On the Question of Radiative Losses in the Frame of Classic and Quantum Formalism

Abstract

The classical consideration of the radiative losses for a free dipole (with dynamics described by in the frame of harmonic oscillator (HO) equation) leads to a well-known extra term in the dynamic equation proportional to the third time derivative.

Appendices

Appendix 1: DM Versus HO Equation

Dynamics of 2-level system is described by the following system of equations:

$$ \left\{ {\begin{array}{*{20}l} {\frac{{\text {d}}{\rho_{12} }}{{\text {d}}t} - i\omega_{21}\rho_{12} + \frac{{\rho_{12} }}{{\tau_{1} }} = - \frac{{iH_{12} \left( {\rho_{22} -\rho_{11} } \right)}}{\hbar }} \hfill \\ {\frac{{\text {d}}{\rho_{22} }}{{\text {d}}t} + \frac{{\rho_{22} }}{{\tau_{2} }} = - \frac{{iH_{12} \left( {\,\rho_{12} -\rho_{12}^{ * } \,} \right)}}{\hbar } + W} \hfill \\ {\rho_{11} +\rho_{22} = 1} \hfill \\ \end{array} } \right. $$

(14.41)

Here ρ describes population (diagonal) and polarisation (non diagonal) dynamics, W is a pump rate, and H is a hamiltonian of interaction (for example, for the interaction with an external electric field it becomes $H_{12} = - \mu * E$.

Sometimes it is more convenient to reduce the dynamics to other variables (Bloch equations)

$$ \left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}Q}}{{{\text{d}}t}} + \frac{Q}{{\tau _{1} }} - i\omega _{{21}} P = \frac{{2iH_{{12}} n}}{\hbar }} \hfill \\ {\frac{{{\text{d}}P}}{{{\text{d}}t}} + \frac{P}{{\tau _{1} }} = i\omega _{{21}} Q} \hfill \\ {\frac{{{\text{d}}n}}{{{\text{d}}t}} + \frac{{1 + n}}{{\tau _{2} }} = - \frac{{2iH_{{12}} Q}}{\hbar } + 2W} \hfill \\ \end{array} } \right. $$

(14.42)

Here ${P} =\rho_{12} +\rho_{12}^{ * } ,\quad {Q} =\rho_{12} -\rho_{12}^{ * } ,\quad N =\rho_{22} -\rho_{11}$

Taking Q from the first equation and substituting it into the second one, the system in Bloch variables takes the following form

$$ \left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}^{2} P}}{{{\text{d}}t^{2} }} + \frac{2}{{\tau _{1} }}\frac{{{\text{d}}P}}{{{\text{d}}t}} + \left( {\frac{1}{{\tau _{1}^{2} }} + \omega _{{21}}^{2} } \right)P = \frac{{2\omega _{{21}} H_{{12}} n}}{\hbar }} \hfill \\ {\frac{{{\text{d}}n}}{{{\text{d}}t}} + \frac{{1 + n}}{{\tau _{2} }} = - \frac{{2H_{{12}} }}{{\hbar \omega _{{21}} }}\left( {\frac{{{\text{d}}P}}{{{\text{d}}t}} + \frac{P}{{\tau _{1} }}} \right) + 2W} \hfill \\ \end{array} } \right. $$

(14.43)

It is clear that the first equation in (14.43) is rather far from a trivial HO one. Nevertheless, if it is assumed that N is not changing too much, then the equations become identical:

$$ \frac{{{\text{d}}^{2} P}}{{{\text{d}}t^{2} }} + \frac{2}{{\tau _{1} }}\frac{{{\text{d}}P}}{{{\text{d}}t}} + \left( {\frac{1}{{\tau _{1}^{2} }} + \omega _{{21}}^{2} } \right)P = \frac{{2\omega _{{21}} H_{{12}} n_{0} }}{\hbar } $$

(14.44)

For example, in case of interaction with the electric field and low intensity (low intensity means that n can be substituted by just −1 corresponding to the situation with all molecules on the lower level ρ₁₁)

$$ \frac{{{\text{d}}^{2} P}}{{{\text{d}}t^{{\text{2}}} }} + \frac{2}{{\tau_ {1}}}\frac{{{\text{d}}P}}{{{\text{d}}t}} + \left( {\frac{1}{{\tau _{1}^{2} }} + \omega _{{21}}^{2} } \right)P = \frac{{2\omega _{{21}} \mu }}{\hbar }E $$

(14.45)

The second possibility to reduce the system to a HO equation is to assume, that due to the pump W the population difference n is kept constant. In order to prove it mathematically, we have to realize that the first term in right side of the second equation in (14.43) is a multiplication of two fast oscillating functions, while n has much slower dynamics. The math procedure is in substitution both P and H (or, in case of an electric field E) by an anzatz with fast oscillating parts, namely:

$$ \begin{aligned} E(t) = \frac{1}{2}\left( {A(t)^{*} \exp (i\omega{\kern 1pt} t) + A(t)^{*} \exp ( - i\omega{\kern 1pt} t)} \right) \hfill \\ P(t) = \frac{1}{2}\left( {p(t)^{*} \exp (i\omega{\kern 1pt} t) + p(t)^{*} \exp ( - i\omega{\kern 1pt} t)} \right) \hfill \\ \end{aligned} $$

(14.46)

and then reduce the system (14.43) into one equation for slowly varying component p. In this case we have to accept basically Slowly Varying Approximation for the slow amplitudes p and A, which gives us equation for p, but with the first derivative. In this case the system (14.43) anyway cannot be reduced to a HO equation.

