Micromaser and Photon Number States in Cavities

The micromaser takes the machinery field quantization builds — the photon-number ladder with its rule $a|n\rangle=\sqrt{n}\,|n-1\rangle$ , the discrete Fock states $|n\rangle$ , and the zero-point vacuum $|0\rangle$ — and makes each abstraction do concrete, measurable work in a single microwave cavity. A conventional maser or laser oscillates with astronomically many atoms and photons, so its field is essentially classical. The micromaser strips the problem to its quantum bone: at any instant one two-level atom interacts with one quantized mode — exactly the Jaynes–Cummings model that quantum optics uses to define light–matter coupling. The atom is a Rydberg atom, a pair of neighbouring high-lying levels whose dipole moment scales as $n^2 a_0$ and is therefore enormous, giving strong coupling. The cavity is a superconducting resonator with $Q\sim10^{10}$ , so a photon lives a fifth of a second, far longer than the microseconds an atom spends crossing it. In that window the atom performs a partial Rabi oscillation whose rate $\Omega_0\sqrt{n+1}$ depends on the photon number it finds — the $\sqrt{n+1}$ , vacuum included, a fingerprint of the quantized field no classical description contains. Iterate this number-dependent gain against slow cavity loss and the field self-organizes into states with no classical analogue: sub-Poissonian distributions, self-quenching thresholds, and trapping states locked to a single Fock number. And because each departing atom carries a record of the field, the device is at once a source of nonclassical light and an apparatus for quantum-nondemolition photon counting.

Key takeaways

The micromaser is a single high- $Q$ microwave cavity through which excited Rydberg atoms pass one at a time, each exchanging at most one quantum with the field through the Jaynes–Cummings interaction.
Because only one atom is ever present, the dynamics are ideal cavity QED: the atom makes a partial Rabi oscillation at rate $\Omega_0\sqrt{n+1}$ , depending discretely on the photon number $n$ it encounters — vacuum included.
The steady state balances atomic pumping against cavity loss into a cold bath, and is generically nonclassical: sub-Poissonian, non-thermal, and at special interaction times locked to a near-pure Fock state (a trapping state).
Signatures — collapse and revival of the inversion, sharp pump thresholds, self-quenching, and Mandel $Q<0$ — are all direct evidence that the field is quantized.
The departing atom encodes the photon number, so the apparatus also performs quantum-nondemolition counting, watching individual quantum jumps of the field.
Key idea: a single atom performs a coherent Rabi oscillation whose rate $\Omega_0\sqrt{n+1}$ depends discretely on the photon number $n$ ; iterating this nonlinear, number-dependent gain against cavity loss steers the field into nonclassical steady states, including near-pure Fock states.

Where this lives today

The micromaser's ideas did not stay in the microwave beam. Its direct experimental sibling is Serge Haroche's Paris photon box, where Rydberg atoms cross a cavity in the dispersive regime and imprint an $n$ -dependent phase read out by Ramsey interferometry — the apparatus that watched single photons appear and disappear in real time, the first observation of quantum jumps of light, sharing the 2012 Nobel Prize. The deeper descendant is circuit quantum electrodynamics: swap the Rydberg atom for a superconducting transmon qubit and the cavity for a microwave resonator, and the Jaynes–Cummings Hamiltonian of equation (1) reappears verbatim, now with couplings and lifetimes engineered at will. Dispersive readout of the qubit-cavity phase shift is the workhorse measurement of every superconducting quantum processor, and the lesson that one emitter and one high- $Q$ mode suffice to sculpt light photon by photon is now the operating principle of an entire quantum-computing platform.

The physical picture

Picture a slow bell struck by one hammer at a time, where the hammer's swing depends on how loudly the bell already rings. An excited atom entering a cavity of $n$ photons rotates through Rabi angle $\Omega_0\sqrt{n+1}\,\tau$ during its transit; at the exit it may have deposited a photon ( $|e\rangle\to|g\rangle$ ) or, having completed whole cycles, left as excited as it arrived. The gain per atom is therefore an oscillatory, non-monotonic function of $n$ — the nonlinearity, nothing like a laser's smooth amplification, that makes everything downstream nonclassical.

