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Adiabatic invariant of the quantum mesoscopic L–C circuit
Optik 126 (2015) 3801–3802
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Adiabatic invariant of the quantum mesoscopic L–C circuit夽 Hong-yi Fan a,b,∗ , Jun-hua Chen c,d,b,e a
Institute of Intelligent Machines, Chinese Academy of Sciences, Hefei 230031, China Department of Material Science and Engineering, USTC, Hefei 230026, China c Hefei Center for Physical Science and Technology, Hefei 230026, China d CAS Key Laboratory of Materials for Energy Conversion, Hefei 230026, China e Synergetic Innovation Center of Quantum Information and Quantum Physics, USTC, Hefei, China b
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 18 June 2014 Accepted 18 July 2015
By analyzing that abrupt change of electric charge Q on the capacity C and abrupt change of on the inductance are not allowed, we derive diabatic invariant of the quantum mesoscopic L–C circuit. This enriches Louisell’s quantization theory of mesoscopic circuit. © 2015 Elsevier GmbH. All rights reserved.
Keywords: Adiabatic invariant L–C circuit Quantization of mesoscopic system
1. Introduction
(note that dE = vdp is valid for any x), thus
Quantized mesoscopic circuit is a hot topic in quantum optics and quantum computer theory. In this work we shall discuss the topic of adiabatic invariant for quantized circuit. The concept of adiabatic invariants was introduced into quantum mechanics by some pioneers of quantum theories, e.g. Lorentz, Einstein, Bohr, etc. [1]. It is usually recapitulated in the following way. In classical x–p phase space, the possible path of a particle is confined in the curve
(E) =
p(E, x) = {2m(E − V )}1/2
=
(1)
this curve surrounds an area (E) = p(E, x)dx
(2)
the frequency of this motion is = 0
(3)
where the basic frequency is 1 = 0
dt =
dx
v
=
∂p d (E) dx = dE ∂E
(4)
pdx =
2Ekinetic dt =
2Ekinetic
(5)
so the frequency is =
dE . d(E)
(6)
For a harmonic oscillator is independent of E, E
(7)
= E/ is an adiabatic invariant quantity. As the first example of adiabatic invariant quantity, Lorentz considered a quantum pendulum whose string length l is shortened very slowly (adiabatically), Einstein pointed out that although the energy and the frequency of the pendulum are both changed √ during the procedure, ıE/E = −ıl/ (2l) , so their ratio E/∼E l is a constant. This observation suggests that the quantities to be quantized (good quantum numbers) must be adiabatic invariants. Sommerfeld finally extended this discussion into a general theory: the quantum number of an arbitrary mechanical system is given by the adiabatic action variable. Since the action variable for the harmonic oscillator is an integer, the general condition is:
夽 Work supported by the National Natural Science Foundation of China under grant: 11175113, 11105133, 61203061, 61374091 and 61403362. ∗ Corresponding author at: Institute of Intelligent Machines, Chinese Academy of Sciences, Hefei 230031, China. Tel.: +86 551 63601696. E-mail address: [email protected] (H.-y. Fan). http://dx.doi.org/10.1016/j.ijleo.2015.07.146 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
pdq = nh.
(8)
Eq. (1) is the foundation of the “Old Quantum Theory”. Although this condition is not exact for the small quantum number n, and the
3802
H.-y. Fan, J.-h. Chen / Optik 126 (2015) 3801–3802
whole theory is still semi-classical, it still gave good first thought to the correct way of quantization. On the other hand, the quantum L–C circuit was first introduced by W.H. Louisell [2]. In this theory, the circuit is composed of a capacity C and an inductance L connected to each other. The charge Q on the capacity C is chosen to be the canonical variable and the magnetic flux = LI on the inductance L is the conjugate “momentum”. The Hamiltonian for a L–C circuit (without electricity source) is H=
Q2 LI 2 + 2C 2
(9)
or Q2 2 H= + 2C 2L
(10)
flux in L, the resonance frequency of this where is the magnetic √ circuit is ω = 1/ LC. The number-phase quantization scheme of L–C circuit is explained in [3], the number-phase uncertainty relationship for the nonlinear number-phase squeezed state is recently discussed in [4]. Since quantization is of necessity for mesoscopic electric circuits, some papers have been published on this subject, say Refs. [5–7]. An interesting question thus arises: what is the adiabatic invariant of the quantum L–C circuit. To the best of our knowledge, this fundamental problem has not been touched in the literature before. In the following we shall discuss it.
