Quantum Optics Modules

The quantum optics modules solve the four basic quantum mechanical wave equations under time dependent excitation by an arbitrary electromagnetic field. The primary application is to solve for the response of an atom to a strong laser field. Admittedly, “quantum optics” is strictly a misnomer, since the radiation field is treated classically.

Wave Equations

The wave equations that are treated are the Schroedinger equation, the Pauli equation, the Klein-Gordon equation, and the Dirac equation. These can be presented in a 2\times 2 table which categorizes the spin and the energy-momentum (dispersion) relation

Table I. Wave Equations.

spin = 0

spin = \hbar/2

\hbar\omega = (\hbar k)^2/2m

Schroedinger

Pauli

(\hbar\omega)^2 = (mc^2)^2 + (c\hbar k)^2

Klein-Gordon

Dirac

In the case of the relativistic equations, it is sometimes useful to consider cylindrical atoms which allow for 2D geometry. This should not be confused with the use of cylindrical geometry, which can be used to simulate a spherical atom in the non-relativistic case, in the dipole approximation. In contrast, relativistic cylindrical atoms are simulated in 2D Cartesian coordinates, with the ignorable coordinate lying along the atom’s symmetry axis.

Quantum Units

For the non-relativistic wave equations, atomic units are used for all input file parameters and output file data. For the relativistic wave equations, natural units are used for all input file parameters and output file data. Tables II and III give the unit conversions. To convert an atomic or natural unit to Gaussian or SI units, multiply by the formula in the cgs or mks column, respectively.

Table II. Atomic Units to Gaussian or SI Units.

Quantity

Unit

cgs

mks

Charge

electronic charge

|e|

|e|

Mass

electronic mass

m

m

Time

atomic time

t_a = \hbar/m(\alpha c)^2

t_a = \hbar/m(\alpha c)^2

Space

Bohr Radius

a_0 = \hbar/m(\alpha c)

a_0 = \hbar/m(\alpha c)

Velocity

\alpha c

\alpha c

Energy

Hartree

Ha = m(\alpha c)^2

Ha = m(\alpha c)^2

Momentum

m\alpha c

m\alpha c

Scalar Potential

Ha/|e|

Ha/|e|

Vector Potential

Ha/\alpha|e|

Ha/\alpha|e|c

Electric Field

Ha/a_0|e|

Ha/a_0|e|

Magnetic Field

Ha/\alpha a_0|e|

Ha/\alpha a_0|e|c

It should be mentioned there is another way to normalize the vector potential. In the scheme of table II a factor of \alpha is eliminated from the Schroedinger equation, compared with the other possible scheme. The way to understand this is to look at the Schroedinger equation written in various units:

cgs — i\hbar\partial\psi/\partial t = \left[\left(i\hbar\nabla + Q{\bf A}/c\right)^2/2M + Q\Phi\right]\psi

mks — i\hbar\partial\psi/\partial t = \left[\left(i\hbar\nabla + Q{\bf A}\right)^2/2M + Q\Phi\right]\psi

TW — i\partial\psi/\partial t = \left[\left(i\nabla + Q{\bf A}\right)^2/2M + Q\Phi\right]\psi

Here, Q and M are the charge and mass of the simulated particle, written in capital letters to distinguish them from the electronic charge and mass. By including the factor of \alpha in the normalization of the vector potential, it disappears from the TW Schroedinger equation. If this had not been done, then the TW equation would resemble more the cgs equation, with \alpha playing the part of the speed of light.

Table III. Natural Units to Gaussian or SI Units.

Quantity

Unit

cgs

mks

Charge

Expanded charge

q = \alpha^{-1/2}|e|

q = \alpha^{-1/2}|e|

Mass

electronic mass

m

m

Time

QED time

\hbar/mc^2

\hbar/mc^2

Space

Reduced Compton Wavelength

\Lambda = \hbar/mc

\Lambda = \hbar/mc

Velocity

Speed of Light

c

c

Energy

mc^2

mc^2

Momentum

mc

mc

Scalar Potential

mc^2/q

mc^2/q

Vector Potential

mc^2/q

mc/q

Electric Field

mc^2/\Lambda q

mc^2/\Lambda q

Magnetic Field

mc^2/\Lambda q

mc/\Lambda q

To illustrate the effect of natural units on a wave equation, the Klein-Gordon equation is presented in various units:

cgs — \left[\left(i\hbar\partial/\partial t - Q\Phi\right)^2 - \left(i\hbar c\nabla + Q{\bf A}\right)^2 - (Mc^2)^2\right]\psi = 0

mks — \left[\left(i\hbar\partial/\partial t - Q\Phi\right)^2 - \left(i\hbar c\nabla + Qc{\bf A}\right)^2 - (Mc^2)^2\right]\psi = 0

TW — \left[\left(i\partial/\partial t - Q\Phi\right)^2 - \left(i\nabla + Q{\bf A}\right)^2 - M^2\right]\psi = 0

The question of how to normalize the vector potential does not come up because in natural units c=1, whereas in atomic units c=\alpha.

Note

The Schwinger field is not the unit of electric field, because of the use of the expanded charge in the unit of electric field. In natural units the Schwinger field is q/|e| = \alpha^{-1/2}.

Quantum Numbers

In order to initialize most quantum optics simulations, it is necessary to create one or more stationary states. Each stationary state is associated with a given static field. Typical static fields include the Coulomb field of the nucleus, an approximation of the Coulomb field called a soft-core potential, or a nuclear field superposed with a uniform magnetic field.

The quantum numbers identifying a given stationary state differ depending on the wave equation that is selected. In order to bring as much uniformity as possible to the situation, we adopt a nomenclature that is based on the most general of the four equations, the Dirac equation. The four quantum numbers that are used to characterize any turboWAVE quantum state are n_r, j, l, m. The values they take in various situations are listed in table IV. There is also an auxiliary number, \kappa, that can be helpful in formulating the state.

Table V. Quantum Number Variable Mapping

Symbol

Code Variable

n_r

nr

j

Jam

l

Lam

m

jzam