An investigation of algebraic quantum dynamics for mesoscopic coupled electric circuits with mutual inductance

Physica B 495 (2016) 123–129

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An investigation of algebraic quantum dynamics for mesoscopic coupled electric circuits with mutual inductance H. Pahlavani n, E. Rahmanpour Kolur Department of Physics, University of Qom, Ghadir Blvd., Qom 371614611, I.R. Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 2 September 2015 Received in revised form 10 February 2016 Accepted 10 May 2016 Available online 13 May 2016

Based on the electrical charge discreteness, the Hamiltonian operator for the mutual inductance coupled quantum mesoscopic LC circuits has been found. The persistent current on two driven coupled mesoscopic electric pure L circuits (two quantum loops) has been obtained by using algebraic quantum dynamic approach. The influence of the mutual inductance on energy spectrum and quantum fluctuations of the charge and current for two coupled quantum electric mesoscopic LC circuits have been investigated. & 2016 Elsevier B.V. All rights reserved.

Keywords: Mesoscopic circuit Mutual inductance Charge discreteness Algebraic quantum dynamic

1. Introduction The progress in nanotechnology, telecommunication and quantum computer has been caused by the integrated circuits and components are miniaturized toward atomic scale dimensions in the manufacture of electronic devices. Quantum electrical circuits larger than atomic systems are known as mesoscopic electrical circuits. When the transport dimension of electric devices in the electric circuits reaches to coherence length, the quantum mechanical effects must be taken into account. The classical equation of motion for an electric circuit of LC (inductance L and capacitance C) design is the same as that for a harmonic oscillator, whereas the coordinate x means electric charge q. The quantization of the circuit LC was carried out in the same way as that of a harmonic oscillator for the first time by Louisell [1], where the electric charge was treated as a continuous variable. The equation of motion RLC (resistance–inductance–capacitance) circuit with timedependent power source has been quantized [2]. Recently, the timedependent quantum treatment of a mesoscopic RLC circuit with timedependent resistance, inductance, capacitance and power source are investigated by Lewis and Riesenfeld quantum invariant method in [3,4]. For classical and quantum collective processes, requires that time behavior of coupled harmonic oscillators be considered. Recently the time-dependent coupled harmonic oscillators are studied in [5,6]. n

Corresponding author. E-mail addresses: [email protected], [email protected] (H. Pahlavani). http://dx.doi.org/10.1016/j.physb.2016.05.009 0921-4526/& 2016 Elsevier B.V. All rights reserved.

The model of coupled harmonic oscillators has been widely used to study the quantum effects in mesoscopic coupled electric circuits. Therefore, based on Louisell's work, the quantization of two coupled LC circuits with mutual inductance is discussed, which turns out to resemble a pair of harmonic oscillators with a kinetic coupling term. Mutual inductance is the ability of one inductor to induce a voltage across a neighboring inductor. The quantum mechanical effects of mesoscopic electrical LC circuits with coupled inductor term are investigated in [7–10]. Under the mesoscopic scale, a theory for electric circuits was proposed by Li and Chen [11], in which a charge discreteness is first introduced in the quantization of quantum electric circuits. In this theory the discreteness of the electric charge plays an important role and has a fundamental concept in electronic devices. Therefore, a lot of interest researches have grown in the study of quantum mesoscopic electronic circuits LC with discrete charge [12–26]. With the dramatic development electronic devices, including quantum computers and nano-devices, the research on the fields of role the mutual inductance in two coupled mesoscopic electric LC circuits with charge discrete are paid close attentions. In this paper, we consider discreteness of electrical charge and generalize Li and Chen quantum theory for two coupled quantum mesoscopic electric circuits. We obtain the persistent current on two driven coupled electric pure L circuits (two quantum loops) by algebraic quantum dynamic approach. Based on quantum algebraic structures, we also study spectrum energy and quantum fluctuations of the charge and current for two coupled quantum mesoscopic LC circuits.

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H. Pahlavani, E.R. Kolur / Physica B 495 (2016) 123–129

2. Two quantum electric circuits with mutual-inductance

2.1. The eigenvalues and eigenvectors two coupled quantum electric LC circuits with mutual-inductance

We consider two LC circuits coupled by mutual inductance drawn in Fig. 1, where ε1 ( t ) and ε2 ( t ) are external fields, L1 and L2 are the self-inductance coefficients, C1 and C2 are the capacitances and m is the mutual inductance. The classical Lagrangian of this system is given by [9–27] 2 ⎞ q2 ⎞ 1⎛ 1⎛q L = ⎜⎜ L1 q1̇ 2 + L2 q2̇ 2 ⎟⎟ + mq1̇ q2̇ − ⎜⎜ 1 + 2 ⎟⎟ − ε1 (t ) q1 − ε2 (t ) q2, 2⎝ 2 ⎝ C1 C2 ⎠ ⎠ (1)

Eq. (4) represents a pair of quantized LC circuits with kinetic energy coupling. Let us consider the Hamiltonian (4) for when the external fields are zero, L1 = L2 = L and C1 = C2 = C as

^ H=

1 2AL

⎛ ^ 2 ^ 2⎞ m 1 ⎛ ^2 ^2⎞ ⎜ p1 + p2 ⎟ − 2 p^1 p^2 + ⎜ q + q2 ⎟, ⎝ ⎠ AL ⎠ 2C ⎝ 1

∂ ∂ where p^1 = − i= ∂q and p^2 = − i= ∂q . In order to find the energy 1

where the variables q1 and q2 are the electrical charges on the two capacitances, they are introduced instead of the conventional coordinates, q1̇ ( t ) and q2̇ ( t ) are the currents through the two inductances. The conjugate variables of the electrical charge are given by

p1 (t ) = p2 (t ) =

∂L = L1 q1̇ + mq2̇ , ∂q1̇ ( t ) ∂L = L2 q2̇ + mq1̇ . ∂q2̇ ( t )

From Eqs. (1) and (2), the classical Hamiltonian of the system in charge representation can be written as follows:

H=

1 2A

2 ⎛ p2 p2 ⎞ q2 ⎞ m 1⎛q ⎜⎜ 1 + 2 ⎟⎟ − p1 p2 + ⎜⎜ 1 + 2 ⎟⎟ + ε1 (t ) q1 L2 ⎠ AL1 L2 2 ⎝ C1 C2 ⎠ ⎝ L1

