this method is its perturbative technique known as Feynman diagrams which extends the applicability of the method to the era of non-quadratic Lagrangians. The path integral technique has been applied to oscillator-bath system in [6–12]. Here we follow an alternative approach to find the quantum propagator. This approach which we will describe in detail is based on the position and momentum operators in Heisenberg picture.

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E-mail address: [email protected]. https://doi.org/10.1016/j.physleta.2018.09.030 0375-9601/© 2018 Published by Elsevier B.V.

In this scheme, using elementary quantum mechanical relations, two independent partial differential equations are found that quantum propagator satisfy in. The solutions of these partial differential equations are easily found and unknown functions are determined from basic properties of quantum propagators. The first message of the present paper is that this method compared to other methods to derive the quantum propagator of an oscillator-bath system with a linear coupling is easier to apply and in particular, comparing with path integral technique, there is no need to introduce more advanced mathematical notions like infinite integrations, operator determinant and Weyl ordering. The second message is that since we will find a closed form for the total quantum propagator, we will find a closed form density matrix describing the combined oscillator-bath system. Also, by tracing out the bath degrees of freedom, we find a reduced density matrix describing the main oscillator in any time. In the following, we will generalise the oscillator-bath model by including external classical sources in Hamiltonian, and find the modified quantum propagator under the influence of classical forces. The modified quantum propagator can be interpreted also as a generating functional from which timeordered correlation functions among different position operators can be determined [13]. The basic ingredient of the approach is a symmetric time-independent matrix B (Eq. (15)), depending on natural frequencies of the bath oscillators and coupling constants. Therefore, from numerical or simulation point of view, the only challenge is finding the inverse of the matrix B or equivalently diagonalizing it.

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F. Kheirandish / Physics Letters A ••• (••••) •••–•••

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5 6 7 8 9 10

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In this section, we set the stage for what will be investigated in the following sections. We start with a total Lagrangian describing an interacting oscillator-bath system. Then from the corresponding Hamiltonian and Heisenberg equations of motion, we find explicit expressions for position and momentum operators as the main ingredients of an approach that will be applied in the next section. The Lagrangian describing a main oscillator interacting linearly with a bath of oscillators is given by [14]

21 22 23

1

L=

2

x˙ − 2

1 2

2 2 0x

ω

+

N  i =1

24 25 27 28

N 1

L=

2

29 30 31

34 35 36

−ω

2 2 i Xi ) +

g i X i x,

i =1

2 2 2 (Y˙ μ − ωμ Y μ) +

N 1 

⎛

0 g1 g2

⎜ ⎜ ⎜ 2 μν = ⎜ ⎜ .. ⎝ .

gN

2

Y μ 2μν Y ν ,

(2)

μ,ν =0

g1 0 0

g2 0 0

.. .

.. .

0

0

··· ··· ··· .. . ···

gN 0 0

⎟ ⎟ ⎟ ⎟, .. ⎟ . ⎠

(3)

0

and

H=

47

1 2

N 

2

2

2

( P μ + ωμ Y μ ) −

μ=0

1 2

N 

2

Y μ μν Y ν ,

μ,ν =0

(5)

where P μ = Y˙ μ is the canonical conjugate momentum correspond-

49

ing to the canonical position Y μ . The system is quantized by imposing the equal-time commutation relations

51 52 53 54 55 56 57 58

[Yˆ μ , Pˆ ν ] = ih¯ δμν , [Yˆ μ , Yˆ ν ] = [ Pˆ μ , Pˆ ν ] = 0,

61 62 63 64 65 66

(6)

and from Heisenberg equations of motion one finds

 ¨ 2 ˆ Yˆ μ + ωμ Yμ = 2μν Yˆ ν . ν

[

−1

(12)

− g2

0

s +ω

.. .

.. .

··· ··· ··· .. .

0

0

···

0 2 2

−gN 0 0

.. . 2 s + ω2N

⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

84 85 86 87

(13)

0

88 89 90 91

(14)

94 95 96

− g1 − g2 · · · ω12 0 ··· 0 ω22 · · · .. .. .. . . .

ω

82

93

( s ) = s I + B ,

⎜ ⎜ − g1 ⎜ ⎜ B = ⎜ − g2 ⎜ . ⎜ . ⎝ . −gN

81

92

2

2 0

78

83

− g1 2 s + ω12

2

77

80

The matrix  is explicitly given by

⎜ ⎜ − g1 ⎜ ⎜ ( s ) = ⎜ − g 2 ⎜ .. ⎜ . ⎝ −gN

76

79

(s)]μν .

s2 + ω02

75

−gN

⎟ 0 ⎟ ⎟ 0 ⎟ ⎟. .. ⎟ ⎟ . ⎠

···

0

97

⎞

98 99 100

(15)

101 102 103

ω2N

104 105

1

 −1 ( s ) =

s2 I + 1

=

B



I−

s2

n =0

s2

1

s2n+2

107

1

I+

Bn

1 s2

108

B

1

B+

s2

∞  (−1)n

=

1

=

106

s

109



110

B2 − · · · 4

111 112 113

( B 0 = I).

