Quantum squeezing effect of mesoscopic capacitance–inductance–resistance coupled circuit

4 March 2002

Physics Letters A 294 (2002) 319–326 www.elsevier.com/locate/pla

Quantum squeezing effect of mesoscopic capacitance–inductance–resistance coupled circuit Shou Zhang a,b,∗ , Jeong-Ryeol Choi b , Chung-In Um b , Kyu-Hwang Yeon c a Department of Physics, College of Science and Engineering, Yanbian University, Yanji, Jilin, 133002, PR China b Department of Physics, College of Science, Korea University, Seoul, 136-701, South Korea c Department of Physics, College of Natural Sciences, Chungbuk National University, Cheongju, Chungbuk, 361-763, South Korea

Received 5 November 2001; received in revised form 17 December 2001; accepted 14 January 2002 Communicated by L.J. Sham

Abstract Quantum fluctuations and squeezing effect of charges and currents of mesoscopic capacitance–inductance–resistance coupled circuit are investigated using canonical transformation and unitary transformation method. Even if the resistance of the mesoscopic circuit is zero, the uncertainty relation between charges and those conjugate currents do not satisfy minimum uncertainty relation. We confirmed that the uncertainties of charge can be reduced by paying the cost that the uncertainties of currents becoming larger relatively, or vice versa.  2002 Elsevier Science B.V. All rights reserved. PACS: 73.23.-b; 03.65.Sq

1. Introduction According to the rapid development of nanophysics and nanoelectronics [1–3], the size of electric devices gradually becomes smaller and, in the future, are expected to approach nanometer scale. Mesoscopic expressions for electric devices such as capacitances and inductances are formulated by Büttiker [4]. The achievement of the technological progress that minimize the integrated circuit and its components toward atomic size scale may be necessary to become a class of quantum computer [5,6] buildable, as well as to be demonstrate that they can be scalably and reliably controlled. When the scale of fabricated electric materials reached to a characteristic dimension, namely, Fermi wavelength, quantum mechanical properties of mesoscopic physics [7,8] become important, since the charge-carriers such as electrons exhibit quantum properties while the application of classical mechanics fails. The study for quantum properties of electric devices are stimulated by made clear of quantum properties of noise in the vacuum state in 1973 by Louisell [9]. The quantum mechanical properties of RLC (resistance–inductance–capacitance) circuit with time-dependent power source are studied by Chen et al. [10]. They obtained the quantum fluctuations of charge and current in the vacuum state when circuit has no power * Corresponding author.

E-mail addresses: [email protected], [email protected] (S. Zhang). 0375-9601/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 0 6 2 - 2

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Fig. 1. Diagram of mesoscopic capacitance–inductance–resistance coupled circuit.

source by introducing the complex formal charge and current. Recently, quantum mechanical effect of mesoscopic circuit with coupled inductance [7,11] and capacitance [12] separately or together [13] has been attracted interest in the literature. It is well known that the classical equation of motion of the RLC circuit is exactly same with that of a damped harmonic oscillator [14–17]. We will use canonical transformation [7] and unitary transformation [18– 21] method to solve the quantum mechanical solutions of mesoscopic capacitance–inductance–resistance coupled circuit. In this Letter, the quantum fluctuation and squeezing effect of the mesoscopic circuit are studied. In Section 2, we use canonical transformation to find the matrix element which transforms the Hamiltonian of mesoscopic capacitance–inductance–resistance coupled circuit to that of simple harmonic oscillator. In Section 3, we will find the unitary operator connecting two quantum systems, discuss quantum fluctuations and obtain the uncertainty relation between charges and those conjugate currents of the circuit. In Section 4, we investigate the squeezing effect of the circuit. Finally, in Section 5, we provide the summary and conclusion by making use of the result obtained in previous sections.

