AC Stark Shift and Dressed States

A two-level atom left alone has two sharp energy levels; shine a near-resonant laser on it and those levels move, split, and stop being the atom's own states at all. This is the dressed-atom picture, where the quantum mechanics of a driven two-state system — the same $2\times2$ problem that governs spin-½ in a rotating field — meets the electromagnetic reality of an intense light beam. The prerequisites supply the pieces: the electric-dipole approximation gives the coupling $-\mathbf d\cdot\mathbf E$ , the transition dipole moment $d_{eg}$ its strength, and the natural linewidth $\Gamma$ tells us when a shift is a shift and not just a blur. This article assembles them into the coupled atom-plus-photon system and asks for its eigenstates. Two things happen when the coupling turns on. The levels repel: each is pushed away from the state it couples to, by an amount that far off resonance is second-order in the field — the AC Stark shift, or light shift, the mechanism behind every optical dipole trap and every optical-lattice clock's biggest systematic. And near resonance the levels genuinely mix: the stationary states become superpositions of ground and excited — the dressed states — split by the generalized Rabi frequency $\Omega'=\sqrt{\Omega^2+\delta^2}$ . Diagonalize the driven Hamiltonian once, in a frame rotating with the laser, and both shift and splitting fall out of the same eigenvalue problem — the one the Autler-Townes doublet and Mollow triplet make visible.

Key takeaways

A strong near-resonant field mixes an atom's ground and excited states into new eigenstates of the coupled atom-field system, the dressed states, split by $\Omega'=\sqrt{\Omega^2+\delta^2}$ .
The AC Stark (light) shift is each level's displacement from its bare energy; far off resonance it scales as $\hbar\Omega^2/4\delta$ — linear in intensity and sign-flipped by the detuning.
Red detuning ( $\delta<0$ ) lowers the ground level where the light is brightest: this is why a focused red beam traps atoms, the optical dipole trap.
At resonance the dressed states are $(|e\rangle\pm|g\rangle)/\sqrt2$ split by the bare Rabi energy $\hbar\Omega$ ; the Autler-Townes doublet and Mollow triplet are direct images of the splitting.
Whether the doublet is resolved is a race between $\Omega'$ and the natural linewidth $\Gamma$ : only for $\Omega'\gg\Gamma$ do the sidebands separate.
Key idea: diagonalizing the driven two-level Hamiltonian in the rotating frame replaces the bare states with dressed states split by $\hbar\Omega'=\hbar\sqrt{\Omega^2+\delta^2}$ , and each level's displacement from its bare energy is the AC Stark shift.

The core insight

Diagonalizing the driven two-level Hamiltonian in the rotating frame replaces bare atomic states with dressed states split by $\hbar\Omega'=\hbar\sqrt{\Omega^2+\delta^2}$ ; the shift of each level relative to its bare energy is the AC Stark shift.

Where this lives today

Dressed states and the light shift are laboratory infrastructure. Every optical dipole trap and lattice is a landscape of AC Stark shifts: a far-red-detuned laser sculpts a potential $U_{\mathrm{dip}}\propto I/\delta$ whose valleys hold neutral atoms, and retro-reflected beams pin a quantum gas into a crystal of light — the platform for quantum simulation, neutral-atom computing, and the coldest matter ever made. The same physics is the enemy in the best clocks: an optical-lattice clock's trapping light shifts the two clock levels unequally, so metrologists tune the lattice to a magic wavelength where the differential shift vanishes, then chase residual hyperpolarizability shifts toward fractional uncertainties below $10^{-18}$ . And the splitting is read out spectroscopically every day: Autler-Townes splitting calibrates Rabi frequencies and underlies electromagnetically induced transparency, while the Mollow triplet — first seen in atomic beams in the 1970s, now routine in trapped ions and quantum dots — remains the cleanest confirmation that a strongly driven emitter genuinely lives in its dressed basis.

