Lasers as Quantum Technology

Conventional atomic lasers combine stimulated emission, a pumped population inversion, and optical feedback to produce light concentrated in selected modes. This page develops that standard mechanism; other gain media and specialized lasing schemes need additional models. They also became one of the main tools for making quantum mechanics experimentally precise, from laser spectroscopy and atom cooling to optical clocks, ion traps, quantum optics, and coherent control.

Helpful background. Spectroscopy explains transition frequencies and linewidths. Coherent light develops the quantum state description of a laser mode.

Stimulated Emission

An atom, molecule, ion, or solid-state emitter with upper and lower energy levels can emit radiation by two conceptually different processes. In spontaneous emission, an excited system emits without an already present resonant field. In stimulated emission, an incoming field at the transition frequency induces emission into the same optical mode.

For a transition with

$$E_2-E_1 = \hbar\omega,$$

emission into a mode with $n$ photons has a rate proportional to $n+1$: the $n$ term is the stimulated contribution and the $1$ term is the spontaneous contribution into that mode. The emitted excitation occupies the same mode; it is not a distinguishable little copy of an incoming photon. Mode-selective gain supports coherence, but this statement does not assign a definite phase to a number state or eliminate amplifier noise.

A schematic rate description uses Einstein coefficients:

$$R_{\rm abs} = B_{12}\rho(\omega)N_1, \qquad R_{\rm stim} = B_{21}\rho(\omega)N_2,$$

where $\rho(\omega)$ is the spectral energy density and $N_1,N_2$ are level populations. The details depend on degeneracy, linewidth, and the medium, but the central competition is clear: absorption scales with the lower-state population, while stimulated emission scales with the upper-state population.

A laser medium is pumped to create population inversion. Stimulated emission amplifies a selected optical mode, while the cavity provides feedback and frequency selection. Real lasers include losses, linewidths, saturation, and noise beyond this schematic.

Population Inversion

At positive temperature, each upper sublevel has a smaller thermal occupation than a lower sublevel. For two nondegenerate levels with energies $E_2>E_1$,

$$\frac{N_2}{N_1} = e^{-(E_2-E_1)/(k_BT)} < 1.$$

Such a medium absorbs more than it amplifies. Laser action requires a non-equilibrium population inversion, roughly

$$N_2>N_1$$

for a simple nondegenerate transition. For degeneracies $g_1,g_2$ with equally populated sublevels and the corresponding Einstein coefficients, $g_1B_{12}=g_2B_{21}$. The gain condition is therefore $N_2/g_2>N_1/g_1$. These rate equations describe ordinary inversion-based gain; coherent interference schemes can operate outside their assumptions.

This is why practical lasers use pumping schemes. A pump moves population into an upper manifold. In a three-level laser, fast decay feeds a long-lived upper laser level and inversion requires depopulating the ground level sufficiently. A four-level scheme instead empties the lower laser level rapidly into a separate ground level, making inversion easier. Three-level, four-level, semiconductor, fiber, gas, dye, and solid-state lasers implement this logic in different ways.

Population inversion is not a violation of thermodynamics. It is a driven non-equilibrium state maintained by an external pump and accompanied by heat, loss, and entropy production elsewhere.

Cavity and Threshold

In the conventional laser oscillator considered here, an optical cavity supplies feedback and selects modes. A gain medium without this feedback is an amplifier. In a simple two-mirror cavity of length $L$, longitudinal mode frequencies are approximately

$$\nu_q \simeq \frac{q c}{2L}, \qquad q=1,2,\ldots,$$

for vacuum propagation and idealized mirrors. The free spectral range is

$$\Delta\nu_{\rm FSR} = \frac{c}{2L}.$$

Laser threshold occurs when round-trip gain exceeds round-trip loss. A schematic condition is

$$\text{gain} \gtrsim \text{mirror loss} + \text{internal loss}.$$

Below threshold, the device may fluoresce or amplify weakly. Above threshold, stimulated emission into cavity modes dominates and the output becomes narrowband, directional, and highly coherent compared with ordinary thermal light.

The word “threshold” should not be overinterpreted as an infinitely sharp ideal transition in every device. Finite size, noise, multimode operation, gain saturation, and measurement bandwidth all matter.

Coherence

Lasers are valued because they can produce light with high temporal and spatial coherence. Temporal coherence means the phase remains predictable over a relatively long time or path difference. Spatial coherence means fields at different transverse points have stable phase relations.

In a rough first-order model, high visibility in an interferometer signals coherence:

$$I \propto 1+\mathcal V\cos\Delta\phi.$$

Lasers can have $\mathcal V$ close to one in well-controlled interferometers, but no real laser is perfectly monochromatic or noiseless. Phase noise, amplitude noise, spontaneous-emission noise, technical vibrations, thermal drift, and mode competition all affect coherence.

