Quantum Optics and the Photon
Light is the cleanest quantum-statistical laboratory there is. A mode of the electromagnetic field is exactly a harmonic oscillator (Oscillations and Mechanical Waves), its excitations are bosons (Quantum Statistics), and detectors count them one at a time; so the statistical questions that are awkward to pose for a gas of atoms — what is the distribution of occupation numbers, what are the correlations between successive detections, what states have no classical description at all — can here be asked and answered with photodetectors and coincidence counters. This chapter takes photon statistics, coherence, antibunching, squeezing and cavity electrodynamics in that order, and in each case names the measurement that settled the matter. Its running theme is the one question that took sixty years to close: which optical phenomena actually require the field to be quantized, and which are reproduced by a classical wave falling on quantized matter.
It sits deliberately inside the statistical part rather than in Part IX — Quantum Mechanics: the objects of study are not single-atom amplitudes but the correlation functions of a many-boson field, and the Bose statistics of Quantum Statistics is what makes photons bunch. The field quantization it uses is set up in Generalized Classical Field Theory and treated dynamically in Quantum Electrodynamics and Renormalization; the entanglement of photon pairs is the material of Entanglement and Bell Tests and Experiment: Bell Tests, and the coherence measurements here are the optical counterpart of the atomic ones in Experiment: Bose–Einstein Condensation. Standard monographs are [Mandel:1995] [Loudon:2000].
Quantum Optics and the Photon: all derivations of this chapter are pending.
The quantized radiation field
Modes, Fock states and the vacuum
[Reserved: Dirac's quantization of the free field as a set of independent oscillators, and the first derivation of the emission and absorption rates from it [Dirac:1927]; annihilation and creation operators with \(\comm{a}{a^{\dagger}}=1\), number states \(\ket{n}\), and the vacuum energy \(\tfrac{1}{2}\hbar\omega\) per mode; the quadrature operators and their commutator as the origin of every uncertainty statement in this chapter; the field-theoretic setting of Generalized Classical Field Theory.]
Einstein A and B coefficients
[Reserved: Einstein's statistical argument that detailed balance between matter and blackbody radiation requires stimulated emission in addition to absorption and spontaneous emission, with \(A/B=8\pi h\nu^{3}/c^{3}\) [Einstein:1917b]; the momentum transfer \(h\nu/c\) in the same paper, later the basis of laser cooling (Experiment: Bose–Einstein Condensation); the relation to the Planck spectrum of Black-Body Radiation and Planck's Hypothesis; why stimulated emission makes the laser of Section 122.7 possible.]
Is the photon necessary?
[Reserved: the honest accounting — the photoelectric effect (The Photon: Photoelectric and Compton Effects) and blackbody spectrum admit semiclassical explanations with quantized matter and a classical field, so neither proves field quantization; the experiments that do: the photoelectric coincidence test that violates a classical inequality [Clauser:1974a], the beam-splitter anticorrelation with a single-photon input [Grangier:1986], and the antibunching of Section 122.4; Compton scattering as the momentum-space argument.]
Photon statistics
Photodetection and photocount distributions
[Reserved: the photodetection formalism and the normally ordered counting formula [Glauber:1963a]; the Poissonian distribution \(P(n)=\ee^{-\bar{n}}\bar{n}^{n}/n!\) for a stable classical wave against the Bose–Einstein distribution \(P(n)=\bar{n}^{n}/(1+\bar{n})^{n+1}\) for a single thermal mode, derived in Quantum Statistics; the measurement that exhibited both distributions from one apparatus by driving a laser above and below threshold [Arecchi:1965]; dead time and quantum efficiency as the systematic limits.]
Poissonian, super- and sub-Poissonian light
[Reserved: the variance of the photon number as a classifier; the Mandel parameter \(Q=(\langle(\Delta n)^{2}\rangle-\bar{n})/\bar{n}\), negative only for states with no positive-definite classical distribution [Mandel:1995]; the first observation of sub-Poissonian light, in the fluorescence of a single atom [Short:1983]; photon bunching and antibunching as the temporal statement of the same fact.]
The Hanbury Brown–Twiss experiment
[Reserved: intensity interferometry as a way of measuring stellar angular diameters with baselines immune to atmospheric phase noise, first demonstrated at radio wavelengths [HanburyBrown:1954]; the laboratory experiment with a mercury arc, showing a positive correlation between the currents of two photodetectors on the two arms of a beam splitter [HanburyBrown:1956]; the measurement of the angular diameter of Sirius, \(3.1\times 10^{-8}\) radians — about \(6.3(5)\) milliarcseconds [HanburyBrown:1956b]; the objection that photons could not correlate at all, and Purcell's resolution in terms of Bose statistics [Purcell:1956], which makes this the founding measurement of the chapter.]