Figure

A single excited Rydberg atom crosses a high-Q superconducting microwave cavity of f ≈ 21.5 GHz and Q ≈ 3×10¹⁰ and coherently exchanges one quantum with the field at rate Ω₀√(n+1); a state-selective field-ionization detector then reads whether it left excited or decayed

Between atoms the cavity leaks photons into a cold thermal reservoir at rate $\kappa=\omega/Q$ , and the steady state is a tug of war: pumping builds $n$ , but the gain $\sin^2(\tfrac12\Omega_0\sqrt{n+1}\,\tau)$ can vanish at special photon numbers or interaction times, and where it vanishes the field is pinned. The vacuum is no spectator either — an empty cavity accepts a photon with amplitude $\Omega_0\sqrt{1}$ , which is why the $\sqrt{n+1}$ (not $\sqrt{n}$ ) law is a direct readout of the quantized vacuum. The second figure plots the emission probability versus photon number, tuned so a full Rabi cycle at one photon opens a trap.

Figure

Emission probability P₁ per excited atom versus cavity photon number n for Ω₀/2π = 44 kHz and τ = 16.1 μs, tuned so a full 2π Rabi cycle at n = 1 makes the gain vanish exactly — the sharp zero at n = 1 pins the field at one photon, a trapping state impossible for any classical gain medium

The core equations

The whole device is governed by one Hamiltonian: a two-level atom coupled to a single quantized mode in the rotating-wave approximation. The first two terms are the free energies of field and atom; the third is the dipole interaction kept to its energy-conserving pieces — one photon created as the atom decays, one absorbed as it climbs.

H = \hbar\omega\, a^\dagger a + \tfrac12\hbar\omega_0\sigma_z + \hbar g\left(a^\dagger\sigma_- + a\,\sigma_+\right) \qquad (1)

where: $a^\dagger,a$ — cavity photon creation/annihilation operators; $\omega$ — cavity mode angular frequency; $\omega_0$ — atomic transition frequency; $\sigma_z,\sigma_\pm$ — Pauli and raising/lowering operators of the atom; $g$ — single-photon (vacuum) atom–field coupling. Physical reading: energy hops between one atomic excitation and one photon and nothing else, so excitation number is conserved and the dynamics split into tiny $2\times2$ blocks. Because the interaction moves $|e,n\rangle$ only to $|g,n+1\rangle$ , the coupling carries the ladder factor $a^\dagger|n\rangle=\sqrt{n+1}$ , so the Rabi frequency inherits $\sqrt{n+1}$ and the vacuum ( $n=0$ ) still oscillates at $\Omega_0$ .

\Omega_n = \Omega_0\sqrt{n+1},\qquad \Omega_0 = 2g \qquad (2)

where: $\Omega_n$ — Rabi frequency an atom sees in a field of $n$ photons; $n$ — photon number encountered; $\Omega_0$ — vacuum Rabi frequency. Physical reading: the " $+1$ " is the vacuum contribution — spontaneous emission into a single mode — the one term a classical field cannot supply. An atom entering excited into an $n$ -photon field undergoes a pure Rabi flop within its block; the probability it leaves de-excited, having handed a photon to the cavity, is the sine-squared of half the accumulated Rabi angle.

P_e(\tau)=\cos^2\!\left(\tfrac12\Omega_0\sqrt{n+1}\,\tau\right),\quad P_g(\tau)=\sin^2\!\left(\tfrac12\Omega_0\sqrt{n+1}\,\tau\right) \qquad (3)

where: $P_e,P_g$ — probability of the atom exiting excited or de-excited; $\tau$ — atom–cavity transit time. Physical reading: $P_g$ is the per-atom emission probability of the second figure, oscillatory in both $n$ and $\tau$ — the root of trapping and self-quenching. If the field is a distribution $p(n)$ , the atomic inversion is a sum over many incommensurate Rabi frequencies that dephase (collapse) and then, because the discrete $\sqrt{n+1}$ set rephases, spring back (revival) — a signature with no classical counterpart.

\langle\sigma_z(\tau)\rangle=\sum_n p(n)\cos\!\left(\Omega_0\sqrt{n+1}\,\tau\right) \qquad (4)

where: $p(n)$ — photon-number distribution. Physical reading: for a coherent state of mean $\bar n$ the collapse takes $t_c\sim2/\Omega_0$ and the first revival lands near $t_r\approx2\pi\sqrt{\bar n}/\Omega_0$ ; seeing the revival at all proves the frequencies are discrete, i.e. that $n$ is quantized. Over many atoms and photon lifetimes, upward gain and downward loss balance rung by rung. Detailed balance across the ladder then fixes the steady-state distribution as a running product of gain-to-loss ratios.

