# Nonstandard physics/Quantum Electromagnetic Resonator

Quantum Electromagnetic Resonator (QER) – closed topological object of the three dimensional space, in the general case – ‘’cavity’’ of arbitrary form, which has definite ‘’surface’’ and ‘’thickness’’. To the contrary to the classical case, there are no any electromagnetic ‘’waves’’ and radiation losses in the QER, but “infinite” phase shifted oscillation of electromagnetic field, due to the quantum properties of QER.

## History

It is happened, that such physical values as capacitance and inductance have no interest in the modern Quantum electrodynamics. Furthermore, they are neglected even in classical electrodynamics, where electric and magnetic fields are dominated. The point is that, the last are not included in the evident form in the Maxwell equations, and so the resulting solutions includes fields only. Yes, sometimes these coefficients were obtained from the solutions of Maxwell equations, but it were very rarely, and relation to them were consideringly low. It is known too, that s.c. “field approach” in electrodynamics that considered “point charges” leads to the “fool infinities”, when the interaction radius trends to zero. Furthermore, these “fool infinities” are presented in the quantum electrodynamics too, where power methods are developed to compensate them. Contrary to the theoretical physics, the applied physics are widely used reactive parameters, such as capacitance and inductance, firstly in the electrotechnics and then in the applied radiotechnics. Now reactive parameters are widely used in the information technologies, which are based on the generation, transmission and radiation of electromagnetic waves of different frequencies.

The present day situation (without proper development of the theory of reactive parameters such as inductance, capacitance and electromagnetic resonator) brakes developments of information technologies and quantum computing. Note that, mechanical harmonic oscillator was considered in quantum mechanics in the early 30-ies of 20-th century, when the quantum theory is developed. However, the quantum consideration of the $LC - \$ circuit was started only by Louisell(1973) [1]. Since then, there were no practical examples of the quantum capacitance and inductance, therefore this approach did not obtain proper consideration. Theoretically correct introduction of the quantum capacitance, based on the density of states, first was presented by Luryi (1988) [2] for QHE. However, Luryi did not introduce the quantum inductance, and this approach was not considered in the quantum LC circuit and resonator. Year later, Yakymakha (1989) [3] considered an example of the series and parallel quantum $LC - \$ circuits (its characteristic impedances) during QHE explanation (integer and fractional). However in this work did not considering the Schrodinger equation for the quantum LC circuit.

For the first time, both quantum values, capacitance and inductance, were considered by Yakymakha (1994) [4], during spectroscopic investigations of MOSFETs at the very low frequencies (sound range). The flat quantum capacitances and inductances here had thicknesses about Compton wave length of electron and its characteristic impedance – the wave impedance of free space. And three year later, Devoret (1997) [5] presented complete theory of the Quantum LC Circuit (applied to the Josephson junction). Possible application of the quantum LC circuits and resonators in the quantum computation are considered by Devoret (2004) [6].

## Classical electromagnetic resonator

In the general case classical electromagnetic resonator (CER) is the cavity in the 3D-space. Therefore, CER has infinite resonance frequencies, due to the three dimensions. For example, the rectangular CER has the following resonance frequencies:

$\omega_{mnp} = \frac{1}{\sqrt{\epsilon_0 \epsilon \mu_0 \mu}}\sqrt{(\frac{m\pi}{a})^2 + (\frac{n\pi}{b})^2 + (\frac{p\pi}{l})^2}, \$

where $m,n,p = integer. \$; $a, b, l - \$ are the width, thickness and length correspondingly, $\epsilon_0- \$ is the dielectric constant, $\epsilon- \$ is the relative permittivity, $\mu_0 - \$ is the magnetic constant, $\mu - \$ relative permeability. To the contrary of classical LC circuit, both electric and magnetic fields are displaced in the same volume of CER. These oscillating electromagnetic fields in the classical case are like standing waves, that form electromagnetic waves, that could be radiated in the external world. Now CER are widely used in the radio frequency range (centimeter and decimeter diapason). Furthermore, CER are also used in the quantum electronics, which deal with the monochromic light waves.

## Quantum general approach

### Quantum LC circuit oscillator

In the classical physics we have the following correspondence between mechanical and electrodynamics physical parameters: magnetic inductance and mechanical mass:

$L \leftrightarrow m \$;

electric capacitance and reverse elasticity:

$C \leftrightarrow 1/k \$;

electric charge and coordinate displacement:

$q \leftrightarrow x \$.