(16)

115 116

Therefore, from Eq. (12) we have

F μν (t ) =

∞  (−1)n t 2n+1 n =0

F˙ μν (t ) =

(2n + 1)!

 dF

(7)

dt μν

=

114

118

( B n )μν ,

119 120

∞  (−1)n t 2n n =0

117

(2n)!

121

n

( B )μν .

(17)

122 123 124

The equations (17) can be formally written as

125

Note that (Yˆ 0 , Pˆ 0 ) refer to the position and momentum of the main oscillator and (Yˆ k , Pˆ k ) (k = 1, · · · , N) refer to position and momentum operators of bath oscillators. Taking the Laplace transform from both sides of Eq. (7) we find

√ 1 F (t ) = √ sin( B t ),

126



From Eqs. (17) we deduce that the matrices F μν (t ) and F˙ μν (t ) are odd and even in t, respectively.

59 60

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F μν (t ) = L

CO RR

(4)

The corresponding Hamiltonian is

44 46

where we defined −1

(11)

The inverse matrix can be formally written as

Y 0 = x, Y k = X k , k = 1, · · · , N .

73 74

Yˆ μ (t ) = F˙ μν (t )Yˆ ν (0) + F μν (t ) Pˆ ν (0),

⎛

⎞

42

45

ν

wherein

40

43

72

(10)

which can be rewritten as

where the matrix μν is given by

38

41

(1)

2

37 39

2

( X˙ i2

μ=0

32 33

1

N 

Eq. (1) can be rewritten in a more compact form as

26

71

 ˆ 1 −1 ˆ ˆ Y˜ μ (s) = [ s − μν (s)Y ν (0) + μν (s) P ν (0)],

⎛

69 70

Therefore, applying the inverse matrix, we find

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(9)

and a formal solution is obtained by inverse Laplace transform as

2. Lagrangian

11 12

67

2 μν (s) = [(s2 + ωμ )δμν − 2μν ].

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where the N + 1-dimensional matrix  is defined by

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The efficiency of the method introduced here in determining the exact form of the quantum propagator for quadratic Lagrangians, inspires the idea of developing a perturbative approach to include non-quadratic Lagrangians too. The process presented to determine the quantum propagator, suggest that these perturbative techniques may be based on perturbative solutions of nonlinear partial differential equations. This development deserves to be investigated in an independent work.

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1

ν

μν (s)Yˆ˜ μ (s) = s Yˆ μ (0) + Pˆ μ (0),

(8)

127

B

√ F˙ (t ) = cos( B t ).

128

(18)

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F. Kheirandish / Physics Letters A ••• (••••) •••–•••

G (ω) =

2.1. Connection to the previous works

2

9 10 11

t

2

( X B X )α β = zα δα β ,

13 14

F μν (t ) =

17 18 19 20 21 22 23 24 25 26 27 28 29

X μα X να

34 35 36 37 38

ω2 − z2 − g1 − g2

0

2 2 ω1 − z 0

− g1

2

− g2 0 ω 2 ⇒

.. .. ..

. . .

−gN 0 0

g ( z) = z2 − ω02 −

43

¨ Yˆ 0 (t ) −

48

51 52 53 54 55 56

χ (t ) =

t



N 

gk2

sin(ωk t )

ωk

ˆ (t ) = ϒ

N 

gk



 sin(ωk t ) cos(ωk t )Yˆ k (0) + Pˆ k (0) ,

ωk

k =1

1 s2

− χ˜ (s) + ω02

where

63 64 65 66

χ˜ (s) =

N 

gk2

k =1

s2 + ωk2

=

1 s2 + ω02 −

N 

k =1

gk2

,

F˙ μν (t ) y ν − ih¯ F μν (t )





(32)

82 83 84 85 86 87 88

92 93 95 96 97 99 100

(33)

103

Eq. (31) can be rewritten as

i

∂

ln K(y |y, t ) = ∂ y ν h¯

yμ −



104



F˙ μν (t ) y ν .

105

(34)

ν

N 

A μ (y, t ) y μ +

μ=0

N 1 

2

106 107

The right hand side of Eq. (34) is linear in y μ , so the following quadratic form can be assumed for ln K

ln K(y |y, t ) = A (y, t ) +

101 102

108 109 110 111

y μ C μν (y, t ) y ν ,

μ,ν =0

(35)

112 113 114 115

where C μν = C νμ . By inserting Eq. (35) into Eq. (34), we easily find

116

N i  −1 Aμ( y, t ) = F μν (t ) y ν , h¯

118 120

N i  −1 C μν (t ) = − F μσ (t ) F˙ σ ν (t ), h¯ σ =0

K(y |y, t )

i  −1 i  −1  ˙ = e A (y,t ) e h¯ y ·F (t )·y e − 2h¯ y ·F (t )F(t )·y .

121

(36)

ˆ

K(y |y, t ) = y |y, t  = y |e h¯ H |y.