The physical picture

Picture two coupled pendulums. Identical and uncoupled, they swing at one frequency — a degeneracy. Add a spring and it splits into in-phase and out-of-phase normal modes, one raised and one lowered. A driven atom is exactly this: the laser is the spring, the "two pendulums" are the ground state with $n$ photons and the excited state with $n-1$ , near-degenerate close to resonance. The coupling lifts the degeneracy, and the true stationary states — the normal modes — are no longer bare atomic states but symmetric and antisymmetric mixtures: the dressed states. Level repulsion is the whole story. Two coupled levels push apart, each away from its partner: the ground level down when tuned below the transition ( $\delta<0$ , red), up when above ( $\delta>0$ , blue). Far off resonance the mixing is negligible but the repulsion survives as a second-order shift $\sim\Omega^2/4\delta$ — the light shift. Because $\Omega^2$ tracks the local intensity, a level's energy becomes a map of the light field: bright regions are valleys or hills by the sign of $\delta$ . The first figure plots the eigenvalues against detuning — crossing, repulsion, and minimum gap $\hbar\Omega$ in one picture.

Figure

Dressed-state energies versus detuning for Ω = 2π × 30 MHz. The bare rotating-frame levels ∓ħδ/2 would cross at δ = 0, but the coupling repels them into an avoided crossing of minimum gap ħΩ = h × 30 MHz; far off resonance the dressed branches ± ½√(Ω² + δ²) hug the bare lines

In the fully quantized picture the field is a ladder of photon-number states, each rung repeating the same doublet — the dressed ladder. A weak probe scanning a shared level then sees the two dressed levels: the Autler-Townes doublet. And an atom decaying between adjacent doublets emits at three frequencies — the drive $\omega$ and two sidebands at $\pm\Omega'$ — the Mollow triplet. Both resolve only when the splitting beats the natural linewidth, $\Omega'\gg\Gamma$ ; otherwise the peaks smear into one broadened line. The second figure shows that spectrum.

Figure

The Mollow triplet for Ω = 2π × 30 MHz and δ = 2π × 40 MHz, so Ω′ = 2π × 50 MHz. A strongly driven atom emits at the drive frequency ω and at two sidebands displaced by ± Ω′ = ± 2π × 50 MHz, with integrated intensities in the ratio 1 to 3 to 1 — the direct fingerprint of the dressed doublet

The core equations

The starting point is a two-level atom — $|g\rangle$ and $|e\rangle$ split by $\hbar\omega_0$ — coupled to a classical field through the electric-dipole interaction. The bare atomic term plus the time-dependent coupling make the full Hamiltonian.

H = \frac{\hbar\omega_0}{2}\sigma_z - \mathbf{d}\cdot\mathbf{E}_0\cos(\omega t) \qquad (1)

where: $\omega_0$ — bare transition frequency; $\sigma_z$ — Pauli operator, $+1$ for $|e\rangle$ , $-1$ for $|g\rangle$ ; $\mathbf{d}$ — electric-dipole operator; $\mathbf{E}_0$ — field amplitude vector; $\omega$ — laser frequency. Physically: a two-level splitting plus a term shaking the atom at the optical frequency $\omega$ . Two numbers control everything: the Rabi frequency $\Omega$ packages the coupling strength, the detuning $\delta$ how far the laser sits from resonance. They replace the field amplitude and frequency below.

\Omega \equiv \frac{d_{eg}E_0}{\hbar},\qquad \delta \equiv \omega - \omega_0 \qquad (2)

where: $\Omega$ — (bare) Rabi frequency; $d_{eg}=\langle e|\hat{\mathbf d}\cdot\hat{\boldsymbol\epsilon}|g\rangle$ — transition dipole matrix element; $E_0$ — field amplitude; $\delta$ — detuning ( $\delta>0$ blue, $\delta<0$ red). Since $\Omega\propto E_0$ and $I\propto E_0^2$ , $\Omega^2\propto I$ — the hinge that makes the light shift linear in intensity. Equation (1) is time-dependent. Rotating to the laser frame and dropping the fast counter-rotating term (the rotating-wave approximation) collapses it to a static $2\times2$ matrix — what we actually diagonalize.

