Superposition states for quantum nanoelectronic circuits and their nonclassical properties
Int Nano Lett
Superposition states for quantum nanoelectronic circuits and their nonclassical properties
Jeong Ryeol Choi 0
0 Department of Radiologic Technology, Daegu Health College , Yeongsongro 15, Bukgu, Daegu 41453 , Republic of Korea
Quantum properties of a superposition state for a series RLC nanoelectronic circuit are investigated. Two displaced number states of the same amplitude but with opposite phases are considered as components of the superposition state. We have assumed that the capacitance of the system varies with time and a timedependent power source is exerted on the system. The effects of displacement and a sinusoidal power source on the characteristics of the state are addressed in detail. Depending on the magnitude of the sinusoidal power source, the wave packets that propagate in charge(q)space are more or less distorted. Provided that the displacement is sufficiently high, distinct interference structures appear in the plot of the time behavior of the probability density whenever the two components of the wave packet meet together. This is strong evidence for the advent of nonclassical properties in the system, that cannot be interpretable by the classical theory. Nonclassicality of a quantum system is not only a beneficial topic for academic interest in itself, but its results can be useful resources for quantum information and computation as well.
Nanoelectronic circuit; Displaced number state; Superposition; Interference; Wave function

& Jeong Ryeol Choi
One of the greatest challenges for modern electronic
science is miniaturizing electronic devices packed in IC chips
towards an atomic scale. From fundamental quantum
theories supported by elaborate experiments, it is well known
that quantum effects are prominent as the transport
dimension becomes small beyond the Fermi wavelength
[1, 2]. Hence, the understanding of quantum characteristics
of nano systems is important in order for developing future
technologies in the electronic industry relevant to nano
dimension. As the scale of metallic electronic devices,
whose electronenergy levels are continuous, reaches a
nanometer, the energy levels may no longer be allowed to
remain continuous but become discrete instead. Then, the
devices may look like a low dimensional quantum systems
in parts.
While the time behavior of charges in ideal electronic
circuits, such as an LC circuit, is represented by a simple
harmonic oscillator, a large part of intricate nanoelectronic
circuits may belong to timevarying systems that are
described by timedependent Hamiltonians [3–5]. Rigorous
mathematical techniques are crucial for exact treatment of
timedependent Hamiltonians. In a previous research [2],
Choi et al. have investigated displaced squeezed number
states of a twodimensional nanoelectronic circuit. The
extension of such research to superposed quantum states
may not only be interesting but also has many useful
applications in science [6–10]. According to this,
superposition states composed of two displaced number states
(DNSs) with an opposite or an arbitrary phase difference
for quantum nanoelectronic circuits will be studied in this
work. We consider a series RLC nanoelectronic circuit
driven by a timedependent power source. One may
possibly treat a general series RLC nanoelectronic circuit,
where R, L, and C vary with time. However, for the
difficulty of mathematical treatment of such a complicated
system, we regard the case that only the capacitance C is an
arbitrary time function while R and L are constants. As well
as it is more easier to vary the capacitance than to vary
resistance and/or inductance, the electronic circuits that
involve a timevarying capacitance have several
applications in science and technology [11–15].
At first, quantum characteristics of the system will be
studied regarding the displacement of number states. Then,
the superposition of two DNSs [16] of the system will be
investigated. Energy eigenvalues in number states for a
quantized RLC nanoelectronic circuit are discrete and the
corresponding energies dissipate like a classical state due
to the existence of a resistor R which roles as a damping
factor [17]. A class of interesting quantum states for a
harmonic oscillator is superpositions (Schro¨dinger’s cat
states) of two DNSs of the same amplitude with opposite
phases. A novel application of DNSs is their use as a
resource for establishing (single) qubit operations in
quantum computations [18]. Displaced number states can
also be implemented to realizing an irreversible analog of
quantum gates, such as the Hadamard gate, and to
optimizing such gates [19]. The DNSs follow subPoissonian
statistics [20] and exhibit several pure quantum effects,
such as the revivalcollapse phenomenon [21] and the
interference in the phase space [22].
The success of experimental setups of superposition states
[23] provides evidence for a remarkable fact that a particular
system could take two or several separate quantum states
simultaneously. In general, superposition states exhibit
nonclassical characters. Such characters can be potentially
exploited to be essential resources in various quantum
information processing, such as quantum computation [6],
quantum teleportation [7], quantum communication [8],
quantum cryptography [9], and densecoding [10]. All these
applicabilities of the nonclassical states are important in
future technology of information science. However, there is a
difficulty for maintaining such nonclassicality of a system due
to the appearance of decoherence of states [24]. Various
quantum properties of the system including nonclassicality
associated with DNSs will be investigated here.