122 123 124 125

(37)

The form of A (y, t ) can be determined from the properties of propagators. Since the Hamiltonian Eq. (5) is time-independent, we can write it

117 119

ν =0

therefore, in dyadic notation, we can write

The Green’s function in frequency space (G (ω)) can be obtained from the Laplace transformed Green’s function G˜ (s) using the identity G (ω) = G˜ (i ω), therefore,

81

98

(25)

(27)

79

94

y |Yˆ μ (0) = y μ y |, ∂ y | Pˆ μ (0) = −ih¯ y |. ∂ y μ

ν =0

78

91

(31)



F μν (t )

77

90

K(y |y, t ) = y μ K(y |y, t ),

∂ y ν

76

89



and made use of the identities

N 

s2 +ωk2

∂

y |y, t  = K(y |y, t ),

(23)

(26)



where we have defined the function K as



.



ν =0

equation Eq. (23), we will find the Laplace transform of the corresponding Green’s function as

60 62

N 

(21)

where Yˆ k (0) and Pˆ k (0) are the position and momentum operators at initial time (t = 0). Taking the Laplace transform of the Langevin

G˜ (s) =

(30)

Multiplying Eq. (29) from the left by y | and using Eq. (11), we find

(24)

,

and the noise operator by

58

61

|y, t  = | y 0 , t  ⊗ | y 1 , t  ⊗ · · · ⊗ | y N , t  = | y 0 , · · · , y N , t .

(22)

dt χ (t − t ) Yˆ 0 (t  ) + ω02 Y 0 (t ) = ϒ(t ), 

k =1

57 59

z2 − ωn2

= 0.

where the susceptibility of the environment is defined by

49 50

gn2

0

46 47

···

(29)

where for notational simplicity the tensor product is abbreviated as

−gN

By making use of Eq. (7), we find the following quantum Langevin equation for the main oscillator

44 45

 n

41 42



0

0

= 0,

..

.

2 2 ωN − z

··· ··· ··· .. .

72

80

Yˆ μ (t ) |y, t  = y μ |y, t ,

the determinant can be evaluated using the mathematical induction leading to the following characteristic equation

39 40

zα

71

75

(20)

det( B − z2 I) = 0

32 33

sin zα t .

70

74

In this section a novel scheme to derive the quantum propagator of the combined oscillator-bath system is introduced in detail. Let | y 0  be an eigenket of Yˆ 0 and | yk  an eigenket of Yˆ k , then in Heisenberg picture, we can write

2 The eigenvalues zα of the matrix B, satisfy the characteristic equation

30 31

1

69

73

(19)

α =0

15 16

N 

67 68

3. Quantum propagator

and using the first equation of Eq. (18), we find [15]

12

(28)

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−1 , g (ω)

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6

=

that is, the roots of the characteristic equation g ( z) = 0 are the poles of the Green’s function G (ω) in frequency domain.

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5

gk2 2 ω −ωk2

k =1

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The Eq. (11) has been appeared in [15] with a minor change of notation in the framework of Ullersma diagonalisation technique [16]. Let the matrix X μν be a unitary matrix that diagonalizes the orthogonal matrix B given by Eq. (15) with corresponding eigen2 (α = 0, 1, · · · , N). Therefore, in matrix notation we have values zα

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−1 N  ω2 − ω02 −

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3

126 127 128 129 130 131

(38)

132

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F. Kheirandish / Physics Letters A ••• (••••) •••–•••

3 4

K(y |y, t ) = K∗ (y|y , −t ),

(39)

5

leading to

7 8

e A (y,t ) = e ϕ (t ) e

9

∗

12 13 14

ϕ (−t ) = ϕ (t ).

(40)

Note that in Section 6, the Hamiltonian will be time-dependent and to find A (y, t ) we can not use these transformations and we will follow another approach. Up to now the form of the propagator is as follows

15 16

i

K(y |y, t ) = e ϕ (t ) e − 2h¯ y·F

17

= e ϕ (t ) e

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

lim F−1 (t ) ≈

t →0

1 t

(42)

t →0

By inserting these asymptotic behaviours into Eq. (41) we find

lim K(y |y, t ) = δ(y − y) = lim e ϕ (t ) e

t →0

t →0



lim

A

A  2 e − t (x−x ) = δ(x − x ),

πt

lim e ϕ (t ) =

i

43

so we can assume



44

e ϕ (t ) =

47

N +2 1

i 2π h¯ t

55 56 57 58 59 60 61 62 63

66

e λ(t ) ,

∂ K (y, t ; 0, 0) = ∂t 2



i

λ(t )

2π h¯ t

N2 e

− 2ih¯ [y·F−1 F˙ ·y+y ·F−1 F˙ ·y −2y ·F−1 ·y]

δ(y − y) =

e

=

t

dy K( y  | y  , t ) K∗ ( y | y  , t ),

N +1 2

| det F(t )|

iθ

e ,

∂ 2 2 + ωμ yμ ∂ y 2μ

81 82 83 84 85



86 87 88



89

2

y μ μν y ν K (y, t ; 0, 0),

(53)

90 91 92





ω0 ω1 ωN , F−1 (t ) = diag , ,··· , sin(ω0 t ) sin(ω1 t ) sin(ω N t ) F˙ (t ) = diag(cos(ω0 t ), cos(ω0 t ), · · · , cos(ω0 t )).

