Case Study 1: The Laser — Coherent States in Action

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Case Study 1: The Laser — Coherent States in Action

The Physical Situation

The laser (Light Amplification by Stimulated Emission of Radiation) is the most important light source of the modern era. From fiber-optic telecommunications to eye surgery, from barcode scanners to gravitational wave detection, lasers are ubiquitous precisely because they produce light with extraordinary coherence — a nearly perfect classical electromagnetic wave.

But what exactly is the quantum state of laser light? The naive answer — "a Fock state $|n\rangle$ with a very large $n$" — is wrong. A Fock state has zero mean electric field, completely uncertain phase, and sub-Poissonian statistics. Laser light, by contrast, has a well-defined amplitude and phase, oscillating electric field, and (very nearly) Poissonian photon statistics. The correct description is a coherent state $|\alpha\rangle$ — the quantum state Glauber introduced in 1963 specifically to describe laser radiation.

This case study traces the physics from Einstein's prediction of stimulated emission (1917) through the quantum description of a laser operating above threshold.

Setting Up the Model

Einstein's A and B Coefficients

Einstein's 1917 analysis of thermal radiation introduced three processes for the interaction of light with a two-level atom (ground state $|g\rangle$, excited state $|e\rangle$, energy gap $\hbar\omega$):

Absorption: An atom in $|g\rangle$ absorbs a photon and transitions to $|e\rangle$. Rate: $B_{ge}\,\rho(\omega)$, where $\rho(\omega)$ is the spectral energy density.
Stimulated emission: An atom in $|e\rangle$, in the presence of a photon of frequency $\omega$, emits an additional identical photon and drops to $|g\rangle$. Rate: $B_{eg}\,\rho(\omega)$.
Spontaneous emission: An atom in $|e\rangle$ spontaneously emits a photon and drops to $|g\rangle$, even in the absence of any external field. Rate: $A_{eg}$.

Einstein showed that thermal equilibrium (Planck distribution) requires $B_{ge} = B_{eg} \equiv B$ and:

$$\frac{A}{B} = \frac{\hbar\omega^3}{\pi^2 c^3}$$

Critically, spontaneous emission is required — without it, the Planck distribution cannot be derived. Einstein recognized this but could not explain why an excited atom emits spontaneously. The answer, as we now know, is vacuum fluctuations of the quantized electromagnetic field (Section 27.1).

The Laser: Population Inversion and Gain

For stimulated emission to dominate over absorption, we need more atoms in $|e\rangle$ than in $|g\rangle$ — a population inversion. This never occurs in thermal equilibrium (the Boltzmann distribution always favors the lower state), so lasers require an external pump to maintain the inversion.

The quantum-mechanical description of a laser above threshold gives the steady-state intracavity field as a coherent state. The argument, due to Glauber, Sudarshan, and Scully, proceeds in three steps:

Step 1: Gain medium creates photons. Each stimulated emission event adds a photon to the cavity mode coherently — in phase with the existing field. This is the quantum description of amplification.

Step 2: Cavity selects a single mode. The optical resonator (two mirrors separated by distance $L$) supports only modes with frequencies $\omega_q = q\pi c/L$. The gain medium amplifies only the mode(s) near the atomic transition frequency. A well-designed laser oscillates in a single longitudinal mode.

Step 3: Steady state is a coherent state. In steady state, gain equals loss, and the intracavity field is described by a coherent state $|\alpha\rangle$ whose amplitude satisfies:

$$|\alpha|^2 = \bar{n} = \frac{g - \ell}{\ell}\,n_{\text{sat}}$$

where $g$ is the single-pass gain, $\ell$ is the single-pass loss (mirror transmission, scattering, absorption), and $n_{\text{sat}}$ is the saturation photon number. Above threshold ($g > \ell$), the mean photon number is macroscopic: $\bar{n} \sim 10^{8}$–$10^{12}$ for typical lasers.

The Analysis

Why Coherent, Not Fock?

The key question is: why does the laser produce a coherent state rather than a Fock state?

The answer involves phase. Each stimulated emission event coherently adds a photon in phase with the existing field. If the field is in a coherent state $|\alpha\rangle$, the addition of one photon (by $\hat{a}^\dagger$) followed by amplitude stabilization through gain saturation keeps the state coherent. The phase of $\alpha$ is set by the initial spontaneous emission event that seeds the laser — it is random but then locked.