Inductance momentum quantum operator in the electric charge space could be presented in the following form:

$\hat p_{Lq} = -i\hbar \frac{d}{dq}, \quad \quad \quad \quad \quad \hat p_{Lq}^* = i\hbar \frac{d}{dq},\quad \quad \quad \quad \quad (1a) \$

where $\hbar- \$ is reduced Plank constant, $\hat p_L^*- \$ is the complex-conjugate momentum operator. Capacitance momentum quantum operator in the magnetic charge space could be presented in the following form:

$\hat p_{C\phi} = -i\hbar \frac{d}{d\phi}, \quad \quad \quad \quad \quad \hat p_{C\phi}^* = i\hbar \frac{d}{d\phi},\quad \quad \quad \quad \quad (1b) \$

where $\phi \$ is the induced magnetic charge. Considering the fact, that there are no free magnetic charges, but it could be immitated by electric current ($i \$):

$\phi = L\cdot i, \$

we can introduce the third momentum quantum operator in the current form:

$\hat p_{Ci} = -\frac{i\hbar}{L} \frac{d}{di}, \quad \quad \quad \quad \quad \hat p_{Ci}^* = \frac{i\hbar}{L} \frac{d}{di},\quad \quad \quad \quad \quad (1c) \$

These quantum momentum operators defines three Hamilton operators:

$\hat H_{Lq} = -\frac{\hbar^2}{2L}\cdot \frac{d^2}{dq^2} + \frac{L\omega_0^2}{2}\hat q^2 \quad \quad \quad \quad \quad (2a) \$
$\hat H_{C\phi} = -\frac{\hbar^2}{2C}\cdot \frac{d^2}{d\phi^2} + \frac{C\omega_0^2}{2}\hat \phi^2 \quad \quad \quad \quad \quad (2b) \$
$\hat H_{Ci} = -\frac{\hbar^2\omega_0^2}{2L}\cdot \frac{d^2}{di^2} + \frac{L\omega_0}{2}\hat i^2, \quad \quad \quad \quad \quad (2c) \$

where $\omega_0 = \frac{1}{\sqrt{LC}} \$ is the resonance frequency. We consider the case without dissipation ($R = 0 \$). The only difference of the charge spaces and current spaces from the traditional 3D- coordinate space is that it are one dimensional (1D). Schrodinger equation for the quantum LC circuit could be defined in three form:

$-\frac{\hbar^2}{2L}\frac{d^2 \Psi}{dq^2} + \frac{L\omega_0^2}{2}q^2\Psi = W\Psi \quad \quad \quad \quad \quad (3a) \$
$-\frac{\hbar^2}{2C}\frac{d^2 \Psi}{d\phi^2} + \frac{C\omega_0^2}{2}\phi^2\Psi = W\Psi \quad \quad \quad \quad \quad (3b) \$
$-\frac{\hbar^2\omega_0^2}{2L}\frac{d^2 \Psi}{di^2} + \frac{L\omega_0}{2}i^2\Psi = W\Psi. \quad \quad \quad \quad \quad (3c) \$

To solve these equations we should to introduce the following dimensionless variables:

$\xi_q = \frac{q}{q_0}; \quad \quad q_0 = \sqrt{\frac{\hbar}{L\omega_0}}; \quad \quad \lambda_q = \frac{2W}{\hbar\omega_0} \quad \quad (4a) \$
$\xi_{\phi} = \frac{\phi}{\phi_0}; \quad \quad \phi_0 = \sqrt{\frac{\hbar}{C\omega_0}}; \quad \quad \lambda_{\phi} = \frac{2W}{\hbar\omega_0} \quad \quad (4a) \$
$\xi_i = \frac{i}{i_0}; \quad \quad i_0 = \sqrt{\frac{\hbar \omega_0}{L}}; \quad \quad \lambda_i = \frac{2W}{\hbar\omega_0}. \quad \quad (4a) \$

where $q_0 \$ is scaling "induced electric charge"; $\phi_0 \$ is scaling "induced magnetic charge" and $i_0 \$ is scaling "induced electric current". Then the Schrodinger equation will take the form of the differential equation of Chebyshev-Ermidt:

$(\frac{d^2}{d\xi^2} + \lambda - \xi^2)\Psi = 0. \$

The eigen values of the Hamiltonian will be:

$W_n = \hbar \omega_0(n + 1/2), \$

where at $n = 0 \$ we shall have zero oscillation:

$W_0 = \hbar \omega_0/2. \$

In the general case the scaling charges could be rewritten in the form:

$q_0 = \sqrt{\frac{\hbar}{L\omega_0}} = \frac{q}{\sqrt{4\pi \alpha }} \$
$\phi_0 = \sqrt{\frac{\hbar}{C\omega_0}} = \sqrt{\frac{\alpha}{\pi}}\cdot \frac{h}{e}, \$

where $\alpha - \$ is the fine structure constant. Furthermore, the scaling current and voltage will be here:

$I_0 = \frac{\phi_0}{L} = \sqrt{\frac{\pi}{\alpha}}\cdot \frac{ec}{\lambda_0} \$
$V_0 = \frac{q_0}{C} = \sqrt{4\pi \alpha}\cdot \frac{ch}{e\lambda_0}, \$

where $\lambda_0 = \frac{h}{m_0c}$ is the particle wavelength. Note that these scaling parameters were obtained by using the following quantum resonator parameters:

$\rho_0 = 2\alpha R_H \quad \quad (5a) \$

for characteristic impedance, and

$\omega_0 = \frac{2\pi c}{\lambda_0} \quad \quad (5b) \$

for resonance frequency.

These three equations (3) form the base of the nonrelativistic quantum electrodynamics, which considers elementary particles from the intrinsic point of view. Note that, the standard quantum electrodynamics conciders elementary particles from the external point of view.

#### Electron as quantum oscillator

Suppose that electron has its mass defined by the quantum oscillations. Thus, its mass will be equivalent to the zero oscillations of the quantum LC circuit:

$0.5\hbar \omega_0 = m_0c^2. \$

So, the oscillation length will be:

$x_0 = \frac{\lambda_0}{2\pi\sqrt{2n}}, \$

where $\lambda_0 - \$ is the Compton wavelength of electron. Further, let us consider that this mass is uniformly distributed on the sphere:

$S_0 = 4\pi x_0^2 = \lambda_0^2/2\pi. \$.

Then we can find out the density of states for this mass:

$D_0 = \frac{1}{S_0}\frac{1}{m_0c^2} = \frac{m_0}{2\pi \hbar^2}. \$

Thus, presentation of the electronic mass as the zero oscillation of the harmonic oscillator yields to the uniform its distribution on the sphere or the $\lambda_0 /2\pi \sqrt{2}$ radius.

#### Photon as quantum oscillator

As is known, photon momentum is defined as:

$p_p = \frac{\hbar \omega_p}{c}, \$

where $c - \$ is velocity of light. So, the "effective (energy) photon mass" could be defined as:

$m_p = \frac{\hbar \omega_p}{c^2}. \$

Then the length scaling parameter of harmonic oscillator will be:

$x_0 = \sqrt{\frac{\hbar}{m_p\omega_p}} = \frac{\lambda_p}{2\pi}, \$

where $\lambda_p = \frac{2\pi c}{\omega_p}- \$ is the photon wavelength.

### Resonator as quantum LC circuit

Luryi density of states (DOS) approach defines quantum capacitance as:

$C_{QR} = q_R^2\cdot D_{2D}\cdot S_R, \$

and quantum inductance as:

$L_{QR} = \phi_R^2\cdot D_{2D}\cdot S_R, \$

where $S_R - \$ resonator surface area, $D_{2D} = \frac{m}{\pi \hbar^2} - \$two dimensional (2D) DOS, $q_R - \$ electric charge (or flux), and $\phi_R - \$ magnetic charge (or flux). Note that these fluxes should be defined afterward.

Energy stored on quantum capacitance:

$W_{CR} = \frac{q_R^2}{2D_{2D}S_R}. \$

Energy stored on quantum inductance:

$W_{LR} = \frac{\phi_R^2}{2D_{2D}S_R} = W_{CR}. \$

Resonator angular frequency:

$\omega_{QR} = \frac{1}{\sqrt{L_{QR}C_{QR}}} = \frac{1}{q_R\phi_RD_{2D}S_R}. \$

Energy conservation law:

$W_{QR} = \hbar \omega_R = \frac{1}{q_R\phi_RD_{2D}S_R} = W_{CR} = W_{LR}. \$

This equation can be rewritten as:

$q_R\phi_R = \hbar, \$

from which it is evident that these "charges" are the fluxes, but not metallurgic charges.