93 94 95 96 97 98 99 100 101 102

(54)

103 104



K (y, t ; y , 0) = e

−i θ

N 



μ=0

×e

105 106

ωμ

107

2π ih¯ sin(ωμ t )

108

 2  i ωμ ( y μ + y  2μ ) cos(ωμ t )−2 y μ y μ 2h¯ sin(ωμ t )

109

,

(55)

1



×e



1



111 112 113 114 115 116

N +1 2

117

2π ih¯

i [y·F−1 F˙ ·y+y ·F−1 F˙ ·y −2y ·F−1 ·y] 2h¯

110

118 119

.

(56)

120 121 122 123

(49)

which can be easily checked using the definition of K, Eq. (32). By inserting Eq. (48) and its complex conjugation into Eq. (49) and doing the integral we will find λ(t )

2

4. Density matrix

To find λ(t ) we make use of the following identity



(52)

.

which is the propagator of N noninteracting oscillators if we set θ = 0. Finally, we find the quantum propagator of oscillator-bath system as

. (48)

73

80

μ,ν =0

K (y, t ; y , 0) = √ det F (t )

The function K now has the form

K(y |y, t ) = e

2

(47)

t →0



μ=0

N 

1

− h¯ 2

72

79

after spatial differentiations we set y = 0 and by comparing both sides of Eq. (53) we find that θ is a constant. To find the constant θ , we turn off the coupling constants, (g 1 = g 2 = · · · = g N = 0), and from consistency condition we should recover the quantum propagator of N noninteracting oscillators. When the coupling constants are turned off, we have

(46)

lim λ(t ) = 0.

64 65

(45)

where the unknown function λ(t ) satisfies

53 54

ih¯

71

78

Now we set y  = 0 and require that K (y, t ; 0, 0) satisfy the Schrödinger equation

  N 1

70

77

| det F (t )| 2π ih¯

i [y·F−1 F˙ ·y+y ·F−1 F˙ ·y −2y ·F−1 ·y] 2h¯

69

76

N +2 1

1

68

75

Inserting Eqs. (54) into Eq. (52) we find

,

2π h¯ t

t →0

52

(43)

(44)

N +2 1

42

51

,

comparing Eq. (43) with the following one-dimensional representation of Dirac delta function

40

50

− 2hi¯ t (y −y)2



e −i θ

−

we deduce immediately

49

K (y, t ; y , 0) = 

I,

lim F˙ (t ) ≈ I.

39

48

(41)

(51)

therefore, Feynman propagator is given by

lim F(t ) ≈ t I,

38

46

74

t →0

t →0

45

.

K (y, t ; y , 0) = y, t |y , 0 = y |y, t ∗ = K∗ (y |y, t ),

×e

i  −1 i  −1  ˙ e h¯ y ·F (t )·y e − 2h¯ y ·F (t )F(t )·y ,

− 2ih¯ [y·F−1 F˙ ·y+y ·F−1 F˙ ·y −2y ·F−1 ·y]

37

41

−1 (t )F ˙ (t )·y

From Eqs. (18) we find the following asymptotic behaviours of matrices F, F−1 , and F˙

35 36

,

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11

y·F−1 (t )F˙ (t )·y

67

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10

− 2ih¯

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6

where θ is a real function that will be determined from a limiting case where the coupling constants are turned off (g 1 = · · · = g N = 0) and also the fact that the propagator should satisfy the Schrödinger equation. It should be noted that according to the definition Eq. (32), the Feynman propagator has the following relation to the function K

OF

Eq. (38) is invariant under successive transformations: (i) complex conjugation, (ii) y ↔ y , (iii) t → −t, therefore,

DP

2

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1

RO

4

(50)

In this section we will find the density matrix for the oscillatorbath system using the explicit form of the quantum propagator Eq. (56) of the combined system. If we denote the evolution operator by Uˆ (t ) then the density matrix at time t can be obtained from the initial density matrix at t = 0 as

124

ρˆ (t ) = Uˆ(t )ρˆ (0)Uˆ † (t ),

130

in position representation we have

(57)

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F. Kheirandish / Physics Letters A ••• (••••) •••–•••

ρ (y, y ; t ) = y|ρ (t )|y 

4



=

5 6

dy1 dy2 y|Uˆ (t )|y1 y1 |ρ (0)|y2 y2 |Uˆ † (t )|y , ∗



dy1 dy2 K (y, t ; y1 , 0)ρ (y1 , y2 ; 0) K (y , t ; y2 , 0).

