Radiation Pressure

Radiation pressure is mechanical momentum transfer from an electromagnetic field to matter. For a freely moving atom, a photon absorbed from a traveling wave changes the atomic momentum by $+\hbar\mathbf k$. Subsequent spontaneous emission gives a recoil $-\hbar\mathbf k_s$, so one complete scattering cycle changes the atomic momentum by

$$\Delta\mathbf p = \hbar \left( \mathbf k-\mathbf k_s \right).$$

In free space, the mean spontaneous-emission direction often vanishes. The mean cycle impulse is then $\hbar\mathbf k$, directed along the incident beam. The emitted direction still fluctuates, so the same cycles that produce a mean force also produce momentum diffusion and recoil heating.

For a closed two-level atom driven by one traveling wave, the steady scattering force is

$$\mathbf F_{\mathrm{sc}} = \hbar\mathbf k \frac{\Gamma}{2} \frac{ s_0 }{ 1+s_0+ \left( 2\Delta_v/\Gamma \right)^2 }.$$

Here $\Gamma$ is the excited-state population decay rate, $s_0=2|\Omega|^2/\Gamma^2$ is the on-resonance saturation parameter, and $\Delta_v$ is the Doppler-shifted detuning. This compact expression is powerful, but its assumptions matter: a closed transition, a stationary internal state, one traveling mode, controlled recoil averaging, and center-of-mass motion slow enough that the internal steady state can follow.

Radiation pressure is not synonymous with every optical force. The steady single-beam scattering force discussed here follows the beam’s phase gradient and relies on spontaneous emission to reset the atom. A conservative dipole force follows an adiabatic light-shift gradient. Coherent exchange between different traveling modes can also transfer momentum without spontaneous emission; it obeys different force limits.

Required background. Photon Momentum gives the momentum per traveling-wave photon, Optical Bloch Equations gives the driven steady state, and Spontaneous Emission explains decay rates and angular patterns.

From Energy Flux to Pressure

Classical benchmark

A plane wave of intensity $I$ carries energy flux $I$ and momentum flux $I/c$ in vacuum. At normal incidence, an ideal absorbing surface therefore feels pressure

$$P_{\mathrm{abs}} = \frac{I}{c}.$$

A perfectly reflecting surface reverses the normal momentum and feels

$$P_{\mathrm{refl}} = \frac{2I}{c}.$$

Radiation pressure therefore does not by itself prove that the field is in a photon-number state. Classical Maxwell fields carry momentum. The photon description becomes especially useful when individual absorption and emission recoils, counting fluctuations, and internal transitions must be tracked.

Atomic effective area

An atom does not present a hard geometric disk. Near an allowed resonance, its optical response defines a frequency-, polarization-, state-, and intensity-dependent effective cross section. In the weak-field limit, the force can be written

$$F_{\mathrm{sc}} = \frac{I}{c} \sigma(\Delta_v).$$

This has the same momentum-flux structure as macroscopic pressure, but the atomic cross section can be much larger than the atom’s geometric area. Saturation eventually makes the response nonlinear in $I$.

Momentum in an Absorption–Emission Cycle

Absorption

For a traveling mode with wave vector $\mathbf k$, absorption removes one field quantum of momentum $\hbar\mathbf k$. Momentum conservation gives

$$\Delta\mathbf p_{\mathrm{atom}}^{\mathrm{abs}} = +\hbar\mathbf k.$$

The statement assumes that any momentum supplied to an optical element, substrate, or second field has already been included in the chosen system boundary.

Stimulated emission

Stimulated emission into the same traveling mode adds a photon with momentum $\hbar\mathbf k$ to that mode. The atom receives

$$\Delta\mathbf p_{\mathrm{atom}}^{\mathrm{stim}} = -\hbar\mathbf k.$$

Absorption and stimulated emission therefore push in opposite directions. It is incorrect to count every coherent excitation as a permanent $+\hbar\mathbf k$ impulse while ignoring stimulated return.

Spontaneous emission

If the emitted photon has wave vector $\mathbf k_s$, the atomic recoil is

$$\Delta\mathbf p_{\mathrm{atom}}^{\mathrm{sp}} = -\hbar\mathbf k_s.$$

For one absorption–spontaneous-emission cycle,

$$\Delta\mathbf p_{\mathrm{cycle}} = \hbar\mathbf k - \hbar\mathbf k_s.$$

The emitted photon is not generally isotropic. Electric-dipole radiation has an angular pattern fixed by the transition and polarization. In inversion-symmetric free space one often has

$$\left\langle \mathbf k_s \right\rangle = 0,$$

even though the second moment is nonzero. Cavities, waveguides, nearby surfaces, collective emission, or chiral coupling can make the mean emission recoil directional.

