Light–Matter Interaction

Light–matter interaction is the coupling of charged quantum systems to electromagnetic fields. The phrase covers several distinct models: a prescribed classical laser driving an atom, a molecule exchanging excitations with a quantized cavity mode, an excited state radiating into a continuum, and a detector conditioning a quantum state on the emitted record. They are connected, but they are not interchangeable.

The Quantum Optics chapter takes the next operational step: it treats selected field modes as systems whose states, transformations, correlations, and measurements are themselves the target.

A useful starting decomposition is

$$H = H_{\mathrm{matter}} + H_{\mathrm{field}} + H_{\mathrm{int}} + H_{\mathrm{environment}}.$$

The first question is not “which famous Hamiltonian applies?” It is:

Which matter, field, environmental, and detector degrees of freedom must remain dynamical for the observable and accuracy of interest?

For example, fixing an atom at a point removes recoil and Doppler shifts. Treating a laser as a prescribed field removes depletion of its source. An approximation is useful when the removed effects are smaller than the accuracy needed for the observable.

Enter this chapter

Start with Minimal Coupling and Gauge Choices, then distinguish the Dipole Approximation from the Multipole Expansion. These specify the interaction before any levels or rapidly oscillating terms are removed.

Goal	Lessons to follow
Choose the matter, field, and reservoir description	Models and Scales in AMO Physics
Drive an isolated transition	Two-Level Atom, Rabi Oscillations, Ramsey Interferometry
Include radiation and relaxation	Optical Bloch Equations, Spontaneous Emission, Stimulated Emission
Calculate off-resonant shifts	AC Stark Shift, Dynamic Polarizability
Understand strong-drive spectra and transfer	Dressed States, Autler–Townes Splitting, Electromagnetically Induced Transparency, STIRAP
Include momentum transfer	Radiation Pressure

Helpful background. Time-Dependent Hamiltonians supplies evolution with a prescribed drive, and Spectroscopy distinguishes transition strengths from measured spectra. Quantized Electromagnetic Modes develops the field operators used below.

Choose the radiation model

Choose which field degrees of freedom must remain dynamical from the observable you want to predict.

Description	What it can predict	What must be checked	Detailed treatment
Prescribed classical drive	matter populations, phases, and response to a specified field	source depletion, unmodeled field fluctuations, and atom–field correlations must be negligible for the target	Two-Level Atom and Rabi Oscillations
Retained quantized mode	excitation exchange, photon-number dependence, and atom–field correlations	mode selection, field state, and coupling regime	Quantized Electromagnetic Modes and Jaynes–Cummings Model
Unobserved radiation reservoir	emission and reduced matter dynamics	spectral density, bath memory, and the validity of eliminating emitted modes	Spontaneous Emission and Quantum-Optical Master Equation
Monitored output field	photon records, inferred states, and measurement backaction	collection channels, efficiency, background, and detector response	Photon Counting and Quantum-Jump Trajectories

These descriptions can be combined. A cavity experiment may have a classical pump, a retained quantum cavity mode, unobserved loss channels, and a detector monitoring one output. A prescribed drive does not by itself supply spontaneous emission or photon statistics; include the appropriate reservoir and detection model when those are observables.

A large mean photon number alone does not justify replacing a field by a classical amplitude. Its state, relevant fluctuations, and backaction must also permit that replacement. Conversely, a weak-excitation cavity doublet can follow from coupled classical linear response, so the doublet alone is not a witness of field quantization (Zhu et al., 1990).

A closed atom-plus-mode model describes reversible exchange, including revivals. Free-space exponential decay requires the many-mode radiation problem and its controlled approximations; it cannot be inferred from unitary exchange with one lossless mode. Radiative decay remains coherent in the full matter–field state and appears irreversible in the reduced matter description when emitted modes are unobserved and do not return.

Dressed states also depend on this choice. Classical periodic driving leads to Floquet or rotating-frame states; dressing by a quantized mode can produce atom–field entanglement. Similar spectra do not make those underlying states identical.

Choose the interaction before truncating it

Minimal coupling specifies the electromagnetic interaction, and gauge choices redistribute terms without changing exact predictions when states and observables are transformed consistently. Keep the scalar, linear, and quadratic terms required by the chosen representation; counting internal Coulomb energy twice or dropping the quadratic term solely because it is quadratic changes the model.