9 10 11 12 13

where y1 = ( y 10 , y 1 ) and y2 = ( y 20 , y 2 ). To find the reduced den-

15

sity matrix of the main oscillator, we should take trace over the degrees of freedom of the bath oscillators. Straightforward calculations lead to

20

ρred ( y 0 , y 0 ; t ) =

21 23

ia( y 2 − y 2 ) ib( y 0 y 01 − y  y 02 ) 0 01 02 − 2h¯ h¯

31

I ( y 0 , y 0 ; y 01 , y 02 ) =

34

·e

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

d x e

ρred ( y 01 , y 02 ; 0)

·e

i 2h¯

−i 2h¯

 d y e

i h¯

N   k =1



2h¯ sinh(β h¯ ωk )

77

( y 21k + y 22k ) cosh(β h¯ ωk )−2 y 1k y 2k



78

, (67)

N 

xk kl xl +

N 

k,l=1

k =1

jk xk

= (2π )

N /2

N  k,l=1 N  k,l=1

y 1k A kl y 1l

e y 2k A kl y 2l

e

N  

i h¯

k =1

−i h¯

y 01 B k − y 0 C k −

N 

l =1

N  



D kl yl y 1k

N







dy 01 dy 02 J ( y 0 , y 0 ; t | y 01 , y 02 ) ρred ( y 01 , y 02 ; 0),

where

J ( y 0 , y 0 ; t | y 01 , y 02 ) 1

1 h ) N +1

e

| det F | (2π ¯ × I ( y 0 , y 0 ; y 01 , y 02 ).

2 ia( y 2 − y  ) 0 0 2h¯

e

(61)

(62)

ia( y 2 − y 2 ) ib( y 0 y 01 − y  y 02 ) 0 01 02 − 2h¯ h¯

−1/2

(det )

e

N  k,l=1

84

−1

jk kl jl

,

(68)

∂t

The function J ( y 0 , y 0 ; t | y 01 , y 02 ), which can be interpreted as a

 2

a(t ) = ( F

F )00 ,

b(t ) = ( F −1 )00 , C k (t ) = ( F

−1

87

90

93 94 95 96 97 98

b3 2π

e

100

ib1 X ξ +ib2 X 0 ξ −ib3 X ξ0 −ib4 X 0 ξ0

101

−a11 ξ 2 −a12 ξ ξ0 −a22 ξ02

,

(69)

102 103 104 105 106 107 108 109 110 111 112 113 114

(63)

−1 ˙

86

89

=  y 0 |[ H ren , ρred ]| y 0 

− i γ (t )( y 0 − y 0 )(

reduced kernel, has been expressed in path integral language in terms of the Feynman–Vernon influence functional [6,7,10]. Here we have obtained the reduced kernel in terms of the quadratic integrals. The time dependent functions (a, b), vectors (C k , B k ) and matrices ( A kl , D kl ) are defined by

85

88

where for later convenience, we have chosen the same notation for the time-dependent coefficients bk (t ) and ai j (t ) introduced by Paz in [17] following the path integral technique. These coefficients can be obtained in terms of the functions given by Eqs. (64) or in terms of the environment properties described in [17]. Following the same process described by Paz in [17], we recover the master equation (¯h = 1) for the reduced density matrix as

i

82

92

×e

∂ ρred ( y 0 , y 0 , t )

81

91

J ( y 0 , y 0 ; y 01 , y 02 ) =

.

80

99





 y 02 B k − y 0 C k − D kl yl y 2k k =1 l =1

79

83 1 2

The main ingredient quantity in open quantum system theory is the master equation. To find the master equation satisfied by the reduced density matrix ρred , we insert the initial bath state Eq. (67) into Eq. (61) and take the integrals over y , y 1 and y 2 , after straightforward but tedious calculations we will find the following expression for the reduced kernel defined in Eq. (63)

( y 0 − y 0 ) B k −( y 01 − y 02 )C k yk

ρred ( y 0 , y 0 ; t )

=

− 12

(60)

The Eq. (60) can be rewritten as

=

k =1

72

76

where  is a positive, symmetric matrix.

d y 1 d y 2 ρ B ( y 1 , y 2 ; 0)

32 33





·



ωk

71

75

then, the integrals over y 1 , y 2 and y in Eq. (61), will be Gaussian type integrals and can be obtained using the generic formula [5]

where

29 30

×e

−

N 

70

74



ωk

69

73

4.1. Master equation

27 28

dy 01 dy 02 e

(66)

2π h¯ sinh(β h¯ ωk )

k =1

2 ia( y 2 − y  ) 0 0 2h¯

· I ( y 0 , y 0 ; y 01 , y 02 ),

24 26



1

| det F | (2π h¯ ) N +1

· e

22

25

1

68

(65)

,

DP

19

ρ B ( y 1 , y 2 ; 0) =

 N 

EC

18

CT D

b C

Let the initial state of the bath be a thermal state given by

CO RR

17

UN

16

, F −1 =

67



T

(59)

14

BT A



B = [ B 1 , B 2 , · · · , B N ], C = [C 1 , C 2 , · · · , C N ].