Due to the timedependence of the Hamiltonian of the
system, a conventional technique for quantizing the system,
which is the separation of variables method, is unapplicable
in this case. Hence, special techniques for quantizing the
system in the superposition states are necessary. The
invariant operator method and the unitary transformation
method will be adopted for this purpose. The underlying idea
for the invariant operator method is that the Schro¨dinger
solutions of a timevarying system is represented in terms of
the eigenstates of an invariant operator [25]. For this reason,
it is necessary to derive eigenstates of the invariant operator
in order to study quantum features of the system. We will
introduce a quadratic invariant operator that can be obtained
from its fundamental definition. The original invariant
operator may be not a simple form due to the
timedependence of the system. For this reason, we will transform the
original invariant operator to a simple form that does not
contain time functions by adopting a unitary transformation
technique. Then, the eigenstates of the transformed invariant
operator may be easily identified due to their simplicities.
The eigenstates of the transformed invariant operator will be
inversely transformed to those in the original system in order
to obtain the full wave functions in the superposition state.
This is the main strategy that we will adopt in this work.
Results and discussion
Hamiltonian dynamics
We consider the series RLC nanoelectronic circuit driven by
a timedependent electromotive force EðtÞ and assume that
the capacitance in the circuit varies with time. A common
example of a varying capacitance can be seen from the
turning of a radio dial for the purpose of receiving a
particular radio wave. The equation of amount for charge stored
in the capacitor can be derived by applying Kirchhoff’s law
in the circuit. Then, the corresponding Hamiltonian can be
easily identified from basic Hamiltonian dynamics. By
replacing classically represented variables of the
Hamiltonian with the counterpart quantum operators, we have
quantum Hamiltonian of the system, that is given in the form
where canonical variables q^ and p^ represent charge stored
in the capacitor and canonical current defined as
p^ ¼ iho=oq, respectively.
The energy operator of a timedependent Hamiltonian
system (TDHS) is different from the Hamiltonian itself.
The role of the Hamiltonian in the TDHS is limited to be
the only one in that it generates the classical equation of
motion [26]. For the present system, the energy operator is
represented as [27]
As you can see, the Hamiltonian given in Eq. (1) is a
timedependent form. It is known that quantum solutions of a
TDHS is represented in terms of classical solutions of the
system (or of a system similar to the given one) [25]. The
classical definition of canonical current is p ¼
LeðR=LÞtdq=dt and the classical equations of motion for
charge and current are
If we denote general classical solutions of Eqs. (3) and (4)
as Q(t) and P(t), respectively, they are in general
represented as QðtÞ ¼ QcðtÞ þ QpðtÞ and PðtÞ ¼ PcðtÞ þ PpðtÞ,
where QcðtÞ and PcðtÞ are complementary functions and
QpðtÞ and PpðtÞ are particular solutions.
When investigating a quantum system that is described
by a timedependent Hamiltonian, it is useful to introduce
an invariant operator [25] as mentioned previously. From
dI^=dt ¼ oI^=ot þ ½I^; H^ =ðihÞ ¼ 0, we obtain a quadratic
invariant operator for the system as
eðR=LÞtLq_ðtÞ½q^
where X0 ¼ 1=pffiLffiffiCffiffiffiffiffiffiffiffiÞffi and qðtÞ is a function of time
ð0
which satisfies the differential equation
Here q0 is an arbitrary real constant which has the same
dimension with qðtÞ. Equation (6) is a modified form of the
Milne–Pinney equation [28–30].
Because the invariant operator given in Eq. (5) is
somewhat complicated, it is necessary to simplify it for the
convenience for further treatment. For this purpose, we use
the unitary transformation technique. We introduce a
suitable unitary operator which is [31]
U^ ¼ U^1U^2U^3;
iLq_ðtÞeðR=LÞtq^2
I^0 ¼U^ 1I^U^;
p ¼ eðR=LÞt
H^0 ¼U^ 1H^U^
We easily see through a standard evaluation using Eqs. (7)–
(10) that this transformation yields
where LpðtÞ is a time function of the form
2 CðtÞ
The transformed Hamiltonian H^0 is very simple and
represented in terms of the Hamiltonian of the simple
harmonic oscillator. But H^0 is still dependent on time. By
performing a basic algebra with the use of Eq. (14), we see
that the classical equations of motion for charge and
current in the transformed system are given by
Because these equations do not involve driving power
source terms, the classical solutions in the transformed
system consist of only complementary functions. Let us
denote them as Qt;cðtÞ and Pt;cðtÞ, respectively for charge
and current. The quantum description in the transformed
system can also be carried out in terms of these solutions.
