Quantum optics, from the photon to stored light

These notes exist for one reason: so that anyone in the group can read the Zeeman EIT guide and understand why every part of the bench is there and what the numbers in it mean. They start from first-year physics and climb, step by step, to the collective light–matter physics behind the project. They do not replace a textbook. Every section ends with real sources (all references in the bibliography were checked against their publisher records, and where a statement comes from a particular paper or data sheet it says so).

The notes have three layers. Primers P1–P5 give the history, the motivation, and the mathematics, quantum mechanics and quantum optics you need. Lectures L1–L12 build the physics of the experiment, from the photon to stored light and strong coupling. Lecture L13 turns to the bench itself: polarization, Gaussian beams, acousto-optic modulators, laser locking, cells and shielding, detectors and laser safety.

The physics in this project did not arrive in one step. It is the meeting of three older lines: quantization of light and matter (1900–1930), coherent control of two-level systems by oscillating fields (1937 onward, first with spins, then with lasers), and the laser itself (1960). Electromagnetically induced transparency was observed in 1991, and the step to slow, stopped and stored light followed within a decade. The timeline lists the landmark papers; every entry links to its source in the bibliography.

Light is the natural carrier of quantum information: photons travel fast, interact weakly with their surroundings and can be sent through fibre or free space. Quantum key distribution (the 1984 proposal of Bennett and Brassard, reprinted in Bennett & Brassard 2014), quantum communication in general (Gisin & Thew 2007) and the proposals for a quantum internet (Kimble 2008) all rely on photons as “flying qubits”. But weak interaction is also the problem: photons are hard to hold, to delay or to make interact. Matter is the natural place to store and process information. Atoms, ions, spins in crystals and superconducting circuits keep a state for a useful time and can be controlled precisely, but are poor at travelling. A quantum memory is the interface: it takes in a photon’s quantum state, holds it, and gives it back on demand (Lvovsky et al. 2009; Hammerer et al. 2010).

Why can the photon not simply be amplified and sent on? Because an unknown quantum state cannot be copied (the no-cloning theorem). The only way to cover a long distance is to split it into segments, create entanglement across each segment, and join the segments. The segments succeed at random times, so each must be held until its neighbours also succeed. That holding device is a quantum memory, and the scheme is the quantum repeater (Briegel et al. 1998; Sangouard et al. 2011). The DLCZ proposal showed how atomic ensembles and linear optics could do this (Duan et al. 2001).

Quantum optics uses a small set of mathematical ideas over and over. This primer collects them, with the lab numbers they will be applied to. If you studied linear algebra and complex numbers recently, skim; otherwise work the exercises at the end of each part.

Every oscillation in the notes is written with a complex exponential. A complex number z = a + ib has a magnitude |z| and a phase φ, and Euler’s formula links the two descriptions. Multiplying by e^iφ rotates z by φ in the complex plane without changing its length: this is what a phase shift is. Quantum amplitudes are complex numbers; probabilities are their squared magnitudes, so a common phase never matters but a relative phase does (L2).

A quantum state of an n-level system is a vector with n complex components. The inner product of two vectors measures their overlap; two states with zero overlap are orthogonal (perfectly distinguishable). A set of mutually orthogonal, normalised vectors is a basis; any state can be written as a weighted sum of basis vectors, and the weights are the amplitudes. For the two ground states |1⟩ and |2⟩ of the Λ system, or the lower and upper states |g⟩ and |e⟩, the basis is just “which level”.

Physical quantities (energy, spin, polarization) are represented by matrices. The special vectors that a matrix only stretches, never rotates, are its eigenvectors, and the stretch factors are the eigenvalues. For the Hermitian matrices of quantum mechanics the eigenvalues are real (they are the possible measurement results) and the eigenvectors are orthogonal. Time evolution under a constant Hamiltonian is a matrix exponential: each energy eigenstate simply acquires a phase.

Dirac’s bra–ket notation is shorthand for vectors and their conjugate transposes: |ψ⟩ is a column vector (a “ket”), ⟨ψ| the corresponding conjugate row (a “bra”), ⟨φ|ψ⟩ an inner product (a number), and |ψ⟩⟨φ| an outer product (a matrix). The projector |e⟩⟨e| picks out the excited-state part of a state; the sum of projectors over a full basis is the identity (completeness). Operators like the lowering operator σ̂ = |g⟩⟨e| are written as outer products, which makes their action obvious: take the part in |e⟩ and put it in |g⟩.