Top: the single-beam force approaches $F_{\max}=\hbar k\Gamma/2$ as intensity increases, while its frequency profile broadens. Bottom: two equal weak beams at $\Delta=\Gamma/2$ produce a force opposing velocity. Here $s_0$ is the saturation parameter of each beam; the dashed tangent has dimensionless slope $-2$, equivalent to $\alpha=\hbar k^2s_0$. The curves describe mean force; momentum fluctuations require a separate counting and emission-angle model.

Force as Momentum Transfer Rate

Ehrenfest form

For a neutral point atom in the electric-dipole approximation, take center-of-mass kinetic energy $\mathbf P^2/(2m)$ and neglect velocity-dependent Röntgen terms. Then $\mathbf P=m\dot{\mathbf R}$. For the retained drive interaction $H_{\mathrm{int}}(\mathbf R)$, the Hamiltonian force operator is

$$\widehat{\mathbf F} = - \boldsymbol\nabla_{\mathbf R} H_{\mathrm{int}}.$$

Ehrenfest’s theorem gives

$$\frac{d}{dt} \left\langle \mathbf P \right\rangle = \left\langle \widehat{\mathbf F} \right\rangle$$

when other forces and the mean reservoir recoil vanish. After emission modes have been eliminated, the general master-equation relation includes $\sum_j\langle\mathcal D^\dagger[L_j]\mathbf P\rangle$ on the right. A recoil jump $L_{\mathbf k_s}\propto e^{-i\mathbf k_s\cdot\mathbf R} |g\rangle\langle e|$ contributes the emitted momentum $-\hbar\mathbf k_s$. This term has zero mean for symmetric free-space emission but still contributes to momentum variance.

A traveling wave can contain both an amplitude gradient and a phase gradient. Differentiating the drive Hamiltonian separates:

• an amplitude-gradient contribution associated with the reactive dipole force;
• a phase-gradient contribution associated with directed momentum transfer.

The cycle picture and the force-operator picture are two descriptions of the same momentum conservation. The cycle picture is particularly transparent after spontaneous events can be treated as resolved or coarse-grained jumps.

Steady-state balance

Let $R_{\mathrm{abs}}$ and $R_{\mathrm{stim}}$ denote the upward and stimulated downward rates in a rate description. For a closed two-level transition,

$$\dot\rho_{ee} = R_{\mathrm{abs}} - R_{\mathrm{stim}} - \Gamma\rho_{ee}.$$

At steady state,

$$R_{\mathrm{abs}} - R_{\mathrm{stim}} = \Gamma\rho_{ee}^{\mathrm{ss}}.$$

The net rate at which directed photons are removed from the traveling beam equals the spontaneous-emission rate. If the mean spontaneous recoil is zero,

$$\mathbf F_{\mathrm{sc}} = \hbar\mathbf k \Gamma\rho_{ee}^{\mathrm{ss}}.$$

In a coherently driven atom, absorption and stimulated emission need not admit a unique trajectory decomposition at every instant. The steady-state balance above is sufficient for the mean force. For a uniform traveling wave it also gives the transient identity

$$\mathbf F(t)=\hbar\mathbf k\,[\dot\rho_{ee}(t)+\Gamma\rho_{ee}(t)]$$

when mean emission recoil vanishes. The population-derivative term matters for short coherent pulses even if little fluorescence occurs.

Closed Two-Level Scattering Force

Conventions

Use a closed transition $|g\rangle\leftrightarrow|e\rangle$ with angular frequency $\omega_0$, population decay rate $\Gamma$, and a plane-wave laser with angular frequency $\omega_L$ and wave vector $\mathbf k$. This chapter uses atom-minus-laser detuning:

$$\Delta = \omega_0-\omega_L.$$

Red detuning has $\Delta>0$. Frequency variables use angular units; $\Gamma$ is an inverse lifetime, corresponding to the natural optical angular-frequency FWHM in this lifetime-limited model. Pure dephasing is absent from the two-level formulas below.

The Rabi Hamiltonian contains the off-diagonal coefficient $\hbar\Omega/2$. Define the on-resonance saturation parameter

$$s_0 = \frac{ 2|\Omega|^2 }{ \Gamma^2 }.$$

Doppler detuning

For a nonrelativistic atom with velocity $\mathbf v$, the laser frequency in the atomic rest frame is

$$\omega_L' \simeq \omega_L - \mathbf k\mathbin{\cdot}\mathbf v.$$

The effective detuning is therefore

$$\Delta_v = \Delta + \mathbf k\mathbin{\cdot}\mathbf v.$$

Writing the optical phase and transforming it is safer than memorizing a Doppler sign. A reference using laser-minus-atom detuning displays the opposite sign. The semiclassical formula neglects a resolved single-photon recoil shift: absorption between free momentum eigenstates instead has detuning $\Delta+\mathbf k\cdot\mathbf v+\omega_r$. Use quantized motional transitions when $\omega_r$ is comparable to the relevant linewidth or resolution.

