Fan Hong-yi, Wu Ze. Radiating frequency of three-loop mesoscopic LC circuit with mutual inductance obtained by IEO method
. Chinese Physics B, 2018, 27(8): 080301
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Radiating frequency of three-loop mesoscopic LC circuit with mutual inductance obtained by IEO method
Fan Hong-yi1, †, Wu Ze2
Department of Material Science and Engineering, University of Science and Technology of China, Hefei 230026, China
Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
† Corresponding author. E-mail: fhym@ustc.edu.cn
Project supported by the National Natural Science Foundation of China (Grant No. 11775208).
Abstract
Instead of normally tackling electric circuits by virtue of the Kirchhoff’s theorem whose aim is to derive voltage, electric current, and electric impedence, our aim in this paper is to derive the characteristic frequency of a three-loop mesoscopic LC circuit with three mutual inductances, e.g., for the radiating frequency of the three-loop LC oscillator, we adopt the invariant eigen-operator (IEO) method to realize our aim.
In condensed matter physics theory, when the transport dimension reaches a characteristic dimension, namely, the charge carrier inelastic coherence length, quantum effects in electric circuits must be taken into account and the circuit is named mesoscopic. The trend in the miniaturization of integrated circuits and devices towards atomic scale dimensions stimulates theoretical progress in quantization of mesoscopic circuits.[1] In history, a single LC (inductance–capacitance) circuit, as a fundamental cell in electric circuits, was quantized by Louisell in 1973;[2] he considered the charge q as a dynamic canonical coordinate, while taking electric current I = dq/dt multiplied by the inductance L as canonical momentum, p = L(dq/dt), by further quantizing (q,p) with the quantization condition [q,p] = ih, the Hamiltonian of the LC circuit is quantized. In this work we consider what the energy level of a quantized two-loop mesoscopic LC circuit with mutual inductance M is, in other words, we want to know the characteristic frequency of two mesoscopic LC circuits when mutual inductance exists between them, this is equivalent to knowing the emitting frequency of this complex L–C oscillator. To the best of our knowledge, such a topic has not been touched on in the literature before. In the following we shall solve this problem by virtue of the invariant eigen-operator (IEO) method, which we briefly review in Section 2.
2. Brief review of the invariant eigen-operator method
Recall the original idea of Schrödinger’s quantization scheme,[3] he took the identification: (Hamiltonian) ↔ i(d/dt) (in this paper we assume ℏ = 1 for simplicity), so i(d/dt) is named as the Schrödinger operator in many references. Thus one naturally has
now we set up an equation for an operator :
when n = 1, it looks similar in form to the Schrödinger equation . Thus we call Eq. (1) as n-order invariant eigen-operator equation, and λ as n-order eigenvlaue. Further, using the Heisenberg equation:
we can write Eq. (1) as
If we can find such a that satisfies Eq. (3) we say that is the energy-level gap. To clarify this point of view, we take n = 2 in Eq. (3) for example, and assuming |ψa〉 and |ψb〉 are two different stationary eigen-states of the Hamiltonian with eigen-values Ea and Eb, respectively, then we have:
whenever is a nonzero matrix element, the energy gap between |ψa〉 and |ψb〉 can be obtained as . We name
as the invariant eigen-operator equation, and this IEO method can be used to derive the energy eigenvalues of some quantum systems.[2–5] The IEO equation is a natural result of combining both the Schrödinger operator and the Heisenberg equation. In the following we apply the IEO method to solving the energy level of a two-loop LC mesoscopic circuit with mutual inductance according to its quantized Hamiltonian.
3. Quantization of the two-loop LC mesoscopic circuit with mutual inductance
Supposing there is a mesoscopic circuit as shown in Fig. 1, where m represents the mutual inductance between the two loops.
Fig. 1. (color online) Two LC mesoscopic circuits with mutual inductance.
The classical Lagrangian of the whole electric system is
where ε1(t) is an external source in loop 1, mI1I2 denotes the mutual inductance between l1 and l2, in the case of leaking magnetism, . Taking the charge q1, q2 as canonical coordinates, their conjugates are
The Hamiltonian is
where
with
Imposing the quantization condition [qi,Pj] = iℏδij, H is an operator. The term (m/Al1l2)P1P2 may causes quantum entanglement.
4. Characteristic frequency of a two-loop LC mesoscopic circuit with mutual inductance
We now employ the IEO method to derive the characteristic frequency of the two-loop LC mesoscopic circuit with mutual inductance. Assuming the invariant eigen-operator is
where g is to be determined. Using the Heisenberg equation we have
and the corresponding IEO equation is
Comparing Eq. (12) with Eq. (10) we see
which leads to the following expression
Its solution is
For convenience, we let
Substituting Eq. (15) into the right-hand side of Eq. (12) we know
this is the square of characteristic frequency of the two-loop LC mesoscopic circuit with mutual inductance.
Consider the identical circuits which means l1 = l2 ≡ l and c1 = c2 ≡ c, we have
5. Characteristic frequency of a three-loop LC mesoscopic circuit with mutual inductance
As an extension of the two-loop LC mesoscopic circuit, we consider the three-loop LC circuit with mutual inductance as shown in the following Fig. 2.
Fig. 2. (color online) Three LC mesoscopic circuits with mutual inductance.
The classical Lagrangian of the whole electric system is
For the convenience of calculations, we consider the identical loops from the very beginning, letting l1 = l2 = l3 ≡ l and c1 = c2 = c3 ≡ c.
Taking the charge q1,q2,q3 as canonical coordinate, their conjugates are
The Hamiltonian is
where
For convenience we make a variable substitution:
which makes
We now employ the IEO method to derive the energy level gap of H0. Assuming the invariant eigen-operator is
where g1,g2 is to be determined. Using the Heisenberg equation we have
and the corresponding IEO equation is
Comparing Eq. (25) with Eq. (27) we obtain two coefficient equations:
There are two solutions of this equation set:
Each solution corresponds to an energy level gap. For g1 = g2 = 1, which can be substituted into Eq. (27), we have
For g1 = −1 − g2, we have
Using Eq. (23) to express ω1 and ω2 in terms of the original variables, we have
This is the characteristic frequency of the three-loop LC mesoscopic circuit with mutual inductance.
Since deriving characteristc frequency means energy quantization, the method introduced in this paper can be applied to quantizing other more complex mesoscopic electric circuits.