A signal in time and its spectrum in frequency are two views of the same thing, related by the Fourier transform. A short pulse has a broad spectrum; a long, steady wave has a narrow one. Two results are used constantly: a Gaussian pulse of FWHM duration Δt has FWHM bandwidth Δν = 0.441/Δt, and a field whose amplitude decays as e^−Γt/2 has a Lorentzian power spectrum of FWHM Γ (in rad/s), that is Γ/2π in Hz. The second is the natural line shape of an atom (L4); the first limits how fast a pulse can be before it no longer fits inside the EIT window (L8).

Two equations cover most of what follows. Exponential decay, dx/dt = −γx, gives x = x₀e^−γt: populations and coherences decay this way. Harmonic oscillation, d²x/dt² = −ω²x, gives cos(ωt) and sin(ωt): a driven two-level atom oscillates between its states in just this way, as do the electromagnetic field and the polariton. A rotating frame is a change of variables that removes a fast oscillation (such as the 377 THz optical carrier) so that only slow changes remain: this is the idea behind the rotating-wave approximation (L3).

Quantum mechanics rests on a few rules (Griffiths & Schroeter; Sakurai & Napolitano; Feynman et al.). State: a system is described by a normalised vector |ψ⟩ in a complex vector space. Observables: each measurable quantity (energy, position, spin component) is a Hermitian operator; its eigenvalues are the possible results. Measurement: the probability of result a is |⟨a|ψ⟩|² (the Born rule, Born 1926), and afterwards the system is in the state |a⟩. Evolution: between measurements the state evolves smoothly under the Schrödinger equation (Schrödinger 1926). Combining systems: the state space of two systems is the tensor product of their spaces, which allows entanglement.

Operators do not always commute: the order in which you apply them matters. The commutator [Â, B̂] = ÂB̂ − B̂Â measures how much. Position and momentum have the fundamental commutator [x̂, p̂] = iħ, which leads to the uncertainty relation: no state has both a sharply defined position and a sharply defined momentum (Griffiths & Schroeter). For atoms the more useful consequence is a Fourier relation between lifetime and linewidth: a level that lives for a time τ has a spectral width of order 1/(2πτ) (L4).

Angular momentum is quantized: the length of an angular momentum J has the value √(J(J+1))ħ and its projection on any axis takes only the 2J+1 values mħ, with m from −J to +J (Gerlach & Stern 1922). A spin-½ (an electron, a proton, or an effective two-level system) has just two projections, ±ħ/2, and is described by the Pauli matrices. Every pure state of a two-level system is a point on a sphere, the Bloch sphere (L2; Feynman et al. 1957). Rubidium’s valence electron has spin ½ and its nucleus has spin 3/2, and the sublevels of the experiment are the projections of the total angular momentum F (L5).

The quantum harmonic oscillator is the most important model system in quantum optics because each mode of the electromagnetic field is one (Dirac 1927). Its energies are equally spaced, E_n = ħω(n + ½), with a zero-point energy ħω/2. Two operators, the lowering and raising (annihilation and creation) operators â and â†, move between levels. In the field picture they remove and add a photon of energy ħω, which is why the next primer can quantize light with the same algebra.

In a hydrogen-like atom the energy depends mainly on the principal quantum number n (E_n ≈ −13.6 eV/n²). The electron’s orbital angular momentum l and spin ½ combine to J (fine structure), and J combines with the nuclear spin I to F (hyperfine structure). Alkali atoms such as rubidium have one valence electron outside closed shells, so they behave much like hydrogen with adjusted energies (Foot 2005). Light couples states through the electric-dipole interaction, which obeys selection rules: the orbital quantum number changes by one (Δl = ±1), and the magnetic quantum number changes by 0 or ±1 depending on the light’s polarization (L5). That is why the D lines of rubidium (5S → 5P) are strong and why σ⁺ and σ⁻ light connect different sublevels.

The semiclassical picture, in which atoms are quantum but light is a classical wave, explains most of what follows (Rabi flopping, EIT, slow light). But three effects need the quantum light field: spontaneous emission, photon counting statistics, and the quantum state of a stored pulse (L8, L10, L11). The route is short: treat each mode of the electromagnetic field as a harmonic oscillator (P4.3) whose quanta are photons (Dirac 1927; Loudon; Gerry & Knight; Fox 2006).