+ ε2 (t ) q2,

(3)

where A = 1 − m2 /L1 L2. We consider the case of m < L1 L2 , which means that there exists magnetic leakage. The classical Hamiltonian (3) is the same as that for a pair of harmonic oscillators with a kinetic coupling term. As is clear from Eq. (2), in the charge representation the dimensional of the p is the same with the magnetics flux ϕ. According to the standard quantization principle, a pair of observable quantities qi and pi can be replaced with a pair of linear Hermitian operators, named q^i and p^i respectively. They satisfy the commutation relation ⎡ q^ , p^ ⎤ = i=δ . Thus the classical Hamiltonian (3) after the quantization ij ⎣ i j⎦ is represented as 2⎞ 2⎞ ⎛ 2 ⎛ 2 p^ q^ m ^ ^ 1 ⎜ p^1 1 q^ ^ H= + 2⎟− p1 p2 + ⎜ 1 + 2 ⎟ + ε1 (t ) q^1 L2 ⎟⎠ AL1 L2 C2 ⎟⎠ 2A ⎜⎝ L1 2 ⎜⎝ C1 + ε (t ) q^ . 2

2

(4)

2

spectrum and eigenvectors of Hamiltonian (5), we introduce the generalized coordinates x^1 and x^2 as

q^1 =

1

q^2 =

1

Then

(2)

(5)

2 2

( x^ + x^ ), ( x^ − x^ ),

∂2 ∂x12

1

2

1

2

and

∂2 ∂x22

(6) are given by

∂2 ∂2 1 ⎛ ∂2 ∂2 ⎞ ⎟⎟, = ⎜⎜ 2 + 2 + 2 ∂q1 ∂q2 2 ⎝ ∂q1 ∂x1 ∂q22 ⎠

(7)

∂2 1 ⎛ ∂2 ∂2 ∂2 ⎞ ⎟⎟, = ⎜⎜ 2 − 2 + ∂q1 ∂q2 2 ⎝ ∂q1 ∂x22 ∂q22 ⎠

(8)

and so 2 2 2 2 q^1 + q^2 = x^1 + x^2 .

(9)

Substituting Eqs. (6)–(8) into (5) yields

m ⎞ ∂2 m ⎞ ∂2 =2 ⎛ =2 ⎛ 1 ⎛ ^2 ^2 ⎞ ^ H= − ⎜1 − ⎟ 2 − ⎜1 + ⎟ 2 + ⎜ x1 + x2 ⎟. 2AL ⎝ L ⎠ ∂x1 2AL ⎝ L ⎠ ∂x2 2C ⎝ ⎠ (10) The time-independent Schrodinger equation Hψ ( t ) = Eψ ( t ) for the Hamiltonian operator (10) can be written as

−

L = 2 ∂ 2ψ1 (x1 ) + ω1′ 2 x12 ψ1 (x1 ) = E1 ψ1 (x1 ), 2L ∂x12 2

(11)

−

2 L = 2 ∂ ψ2 ( x2 ) + ω2′ 2 x22 ψ2 (x2 ) = E2 ψ2 (x2 ), 2L 2 ∂x22

(12)

Fig. 1. Two LC circuits with mutual inductance.

H. Pahlavani, E.R. Kolur / Physica B 495 (2016) 123–129

⎞ = 2 ⎛⎜ ^ ^† ^ H= − Q + Q − 2⎟⎟. 2 ⎜ 2Lqe ⎝ ⎠

where

1 ⎛ m⎞ ⎜ 1 + ⎟, LC ⎝ L⎠ ⎛ ⎞ 1 m ⎜ 1 − ⎟. ω22 ′ = LC ⎝ L⎠ ω12 ′ =

(13)

The eigenvectors and eigenvalues of energy equations (11) and (12) are given by

ψn1 (x1 ) = ψn2 (x2 ) =

1 2n1 n1 !

λ1 π

⎛ λ x2 ⎞ exp ⎜⎜ − 1 1 ⎟⎟ Hn1 ( λ1 x1 ), 2 ⎠ ⎝

⎛ λ x2 ⎞ 1 λ2 exp ⎜⎜ − 2 2 ⎟⎟ Hn2 ( λ2 x2 ), 2n2 n2 ! π 2 ⎠ ⎝

^ I =

= ⎛^ ^ †⎞ ⎜ Q − Q ⎟. 2iqe L ⎝ ⎠

En2

2

2

1 2

1 2

Iφ =

⎛q ⎞ = sin ⎜ e φ⎟, Lqe ⎝= ⎠

(15)

3. The discreteness of electronic charge In this section, we review the quantum theory of mesoscopic LC circuits. In order to take into account the discreteness of electronic charge, Li and Chen impose that the eigenvalues of the self-adjoins operator q^ (electric charge) take discrete values i.e.

(16)

where n ∈ Z (set of integers) and qe = 1.602 × 10−19C , the elementary electric charge [11]. Since the spectrum of charge is discrete, the inner product in charge representation will be a sum instead of the usual integral and the electric current operator will be defined by the discrete derivatives

∇qe =

^ Q −1 , qe

^† ¯ qe = 1 − Q , ∇ qe

(17)

^ ^ where Q = eiqe p / = is a minimum shift operator. The physical di^ mension of qe p introduced by Li and Chen should be the same as ^ ^† the planck constant = . Operators Q , Q act on the eigenvectors of and q^ and satisfy the following commutation relations:

⎡ ^ ^⎤ ^ ⎢⎣ q , Q ⎥⎦ = − qe Q ,

(23)

where φ ∈ R is a real quantum number with continuum eigenvalues 0 ≤ φ ≤ 2π , which may be called the pseudo-flux. The magnitude of electric current is bounded between −= /Lqe and = /Lqe . Eq. (22) shows that the persistent current in a mesoscopic L design is an observable quantity periodically depending on the flux φ. Therefore this equation is valid for persistent current on a single mesoscopic ring.

