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Use the Driven two-level Rabi model in the ordered basis $(|e⟩,|g⟩)$, with Pauli matrices satisfying ${\sigma }_z=\mathrm{diag}(1,-1)$. Let the Rabi detuning be $\delta ={\omega }_0-\omega =2\pi ({\nu }_0-\nu )$, in angular frequency, while ${\nu }_0$ and $\nu$ are ordinary frequencies in hertz. Assume two identical square pulses, equal drive oscillation phases, constant positive Rabi angular frequency $\Omega$, constant $\delta$ during pulses and flight, and no quantum decoherence or loss. The rotating-wave approximation and isolation of the chosen two-level system are required, just as for the Driven two-level Rabi model. Free flight means no transverse drive; the reference oscillator continues running.

The pulse and free-flight Hamiltonian operators in the rotating frame areThe Pauli matrix multiplication law gives $(\delta {\sigma }_z+\Omega {\sigma }_x{)}^2=q^{2}I$. Thus the exact special unitary group $\mathrm{SU}(2)$ pulse matrix for this effective Hamiltonian operator, and the flight matrix, areThe sequence is $U_p{U}_f{U}_p$. Starting in $|g⟩$, matrix multiplication and the Born rule giveThis retains the tilted rotation axis and quantum phase accumulated during each finite pulse. At resonance it gives $P_{g\to e}(0)={\sin }^2(\Omega \tau )$; with no flight it reduces to the single square-pulse response for duration $2\tau$. Both checks would fail if arbitrary ideal $\pi /2$ rotations were silently substituted at nonzero Rabi detuning.

For resonant pulse area $\Omega \tau =\pi /2$, $|\delta |/\Omega \ll 1$ and $\tau \ll T$, the central fringe approachesIts half-height points are $|\nu -{\nu }_0|=1/(4T)$, so the central peak's full width at half maximum is $1/(2T)$ in hertz, or $\pi /T$ in angular frequency. This width assumes an effectively flat pulse envelope across that central peak, one flight time and unit contrast; it is neither a universal finite-pulse width nor a frequency uncertainty. Longer flight lets a smaller frequency difference accumulate the same observable quantum phase.

The two pulses here have the same sign and the plotted outcome is transfer to $|e⟩$. The earlier Ramsey interferometry example uses an inverse analysis pulse and reports return to its initial label. Specifying the measured outcome and pulse oscillation phases reconciles the two conventions. In a beam the lengths and speed determine $\tau$ and $T$; in the Quantronium Ramsey interference experiment electronics set the pulse times for a stationary circuit.

In the 2007 cavity quantum electrodynamics experiment of Christine Guerlin and collaborators, including Sébastien Gleyzes, Michel Brune, Jean-Michel Raimond and Serge Haroche, an injected coherent state occupied a 51.1 GHz microwave cavity with niobium-coated mirrors exhibiting superconductivity. Velocity-selected rubidium chemical atoms were prepared as circular Rydberg atoms in levels with principal quantum numbers 50 and 51. Repeated atomic readouts progressively narrowed the inferred photon number to a Fock state; later records revealed jumps. The cavity damping time was $T_c=0.130\mathrm{s}$. Empty-cavity Ramsey interferometry scans fixed the analysis-phase origin; weak injected fields refined the response calibration. Finite fringe contrast required calibrated likelihoods; photon losses could interrupt convergence.^[ref]

The functional sequence is preparation, first Ramsey interferometry zone $R_1$, dispersive cavity $C$, second zone $R_2$, and state-selective detection. The first $\pi /2$ microwave pulse prepares an atomic quantum superposition; dispersive atom-cavity coupling adds a photon number-dependent relative quantum phase while suppressing energy exchange. The second pulse converts that quantum phase into atomic populations; state-selective field ionization reads $e$ or $g$. Successive chemical atoms carry the meter records away while the same field remains available.^[ref]

This arrangement makes quantum nondemolition measurement useful for following a field repeatedly and preparing a Fock state by conditioning. The photon number is inferred from a sequence of physical records; it is not a directly visible label attached to a photon. Each atomic probe makes a weak, partial measurement in quantum mechanics of photon number, and the accumulated records can approach a resolved count. A single binary meter need not identify one among many possible Fock states, as perfect discrimination of pure states requires orthogonality.