A number (Fock) state |n⟩ has exactly n photons and a completely undefined phase. A coherent state |α⟩ is the quantum version of an ideal classical wave: it is the eigenstate of the lowering operator, â|α⟩ = α|α⟩, with a mean photon number |α|² and a Poisson photon-number distribution (Glauber 1963). A well-stabilised laser emits light very close to a coherent state, which is why attenuating it never produces a single-photon source (L11). The price of a well-defined phase is an uncertain photon number: for a coherent state ΔN = √N̄ and the phase uncertainty is about 1/(2√N̄).

Einstein (Einstein 1917) described the exchange of energy between atoms and radiation with three processes. An atom in the lower level can absorb a photon (rate ∝ B₁₂ times the radiation density). An atom in the upper level can be stimulated to emit a photon identical to the incoming one (rate ∝ B₂₁ times the same density), and can also decay on its own by spontaneous emission at rate A₂₁. Consistency with thermal radiation fixes the ratios between the coefficients. Spontaneous emission is not an extra assumption in quantum optics: it can be understood as emission stimulated by the vacuum fluctuations of P5.1.

Classically, a laser beam is an oscillating electric field. At 795 nm it oscillates about 3.8 × 10¹⁴ times per second, far too fast for any detector to follow, so detectors measure time-averaged power. Quantum mechanically, that energy arrives in indivisible packets, photons, each carrying hν. Both pictures are right. The wave picture explains interference and polarization (the waveplates of Zeeman EIT); the photon picture explains why a detector clicks and why a quantum memory must preserve single quanta.

A quantum state is a vector |ψ⟩. For a two-state system with basis {|g⟩, |e⟩}, any state is a superposition with complex amplitudes. Measurement in that basis gives g or e with probabilities |c_g|² and |c_e|², and leaves the system in the outcome state. The relative phase between c_g and c_e does not change those probabilities, but it decides how the state interferes later. EIT is built entirely on such a phase (L7).

An atom with transition frequency ω₀ in a laser field E cos ωt feels the energy −d·E. Moving to a frame rotating at ω and dropping terms oscillating at 2ω (the RWA) leaves a simple 2×2 problem that depends only on the detuning Δ = ω − ω₀ and the Rabi frequency Ω. Here Ω = d·E₀/ħ is the full Rabi frequency, the rate of population cycling on resonance. The EIT paper defines Ω = µẼ/ħ with its own field convention; always check factors of 2 when comparing sources.

Real atoms leak energy (spontaneous emission) and lose phase (collisions, fields, motion). A state vector cannot describe a statistical mixture, but a density matrix can. Diagonal elements are populations (how many atoms are where); off-diagonal elements are coherences (how well-defined the relative phase is across the ensemble). EIT and memories live in coherences, so coherence lifetimes are the central figure of merit of the whole project.

Rubidium’s single valence electron has orbital L and spin S = ½, which combine to J (fine structure: 5P₁/₂ vs 5P₃/₂, the D1 vs D2 lines). The nucleus has spin I = 3/2 for ⁸⁷Rb, and J and I combine to F (hyperfine). In a field B each F splits into 2F+1 sublevels m_F. Light carries angular momentum: σ⁺ adds +1 to m_F, σ⁻ subtracts 1, and π (polarized along B) adds 0.

Many atoms together respond to light with a polarization P = ε₀χE. The real part of χ slows the phase (refractive index); the imaginary part removes energy (absorption). Beer’s law follows: intensity falls exponentially with distance, and the exponent at resonance is the optical depth.

Add a second ground state and a second laser. Two excitation routes now lead to the same excited state, and their amplitudes can cancel. The cancelling superposition is the dark state. If the control is strong and the probe weak, the probe sees a narrow transparency window: EIT. With two comparable fields, atoms are trapped in the dark state: CPT. A strong control also splits the excited state into an Autler–Townes doublet (at ±Ω_c in the paper’s convention, ±Ω_c/2 in ours: see the note in “Go deeper”), which also reduces absorption at line centre, but without interference, so some residual absorption remains (paper refs [29], [30]).

The steep dispersion inside the EIT window makes the group index huge, so a pulse crawls through the cell. In the Fleischhauer–Lukin picture (paper ref. [11]) the propagating excitation is a dark-state polariton, a superposition of the probe field and the collective spin coherence. Its mixing angle θ is set by the control. Lower the control adiabatically to zero and θ → 90°: the excitation becomes purely atomic and stops. Raise it again and the light returns.