Coherence
Correlation functions and the Glauber hierarchy
[Reserved: Glauber's normally ordered correlation functions \(G^{(n)}\) and the definition of coherence to all orders, replacing the first-order (interferometric) notion inherited from classical optics [Glauber:1963a]; the normalized second-order function \(g^{(2)}(\tau)\), with \(g^{(2)}(0)=2\) for thermal light, \(1\) for a coherent state and \(0\) for a single photon; the classical inequalities \(g^{(2)}(0)\geq1\) and \(g^{(2)}(0)\geq g^{(2)}(\tau)\) whose violation is the operational definition of nonclassical light.]
Coherent states
[Reserved: the eigenstates of the annihilation operator, \(a\ket{\alpha}=\alpha\ket{\alpha}\), their Poissonian number distribution, minimum-uncertainty quadratures and non-orthogonal overcompleteness [Glauber:1963]; why they are the quantum state closest to a classical field and the state a laser well above threshold actually produces; the displacement operator and the coherent state as a displaced vacuum, connecting to the harmonic oscillator of Elementary Quantum Systems.]
The optical equivalence theorem
[Reserved: the diagonal coherent-state representation of an arbitrary density operator, introduced independently by Sudarshan [Sudarshan:1963] and Glauber [Glauber:1963]; the theorem that every normally ordered expectation is a classical average over the weight \(P(\alpha)\); the precise sense in which a state is nonclassical — \(P\) fails to be a probability density — and why that criterion, not “smallness” of the field, is the right one; the standing dispute over naming, recorded honestly.]
Nonclassical light
[Reserved: the catalogue of states with no classical analogue and the signature by which each is recognized — number states (sub-Poissonian statistics), antibunched fluorescence (Section 122.4), squeezed states (Section 122.5) and entangled photon pairs (Entanglement and Bell Tests); quantum state tomography by balanced homodyne detection as the modern way to exhibit a negative quasiprobability; the reconstruction of Fock states and their Wigner functions [Lvovsky:2001].]
Antibunching and single photons
Resonance fluorescence and photon antibunching
[Reserved: the prediction that light scattered by a single atom must show \(g^{(2)}(0)<g^{(2)}(\tau)\), because the atom cannot emit again until it has been re-excited [Carmichael:1976]; the measurement on a dilute sodium beam that observed it [Kimble:1977], the first phenomenon with no classical wave description whatever; the Mollow triplet and Rabi oscillations visible in the correlation function; residual multi-atom events as the systematic that had to be extrapolated away.]
Single-photon sources
[Reserved: the progression from attenuated lasers (which remain Poissonian and are not single-photon sources) to true antibunched emitters — single molecules at room temperature [Lounis:2000], nitrogen-vacancy centres in diamond [Kurtsiefer:2000], semiconductor quantum dots in a turnstile device [Michler:2000], and deterministic emission from a single atom in a cavity [Kuhn:2002]; the figures of merit — purity \(g^{(2)}(0)\), indistinguishability, and collection efficiency — with their currently measured values.]
Two-photon interference
[Reserved: the Hong–Ou–Mandel effect, in which two indistinguishable photons entering opposite ports of a balanced beam splitter always leave together, producing a coincidence dip whose width measures their wavepacket overlap to subpicosecond precision [Hong:1987]; the dip as an interference of amplitudes, not of intensities, and its use as the standard test of photon indistinguishability; the connection to boson exchange symmetry in Identical Particles.]
Squeezed light
Quadratures and the uncertainty ellipse
[Reserved: the quadratures \(X_{1},X_{2}\) with \(\langle\Delta X_{1}^{2}\rangle\langle\Delta X_{2}^{2}\rangle\geq1/16\); squeezing as redistribution of vacuum fluctuations between them at fixed product; the review that set the programme [Walls:1983]; Caves's demonstration that the shot-noise limit of an interferometer is set by vacuum fluctuations entering the unused port, and can be beaten by injecting squeezed vacuum there [Caves:1981] — the theoretical origin of Section 122.5.3.]
Generation and detection
[Reserved: the first observation of squeezed light, by four-wave mixing in a sodium-vapour cavity, with a noise reduction of about \(7\,\mathrm{\%}\) below vacuum [Slusher:1985]; parametric down-conversion in an optical parametric oscillator, which reached more than \(60\,\mathrm{\%}\) and remains the standard method [Wu:1986]; balanced homodyne detection with a local oscillator as the phase-sensitive measurement; loss as the enemy, since every inefficiency mixes vacuum back in.]