In classical theory, abrupt change of Q on the capacity C requires a pulse current I, which would generate an infinitely large magnetic field on the inductance L, thus such kind of change is unphysical and impossible. In quantum case we expect to have the same constrain. Also, according to Faraday’s law of induction, abrupt change of on the inductance would generate an infinitely large induced electric field, which is impossible either. So under any circumstances, Q on the capacity and on the inductance are not allowed to have abrupt change. Therefore, when the parameter values L and C of a real L–C circuit vary infinitesimally by external force or some other practical ways, L → L + ıL, C → C + ıC, the change of energy in the circuit is given by
Q2 2C
+ 2 ı
+ı
1 2L
2 2L
=−
= Q 2ı
1 2C
2 ıL Q 2 ıC − 2C C 2L L
(11)
If parameters are changed adiabatically, the oscillation in circuit must have undergone many periods before L, C make noticeable changes, so the change of energy of the circuit is √ ı LC ıL Q 2 ıC 2 ıL H ıC , ıH = − = −H √ (12) − =− + 2C C 2L L 2 C L LC where the superscript “–” means taking average, H denotes the energy of the circuit and we have used the property that Q2 2 H = = 2C 2L 2
(14)
so √ H H LC = = const ω
(15)
Eq. (15) gives the correct adiabatic invariant of the quantum L–C circuit, since it is similar to Eq. (7). 3. Case for adiabatic force acting on the plate of capacitance Let us consider a specific case of the above general discussion. In classical electromagnetic theories, supposing the capacity in the circuit is a parallel-plate capacitor with plates of area A, separated by distance D, filled with material with permittivity ε, then the capacity is C=
εA D
(16)
The attractive force experienced of one plate by another plate is F = QE
(17)
where E = Q/2Aε is the electric field generated by one plate, so F=
Q2 2Aε
(18)
This is exactly the force need to pull the two plates apart. Since we are pulling the plates very slowly (adiabatically), the real force that is needed is
2. Adiabatic invariant of the quantum L–C circuit
ıH = ı
Integrating Eq. (12), we have √ − ln H = ln LC
(13)
F=
Q H 1 Q = CH = , 2Aε 2Aε 2D
(19)
where we have used (13), so ıH = FıD =
H ıD. 2D
(20)
Integrating Eq. (20) we obtain √ ln H = ln D + const, (21) √ so H/ D = const. Since D is the distance between the two plates of √ the capacity, the larger D is, the smaller the capacity, so ω = √ 1/ LC ∝ D, we again have H/ω =const. Thus we have found the adiabatic invariant of the quantum L–C circuit, which is similar in form to the corresponding theory for a pendulum. This work enriches the adiabatic invariant theory in quantum mechanics and develops Louisell’s quantization theory of mesoscopic circuit. References [1] See e.g. F. Hund, The Development of Quantum Theory, 1967, Berlin. [2] W.H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1973. [3] Hong-yi Fan, Bao-long Liang, Ji-suo Wang, Commun. Theor. Phys. 48 (6) (2007) 1038. [4] Rui He, Hong-yi Fan, Optik 125 (2014) 2426. [5] Ji-suo Wang, et al., Int. J. Theor. Phys. 37 (1998) 1213. [6] Hong-yi Fan, Xiaoyin Pan, Chin. Phys. Lett. 15 (1998) 625. [7] Xing-lei Xu, Hong-qi Li, Ji-suo Wang, Commun. Theor. Phys. 16 (1991) 123.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Adiabatic invariant of the quantum mesoscopic L–C circuit夽 Hong-yi Fan a,b,∗ , Jun-hua Chen c,d,b,e a
Institute of Intelligent Machines, Chinese Academy of Sciences, Hefei 230031, China Department of Material Science and Engineering, USTC, Hefei 230026, China c Hefei Center for Physical Science and Technology, Hefei 230026, China d CAS Key Laboratory of Materials for Energy Conversion, Hefei 230026, China e Synergetic Innovation Center of Quantum Information and Quantum Physics, USTC, Hefei, China b
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 18 June 2014 Accepted 18 July 2015
By analyzing that abrupt change of electric charge Q on the capacity C and abrupt change of on the inductance are not allowed, we derive diabatic invariant of the quantum mesoscopic L–C circuit. This enriches Louisell’s quantization theory of mesoscopic circuit. © 2015 Elsevier GmbH. All rights reserved.