4. Extended the quantum theory Li and Chen for two coupled quantum mesoscopic electric LC circuits As a matter of fact, that the electric charge is discrete, in the process quantization of the mesoscopic circuits, the eigenvalues of the self adjoint operator q^ i = 1, 2 should take discrete values as i

(

)

follows:

q^1 |q1, q2 〉 = n1 qe |q1, q2 〉,

(24)

q^2 |q1, q2 〉 = n2 qe |q1, q2 〉,

(25)

where n1,n2 are the set of integer ( n1, n2 ∈ Z ) in Fock space. |q1, q2 〉 = |q1 〉 ⊗ |q2 〉 stand for eigenvectors of charge operator for circuits 1 and 2 respectively. These allow us to introduce minimum ^ ^ ^ ^ shift operators Q 1 = eiqe p1/ = and Q 2 = eiqe p2 / = which is shown to satisfy the following commutation relations:

⎡^ ^ ⎤ ^ ⎣⎢ qi , Q i ⎥⎦ = − qe Q i, ⎡ ^ ^ †⎤ ^+ q, Q = qe Q i , ⎣⎢ i i ⎥⎦

i = 1, 2. (26)

For q^1 |n1, n2 〉 = n1 qe |n1, n2 〉 and q^2 |n1, n2 〉 = n2 qe |n1, n2 〉, the algebraic structure (26) is driven by

^ †⎤

(18)

Then we can write down the ‘momentum’ operator which is also the ‘current’ operator apart from the inductance factor

⎛ †⎞ = ^ ¯ qe ) = = ⎜ Q^ − Q^ ⎟. P= (∇qe + ∇ 2i 2iqe ⎝ ⎠

qe

ein = φ |n〉,

^ ^† ^†^ Q i Q i = Q i Q i = 1.

⎡ ^ ^ †⎤ ^† ⎢⎣ q , Q ⎥⎦ = qe Q , ⎡^ Q , Q ⎥ = 0. ⎣⎢ ⎦

∑ n =−∞

where λ1 = Lω1′/= and λ2 = Lω2′ /= .

q^|n〉 = nqe |n〉,

(22)

(14)

n1 = 0, 1, 2, … n2 = 0, 1, 2, …

(21)

The eigenvalues Iφ and eigenvectors |Iφ 〉 of the current operator are

n =∞

( ), = =ω ′ ( n + ).

(20)

^ The definition of physical current I arises from the Heisenberg ⎤ ⎡ ^ ^ equation I = dq^ /dt = 1/i= ⎣⎢ q^ , H ⎦⎥ [11,17,26]:

|Iφ 〉 = En1 = =ω1′ n1 +

125

^ Q i |ni , nj 〉 = |ni − 1, nj 〉, ^† Q i |ni , nj 〉 = |ni + 1, nj 〉.

(27)

These states fulfill ∑n , n

1 2∈ Z

(19)

The free Hamiltonian of a discrete-charge mesoscopic quantum circuit is described by

|n1, n2 〉〈n1, n2 | = 1 the completeness and

〈n1, n2 |m1, m2 〉 = δ n1, n2 δ m1, m2 the orthogonality conditions. Using commutation relations (26) and the Heisenberg equation of motion two important self-adjoint momentum operators are obtained as

126

H. Pahlavani, E.R. Kolur / Physica B 495 (2016) 123–129

^ ^ P1 = L1 I1 =

^ ^ P2 = L2 I2 =

⎞ i= ⎧ ⎛ ^ † m ⎛ ^† ^ ⎞⎛ ^ † ^ ⎞⎫ ⎨ ⎜ Q 1 − Q^1 ⎟ − ⎜ Q + Q1 ⎟ ⎜ Q 2 − Q 2 ⎟ ⎬, ⎠⎭ ⎠⎝ ⎠ 2L2 ⎝ 1 2Aqe ⎩ ⎝

⎞ i= ⎧ ⎛ ^ † m ⎛ ^† ^ ⎞⎛ ^ † ^ ⎞⎫ ⎨ ⎜ Q 2 − Q^ 2 ⎟ − ⎜ Q1 − Q1 ⎟ ⎜ Q 2 + Q 2 ⎟ ⎬. ⎠⎭ ⎠⎝ ⎠ 2L1 ⎝ 2Aqe ⎩ ⎝

inductance

(28)

and ε2 ( t ) with mutual inductance, is

(29)

The operator P^i is proportional with the physical quantity, electric current (apart from a factor 1 ), differs from the operator p^i as long L as the discreteness of charge is taken into account. In the limit ^ q → 0, P will become the usual p^ and the charge is treated as a i

e

Time dependence of the Hamiltonian (32), for two pure L circuits in change representation under the external potentials ε1 ( t )

i

⎧ ⎫ ⎞ ⎞ † † †⎞ ⎛ †⎞ ⎪ =2 ⎪ 1 ⎛ ^ 1 ⎛^ m ⎛^ ⎨ ⎜ Q1 + Q^1 − 2⎟ + ⎜ Q 2 + Q^ 2 − 2⎟ − ⎜ Q1 − Q^1 ⎟ ⎜ Q^ 2 − Q^ 2 ⎟ ⎬ L2 ⎝ ⎠ ⎠ 2L1L2 ⎝ ⎠⎝ ⎠⎪ 2Aqe2 ⎪ ⎩ L1 ⎝ ⎭ + ε1 (t ) q^1 + ε 2 (t ) q^2 .

^ HL = −

Using the Heisenberg equation of motion, the current operators can be determined as

continuous variable i.e.

m ⎞ 1⎛ ^ p^1 = L1 I1 = ⎜ p^1 − p^2 ⎟, A⎝ L2 ⎠

m ⎞ 1⎛ ^ p^2 = L2 I2 = ⎜ p^2 − p^1 ⎟. A⎝ L1 ⎠

⎧ ⎫ ⎞ ⎞ + † †⎞ ⎛ †⎞ ⎪ =2 ⎪ 1 ⎛ ^ 1 ⎛^ m ⎛^ ⎨ ⎜ Q1 + Q^1 − 2⎟ + ⎜ Q 2 + Q^ 2 − 2⎟ − ⎜ Q1 − Q^1 ⎟ ⎜ Q^ 2 − Q^ 2 ⎟ ⎬ . L2 ⎝ ⎠ ⎠ 2L1L2 ⎝ ⎠⎝ ⎠⎪ 2Aqe2 ⎪ ⎭ ⎩ L1 ⎝

i = 1, 2

⎤ ⎡ ^ Pi ^ ^⎥ ⎢H , 0, qi = − i= ⎥⎦ ⎢⎣ Li

(42)