2. Canonical transformation We consider two loop of RLC circuit that coupled via inductance, capacitance and resistance as in Fig. 1. There is a power source ε(t) in the left side loop of the circuit. The classical equation of motion for charges can be obtained from Kirchoff’s law as
2 
 q1 d q1 d 2 q2 dq1 dq2 d 2 q1 dq1 (q1 − q2 ) + − = ε(t), +L − + R + L1 2 + R1 (2.1) dt dt C1 dt 2 dt 2 dt dt C
2 
 q2 d 2 q2 dq2 d q1 d 2 q2 dq1 dq2 (q1 − q2 ) L2 2 + R2 (2.2) + − = 0, −L − −R − 2 2 dt dt C2 dt dt dt dt C where qi (i = 1, 2; hereafter we apply this convention for all subscript) represent electric charges, Li inductances and Ci capacitances of the loops, and L, C and R the coupling inductance, capacitance and resistance of the mesoscopic circuit, respectively. To simplify the problem, let us suppose that R1 R2 R = = ≡ γ, (2.3) L1 L2 L and ε(t) = 0. The equation of motion, Eqs. (2.1) and (2.2), can be formulated from the following Hamiltonian [22]:  
 
2 2 q1 p22 q2 LL1 L2 p1 1 p2 2 (q1 − q2 )2 −γ t p1 Hq = e + eγ t . + − − + 2 + 2L1 2L2 2 LL1 + LL2 + L1 L2 L1 L2 2C1 2C2 2C (2.4) Note that qi stand for electric charges and pi electric currents instead of the conventional ‘coordinates’ and its conjugate ‘momenta’, respectively.

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To simplify Hamiltonian (2.4), let us transform the variables (qi , pi ) to the variables (Qi , Pi ) as


 ϕ −ν sin ϕ2 q1 Q1 (γ /2)t µ cos 2 (2.5) =e , ϕ ϕ Q2 µ sin 2 ν cos 2 q2
1 

γ 


 ϕ − ν1 sin ϕ2 α1 0 p1 µ cos ϕ2 −ν sin ϕ2 q1 P1 µ cos 2 −(γ /2)t (γ /2)t 2 =e +e , γ ϕ ϕ ϕ ϕ 1 1 P2 0 p2 µ sin 2 ν cos 2 q2 sin cos 2 α2 µ

2

ν

2

(2.6) where µ, ν, α1 , α2 and ϕ are given by  L1 /C − L/C1 , µ= 4 L2 /C − L/C2  L2 /C − L/C2 , ν= 4 L1 /C − L/C1 α1 = α2 =

(2.7)

(2.8)

LL1 + LL2 + L1 L2 (L + L1 )ν 2 sin2

ϕ 2

+ (L + L2 )µ2 cos2 LL1 + LL2 + L1 L2

ϕ 2

− µνL sin ϕ

,

(2.9)

, (2.10) + (L + L2 )µ2 sin2 ϕ2 + µνL sin ϕ µν/C Lµν tan ϕ = (2.11) = . 2 2 2 (1/C + 1/C1 )ν − (1/C + 1/C2 )µ (L + L1 )ν − (L + L2 )µ2 From Eqs. (2.7) and (2.8), we can confirm that µ and ν are dependent on the capacitance and inductance of the coupled circuit, respectively. Eqs. (2.5) and (2.6) do not always represent canonical transformations [22,23] between the variables (qi , pi ) and variables (Qi , Pi ). If (Qi , Pi ) are canonical coordinates, there should exit a new Hamiltonian which is determined only by Hamiltonian (2.4) and the linear transformations (2.5) and (2.6). These variables (qi , pi ) and (Qi , Pi ) in two representations must satisfy the following relation [22]: 2 

(L + L1 )ν 2 cos2

Pi Q˙ i − HQ =

i=1

2 

ϕ 2

pi q˙i − Hq +

i=1

dF , dt

(2.12)

where F , called as generating function, is possibly a time-dependent function in phase space. Because the variables ˙ i , P˙i ) we can get the relations (Qi , Pi ) are independent mutually, by identifying the coefficients of (Q Pj −