H_{\mathrm{RWA}} = \frac{\hbar}{2}\begin{pmatrix} -\delta & \Omega \\ \Omega & \delta \end{pmatrix} \qquad (3)

where: $H_{\mathrm{RWA}}$ — time-independent rotating-frame Hamiltonian in the basis $\{|g\rangle,|e\rangle\}$ ; the diagonal $\mp\hbar\delta/2$ are the bare levels in that frame, the off-diagonal $\hbar\Omega/2$ the coupling. It is now a Hermitian $2\times2$ eigenvalue problem, like a spin in a static field. Diagonalizing (3) gives two eigenvalues symmetric about the rotating-frame origin, split by $\Omega'$ — the dressed-state energies.

E_\pm = \pm\frac{\hbar}{2}\sqrt{\Omega^2+\delta^2} = \pm\frac{\hbar\Omega'}{2} \qquad (4)

where: $E_\pm$ — dressed eigenvalues about the mid-point; $\Omega'\equiv\sqrt{\Omega^2+\delta^2}$ — generalized Rabi frequency. The gap $E_+-E_-=\hbar\Omega'$ is never below $\hbar\Omega$ (reached at resonance) — the avoided crossing of the first figure. The eigenvectors are $\delta$ -dependent mixtures of the bare states, set by one mixing angle $\theta$ — where the states become genuinely dressed.

|+\rangle = \cos\tfrac{\theta}{2}|e\rangle + \sin\tfrac{\theta}{2}|g\rangle,\quad |-\rangle = -\sin\tfrac{\theta}{2}|e\rangle + \cos\tfrac{\theta}{2}|g\rangle \qquad (5)

where: $|\pm\rangle$ — dressed eigenstates; $\theta$ — mixing angle, $\tan\theta=\Omega/\delta$ , $0<\theta<\pi$ . At resonance $\theta=\pi/2$ and the states are half-and-half; far off resonance $\theta\to0$ or $\pi$ and they relax into nearly pure bare states. Subtracting each dressed energy from the bare level it descends from gives the AC Stark shift — the topic's working result, exact and far-detuned forms together.

\Delta E_{\mathrm{light}} = \frac{\hbar}{2}\left(\sqrt{\Omega^2+\delta^2}-|\delta|\right)\;\xrightarrow{\ \Omega\ll|\delta|\ }\;\frac{\hbar\Omega^2}{4|\delta|} \qquad (6)

where: $\Delta E_{\mathrm{light}}$ — displacement of a level from its bare energy. Excited and ground shift oppositely; the ground shift $+\hbar\Omega^2/4\delta$ is negative for red detuning ( $\delta<0$ ), positive for blue. Because $\Omega^2\propto I$ , the ground-state light shift is a potential proportional to intensity — a real conservative force, the optical dipole potential.

U_{\mathrm{dip}}(\mathbf r) = \frac{\hbar\Omega^2(\mathbf r)}{4\delta} = \frac{3\pi c^2}{2\omega_0^3}\,\frac{\Gamma}{\delta}\,I(\mathbf r) \qquad (7)

where: $U_{\mathrm{dip}}$ — conservative light-shift potential; $\Gamma$ — natural linewidth; $I(\mathbf r)$ — local intensity. The potential scales as $1/\delta$ while residual scattering scales as $1/\delta^2$ , so large detuning buys deep traps with little heating. Finally, the dressed splitting is directly observable: fluorescence sidebands and Autler-Townes components sit a full generalized Rabi frequency from the drive.

\omega_{\pm} = \omega \pm \Omega',\qquad \text{peaks at } \ \omega,\ \omega\pm\Omega' \qquad (8)

where: $\omega_\pm$ — Mollow sideband frequencies; a central peak at $\omega$ plus sidebands at $\pm\Omega'$ . In a probe experiment the same $\Omega'$ is the Autler-Townes splitting — the spacing directly reads out $\sqrt{\Omega^2+\delta^2}$ .