^ I2 (t ) =

⎞ i= ⎧ 1 ⎛ ^ † m ⎛^† ^ ⎞⎛ ^ † ^ ⎞⎫ ⎨ ⎜ Q 2 − Q^ 2 ⎟ − ⎜ Q 2 + Q 2 ⎟ ⎜ Q1 − Q1 ⎟ ⎬. ⎠⎭ ⎠⎝ ⎠ 2L1 L2 ⎝ 2Aqe ⎩ L2 ⎝

(43)

The time-independent eigenvectors |Iφ1, Iφ2 〉 of the current operators are

(32)

(33)

i = 1, 2

⎡ i= ⎛ ^ m ⎛^ ^ †⎞ ^ †⎞⎛ ^ ^ †⎞ ^⎤ {⎜ Q 1 + Q 1 ⎟ − ⎜ Q1 − Q1 ⎟ ⎜ Q 2 − Q 2 ⎟ }, ⎢ q^1, P1 ⎥ = ⎣ ⎦ 2A ⎝ ⎠ ⎠⎝ ⎠ 2L2 ⎝

⎞ i= ⎧ 1 ⎛ ^ † m ⎛ ^† ^ ⎞⎛ ^ † ^ ⎞⎫ ⎨ ⎜ Q 1 − Q^1 ⎟ − ⎜ Q + Q1 ⎟ ⎜ Q 2 − Q 2 ⎟ ⎬, ⎠⎭ ⎠⎝ ⎠ 2L1 L2 ⎝ 1 2Aqe ⎩ L1 ⎝

(31)

This Hamiltonian is equivalent to a pair of quantum mesoscopic L circuit (the pure L-design) with inductance coupling term(mutual inductance). In order to understand the main conclusions quantum mechanical for two coupled quantum mesoscopic electrical circuits, we obtain the following commutation relations as:

⎡ ^ ^⎤ ⎢⎣ H0, Pi ⎥⎦ = 0,

^ I1 (t ) = (30)

The Hamiltonian operator for such system is given by ^ H0 = −

(41)

(34)

∑

|Iφ1, Iφ 2 〉 =

e

in1qe in2 qe φ1 = e = φ 2 |n1,

n2 〉,

n1, n2

(44)

where ( φ1, φ2 ) ∈ R are two real quantum numbers with continuum eigenvalues 0 ≤ φ1 ≤ 2π and 0 ≤ φ2 ≤ 2π , which may be called the pseudo-fluxes. In this section, we will obtain the quantum current relations of the system based on an algebra approach. In order to prove, we find the time-evolution operator U^ (t ) for the Hamiltonian (41) using the method introduced in Ref [28]. For this purpose, let us decompose the Hamiltonian (41) to ^ HR (t ) = −

⎧ ⎫ ⎞ ⎞ † † †⎞ ⎛ †⎞ ⎪ =2 ⎪ 1 ⎛ ^ 1 ⎛^ m ⎛^ ⎨ ⎜ Q1 + Q^1 − 2⎟ + ⎜ Q 2 + Q^ 2 − 2⎟ − ⎜ Q1 − Q^1 ⎟ ⎜ Q^ 2 − Q^ 2 ⎟ ⎬ , L2 ⎝ ⎠ ⎠ 2L1L2 ⎝ ⎠⎝ ⎠⎪ 2Aqe2 ⎪ ⎭ ⎩ L1 ⎝

^ HS (t ) = ε1 (t ) q^1 + ε2 (t ) q^2,

(45)

(46)

(35) and

⎡ i= ⎛ ^ m ⎛^ ^ †⎞ ^ †⎞⎛ ^ ^ †⎞ ^ ⎤ {⎜ Q 2 + Q 2 ⎟ − ⎜ Q1 − Q1 ⎟ ⎜ Q 2 − Q 2 ⎟ }. ⎢ q^2, P2 ⎥ = ⎣ ⎦ 2A ⎝ ⎠ ⎠⎝ ⎠ 2L1 ⎝

i= (36)

(47)

The operator U^S (t ) can be found easily as

The uncertainty relation namely

= ⎛ ^† ^ ⎞ m ⎛ ^† ^ ⎞⎛ ^ † ^ ⎞ ^ Δq^1 ΔP1 ≥ {⎜ Q 1 + Q 1 ⎟ − ⎜ Q1 − Q1 ⎟ ⎜ Q 2 − Q 2 ⎟ }, ⎠ ⎠⎝ ⎠ 2L2 ⎝ 4A ⎝

^ dUS ^ ^ = HS US , dt

(37)

i i ^ ^ ^ US (t ) = e− = η1 (t ) q1 e− = η2 (t ) q2 ,

(48)

where

= ⎛^† ^ ⎞ m ⎛ ^† ^ ⎞⎛ ^ † ^ ⎞ ^ Δq^2 ΔP2 ≥ {⎜ Q 2 + Q 2 ⎟ − ⎜ Q1 − Q1 ⎟ ⎜ Q 2 − Q 2 ⎟ }. ⎠ ⎠⎝ ⎠ 2L1 ⎝ 4A ⎝

(38)

t

η1 (t ) =

∫0

η2 (t ) =

∫0

ε1 (t′) dt′,

(49)

ε2 (t′) dt′.

(50)

In the limit ( qe → 0), we have

= Δq^i Δp^i ≥ , 2A

i = 1, 2

(39)

The uncertainty relation (39) recovers the usual Heisenberg uncertainty relation for every circuit, if m ¼0 or m2 < L1 L2 i.e. magnetic leakage, therefore

Δq^i Δp^i ≥

= , 2

i = 1, 2.