Characteristic resonator impedance:

$\rho_{QR} = \sqrt{\frac{L_{QR}}{C_{QR}}} = \frac{\phi_R}{q_R} = \begin{cases} 2\frac{\phi_0}{e}=2R_H, & \mbox{for QHE } \\ 2\alpha \frac{\phi_0}{e}=2\alpha R_H, & \mbox{for other } \end{cases} \$

where $\phi_0 = h/e - \$ magnetic flux quantum. Considering above equations, we can find out the following electric and magnetic set of fluxes:

$q_R = \begin{cases} q_{R1} = \frac{e}{\sqrt{2\pi}}, & \mbox{for QHE } \\ q_{R2} = \frac{e}{\sqrt{2\pi \alpha}}, & \mbox{for other } \end{cases} \$
$\phi_R = \begin{cases} \phi_{R1} = \frac{\phi_0}{\sqrt{\frac{2}{\pi}}}, & \mbox{for QHE } \\ \phi_{R2} = \frac{\phi_0}{\sqrt{\frac{2\alpha}{\pi}}}, & \mbox{for other } \end{cases} \$

Note, that these values are not the real "metallurgic charges", but the maximal fluxes, that maintain the energy balance between resonator oscillation energy and total energy on capacitance and inductance

$\hbar \omega_{QR} = W_{QL}(t) + W_{QC}(t). \$

Since capacitance oscillations are phase shifted ($\psi = \pi /2 \$) with respect to inductance oscillations, therefore we get:

$W_{QL} = \begin{cases} 0, & \mbox{at }t=0; \psi=0\mbox{ and} t=\frac{T_{QR}}{2};\psi=\pi \\ W_{QL}, & \mbox{at }t=\frac{T_{QR}}{4};\psi=\frac{\pi}{4} \mbox{ and}t=\frac{3T_{QR}}{4};\psi=\frac{3\pi}{4} \end{cases} \$
$W_{QC} = \begin{cases} W_{QC}, & \mbox{at }t=0; \psi=0\mbox{ and} t=\frac{T_{QR}}{2};\psi=\pi \\ 0, & \mbox{at }t=\frac{T_{QR}}{4};\psi=\frac{\pi}{4} \mbox{ and}t=\frac{3T_{QR}}{4};\psi=\frac{3\pi}{4} \end{cases} \$

where $T_{QR} = \frac{2\pi}{\omega_{QR}}- \$ is oscillation period.

## De Broglie electromagnetic resonator

De Broglie matter wave[7] could be considered for the charges (electric and magnetic). Actually, using approach de Broglie in the form of Blohintzev[8] we could derive the following charge quantum resonator (de Broglie charge bubble).

### De Broglie electric charge

Electric charge in the de Broigle wave matter approach could be presented by the following wave function:

$\Psi(q_E,t) = A_E exp [i(\frac{\phi_Mq_E}{\hbar} - \frac{W_Lt}{\hbar})] = A_E exp [i(\gamma_E)] \$

with

$W_L = \hbar \omega_L \$
$\phi_M = \hbar k_E, \$

where $k_E = \frac{2\pi}{q_{\lambda}}$ is electric charge "wave vector" and $q_{\lambda}$ is electric charge "wave length". The phase function

$\gamma_E = k_Eq_E - \omega_Lt \$

have differential

$d\gamma_E = k_Edq_E - \omega_Ldt = 0. \$

So, the phase current will be:

$i_{ph} = \frac{dq_E}{dt} = \frac{\omega_L}{k_E} \$

Considering the wave energy in the form

$W_L = \frac{\phi_M^2}{2L} = \frac{\hbar^2k_E^2}{2L} = \hbar \omega_L, \$

where $L \$ is de Broglie quantum inductance, and frequency

$\omega_L = \frac{\hbar k_E^2}{2L} \$

De Broglie "group current" will be:

$i_g = \frac{d\omega_L}{dk_E} = \frac{\hbar k_E}{L}. \$

From the other hand we have the following limit for the group current:

$i_g = \frac{ec}{\lambda}. \$

Equating group currents, we can derive the de Broglie quantum inductance in the evident form:

$L = \frac{h}{eq_{\lambda}}\cdot \frac{\lambda}{c}. \$

### De Broglie magnetic charge

Magneticic charge in the de Broigle wave matter approach could be presented by the following wave function:

$\Psi(q_M,t) = A_M exp [i(\frac{\phi_Eq_M}{\hbar} - \frac{W_Ct}{\hbar})] = A_M exp [i(\gamma_M)] \$

with

$W_C = \hbar \omega_C \$
$\phi_E = \hbar k_M, \$

where $k_M = \frac{2\pi}{q_{M\lambda}}$ is magnetic charge "wave vector" and $q_{M\lambda}$ is magnetic charge "wave length". The phase function

$\gamma_M = k_Mq_M - \omega_Ct \$

have differential

$d\gamma_M = k_Mdq_M - \omega_Cdt = 0. \$

So, the phase voltage will be:

$v_{ph} = \frac{dq_M}{dt} = \frac{\omega_C}{k_M} \$

Considering the wave energy in the form

$W_C = \frac{\phi_E^2}{2C} = \frac{\hbar^2k_M^2}{2C} = \hbar \omega_C, \$

where $C \$ is de Broglie quantum cdpacitance, and frequency

$\omega_C = \frac{\hbar k_M^2}{2C} \$

De Broglie "group voltage" will be:

$v_g = \frac{d\omega_C}{dk_M} = \frac{\hbar k_M}{C}. \$

From the other hand we have the following limit for the group voltage:

$v_g = \frac{hc}{e}\cdot \frac{1}{\lambda}. \$

Equating group voltages, we can derive de Broglie capacitance in the evident form:

$C = \frac{e}{q_{M\lambda}}\cdot \frac{\lambda}{c}. \$

### De Broglie LC circuit

Note that, using separatedly electric and magnetic de Broglie charge approaches, we should to consider the pilot wave approach to maintane electric and magnetic charges in the stable form. However, when we consider the composite de Broglie wave function in the following form:

$\Psi(q_M,t) = A_MA_E exp [i(\frac{\phi_Mq_E}{\hbar} - \frac{W_Lt}{\hbar})] exp [i(\frac{\phi_Eq_M}{\hbar} - \frac{W_Ct}{\hbar})] \$

with

$W_C = \hbar \omega_C = W_L = \hbar \omega_L \$
$\phi_E = \hbar k_M, \$
$\phi_M = \hbar k_E, \$

then separated electric and magnetic charges produces the composite quantum resonator. Considering that the de Broglie particle should be consistent with the quantum resonator, so we should to find out the electric and magnetic charges induced in the LC circuit:

$q_{\lambda} = \gamma_q e \$
$q_{M\lambda} = \gamma_{\phi} \frac{h}{e}. \$

The reactive parameters in that case will be:

$L(\lambda) = \frac{R_H}{\gamma_q}\cdot \frac{\lambda}{c} \$
$C(\lambda) = \frac{1}{\gamma_{\phi}R_H}\cdot \frac{\lambda}{c}. \$

Characteristic impedance for the de Broglie charged particle resonator:

$\rho_{\lambda} = \sqrt{\frac{L(\lambda)}{C(\lambda)}} = R_H\sqrt{\frac{\gamma_{\phi}}{\gamma_q}} = 2\alpha R_H. \$

Resonance frequency for the de Broglie charged particle resonator:

$\omega(\lambda) = \frac{1}{\sqrt{L(\lambda)C(\lambda)}} = \frac{c}{\lambda}\sqrt{\gamma_{\phi}\gamma_q} = \frac{2\pi c}{\lambda}. \$

Thus, we have the following algebraic system for uncknown parameters:

$\sqrt{\gamma_{\phi}\gamma_q} = 2\pi \$
$\sqrt{\frac{\gamma_{\phi}}{\gamma_q}} = 2\alpha. \$

The solution of this system will be as follows:

$\gamma_q = \frac{\pi}{\alpha} \quad \quad \gamma_{\phi} = 4\pi \alpha \$

Then, induced electric and magnetic charges could be rewritten as:

$q_{\lambda} = \gamma_q e \ = \frac{\pi}{\alpha}\cdot e \$
$q_{M\lambda} = \gamma_{\phi} \frac{h}{e} = 4\pi \alpha \cdot \frac{h}{e}, \$

and reactive parameters will be:

$L(\lambda) = \frac{\alpha R_H}{\pi}\cdot \frac{\lambda}{c} \$
$C(\lambda) = \frac{1}{4\pi \alpha R_H}\cdot \frac{\lambda}{c}. \$

So, the quantum oscillations of electric and magnetic charges make charged de Broglie buble stable in time.