Now, let us consider the following Schro¨dinger
equations in the transformed system
From this equation, we easily confirm that the Schro¨dinger
solutions at initial time are given by
LX0 1=4
where Hn are Hermite polynomials of order n. In Eq. (19),
it is assumed for convenience that the global phase at initial
time is zero. The annihilation and the creation operators in
the transformed system is represented as
kc½D^ðaÞ þ D^ð aÞ w0nðq; 0Þ;
¼ j jeiu and kc is a normalization constant of the
which correspond to those of the simple harmonic
oscillator.
It may be worthy to find quantum states that oscillate with
time like classical ones. These states correspond to a class of a
displaced state and are obtained by displacing number states
with a displacement operator. We can put the displacement
operator in terms of a^ and a^y, at initial time, in the form
D^ðaÞ ¼ expðaa^y
where a is an eigenvalue of a^ and is given by
iPt;cð0Þ :
2h Qt;cð0Þ þ pffi2ffiffihffiffiLffiffiffiXffiffiffi0ffiffi
Using Eqs. (20) and (21), it is possible to show that
Eq. (22) is represented in terms of q^ and p^. Then, after
decoupling the exponential function into q^ and p^terms,
we have [32]
i Qt;cð0ÞPt;cð0Þ exp i Pt;cð0Þq^
2h h
With the use of this operator, the initial wave functions can
be made to be displaced. Then, we have a DNS which
oscillates like a classical state. By acting a time evolution
operator on the wave functions of such displaced states,
one can find the time evolution of the wave functions. The
degree of displacement is determined by the scale of a, i.e.,
the values of Qt;cð0Þ and Pt;cð0Þ. In the next subsection, we
will investigate the time behavior of superposition of two
individual DNSs.
Superposition of displaced number states
It is interesting to study superpositions of two different
quantum states on account of their widely acknowledged
nonclassical properties. Amplitude interference that
appears in the superposition states (Schro¨dinger cat states)
is one of the most novel characteristics of quantum
mechanics that has no analogue in classical mechanics.
While superpositions of pure number states seldom share
the coherence properties that are necessary in both
fundamental experiments and practical implementations
applicable to science and technology, a superposition of DNSs
exhibits coherence properties and other interesting
quantum statistical properties such as unusual oscillations in the
quantum number distribution [33].
Consider a superposition of two DNSs, D^ðaÞw0nðq; 0Þ
and D^ð aÞw0nðq; 0Þ:
jkcj2 ¼ ½1 þ j j2 þ 2j j expð 2jaj2ÞLnð4jaj2Þ cos u
where Ln are Laguerre polynomials of order n. In the
earlier work of Cahill and Glauber, we can find the idea of
the definition of DNS, where they have considered it as the
eigenstate of D^ a
ð Þ [34]. For the methods of generating
DNSs and how to reconstruct them, one can refer to Ref.
[35]. The generation of superposition of DNSs given in
Eq. (25) is given in Ref. [36]. For ¼ 1 with n ¼ 0,
Eq. (25) becomes even and odd coherent states respectively
[24].
Using Eq. (24), we can easily evaluate Eq. (25) to be
w0c;nðq; 0Þ ¼ kc½w0c;n;þðq; 0Þ þ w0c;n; ðq; 0Þ ;
Now, let us consider the following time evolution operator
defined in the transformed system
T^0ðq^; p^; tÞ ¼ exp
H^0ðq^; p^; sÞdsÞ:
If we use a useful identity that is given in Eq. (A1) in
Appendix A, the time evolution operator becomes
where XðtÞ is given by
Z t e ðR=LÞs
w0c;nðq; tÞ ¼ T^0w0c;nðq; 0Þ:
The time evolution of the wave functions in the
transformed system is obtained by acting T^0 on Eq. (27), i.e.,
Using Eq. (30) with the relations given in Eqs. (A2) and
(A3) in Appendix A, we see that Eq. (32) is easily
evaluated to be
w0c;nðq; tÞ ¼ kc½w0c;n;þðq; tÞ þ w0c;n; ðq; tÞ ;
L2Xh0 ðq Qt;cðtÞÞ2 2iqPLt;Xcð0tÞþiQt;cðtÞPLt;Xcð0tÞ
Notice that the time evolutions of Qt;cðtÞ and Pt;cðtÞ that
appeared in Eq. (34) are given by
Qt;cðtÞ ¼ Qt;cð0Þ cos XðtÞ þ PLt;cXð00Þ sin XðtÞ;
Pt;cðtÞ ¼ Pt;cð0Þ cos XðtÞ
LX0Qt;cð0Þ sin XðtÞ:
Thus, we have identified the complete quantum solutions
associated to the DNS in the transformed system. We see
from Eqs. (35) and (36) that, if the initial condition,
ðQt;cð0Þ; Pt;cð0ÞÞ, is determined and the time evolution of
XðtÞ is known, we can easily deduce the time evolutions of
Qt;cðtÞ and Pt;cðtÞ.