(40)

5. The persistent current of two quantum rings with mutual

t

Using Levi–Malcev theory [28], we can write differential equation for the part R of the Hamiltonian (45) as

i=

⎛ ⎞ ^ dUR ⎜ ^ −1 ^ ^ ^ = ⎜ US (t ) HR (t ) US (t ) ⎟⎟ UR (t ), dt ⎝ ⎠

The time evolution operator U^R (t ) can be obtained by

(51)

H. Pahlavani, E.R. Kolur / Physica B 495 (2016) 123–129

⎡ i⎛ ⎞⎤ ^ ^† ^ UR = exp ⎢ − ⎜ α (t ) Q 1 + α† (t ) Q 1 + λ1 ⎟ ⎥ exp ⎠⎦ ⎣ =⎝

Iφ1

= −

⎡ i⎛ ⎞⎤ ^ ^† ⎢ − ⎜ β (t ) Q 2 + β † (t ) Q 2 + λ2 ⎟ ⎥ × exp ⎠⎦ ⎣ =⎝ ⎡i ⎢ ⎣=

qe ⎛ ⎞ m ⎡ ⎜ η (t ) + η (t ) − ( φ + φ ) ⎟ ⎢ sin 2 1 2 ⎠ = ⎝ 1 2L1 L2 ⎣ q ⎛ ⎞⎤⎫ + sin e ⎜ η2 (t ) − η1 (t ) + ( φ1 − φ2 ) ⎟ ⎥ ⎬ , ⎠⎦⎭ = ⎝

Iφ 2 = −

=2 , 2Aqe2 L1

g2 = −

=2 , 2Aqe2 L2

k=

∫0

t

β (t ) =

∫0

χ (t ) =

∫0

t

t

t

ξ (t ) =

∫0

λ1 (t ) =

∫0

The second term of these expressions shows the coupling effects between two circuits evidently. One of the results in these equations is when the parameter η (t ) = 0, Eqs. (56) and (57) reduced into the persistent current on the two coupled quantum mesoscopic rings with mutual inductance

λ2 (t ) =

∫0

⎛ i ⎛ ⎞ ⎞ g1 exp ⎜ − η1 ⎜ t′⎟ qe ⎟ dt′, ⎝ = ⎝ ⎠ ⎠ ⎛ i ⎛ ⎞ ⎞ g2 exp ⎜ − η2 ⎜ t′⎟ qe ⎟ dt′, ⎝ = ⎝ ⎠ ⎠ ⎛ ⎛ ⎞⎞ ⎞ i⎛ ⎛ ⎞ k exp ⎜ − ⎜ η1 ⎜ t′⎟ − η2 ⎜ t′⎟ ⎟ qe ⎟ dt′, ⎝ ⎝ ⎠⎠ ⎠ =⎝ ⎝ ⎠ ⎛ ⎛ ⎞⎞ ⎞ i⎛ ⎛ ⎞ k exp ⎜ − ⎜ η1 ⎜ t′⎟ + η2 ⎜ t′⎟ ⎟ qe ⎟ dt′, ⎝ ⎝ ⎠⎠ ⎠ =⎝ ⎝ ⎠

t

⎞ ⎛ ⎜ − 2g1 ⎟ dt′ = − 2g1 t , ⎠ ⎝

t

⎛ ⎞ ⎜ − 2g2 ⎟ dt′ = − 2g2 t . ⎠ ⎝

Iφ1 =

⎛ q φ ⎞⎫ ⎛q φ ⎞ q φ = ⎧1 m ⎨ sin e 1 − cos ⎜ e 1 ⎟ sin ⎜ e 2 ⎟ ⎬ , ⎝ = ⎠⎭ ⎝ = ⎠ Aqe ⎩ L1 L1 L2 =

(58)

Iφ 2 =

⎛ q φ ⎞⎫ ⎛q φ ⎞ q φ = ⎧ 1 m ⎨ sin e 2 − sin ⎜ e 1 ⎟ cos ⎜ e 2 ⎟ ⎬ . ⎝ = ⎠⎭ ⎝ = ⎠ Aqe ⎩ L2 = L1 L2

(59)

Eqs. (58) and (59) exhibit that the persistent currents on system are an observable quantity periodically depending on the fluxes φ1 and φ2. In the limit qe → 0 (continuum charge) Eqs. (56) and (57) are

(53)

Iφ1 = −

⎞⎫ ⎞ m ⎛ 1⎧ 1 ⎛ ⎨ ⎜ η (t ) − φ1 ⎟ − ⎜ η (t ) − φ2 ⎟ ⎬ , ⎠⎭ ⎠ L1 L2 ⎝ 2 A ⎩ L1 ⎝ 1

(60)

Iφ 2 = −

⎞⎫ ⎞ m ⎛ 1⎧ 1 ⎛ ⎨ ⎜ η2 (t ) − φ2 ⎟ − ⎜ η1 (t ) − φ1 ⎟ ⎬ . ⎠⎭ ⎠ L1 L2 ⎝ A ⎩ L2 ⎝

(61)

Then the time evolution operator for the Hamiltonian (41) can be found as a product

⎡ i ⎤ ⎡ i ⎤ ^ ^ ^ U (t ) = US (t ) UR (t ) = exp ⎢ − η1 (t ) q^1 ⎥ exp ⎢ − η2 (t ) q^2 ⎥ × exp ⎣ = ⎦ ⎣ = ⎦ ⎡ i⎛ ⎞⎤ ^ ^† ⎢ − ⎜ α (t ) Q 1 + α† (t ) Q 1 + λ1 ⎟ ⎥ exp ⎠⎦ ⎣ =⎝

6. Two mesoscopic electric LC circuits with mutual-inductance

⎡ i⎛ ⎞⎤ ^ ^† ⎢ − ⎜ β (t ) Q 2 + β † (t ) Q 2 + λ2 ⎟ ⎥ × exp ⎠⎦ ⎣ =⎝ ⎡i⎛ ^ ^† ^ † ^ ⎞⎤ ⎢ ⎜ χ (t ) Q 1 Q 2 + χ † (t ) Q 1 Q 2 ⎟ ⎥ exp ⎝ ⎠⎦ ⎣= ⎡ i ⎢− ⎣ =

⎛ ^ ^ ^ † ^ †⎞⎤ ⎜ ξ (t ) Q 1 Q 2 + ξ † (t ) Q 1 Q 2 ⎟ ⎥. ⎝ ⎠⎦

q qe ⎛ = ⎧ 1 m ⎡ ⎜ η ( t ) + η (t ) ⎨ sin e (η2 (t ) − φ 2 ) − ⎢ sin 2 = = ⎝ 1 Aqe ⎩ L2 2L1 L2 ⎣ q ⎛ ⎛ ⎞ ⎛ ⎞⎤⎫ − ⎜ φ1 + φ 2 ⎟ + sin e ⎜ η1 (t ) − η2 (t ) − ⎜ φ1 − φ 2 ⎟ ⎥ ⎬. ⎝ ⎠ ⎝ ⎠ ⎦ ⎭ (57) = ⎝

m= 2 , 4Aqe2 L1 L2

α (t ) =

(56)