Squeezed light in gravitational-wave detectors
[Reserved: the application that made squeezing an engineering tool — injection of squeezed vacuum into the dark port of the kilometre-scale interferometers of Advanced LIGO [Aasi:2015], first demonstrated in LIGO at about \(2\) decibels of shot-noise reduction [Aasi:2013] and now routine at roughly \(3\) decibels across the observing band [Tse:2019]; the resulting increase in detection rate; radiation-pressure back-action and frequency-dependent squeezing as the response; links to Experiment: Gravitational Waves.]
Cavity quantum electrodynamics
The Jaynes–Cummings model
[Reserved: one two-level atom coupled to one field mode under the rotating-wave approximation, and the comparison with the semiclassical theory it was written to test [Jaynes:1963]; the dressed-state ladder with splitting \(2g\sqrt{n+1}\); collapse and revival of the Rabi oscillation as a signature that the field is quantized, observed in the one-atom maser [Rempe:1987] and resolved into individual Fock components in the microwave regime [Brune:1996a].]
Modified spontaneous emission
[Reserved: spontaneous emission as a property of the mode structure rather than of the atom alone; Purcell's enhancement factor \(F=3Q\lambda^{3}/4\pi^{2}V\) for an atom in a resonant cavity [Purcell:1946b]; the observed enhancement for a Rydberg atom between mirrors [Goy:1983] and the complementary inhibition when the cavity is detuned [Hulet:1985]; the practical consequence for the single-photon sources of Section 122.4.2.]
Strong coupling and vacuum Rabi splitting
[Reserved: the strong-coupling condition \(g>\kappa,\gamma\) in which a single quantum is exchanged coherently before it is lost; the observation of normal-mode splitting for one atom in an optical cavity [Thompson:1992], the optical counterpart of the microwave work; the extension to solid-state systems; why the splitting scales as \(\sqrt{N}\) with atom number, and what a single-atom measurement therefore has to control.]
Photon counting without absorption
[Reserved: quantum non-demolition measurement of photon number by the phase shift imprinted on a Rydberg atom crossing the cavity; the progressive collapse of a coherent field onto a Fock state, observed photon by photon, and the subsequent quantum jumps as photons decay [Guerlin:2007]; the same apparatus used to watch a mesoscopic superposition decohere at a rate proportional to its size [Brune:1996], an experimental entry point to Open Quantum Systems and Decoherence; the first-hand account [Haroche:2013].]
The laser
Stimulated emission, inversion and the maser
[Reserved: population inversion as the condition for net gain, and why it is forbidden in thermal equilibrium (Statistical Mechanics); the ammonia beam maser, the first device to oscillate on stimulated emission [Gordon:1954]; three- and four-level pumping schemes; the inverted medium as a negative-temperature system, and the care that statement requires.]
The optical maser and its realization
[Reserved: the proposal to extend maser action to optical frequencies using a Fabry–Pérot resonator to select modes, with the threshold condition and the predicted linewidth [Schawlow:1958]; the first working laser, a flash-pumped ruby rod emitting at \(694.3\,\mathrm{nm}\) [Maiman:1960]; the semiclassical theory of the laser and its mode competition [Lamb:1964]; the helium–neon and semiconductor lasers that made the technique general.]
Laser linewidth and coherence
[Reserved: the Schawlow–Townes limit \(\Delta\nu=\pi h\nu(\Delta\nu_{c})^{2}/P\), spontaneous emission into the lasing mode as the irreducible source of phase diffusion [Schawlow:1958]; the output as a coherent state with a slowly diffusing phase, and its measured \(g^{(2)}(0)=1\) [Arecchi:1965]; the laser transition as a nonequilibrium phase transition with the field amplitude as order parameter (Phase Transitions and Critical Phenomena); modern stabilized lasers and the frequency comb, whose metrology belongs to Experiment: Precision Spectroscopy and Atomic Clocks.]
Zewail: femtosecond probing of the transition state (1987)
The laser of Section 122.7 enters here not as the object of study but as a shutter. Mode locking compresses its output into packets shorter than one vibration of a chemical bond, and two such packets — one to start a reaction, one to photograph it — resolve the motion of the nuclei themselves. What that buys is the configuration a reacting molecule occupies on its way from reactants to products: the activated complex of Eyring's rate theory [Eyring:1935], which for half a century had been an inference drawn from measured rate constants and never an observation, because it endures only as long as it takes two atoms to move apart by about their own diameter. Femtochemistry is the demonstration that this interval is long enough to be timed [Dantus:1987] [Rosker:1988].