Steady excited-state population

The optical Bloch steady state gives

$$\rho_{ee}^{\mathrm{ss}} = \frac12 \frac{ s_0 }{ 1+s_0+ \left( 2\Delta_v/\Gamma \right)^2 }.$$

It is also useful to define the detuning-dependent saturation parameter

$$s(\Delta_v) = \frac{ s_0 }{ 1+ \left( 2\Delta_v/\Gamma \right)^2 }.$$

Then

$$\rho_{ee}^{\mathrm{ss}} = \frac{ s }{ 2(1+s) }.$$

Scattering rate

The spontaneous scattering rate is

$$R_{\mathrm{sc}} = \Gamma\rho_{ee}^{\mathrm{ss}},$$

or

$$R_{\mathrm{sc}} = \frac{\Gamma}{2} \frac{ s_0 }{ 1+s_0+ \left( 2\Delta_v/\Gamma \right)^2 }.$$

The corresponding mean force is

$$\mathbf F_{\mathrm{sc}} = \hbar\mathbf k R_{\mathrm{sc}}.$$

This result assumes the mean emitted momentum vanishes. If it does not, replace the impulse per cycle:

$$\mathbf F = \hbar \left( \mathbf k - \left\langle \mathbf k_s \right\rangle \right) R_{\mathrm{sc}}.$$

Saturation and Force Limits

Maximum rate and force

On resonance,

$$R_{\mathrm{sc}}(0) = \frac{\Gamma}{2} \frac{ s_0 }{ 1+s_0 }.$$

As $s_0\to\infty$,

$$\rho_{ee}^{\mathrm{ss}} \longrightarrow \frac12,$$

and

$$R_{\mathrm{sc}} \longrightarrow \frac{\Gamma}{2}.$$

The steady force of this single continuously driven traveling-wave model is bounded by

$$F_{\max} = \frac{ \hbar k\Gamma }{ 2 }.$$

The associated maximum acceleration is

$$a_{\max} = \frac{ \hbar k\Gamma }{ 2m }.$$

Increasing intensity far beyond saturation does not make a closed two-level atom scatter arbitrarily fast. The atom spends approximately half its time excited, and stimulated emission competes with absorption. This is not a universal optical-force bound: alternating absorption and stimulated emission into an opposite mode can exceed it, as demonstrated with bichromatic forces by Liebisch and colleagues (2012).

Power broadening

The force profile is Lorentzian in $\Delta_v$ with a saturated half-width

$$\Delta_{\mathrm{HWHM}} = \frac{\Gamma}{2} \sqrt{ 1+s_0 }.$$

Intensity therefore changes both the peak and the velocity or frequency range over which the force is substantial. A broadened force profile is not evidence that $\Gamma$ itself increased.

Response-time condition

The steady formula assumes that position, velocity, intensity, detuning, and polarization vary slowly enough for the internal state to relax toward the local steady state. A short pulse or rapid transit requires time-dependent optical Bloch evolution and the impulse

$$\Delta\mathbf p = \int_{t_i}^{t_f} \mathbf F(t)\,dt.$$

A coherent $\pi$ pulse can transfer approximately one photon recoil, but repeated dissipative force requires a mechanism that resets the internal state.

Resonant Cross Section

Weak-field form

The photon flux in a plane wave is

$$\Phi_\gamma = \frac{ I }{ \hbar\omega_L }.$$

In the weak-field limit,

$$R_{\mathrm{sc}} = \Phi_\gamma \sigma(\Delta_v).$$

For an ideal closed electric-dipole transition with the appropriate polarization and degeneracy assumptions,

$$\sigma_0 = \frac{ 3\lambda^2 }{ 2\pi },$$

and

$$\sigma(\Delta_v) = \frac{ \sigma_0 }{ 1+ \left( 2\Delta_v/\Gamma \right)^2 }.$$

Then

$$F_{\mathrm{sc}} = \frac{I}{c} \sigma(\Delta_v).$$

This directly connects the photon-cycle result to incident momentum flux.

Saturation intensity

Define $I_{\mathrm{sat}}$ by

$$s_0 = \frac{I}{I_{\mathrm{sat}}}.$$

For the same ideal cycling transition and conventions,

$$I_{\mathrm{sat}} = \frac{ \pi h c\Gamma }{ 3\lambda^3 }.$$

At finite intensity one may write an effective nonlinear cross section

$$\sigma_{\mathrm{eff}} = \frac{ \sigma_0 }{ 1+s_0+ \left( 2\Delta_v/\Gamma \right)^2 },$$

so that $F=I\sigma_{\mathrm{eff}}/c$. This effective cross section describes nonlinear saturation and depends on the incident intensity.