An explicit original ideal meter model explains how photon number survives while field quantum coherence changes. Let $\hat{N}$ be the number operator and prepare an atomic qubit in $|+⟩=(|e⟩+|g⟩)/\sqrt{2}$. The dispersive unitary operatorcommutes with $\hat{N}\otimes I$. For a field quantum superposition ${\sum }_n{c}_n|n⟩$, it produces ${\sum }_n{c}_n|n⟩|a_n⟩$, where $|a_n⟩=(e^{-in\chi /2}|e⟩+e^{in\chi /2}|g⟩)/\sqrt{2}$. The atomic meter states encode $n$ through relative quantum phase and can become entangled with the field. The second Ramsey interferometry pulse and detector turn those correlations into the outcome probabilities above when $C=1$.

Tracing out the atom multiplies a field density matrix entry ${\rho }_{nm}$ by $⟨a_m|a_n⟩=\cos ((n-m)\chi /2)$. Hence the nonselective quantum measurement preserves every photon number population ${\rho }_{nn}$ while changing off-diagonal quantum coherence. For $\chi =\pi /2$, coherence between $|0⟩$ and $|1⟩$ is multiplied by $1/\sqrt{2}$ per fresh meter, even with no photons absorbed. Field quantum phase information can therefore be disturbed by an energy-preserving probe. No universal self-adjoint phase operator or naive canonical number–phase commutator is needed for this conclusion.

One possible finite-contrast quantum instrument makes the distinction sharper. Define diagonal Kraus operators $K_e=\cos ((\varphi +\chi \hat{N})/2)$ and $K_g=\sin ((\varphi +\chi \hat{N})/2)$. After their ideal readout, flip the reported label with probability $r=(1-C)/2$. The unnormalized selective maps areTheir traces reproduce $L_e,L_g$ on a Fock state, and their sum reproduces the displayed dephasing quantum channel, independently of $r$. At $C=0$ the recorded labels carry no information although the physical meter interaction can still cause measurement backaction. This is an illustrative instrument, not a reconstruction of all laboratory imperfections; equal likelihoods alone do not determine the disturbance.

The 2002 experiment described in Manipulating the Quantum State of an Electrical Circuit demonstrates Ramsey interferometry in a circuit using superconductivity. An aluminum island between two small Josephson junctions and a larger readout Josephson junction formed the loop; a dilution refrigerator cooled it. Manipulation used the charge/flux bias point least sensitive to first-order fluctuations. Microwave pulse amplitude and duration were calibrated using Rabi oscillations. Two phase-coherent $\pi /2$ pulses then enclosed a variable free interval. A current pulse switched the large Josephson junction to finite voltage with state-dependent probability; a preamplifier and threshold discriminator counted repeated trials.^[ref]

As a readout control, switching was tested against pulse height at thermal equilibrium. The Ramsey interferometry oscillation period agreed with inverse frequency detuning, supporting relative quantum phase accumulation against the external oscillator. The decaying envelope indicated finite quantum coherence, while reduced contrast was attributed provisionally to population relaxation during readout. This supports controlled coherent evolution over a finite interval, without establishing perfect isolation or identifying one exclusive noise mechanism.^[ref]

A Rabi oscillation is a coherent oscillation of populations in a driven two-level quantum system.

A resonant drive can transfer a quantum state from one energy eigenstate to another and then bring its population back. This is unitary time evolution: the drive exchanges energy with the selected two-level system, while total probability stays one. One complete Rabi cycle has population period $2\pi /\Omega$; halfway through, a Rabi pi pulse transfers the whole population in the ideal model. This makes Rabi oscillations useful for calibrating resonant quantum control, preparing quantum superpositions and supplying the pulses of Ramsey interferometry.