(52)

where

g1 = −

⎞ q ⎛ = ⎧1 ⎨ sin e ⎜ η1 (t ) − φ1 ⎟ ⎠ = ⎝ Aqe ⎩ L1

−

⎛ ^ ^† ^ † ^ ⎞⎤ ⎜ χ (t ) Q 1 Q 2 + χ † (t ) Q 1 Q 2 ⎟ ⎥ exp ⎝ ⎠⎦

⎡ i⎛ ^ ^ ^ † ^ †⎞⎤ ⎢ − ⎜ ξ (t ) Q 1 Q 2 + ξ † (t ) Q 1 Q 2 ⎟ ⎥, ⎠⎦ ⎣ =⎝

127

(54)

Eq. (54) is the time evolution driven two coupled quantum mesoscopic rings with mutual inductance. Using the time evolution operator, we can find the current through of the system easily. Let |Iφ1, Iφ2 〉 be eigenvectors of the current operator in t¼ 0 then at t ≠ 0 these states are

Quantum circuits LC with discrete charge have been intensely studied in the past decades due to their potential technological, nano-electronic, quantum computation and quantum information applications [11–24,29–33]. Therefore, the study of two coupled mesoscopic electric circuits LC with mutual inductance is very important both theoretically and experiment. So in this section, we will consider the Hamiltonian operator (31) for two quantum mesoscopic electric LC circuits with inductance coupling term (mutual inductance) i.e. ^ HLC ⎧ † ⎞ ⎞ 1 ⎛^ =2 ⎪ 1 ⎛ ^ ^ † ⎨ ⎜ Q 1 + Q^ 1 − 2⎟ + ⎜ Q 2 + Q 2 − 2⎟ 2 ⎝ ⎠ ⎝ ⎠ L L 2Aqe ⎪ 2 ⎩ 1 2 2 ⎫ q^ q^ m ⎛^ ^ †⎞ ⎛ ^ ^ †⎞ ⎪ − ⎜ Q1 − Q1 ⎟ ⎜ Q 2 − Q 2 ⎟ ⎬ + 1 + 2 . ⎠⎝ ⎠ ⎪ 2C1 2L1 L2 ⎝ 2C2 ⎭

= −

^ ^ |Iφ1, Iφ 2, t〉 = US (t ) UR (t ) |Iφ1, Iφ 2 〉.

(55)

Therefore using Eqs. (54) and (55) and applying Eqs. (42) and (43) after some calculations, we can find the persistent currents on two coupled quantum ring under the external fields as

(62) Then will obtain the energy spectrum and find the time evolution of such system.

128

H. Pahlavani, E.R. Kolur / Physica B 495 (2016) 123–129

6.1. The energy spectrum of coupled two LC mesoscopic circuits The study of the energy spectral properties of the Hamiltonian (62) is difficult because of the nonlinear term associated with the magnetic energy of two circuits. Therefore, when qe = 0, we are in the continuous regime and for a finite value of qe, namely 1.6 × 10−19C , so it is expected for us to choose qe as perturbative ^ ^† ^ ^† parameter and expand the shift operators Q 1, Q 1 , Q 2 and Q 2 in the powers of qe. Here we expand these operators up to the order q4e , i.e. p^1 eiqe =

^ Q1 =

the Hamiltonian H0. The energy spectrum of two LC mesoscopic circuits with mutual inductance in the grand state using Eq. (15) is given by

( 1) + ⋯ = =ω1′ + =ω2′ E = E0,1 + E0,2 + E0,0 2 2

+

+

2 3 4 p^ p^ ⎞ p^ ⎞ p^ ⎞ 1 ⎛⎜ 1 ⎛⎜ 1 ⎛⎜ = 1 + iqe 1 + iqe 1 ⎟⎟ + iqe 1 ⎟⎟ + iqe 1 ⎟⎟ ⎜ ⎜ ⎜ 2! ⎝ 3! ⎝ 4! ⎝ = =⎠ =⎠ =⎠

⎡ ⎪ qe2 L ⎧ 1 m⎢ 2 ⎨ − ( ω1′ + ω2′ )2 + ω2′ − ω1′ 2 ⎪ 8A ⎩ 4 L ⎢⎣ qe2 L ⎛ ⎜⎜ 5 ω1′ 3 − ω2′ 3 + 9 ω1′ 2 ω2′ − ω1′ ω2′ 2 24= ⎝

(

)

(

⎞⎤⎫ ⎪

) ⎟⎟⎠ ⎥⎥⎦ ⎬. ⎪

⎭

(68)

+ ⋯, 2 3 4 p^ 2 p^ p^ ⎞ p^ ⎞ p^ ⎞ 1 ⎛⎜ 1 ⎛⎜ 1 ⎛⎜ ^ Q 2 = eiqe = = 1 + iqe 2 + iqe 2 ⎟⎟ + iqe 2 ⎟⎟ + iqe 2 ⎟⎟ ⎜ ⎜ ⎜ 2! ⎝ 3! ⎝ 4! ⎝ = =⎠ =⎠ =⎠

+ ⋯, ^† Q1 =

p^1 e−iqe =

In order to study the quantum fluctuations of the charge and current for two coupled quantum mesoscopic LC circuits, let us obtain ̇ ̇ the time-evolution equations for the operators q^ and q^ in the Hei-

2 3 p^ p^ ⎞ p^ ⎞ 1 ⎛⎜ 1 ⎛⎜ = 1 − iqe 1 + − iqe 1 ⎟⎟ + − iqe 1 ⎟⎟ ⎜ ⎜ 2! ⎝ 3! ⎝ = =⎠ =⎠

1

p^ 1 ⎛⎜ − iqe 1 ⎟⎟ + ⋯, ⎜ 4! ⎝ =⎠

⎛ ^† 1 ⎡ ^̇ ^ ⎤ 1 ⎧ 1 ⎛ ^† ⎞ ¨ ⎢ q1, H ⎥ = ⎨ q^1 = ⎜ −qe ⎜ q^1 Q 1 + Q 1 q^1 ⎟ ⎝ ⎠ i= ⎣ ⎦ 2Aqe ⎩ 2L1 C1 ⎝ ⎪