Real-atom corrections

The familiar $\sigma_0$ and $I_{\mathrm{sat}}$ require a declared transition. Real corrections include:

• Clebsch–Gordan coefficients and laser polarization;
• ground- and excited-state degeneracy;
• branching into states outside the driven cycle;
• optical pumping among Zeeman or hyperfine levels;
• imperfect spatial mode and intensity calibration;
• laser linewidth and frequency noise;
• nearby transitions and interference;
• collective and propagation effects.

Quoting a universal saturation intensity without the transition, polarization, and state preparation is incomplete.

A Rubidium-Like Estimate

Consider a cycling transition with rounded parameters

$$\begin{aligned} \lambda &= 780\ \mathrm{nm}, \\ \frac{\Gamma}{2\pi} &= 6.0\ \mathrm{MHz}, \\ m &= 1.44\times10^{-25}\ \mathrm{kg}. \end{aligned}$$

Take

$$s_0=2, \qquad \Delta=\frac{\Gamma}{2}, \qquad v=0.$$

The force denominator is

$$1+s_0+ \left( 2\Delta/\Gamma \right)^2 = 4,$$

so

$$R_{\mathrm{sc}} = \frac{\Gamma}{4} \simeq 9.4\times10^6\ \mathrm{s^{-1}}.$$

The photon recoil momentum is

$$\hbar k = \frac{h}{\lambda} \simeq 8.5\times10^{-28}\ \mathrm{kg\,m\,s^{-1}}.$$

Hence

$$\begin{aligned} F_{\mathrm{sc}} &\simeq 8.0\times10^{-21}\ \mathrm N, \\ a &\simeq 5.6\times10^4\ \mathrm{m\,s^{-2}}. \end{aligned}$$

The force is microscopic, but the mass is smaller still. During $10\ \mathrm{\mu s}$, a constant-force estimate gives approximately

$$N_{\mathrm{sc}} = R_{\mathrm{sc}}t \simeq 94$$

cycles and a speed change

$$\Delta v \simeq N_{\mathrm{sc}} \frac{\hbar k}{m} \simeq 0.56\ \mathrm{m\,s^{-1}}.$$

The changing Doppler shift tests this constant-force approximation. At the predicted final speed,

$$\frac{k\Delta v}{\Gamma} \simeq 0.12.$$

The atom has moved measurably across the force profile, so even this short constant-force estimate is only approximate. Over longer slowing times, laser chirping or a position-dependent Zeeman shift is needed to maintain resonance.

Recoil Scales

Single-photon recoil

Define

$$p_r = \hbar k,$$

the recoil velocity

$$v_r = \frac{ \hbar k }{ m },$$

and the recoil energy

$$E_r = \frac{ \hbar^2k^2 }{ 2m }.$$

The recoil angular frequency is

$$\omega_r = \frac{E_r}{\hbar}.$$

These scales determine when center-of-mass motion must be quantized rather than treated as a continuous classical trajectory.

Energy per scattering cycle

For absorption from fixed $\mathbf k$ followed by free-space emission with $|\mathbf k_s|\simeq k$ and $\langle\mathbf k_s\rangle=0$,

$$\left\langle |\Delta\mathbf p|^2 \right\rangle = 2\hbar^2k^2.$$

The corresponding recoil contribution to kinetic energy is

$$\frac{ \left\langle |\Delta\mathbf p|^2 \right\rangle }{ 2m } = 2E_r$$

per cycle as the quadratic term in $\Delta E=\mathbf v\cdot\Delta\mathbf p+|\Delta\mathbf p|^2/(2m)$. This raw second moment is not the variance of a fixed number of cycles: the absorption kick is deterministic if the cycle count is fixed. For independent Poisson event times, it does enter the diffusion term in the kinetic-energy balance:

$$\dot E_{\mathrm{kin}} = \mathbf F\mathbin{\cdot}\mathbf v + \text{diffusive heating}.$$

Directed acceleration and random heating are different moments of the same recoil process.

Momentum Diffusion

Covariance growth

Let $\mathbf u$ be a unit vector along a measured axis. Assume independent Poisson event times, independent emitted directions with zero mean, fixed incident momentum, and negligible Doppler feedback during the interval. Then the coarse-grained variance growth is

$$\frac{d}{dt} \operatorname{Var}(p_u) = \hbar^2R_{\mathrm{sc}} \Big[ \left( \mathbf u\mathbin{\cdot}\mathbf k \right)^2 + \left\langle \left( \mathbf u\mathbin{\cdot}\mathbf k_s \right)^2 \right\rangle \Big].$$

The first term is absorption-number noise along the incident beam. The second is random spontaneous-emission recoil. If a Fokker–Planck equation uses

$$\frac{d}{dt} \operatorname{Var}(p_u) = 2D_{p,u},$$

then

$$D_{p,u} = \frac{\hbar^2R_{\mathrm{sc}}}{2} \Big[ \left( \mathbf u\mathbin{\cdot}\mathbf k \right)^2 + \left\langle \left( \mathbf u\mathbin{\cdot}\mathbf k_s \right)^2 \right\rangle \Big].$$

Authors also use a convention without the factor of two. A quoted momentum diffusion coefficient is ambiguous until its defining equation is stated.