The section sits in this chapter as the ultrafast limit of the material around it. The pulse doing the timing is a coherent state (Section 122.3.2) whose duration and spectral width are conjugate through the Fourier relation of Fourier Analysis and Integral Transforms, so a shutter short enough to resolve nuclear motion is necessarily broad enough to span the vibrational structure it interrogates; that structure is the molecular spectroscopy of Atoms and Molecules.
Apparatus
A colliding-pulse mode-locked ring dye laser [Fork:1981], the source that first delivered pulses shorter than \(100\,\mathrm{fs}\), amplified and frequency-converted into two synchronized beams: an ultraviolet pump lying inside the dissociative absorption continuum of the parent molecule, and a tunable probe resonant with an electronic transition of one of the fragments. The two travel separate arms of a Michelson-type splitter, the probe arm carrying a retroreflector on a computer-driven translation stage. Sample: a collision-free beam of the parent molecule, so that what is timed is one molecule's internal motion and not an encounter with a neighbour. Detection: laser-induced fluorescence excited by the probe, collected by a photomultiplier and gated so that only light following the probe is counted.
The translation stage is the clock, and its arithmetic is why the technique is possible at all. Light travels \(0.3\,\mu\mathrm{m}\) in \(1\,\mathrm{fs}\), so a delay step of \(10\,\mathrm{fs}\) is \(1.5\,\mu\mathrm{m}\) of stage travel in a double-pass arm — an ordinary mechanical tolerance. No electronic gate resolves \(100\,\mathrm{fs}\); a screw thread does.
Procedure
The pump promotes the molecule to a dissociative surface and thereby fixes the zero of time, to within the pulse duration. The probe arrives a delay \(\tau\) later and lifts whatever the system has become to a fluorescing state; the fluorescence yield, recorded against \(\tau\), is the signal. Two probe settings are used, and they answer different questions. Tuned to a transition of the free fragment, the probe counts finished products, and the curve is a rise to a plateau whose half-rise time is the appearance time of those products. Detuned from that transition, so that it is resonant only while the departing fragments still perturb each other's levels, the probe counts molecules caught in transit, and the curve is a transient that rises and falls. The delay is scanned in steps, many shots are averaged at each step, and the pump–probe cross-correlation, measured in a nonlinear crystal, fixes both the time resolution and the origin.
[Reserved: the systematic checks — pump and probe intensity dependence, confirming one photon of each; beam density low enough to exclude collisions inside the observation window; calibration of zero delay and of the instrument response function; the deconvolution by which a transient narrower than the cross-correlation is recovered.]
Observations and data
The vibrational fundamentals of molecules, read off infrared absorption and Raman scattering, occupy a band of wavenumbers running from about \(100\,/\mathrm{cm}\) for the bending of heavy atoms to about \(4400\,/\mathrm{cm}\) for the stretch of the hydrogen molecule [Herzberg:1950]. The period of the corresponding nuclear motion,
therefore lies between about \(330\,\mathrm{fs}\) and \(8\,\mathrm{fs}\). A configuration the nuclei pass through once and do not return to — the transition state of a dissociation — endures for a time of this same order, and is invisible to any probe slower than it.
Derivation. A wavenumber is a frequency divided by the speed of light, so \(\nu=c\tilde{\nu}\) and the period is \(T=1/(c\tilde{\nu})\), which is Equation (122.1). A typical stretching fundamental at \(1000\,/\mathrm{cm}\), that is \(10^{5}\,/\mathrm{m}\), gives a frequency of \(3.0\times 10^{13}\,\mathrm{Hz}\) and a period of \(33\,\mathrm{fs}\); the two ends of the observed band give \(330\,\mathrm{fs}\) and about \(8\,\mathrm{fs}\). A second and independent estimate uses only kinematics: the fragments of a dissociation stop interacting once they have separated by a distance of order the bond length itself, about \(0.1\,\mathrm{nm}\), and they separate at the recoil speed set by the energy released, of order \(1\,\mathrm{km}/\mathrm{s}\), so the transit time \(d/v\) is of order \(100\,\mathrm{fs}\). Both routes give the same order of magnitude, and the order is what matters: a probe of duration \(\Delta t\) cannot resolve structure in time shorter than \(\Delta t\), so an apparatus that is to watch nuclei move must carry a shutter of a few tens of femtoseconds. Nothing electronic is that fast. The shutter therefore has to be the light pulse itself, and the delay has to be generated as a difference of optical path.