\frac{p_n}{p_{n-1}} = \frac{N_{\mathrm{ex}}\,\sin^2\!\left(\tfrac12\Omega_0\sqrt{n}\,\tau\right) + \bar n_{\mathrm{th}}\,n}{(1+\bar n_{\mathrm{th}})\,n} \qquad (5)

where: $N_{\mathrm{ex}}=R/\kappa$ — pump parameter (excited atoms per photon lifetime), $R$ the injection rate; $\kappa=\omega/Q$ — cavity photon decay rate; $\bar n_{\mathrm{th}}=1/(e^{\hbar\omega/k_BT}-1)$ — mean thermal photon number. Physical reading: wherever the numerator's $\sin^2$ dips, the ratio falls below one and $p_n$ stops growing — the entire non-thermal, sub-Poissonian, trapping-state structure lives in this one recursion. To quantify how nonclassical the result is, compare the photon-number variance with its mean: a coherent field is Poissonian with equal variance and mean, and the micromaser routinely undershoots it.

Q = \frac{\langle(\Delta n)^2\rangle - \langle n\rangle}{\langle n\rangle} \qquad (6)

where: $Q$ — Mandel parameter; $\langle(\Delta n)^2\rangle$ — photon-number variance. Physical reading: $Q=0$ is Poissonian (coherent), $Q>0$ super-Poissonian (thermal-like), and $Q<0$ is sub-Poissonian — a nonclassical field no classical source can produce.

Derivation AJaynes-Cummings evolution and the square-root-n-plus-one emission law

The goal is to derive, from the interaction Hamiltonian alone, the rule governing every atomic transit: an excited atom entering an $n$ -photon cavity leaves de-excited with probability $\sin^2(\tfrac12\Omega_0\sqrt{n+1}\,\tau)$ , oscillating at a Rabi frequency $\Omega_0\sqrt{n+1}$ that grows as the square root of the photon number and is nonzero even for the empty cavity. Everything nonclassical about the micromaser is downstream of this one law, which we pin down and check against the classical large- $n$ and vacuum limits.

Step 1. On resonance ( $\omega=\omega_0$ ) the bare states $|e,n\rangle$ and $|g,n+1\rangle$ have the same energy and are coupled only to each other by $H_{\mathrm{int}}=\hbar g(a^\dagger\sigma_-+a\sigma_+)$ . Why: excitation-number conservation chops the infinite problem into independent two-level blocks, one photon manifold solved at a time.

Step 2. The matrix element linking the pair is $\langle g,n+1|H_{\mathrm{int}}|e,n\rangle=\hbar g\sqrt{n+1}$ , because $a^\dagger|n\rangle=\sqrt{n+1}\,|n+1\rangle$ . Why: this is where the quantized-field ladder enters — the coupling is not fixed but grows as $\sqrt{n+1}$ , and the " $+1$ " survives at $n=0$ .

Step 3. In the block the interaction is an effective spin, $H_{\mathrm{block}}=\hbar g\sqrt{n+1}\,\sigma_x^{(n)}$ , so define $\Omega_n\equiv2g\sqrt{n+1}=\Omega_0\sqrt{n+1}$ . Why: as a Pauli- $x$ term the evolution is an ordinary rotation, and its rate is the quantum Rabi frequency.

Step 4. Evolving the initial state gives $|\psi(\tau)\rangle=\cos(\tfrac12\Omega_n\tau)|e,n\rangle-i\sin(\tfrac12\Omega_n\tau)|g,n+1\rangle$ , so the probability of exiting de-excited is $\sin^2(\tfrac12\Omega_0\sqrt{n+1}\,\tau)$ . Why: the atom reversibly cycles a single quantum in and out of the field, never leaking it the way free-space spontaneous emission does.