2 3 p^ 2 p^ p^ ⎞ p^ ⎞ 1 ⎛⎜ 1 ⎛⎜ ^† − iqe 2 ⎟⎟ + − iqe 2 ⎟⎟ Q 2 = e−iqe = = 1 − iqe 2 + ⎜ ⎜ 2! ⎝ 3! ⎝ = =⎠ =⎠

+

4 p^1 ⎞ 1 ⎛⎜ ⎟ − + ⋯. iq e 4! ⎜⎝ = ⎟⎠

† ⎞ † ⎛ ^ m ⎛ qe ⎧ ⎛ ^ ^ † ^ ⎞⎞ ⎨ ⎜ q1 Q 1 + Q^1 q^1 ⎟ Q^ 2 − qe ⎜ q^1 Q 1 + Q 1 q^1 ⎟ ⎟ − ⎜− ⎝ ⎠ ⎠ 2L1 L2 ⎝ 2C1 ⎩ ⎝ ⎠

⎛ ^ ⎛ ^† ^† ⎞ ^ ^ ⎞^† ⎛ ^ ^ ⎞^ ⎫ − ⎜ q^1 Q 1 + Q 1 q^1 ⎟ Q 2 − ⎜ q^1 Q 1 + Q 1 q^1 ⎟ Q 2 + ⎜ q^1 Q 1 + Q 1 q^1 ⎟ Q 2 ⎬ ⎠ ⎭ ⎝ ⎠ ⎝ ⎠ ⎝

(63)

qe ^ † ⎛ ^ ^ † ^† ⎞ ^ †⎛ ^ † ^† ⎞ {Q 1 ⎜ q2 Q 2 + Q 2 q^2 ⎟ + Q 1 ⎜ q^2 Q 2 + Q 2 q^2 ⎟ ⎠ ⎝ ⎠ ⎝ 2C2 ⎫ ⎞ ⎞ ⎞ ^ ⎛ ^ ^ ^ ⎛ ^ ^ + Q 1 ⎜ q^2 Q 2 + Q 2 q^2 ⎟ + Q 1 ⎜ q^2 Q 2 + Q 2 q^2 ⎟ } ⎟ ⎬ . ⎠ ⎠⎭ ⎝ ⎠ ⎝

Substituting these equations into Eq. (62), we find

^ HLC =

−

⎛ 2 ⎞ qe2 ⎛ ^ 4 ^ 4 ⎞ 2 ⎜⎜ p^1 + p^2 ⎟⎟ − ⎜⎜ p1 + p2 ⎟⎟ 2 ⎝ ⎠ 24= AL ⎝ ⎠ ⎡ 2 ⎛ ⎞ 3 3 q m 1⎛ ⎞ 1 ⎛⎜ qe ⎞⎟4 ^ 3 ^ 3 ⎤ − 2 ⎢ p^1 p^2 − ⎜ e ⎟ ⎜ p^1 p^2 + p^1 p^2 ⎟ + p p ⎥ ⎝ = ⎠ 1 2 ⎥⎦ AL ⎢⎣ 6⎝=⎠ ⎝ 36 ⎠ 1 2AL

+

1 2C

⎛ 2 ⎞ 2 ⎜⎜ q^1 + q^2 ⎟⎟, ⎝ ⎠

⎪

^ H1 =

1 2AL

(64)

⎛ ^ 2 ^ 2⎞ m 1 ⎛ ^2 ^2⎞ ⎜ p1 + p2 ⎟ − 2 p^1 p^2 + ⎜ q + q2 ⎟, ⎝ ⎠ AL ⎠ 2C ⎝ 1

4⎞ 1 ⎛⎜ qe ⎞⎟2 ⎧ 1⎛ 4 ⎨ − ⎜ p^1 + p^2 ⎟ ⎝ ⎠ 6AL = ⎩ 4⎝ ⎠ 3 q ⎛ ⎞2 3 3 ⎤ ⎫ m ⎡^3^ 1 + ⎢ p p + p^1 p^2 − ⎜ e ⎟ p^1 p^2 ⎥ ⎬ . L ⎣ 1 2 6⎝=⎠ ⎦⎭

(65)

(66)

⎤ ⎡ ^ 1 M 2 ^ ⎥ ¨ 2 ^ ^ q^1 = ⎢ −ω10 M1 q1 + 2 ω20 q2 , ⎥⎦ A ⎢⎣ L1

ψn1 (x1 ), ψn2 (x2 )

⎛q ⎞ ⎛ q ⎞ −1 1⎧ ^ M1 = ⎨ q^1 cos ⎜ e p^1 ⎟ q^1 + cos ⎜ e p^1 ⎟ ⎝= ⎠ ⎝= ⎠ 2⎩ ⎡ ⎛ ⎛ qe ⎞ ⎛ qe ⎞ ⎞ −1 m ^ + ⎢ q1 ⎜ sin ⎝⎜ p^1 ⎠⎟ sin ⎝⎜ p^2 ⎠⎟ ⎟ q^1 ⎣ ⎝ ⎠ = = L2 ⎛ ⎛q ⎞ ⎛q ⎞⎞⎤⎫ + ⎜ sin ⎜ e p^1 ⎟ sin ⎜ e p^2 ⎟ ⎟ ⎥ ⎬ , ⎝= ⎠ ⎝ = ⎠⎠⎦⎭ ⎝ ⎛q ⎞ ⎛q ⎞ ⎞ −1 m⎧^ ⎛ ^ ⎨ q2 ⎜ cos ⎜ e p^1 ⎟ cos ⎜ e p^2 ⎟ ⎟ q^2 M2 = ⎝ ⎠ ⎝ ⎝ = = ⎠⎠ 2 ⎩ ⎛ ⎛q ⎞ ⎛q ⎞⎞⎫ + ⎜ cos ⎜ e p^1 ⎟ cos ⎜ e p^2 ⎟ ⎟ ⎬ , ⎝= ⎠ ⎝ = ⎠⎠⎭ ⎝

(71)

and

1 ⎛⎜ qe ⎞⎟2 6AL ⎝ = ⎠

⎧ 4⎞ 3 m ⎡^3^ 1⎛ 4 1 ⎛ q ⎞2 3 3 ⎤ ⎫ ⎨ − ⎜ p^1 + p^2 ⎟ + ⎢ p1 p2 + p^1 p^2 − ⎜ e ⎟ p^1 p^2 ⎥ ⎬ L ⎣ 4⎝ 6⎝=⎠ ⎦⎭ ⎩ ⎠ |ψn1 (x1 ), ψn2 (x2 ) ,