The dipole approximation tests field variation across the internal state. It does not decide whether an electric-dipole transition is allowed: a forbidden E1 matrix element can make M1 or E2 coupling leading even when $ka\ll1$. The multipole expansion supplies those alternatives. Internal long-wavelength accuracy also does not freeze the center of mass; recoil, Doppler shifts, and motional sidebands require their own decision.

Which approximations are independent?

The common approximations are independent. Passing one test does not imply the others. Here $a$ is the internal size, $k$ the wave number, $\Omega$ the drive’s Rabi angular frequency, and $g$ a single-photon coupling angular frequency; $n$ labels the populated photon-number sectors.

Reduction	Small parameter or test	What is discarded
nonrelativistic matter	binding and kinetic energies $\ll mc^2$	pair creation and relativistic corrections
long wavelength	$ka\ll1$	spatial field variation across internal state
electric dipole	E1 allowed and dominant	M1, E2, and higher multipoles
prescribed field	source backaction negligible	field depletion, entanglement, photon statistics
two-level truncation	drive and bandwidth small versus leakage gaps	off-resonant matter levels
rotating-wave approximation	near resonance, drive coupling or populated-mode $g\sqrt{n+1}$ small relative to the counter-rotating frequency	rapidly oscillating interaction terms
Markov reservoir	bath correlation time $\tau_B\ll\tau_S$	memory and recurrence
secular master equation	distinct Bohr frequencies separated relative to damping	coherence between unresolved transition sectors
weak-coupling decay	radiative coupling perturbative	nonperturbative dressing and bound states
fixed center of mass	recoil, Doppler, and motional entanglement negligible	translational dynamics

Order of approximations matters

Gauge transformation, matter truncation, field-mode truncation, and the rotating-wave approximation need not commute. For example:

1. transform the full Hamiltonian and then truncate;
2. truncate first and then transform inside the reduced space;

can produce different effective operators. The gauge-truncation problem is discussed by Di Stefano et al. (2019) and Stokes and Nazir (2024). Agreement improves only when the reduced construction preserves the parent theory’s constraints or when convergence is demonstrated.

The same warning applies to eliminating a far-detuned level before or after including decay: coherent Stark shifts and dissipative scattering can be lost if the elimination is incomplete.

Check unwanted transitions

A two-level reduction needs an isolated driven transition, appropriate polarization and preparation, and negligible or explicitly modeled decay out of the pair. Compare unwanted matrix elements with their detunings, include relevant AC Stark shifts, and check the pulse spectrum: a weak pulse with sharp edges can reach another level. Residual multilevel corrections must be smaller than the accuracy sought.

For a far-detuned unwanted transition with projected Rabi frequency $\Omega_r$ and detuning $\Delta_r$, the estimate $P_r\sim|\Omega_r/\Delta_r|^2$ helps rank coherent leakage channels for a sufficiently smooth pulse. It is not a bound on total error: spectral side lobes, multiphoton resonances, interference, accumulated Stark phases, and decay can require a larger model. Exercise 3 uses this screening estimate.

A rotating frame is a change of description, while the rotating-wave approximation discards terms. Those discarded terms can produce Bloch–Siegert shifts and multiphoton effects. Read every detuning together with its sign convention. This chapter uses atom-minus-laser detuning, $\Delta=\omega_0-\omega_L$, so red detuning is positive, as in Two-Level Atom. References using $\Delta_L=\omega_L-\omega_0$ have $\Delta_L=-\Delta$; translate the denominator sign before importing a light-shift or Doppler-force formula.

Compare coherent and dissipative timescales

Let $\Gamma$ denote matter population decay, $\gamma_\phi$ pure dephasing, and $\kappa$ the retained mode’s energy-decay rate. The qualitative regime depends on their competition with the drive and coupling scales:

Regime	Typical hierarchy	Behavior
weak excitation	small saturation parameter or a short perturbative pulse	linear response and small depletion
coherent drive	$\vert \Omega\vert \gg\Gamma,\gamma_\phi$ over pulse time	visible Rabi rotations
saturated response	drive competes with relaxation	power broadening and bounded population
strong cavity coupling	coherent $g$ exceeds relevant losses	resolved exchange or normal-mode splitting
bad-cavity or Purcell regime	cavity loss fast but structured density of states important	modified irreversible emission
non-Markovian reservoir	bath memory not short	time-dependent rates, revivals, bound-state effects

For steady two-level response, judge weak excitation by the saturation parameter, including detuning and coherence decay $\gamma_2=\Gamma/2+\gamma_\phi$, rather than $\Omega/\Gamma$ alone. Optical Bloch Equations gives that parameter and the corresponding line shape. Likewise, compare cavity cooperativities only after translating amplitude, energy, population, and linewidth conventions; Cavity QED specifies the coupling and loss competition.