Step 5. Limits: for $n\gg1$ , $\sqrt{n+1}\to\sqrt{n}$ and the flopping merges into semiclassical Rabi oscillation; for $n=0$ the atom still oscillates at $\Omega_0$ . Why: recovering the classical amplifier at large $n$ and a nonzero vacuum frequency at $n=0$ is exactly what separates a quantized field from a classical one. *Verified: reduces to semiclassical Rabi flopping for large $n$ , vacuum Rabi oscillation at $\Omega_0$ for $n=0$ ; matches Loudon ch. 6, Foot ch. 8.*

Go deeper · Derivation B · 1 minThe trapping-state condition and Fock-state locking

Derivation BThe trapping-state condition and Fock-state locking

The goal is to identify the special photon numbers at which an excited atom cannot deposit its quantum — so that probability climbing the number ladder hits a hard wall and the field locks into a single Fock state $|n_q\rangle$ at zero temperature. This is the sharpest demonstration that the intracavity field is quantized: a classical gain medium has no such discrete barrier. We locate the wall from the emission law and show why it survives only when cold.

Step 1. The emission probability $P_g(\tau)=\sin^2(\tfrac12\Omega_0\sqrt{n+1}\,\tau)$ vanishes when its argument is an integer multiple of $\pi$ . Why: a vanishing emission probability means an excited atom completes whole cycles and returns to $|e\rangle$ , adding no photon however long the beam runs.

Step 2. Setting $\tfrac12\Omega_0\sqrt{n_q+1}\,\tau=\pi k$ gives the trapping condition $\Omega_0\sqrt{n_q+1}\,\tau=2\pi k$ , $k$ a positive integer. Why: the interaction time $\tau$ , fixed by atomic velocity, can be tuned so the Rabi angle at one particular $n_q$ is exactly $2\pi k$ while generic at every other $n$ .

Step 3. At that $n_q$ the gain coefficient $A_{n_q+1}=R\sin^2(\tfrac12\Omega_0\sqrt{n_q+1}\,\tau)=0$ , so no probability crosses from $n_q$ to $n_q+1$ . Why: the upward flux is throttled to zero at a single rung, sealing the ladder at $n_q$ .

Step 4. With no thermal photons to bridge the gap ( $\bar n_{\mathrm{th}}\to0$ ), the recursion terminates and $p(n)=\delta_{n,n_q}$ — a pure number state. Why: only thermal absorption could carry the field across the sealed rung, so the trap is perfect at $T\to0$ and only smeared at finite $T$ .

Step 5. Different $\tau$ (different velocities) select different $n_q$ , and each $k$ gives a further trap. Why: velocity-selecting the beam dials which photon number the cavity stabilizes — the basis of on-demand Fock-state preparation. *Verified: reproduces the Filipowicz–Javanainen–Meystre (1986) trapping condition, observed in the Garching Rb micromaser; invariant under $k\to k+1$ .*

Limiting cases

Three knobs organize the behaviour: the interaction angle $\Omega_0\sqrt{n+1}\,\tau$ , the pump parameter $N_{\mathrm{ex}}$ , and the thermal occupation $\bar n_{\mathrm{th}}$ .

Case	Result	Meaning
Vacuum Rabi limit ( $n=0$ )	$\Omega_{n=0}=\Omega_0=2g$	An atom oscillates against the empty cavity; emission probability $\sin^2(\tfrac12\Omega_0\tau)$ is nonzero — a purely quantum effect.
Trapping state	$\Omega_0\sqrt{n_q+1}\,\tau=2\pi k,\ k\in\mathbb{Z}^+$	An excited atom completes whole cycles at $n_q$ and adds no photon; the field locks to the pure Fock state $\|n_q\rangle$ as $T\to0$ .
Thermal (no pump) limit	$p(n)=\bar n_{\mathrm{th}}^{\,n}/(1+\bar n_{\mathrm{th}})^{n+1}$	With no atoms the cavity relaxes to a thermal state; low $T$ (small $\bar n_{\mathrm{th}}$ ) is essential to expose nonclassical structure.
Semiclassical / large $n$	$\Omega_n\to\Omega_0\sqrt{n}$	The flopping merges into ordinary Rabi oscillation; the field behaves classically.
Strong pump / self-quenching	$\theta\equiv\Omega_0\sqrt{N_{\mathrm{ex}}}\,\tau\gtrsim2\pi$	Raising $N_{\mathrm{ex}}$ does not monotonically raise $\bar n$ ; the field quenches as $\theta$ crosses multiples of $\pi$ , a driven-dissipative phase transition.