Derivation AFrom the driven atom to the dressed eigenstates

The goal is to travel from the time-dependent Hamiltonian (1) to the dressed energies (4) and states (5), making explicit the two moves textbooks often blur: transforming into the rotating frame to kill the explicit time dependence, then diagonalizing the static $2\times2$ matrix. Getting this once, cleanly, lets every later result — light shift, Mollow sidebands, Autler-Townes doublet — be read off the eigenvalues rather than re-derived.

Step 1. Expand the state as $|\psi\rangle = c_g\,e^{+i\omega t/2}|g\rangle + c_e\,e^{-i\omega t/2}|e\rangle$ , pulling the rapid optical oscillation into the phases. Why: this frame is chosen so a laser on resonance leaves the amplitudes $c_g,c_e$ constant — only the slow physics of detuning and coupling survives.

Step 2. Insert into the Schrödinger equation, write $\cos\omega t=\tfrac12(e^{i\omega t}+e^{-i\omega t})$ , and keep only the term whose phase nearly cancels the atomic phase, dropping the piece at $\omega+\omega_0$ . Why: the counter-rotating term rotates about twice as fast as everything else and averages to nearly zero over a cycle; discarding it is the rotating-wave approximation, valid when $\Omega,|\delta|\ll\omega_0$ .

Step 3. What survives is the static Hamiltonian (3), with characteristic equation $\big(-\tfrac{\hbar\delta}{2}-E\big)\big(\tfrac{\hbar\delta}{2}-E\big)-\big(\tfrac{\hbar\Omega}{2}\big)^2=0$ . Why: the time-dependent atom is now a Hermitian matrix — finding its stationary states is ordinary linear algebra.

Step 4. This gives $E^2=\tfrac{\hbar^2}{4}(\delta^2+\Omega^2)$ , hence

E_\pm=\pm\frac{\hbar}{2}\sqrt{\Omega^2+\delta^2}=\pm\frac{\hbar\Omega'}{2}.

Why: the roots are symmetric about the rotating-frame zero, and their separation $\hbar\Omega'$ is the dressed splitting — minimized at $\hbar\Omega$ on resonance, growing as $\hbar|\delta|$ far off.

Step 5. Feeding $E_+$ back and normalizing yields $\tan\theta=\Omega/\delta$ and the states (5). At $\delta=0$ , $\theta=\pi/2$ gives $(|e\rangle\pm|g\rangle)/\sqrt2$ ; as $\Omega\to0$ with $\delta>0$ , $\theta\to0$ so $|+\rangle\to|e\rangle$ , $|-\rangle\to|g\rangle$ . Why: strong driving maximally mixes the states, zero driving returns the bare atom — the check Foot ch. 7 requires.

Go deeper · Derivation B · 1 minThe AC Stark shift as second-order level repulsion

Derivation BThe AC Stark shift as second-order level repulsion

The goal is to show that the far-detuned light shift $\hbar\Omega^2/4\delta$ is nothing more exotic than second-order perturbation theory: two coupled levels repel, each by the coupling squared over their energy gap. Deriving it this way makes the sign transparent — which way each level moves, why red versus blue detuning matters — and independently reproduces the small- $\Omega$ limit of the exact eigenvalue, a genuine cross-check.

Step 1. In the rotating frame the unperturbed levels sit at $E_g^{(0)}=-\hbar\delta/2$ and $E_e^{(0)}=+\hbar\delta/2$ , coupled by $V$ with $\langle e|V|g\rangle=\hbar\Omega/2$ . Treat $V$ as a perturbation, valid for $\Omega\ll|\delta|$ . Why: far from resonance the coupling is small compared with the gap $\hbar\delta$ , so perturbation theory is the honest tool.

Step 2. The second-order shift of the ground level is

\Delta E_g=\frac{|\langle e|V|g\rangle|^2}{E_g^{(0)}-E_e^{(0)}}=\frac{(\hbar\Omega/2)^2}{-\hbar\delta}=+\frac{\hbar\Omega^2}{4\delta}.