A Fock state has completely uncertain phase. If you tried to build up a Fock state by adding photons one at a time, each with a random phase, you would not get $|n\rangle$ — you would get a statistical mixture described by a density matrix diagonal in the Fock basis. The coherent state arises because stimulated emission adds photons in phase, producing the specific superposition of Fock states that defines $|\alpha\rangle$.

Mathematically: the laser above threshold is described (to good approximation) by the density matrix $\hat{\rho} = |\alpha\rangle\langle\alpha|$, not $|n\rangle\langle n|$. The phase is well-defined (though it diffuses slowly due to spontaneous emission — the Schawlow-Townes linewidth).

Measuring the Coherent State Properties

Poisson statistics. A photon-counting experiment on laser light measures the photon number distribution $P(n) = e^{-\bar{n}}\bar{n}^n/n!$. The variance equals the mean: $(\Delta n)^2 = \bar{n}$. This was first verified by Arecchi, Berné, and Sona in 1966, who measured $P(n)$ for a single-mode He-Ne laser and confirmed Poissonian statistics to high precision.

First-order coherence. The first-order correlation function $|g^{(1)}(\tau)|$ remains close to unity for delays $\tau$ much shorter than the coherence time $\tau_c \sim 1/\Delta\nu$, where $\Delta\nu$ is the laser linewidth. For a stabilized single-mode laser, $\tau_c$ can exceed seconds — coherence lengths of hundreds of kilometers. This extraordinary coherence is what makes laser interferometry (including LIGO) possible.

Second-order coherence. The second-order correlation function $g^{(2)}(0) = 1$ for an ideal coherent state. Experimentally, lasers operating well above threshold achieve $g^{(2)}(0) = 1.00 \pm 0.01$, confirming the coherent state model. Near threshold, excess noise from spontaneous emission produces $g^{(2)}(0) > 1$ (partial bunching).

The Schawlow-Townes Linewidth

Even an ideal single-mode laser is not perfectly monochromatic. Spontaneous emission into the lasing mode introduces random phase kicks that cause the phase of $\alpha$ to diffuse. The resulting Schawlow-Townes linewidth is:

$$\Delta\nu_{\text{ST}} = \frac{\pi h\nu(\Delta\nu_c)^2}{P_{\text{out}}}$$

where $\Delta\nu_c$ is the "cold cavity" linewidth (cavity decay rate divided by $2\pi$) and $P_{\text{out}}$ is the output power.

For a 5 mW He-Ne laser at 632.8 nm with a cavity finesse of $\sim 100$ and cavity length $L = 30$ cm: $\Delta\nu_c \approx 5$ MHz, and $\Delta\nu_{\text{ST}} \approx 10^{-3}$ Hz — a fantastically narrow line. In practice, technical noise (vibrations, thermal fluctuations) broadens the linewidth to $\sim 1$ MHz, but the Schawlow-Townes limit represents the fundamental quantum noise floor.

Connection to Main Chapter

This case study illustrates several key ideas from the chapter:

Coherent states as the natural description of laser light — not Fock states, not thermal states.
The role of vacuum fluctuations — spontaneous emission (driven by vacuum fluctuations) seeds the laser, sets the random phase, and ultimately limits the linewidth.
Photon statistics as a diagnostic — $g^{(2)}(0) = 1$ (Poissonian) confirms coherent state; deviations reveal the quantum nature of the light source.
The bridge from quantum to classical — a coherent state with $\bar{n} \sim 10^{12}$ has relative fluctuations $\Delta n/\bar{n} \sim 10^{-6}$, making it essentially classical. The laser is the device that converts quantum vacuum fluctuations into macroscopic coherent light.

Further Questions

Laser threshold. Below threshold, the laser output is essentially amplified spontaneous emission — thermal-like light with $g^{(2)}(0) = 2$. At threshold, $g^{(2)}(0)$ drops from 2 to 1. Describe qualitatively how this transition occurs.
Amplitude squeezing. A laser can be made to produce sub-Poissonian light ($\Delta n < \sqrt{\bar{n}}$, hence $g^{(2)}(0) < 1$) by stabilizing the pump intensity. This is called an "amplitude-squeezed" laser. What is the relationship between this and the squeezed states of Section 27.5?
Microlasers. A microlaser consists of a single atom coupled to a high-finesse optical cavity. The output can exhibit antibunching ($g^{(2)}(0) < 1$) — photons are emitted one at a time. Is this still a "laser" in the Glauber sense? What quantum state describes the output?
Laser linewidth and quantum information. Why does the Schawlow-Townes linewidth matter for quantum key distribution protocols that use coherent states (e.g., the BB84 protocol with weak coherent pulses)?