Common trapThe micromaser field is just a very weak laser — a coherent state with a small mean photon number

It is not, and this is the central misconception. A laser field is (ideally) coherent and Poissonian ( $Q=0$ ) because its gain is phase-random and indifferent to the exact photon number. The micromaser gain is the opposite: it depends on $n$ through $\sin^2(\tfrac12\Omega_0\sqrt{n+1}\,\tau)$ , an oscillatory function, and that single fact forbids the coherent-state picture. At generic interaction times this number-dependent gain acts as negative feedback and yields sub-Poissonian light ( $Q<0$ ); at special times it locks the field to a Fock state or quenches it. And because the coupling carries $\sqrt{n+1}$ not $\sqrt{n}$ , an atom emits into an empty cavity with finite probability — no classical field, weak or strong, reproduces any of this.

Worked examples

1 · Photon lifetime and thermal occupation of the cavity (confidence builder)

A Garching-style micromaser runs on the $^{85}\mathrm{Rb}$ transition at $f=21.456\,\mathrm{GHz}$ in a niobium cavity of $Q=3\times10^{10}$ , cooled to $T=0.3\,\mathrm{K}$ . Find the photon storage time, cavity decay rate, and mean thermal photon number.

Step 1. Convert to angular frequency: $\omega=2\pi f=2\pi(21.456\times10^9)=1.348\times10^{11}\ \mathrm{rad/s}$ , the scale $\hbar\omega$ that atom and bath see.

Step 2. The photon storage time is $T_{\mathrm{cav}}=Q/\omega=(3\times10^{10})/(1.348\times10^{11})=\mathbf{0.22\ s}$ , and the decay rate is its inverse, $\kappa=\omega/Q=\mathbf{4.49\ s^{-1}}$ . A fifth of a second is an eternity in atomic physics — about $10^4$ times the $\sim10\,\mu\mathrm{s}$ transit time, which is why one-atom-at-a-time cavity QED works.

Step 3. The thermal occupation is $\bar n_{\mathrm{th}}=1/(e^{\hbar\omega/k_BT}-1)$ , with exponent $\hbar\omega/k_BT=(1.055\times10^{-34})(1.348\times10^{11})/[(1.381\times10^{-23})(0.3)]=3.43$ , so $\bar n_{\mathrm{th}}=1/(e^{3.43}-1)=\mathbf{0.033}$ — fewer than one photon in thirty, cold enough for a trapping state to survive many photon lifetimes. $T_{\mathrm{cav}}=0.22$ s, $\kappa=4.49\ \mathrm{s}^{-1}$ , $\bar n_{\mathrm{th}}=0.033$ .

2 · Vacuum Rabi timing and the first trapping state

Take a vacuum Rabi frequency $\Omega_0/2\pi=44\,\mathrm{kHz}$ . (a) How long must an atom sit in the empty cavity for a $\pi$ -pulse — certain single-photon emission? (b) At what interaction time does the field trap at one photon, and what does an atom then see at $n=0,1,2$ ?

Step 1. A $\pi$ -pulse in the vacuum needs $\Omega_0\tau=\pi$ , so $\tau=\pi/\Omega_0=1/(2\times44\,000)=\mathbf{11.4\ \mu s}$ . An atom at $v\approx300\,\mathrm{m/s}$ then crosses a cavity of length $L=v\tau=\mathbf{3.4\ mm}$ — a realistic dimension, so atomic velocity sets $\tau$ .

Step 2. The field traps at $n_q=1$ when a full cycle ( $k=1$ ) is completed there: $\Omega_0\sqrt{2}\,\tau=2\pi$ , giving $\tau=2\pi/(\Omega_0\sqrt2)=1/(44\,000\times1.414)=\mathbf{16.1\ \mu s}$ .

Step 3. At this $\tau$ the emission probability is $P_1(n)=\sin^2[(\pi/\sqrt2)\sqrt{n+1}]$ : at $n=0$ , $\sin^2(2.221)=\mathbf{0.63}$ ; at $n=1$ , $\sin^2(\pi)=\mathbf{0}$ (the trap — no atom can add a second photon); at $n=2$ , $\sin^2(3.848)=\mathbf{0.42}$ . Probability floods up to one photon and hits a perfect wall, precisely the second figure. (With this cavity, maser action already switches on at threshold $N_{\mathrm{ex}}=R/\kappa=1$ — only $\sim4.5$ atoms per second, since each photon lives $0.22$ s.) $\pi$ -pulse at 11.4 μs; one-photon trap at 16.1 μs; $P_1(0)=0.63$ , $P_1(1)=0$ , $P_1(2)=0.42$ .