(67)

( )

(70)

where

The first order energy shift is obtained using the perturbation theory as 1 En(1,)n2 =

(69)

^ ^ Substituting Q j = eiqe pj / = , j = 1, 2 into Eq. (69), we find

where L1 = L2 = L and C1 = C2 = C . The Hamiltonian (64) can be split into two parts, as

^ H0 =

2

senberg representation. Using the algebraic structures (26), we have

⎞4

+

6.2. The time evolution of two coupled quantum mesoscopic circuits LC with mutual inductance

( )

According to Eq. (14), |ψn1 x1 〉 and |ψn2 x2 〉 are eigenvectors of

2 ω10 =

1 , L1 C1

(72)

2 ω20 =

1 . L2 C2

(73)

Also in a similar manner, we have

H. Pahlavani, E.R. Kolur / Physica B 495 (2016) 123–129

⎤ ⎡ ^ M′ 2 ^ ⎥ 1 ¨ 2 ^ ′^ q^2 = ⎢ −ω20 M2 q2 + 1 ω10 q1 , ⎥⎦ A ⎢⎣ L2

7. Conclusions

(74)

where ⎛ ⎛q ^ ⎞ ⎛ qe ^ ⎞ ⎞ ^ −1 ⎛ ⎛q ^ ⎞ ⎛ qe ^ ⎞ ⎞ ⎫ ^′ m⎧ ⎟ cos ⎜ ⎟ cos ⎜ ⎨ q^ ⎜ cos ⎜ e p M1 = p ⎟ ⎟ q + ⎜ cos ⎜ e p p ⎟ ⎟ ⎬, ⎝ = 1⎠ ⎝ = 2⎠ ⎠ 1 ⎝ = 1⎠ ⎝ = 2⎠ ⎠ ⎭ ⎝ 2 ⎩ 1⎝ ^′ M2 =

⎛q ⎞ 1⎧^ ^ ⎟ ⎨ q cos ⎜ e p 2 2⎩ 2 ⎝ = ⎠ ⎪

⎛q ⎞ −1 ^ ⎟ q^2 + cos ⎜ e p 2 ⎝ = ⎠ +

⎛q ^ ⎞ ⎛ qe ^ ⎞ ⎞ ⎤ ⎫ ⎛q ^ ⎞ ⎛ qe ^ ⎞ ⎞ ^ −1 ⎛ m⎡^ ⎛ ⎟ sin ⎜ ⎟ sin ⎜ p ⎟ ⎟ q + ⎜ sin ⎜ e p p ⎟ ⎟ ⎥ ⎬. ⎢ q ⎜ sin ⎜ e p ⎝ = 1⎠ ⎝ = 2⎠ ⎠ 2 ⎝ = 1⎠ ⎝ = 2⎠ ⎠ ⎦ ⎭ ⎝ L1 ⎣ 2 ⎝ ⎪

(75)

Eqs. (70) and (74) are the coupled quantum Kirchhoff equations for two quantum mesoscopic circuits LC with mutual inductance. In order to solve the above equations, we take average of Eqs. (70) and (74) i.e.

⎞ ω2 ^ d2 ^ 1⎛ 2 ^ ^ 〈q1 〉 + ⎜ ω10 〈M1 q1 〉 − 20 〈M2 q^2 〉⎟ = 0, 2 dt A⎝ L1 ⎠

(76)

⎞ ω2 ^ 1⎛ 2 ^ ′^ 〈q^2 〉 + ⎜ ω20 〈M2 q2 〉 − 10 〈M1′ q^1 〉⎟ = 0, 2 dt A⎝ L2 ⎠ d2

(77)

which can be solved easily as

()

(

)

()

(

)

〈q^1 〉 = 〈q^1 0 〉 cos ω+t + ϕ1 + 〈q^2 0 〉 cos ω−t + ϕ2 ,

〈q^2 〉 =

(78)

^ 2 〈M1 〉ω10 − Aω+2 ^ 〈q1 (0) 〉 cos (ω+t + ϕ1 ) ^ 〈M ′ 〉ω 2 1

20

^ 2 〈M1 〉ω10 − Aω−2 ^ + 〈q2 (0) 〉 cos (ω−t + ϕ2 ), ^′ 2 〈M 〉ω 1

20

(79)

where 2= ω±

⎧⎛ ⎞ ⎪⎜ ^ ^′ 2 ⎟ 2 1 ⎪ ⎜ 〈M1〉ω10 + 〈M2 〉ω 20 ⎟ ⎨⎜ ⎟ 2⎪ A ⎟⎟ ⎪ ⎜⎜ ⎠ ⎩⎝

±

⎛ ^ ^ ′ 2 ⎞2 2 ω2 ω2 ⎜ 〈M1〉ω10 + 〈M2 〉ω 20 ⎟ 10 20 ⎟⎟ − 4 ⎜⎜ A A2 ⎠ ⎝

⎛ ⎞⎫ ⎜ ⎟⎪ ^ ^′ ^′ ^′ ⎪ ⎜ 〈M 1〉〈M2 〉 − 〈M1 〉〈M2 〉⎟ ⎬. ⎜ ⎟⎪ ⎜ ⎟ ⎝ ⎠⎪ ⎭

(80)

In the limit qe → 0 Eqs. (71) and (75) reduce to

^ M1 = 1, ^ M1′ = m,

^ M2 = m, ^ M2′ = 1.

129

(81)

Eqs. (78) and (79) show the quantum oscillation charge for two coupled quantum electric circuit LC.

In this paper, using an advanced quantum theory to mesoscopic circuits developed by Li and Chen, which is based on the fundamental fact that the electric charge takes discrete value, we generalized the quantum theory mesoscopic electric circuits, for two coupled quantum mesoscopic circuits LC with mutual inductance. We obtained energy spectrum and the fluctuations of charge and current this system and showed the special role mutual inductance. With the dramatic development in nanotechnology and nanoelectronics, in the manufacture all practical quantum computer that is composed of many mesoscopic LC circuits including capacitors, inductors, electric resistor, etc. [34,35], calculating quantum noise of two mesoscopic LC electric circuits coupled by mutual inductance is very important in both theoretically and experimentally. Studies in such fields are promising, as they produce ideas for further technological developments.

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