Why: the coupling squared over the denominator $E_g^{(0)}-E_e^{(0)}=-\hbar\delta$ delivers the shift with its correct sign — the level is pushed away from the state it couples to.

Step 3. By symmetry $\Delta E_e=-\hbar\Omega^2/4\delta$ . For red detuning ( $\delta<0$ ) the ground shift is negative: the level drops most where $\Omega^2\propto I$ is largest. Why: this sign is why a focused red beam traps — atoms roll downhill toward the intensity maximum.

Step 4. Cross-check by Taylor-expanding the exact eigenvalue (6):

\frac{\hbar}{2}\Big(\sqrt{\Omega^2+\delta^2}-|\delta|\Big)=\frac{\hbar|\delta|}{2}\Big(\sqrt{1+\Omega^2/\delta^2}-1\Big)\approx\frac{\hbar\Omega^2}{4|\delta|}.

Why: expanding to lowest order in $\Omega/\delta$ reproduces the perturbative result — the light shift is the weak-coupling face of the dressed spectrum.

Go deeper · Derivation C · 1 minFrom light shift to optical dipole potential

Derivation CFrom light shift to optical dipole potential

The goal is to convert the abstract level shift $\hbar\Omega^2/4\delta$ into the concrete, engineerable potential $U_{\mathrm{dip}}(\mathbf r)$ used to build traps and lattices, and to expose the two scalings that govern every dipole-trap design: the potential falls as $1/\delta$ but the photon-scattering heating falls faster, as $1/\delta^2$ . That competition, why real traps run hundreds of linewidths detuned, follows from re-expressing $\Omega^2$ in intensity and linewidth.

Step 1. Take the ground-state light shift $U_{\mathrm{dip}}=\hbar\Omega^2/4\delta$ as a position-dependent potential, since $\Omega^2(\mathbf r)$ tracks the local field. Why: a ground-state atom feels this as a real conservative landscape — its gradient is a force, so the light-shift map is a potential map.

Step 2. Relate $\Omega^2$ to intensity via $\Omega^2=\Gamma^2 I/2I_{\mathrm{sat}}$ , $I_{\mathrm{sat}}=\hbar\omega_0^3\Gamma/12\pi c^2$ , giving

U_{\mathrm{dip}}(\mathbf r)=\frac{\hbar\Omega^2(\mathbf r)}{4\delta}=\frac{3\pi c^2}{2\omega_0^3}\,\frac{\Gamma}{\delta}\,I(\mathbf r).

Why: this trades the microscopic $\Omega$ for laboratory quantities — $\Gamma$ , $\omega_0$ , intensity $I$ — so trap depth follows from a data sheet and a power meter.

Step 3. The scattering rate carries an extra $\Gamma/\delta$ : $\Gamma_{\mathrm{sc}}=\dfrac{3\pi c^2}{2\hbar\omega_0^3}\big(\dfrac{\Gamma}{\delta}\big)^2 I=\dfrac{\Gamma}{\hbar\delta}\,U_{\mathrm{dip}}$ . Why: the useful potential goes as $1/\delta$ but destructive scattering as $1/\delta^2$ , so detuning farther (with more power to hold $U$ fixed) suppresses heating — the central trade of every far-off-resonance trap.

Limiting cases

The dressed spectrum $E_\pm=\pm(\hbar/2)\sqrt{\Omega^2+\delta^2}$ interpolates between four regimes:

Case	Result	Meaning
Resonance $\delta=0$	$E_\pm=\pm\tfrac12\hbar\Omega$	equal superpositions $(\|e\rangle\pm\|g\rangle)/\sqrt2$ , split symmetrically by $\hbar\Omega$
Weak / far-detuned $\Omega\ll\|\delta\|$	$\Delta E_g\approx+\dfrac{\hbar\Omega^2}{4\delta},\ \Delta E_e\approx-\dfrac{\hbar\Omega^2}{4\delta}$	second-order light shift; ground level drops for red detuning ( $\delta<0$ ) — the basis of dipole traps
Strong driving $\Omega\gg\|\delta\|$	$\Omega'\approx\Omega+\dfrac{\delta^2}{2\Omega}$	splitting dominated by $\Omega$ ; nearly the resonant superpositions with a small correction
Intensity scaling	$\Omega\propto E_0\propto\sqrt I\Rightarrow\Delta E\propto I$	far-detuned shift linear in intensity, giving $U(\mathbf r)\propto I(\mathbf r)/\delta$

Worked examples

1 · Dressed splitting, mixing, and light shift (confidence builder)

A two-level atom is driven at $\Omega=2\pi\times30$ MHz with blue detuning $\delta=2\pi\times40$ MHz. Find the generalized Rabi frequency, the dressed splitting, the composition of the upper dressed state, and the light shift — exactly and approximately.

Step 1. Generalized Rabi frequency: $\Omega'=2\pi\sqrt{30^2+40^2}=2\pi\times50$ MHz — a clean 3-4-5 triangle. The dressed splitting $E_+-E_-=\hbar\Omega'$ is a frequency spacing of 50 MHz — exactly the Mollow sideband offset of the second figure.

Step 2. Mixing angle: $\tan\theta=\Omega/\delta=0.75$ , so $\theta=36.87^\circ$ . The upper state $|+\rangle=\cos(\theta/2)|e\rangle+\sin(\theta/2)|g\rangle$ has $\cos^2(\theta/2)=\tfrac12(1+0.8)=0.90$ , so $|+\rangle$ is 90% excited, 10% ground — pulled toward $|e\rangle$ because the laser is tuned above resonance.

Step 3. Exact light shift from (6): $\Delta E_{\mathrm{light}}/h=\tfrac12(50-40)=\mathbf{5.0\ MHz}$ . Far-detuned approximation $\Omega^2/4\delta=30^2/(4\times40)=\mathbf{5.6\ MHz}$ .

Step 4. The approximation overshoots by about 12% because $\Omega/\delta=0.75$ is not small; at $\delta=2\pi\times300$ MHz the two agree to under 1%. Exact shift 5.0 MHz; perturbative 5.6 MHz — close but not yet in its comfort zone.

2 · Depth of a real optical dipole trap for rubidium

A $P=100$ mW beam at $\lambda=850$ nm is focused to a waist $w_0=25\ \mu$ m to trap $^{87}$ Rb (D2: $\lambda_0=780.24$ nm, $\Gamma=2\pi\times6.07$ MHz). Estimate the trap depth and scattering rate in the two-level model.

Step 1. Peak intensity: $I_0=\dfrac{2P}{\pi w_0^2}=\dfrac{2(0.100)}{\pi(25\times10^{-6})^2}=1.02\times10^{8}\ \mathrm{W\,m^{-2}}$ .

Step 2. Detuning: $\omega_0=2.416\times10^{15}$ rad/s, $\omega=2.218\times10^{15}$ rad/s, so $\delta=-1.98\times10^{14}$ rad/s ( $=-2\pi\times31.5$ THz) — red-detuned, so the trap is attractive.

Step 3. Dipole potential (7): with $\dfrac{3\pi c^2}{2\omega_0^3}=3.00\times10^{-29}$ and $\dfrac{\Gamma}{\delta}=-1.93\times10^{-7}$ ,

U_{\mathrm{dip}}=\big(3.00\times10^{-29}\big)\big(-1.93\times10^{-7}\big)\big(1.02\times10^{8}\big)=-5.9\times10^{-28}\ \mathrm{J}.

Dividing by $k_B$ : $U_{\mathrm{dip}}/k_B=-\mathbf{43\ \mu K}$ (light shift $U/h=-0.89$ MHz) — tens of microkelvin, the laser-cooled regime.

