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Equivalent Circuits for Two–Fermion Four–State Quantum Systems
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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: [email protected]
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
574
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)
575
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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[3] P. P. Civalleri, M. Gilli and M. Bonnin, “Circuit Models for Small SignalPerformance of Nano-Devices Based on Two–State Quantum Systems”,paper no. 004, Proceedings of the 2005 European Conference on Circuit
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