From the inverse transformation of the solutions given
in Eq. (33), it is also possible to obtain the complete
solutions in the original system:
wc;nðq; tÞ ¼ U^w0c;nðq; tÞ:
Hence, using Eq. (7), we now have
wc;nðq; tÞ ¼ kc½wc;n;þðq; tÞ þ wc;n; ðq; tÞ ;
r4ffiLffihffiXffipffiffiffi0ffirffiqffiqffiðffiffi0tffiÞffiffi pffi21ffiffinffiffinffiffi!ffi Hn½n ðq; tÞ exp hi PpðtÞq
Although we have chosen q0 as an arbitrary constant, the
magnitude of q0 does not affect to the results. If we
represent qðtÞ as q0f ðtÞ without loss of generality, all q0 in
Eqs. (39) and (40) are canceled out and, as a consequence,
the final results are independent of q0.
The full wave functions, Eq. (38) with Eqs. (39) and (40),
are very useful for studying the superposition properties of
DNSs in the original system. It is well known that the wave
function is a probability function that enables us to
understand the characteristics of the nanoscale world and its
concept constitutes the heart of quantum mechanics. We can
estimate subsequent time behavior of charge carriers of the
nanoelectronic circuit using the wave functions with some
degree of certainty as far as quantum mechanics allows.
To see the time behavior of the state given in Eq. (38), let
us consider a solvable case that the timedependence of the
capacitance and the electromotive force is given by
where C0½¼ Cð0Þ , b, Q, and x1 are real constants, and we
put R ¼ 0 in this example for simplicity. Then, it is easily
verified that the solutions of Eqs. (6) and (31) are given by
qðtÞ ¼ q0ð1 þ btÞ;
QpðtÞ ¼ Q sinðx1tÞ;
PpðtÞ ¼ LQx1 cosðx1tÞ:
and the particular solutions of Eqs. (3) and (4) are given by
It is important to note that the superposition state composed
of the two DNSs exhibits very distinct characteristics
compared to those shown by their components. The
probability density, jwc;nðq; tÞj2 which is the absolute square of
Eq. (38), is plotted in Fig. 1 as a function of q and t under
the same choice of parameters as given from Eq. (41) to
Eq. (44) without considering a power source. We see from
this figure that the effects of displacement become more
conspicuous as the displacing parameters Qt;cð0Þ and
Pt;cð0Þ grow. Hence, the amplitude of the oscillation of
each component increases as the initial values Qt;cð0Þ and
Pt;cð0Þ become large. We can confirm from Fig. 1c, which
reveals the highest displacement among Fig. 1a–c, that
there appear interference structures when the two
components of packets meet together. This is a signature of the
nonclassicality of the system. The effects of the sinusoidal
power source on charge can be identified from Fig. 2. The
comparison of this figure with Fig. 1 reveals that the wave
packets are distorted somewhat significantly by the driving
power source. Figure 2a is the case of a higher driving
Fig. 1 Probability density jwc;nðq; tÞj2 [the absolute square of
Eq. (38)] for the system that has parameters illustrated between
Eqs. (41) and (46), plotted as a function of q and t. Here, the driving
electromotive force is not considered [ðQ; x1Þ ¼ ð0; 0Þ]. Displacing
parameters ðQt;cð0Þ; Pt;cð0ÞÞ are (1,1) for a, (2,2) for b, and (5,5) for c.
Other values taken here are L ¼ 1, C0 ¼ 1, X0 ¼ 1, ¼ ð1 þ iÞ=pffi2ffi,
h ¼ 1, b ¼ 0:1, and n ¼ 3. All values are taken to be dimensionless
for the sake of convenience. This convention will also be used in all
subsequent figures.
frequency while Fig. 2c is that of a relatively low driving
frequency. The effects of a higher value of b on packets are
shown in Fig. 3. As b increases, the displacing of packets
becomes less prominent.
Fig. 2 The same as Fig. 1c, but for the case that the driving
electromotive force is not zero. The parameters ðQ; x1Þ associated
with the electromotive force are (0.3, 15) for a, (1, 3) for b, and
(10, 0.3) for c.