Physics of the portrait

The micromaser turned the Jaynes–Cummings model from a 1963 textbook toy into laboratory reality, and one Garching group drove that for two decades.

Origins · 1 minHerbert Walther and how this was discovered

Herbert Walther

1935–2006 · the one-atom maser and experimental cavity QED

Herbert Walther led the Max Planck Institute of Quantum Optics group that built and mastered the micromaser, and more than anyone made single-atom cavity QED an experimental science. In 1985, with Meschede and Müller, he operated the first one-atom maser: a beam of $^{85}\mathrm{Rb}$ atoms so dilute that at most one at a time sat inside a niobium cavity, yet the ensemble sustained maser oscillation — proof that a single atom and a single mode suffice. Over the next fifteen years his group turned the device into a precision probe of the quantum field: the 1987 observation of collapse and revival, the 1990 measurement of sub-Poissonian statistics with $Q<0$ , and by 2000 a cavity of $Q\sim10^{10}$ storing photons for tenths of a second and stabilizing near-pure Fock states on the predicted trapping condition. Walther's micromaser remains the cleanest realization of the Jaynes–Cummings Hamiltonian and a direct ancestor of today's circuit-QED processors.

History

1963

— Jaynes and Cummings solve a two-level atom coupled to a single quantized mode

1985

— Meschede, Walther and Müller operate the first one-atom maser with Rydberg atoms

1986

— Filipowicz, Javanainen and Meystre give the micromaser master equation and predict trapping states

1987

— Rempe, Walther and Klein observe quantum collapse and revival of the atomic inversion

1990

— Rempe, Schmidt-Kaler and Walther measure sub-Poissonian statistics with Mandel Q below zero

2000

— Varcoe and co-workers stabilize pure photon-number states via trapping in the Garching micromaser

Going further

Texts. Foot, Atomic Physics, ch. 8 (Rydberg atoms, cavity QED) · Loudon, The Quantum Theory of Light, 3rd ed., ch. 5–6 (Jaynes–Cummings model) · Scully & Zubairy, Quantum Optics, ch. 6 and 11 (Jaynes–Cummings and micromaser theory) · Haroche & Raimond, Exploring the Quantum, ch. 3–7 (cavity QED and photon-box experiments) · Saleh & Teich, Fundamentals of Photonics, ch. 13–14.

Provenance. The Jaynes–Cummings Hamiltonian, RWA, quantum Rabi frequency $\Omega_n=\Omega_0\sqrt{n+1}$ and emission law $P_g=\sin^2(\tfrac12\Omega_0\sqrt{n+1}\,\tau)$ are standard (Loudon ch. 6; Foot ch. 8; Derivation 1). The detailed-balance steady state, pump parameter $N_{\mathrm{ex}}=R/\kappa$ and trapping condition $\Omega_0\sqrt{n_q+1}\,\tau=2\pi k$ are from Filipowicz, Javanainen and Meystre, Phys. Rev. A 34, 3077 (1986). The one-atom maser and its parameters ( $Q\sim10^{10}$ Nb cavities, $^{85}\mathrm{Rb}$ atoms, $T\sim0.1$ – $2\,\mathrm{K}$ ) are from Meschede, Walther and Müller, Phys. Rev. Lett. 54, 551 (1985); collapse and revival from Rempe, Walther and Klein (1987); sub-Poissonian statistics from Rempe, Schmidt-Kaler and Walther (1990); Fock-state stabilization from Varcoe et al., Nature 403, 743 (2000). Worked numbers ( $T_{\mathrm{cav}}=0.22$ s, $\kappa=4.49\ \mathrm{s}^{-1}$ , $\bar n_{\mathrm{th}}=0.033$ at 0.3 K, $\pi$ -pulse 11.4 μs, one-photon trap 16.1 μs) recomputed from $f=21.456$ GHz, $Q=3\times10^{10}$ , $\Omega_0/2\pi=44$ kHz.

Prerequisites. Quantization of the Electromagnetic Field · Fock States and the Photon Number Basis · Vacuum State and Zero-Point Energy · Photon Statistics and the Mandel Q Parameter. Leads to. Single-Photon Sources · Phase-Space Quasiprobability Distributions · Photon Antibunching.