But let us just assume that due to some reasons the n is kept on some constant level. In this case (14.44) becomes:

$$ \frac{{{\text{d}}^{2} P}}{{{\text{d}}t^{{\text{2}}} }} + \frac{2}{{\tau_ {1}}}\frac{{{\text{d}}P}}{{{\text{d}}t}} + \left( {\frac{1}{{\tau _{1}^{2} }} + \omega _{{21}}^{2} } \right)P = - \frac{{2\omega _{{21}} \mu n_{0} }}{\hbar }E $$

(14.47)

and positive N (inversion) should give us an amplification effect, in this case

$$ \frac{{{{\text{d}}^{2} P}}}{{{{\text{d}}}{t}^{{\text{2}}} }} + \frac{2}{{\tau_ {1}}}\frac{{{{\text{d}}P}}}{{{{\text{d}}t}}} + \left( {\frac{1}{{\tau _{1}^{2} }} + \omega _{{21}}^{2} } \right)P = - \frac{{2\omega _{{21}} \mu \left| {n_{0} } \right|}}{\hbar }E $$

(14.48)

This equation by no means can be equivalent to the equation of HO with negative absorption. Absorption remains the same (first derivative with some coefficient giving typical relaxation time), and effect of amplification is in a term in right side of (14.48), which is equivalent to a kind of external force.

• Conclusion:

• 1. An absorption coefficient in a HO equation can only have a sign, corresponding to an energy losses.

• 2. Rigorous quantum description can be reduced to a HO equation only in case of low, unsaturated losses in a passive (unpumped) quantum system.

• 3. Quantum dynamics, associated with the amplification process (14.43) can NOT be described by a HO equation with an inverted sign of losses anyway.

• 4. Equation (14.48), which is actually a HO equation with an external force, can be taken to some extend as an equation for polarizability, but even this equation can NOT be obtained through a rigorous math from the DM approach.

There are basically no reasons to use a HO equation for amplification, the system (14.41) is not too complicated and can be analyzed analytically.

Appendix 2: Maxwell Equations and Density Matrix Formalism

A starting point for the consideration is the Helmhoz equation:

$$ \begin{aligned} & \Delta A + k^{2} A = \frac{{4\pi \omega ^{2} }}{{c^{2} }}P \\ & P =\mu\left( {\left\langle {\left. {\psi_{2} } \right|\mu{\kern 1pt} A\left| {\psi_{1} } \right.} \right\rangle + \left\langle {\left. {\psi_{2} } \right|\mu{\kern 1pt} A\left| {\psi_{1} } \right.} \right\rangle^{*} } \right) \\ & H\psi = E\psi \\ & H = H_{0} +\mu{\kern 1pt} A;\quad H_{0}\psi_{k} = E_{k}\psi_{k} \\ & A = A_{\text {regular}} + A_{\text {zero}\,\text {fluctuatins}} + A_{\text {stochastic}} \Rightarrow \\ & \Rightarrow H = H_{0} +\mu\left( {A_{\text {regular}} + A_{\text {zero}\,\text {fluctuatins}} + A_{\text {stochastic}} } \right) \\ & \quad \quad \quad + V_{\text {stochastic}} = H_{0} +\mu{\kern 1pt} A_{\text {regular}} +\mu\left( {A_{\text {zero}\,\text {fluctuatins}} + A_{\text {stochastic}} } \right) \\ \end{aligned} $$

(14.49)

A_{zero fluctuations} are the zero (vacuum) fluctuations of the electric field.