Isotropic-emission example

For isotropic emission,

$$\left\langle \left( \mathbf u\mathbin{\cdot}\mathbf k_s \right)^2 \right\rangle = \frac{k^2}{3}.$$

Along the incident beam,

$$\frac{d}{dt} \operatorname{Var}(p_\parallel) = \frac43 \hbar^2k^2 R_{\mathrm{sc}}.$$

Electric-dipole emission is not generally isotropic, so the factor $1/3$ must be replaced by the angular moment of the actual radiation pattern.

Fixed and fluctuating photon counts

Let $N$ be the number of completed cycles and $\xi=\langle n_{s,z}^2\rangle$ the emission angular moment along the incident beam. Independent emission directions, independent of $N$, give

$$\operatorname{Var}(p_z) =(\hbar k)^2\left[\operatorname{Var}N+\xi\langle N\rangle\right].$$

For fixed $N$, only emission angles fluctuate. Poisson counting gives $\operatorname{Var}N=\langle N\rangle$; correlated fluorescence need not. Thus independence of recoil directions alone does not justify Poisson absorption noise. During a finite unfinished cycle, internal excitation also contributes a bounded endpoint momentum term.

Beyond independent cycles

At high saturation or in structured fields, the simple Poisson-cycle model can miss:

• antibunching and temporal correlations in resonance fluorescence;
• fluctuations of the coherent dipole force;
• standing-wave position dependence;
• internal-state switching;
• cavity-modified emission;
• collective scattering and reabsorption.

The mean force can remain accurate while its noise model fails. Mechanical heating, linewidths, and trap loss depend on the second moments.

Scattering Force and Dipole Force

Reactive optical potential

For a slowly varying two-level field with $\Gamma,|\Omega|\ll|\Delta|\ll\omega_0$, the ground-connected adiabatic light shift in this chapter’s detuning convention is

$$U_g(\mathbf r) \simeq - \frac{ \hbar|\Omega(\mathbf r)|^2 }{ 4\Delta }.$$

The conservative dipole force is

$$\mathbf F_{\mathrm{dip}} = - \boldsymbol\nabla U_g.$$

For red detuning, $\Delta>0$, this ideal ground state is attracted toward larger intensity.

Dissipative rate

In the same weak, far-detuned limit,

$$R_{\mathrm{sc}} \simeq \frac{ \Gamma|\Omega|^2 }{ 4\Delta^2 }.$$

The light shift scales as $I/\Delta$, while scattering scales as $I/\Delta^2$. Increasing detuning at fixed trap depth reduces the scattering cost in the ideal one-line model, but finite power and multilevel structure limit that strategy.

Operational distinction

Property	Scattering force	Dipole force
response	absorptive or dissipative	dispersive or reactive
direction	optical phase gradient, often $\mathbf k$	intensity or dressed-energy gradient
steady origin	irreversible photon redistribution	position-dependent light shift
saturation	bounded by the internal cycling rate	depends on dressed potential and adiabaticity
fluctuations	photon recoil and force noise are intrinsic	technical and quantum fluctuations can remain
uniform plane wave	generally nonzero near resonance	zero if amplitude is uniform

A focused beam can exert both forces simultaneously. Calling the total force “radiation pressure” without specifying its dissipative and conservative parts can hide the relevant physics.

Counterpropagating Beams

Weak-saturation model

Consider two equal, incoherently added traveling waves along $\pm x$, each with on-resonance saturation parameter $s_0\ll1$. For velocity $v$ along $+x$, their detunings are

$$\Delta_+ = \Delta+kv,$$

and

$$\Delta_- = \Delta-kv.$$

The low-saturation net force is

$$F(v) = \frac{\hbar k\Gamma s_0}{2} \Bigg[ \frac{ 1 }{ 1+ \left( 2\Delta_+/\Gamma \right)^2 } - \frac{ 1 }{ 1+ \left( 2\Delta_-/\Gamma \right)^2 } \Bigg].$$

At $v=0$, the two forces cancel.

Low-velocity damping

For $|kv|\ll\Gamma$, expand:

$$F(v) \simeq -\alpha v.$$

The friction coefficient is

$$\alpha = \frac{ 8\hbar k^2s_0 \left( \Delta/\Gamma \right) }{ \left[ 1+ \left( 2\Delta/\Gamma \right)^2 \right]^2 }.$$

With this page’s convention, red detuning has $\Delta>0$, so $\alpha>0$ and the force opposes small velocities. Blue detuning reverses the slope and produces antidamping.