Atoms in a warm cell move at hundreds of metres per second. Each motional effect is a time-of-flight T across a length l, giving broadening ≈ 1/T (paper Eq. 29). Over an optical wavelength this is Doppler broadening; across the beam it is transit broadening; across the two-photon wavelength it determines whether EIT survives motion.

A quantum memory stores a quantum state of light and releases it on demand. Photon loss in fibre grows exponentially with distance and a quantum state cannot be copied, so long-distance links need repeaters. Repeaters create entanglement over short segments and connect them, and they need memories to hold one segment’s success while the others try. The DLCZ proposal (paper ref. [47]) used atomic ensembles for exactly this. Published Rb and Cs ensemble memories report high efficiencies and fidelities (e.g. Hsiao et al. 2018, paper ref. [114]); read the originals for the numbers.

A laser, however strongly attenuated, emits a coherent state: the photon number in a pulse is Poisson-distributed. Attenuation lowers the mean but never removes the chance of two photons. A true single-photon source (e.g. the α-MoO₃ emitters planned for Objective 3 in the project plan; see the source note in L12) ideally emits a Fock state |1⟩ with zero chance of two. The Hanbury Brown–Twiss measurement of g⁽²⁾(0) tells them apart.

Put an emitter in an optical cavity. If the emitter–photon coupling g beats both the cavity loss κ and the emitter decay γ, energy oscillates back and forth and the spectrum splits into two polariton peaks separated by the vacuum Rabi splitting. N emitters coupled to the same mode act collectively with g√N. The EIT paper makes the same point for vapours: strong light–matter coupling arises from collective enhancement of a large ensemble, and “no high-quality cavity is required”.

Objective 3 concerns solid-state systems. The solid-state literature includes studies of P3HT under ultrastrong coupling (Nagarajan et al. 2020) and of low-loss phonon polaritons in α-MoO₃ (Ma et al. 2018). Whether such systems can support EIT-like storage at room temperature is precisely the open question the calibrated Rb platform is meant to help test. Nothing here should be read as a claim that they do (see the source note in “Go deeper”).

Lectures L1–L12 explain the atomic physics. This lecture explains the bench: the optical and electronic tools that turn the physics into a measurement, in the same order that the 3D bench simulator lays them out. Each part ends with what to check on the real bench, because every number in the model has to be measured, not assumed.

Light that is fully polarized is described by a two-component Jones vector (the amplitudes along horizontal and vertical, with their relative phase; Jones 1941). Optical elements are 2×2 matrices acting on it. A half-wave plate at angle θ to horizontal reflects the polarization about its fast axis: linear light at angle φ leaves at 2θ − φ. A quarter-wave plate turns linear light at 45° to its axis into circular light and vice versa. A polarizing beamsplitter transmits H and reflects V. Zeeman EIT uses all three: an HWP and PBS to set the probe/control split, a QWP to make σ⁺ and σ⁻, and a Glan–Taylor polarizer, whose extinction is of order 10⁵ (the paper), to remove the control after the cell (a PBS manages only about 10³).

A laser beam in free space is a Gaussian beam: its intensity across the beam is a Gaussian, and its radius w(z) (where the intensity has fallen to 1/e² of the peak) is smallest at the waist w₀ and grows on either side (Kogelnik & Li 1966; Siegman 1986; Saleh & Teich). The distance over which the beam stays roughly collimated is the Rayleigh range z_R. These few formulas decide four things on the bench: the intensity that sets the Rabi frequency (L3), the time atoms spend in the beam (L9), whether the beam fits through an AOM crystal, and how to focus into the cat’s-eye of a double pass.

In an AOM a radio-frequency signal drives a transducer that launches a sound wave through a crystal. The sound is a moving diffraction grating: light entering at the Bragg angle is diffracted into a first order that is shifted in frequency by the sound frequency (Siegman 1986; Saleh & Teich). The deflection angle depends on the RF frequency, which is a nuisance when the beam must stay aligned. The double pass (Donley et al. 2005) fixes it: the first order is focused by a lens onto a mirror at the lens focus (a cat’s-eye), which sends it back through the AOM along its own path. The returning beam is diffracted a second time onto the original axis, so the output direction no longer depends on the RF frequency, and the total shift is doubled. A quarter-wave plate crossed twice turns the polarization by 90°, so a polarizing beamsplitter can lift the returning beam out of the input path. Zeeman EIT uses one such double pass per arm.

