Seventeen metres per second — but what is stopped is not light
Glass with \(n=1.5\) gives \(c/1.5\); diamond gives \(c/2.4\). So how slow can it get? The answer is "arbitrarily," and in 1999 Hau and collaborators brought light down to 17 metres per second in a sodium Bose–Einstein condensate. By 2001 they had stopped it completely and read it back out.
There is one trap here and one gift. The trap: slowing light a great deal needs steep dispersion, and steep dispersion was supposed to come packaged with strong absorption (Episode 5). How was that avoided? The gift: once you see how, you see that Episode 5's Kramers–Kronig relations do not forbid slow light — they demand it. And finally, one thing that has to be said honestly — what is stopped is not light.
A wave packet travels at the group velocity, and when the index depends on frequency,
$$v_g=\frac{c}{n_g},\qquad n_g=n+\omega\frac{dn}{d\omega}$$Glass's \(n=1.5\) is the first term. But the second term, \(\omega\,dn/d\omega\), can be made arbitrarily large if the index changes sharply.
Getting 17 m/s requires a group index of \(n_g=c/17\approx1.8\times10^7\) — ten million times glass. Not by making \(n\) itself eighteen million, of course, but by harvesting slope.
Steep \(dn/d\omega\) requires a sharp resonance. And Kramers–Kronig says a sharp resonance brings sharp absorption with it.
Approach the resonance naively and the light is absorbed before it can be slowed. That is why slow light was hard until the 1990s.
The way around was proposed by Harris in 1990 and demonstrated by Boller, Imamoğlu and Harris in 1991: electromagnetically induced transparency (EIT).
Take an atom with two lower levels and one upper level — a "lambda" configuration — and keep a strong control beam shining on it. The signal beam now has two routes to the upper level.
The two routes cancel exactly out of phase, so absorption vanishes right at the centre of the resonance. The atom falls into a superposition called the "dark state," which does not couple to the light.
Result: a narrow transparent window opens in the middle of an opaque absorption line.
What does Kramers–Kronig say when you punch a narrow hole in the absorption?
The integral \(n(\omega)-1=\frac{2}{\pi}\mathrm{P}\int\frac{\omega'\kappa(\omega')}{\omega'^2-\omega^2}d\omega'\) translates a sharp change in \(\kappa\) into a sharp change in \(n\). The narrower the hole, the steeper \(dn/d\omega\) inside it.
So slow light does not violate Kramers–Kronig. Kramers–Kronig produces it. The narrowness of the absorption window is the size of the group index.
A quick estimate — with control Rabi frequency \(\Omega_c\) and upper-level decay \(\Gamma\),
window width \(\ \sim\ \Omega_c^2/\Gamma\), group index \(\ n_g\ \propto\ 1/\Omega_c^2\).
Weaken the control beam and the window narrows and the light slows. These are the same single knob.
| Year | System | Group velocity | Note |
|---|---|---|---|
| 1999 | sodium Bose–Einstein condensate (435 nK) | 17 m/s | Hau, Harris et al.; \(n_g\approx1.8\times10^7\) |
| 1999 | hot rubidium vapour | 90 m/s | Kash et al.; no cooling needed |
| 1999 | room-temperature rubidium vapour | 8 m/s | Budker et al. |
| 2001 | sodium condensate / rubidium vapour | 0 | stopped completely and read back out |
| 2013 | Pr:YSO crystal (a solid) | 0 | storage time one minute |
A condensate is not required. Room-temperature rubidium vapour reaches 8 m/s. What the condensate bought was the absence of Doppler broadening, which let the window be made narrower.
"Light was stopped" is not accurate. Here is what happens.
What travels through the medium is neither light nor atoms but a mixture:
$$\Psi=\cos\theta\cdot\underbrace{E}_{\text{light}}\ -\ \sin\theta\cdot\underbrace{S}_{\text{atomic coherence}}, \qquad \tan^2\theta\ \propto\ \frac{1}{\Omega_c^2}$$and it travels at \(v=c\cos^2\theta\). Weaken the control field \(\Omega_c\) and \(\theta\to90^\circ\), so \(\Psi\) becomes almost entirely atomic.
Take \(\Omega_c\) to zero and \(v=0\). It stops. But at that moment \(\cos\theta=0\) — the light component is zero. What is there is nothing but a phase pattern in the atomic spins.