∎Photodissociation of a triatomic — the worked case is \(\mathrm{ICN}\to\mathrm{I}+\mathrm{CN}\) — started by an ultraviolet pump and interrogated by a delayed probe yields two distinguishable signals. With the probe tuned to a transition of the free CN fragment, the fluorescence rises from zero and levels off: the products appear after a delay rather than at once. With the probe detuned from that transition, so that it is resonant only while the departing fragments still perturb each other's levels, the signal instead rises and falls again — a transient that exists only during the passage through the intermediate configuration. Both the delay and the width of the transient are of order \(100\,\mathrm{fs}\) [Dantus:1987] [Rosker:1988].
The shape of the pump–probe signal for two fragments separating on a repulsive potential energy surface: the classical clocking relation that maps probe detuning onto internuclear separation, the wave-packet computation of the fluorescence yield as a function of delay, and the extraction of the free-fragment appearance time from the saturating on-resonance curve
When the surface reached by the pump is not simply repulsive but is held together at large separation by an avoided crossing with a second surface, the same measurement returns a train of transients instead of one. In sodium iodide, whose covalent and ionic curves cross at several times the equilibrium bond length, the nuclear wave packet prepared by the pump oscillates in the resulting well and returns to the same internuclear separation once per vibrational period, a fixed fraction of it escaping to free atoms at each pass through the crossing. The probe accordingly records an oscillation whose period is that of the trapped motion and whose envelope decays, while the signal from free sodium rises in steps synchronized with it [Rose:1988].
The two-state model of alkali-halide dissociation: the diabatic covalent and ionic curves and the adiabatic well their crossing produces, the Landau–Zener probability of remaining on the adiabatic branch at each pass, and the resulting periodic recurrences with a geometrically decaying envelope and a step-wise product signal
Interpretation
Rate theory describes a reaction through an activated complex in equilibrium with the reactants and reports one number, the rate constant, which is an average over an enormous number of uncorrelated barrier crossings [Eyring:1935]. The pump–probe measurement removes that average: preparing every molecule in the sample at a common instant makes the subsequent evolution of the ensemble a function of time, and it is that evolution, not its integral, which is recorded. Three consequences follow. The transition state moves from the class of inferred quantities to the class of observed ones, since it now has a duration and the duration is what appears on the chart. The reaction coordinate is followed rather than postulated, because the detuning of the probe selects the internuclear separation at which the system is counted. And the assumption on which rate theory rests, that the barrier region is crossed once, becomes testable — in the alkali halide of Phenomenon 122.3 it visibly fails, the system crossing and recrossing many times before it commits.
Two limits should be stated with the result. The measured interval is a transit time, not the lifetime of a stationary state, so it has no meaning independent of the observable used to define it; and the zero of time and the resolution are both fixed by the pump–probe cross-correlation, so every quoted duration is only as good as the characterization of that correlation.
[Reserved: the extension of the method to bimolecular reactions initiated inside a van der Waals complex, to the condensed phase where the solvent turns the problem into one of Open Quantum Systems and Decoherence, and to ultrafast electron and X-ray diffraction, which replace the spectroscopic probe by one that reads geometry directly; coherent control as the inverse problem, in which the pulse is shaped to steer the outcome rather than to observe it.]
Primary references
[Dantus:1987] [Rosker:1988] [Rose:1988]. The light source that made the timescale accessible is [Fork:1981]; the transition-state concept under test is [Eyring:1935]; Zewail's own survey of the field, covering both measurements above and much besides, is [Zewail:2000].
Which-path information and the quantum eraser
Complementarity and the duality relation
[Reserved: the quantitative duality relation \(D^{2}+V^{2}\leq1\) between which-path distinguishability and fringe visibility [Englert:1996], replacing the qualitative slogan; Wheeler's delayed-choice configuration realized with a single-photon source and a randomly switched output beam splitter [Jacques:2007]; why momentum transfer is not the mechanism, and what remains of the uncertainty-principle account.]
The quantum eraser
[Reserved: the proposal to mark the path with an internal atomic state and then erase the mark, restoring interference in coincidence [Scully:1982]; the delayed-choice realization with down-converted photon pairs, in which the decision to erase is taken after the signal photon has been detected [Kim:2000]; the essential and often-omitted point that no fringes appear in the unconditioned signal, so no signalling is possible; the interpretive questions deferred to Interpretations (Evidence-Anchored).]