Step 4. Scattering rate: $\Gamma_{\mathrm{sc}}=(\Gamma/\hbar\delta)\,U_{\mathrm{dip}}\approx\mathbf{1.1\ photons/s}$ — the $1/\delta$ -vs- $1/\delta^2$ split of Derivation 3, and why such traps hold atoms for seconds. A 43 µK trap scattering about one photon per second.

Physics of the portrait

The dressed-atom picture is not merely a change of basis; it turned a zoo of strong-field phenomena — light shifts, Autler-Townes doublets, Mollow triplets, adiabatic passage — into one diagram. It is associated above all with the physicist who built it into a language and used it to cool atoms.

Origins · 1 minClaude Cohen-Tannoudji and how this was discovered

Claude Cohen-Tannoudji

born 1933 · dressed-atom theory, light shifts, and laser cooling

Claude Cohen-Tannoudji learned the light shift the hard way — by measuring it. In his early-1960s work on optical pumping he found that off-resonant light not only pumps atoms between sublevels but visibly displaces their energies, an effect his group christened the déplacement lumineux when it was still more nuisance than resource. Over two decades he inverted that view. With Serge Reynaud he developed the dressed-atom formalism into a complete framework for the strongly driven atom, showing that the Mollow triplet, the Autler-Townes splitting, and the atom's probe response all read off the dressed-state ladder, with spontaneous emission cascading the atom down it. The picture proved unexpectedly practical: how the dressed energies vary in space and with polarization underlies Sisyphus cooling and the sub-Doppler temperatures that let atoms be trapped in light shifts at all. For laser cooling and trapping, Cohen-Tannoudji shared the 1997 Nobel Prize with Chu and Phillips; his Atom-Photon Interactions is where most physicists first meet the dressed atom as more than a $2\times2$ matrix.

History

1955

— Autler and Townes observe a strong radiofrequency field splitting a molecular line, the first Autler-Townes doublet

1961

— Cohen-Tannoudji measures light shifts of atomic ground-state sublevels

1969

— Mollow predicts the three-peak resonance-fluorescence spectrum

1977

— Cohen-Tannoudji and Reynaud recast resonance fluorescence in the dressed-atom basis

1997

— Cohen-Tannoudji shares the Nobel Prize for laser cooling and trapping of atoms

Going further

Texts. Foot, Atomic Physics, ch. 7 (dressed atom, AC Stark effect) · Loudon, The Quantum Theory of Light, 3rd ed., ch. 2 and 8 (resonance fluorescence, Mollow triplet) · Cohen-Tannoudji, Dupont-Roc and Grynberg, Atom-Photon Interactions, ch. VI · Grimm, Weidemüller and Ovchinnikov, Optical Dipole Traps for Neutral Atoms, Adv. At. Mol. Opt. Phys. 42 (2000) · Demtröder, Laser Spectroscopy, Vol. 1, ch. 2.

Provenance. The dressed eigenvalues $E_\pm=\pm(\hbar/2)\sqrt{\Omega^2+\delta^2}$ and eigenvectors follow by diagonalizing the rotating-frame $2\times2$ matrix (Foot ch. 7; Loudon ch. 8). The light shift $\hbar\Omega^2/4\delta$ is checkable by second-order perturbation theory or by Taylor-expanding the exact eigenvalue (Cohen-Tannoudji et al. ch. VI). The dipole potential $U_{\mathrm{dip}}=(3\pi c^2/2\omega_0^3)(\Gamma/\delta)I$ and the $1/\delta$ -vs- $1/\delta^2$ scaling are from the Grimm-Weidemüller-Ovchinnikov review. Worked numbers use $^{87}$ Rb D2 data ( $\lambda_0=780.24$ nm, $\Gamma=2\pi\times6.07$ MHz); the 3-4-5 example and 43 µK trap depth are independently reproducible.

Prerequisites. Electric Dipole Approximation · Transition Dipole Moment · Stimulated Emission · Radiative Lifetime and Natural Linewidth. Leads to. Lorentzian Line Shape · Spontaneous Emission · Oscillator Strength.