This is the mechanical seed of Doppler Cooling. It is not a complete temperature theory: the same scattering events generate momentum diffusion, and equilibrium follows from the balance between damping and noise. Multilevel polarization-gradient mechanisms can operate below the simple two-level Doppler scale.

Why one beam does not make molasses

A single beam opposing the initial motion can slow a selected velocity class, but it also gives nonzero mean acceleration. Two opposite beams cancel at zero velocity while retaining a velocity-odd restoring force in momentum space. Spatial confinement requires additional position dependence, such as magnetic-field-induced Zeeman shifts or a conservative potential.

Multiple Beams and Coherent Fields

Independent-rate limit

When every beam is weak and mutual coherences average away, the total force can be approximated by

$$\mathbf F \simeq \sum_j \hbar\mathbf k_j R_j.$$

Each rate uses its own Doppler detuning and polarization coupling.

When forces do not add independently

Independent Lorentzians fail when:

• the total saturation is appreciable;
• coherent beams form standing waves;
• polarization varies in space;
• several ground states support dark or bright superpositions;
• stimulated processes couple the modes;
• optical pumping changes the addressed state;
• beat notes are slow on the internal timescale.

Then the force must be obtained from a multilevel master equation or equivalent dressed-state treatment using the full spatial field.

Moving standing waves

Two coherent beams with a frequency difference form a moving intensity and phase pattern. The atom can exchange momentum coherently between modes without a spontaneous event. Bragg diffraction, Raman transitions, and Bloch oscillations are not captured by adding independent scattering rates.

Multilevel Atoms and Molecules

Branching and dark states

If the excited state decays to an undriven level, the nominal cycling rate is temporary. After roughly the inverse leakage probability in photon cycles, population accumulates outside the addressed transition. Repump lasers may restore cycling, but they add:

• additional recoil directions;
• new excited-state fractions;
• coherences and dark states;
• branching uncertainty;
• frequency and polarization requirements.

The measured force is determined by the complete steady-state population flow, not by the strongest line alone.

Zeeman and hyperfine structure

Magnetic sublevels have different Clebsch–Gordan coefficients and polarization selection rules. A changing quantization axis or imperfect polarization can make $s_0$, branching, and the emission pattern depend on position and velocity.

Molecules

Molecules often have many rovibrational decay channels. Strong radiation pressure requires highly diagonal branching and multiple repumps. A large absorption cross section does not guarantee many usable cycles.

Dense samples

In an optically thick cloud, incident intensity changes through the sample and emitted photons can be reabsorbed. Multiple scattering produces collective forces and extra diffusion. The independent-atom, undepleted-beam formula is then incomplete.

Directional Emission and Structured Reservoirs

Cavity recoil

For emission into a specified traveling cavity output, the mean emitted momentum need not vanish. If resolved incident and outgoing photon momenta describe the complete cycle, its mean force is

$$\mathbf F = \hbar \left( \mathbf k_{\mathrm{in}} - \left\langle \mathbf k_{\mathrm{out}} \right\rangle \right) R.$$

A standing cavity mode does not have one definite traveling momentum. Include the cavity mirrors, drive, intracavity field gradients, and output channels when applying momentum conservation; a cavity can modify both damping and diffusion.

Waveguides and chiral coupling

A one-dimensional waveguide can favor one propagation direction. Then emission itself supplies a directed average recoil. Such forces are not described by the free-space assumption $\langle\mathbf k_s\rangle=0$.

Nearby surfaces

Surfaces alter the electromagnetic mode density and radiation pattern. They can also exert conservative Casimir–Polder forces. Separating resonant recoil from static or dispersive surface forces requires one consistent environment model.

Experimental Observables

Acceleration and deflection

For a known interaction time,

$$\Delta\mathbf p = \int \mathbf F(t)\,dt.$$

Time of flight, cloud displacement, atomic-beam deflection, and momentum-resolved imaging can infer this impulse. The analysis must include the changing Doppler detuning and spatial intensity.

Fluorescence

Detected fluorescence is proportional to

$$R_{\mathrm{det}} = \eta_{\mathrm{det}} b_{\mathrm{det}} R_{\mathrm{sc}},$$

where $\eta_{\mathrm{det}}$ is the collection and detection efficiency and $b_{\mathrm{det}}$ is the branching fraction into the detected channel. Fluorescence can calibrate the internal scattering rate, but it does not by itself determine the recoil direction or net force.

Momentum width

The growth of momentum variance tests the diffusion model. A force model that fits the mean displacement but underpredicts the width may be missing emission anisotropy, state switching, beam noise, standing-wave structure, or reabsorption.

The Laser Cooling Simulation Notebook combines two-beam friction and a specified recoil model. Quantum Noise develops the correlation functions needed beyond independent events.

Exercises

1. Separate the mean impulse from its variance

An atom absorbs from a beam with wave vector $k\widehat{\mathbf z}$ and spontaneously emits a photon of the same wave-number magnitude in direction $\widehat{\mathbf n}$. Assume an isotropic emission distribution.