## Applications

### Bohr atomic resonator

General information on Bohr atom. Bohr radius of electron:

$a_B = \frac{\lambda_0}{2\pi \alpha}, \$

where $\alpha - \$ electric fine structure constant, $\lambda_0 - \$ Compton wavelength of electron.

Bohr surface scaling parameter:

$S_B = 4\pi a_B^2. \$

Bohr angular frequency:

$\omega_B = \frac{\hbar}{2ma_B^2}, \$

where $\hbar - \$ reduced Plank constant and $m - \$ electron mass.

Bohr density of states:

$D_B = \frac{1}{\hbar \omega_B S_B} = \frac{m}{2\pi \hbar^2}. \$

Standard DOS quantum resonator approach yields the following values for the Bohr atom reactive quantum parameters:

$C_{QRB} = q_{R2}^2D_BS_B = \frac{e^2}{4\pi \alpha}\frac{m}{2\pi \hbar^2}S_B = \frac{\epsilon_0}{\lambda_0}S_B \$
$L_{QRB} = \phi_{R2}^2D_BS_B = \frac{\alpha \phi_0^2m}{2\pi^2 \hbar^2}S_B = \frac{\mu_0}{\lambda_0}S_B, \$

where $\beta = \frac{\phi_0^2}{2\mu_0hc} = \frac{1}{4\alpha} - \$ is considered.

Thus, s.c. "Bohr atom" could be considered as spherical quantum resonator which has radius $a_B \$ and thickness $\lambda_0 \$.

### Electron resonator

$r_e = \frac{\lambda_0}{2\pi \sqrt{2}}. \$

Electron surface scaling parameter:

$S_e = 4\pi r_e^2 = \frac{\lambda_0^2}{2\pi}. \$

Electron angular frequency:

$\omega_e = \frac{mc^2}{\hbar} = \frac{2\pi c}{\lambda_0}, \$

where $c - \$ is the velocity of light. Electron density of states:

$D_e = \frac{1}{S_eW_e} = \frac{m}{2\pi \hbar^2}. \$

Standard DOS quantum resonator approach yields the following values for the electron reactive quantum parameters:

$C_{QRe} = q_{R2}^2D_eS_e = \frac{e^2}{4\pi \alpha}\frac{m}{2\pi \hbar^2}S_e = \frac{\epsilon_0}{\lambda_0}S_e \$
$L_{QRe} = \phi_{R2}^2D_eS_e = \frac{\alpha \phi_0^2m}{2\pi^2 \hbar^2}S_e = \frac{\mu_0}{\lambda_0}S_e. \$

Thus, s.c. "free electron" could be considered as spherical quantum resonator which has radius $r_e \$ and thickness $\lambda_0 \$.

### Photon resonator

$r_p = \frac{\lambda_p}{2\pi}. \$

Photon surface scaling parameter:

$S_p = 4\pi r_p^2 = \frac{\lambda_p^2}{\pi}. \$

Photon angular frequency:

$\omega_p = \frac{2\pi c}{\lambda_p}, \$

where $c - \$ is the velocity of light. Photon density of states:

$D_p = \frac{1}{S_pW_p} = \frac{\pi}{\lambda_phc}. \$

Standard DOS quantum resonator approach yields the following values for the photon reactive quantum parameters:

$C_{QRp} = q_{R2}^2D_pS_p = \frac{e^2}{4\pi \alpha}\frac{\pi}{\lambda_phc}S_e = \frac{\epsilon_0}{2\lambda_p}S_p \$
$L_{QRp} = \phi_{R2}^2D_pS_p = \frac{\alpha}{\pi}\frac{\phi_0^2}{2\mu_0hc}\frac{2\pi \mu_0}{\lambda_p}S_p = \frac{\mu_0}{2\lambda_p}S_e. \$

Thus, s.c. "free photon" could be considered as spherical quantum resonator which has radius $r_p \$ and thickness $2\lambda_p \$.

