How Light Can Cool Atoms: Momentum Bookkeeping and the Doppler Limit — Epoche B2
Introduction: A Counter-Intuitive Claim Everyday experience suggests that shining light on an object can only warm it up. Sunlight heats a roof; a laser can cut steel. Yet since the 1980s, physicists have used laser light to cool clouds of atoms to temperatures below one thousandth of a kelvin (Chu, 1998). The apparent paradox dissolves once one remembers what temperature is. For a dilute gas, temperature is not a substance an object contains but a measure of the spread of its particles' velocities: $\tfrac{3}{2}k_B T = \tfrac{1}{2}m\langle v^2\rangle$, where $k_B$ is Boltzmann's constant, the conversion factor between temperature and energy per particle. To cool a gas is therefore to narrow that spread — to slow the fast atoms — and slowing something is a matter of momentum, not of heat. This essay explains the causal mechanism: cooling works not by transferring energy away in the ordinary sense, but by careful momentum bookkeeping . The push a single photon gives Light carries momentum. A photon of wavelength $\lambda$ carries $p = \hbar k$, where $k = 2\pi/\lambda$ is the wavenumber (the number of radians of wave per metre) and $\hbar$ is Planck's constant divided by $2\pi$. When an atom absorbs a photon it absorbs that momentum too, and recoils. The size of the kick is worth computing, because it determines everything that follows. For sodium, whose strong absorption line lies at $\lambda = 589\,\mathrm{nm}$, we have $k = 2\pi/\lambda = 1.07\times10^{7}\,\mathrm{m^{-1}}$, so $\hbar k = 1.13\times10^{-27}\,\mathrm{kg\,m\,s^{-1}}$. A sodium atom has mass $m = 3.8\times10^{-26}\,\mathrm{kg}$, so one photon changes its velocity by $$v_r = \frac{\hbar k}{m} \approx 3\,\mathrm{cm\,s^{-1}}.$$ At room temperature the same atom moves at about $570\,\mathrm{m\,s^{-1}}$. One photon is thus almost nothing — but an atom can absorb and re-emit again and again. The cycle time is set by the excited state's lifetime, $16\,\mathrm{ns}$ for sodium, so an atom driven hard scatters photons at up to about $3\times10^{7}$ per second. Twenty thousand kicks bring a room-temperature sodium atom to rest, and twenty thousand kicks take under a millisecond. The corresponding deceleration, roughly $10^{6}\,\mathrm{m\,s^{-2}}$, is some $10^{5}$ times the acceleration of gravity. Momentum, unlike heat, can be transferred by light with brutal efficiency. The difficulty is that this force pushes; it does not sort. A beam that slows an atom coming towards it accelerates one moving away. What is needed is a force that always opposes the motion — friction — and friction requires the atom's response to depend on its velocity. Cause: Directional Absorption via the Doppler Effect The velocity dependence is supplied by the Doppler effect, the same shift that raises the pitch of an approaching siren. An atom moving with velocity $v$ towards a beam of angular frequency $\omega_L$ sees it shifted up to $\omega_L + kv$. Atoms absorb strongly only near their resonance frequency $\omega_0$, the frequency matching the energy gap between two of their internal states; the sharpness of that resonance is set by the natural linewidth $\Gamma$, the reciprocal of the excited-state lifetime, which for sodium is $\Gamma = 2\pi \times 9.8\,\mathrm{MHz}$. The key trick, proposed by Hänsch and Schawlow (1975) for neutral atoms and independently by Wineland and Dehmelt (1975) for trapped ions, is to tune the laser slightly below resonance — "red detuning", $\delta = \omega_L - \omega_0 Effect: A Viscous Force The excited atom then returns to its ground state by spontaneous emission , radiating a photon in a direction unrelated to that of the beam. One correction to the usual shorthand is due here: this emission is not strictly isotropic — an atomic dipole radiates in a doughnut-shaped pattern, not equally in all directions — but the pattern is symmetric under reversal, so opposite directions are equally likely and the recoils average to zero over many cycles. That asymmetry between the two halves of the cycle is the whole mechanism: absorption is directional, emission is not. Now shine two identical counter-propagating beams along one axis. Each pushes the atom towards the other, and for a slowly moving atom the imbalance grows linearly with velocity. Expanding the scattering rate to first order in $v$ gives a force proportional to velocity and opposed to it: $$F \approx -\alpha v, \qquad \alpha = -4\hbar k^2 s_0 \, \frac{2\delta/\Gamma}{\left[1 + s_0 + (2\delta/\Gamma)^2\right]^2} > 0 \text{ for } \delta where $s_0$ is the saturation parameter — the beam's intensity in units of the intensity at which the transition begins to be driven as hard as it can be. Read the expression physically: the factor $\hbar k^2$ combines the momentum per photon with the Doppler sensitivity $kv$; the factor $2\delta/\Gamma$ is the detuning measured in linewidths, and it carries the minus sign that makes $\alpha$ positive only for red detuning; and the squared bracket in the denominator is the usual resonance lineshape, which suppresses the force when the light is either far off resonance or so intense that the atom is already scattering as fast as it can. The atom moves as if through a thick liquid — hence "optical molasses", first realised in three dimensions by Chu and colleagues in 1985. Note that molasses damps velocity but does not confine position: it is a viscosity, not a trap, and adding a magnetic field gradient to make the scattering position-dependent as well was the further step that produced the magneto-optical trap (Raab et al., 1987). Where, then, does the energy go? Not directly into the light beams' balance sheet as work, since the average force on a stationary atom vanishes. It leaves in the frequency of the scattered photons. The atom absorbs at the red-detuned laser frequency $\omega_0 + \delta$ but re-emits, on average, at its own resonance $\omega_0$, because spontaneous emission is governed by the atom's internal energy gap and not by the