Quantum mechanically, an ideal coherent state is often a good model for a stabilized laser mode. But “laser light is a coherent state” is a model statement, not a universal fact about every laser under every condition. Photon statistics, linewidth, and phase reference conventions matter.

Spectroscopy and AMO Physics

Lasers transformed spectroscopy because they provided narrow, bright, tunable, and controllable radiation. Compared with lamps or discharge sources, lasers made it possible to:

• drive selected transitions with high spectral resolution;
• resolve narrow Doppler-free features with counterpropagating saturated-absorption beams;
• cool and trap atoms using momentum transfer and velocity-dependent scattering;
• manipulate internal states with pulses;
• stabilize frequencies to atomic or molecular references;
• build optical frequency combs and precision clocks;
• probe weak transitions and small shifts.

The connection to Spectroscopy is direct: line positions, line shapes, transition strengths, and selection rules became measurable with much greater control.

The connection to Atomic Beams and Interferometry is equally important. Laser beams became state selectors, beam splitters, mirrors, phase references, cooling fields, and readout tools.

Quantum Optics Bridge

Lasers helped create quantum optics as an experimental field. They made it possible to study:

• photon counting and shot noise;
• coherence functions and correlation measurements;
• squeezed light;
• cavity quantum electrodynamics;
• single-atom and single-ion control;
• nonlinear optical processes;
• entangled photon-pair generation;
• quantum-limited interferometry.

Lasers also forced a useful distinction. A strong laser field can often be treated as a classical drive in a Hamiltonian such as

$$V(t) = - \mathbf d\cdot\mathbf E_0\cos(\omega t),$$

where $\mathbf d$ is an electric-dipole operator. That semiclassical description is extremely useful for driven transitions. But questions about photon statistics, spontaneous emission, squeezing, and field quantization require a quantum field description. The right model depends on the observable.

Coherent Light gives the quantum-state version of the laser approximation, including Poisson statistics, phase diffusion, reference frames, and finite-bandwidth caveats.

Common pitfalls

Inversion is not the threshold condition by itself. Net gain must overcome cavity and internal losses. Pumping harder also introduces saturation and heating.

A laser need not emit one perfectly sharp frequency. Mode competition, phase diffusion, technical noise, and pulsed operation affect its spectrum. State the timescale over which a phase reference remains useful.

Choose the field model for the question. A strong laser can often be treated as a classical drive when calculating atomic motion or transitions. Photon statistics, squeezing, and spontaneous emission require a description of field fluctuations.

Exercises

1. A simple two-mirror cavity has length $L=0.30\,\mathrm m$. Estimate its free spectral range $\Delta\nu_{\rm FSR}=c/(2L)$.

Solution

Use $c\approx3.00\times10^8\,\mathrm{m/s}$:

$$\Delta\nu_{\rm FSR} = \frac{3.00\times10^8}{2(0.30)}\,\mathrm{Hz} = 5.0\times10^8\,\mathrm{Hz}.$$

The free spectral range is about $500\,\mathrm{MHz}$.

2. Estimate the photon energy of light with wavelength $780\,\mathrm{nm}$ using $hc\approx1240\,\mathrm{eV\,nm}$.

Solution

Use

$$E = \frac{hc}{\lambda} = \frac{1240\,\mathrm{eV\,nm}}{780\,\mathrm{nm}} \approx 1.59\,\mathrm{eV}.$$

3. Why is a two-level atom in thermal equilibrium not enough to make a laser?

Solution

In thermal equilibrium the upper level is less populated than the lower level, so absorption exceeds stimulated emission for the transition. Laser gain requires a non-equilibrium population inversion or an equivalent gain mechanism. For the usual atomic rate-equation scheme, pumping through additional levels can maintain inversion. Resonant pumping of just two levels with the same incoherent field saturates their populations instead of maintaining inversion.

4. When is it reasonable to model a laser as a classical drive?

Solution

It is often reasonable when the laser mode has a large photon occupation, stable phase relative to the experiment, and the observable of interest is a driven transition rate or coherent state evolution of matter. It is not enough for questions about photon counting statistics, squeezing, spontaneous emission, or field quantization itself.

References

• W. Demtröder, Laser Spectroscopy, Vol. 1: Basic Principles, 4th ed., Springer, 2008.
• A. Einstein, “Zur Quantentheorie der Strahlung,” Physikalische Zeitschrift 18, 121-128, 1917.
• C. J. Foot, Atomic Physics, Oxford University Press, 2005.
• T. H. Maiman, “Stimulated Optical Radiation in Ruby,” Nature 187, 493-494, 1960, DOI: 10.1038/187493a0.
• A. L. Schawlow and C. H. Townes, “Infrared and Optical Masers,” Physical Review 112, 1940-1949, 1958, DOI: 10.1103/PhysRev.112.1940.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.
• A. E. Siegman, Lasers, University Science Books, 1986.