1. Find the mean momentum transfer per cycle.
2. Find the variance of the $z$ component of the cycle impulse.
3. Find the mean squared three-dimensional impulse.

Solution

The cycle impulse is

$$\Delta\mathbf p = \hbar k \left( \widehat{\mathbf z} - \widehat{\mathbf n} \right).$$

Isotropy gives

$$\left\langle \widehat{\mathbf n} \right\rangle = 0,$$

so

$$\left\langle \Delta\mathbf p \right\rangle = \hbar k \widehat{\mathbf z}.$$

The fluctuating $z$ component is

$$\Delta p_z - \left\langle \Delta p_z \right\rangle = -\hbar k n_z.$$

Since $\langle n_z^2\rangle=1/3$,

$$\operatorname{Var}(\Delta p_z) = \frac{ \hbar^2k^2 }{ 3 }.$$

Finally,

$$\begin{aligned} \left\langle |\Delta\mathbf p|^2 \right\rangle &= \hbar^2k^2 \left\langle |\widehat{\mathbf z}-\widehat{\mathbf n}|^2 \right\rangle \\ &= \hbar^2k^2 \left( 2 - 2\langle n_z\rangle \right) \\ &= 2\hbar^2k^2. \end{aligned}$$

The mean emitted recoil vanishes, but its fluctuations do not.

2. Exceed the single-beam force ceiling

An ideal sequence absorbs through a π pulse from wave vector $+k\hat x$, then stimulates emission through a π pulse into $-k\hat x$. The pair takes time $T$, with negligible spontaneous decay and appropriately compensated recoil/Doppler detunings. Find its momentum transfer and mean force. When does it exceed $\hbar k\Gamma/2$?

Solution

Absorption gives $+\hbar k$ and stimulated emission into $-k\hat x$ gives another $+\hbar k$. The atom returns to its ground state, so a pair gives $2\hbar k$ and mean force $2\hbar k/T$. This exceeds the spontaneous-cycle ceiling for $T<4/\Gamma$. There is no contradiction: the reset uses a second coherent mode instead of waiting for spontaneous emission. Pulse-area errors and recoil-dependent detunings limit repeated ideal pairs in practice.

3. Count recoils in a short push

Use

$$\begin{aligned} \lambda &= 780\ \mathrm{nm}, \\ \Gamma/(2\pi) &= 6.0\ \mathrm{MHz}, \\ m &= 1.44\times10^{-25}\ \mathrm{kg}. \end{aligned}$$

At resonance and $s_0=1$, calculate:

1. the scattering rate;
2. the force and acceleration;
3. the recoil velocity;
4. the number of cycles and constant-force speed change in $5.0\ \mathrm{\mu s}$.

Solution

At resonance with $s_0=1$,

$$R_{\mathrm{sc}} = \frac{\Gamma}{4}.$$

Numerically,

$$\Gamma = 2\pi \left( 6.0\times10^6 \right) \simeq 3.77\times10^7\ \mathrm{s^{-1}},$$

so

$$R_{\mathrm{sc}} \simeq 9.42\times10^6\ \mathrm{s^{-1}}.$$

The recoil momentum is

$$\hbar k = \frac{h}{\lambda} \simeq 8.49\times10^{-28}\ \mathrm{kg\,m\,s^{-1}}.$$

Therefore

$$\begin{aligned} F &= \hbar kR_{\mathrm{sc}} \simeq 8.00\times10^{-21}\ \mathrm N, \\ a &= \frac{F}{m} \simeq 5.56\times10^4\ \mathrm{m\,s^{-2}}. \end{aligned}$$

The recoil velocity is

$$v_r = \frac{\hbar k}{m} \simeq 5.90\times10^{-3}\ \mathrm{m\,s^{-1}}.$$

In $5.0\ \mathrm{\mu s}$,

$$N = R_{\mathrm{sc}}t \simeq 47.1,$$

and

$$\Delta v = Nv_r \simeq 0.278\ \mathrm{m\,s^{-1}}.$$

The Doppler shift should be checked before extending the constant-force estimate to longer times.

4. Recover the resonant cross section

For an ideal cycling transition, use

$$I_{\mathrm{sat}} = \frac{ \pi h c\Gamma }{ 3\lambda^3 }$$

and the weak resonant rate

$$R_{\mathrm{sc}} \simeq \frac{\Gamma}{2} \frac{I}{I_{\mathrm{sat}}}.$$

Show that $R_{\mathrm{sc}}=\Phi_\gamma\sigma_0$ with $\Phi_\gamma=I/(\hbar\omega)$ and find $\sigma_0$.