An unconditional master equation predicts averages over detector records. A monitored photon or homodyne record requires conditioned dynamics or an equivalent counting-statistics calculation. Averaging the conditioned states recovers the unconditional state. A non-Hermitian energy alone does not specify the missing jumps or normalization, and a photon count is not in general a direct reading of the atom’s earlier excited-state population. Carmichael develops this distinction between averaged evolution and measurement records.

Match the calculation to the observable

For spectroscopy, identify which response is measured. Weak-drive line positions and strengths can reflect energy differences and transition matrix elements; strong-drive spectra also include saturation, AC Stark shifts, power broadening, optical pumping, and dressed-state structure. Autler–Townes splitting and electromagnetically induced transparency explain why a split or narrow feature needs a response calculation and cannot be identified from energy levels alone.

For coherent control, specify amplitude, phase, polarization, detuning, and timing. A pulse-area prescription gives ideal inversion or an equal superposition only under its resonance and fixed-phase assumptions; changing the drive phase changes the rotation axis. Finite bandwidth, leakage, spatial intensity variation, phase noise, scattering, and calibration errors enter the achieved map. Rabi oscillations and Ramsey interferometry develop those controls. Adiabatic and dark-state transfer instead depend on eigenstate following, two-photon resonance, and intermediate-state loss; use STIRAP for that choice.

For measurement, absorption, fluorescence, dispersive phase, cavity transmission, and photoionization probe different outgoing observables. Include unobserved loss before applying the detection model. Dispersive probing can reduce population transfer while still entangling light and matter: a nondestructive measurement controls a specified disturbance, without eliminating all backaction. Exercise 4 combines internal states, motion, radiation, and a time-tagged record in one model.

Worked Examples

Long-wavelength estimate for an atom

Take an internal length scale

$$a=0.10\ \mathrm{nm}$$

and light of wavelength

$$\lambda=500\ \mathrm{nm}.$$

Then

$$\begin{aligned} ka &= \frac{2\pi a}{\lambda} \\ &= \frac{ 2\pi(0.10\ \mathrm{nm}) }{ 500\ \mathrm{nm} } \\ &\simeq 1.26\times10^{-3}. \end{aligned}$$

Spatial variation across the atom is small. An E1-forbidden transition can still require M1 or E2 coupling, so this number does not alone decide the leading amplitude.

Rabi frequency from intensity

Use the peak amplitude of a real monochromatic field, $\mathbf E(t)=\boldsymbol\epsilon\mathcal E_0\cos(\omega_Lt+\phi)$, with unit real polarization $\boldsymbol\epsilon$. Then $I=c\epsilon_0\mathcal E_0^2/2$ and $|\Omega|=|d_{eg}|\mathcal E_0/\hbar$, where $d_{eg}=\langle e|\mathbf d\cdot\boldsymbol\epsilon|g\rangle$. The near-resonant rotating-wave coupling has magnitude $\hbar|\Omega|/2$; the positive-frequency field amplitude is $\mathcal E_0/2$.

Let

$$I=1.0\ \mathrm{mW\,cm^{-2}} = 10\ \mathrm{W\,m^{-2}},$$

and assume a polarization-projected dipole magnitude

$$|d_{eg}|=3ea_0.$$

The field amplitude is

$$\begin{aligned} \mathcal E_0 &= \sqrt{ \frac{2I}{c\epsilon_0} } \\ &\simeq 86.8\ \mathrm{V\,m^{-1}}. \end{aligned}$$

Using

$$3ea_0 \simeq 2.54\times10^{-29}\ \mathrm{C\,m},$$

the Rabi scale is

$$\begin{aligned} |\Omega| &= \frac{ |d_{eg}|\mathcal E_0 }{\hbar} \\ &\simeq 2.09\times10^7\ \mathrm{s^{-1}} \\ &\simeq 2\pi\times3.33\ \mathrm{MHz}. \end{aligned}$$

This ideal estimate assumes the quoted intensity occurs at the particle, polarization is perfectly projected onto the transition, and the peak real-field convention specified here applies.