Ask the tutor: Ask me to walk through diagonalizing the $2\times2$ dressed Hamiltonian for your own $\Omega$ and $\delta$ , or to size an optical dipole trap for any atom, wavelength, and beam power.

What is the difference between the bare Rabi frequency and the generalized Rabi frequency?

The bare Rabi frequency $\Omega=d_{eg}E_0/\hbar$ measures only the coupling strength — how fast the field drives population between $|g\rangle$ and $|e\rangle$ on resonance. The generalized $\Omega'=\sqrt{\Omega^2+\delta^2}$ folds in the detuning and is what you observe: it sets the dressed splitting $\hbar\Omega'$ , the off-resonant Rabi-oscillation frequency, and the Mollow and Autler-Townes spacings. They coincide only at $\delta=0$ ; off resonance $\Omega'>\Omega$ always, so the levels are harder to fully invert but their energy separation grows.

Why can the light shift push a level up for blue detuning but down for red?

Because the shift is level repulsion, which always pushes a level away from the state it couples to. When the laser is red-detuned ( $\delta<0$ ) the ground level's dressed partner sits above it in the rotating frame, so the ground level is pushed down ( $\Delta E_g<0$ ); when blue-detuned it is pushed up. The formula $\Delta E_g=+\hbar\Omega^2/4\delta$ carries this in its sign — flip $\delta$ and the shift flips. That single fact is why red beams trap and blue beams repel.

Is the AC Stark shift a real force or just a change of energy?

It is a real mechanical force. The ground-state light shift $U_{\mathrm{dip}}(\mathbf r)=\hbar\Omega^2(\mathbf r)/4\delta$ depends on position through the intensity, and any position-dependent energy is a potential whose gradient is a force, $\mathbf F=-\nabla U_{\mathrm{dip}}$ . Atoms are pushed toward intensity maxima (red) or minima (blue) and can be held, guided, and accelerated by shaped light — exactly how optical tweezers, dipole traps, and lattices work. The force is conservative; photon scattering is the separate dissipative piece you minimize.

Where do dressed states actually get occupied, versus just being a useful basis?

Both happen. Weakly, the dressed states are a convenient basis you compute in and transform back. But under strong steady driving the atom genuinely settles into a statistical mixture of its dressed states, and spontaneous emission carries it between adjacent doublets — precisely what produces the three Mollow lines. In adiabatic passage (STIRAP) one prepares a single dressed state and slowly deforms the Hamiltonian so the atom rides it to a target bare state. So the dressed basis is both bookkeeping and, under strong or adiabatic driving, the state the atom occupies.

Why does the rotating-wave approximation eventually fail, and what fixes it?

The RWA drops the counter-rotating term at $\omega+\omega_0$ , legitimate only when $\Omega,|\delta|\ll\omega_0$ — that fast term then averages away. For very intense fields ( $\Omega$ approaching $\omega_0$ ) or extreme detunings the discarded term contributes a small but real correction, the Bloch-Siegert shift, nudging the resonance and level positions beyond the clean $2\times2$ prediction. The fix is to keep the full Hamiltonian and treat it with Floquet theory or higher-order averaging. For ordinary near-resonant optics the correction is tiny; it mainly matters in strong-drive and low-frequency (microwave, RF) regimes.

How is the AC Stark shift related to the atomic polarizability?

Directly: far off resonance the light shift is $U=-\tfrac14\,\mathrm{Re}[\alpha(\omega)]\,|E_0|^2$ , where $\alpha(\omega)$ is the atom's dynamic (AC) polarizability. Our two-level $\hbar\Omega^2/4\delta$ is exactly this with the single-resonance $\alpha(\omega)\propto d_{eg}^2/\hbar\delta$ inserted. Real multilevel atoms need the full $\alpha(\omega)$ , which sums many transitions and passes through zero at the magic wavelengths where two states share the same polarizability — and light shift. The dressed-state shift and frequency-dependent polarizability are two descriptions of one physics; the two-level model is the single-resonance case.