### Quantum Hall resonator

General information on Hall effect. Cyclotron frequency:

$\omega_c = \frac{eB}{m}, \$

where $e - \$ electronic charge, $B - \$ magnetic field induction and $m - \$ electron effective mass in solid.

Magnetic length:

$l_B = \sqrt{\frac{\hbar}{eB}}, \$

Scaling area parameter:

$S_H = 2\pi l_B^2 = \frac{h}{eB} = \frac{1}{n_H}, \$

where $n_H - \$ electron surface density.

Density of states:

$D_H = \frac{n_H}{\hbar \omega_c} = \frac{m}{2\pi \hbar^2}. \$

Density of states quantum capacitance:

$C_H = q_H^2 \cdot D_{2D}\cdot S_H = \frac{e^2m}{2h^2}S_H = \alpha \frac{\epsilon_0}{\lambda_0}S_H \$

or in another form:

$C_H = \frac{\epsilon_0}{d_C}S_H, \$

where $d_C = \frac{\lambda_0}{\alpha}- \$ is capacitance thickness.

Density of states quantum inductance:

$L_H = \phi_H^2 \cdot D_{2D}\cdot S_H = \frac{\phi_0^2m}{h^2}S_H = 4\beta \frac{\mu_0}{\lambda_0}S_H \$

or in another form:

$L_H = \frac{\mu_0}{d_L}S_H, \$

where $d_L = \frac{\lambda_0}{4\beta} = \alpha \lambda_0- \$ is inductance thickness.

Thus, quantum Hall resonator has the cylindrical form with radius $l_B \$ and two different thickness for capacitance $d_C \$ and inductance $d_L \$.

### Flat atom resonator

Very low frequency resonance discovered by Yakymakha (1994)[4] with mesoscopic parameters:

$\omega_0 = \frac{1}{\sqrt{L_0C_0} } = 5088 rad/s \$

for angular frequency,

$S_0 = \frac{h}{m\omega_0} = 1.4196\cdot 10^{-7} m^2 \$

for surface scaling parameter,

$C_0 = \frac{1}{\omega_0\rho_0} = 5.1805\cdot 10^{-7} F \$

for quantum capacitance, and

$L_0 = \frac{\rho_0}{\omega_0} = 7.3524 \cdot 10^{-2} H \$

for quantum inductance, could be considered as quantum resonator too. DOS in that case will be:

$D_0 = \frac{1}{\hbar \omega_0 S_0} = \frac{m}{2\pi \hbar^2}. \$

Quantum resonator capacitance:

$C_{QR0} = q_{R2}^2D_0S_0 = \frac{\epsilon_0}{\lambda_0}S_0. \$

Quantum resonator inductance:

$L_{QR0} = \phi_{R2}^2D_0S_0 = \frac{\mu_0}{\lambda_0}S_0. \$

Note that, even in the mesoscopic case we have the reactive parameter thickness about Compton wave length of electron.

## References

1. Louisell W. H. (1973). “Quantum Statistical Properties of Radiation”. Wiley, New York.
2. Serge Luryi (1988). "Quantum capacitance device". Appl.Phys.Lett. 52(6). Pdf
3. Yakymakha O.L.(1989). High Temperature Quantum Galvanomagnetic Effects in the Two- Dimensional Inversion Layers of MOSFET's (In Russian). Kyiv: Vyscha Shkola. p.91. ISBN 5-11-002309-3. djvu</
4. Yakymakha O.L., Kalnibolotskij Y.M. (1994). "Very-low-frequency resonance of MOSFET amplifier parameters". Solid- State Electronics 37(10),1739-1751 Pdf
5. Deboret M.H. (1997). "Quantum Fluctuations". Amsterdam, Netherlands: Elsevier. pp.351-386. Pdf
6. Devoret M.H., Martinis J.M. (2004). "Implementing Qubits with Superconducting Integrated Circuits". Quantum Information Processing, v.3, N1. Pdf
7. Louis de Broglie. The wave nature of the electron, Nobel Lecture, 12, 1929 PDF
8. Блохинцев Д. И. Основы квантовой механики.- М.:ГосИздат, 1949.-588с.

## Reference Books

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