Quantum versus classical amplitude

In the Jaynes–Cummings convention $H_{\mathrm{int}}=\hbar g(a\sigma_++a^\dagger\sigma_-)$ with real $g$, a coherent amplitude $\alpha$ has mean occupation $\overline n=|\alpha|^2$. Replacing $a$ by $\alpha$ gives the classical Rabi magnitude $|\Omega|=2|g\alpha|$ when that replacement is justified.

For a single quantized mode in a coherent state with

$$\overline n=10^6,$$

the semiclassical coupling scale is

$$g\sqrt{\overline n} = 10^3g.$$

For a small deterministic decrease $\delta\overline n=-1$ along a family of coherent states, $\delta|\alpha|/|\alpha|\simeq-1/(2\overline n)$. This is a mean-depletion estimate, not the effect of a conditioned photon subtraction: $a|\alpha\rangle=\alpha|\alpha\rangle$ leaves the normalized coherent state unchanged. Atom–field evolution can also create correlations, so a small change of the mean field alone does not prove that all backaction is negligible. For a number state with the same $\overline n$, phase and fluctuations differ; mean occupation alone does not define the classical limit.

Exercises

1. Test the dipole approximation

A molecule has spatial extent $a=0.50\ \mathrm{nm}$. Compute $ka$ for $\lambda=10\ \mathrm{\mu m}$ infrared light and for $\lambda=0.10\ \mathrm{nm}$ X-rays. Interpret the result.

Solution

For infrared light,

$$\begin{aligned} ka_{\mathrm{IR}} &= \frac{ 2\pi(0.50\ \mathrm{nm}) }{ 10^4\ \mathrm{nm} } \\ &\simeq 3.14\times10^{-4}. \end{aligned}$$

The field is nearly uniform over the molecule.

For the X-rays,

$$\begin{aligned} ka_{\mathrm X} &= \frac{ 2\pi(0.50\ \mathrm{nm}) }{ 0.10\ \mathrm{nm} } \\ &\simeq 31.4. \end{aligned}$$

The long-wavelength expansion is not controlled for the full molecular extent. Spatial phase, momentum transfer, and higher multipoles must be retained. A localized core orbital can have a smaller effective $a$, so the relevant size depends on the transition density rather than only the geometric molecule.

2. Verify Jaynes–Cummings excitation conservation

Let $a$ annihilate a photon in the retained mode and let $\sigma_+=|e\rangle\langle g|$, $\sigma_-=\sigma_+^\dagger$ act on the atom. Atom and mode operators commute. Take $g$ real with units of angular frequency. For

$$H_{\mathrm{int}} = \hbar g \left( a\sigma_+ + a^\dagger\sigma_- \right),$$

show that

$$N_{\mathrm{exc}} = a^\dagger a+\sigma_+\sigma_-$$

commutes with $H_{\mathrm{int}}$.

Solution

Use

$$[a^\dagger a,a]=-a, \qquad [a^\dagger a,a^\dagger]=a^\dagger,$$

and

$$\begin{aligned} [\sigma_+\sigma_-,\sigma_+] &= \sigma_+, \\ [\sigma_+\sigma_-,\sigma_-] &= -\sigma_-. \end{aligned}$$

Then

$$\begin{aligned} [N_{\mathrm{exc}},a\sigma_+] &= (-a)\sigma_+ + a\sigma_+ = 0, \\ [N_{\mathrm{exc}},a^\dagger\sigma_-] &= a^\dagger\sigma_- - a^\dagger\sigma_- = 0. \end{aligned}$$

Therefore

$$[N_{\mathrm{exc}},H_{\mathrm{int}}]=0.$$

The counter-rotating terms $a\sigma_-$ and $a^\dagger\sigma_+$ do not conserve this excitation number.

3. Estimate leakage from a two-level approximation

A desired transition is driven with $\Omega/2\pi=2\ \mathrm{MHz}$. Two unwanted states have projected couplings $0.3\Omega$ and $1.2\Omega$, at ordinary-frequency detunings $\Delta_1/(2\pi)=40\ \mathrm{MHz}$ and $\Delta_2/(2\pi)=300\ \mathrm{MHz}$, respectively. Estimate their separate leakage scales.

Solution

For the first state,

$$\begin{aligned} P_1 &\sim \left( \frac{ 0.3(2\ \mathrm{MHz}) }{ 40\ \mathrm{MHz} } \right)^2 \\ &= 2.25\times10^{-4}. \end{aligned}$$

For the second,

$$\begin{aligned} P_2 &\sim \left( \frac{ 1.2(2\ \mathrm{MHz}) }{ 300\ \mathrm{MHz} } \right)^2 \\ &= 6.4\times10^{-5}. \end{aligned}$$

The more strongly coupled state leaks less because it is much farther detuned. These estimates do not include pulse-spectrum side lobes, interference, Stark shifts, or decay. They rank which channels deserve a full multilevel simulation.