2 

pi

i=1

−

2  i=1

pi

∂qi ∂F = , ∂Qj ∂Qj

∂qi ∂F = , ∂Pj ∂Pj

(2.13)

(2.14)

and HQ = Hq −

2  i=1

pi

∂F ∂qi − . ∂t ∂t

(2.15)

Substituting Eqs. (2.5) and (2.6) into Eqs. (2.13) and (2.14) the generating function is given by 1γ F= αi Q2i . 2 2 2

i=1

(2.16)

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Substituting Eqs. (2.4)–(2.6) and (2.16) into Eq. (2.15), we obtain the transformed Hamiltonian of the mesoscopic circuit as HQ =

P12 P2 β1 β2 + Q21 + 2 + Q22 , 2α1 2 2α2 2

where β1 =

cos2 ϕ2 µ2

sin2 ϕ2 β2 = µ2





1 1 + C C1

1 1 + C C1

 + 

sin2 ϕ2 ν2

cos2 ϕ2 + ν2





1 1 + C C2 1 1 + C C2

(2.17)  +

sin ϕ α1 2 − γ , µν 4

(2.18)

−

sin ϕ α2 2 − γ . µν 4

(2.19)



To manage the system quantum mechanically, we can replace a pair of the observable quantities qi and pi with a pair of linear Hermitian operators, named qˆi and pˆi , respectively. They satisfy the boson commutation relation, [qˆi , pˆ i ] = i h¯ . Then, we can confirm that Hamiltonian (2.17) converted to operator which is evidently time-independent. With quantum version, we can reexpress Eq. (2.17) as Hˆ Q =

2  ˆ2  P

 1 2 ˆ2 + αi ωi Qi , 2αi 2 i

i=1

which is of more compact form, where  βi ωi = . αi

(2.20)

(2.21)

We can say that ω1 and ω2 are the frequencies of the two loops, respectively. Then, Eq. (2.20) represents the sum of two Hamiltonian of the simple harmonic oscillators having the frequencies ω1 and ω2 .

3. Quantum fluctuation Because the two coupled Hamiltonians given in Eq. (2.20) are independent of each other, the transformed quantum mechanical energy spectrum of the coupled electric circuit can be written as  

1 1 En1 ,n2 = n1 + (3.1) hω h¯ ω2 , ¯ 1 + n2 + 2 2 and the corresponding eigenstates are |Ψn1 ,n2 = |n1 ⊗ |n2 ,

(3.2)

where |n1 and |n2 are the eigenstates of individual simple harmonic oscillators with frequencies ω1 and ω2 , respectively. We can write the Schrödinger equation for Hamiltonian (2.4) as i h¯

∂ |ψn1 ,n2 = Hˆ q |ψn1 ,n2 , ∂t

(3.3)

where |ψn1 ,n2 is the wave function in qˆ representation. We can confirm that it is connected with that in Qˆ representation by a unitary transformation operator Uˆ as |ψn1 ,n2 = Uˆ |Ψn1 ,n2 ,

(3.4)

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where Uˆ = Uˆ 1 Uˆ 2 Uˆ 3 , and Uˆ 1 , Uˆ 2 , and Uˆ 3 are given by
 
  i γ

γ

i Uˆ 1 = exp ln µ + t pˆ1 qˆ1 + qˆ1 pˆ 1 exp ln ν + t pˆ 2 qˆ2 + qˆ2 pˆ 2 , 2h¯ 2 2h¯ 2  i ϕ

pˆ 1 qˆ2 − pˆ 2 qˆ1 , Uˆ 2 = exp − h¯ 2  iγ

Uˆ 3 = exp − α1 qˆ12 + α2 qˆ22 . 4h¯

(3.5)

(3.6) (3.7) (3.8)

Substituting Eq. (3.4) into Eq. (3.3), we can obtain a transformed Hamiltonian that represents the coupled simple harmonic oscillator having frequencies ω1 and ω2 , respectively: pˆ 2 pˆ 2 ∂ Uˆ β1 β2 ˜ = 1 + qˆ12 + 2 + qˆ22 . Hˆ q = Uˆ −1 Hˆ q Uˆ − i h¯ Uˆ −1 ∂t 2α1 2 2α2 2