Keywords: Adiabatic invariant L–C circuit Quantization of mesoscopic system
1. Introduction
(note that dE = vdp is valid for any x), thus
Quantized mesoscopic circuit is a hot topic in quantum optics and quantum computer theory. In this work we shall discuss the topic of adiabatic invariant for quantized circuit. The concept of adiabatic invariants was introduced into quantum mechanics by some pioneers of quantum theories, e.g. Lorentz, Einstein, Bohr, etc. [1]. It is usually recapitulated in the following way. In classical x–p phase space, the possible path of a particle is confined in the curve
(E) =
p(E, x) = {2m(E − V )}1/2
=
(1)
this curve surrounds an area (E) = p(E, x)dx
(2)
the frequency of this motion is = 0
(3)
where the basic frequency is 1 = 0
dt =
dx
v
=
∂p d (E) dx = dE ∂E
(4)
pdx =
2Ekinetic dt =
2Ekinetic
(5)
so the frequency is =
dE . d(E)
(6)
For a harmonic oscillator is independent of E, E
(7)
= E/ is an adiabatic invariant quantity. As the first example of adiabatic invariant quantity, Lorentz considered a quantum pendulum whose string length l is shortened very slowly (adiabatically), Einstein pointed out that although the energy and the frequency of the pendulum are both changed √ during the procedure, ıE/E = −ıl/ (2l) , so their ratio E/∼E l is a constant. This observation suggests that the quantities to be quantized (good quantum numbers) must be adiabatic invariants. Sommerfeld finally extended this discussion into a general theory: the quantum number of an arbitrary mechanical system is given by the adiabatic action variable. Since the action variable for the harmonic oscillator is an integer, the general condition is:
夽 Work supported by the National Natural Science Foundation of China under grant: 11175113, 11105133, 61203061, 61374091 and 61403362. ∗ Corresponding author at: Institute of Intelligent Machines, Chinese Academy of Sciences, Hefei 230031, China. Tel.: +86 551 63601696. E-mail address: [email protected] (H.-y. Fan). http://dx.doi.org/10.1016/j.ijleo.2015.07.146 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
pdq = nh.
(8)
Eq. (1) is the foundation of the “Old Quantum Theory”. Although this condition is not exact for the small quantum number n, and the
3802
H.-y. Fan, J.-h. Chen / Optik 126 (2015) 3801–3802
whole theory is still semi-classical, it still gave good first thought to the correct way of quantization. On the other hand, the quantum L–C circuit was first introduced by W.H. Louisell [2]. In this theory, the circuit is composed of a capacity C and an inductance L connected to each other. The charge Q on the capacity C is chosen to be the canonical variable and the magnetic flux = LI on the inductance L is the conjugate “momentum”. The Hamiltonian for a L–C circuit (without electricity source) is H=
Q2 LI 2 + 2C 2
(9)
or Q2 2 H= + 2C 2L
(10)
flux in L, the resonance frequency of this where is the magnetic √ circuit is ω = 1/ LC. The number-phase quantization scheme of L–C circuit is explained in [3], the number-phase uncertainty relationship for the nonlinear number-phase squeezed state is recently discussed in [4]. Since quantization is of necessity for mesoscopic electric circuits, some papers have been published on this subject, say Refs. [5–7]. An interesting question thus arises: what is the adiabatic invariant of the quantum L–C circuit. To the best of our knowledge, this fundamental problem has not been touched in the literature before. In the following we shall discuss it.
