Equivalent Circuits for Two–Fermion Four–State Quantum Systems

Pier Paolo Civalleri, Life Fellow, IEEE, Marco Gilli, Fellow, IEEE, and Michele Bonnin

Department of Electronics, Politecnico di Torino, Italy

E-mail: pier.civalleri@polito.it

Abstract – An equivalent circuit is presented for a quantum

system composed of two spin 1/2 particles. Such a circuit shows

that the entire dynamics of the system, including single–particle

and two–particle annihilation and creation, as well as the single

particle transitions between ground and excited states, can be

described as the superposition of the variables of two uncoupled

resonant circuits.

I. INTRODUCTION

The circuit paradigm has been proved to be very effective in

unifying descriptions of dynamical systems of various kinds.

In particular the advent of nanoscience and nanotechnolo-

gies suggests the use of equivalent circuits to describe such

complex quantum systems processes as the annihilation and

creation of identical particles, as well as their transitions

between various states [1], [2], [4].

We consider a four–state system of two spin 1/2 particles,

in which external interactions cause single–particle and two–

particle annihilation and creation, as well as single–particle

transitions between a ground and an excited state, arising

from the energy splitting due the immersion of the system

in a constant magnetic field. In particular, we study the single

particle dynamics for a system whose state has had in the past

nonzero components in the vacuum and/or the two–particle

states. For such a system an equivalent circuit is presented

consisting of two coupled oscillators. If the latter are decou-

pled by a modal transformation, the complex dynamics of the

physical system can be simply pictured as the superposition of

their state variables. The oscillations take place on the average

(that is in a classical perspective) if and only if the state has

nonzero component in at least one of the vacuum or the two–

particle states. It is however possible, by replacing the usual

particle annihilation and creation operators with their state

correspondents (that can be expressed in a nonlinear form in

terms of them), to derive a simplified equivalent circuit (a

single oscillator), that only freezes when no coupling between

the ground and the excited state has never been established.

The possibility of representing by equivalent circuits such

complex phenomena as those occurring in nanoscience and

even in high energy physics has two main advantages: one is

allow to picture quantum phenomena by analogy in classical

terms, the other is that, even more important, of providing a

description of a quantum system interacting with macroscopic

electrical devices directly in electrical terms, so as to provide

a unified description of the whole. In both cases, use of a

simulator like PSPICE allows to obtain a visual description of

time evolution of all quantum and classical quantities, which

greatly helps an intuitive grasping of the actual dynamics.

II. THE STATE DIAGRAM

We describe the system in the Fock space by using the

occupation number basis. So we represent by ket |0 0〉 the

vacuum state, by kets |1 0〉 and |0 1〉 the single–particle ground

and excited states, by ket |1 1〉 the two–particle state. In each

ket we denote as 1 and 2 respectively the states in the first

and the second position, corresponding to the ground and

the excited states respectively. To each state is associated an

energy. We set to −mc2 the energy of the vacuum (m being the

particle mass); we take symmetrical values −E and E for the

potential energies of the single particle in the magnetic field

for the ground and excited states, corresponding to a magnetic

moment parallel and antiparallel to the field (for a negatively

charged particle the spin direction is obviously opposite to that

of the magnetic moment.); finally the energy of the two particle

state is mc2, because exclusion principle compels them to have

opposite spins in the magnetic field so that one particle has

energy −E, the other +E and the two energies cancel, leaving

the sum of the rest energy of the two particles plus that of the

vacuum. We introduce annihilation operators a1 and a2 and

their creation counterparts a†1 and a

†2 acting on particles in the

ground and the excited state respectively [5], [6]. Precisely

they are defined by the equations

|0 0〉 = a1|1 0〉, |0 1〉 = a1|1 1〉

|1 0〉 = a†1|0 0〉, |1 1〉 = a

†1|0 1〉

|0 0〉 = a2|0 1〉, |1 0〉 = a2|1 1〉

|0 1〉 = a†2|0 0〉, |1 1〉 = a

†2|1 0〉

(1)

To such operators are associated coupling energies that medi-

ate the appropriate transitions; V1 for a1, V2 for a2, V ∗1 for

a†1 and V ∗

2 for a†2. We also consider “compound” transitions

a2a1 from the two–particle to the vacuum state, with coupling

energy V3, a†1a

†2 for the opposite transition with energy V ∗

3

and a†1a2 for the transition from the single–particle excited

state to the ground state, with coupling energy V , and a†2a1,

with coupling energy V ∗, for the opposite transition. All such

energies represent the (parametric) inputs to our system from

the outside world. The whole story is depicted in Figure 1.