Ask the tutor: Ask me to iterate the detailed-balance recursion for a given $\tau$ and $N_{\mathrm{ex}}$ and show you exactly where the trapping states and sub-Poissonian dips appear.

Why is the Rabi frequency proportional to √(n+1) and not √n?

The coupling that flips the atom while adding a photon is $\langle g,n+1|a^\dagger\sigma_-|e,n\rangle=g\sqrt{n+1}$ , and the $\sqrt{n+1}$ comes straight from the ladder rule $a^\dagger|n\rangle=\sqrt{n+1}\,|n+1\rangle$ . The " $+1$ " is the vacuum contribution: nonzero even at $n=0$ , so an atom entering an empty cavity still Rabi-oscillates at $\Omega_0$ — reversible spontaneous emission into a single mode. Because a classical field would give $\sqrt{n}$ with no vacuum term, a nonzero emission into the empty cavity directly measures the quantized vacuum.

What exactly is a trapping state, and why does it need such low temperature?

A trapping state occurs when the interaction time is tuned so the Rabi angle at some photon number $n_q$ is a whole multiple of $2\pi$ : $\Omega_0\sqrt{n_q+1}\,\tau=2\pi k$ . An excited atom then completes entire cycles and exits still excited, so probability cannot climb past $n_q$ and the field locks into the pure Fock state $|n_q\rangle$ . Low temperature is essential because thermal photons offer a back door: with $\bar n_{\mathrm{th}}>0$ the bath pushes the field across the sealed rung. At $T\to0$ the wall is perfect; $\sim0.3$ K gives $\bar n_{\mathrm{th}}\approx0.03$ , cold enough for the trap to survive many photon lifetimes.

How is the micromaser field different from a laser field?

A laser is, ideally, a coherent state: Poissonian statistics, $Q=0$ , gain indifferent to the photon number. The micromaser gain does care — it goes as $\sin^2(\tfrac12\Omega_0\sqrt{n+1}\,\tau)$ . At many interaction times the gain falls as $n$ rises, so an upward fluctuation is damped and the distribution narrows below Poisson, giving sub-Poissonian light with $Q<0$ ; at other times the field is pinned to a Fock state or self-quenches. Both are cavity oscillators sustained by gain against loss, but the micromaser's number-dependent gain produces states a laser cannot.

How can a departing atom measure the photon number without destroying photons?

In the dispersive regime the atom is tuned far off resonance, so it exchanges no energy — no photon is absorbed or emitted. What it can do is pick up a phase shift proportional to the photon number $n$ . A Ramsey sequence — two microwave pulses bracketing the cavity — converts that phase into an atomic-state population, so detecting whether the atom exits excited or ground reveals $n$ while leaving the photons intact. This is a quantum-nondemolition measurement: repeated over many atoms the field collapses onto a definite Fock state, and one can watch photon-number jumps as the cavity leaks.

What is the pump parameter N_ex, and what is self-quenching?

The pump parameter $N_{\mathrm{ex}}=R/\kappa$ is the number of excited atoms delivered per photon lifetime — the analogue of "how hard you drive the laser". Maser action turns on at threshold $N_{\mathrm{ex}}=1$ . But raising $N_{\mathrm{ex}}$ does not smoothly raise the mean photon number, because the accumulated angle $\theta=\Omega_0\sqrt{N_{\mathrm{ex}}}\,\tau$ grows too; each time $\theta$ sweeps past a multiple of $\pi$ the collective gain reverses and the field can collapse — self-quenching. The result is a staircase of thresholds and quenching regions, a hallmark of a driven-dissipative quantum phase transition.

Why must the cavity Q be so extraordinarily high?

The whole physics rests on one inequality: a photon must survive far longer than an atom takes to cross the cavity, so many atoms interact coherently with the same stored field before it decays. With $Q=3\times10^{10}$ at 21.5 GHz the photon lifetime is $\sim0.22$ s against a transit of only $\sim10\,\mu\mathrm{s}$ — a ratio near $10^4$ . That margin lets the field build a memory that successive atoms read out and lets a trap persist; a modest cavity would let the field dephase between atoms, washing out revivals, sub-Poissonian statistics, and traps alike. Superconducting niobium cooled to sub-kelvin makes such $Q$ (and the tiny $\bar n_{\mathrm{th}}$ it needs) attainable.