A series RLC nanoelectronic circuit driven by an arbitrary
power source was considered, where its capacitance is
allowed to vary with time. The Hamiltonian of the system
is constructed from Kirchhoff’s law and the corresponding
quadratic invariant operator is introduced in order to study
quantum characteristics of the system. As you can see from
Fig. 3 The effects of large values of b on time evolution of
probability density jwc;nðq; tÞj2 illustrated in Figs. 1 and 2. The values
of ðb; Q; x1Þ are (0.25, 0, 0) for a, (0.25, 10, 0.3) for b, and (0.50, 10,
0.3) for c. Other values taken here are L ¼ 1, C0 ¼ 1, X0 ¼ 1,
¼ ð1 þ iÞ=pffi2ffi, Qt;cð0Þ ¼ 5, Pt;cð0Þ ¼ 5, h ¼ 1, and n ¼ 3, which are
in fact the same as those of Figs. 1c or 2.
Eq. (5), the invariant operator is a somewhat complicated
form to manage. In this case, we need to simplify it for
further treatment using unitary transformation or canonical
transformation. For this purpose, a unitary operator is
introduced as shown in Eq. (7) with Eqs. (8)–(10). The
transformed invariant operator I^0 is the same as the
Hamiltonian of the simple harmonic oscillator. Because the
transformed Hamiltonian is represented in terms of I^0, we
easily identified the quantum solutions in number states in
the transformed system. Superposition of DNSs at initial
time is considered as given in Eq. (25). The time evolution
of DNS in the transformed system is given in Eq. (33).
Through inverse transformation of this, the DNS in the
original system is evaluated [see Eq. (38) with Eqs. (39)
and (40)].
To promote the understanding of our consequence, our
results are applied to a particular system that the time
dependence of the capacitance is given by Eq. (41). The
wave packet is somewhat distorted when a sinusoidal
power source is exerted on the system. The corresponding
probability densities are illustrated in Figs. 1, 2, and 3 for
several values of displacing parameters ðQt;cð0Þ; Pt;cð0ÞÞ.
When displacing parameters are small (Fig. 1a), the
distortion of the wave packet is not so significant and its form
is near to that of the original number state. As the values of
the displacing parameters increase (Fig. 1b), the distortion
of the packet become more or less significant. From
Fig. 1c, we see the effects of the strong displacement on the
wave function. By comparing Fig. 2c with Fig. 2a, we can
make out the difference of time evolution of the wave
packet between the cases that angular frequency of the
power source is small and large. The effects of
displacement become less significant as the value of b increases.
All RLC circuits with timedependent capacitance C(t) and
power source EðtÞ, founded in an electronic laboratory,
may have this quantum nature under the same situation.
From Figs. 1, 2, and 3, you can see interference
structures that appear when the two components of the
state meet together. This quantum interference is inherent
to superposition states and is strong evidence for the
signature of nonclassicality of the system, that we cannot
find any analogous effects from classical systems
[37, 38]. A scheme for observing quantum interference
via phasesensitive amplification of a superposition state
using a twophoton CEL (correlated emission laser)
amplifier has been suggested by Zubairy and Qamar
[39]. Superposition states are vulnerable to external
interventions caused from the environment; hence, they
can be easily corrupted by noisy or dissipative forces.
This is a stumblingblock for achieving robust quantum
computations on the basis of nonclassical features of
superposition states through encoding logical qubits with
a treatment of the states [40, 41]. A number of proposals
to overcome this major hurdle in quantum computing has
been suggested so far [42–44]. The development of
techniques for protecting quantum information from
decoherence is crucial for realizing universal quantum
computation.
Compliance with ethical standards
Conflict of interest The authors declare no conflict of interests.
Open Access This article is distributed under the terms of the
Creative Commons Attribution 4.0 International License (http://crea
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made.
Appendix A: Mathematical formulae
Some of mathematical formulae that are useful for deriving
several results in the text are provided here.
Mathematical formula 1: A useful identity which is
necessary to perform the integration in Eq. (29) is [45]
a
2hh
i
h
sinh h cosh h
sinh h cosh h
where h ¼ pfficffiffi2ffiffiffiffiffiffiffiaffiffiffibffiffi.
Mathematical formula 2: A mathematical relation
which is required to obtain Eq. (33) from Eq. (32) is [32]
In addition, the following integral formula [46] is also
needed in the same calculation
e ðx yÞ2 HnðaxÞdx ¼ p1=2ð1
Author’s contribution statement
paper and approved it.
J.R.C. wrote the
a2Þ1=2
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