The last scopes in the last equation for A are the combination of the interaction with the vacuum fluctuations and thermo bath. Both are considered as stochastic functions with zero averaged values. As for the Helmholz equation:

$$ \begin{aligned} & \Delta A + k^{2} A = \frac{{4\pi \omega ^{2} }}{{c^{2} }}P \\ & A = A_{{{\text{regular}}}} + A_{{{\text{zero}}{\kern 1pt} {\text{fluctuatins}}}} + A_{{{\text{stochastic}}}} \Rightarrow \left( {\Delta + k^{2} } \right)A_{{{\text{regular}}}} \\ & \quad \quad + \left( {\Delta + k^{2} } \right)\left( {A_{{{\text{zero}}{\kern 1pt} {\text{fluctuatins}}}} + A_{{{\text{stochastic}}}} } \right) = \frac{{4\pi \omega ^{2} }}{{c^{2} }}P \Rightarrow \\ & \left( {\Delta + k^{2} } \right)A_{{{\text{regular}}}} = \frac{{4\pi \omega ^{2} }}{{c^{2} }}P\left( {A_{{{\text{regular}}}} } \right) \\ & \left( {\Delta + k^{2} } \right)\left( {A_{{{\text{zero}}{\kern 1pt} {\text{fluctuatins}}}} + A_{{{\text{stochastic}}}} } \right) = 0 \\ \end{aligned} $$

(14.50)

Substituting A into the equation for polarization we see, that the stochastic part gets canceled:

$$ \begin{aligned} P & = \mu \left( {\left\langle {\left. {\psi _{2} } \right|\mu \left| {\psi _{1} } \right.} \right\rangle + \left\langle {\left. {\psi _{2} } \right|\mu A\left| {\psi _{1} } \right.} \right\rangle ^{*} } \right) \\ A & = A_{{{\text{regular}}}} + A_{{{\text{zero}}{\kern 1pt} {\text{fluctuatins}}}} + A_{{{\text{stochastic}}}} \Rightarrow P = \mu \left( {\left\langle {\left. {\psi _{2} } \right|A_{{{\text{regular}}}} \left| {\psi _{1} } \right.} \right\rangle + \left\langle {\left. {\psi _{2} } \right|A_{{{\text{regular}}}} \left| {\psi _{1} } \right.} \right\rangle ^{*} } \right) \\ & \quad + \mu \left( {\left\langle {\left. {\psi _{2} } \right|\left( {A_{{{\text{zero}}{\kern 1pt} {\text{fluctuatins}}}} + A_{{{\text{stochastic}}}} } \right)\left| {\psi _{1} } \right.} \right\rangle + \left\langle {\left. {\psi _{2} } \right|\left( {A_{{{\text{zero}}{\kern 1pt} {\text{fluctuatins}}}} + A_{{{\text{stochastic}}}} } \right)\left| {\psi _{1} } \right.} \right\rangle ^{*} } \right) \\ & = \mu \left( {\left\langle {\left. {\psi _{2} } \right|A_{{{\text{regular}}}} \left| {\psi _{1} } \right.} \right\rangle + \left\langle {\left. {\psi _{2} } \right|A_{{{\text{regular}}}} \left| {\psi _{1} } \right.} \right\rangle ^{*} } \right)\left( {\Delta + k^{2} } \right)A_{{{\text{regular}}}} = \frac{{4\pi \omega ^{2} }}{{c^{2} }}P\left( {A_{{{\text{regular}}}} } \right) \\ \end{aligned} $$

(14.51)

As for the Schrödinger equation, it gets transferred into the DM equations, where both $A_{\text{zero}\,{\text {fluctuatins}}}$ and $A_{\text{stochastic}}$ (i.e. interaction with vacuum fluctuations and with thermo bath) are packed into the two relaxation times τ₁ and τ₂.

$$ \begin{aligned} H\psi & = E\psi \\ H_{0} \psi _{k} & = E_{k} \psi _{k} \\ H & = H_{0} + \mu \left( {A_{{{\text{regular}}}} + A_{{{\text{zero}}{\kern 1pt} {\text{fluctuatins}}}} + A_{{{\text{stochastic}}}} } \right) \\ & = H_{0} + \mu _{{{\text{regular}}}} + \mu \left( {A_{{{\text{zero}}{\kern 1pt} {\text{fluctuatins}}}} + A_{{{\text{stochastic}}}} } \right) \\ \end{aligned} $$

$$ \left\{ {\begin{array}{*{20}l} {\frac{{{\text {d}}{p}}}{{{\text {d}}{t}}} + p\left( {\frac{1}{{\tau_{2} }} + i\left( {\omega -\omega_{21} } \right)} \right) = \frac{{i\mu^{2} A_{\text {regular}}^{*} n}}{\hbar }} \hfill \\ {\frac{{{\text {d}}{n}}}{{{\text {d}}{t}}} + \frac{{\left( {n - n_{0} } \right)}}{{\tau_{1} }} = \frac{{\,i\left( {A_{\text {regular}} p - A_{\text {regular}}^{*} p^{*} } \right)}}{2\hbar }} \hfill \\ {n = \rho_{22} - \rho_{11} } \hfill \\ {p =\mu\rho_{12} } \hfill \\ {n_{0} = \frac{{\left( {W{\tilde{\tau }}_{1} - 1} \right)}}{{\left( {W{\tilde{\tau }}_{1} + 1} \right)}}} \hfill \\ \end{array} } \right. $$

(14.52)

In DM equations the both relaxation times depend on the environment according to the local density of state which in turn depends on imaginary part of the respective Green function.