We can assume an arbitrary initial state for oscillator-bath system, but for simplicity we assume that the initial state is a product state as

ρ (y1 , y2 ; 0) = ρred ( y 10 , y 20 ; 0) ⊗ ρ B ( y 1 , y 2 ; 0),

a B



T

(58)

7 8

F −1 F˙ =



OF

=

3

which can be rewritten more compactly in matrix form as

RO



2

TE

1

5

∂ ∂ y0

−

∂ ∂ y 0

)ρred ( y 0 , y 0 , t ),



− i D (t )( y 0 − y 0 ) ρred ( y 0 , y 0 , t ) ∂ ∂ + )ρred ( y 0 , y 0 , t ), + f (t )( y 0 − y 0 )( ∂ y 0 ∂ y 0 where H ren is the renormalized Hamiltonian of the main oscillator with the renormalized frequency ωren (t ). To find the connection between the functions ωren (t ), γ (t ), D (t ), f (t ) and coefficients bk (t ), ai j (t ), the interested reader is referred to [17].

115 116 117 118 119 120 121 122 123 124 125 126

5. Thermal equilibrium: fixed point

−1

)k0 = ( F )0k , −1 ˙ B k (t ) = ( F F )0k = ( F −1 F˙ )k0 , A kl = (A)kl = ( F −1 F˙ )kl , D kl = (D)kl = ( F −1 )kl = ( F −1 )lk ,

127 128

(64)

In the equilibrium state, the density matrix of oscillator-bath system can be obtained from the quantum propagator using the correspondence between quantum propagator and partition function as

129 130 131 132

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6

5



8 9

=

10 11 12 13 14 15

1 2



N +1 2

18



1

(71)

and I is a N-dimensional unit matrix. The reduced density matrix of the oscillator is obtained by integrating out the bath degrees of freedom as

ρred ( y 0 , y 0 ; β) =



d y K ( y 0 , y , −ih¯ β; y 0 , y , 0),



19

det( F˙ − I)

=

20 21 22



 i ( y 0 2 + y 0 2 )(a− η2 )−2 y 0 y 0 (b+ η2 ) 2h¯

29 30 31

η=

(72)

,

36 37 38 39 40

43 44

From Eq. (65) we have

F −1 ( F˙ − I) =



a−b B−C

BT − CT A−D



−1

T

= det(A − D) det[a − b − (B − C )(A − D)   η

= det(A − D)(a − b − η),

ρred ( y 0 , y 0 ; β) =

a−b−η ih¯ π

54 55 56 57 58





From Eq. (76) we find the thermal mean square of position and momentum as

ih¯ 2

 y0  = , 2(a − b − η) t =−ih¯ β

a +b

,  p 20  = −ih¯

2

t =−ih¯ β

63 64

ρred ( y 0 , y 0 ; β) = 

1

e

2π  y 20 

65 66

for another derivation, see [14].

 p2  0 2h¯ 2

78

Y μ 2μν Y ν + f (t )Y 0 .

2

( y 0 − y 0 )2 −

(80)

(77)

1 ( y 0 + y 0 )2 8 y 2  0

82

81



 2

(81)

87

90



91

(82)

92 93 94 95 96

(83)

97 98

and the position and momentum operators are respectively given by



99 100 101

F˙ μν (t )Yˆ ν (0) + F μν (t ) Pˆ μ (0) − R μ (t ),

102 103

ν

(84)

ν

104 105 106 107 108 109

dt  F μ0 (t − t  ) f (t  ).

(85)

110 111

0

112

We can rewrite the identity

113

Pˆ μ (t ) = Uˆ † (t ) Pˆ μ (0)Uˆ (t ),

(86)

114 115 116

as

117

Pˆ μ (t )Uˆ † (t ) = Uˆ † (t ) Pˆ μ (0),

(87)

118 119 120 121

(88)

By inserting the momentum operator from the second line of Eqs. (84) into Eq. (88), we easily find

ν

(78)

85

89

By making use of Laplace transform and definitions Eqs. (9), (12), we find the retarded Green tensor as

t

84

88

∂t2 + ωμ δμν − 2μν G να (t − t  ) = δμα δ(t − t  ).



83

86

2μν Yˆ ν = − f (t ) δμ0 .

The Green tensor corresponding to Eq. (81) is defined by

 ,

79

Note that the Hamiltonian is now time-dependent and we can not use Eqs. (38), (39). In this case, we can find another partial differential equation satisfied by K( y  | y , t ) as follows. From Heisenberg equations of motion we find

 y  | Pˆ μ (t )Uˆ † (t )| y  =  y  |Uˆ † (t ) Pˆ μ (0)| y .

therefore, −

1

77

then

61 62

76

where we defined

(76)

59 60

(B − C)], 

UN

53

N 

74 75

  ˙ Pˆ μ = Yˆ μ = F¨ μν (t )Yˆ ν (0) + F˙ μν (t ) Pˆ μ (0) − R˙ μ (t ),

η η i 2  2 e 2h¯ ( y 0 + y 0 )(a− 2 )−2 y 0 y 0 (b+ 2 ) .

50 52

71

80

R μ (t ) =

49 51

70

μ,ν =0

(75)

Eq. (72) can be rewritten as



2 2 2 (Pμ + ωμ Y μ) −

μ=0

Yˆ μ (t ) =

det[ F −1 ( F − I )]

46 48

(74)

,

T

μ,ν =0

G μν (t − t  ) = F μν (t − t  ),

by making use of the identity [18]

45 47

(73)

k,l=1

41 42

2

ν

−1 ( B k − C k )(A − D)kl ( B l − C l )|t =−ih¯ β .