Bring the control field back and \(\theta\) returns, writing light back out of the atomic coherence. You deposited it and withdrew it.
Not a lie, but the subject is wrong.
• What is stopped: the atomic coherence (not photons).
• What is preserved: the quantum state of the light itself — amplitude, phase, even a single photon's quantum superposition, all stored faithfully and retrieved faithfully.
So "the information carried by the light was stored" is entirely correct, while "a photon is sitting still in there" is wrong. In this collection's phrasing, this is not "the same thing" but "a faithful copy."
Below are EIT's absorption (left) and refractive index (right). Grey is the control beam switched off — an ordinary absorption line. Weaken the control beam and the window narrows, the slope inside it steepens, and the light slows. All three from one knob.
Narrow the window and the light slows. But a narrow window means —
light that does not fit inside it cannot get through. If the pulse's bandwidth exceeds the window, the overflowing components are absorbed or distorted. The window \(\sim\Omega_c^2/\Gamma\) narrows at exactly the rate \(n_g\propto1/\Omega_c^2\) grows, so
(delay) × (usable bandwidth) ≈ constant
This is the delay–bandwidth product, bounded only by the medium's optical depth. Holding a short pulse for a long time is impossible in principle. Put physically — the pulse has to be shorter than the delay it receives, or it will not fit inside the medium.
Established: the group index \(n_g=n+\omega\,dn/d\omega\); the EIT proposal (Harris 1990) and demonstration (Boller–Imamoğlu–Harris 1991); the lambda-system dark state and transparency by path interference; 17 m/s in a sodium condensate (Hau et al., Nature 397, 594, 1999), 8–90 m/s in rubidium vapour, and the 2001 storage and revival of light; the dark-state polariton description (Fleischhauer–Lukin 2000) with \(v=c\cos^2\theta\); the delay–bandwidth product bounded by optical depth; one-minute optical storage in Pr:YSO (2013); six-hour spin coherence in Eu:YSO (2015). All peer-reviewed results.
Caveats: (1) The subject of "light was stopped" is not the photon. What is stored is atomic coherence; accurately, the quantum state of the light is faithfully copied and faithfully restored. (2) The figure uses the weak-probe linear susceptibility of a lambda system; strong pulses, multilevel structure, Doppler broadening and atomic motion are not included. (3) The physical constants in the figure are representative values for the sodium D₂ line (\(\Gamma/2\pi\approx9.8\) MHz, \(\lambda=589\) nm) with a representative optical depth and medium length; they do not reproduce any particular experiment. The "Hau 1999" button marks the knob position that lands near 17 m/s; it is not a replication of the paper. (4) The group velocity is not the signal velocity. The front is always \(c\), in EIT as elsewhere. (5) Six hours is a spin coherence time, not a light storage time (as noted in the text). (6) "Constant" in the delay–bandwidth product is an order-of-magnitude estimate; the prefactor differs from system to system.
The group velocity is set by the second term of \(n_g=n+\omega\,dn/d\omega\). The \(n_g\approx1.8\times10^7\) needed for 17 m/s comes from slope, not from the value of the index. But steep slope was supposed to come with strong absorption (Episode 5) — and what broke that was EIT. Path interference into a control-field dark state opens a narrow transparent window right in the middle of the absorption line.
And here is the beautiful part — once the window is open, Kramers–Kronig forces a steep \(dn/d\omega\) inside it. Slow light does not defy causality; causality makes it. The window width \(\sim\Omega_c^2/\Gamma\) and the group index \(n_g\propto1/\Omega_c^2\) are one knob, which is why delay × bandwidth is fixed (a short pulse cannot be held long).
And the honest line — what is stopped is not light. What travels is the dark-state polariton \(\Psi=\cos\theta\,E-\sin\theta\,S\), at \(v=c\cos^2\theta\). Switch off the control beam to stop it and \(\cos\theta=0\) — the light component is zero, and what remains is atomic coherence. What is preserved is the light's quantum state, and it comes back out faithfully. So "the information was stored" is right; "a photon is sitting still" is wrong.
Print / PDF: ⌘+P (Ctrl+P on Windows). Weaken the control beam on screen and watch the transparent window narrow, the index slope steepen and the group velocity fall — three things on one knob. The "Hau 1999" button jumps to around 17 m/s. "Show answer" reveals the solutions.