(3.9)

In this state, we can confirm that the relation given in Eq. (3.4) is right, since the above equation is consistent with Eq. (2.17). From unitary operator (3.5), we can confirm that the operator Uˆ 1 is the squeezed operator, Uˆ 2 is the rotate operator with the angle ϕ/2. Therefore, the uncertainties of charges or currents may have squeezing effect in the state |ψn1 ,n2 . The definitions of uncertainties for canonical variables in number state are given by, respectively,

2

2 ∆qˆi = ψn1 ,n2 |qˆi2 |ψn1 ,n2 − ψn1 ,n2 |qˆi |ψn1 ,n2 ,

2

2 ∆pˆ i = ψn1 ,n2 |pˆ i2 |ψn1 ,n2 − ψn1 ,n2 |pˆ i |ψn1 ,n2 .

(3.10) (3.11)

For ground state, substituting Eq. (3.4) with Eq. (3.5) into Eqs. (3.10) and (3.11), the uncertainties of charges and those conjugate currents of the system can be obtained, respectively, as

 2 ϕ 2 ϕ 2

cos sin ¯ 2 2 h , ∆qˆ1 = e−γ t ν 2 (3.12) + ω1 α1 ω2 α2 2

 2 ϕ 2 ϕ

2 cos sin ¯ 2 2 h ∆qˆ2 = e−γ t µ2 , + ω1 α1 ω2 α2 2 



  2

γ2 γ2 ¯ γt 2 2 ϕ 2 ϕ h + ω2 α2 1 + 2 sin , ∆pˆ 1 = e µ ω1 α1 1 + 2 cos 2 2 2 4ω1 4ω2  

 


2 γ2 γ2 ¯ γt 2 2 ϕ 2 ϕ h ∆pˆ 2 = e ν ω1 α1 1 + 2 sin + ω2 α2 1 + 2 cos . 2 2 2 4ω1 4ω2

(3.13)

(3.14)

(3.15)

From Eqs. (3.12) and (3.13), we can confirm that the uncertainties of charges reduce exponentially as time goes by. This phenomenon takes place by the decay of oscillating amplitude for charges, since the resistances of circuit damp the electromagnetic field.

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Using the above equations, we can obtain the uncertainty relations of the charges and those conjugate currents in vacuum state as  



2

2 ν2 ϕ ν2 4 ϕ ∆qˆ1 ∆pˆ 1 = + 1 + 2 sin4 1 + 2 cos 2 2 4ω1 4ω2 
 ω2 α2 ω1 α1 α1 α2 ν2 ¯2 2 ϕ 2 ϕ h (3.16) + cos , sin + + + ω1 α1 ω2 α2 4ω1 ω2 α2 α1 2 2 4

  

2

2 ν2 ϕ ν2 4 ϕ ∆qˆ2 ∆pˆ 2 = + 1 + 2 cos4 1 + 2 sin 2 2 4ω1 4ω2 
 ω2 α2 ω1 α1 h¯ 2 ϕ ϕ ν2 α1 α2 (3.17) + . + + + sin2 cos2 ω1 α1 ω2 α2 4ω1 ω2 α2 α1 2 2 4 From Eqs. (2.9), (2.19) and (2.21), we can confirm that one frequency ω1 cannot be equal to another frequency ω2 . Therefore, the uncertainty relations do not satisfy minimum uncertainty relation, (∆qˆi ∆pˆ i )min = h/2, even ¯ though the resistance of the circuit is zero.