In classical theory, abrupt change of Q on the capacity C requires a pulse current I, which would generate an infinitely large magnetic field on the inductance L, thus such kind of change is unphysical and impossible. In quantum case we expect to have the same constrain. Also, according to Faraday’s law of induction, abrupt change of on the inductance would generate an infinitely large induced electric field, which is impossible either. So under any circumstances, Q on the capacity and on the inductance are not allowed to have abrupt change. Therefore, when the parameter values L and C of a real L–C circuit vary infinitesimally by external force or some other practical ways, L → L + ıL, C → C + ıC, the change of energy in the circuit is given by
Q2 2C
+ 2 ı
+ı
1 2L
2 2L
=−
= Q 2ı
1 2C
2 ıL Q 2 ıC − 2C C 2L L
(11)
If parameters are changed adiabatically, the oscillation in circuit must have undergone many periods before L, C make noticeable changes, so the change of energy of the circuit is √ ı LC ıL Q 2 ıC 2 ıL H ıC , ıH = − = −H √ (12) − =− + 2C C 2L L 2 C L LC where the superscript “–” means taking average, H denotes the energy of the circuit and we have used the property that Q2 2 H = = 2C 2L 2
(14)
so √ H H LC = = const ω
(15)
Eq. (15) gives the correct adiabatic invariant of the quantum L–C circuit, since it is similar to Eq. (7). 3. Case for adiabatic force acting on the plate of capacitance Let us consider a specific case of the above general discussion. In classical electromagnetic theories, supposing the capacity in the circuit is a parallel-plate capacitor with plates of area A, separated by distance D, filled with material with permittivity ε, then the capacity is C=
εA D
(16)
The attractive force experienced of one plate by another plate is F = QE
(17)
where E = Q/2Aε is the electric field generated by one plate, so F=
Q2 2Aε
(18)
This is exactly the force need to pull the two plates apart. Since we are pulling the plates very slowly (adiabatically), the real force that is needed is
2. Adiabatic invariant of the quantum L–C circuit
ıH = ı
Integrating Eq. (12), we have √ − ln H = ln LC
(13)
F=
Q H 1 Q = CH = , 2Aε 2Aε 2D
(19)
where we have used (13), so ıH = FıD =
H ıD. 2D
(20)
Integrating Eq. (20) we obtain √ ln H = ln D + const, (21) √ so H/ D = const. Since D is the distance between the two plates of √ the capacity, the larger D is, the smaller the capacity, so ω = √ 1/ LC ∝ D, we again have H/ω =const. Thus we have found the adiabatic invariant of the quantum L–C circuit, which is similar in form to the corresponding theory for a pendulum. This work enriches the adiabatic invariant theory in quantum mechanics and develops Louisell’s quantization theory of mesoscopic circuit. References [1] See e.g. F. Hund, The Development of Quantum Theory, 1967, Berlin. [2] W.H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1973. [3] Hong-yi Fan, Bao-long Liang, Ji-suo Wang, Commun. Theor. Phys. 48 (6) (2007) 1038. [4] Rui He, Hong-yi Fan, Optik 125 (2014) 2426. [5] Ji-suo Wang, et al., Int. J. Theor. Phys. 37 (1998) 1213. [6] Hong-yi Fan, Xiaoyin Pan, Chin. Phys. Lett. 15 (1998) 625. [7] Xing-lei Xu, Hong-qi Li, Ji-suo Wang, Commun. Theor. Phys. 16 (1991) 123.