978-1-4244-3828-0/09/$25.00 ©2009 IEEE 573

|1 0〉 |0 1〉

|1 1〉

|0 0〉

a†

2a2

a†

1a1

a†

1

a1a†

2

a2

a†

2a1

a†

1a2

a†

1a†

2a1 a2

Figure 1. The state diagram.

III. TWO–PARTICLE DYNAMICS

The two–particle Hamiltonian has the form

H = −Ea†1a1 + V a†1a2 + V ∗a

†2a1 + Ea

†2a2

+V1a1 + V2a2 + V3a2a1

+V ∗1 a

†1 + V ∗

2 a†2 + V ∗

3 a†2a

†1

(2)

We assume that at some t0 < 0 the system be in state |0 0〉and that the coupling energies have constant values between

t = t0 and t = 0. We use at this stage Schrodinger’s picture.

By integrating Schrodinger’s equation

i~d

dt|ψ(t)〉 = H|ψ(t)〉 (3)

with the initial condition

|ψ(t0)〉 = |0 0〉 (4)

we obtain for |ψ(0)〉 an expression of the form

|ψ(0)〉 = c00|0 0〉+ c10|1 0〉+ c01|0 1〉+ c11|1 1〉 (5)

where by properly choosing time t0 and the values of the

coupling energies all coefficients cij can be made nonzero.

IV. SINGLE–PARTICLE DYNAMICS

We assume now 0 as the new initial time and adopt the

Heisenberg picture. The single–particle dynamics develops

entirely in the subspace spanned by kets |1 0〉 and |0 1〉, by

setting to zero coupling energies V1, V2 and V3. Despite the

fact that states |0 0〉 and |1 1〉 are by now isolated, we still

describe the dynamics in terms of the now time–dependent

operators a1(t) and a2(t) and their adjoints a†1(t) and a

†2(t).

This amounts to describe the transition from the excited to the

ground state through the creation and destruction of a particle

(i.e. via |1 1〉) or through destruction and creation of a particle

(i.e. via |0 0〉).The Heisenberg dynamical equations have the form

da1(t)

dt=

i

~

[

H, a1(t)]

da2(t)

dt=

i

~

[

H, a2(t)]

(6)

andda†1(t)

dt=

i

~

[

H, a†1(t)]

da†2(t)

dt=

i

~

[

H, a†2(t)]

(7)

By using equation (2) and with the definitions

ω0 =E

~, ωV = ω1 − iσ1 =

V

~(8)

they become

da1(t)

dt= iω0a1 − iωVa2

da2(t)

dt= −iω∗

Va1 − iω0a2

(9)

and

da†1(t)

dt= −iω0a

†1 − iω∗

Va†2

da†2(t)

dt= iω∗

Va†1 + iω0a

†2

(10)

We now define normalized coordinates and momenta

Qi =1√2(ai + a

†i ), Pi =

−i√2(ai − a

†i ); i = 1, 2 (11)

which can be identified with the first two components of a

pseudo–spin σ,√

2Qi = σxi,√

2Pi = σyi (12)

By using such quantities, equations (9) and (10) assume the

form

dP1

dt= ω0Q1 −σ1P2 −ω1Q2

dQ1

dt= −ω0P1 +ω1P2 −σ1Q2

dP2

dt= σ1P1 −ω1Q1 −ω0Q2

dQ2

dt= ω1P1 +σ1Q1 +ω0P2

(13)

which clearly represent the dynamics of two symmetrically

coupled equal oscillators.