4. Squeezed effect We introduce the Hermitian amplitude operators to discuss the squeezed effect in mesoscopic circuit,  ωi αi ˆ ˆ Qi , Xi = 2h¯ 1 Yˆi = √ Pˆi , 2h¯ ωi αi

(4.1) (4.2)

where the operators Xˆ i and Yˆi are essentially dimensionless charges and currents operators. The Hermitian operator Xˆ i and Yˆi are now readily seen to be the amplitudes of the two quadratures of the the field having a phase difference π/2. The operators Xˆ i and Yˆj from the relation [qˆi , pˆj ] = i h¯ satisfy the commutation relation   i Xˆ i , Yˆj = δij . 2

(4.3)

The dimensionless charges and currents operators corresponding to qˆi and pˆi are defined as xˆi and yˆi . From Eqs. (2.5), (2.6), (4.1), and (4.2), the uncertainties of the dimensionless charges and currents can be expressed in terms of the amplitudes xˆi and yˆi : 2 e−γ t 2

∆xˆ1 = ν , 4

2 e−γ t 2 µ , ∆xˆ2 = 4  

  2 e γ t 2

γ 2 α12 γ 2 α22 2 ϕ 2 ϕ 1+ µ cos + 1+ sin , ∆yˆ1 = 4 4 2 4 2

    2 e γ t 2

γ 2 α22 γ 2 α12 2 ϕ 2 ϕ ν + 1+ 1+ ∆yˆ2 = sin cos . 4 4 2 4 2

(4.4) (4.5) (4.6)

(4.7)

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Using the above equations, we can obtain the uncertainty relations of the two amplitude operators in vacuum state, when γ = 0, as 1 ∆xˆi ∆yˆi = . 4 If the values of capacitances and inductance of mesoscopic circuit are given to satisfy relation ν 2 < 1,

(4.8)

(4.9)

then we obtain the uncertainties of amplitude operators, when γ = 0, as 2 1

∆xˆ1 < , 4

2 1 ∆yˆ1 > , 4

(4.10) (4.11)

and

2 1 ∆xˆ2 > , 4

2 1 ∆yˆ2 < . 4 These uncertainty relations follow squeezing effect depending on the values of µ and ν.

(4.12) (4.13)

5. Discussion and conclusion We will discuss two limiting cases which are capacitance coupled circuit and inductance–resistance coupled circuit, respectively, using the result written in previous section. First, we suppose that 1/C = 0, then Fig. 1 becomes inductance–resistance coupled circuit and Eqs. (2.7)–(2.11) can be rewritten as  C2 1 , µ= = 4 (5.1) ν C1 √ 2 C1 C2 . tan ϕ = (5.2) C1 (1 + L1 /L) − C2 (1 + L2 /L) This is identical to the values given in Ref. [11]. Next, investigate the case L = R = 0. Then Eqs. (2.7)–(2.11) become  1 L1 µ= = 4 , ν L2 √ 2 L1 L2 . tan ϕ = L2 (1 + C/C1 ) − L1 (1 + C/C2 )

(5.3) (5.4)

This is identical to the values given in Ref. [12]. From Eqs. (5.1)–(5.4), we know that µ and ν are independent of the coupled capacitance and inductance both in capacitance coupled circuit and inductance–resistance coupled circuit. On the other hand, in case of the capacitance–inductance–resistance coupled circuit, it is explicitly dependent on them. In this Letter, we investigated the quantum fluctuations of charges and those conjugate currents of mesoscopic capacitance–inductance–resistance coupled circuit on the basis of classical equation of motions for charges. We

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obtained canonical transformation matrices and unitary transformation operator that transforms the Hamiltonian of mesoscopic capacitance–inductance–resistance coupled circuit to that of mutually independent two-dimensional simple harmonic oscillator. From Eqs. (3.12)–(3.15), we can confirm that the quantum variance of the charges decreases exponentially as time goes by, while that of those conjugate current increases exponentially. Even if the resistances which role as damping factors are zero, the uncertainty relations given by Eqs. (3.16) and (3.17) do not satisfy minimum uncertainties relation. The uncertainties of charges can be reduced by paying the cost that becoming the uncertainties of currents larger, or vice versa.

Acknowledgement This work was supported by the Ministry of Education (Hacksim, BK21 project).

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