34 35

1

 

32 33

H=

N 

EC

28

69

73

(79)

ν

where N 

2

μ=0

Y μ 2μν Y ν − f (t )Y 0 ,

and the corresponding Hamiltonian is

CO RR

27

2

N 1 

2 2 2 (Y˙ μ − ωμ Y μ) +

¨

24 26

N 1

2 ˆ Yˆ μ + ωμ Yμ −

i π h¯ det F det(A − D)

×e

23 25

Now assume that the main oscillator is under the influence of an external classical field f (t ). In this case the total Lagrangian is written as

L=

,

det( F˙ − I) t =−ih¯ β

16 17

67

72

dy 0 d y K ( y 0 , y , −ih¯ β; y 0 , y , 0),

Z (β) =

6. Main oscillator interacts with an external field

68

where β = 1/κ B T is the inverse of temperature and κ B is Boltzmann constant. The function Z (β) is the total partition function

6 7

(70)

OF

4

K ( y 0 , y , −ih¯ β; y 0 , y  , 0),

Z (β)

RO

3

1

DP

2

ρ ( y 0 , y ; y 0 , y  , β) =

TE

1

F¨ μν (t ) y ν − ih¯ F˙ μν (t )

∂ ∂ y ν



122 123 124 125 126

− g˙ μ K(y |y, t ) = y μ K(y |y, t ).

127 128

(89)

129

By making use of Eqs. (31), (43), (49), (89), and following the same process as we did in Section 3, we will find

131

130 132

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(y, t ; y , 0) = 

3 4 5 6 8

11

t

Rˇ μ (t ) =

14 15 16 17 18





ˇ y ·F−1 ·R+y·F−1 ·R

(90)

,

−

1 2

(91)



y μ 2μν y ν + f (t ) y 0

μ,ν =0

ζ (t ) =

1 2h¯

27

0

28

t

K ( f ) (y, t ; 0, 0)

=

29 30

1 h¯

31

ds 0

0

y =0

√ √   sin( Bu ) sin[ B (t − s)] du f (s) f (u ). √ √ B sin( Bt ) 00 (93)

Finally, the quantum propagator for oscillator-bath system under the influence of an external classical force on the main oscillator, is obtained as



36 37 38

K ( f ) (y, t ; y , 0) = 

39

×e

40 41

×e

42

1

1

N +2 1

2π ih¯

| det F (t )|

i 2h¯





y·F−1 F˙ ·y+y ·F−1 F˙ ·y −2y ·F−1 ·y

t   − h¯i y ·F−1 ·R+y·F−1 ·Rˇ − 2ih¯ 0 ds Rˇ (s)·F−2 (s)·Rˇ (s)

e

46 47

6.1. A generalization: generating function

We can generalize the Lagrangian Eq. (79) as

48 49 50 51

L=

N 1

2

N N  1  2 2 2 (Y˙ μ − ωμ Y μ) + Y μ 2μν Y ν − f μ (t )Y μ ,

2

μ=0

μ,ν =0

54 55 56

in this case the quantum propagator is given by Eq. (85) but now the definitions Eqs. (85), (91) have to be replaced by the new definitions

t

57 58 59

R μ (t ) = 0

61

t

63 64 65 66

Rˇ μ (t ) =





dt F μν (t − t ) f ν (t ),

60 62



dt  F μν (t  ) f ν (t  ).

(96)

0

The path integral representation of quantum propagator Eq. (94) is [13]

67 68

75

N

N

δ (ih¯ ) K (0) (y, t ; y , 0) δ f μ1 (t 1 ) · · · δ f μN (t N )

(f)  K (y, t ; y , 0)

70 71 72 73 74 76 77 78 79 80

, f =0

81 82

(98)

83

where Tˆ is a time ordering operator acting on bosonic operators

84



ˆ (t ) Bˆ (t  ), t > t  ; A

ˆ (t ), t  > t . Bˆ (t  ) A

85 86 87

(99)

Using elementary quantum mechanical calculations and basic properties of quantum propagators, an alternative derivation of exact quantum propagator for the oscillator-bath system was introduced. The method compared to other methods to derive quantum propagator of an oscillator-bath system with linear interaction or generally quadratic Lagrangians, was easier to apply and in particular, compared to path integral approach, there was no need to introduce more advanced mathematical notions like infinite integrations, operator determinant and Weyl ordering. From quantum propagator, a closed form density matrix describing the combined oscillator-bath system was obtained from which reduced density matrix could be derived. The problem was generalised to the case where the main oscillator was under the influence of an external classical source. By introducing auxiliary classical fields the modified quantum propagator or generating functional of position correlation functions was found. The basic ingredient of the approach was a symmetric timeindependent matrix B, which was dependent on natural frequencies of the bath oscillators and coupling constants. Therefore, from numerical or simulation point of view, the only challenge was finding the inverse of the matrix B or equivalently diagonalizing it. The efficiency of the method in determining the exact form of the quantum propagator for quadratic Lagrangians, inspired the idea of developing a perturbative approach to include nonquadratic Lagrangians too.