By denoting collectively the four state variables as xi(t)with i assuming the integer values from 1 to 4, we obtain

xi(t) = Ai cosΩt+Bi sin Ωt, i = 1, 2, 3, 4 (14)

where the constants Ai and Bi can be easily determined as

functions of the initial conditions Pi(0) = Pi and Qi(0) = Qi

and

Ω =√

ω20 + |ωV|2 (15)

It is remarkable that the coupling, due to its very particular

form, does not give rise to a split of the resonant frequency.

It must be recognized that the dynamical quantities Pi and

Qi do not have the same physical meaning as for harmonic

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oscillators. Nevertheless they allow to construct equivalent

circuits imbedding important information on the quantum

system performance. As an example, consider the case of a

single oscillator in a mixed state characterized by a density

operator ρ. Then its classical energy can be easily calculated

to be

Ecl =1

2(〈P 〉2 + 〈Q〉2) = |ρ12|2 (16)

Thus such an energy represents the squared coherence mod-

ulus between states 1 and 2 and reaches its maximum value,

the product of populations ρ11 and ρ22, for a pure state.

To discuss the average dynamics it is convenient to go back

to Schrodinger picture. The system state at time t, given the

initial condition of equation (5), is

|ψ(t)〉 = c00|0 0〉 + c10(t)|1 0〉 + c01(t)|0 1〉 + c11|1 1〉 (17)

The constants c00 and c11 do not depend on t since the

coupling between the states |0 0〉 and |1 1〉 and the states |1 0〉and |0 1〉 has been broken at t = 0. From equation (17) we

obtain

〈a1〉(t) = 〈ψ(t)|a1|ψ(t)〉 = c00c∗10(t) + c01(t)c

∗11

〈a2〉(t) = 〈ψ(t)|a1|ψ(t)〉 = c∗00c01(t) + c∗10(t)c11

(18)

The average values of a†1(t) and a

†2(t) are simply expressed

by the conjugate complex of equations (18). It is therefore

seen that if both c00 and c11 are zero, then both average

values vanish and so do the average values of Qi and Pi, as a

consequence of their definition in equation (11). Thus we have

proved that freezing of oscillations can be avoided if and only

if the initial state is not confined to the one–particle subspace.

In particular, if the actual situation involves a single particle,

whose allowed transitions are only between the excited and

the ground state (and viceversa), no oscillation takes place.

However such a performance is an effect of the choice of the

annihilation and creation operators, that connect the single–

particle with the vacuum and the two–particle states. If we

define new annihilation and creation operators1

b = a†1a2, b† = a

†2a1 (19)

and assume that the state ψ(t) is initially confined to the

single–particle subspace

ψ(0) = c10(0)|1 0〉 + c0 1|0 1〉 (20)

then we obtain

〈b〉(t) = 〈ψ(t)|a†1a2|ψ(t)〉 = c∗10(t)c01(t) (21)

the average value of b† being the conjugate of the expression

above. This simply means that the dynamics of state interac-

tion need not to be interpreted in terms of particle annihilation

and creation.

1The product of operators referring to different states 1 and 2 must beunderstood as tensor products, even though this is not explicitly indicated.

V. THE EQUIVALENT CIRCUIT

In order to represent equations (13) it is convenient to

reorder them as follows

dP1

dt= −σ1P2 +ω0Q1 −ω1Q2

dP2

dt= σ1P1 −ω1Q1 −ω0Q2

dQ1

dt= −ω0P1 +ω1P2 −σ1Q2

dQ2

dt= ω1P1 +ω0P2 +σ1Q1

(22)

Then we define normalized magnetic fluxes Φi and electric

charges Q(el)i , (i = 1, 2), as

Φi = Pi, Q(el)i = Qi (23)