A free-running diode laser drifts by tens of MHz; Zeeman EIT needs the one-photon detuning stable to the MHz scale (L9, L7). The reference is the atom itself, read out by saturated-absorption spectroscopy (Preston 1996; MacAdam et al. 1992; Demtröder). A strong pump and a weak probe cross the same vapour from opposite directions. Atoms moving along the beam see the pump and probe at opposite Doppler shifts, so only atoms with nearly zero velocity along the beam are resonant with both. The pump saturates them, and the probe sees a narrow dip (as narrow as the natural width, here a few MHz) in the 500 MHz-wide Doppler absorption. Where two lines share a velocity class, a crossover resonance appears exactly midway between them.

To lock on such a dip you need an error signal that is zero on the line and changes sign across it: the derivative of the dip. A small current dither on the diode, with lock-in detection at the dither frequency, gives it. The lock box feeds the error back to the laser current, and slow drifts are removed by retuning the temperature, since a DFB has no piezo (Wieman & Hollberg 1991; the dither/lock-in and the Pound–Drever–Hall method are compared in Black 2001 and Drever et al. 1983).

The optical depth of the cell rises steeply with temperature, because the density of rubidium vapour follows the vapour pressure, which grows roughly exponentially with T (Steck, Rb-87 data, vapour-pressure model). At 25 °C the vapour pressure is 3.92 × 10⁻⁷ Torr, which with the ideal-gas law gives n = P/k_BT ≈ 1.3 × 10¹⁰ cm⁻³ Steck; at 60 °C the density is roughly 20 to 25 times higher design estimate, in the range 10¹¹–10¹² cm⁻³ that the paper treats as typical (Finkelstein et al. 2023). Heating the cell is therefore the main OD knob.

Magnetic fields are the other. The two-photon resonance of Zeeman EIT moves by 1.40 kHz per milligauss (L5), while the EIT line is only about 7 to 26 kHz wide: a stray field of just 5 mG shifts the resonance by 7 kHz, which is as large as the linewidth. Earth’s field is roughly 0.5 G, a hundred times bigger. That is why the cell sits in a µ-metal shield (three nested layers in the paper) with a solenoid inside to provide a small, known, stable bias field along the beams, and why the shield must be degaussed after assembly: leftover magnetization otherwise sets the residual field.

A photodiode converts power P into a current I = RP with responsivity R = ηeλ/hc, about 0.64 A/W per unit quantum efficiency at 795 nm; silicon photodiodes are around 0.5 A/W near 800 nm (check the data sheet). The current is converted to a voltage by a transimpedance amplifier. Three noise sources compete: shot noise of the photocurrent itself, thermal (Johnson) noise of the feedback resistor, and amplifier noise (Saleh & Teich). At the microwatt power levels of the probe the resistor and shot noise are comparable, and below that the resistor noise wins, so the signal-to-noise ratio improves by averaging (the scope) or by using a lower bandwidth.

The 795 nm beam is in the near infrared: invisible, but focused by the eye onto the retina like visible light. Because you cannot see it, the blink reflex does not protect you, and the damage is permanent. Beam powers of a few to tens of milliwatts in the experiment generally fall in laser classes 3R or 3B under the international standard (IEC 60825-1) and the corresponding American one (ANSI Z136.1); the exact class depends on power and wavelength-dependent limits, so confirm the classification with your institution’s laser safety officer and follow the local rules.

The reference bench in the simulator is not just a diagram: it is a checklist. Three habits prevent most problems. First, fix the master axis with two mirrors and two irises before adding anything else: centre the beam on both irises by adjusting the mirrors alternately; every optic between them is then on axis. Second, measure each stage: the power after the isolator, the pick-off ratio, the control/probe split, the AOM efficiency (first-order power over input) and the double-pass efficiency, the polarization purity after each PBS, and the power at the cell. Compare each with what the simulator predicted for the same settings; a discrepancy is information. Third, change one thing at a time and write it down: plate angle, RF frequency, temperature, bias current. A logbook is the cheapest instrument in the lab.

The same definitions that appear when you hover over a symbol. Click a lecture link to jump to where the symbol is used first.

Fundamental constants are the 2018 CODATA recommended values as tabulated in Steck, Rb-87 data (Table 1); atomic data are from the same document, revision 2.3.4 (8 August 2025). Digits in parentheses are the standard uncertainty in the last digits.

The lecture notes were audited line by line against their sources before the primers and L13 were added. This record lets you see what changed and why.