(95)

52 53

μ=0

UN

45

.

(94)

43 44

(97)

,

EC

35

dτ L

7. Conclusions

CO RR

34

0

y, t | Tˆ [Yˆ μ1 (t 1 )Yˆ μ2 (t 2 ) · · · Yˆ μN (t N )|y , 0 t  i D [x] y μ1 (t 1 ) y μ2 (t 2 ) · · · y μN (t N ) e h¯ 0 dτ L = t  i D [x] e h¯ 0 dτ L

ˆ (t ) Bˆ (t  ) = Tˆ ( A

32 33

t

69

as

(92)

,

ˇ (s) · F−2 (s) · Rˇ (s), ds R s

i

d[x] e h¯

where L is the Lagrangian Eq. (95). Having the closed form expression Eq. (94), we can find ordered correlation functions among position operators of the oscillator-bath system. In this case, the external source f μ (t ) is an auxiliary force that should be set zero at the end of functional derivatives [13], we have

=

as

t





dt  F μ0 (t  ) f (t  ),

N 

24 26

×e

− h¯i

y·F−1 F˙ ·y+y ·F−1 F˙ ·y −2y ·F−1 ·y

μ=0

21

25

×e

i 2h¯





  N 2 ∂ K ( f ) (y, t ; 0, 0)

1 2 ∂ 2 2 − ih¯ = h + ω y ¯ μ μ

∂t 2 ∂ y 2μ y =0

20

23



and the function ζ (t ) can be determined from the Schrödinger equation

19

22

| det F (t )| 2π ih¯

0

12 13

K ( f ) (y, t ; y , 0) =

ˇ as where we have defined R

9 10

N +2 1

1

DP

7



e −i ζ (t )

OF

K



TE

2

(f)

RO

1

7

88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118

References

119 120

[1] W.H. Dickhoff, D.V. Neck, Many-Body Theory Exposed! Propagator Description of Quantum Mechanics in Many-Body Systems, World Scientific, 2005. [2] J. Linderberg, Y. Öhrn, Propagators in Quantum Chemistry, John Wiley, 2004. [3] R.P. Feynman, A.R. Hibbs, D.F. Styer, Quantum Mechanics and Path Integrals, emended ed., McGraw–Hill, 2005. [4] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, fifth ed., World Scientific, 2009. [5] J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005. [6] R. Feynman, F. Vernon Jr., The theory of a general quantum system interacting with a linear dissipative system, Ann. Phys. 24 (1963) 118–173. [7] A.O. Caldeira, A.J. Leggett, Path integral approach to quantum Brownian motion, Physica A 121 (1963) 587–616. [8] V. Hakim, V. Ambegoakar, Quantum theory of a free particle interacting with a linearly dissipative environment, Phys. Rev. A 32 (1985) 423–434.

121 122 123 124 125 126 127 128 129 130 131 132

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[9] C. Morais Smith, A.O. Caldeira, Generalized Feynman–Vernon approach to dissipative quantum systems, Phys. Rev. A 36 (1987) 3509–3511. [10] H. Grabert, P. Schramm, G.-L. Ingold, Quantum Brownian motion: the functional integral approach, Phys. Rep. 168 (1988) 115–207. [11] B.L. Hu, J.P. Paz, Y. Zhang, Quantum Brownian motion in a general environment: exact master equation with nonlocal dissipation and colored noise, Phys. Rev. D 45 (1992) 2843–2861. [12] B.L. Hu, A. Matacz, Quantum Brownian motion in a bath of parametric oscillators: a model for system–field interactions, Phys. Rev. D 49 (1994) 6612–6635. [13] W. Greiner, J. Reinhardt, Field Quantization, Springer-Verlag, 1996.

[14] U. Weiss, Dissipative Quantum Systems, World Scientific, 1993. [15] F. Haake, R. Reibold, Strong damping and low-temperature anomalies for the harmonic oscillator, Phys. Rev. A 32 (1985) 2462–2475. [16] P. Ullersma, An exactly solvable model for Brownian motion: I. Derivation of the Langevin equation, Physica (Utrecht) 32 (1966) 27–55. [17] J.P. Paz, Decoherence in quantum Brownian motion, in: J.J. Halliwell, J. PérezMercader, W.H. Zurek (Eds.), Physical Origins of Time Asymmetry, Cambridge University Press, 1994. [18] F. Zhang, Matrix Theory: Basic Results and Techniques, Springer-Verlag, 2010.

67 68 69 70 71 72 73 74 75 76

11

77

12

78

OF

10

13 14 15 16 17

RO

18 19 20 21 22 23

DP

24 25 26 27 28 29 30

TE

31 32 33 34 35

EC

36 37 38 39 40

43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

UN

42

CO RR

41

79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125

60

126

61

127

62

128

63

129

64

130

65

131

66

132