Solution

Substitution gives

$$\begin{aligned} R_{\mathrm{sc}} &= \frac{\Gamma}{2} I \frac{ 3\lambda^3 }{ \pi h c\Gamma } \\ &= I \frac{ 3\lambda^3 }{ 2\pi h c }. \end{aligned}$$

The photon flux is

$$\Phi_\gamma = \frac{I}{\hbar\omega} = \frac{ I\lambda }{ h c }.$$

Therefore

$$\sigma_0 = \frac{ R_{\mathrm{sc}} }{ \Phi_\gamma } = \frac{ 3\lambda^2 }{ 2\pi }.$$

This equality relies on the same ideal cycling-transition and polarization conventions used to define $I_{\mathrm{sat}}$.

5. Maximize damping at fixed weak intensity

For two equal weak beams, maximize the displayed $\alpha$ over red detuning. Does maximum friction alone imply minimum temperature?

Solution

Set $x=\Delta/\Gamma>0$. Apart from the constant $8\hbar k^2s_0$, the function is $x/(1+4x^2)^2$. Its derivative is $(1-12x^2)/(1+4x^2)^3$, so

$$\Delta_{\alpha\max}=\frac{\Gamma}{2\sqrt3},\qquad \alpha_{\max}=\frac{9}{4\sqrt3}\hbar k^2s_0.$$

The temperature depends on the ratio of diffusion to friction, not friction alone. Its optimum can therefore occur at a different detuning. For the usual ideal Doppler model it is $\Delta=\Gamma/2$ at weak intensity.

6. Compare fixed-count and Poisson recoil heating

Atoms initially at rest absorb along $z$ and emit isotropically. Compare the mean kinetic energy after exactly $N$ completed cycles with the energy after a Poisson-distributed number of mean $\bar N$. Separate mean-motion energy from momentum-spread energy. Neglect motion-dependent scattering.

Solution

Conditioned on $N$, $\langle\mathbf p\rangle=N\hbar k\hat z$ and $\operatorname{tr}\operatorname{Cov}\mathbf p=N(\hbar k)^2$. Thus the energy is $E_r(N^2+N)$: $E_rN^2$ is directed motion, while $E_rN$ comes from random emission angles. For Poisson counts, $\langle N^2\rangle=\bar N^2+\bar N$, giving $E_r(\bar N^2+2\bar N)$. The extra $E_r\bar N$ is absorption-count noise. This explains the $2E_rR_{\rm sc}$ diffusion heating rate in the Poisson model without assigning two recoil energies of random heating to every fixed-count cycle.

7. Separate forces in a Gaussian beam

A far-detuned Gaussian beam has local Rabi frequency

$$|\Omega(r)|^2 = \Omega_0^2 \exp \left( - \frac{2r^2}{w_0^2} \right).$$

Assume $\Delta>0$ and $|\Delta|\gg\Gamma,|\Omega|$.

1. Find the radial dipole force.
2. Find the local scattering rate.
3. State the direction of each force near the beam axis.

Solution

The ground-state potential is

$$U_g(r) = - \frac{ \hbar\Omega_0^2 }{ 4\Delta } \exp \left( - \frac{2r^2}{w_0^2} \right).$$

Differentiating,

$$F_r^{\mathrm{dip}} = - \frac{ \hbar\Omega_0^2 }{ \Delta } \frac{r}{w_0^2} \exp \left( - \frac{2r^2}{w_0^2} \right).$$

For $\Delta>0$ and $r>0$, this is negative: the red-detuned dipole force points toward the beam axis.

The scattering rate is

$$R_{\mathrm{sc}}(r) \simeq \frac{ \Gamma\Omega_0^2 }{ 4\Delta^2 } \exp \left( - \frac{2r^2}{w_0^2} \right).$$

Its mean force points mainly along the beam propagation direction, not radially, if the spontaneous recoil averages to zero. The focused beam therefore provides radial conservative confinement and longitudinal dissipative push at the same time.

8. Compare fluorescence with impulse

One uniform traveling wave drives a closed atom with zero mean emitted momentum. During a pulse, the mean number of spontaneous emissions is $N_{\rm sp}=40$, and the atom begins in its ground state. Derive the allowed mean impulse. Could a final impulse of $20\hbar k$ be explained solely by changing the internal steady population while keeping that emission count?

Solution

Integrating the transient identity gives

$$\Delta p=\hbar k\,[N_{\rm sp}+\rho_{ee}(t_f)-\rho_{ee}(t_i)].$$

With $\rho_{ee}(t_i)=0$, the impulse lies between $40\hbar k$ and $41\hbar k$. Internal population changes cannot halve it at the same actual emission count under these assumptions. A discrepancy that large requires a changed assumption: count or momentum calibration, different atoms contributing to the two measurements, extra beam momenta, directed emission, or another mechanical force. Optical pumping alone reduces both scattering and push; it is not an independent explanation at fixed true count when each completed cycle has the same mean recoil.

References

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