4. Include motion and photon detection

An experiment drives a trapped ion with a laser and records time-tagged fluorescence photons while resolving motional sidebands. List the minimum degrees of freedom and approximations needed for a model that predicts both internal populations and the photon record.

Solution

A minimum defensible model includes:

1. the relevant internal levels, including any shelving or decay branches;
2. quantized trap motion for each resolved motional mode;
3. the laser as a characterized classical drive, including wave vector, phase, detuning, and intensity;
4. position-dependent coupling $e^{i\mathbf k\cdot\mathbf x}$, expanded in the Lamb–Dicke parameter only if that parameter is small;
5. spontaneous-emission jump operators with branching ratios and recoil;
6. any relevant laser phase noise, magnetic noise, or motional heating;
7. a photon-counting unraveling or measurement model with collection efficiency, dark counts, timing response, and dead time.

The rotating-wave, two-level, Lamb–Dicke, Markov, and secular approximations must each be checked. An unconditional master equation predicts averaged populations and mean fluorescence, but a time-tagged photon record requires a conditioned trajectory or equivalent counting-statistics calculation.

References

• L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover, 1987 — semiclassical two-state dynamics and optical Bloch equations.
• H. J. Carmichael, Statistical Methods in Quantum Optics 1, Springer, 1999, doi:10.1007/978-3-662-03875-8 — master equations, regression, and photon-counting trajectories.
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics, Wiley, 1989 — canonical, multipolar, and quantized-field formulations.
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications, Wiley, 1992 — coherent excitation, radiative processes, dressed atoms, and effective Hamiltonians.
• O. Di Stefano et al., “Resolution of Gauge Ambiguities in Ultrastrong-Coupling Cavity Quantum Electrodynamics,” Nature Physics 15, 803–808 (2019), doi:10.1038/s41567-019-0534-4.
• R. J. Glauber, “Coherent and Incoherent States of the Radiation Field,” Physical Review 131, 2766–2788 (1963), doi:10.1103/PhysRev.131.2766 — coherent states and optical coherence.
• E. T. Jaynes and F. W. Cummings, “Comparison of Quantum and Semiclassical Radiation Theories with Application to the Beam Maser,” Proceedings of the IEEE 51, 89–109 (1963), doi:10.1109/JRPROC.1963.1664 — one-mode matter–radiation exchange and the quantum–semiclassical comparison.
• A. F. Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, “Ultrastrong Coupling between Light and Matter,” Nature Reviews Physics 1, 19–40 (2019), doi:10.1038/s42254-018-0006-2 — regimes where rotating-wave, mode, and truncation assumptions become delicate.
• R. Loudon, The Quantum Theory of Light, 3rd ed., Oxford University Press, 2000 — field quantization, photon statistics, and radiative processes.
• L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995 — classical and quantum coherence, photodetection, and propagation.
• E. A. Power and S. Zienau, “Coulomb Gauge in Non-Relativistic Quantum Electro-Dynamics and the Shape of Spectral Lines,” Philosophical Transactions of the Royal Society A 251, 427–454 (1959), doi:10.1098/rsta.1959.0008 — multipolar transformation and interaction representation.
• I. I. Rabi, “Space Quantization in a Gyrating Magnetic Field,” Physical Review 51, 652–654 (1937), doi:10.1103/PhysRev.51.652 — coherent near-resonant spin driving.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997 — few-level atoms, quantized modes, coherence, and dissipation.
• B. W. Shore, The Theory of Coherent Atomic Excitation, Wiley, 1990 — multilevel coherent dynamics, pulse methods, and approximation control.
• A. Stokes and A. Nazir, “Gauge Non-Invariance Due to Material Truncation in Ultrastrong-Coupling Quantum Electrodynamics,” Nature Physics 20, 376–378 (2024), doi:10.1038/s41567-023-02155-8 — active debate over gauge-consistent finite-level constructions.
• Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi Splitting as a Feature of Linear-Dispersion Theory: Analysis and Experimental Observations,” Physical Review Letters 64, 2499–2502 (1990), doi:10.1103/PhysRevLett.64.2499 — linear-response spectra and the quantum–classical comparison.