Finally we construct two–dimensional vectors

ΦL = (Φ1, Φ2)T, Q

(el)C = (Q

(el)1 , Q

(el)2 )T (24)

where apex T means transpose. Thus equations (22) can be

rewritten in the form

−ldILdt

= hLLlIL + hLCcUc

−cdUC

dt= hCLIL + hCCUc

(25)

where l and c are an arbitrary inductance and an arbitrary

capacitance respectively. The minus signs follow from the

fact that the four–port of equations (25) is analyzed with the

convention that all currents enter the ports at the terminals

where the voltages have their pluses. By choosing l = c = 1,

equations (25) can be rewritten as

UL = hLLIL + hLCUC

IC = hCLIL + hCCUC

(26)

where, from equations (22),

hLL = −hTLL

hLC = hTLC = −hCL = −hT

CL

hCC = −hTCC

(27)

The four–port description is brought from the hybrid repre-

sentation of equations (26) to the resistance one,

UL = (hLL + hLCh−1CChLC)IL + hLCh

−1CCIC

UC = −(hLCh−1CC)TIL + h−1

CCIC(28)

by taking into account that

h−1CChLC = −(h−1

CC)ThTLC = −(hLCh

−1CC)T (29)

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The resistance matrix R in equations (28) is clearly skew–

symmetric and can therefore be brought by a congruent

transformation to block diagonal form

R = MTRdM (30)

where

Rd =

0 1 0 0

−1 0 0 0

0 0 0 −1

0 0 1 0

(31)

Thus the equivalent circuit, in which matrix M is represented

by a network of ideal transformers and matrix Rd by a

couple of gyrators of unit gyration resistance, is that shown

in Figure 2. Note that, while the entire derivation was made

1 1

m11

m21

m31

m41

1

1

1

1

m12

m22

m32

m42

1

1

1

1

m13

m23

m33

m43

1

1

1

1

m14

m24

m34

m44

1

1

1

1

1

1

1 1

Figure 2. The equivalent circuit.

in terms of operator terminal quantities, replacement of the

latter with their average values yields for the circuit a classical

interpretation.

However, its rather complicated structure makes it more

suitable for use in simulations than for an intuitive grasping of

the system performance. Thus it is convenient, to gain such an

insight, to simplify it further on, by evidencing the core of its

dynamics out of the intricacies of the frequency independent

network.

To this aim go back to equations (9) and (10), find their

eigenvalues (equal to ∓iΩ and ±iΩ respectively), and trans-

form the equations to their diagonal form using their (unitary)

modal matrix. Then each pair of equations splits into two

independent ones for the annihilation or creation operators

in the newly introduced basis. Using equation (12) the four

equations are rephrased into two sets of two coupled equations

for the coordinate and momentum, or, in electrical terms, the

electric charges and the magnetic fluxes, of the new ground

and excited states

dP ′1

dt= ΩQ′

1

dQ′1

dt= −ΩP ′

1

dP ′2

dt= −ΩQ′

2

dQ′2

dt= +ΩP ′

2

(32)

Such equations have the form of equation (13) with the

coupling terms between the first two and the last two sup-

pressed. Clearly they describe a pair of uncoupled harmonic

oscillators, that, after averaging the operator quantities, can be

represented by two equal LC circuits. Voltages and currents

of the inductors and capacitors at the ports of the frequency

independent network of figure 2 are superpositions of those

in the elements of the two oscillators and have therefore

necessarily the form in equation (14). The procedure that we

have described is based on the assumption that an equation

set where the variables are creation and annihilation operators

can be brought to diagonal form by the usual technique valid

for the scalar case. That this is so can be proved rigorously as

shown in [5], chapter 21.

VI. CONCLUSION

We have shown how classical network theory can be ad-

vantageously used to provide equivalent circuits for physical

systems whose dynamics involves state transitions and creation

and/or annihilation of elementary particles. This should not

only help understanding of such complex phenomena, but

also provide unified descriptions of interacting quantum and

classical systems.

ACKNOWLEDGEMENTS

This research was partially supported by ”Ministero del-

l’Istruzione, dell’Universita e della Ricerca (MIUR)”, Rome,

Italy, under FIRB Project no. RBAU01LRKJ, and by Istituto

Superiore Mario Boella, Turin, Italy.

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