Explaining the laser’s light: classical versus quantum electrodynamics in the 1960s

Abstract

The laser, first operated in 1960, produced light with coherence properties that demanded explanation. While some attempted a treatment within the framework of classical coherence theory, others insisted that only quantum electrodynamics could give adequate insight and generality. The result was a sharp and rather bitter controversy, conducted over the physics and mathematics that were being deployed, but also over the criteria for doing good science. Three physicists were at the center of this dispute, Emil Wolf, Max Born’s collaborator on a canonical text on optics as a branch of classical electromagnetism, Roy J. Glauber, a student of Julian Schwinger and a high-energy particle theorist, and Leonard Mandel, both experimentalist and theorist and versed in the physics of photodetection. The story told here is thus one of three distinct research trajectories and of the explosion that occurred when, pushed into the well-financed field of laser studies, these trajectories collided.

Appendix

Wolf’s mutual coherence function may be written as

$$\begin{aligned} \Gamma (P_1 ,P_2 ,\tau )=\left\langle {V(P_1 ,t+\tau )V^{*}(P_2 ,t)} \right\rangle \end{aligned}$$

where the angle brackets denote an average over time. His complex degree of coherence is then obtained by normalizing this function (Wolf 1955).

$$\begin{aligned} \gamma =\frac{\Gamma (P_1 ,P_2 ,\tau )}{\sqrt{\Gamma (P_1 ,P_1 ,0)\cdot \Gamma (P_2 ,P_2 ,0)}} \end{aligned}$$

If the absolute value of $\gamma$ is unity, it is clear that $\Gamma$ factors into two independent functions, in conformity with Glauber’s prescription for first-order coherence. Wolf’s terminology differs. Since $\Gamma (P_1 ,P_2 ,\tau )$ connects two different space–time points, he calls it a second-order correlation.

In his 1957 paper on the HBT effect, Wolf calculates the correlation of the fluctuations in intensity at two space–time points. He writes the fluctuation at one of these points as

$$\begin{aligned} \Delta I=I(t)-\left\langle {I(t)} \right\rangle \end{aligned}$$

where

$$\begin{aligned} I(t)=V^{2}(t). \end{aligned}$$

The correlation of the fluctuations at two points is then

$$\begin{aligned} \left\langle {\Delta I_1 (t)\cdot \Delta I_2 (t+\tau )} \right\rangle \end{aligned}$$

For any two random functions X(t) and Y(t) that satisfy a two-dimensional Gaussian probability distribution, Wolf proves that

$$\begin{aligned} \langle \Delta X^{2}\cdot \Delta Y^{2}\rangle =2\langle XY\rangle ^{2} \end{aligned}$$

Setting ${X}={V}_{1}$ and ${Y}={V}_{2}$, Wolf is then able to equate the correlation of intensity fluctuations to $2\langle V_1 \left( t \right) V_2 \left( {t+\tau } \right) \rangle ^{2}=2\Gamma ^{2}$ (In this publication, Wolf uses the symbol J for $\Gamma$).

Glauber writes the electric field operator, $\vec {E}$, as $\mathop \int \nolimits _{-\infty }^{+\infty } \varvec{e}\left( {\omega ,{\varvec{r}}} \right) e^{-i\omega t}\hbox {d}\omega .$

Then, $\vec {E}=\int \nolimits _{-\infty }^{0}\varvec{e}\left( {\omega ,{\varvec{r}}} \right) e^{-i\omega t}\hbox {d}\omega +\int \nolimits _{0}^{+\infty }\varvec{e}\left( {\omega ,{\varvec{r}}} \right) e^{-i\omega t}\hbox {d}\omega =\vec {E}^{-}+\vec {E}^{+}$.

He cites Dirac’s Principles of Quantum Mechanics as the authority for interpreting $\hbox {E}^{+}$ as a photon annihilation operator. When it acts on a state with n photons, it produces a state with ($n-1$) photons, while $\hbox {E}^{-}$ yields an ($n+1$) state:

“the probability per unit time that a photon be absorbed by an ideal detector at point r at time t is proportional to

$$\begin{aligned} \sum \limits _f \left| \langle {f\left| {E^{+}_{\mu }\left( {\varvec{r},t} \right) } \right| i\rangle } \right| ^{2}= & {} \sum \limits _f \langle i\left| {E^{-}_{\mu }\left( {\varvec{r},t} \right) } \right| f\rangle \langle i\left| {E^{+}_{\mu }\left( {\varvec{r},t} \right) } \right| i\rangle \\= & {} \langle i\left| {E^{-}_{\mu }\left( {\varvec{r},t} \right) E^{+}_{\mu }\left( {\varvec{r},t} \right) } \right| i\rangle .\hbox {''} \end{aligned}$$

Here ${\vert }{i}>$ is the state of the initial field and ${\vert }{f}>$ is the state of the final field and the sum is over all possible final states ($\mu$ is an index that Glauber used to indicate the photon’s state of polarization) (Glauber 1963b, quotation on p. 2531).

The probability that n photons are absorbed at n photodetectors then becomes

$$\begin{aligned} \langle i | {E^{-}_{\mu }\left( \vec {r_{1}}, t_{1} \right) }\ldots E^{-}_{\mu }\left( {\varvec{r}_0 ,t_0 } \right) E^{+}_{\mu }\left( {\varvec{r}_0 ,t_0 } \right) \ldots E^{+}_{\mu }\left( {{\varvec{r}}_1 ,t_1 } \right) | i \rangle \end{aligned}$$

For the case where the initial state of the field is imperfectly known, Glauber invokes the quantum mechanical density matrix and writes $tr\left\{ {\rho E_{\mu }^{-}\left( {\varvec{r},t} \right) E_{\mu }^{+}\left( {\varvec{r},t} \right) } \right\} =G^{\left( 1 \right) }\left( {\varvec{r}t,\varvec{r}t} \right)$ for the average counting rate of an ideal photodetector. For an n photon coincidence, this becomes $G^{\left( n \right) }=tr\big \{ \rho E^{-}\left( {\varvec{r}_1 ,t_1 } \right) \ldots E^{-}\left( {\varvec{r}_n ,t_n } \right) E^{+}\left( {\varvec{r}_{n+1} ,t_{n+1} } \right) \ldots E^{+}\left( {r_{2n} ,t_{2n} } \right) \big \}.$ Like Wolf, he arrives at a definition of coherence by normalizing these correlation functions. He sets $g^{(n)}=G^{(n)}(x_1\ldots x_{2n} )/\prod \nolimits _j {\left\{ {G^{(1)}(x_j ,x_j )} \right\} ^{1/2}}$, where x stands for $\mathbf{r}$ and t, and the product is evaluated for j ranging from 1 to 2n.

“If the field in question possesses nth-order coherence, it must, therefore, have $g^{(j)}(x_1...x_j ,x_j ...x_1 )=1$ for $j\le n$. It follows from the definition of the $g^{(j)}$ that the corresponding values of the correlation functions ... factorize” (Glauber 1963b, 2534–2535).