For an ideal single transit, let the oscillatory magnetic field occupy a region of length $L$, take the longitudinal speed $v$ constant, and neglect edge fields. Each molecule then experiences one square pulse of duration $\tau =L/v$. This is one contiguous interaction region, distinct from the two separated pulses of Ramsey interferometry. The static magnetic field sets the selected transition gap; the transverse drive sets Rabi angular frequency. A change of RF frequency or static magnetic field changes Rabi detuning.

For an original ideal detector model, suppose every successful transition removes an otherwise accepted molecule and the incident beam occupies the lower state of the selected pair. With accepted no-transition signal $S_0$, constant background $S_{\mathrm{bg}}$, and effective loss contrast $0\le \eta \le 1$, writeFor identical trajectories $\overline{P}=P_{g\to e}(L/v)$, giving an intensity dip where transfer is largest. For real trajectory variation, a normalized joint probability density $w(v,ϵ,{\Omega }^{\prime })$ gives the static-offset modelHere $ϵ$ is the trajectory's transition-frequency offset and ${\Omega }^{\prime }$ its drive strength, each in angular-frequency units. The weighting is for molecules accepted by the apparatus, rather than an assumed unfiltered oven distribution. In particular, designing $\Omega L/v_0=\pi$ does not produce a Rabi pi pulse for every speed: at $0.8v_0$ and $1.2v_0$ the resonant transfer probabilities are respectively ${\sin }^2(5\pi /8)\simeq 0.8536$ and ${\sin }^2(5\pi /12)\simeq 0.9330$.

Spread in transit times, static magnetic fields and drive amplitudes can reduce or smear oscillations without quantum decoherence on any individual trajectory. Even the single-transit square pulse has finite spectral width of order $1/\tau$ in angular frequency. If the magnetic field varies substantially along one path, its Hamiltonian operator changes during the passage and the constant-offset average above must be replaced by a time-ordered exponential. Actual molecules may also have several coupled energy levels; the isolated pair and perfect loss discriminator are explicit idealizations of this model.

Ramsey interferometry uses two coherent pulses with a free interval between them: the first creates a quantum superposition, the interval accumulates relative quantum phase, and the second converts that quantum phase into measurement probabilities. Its attraction is that an ordinary population readout can reveal the quantum coherence hidden from an energy measurement during the free interval.

Here is an ideal qubit model with an explicit pulse convention. Let $R_y(\theta )=\exp (-i\theta {\sigma }_y/2)$, prepare $|0⟩$, apply $R_y(\pi /2)$, evolve freely with $U_{\varphi }=\exp (-i\varphi {\sigma }_z/2)$, and apply the inverse pulse $R_y(-\pi /2)$ before a computational-basis measurement. The first pulse gives $|+⟩$. Direct matrix multiplication givesThus the Born rule gives $P(0)={\cos }^2(\varphi /2)$ and $P(1)={\sin }^2(\varphi /2)$. Before the second pulse, the same computational-basis measurement would give $1/2$ for both outcomes for every $\varphi$. At $\varphi =2\pi /3$, the second pulse changes these probabilities to $1/4$ and $3/4$. If the two pulses have the same sign instead, the output labels interchange: the pulse convention is part of the prediction.

In a pulse experiment the phase is measured relative to the oscillator supplying the coherent pulses. In a rotating frame, write the free Hamiltonian operator as $H_{\mathrm{rot}}=\hslash \delta {\sigma }_z/2$ with signed angular frequency difference $\delta$ in the chosen labeling convention. A free interval $T$ then accumulates $\varphi =\delta T$; changing the second pulse's phase changes the analysis direction. This comparison makes Ramsey interferometry useful for measuring frequency differences and diagnosing loss of quantum coherence.

For example, suppose the first pulse has prepared a density matrix with equal populations and transverse Bloch vector length $0\le V\le 1$:The same inverse analysis pulse yields $P(0)=(1+V\cos \varphi )/2$. The pure state case $V=1$ gives the full ideal fringe; the mixed state $I/2$ gives the flat result $1/2$. This is a controlled mathematical comparison, not a fit to an experiment. Experimental contrast alone does not identify quantum decoherence: imperfect preparation, pulse errors and readout errors can also reduce it.

A moving chemical atom or molecule can experience the two pulses of Ramsey interferometry by flying through two short oscillatory-field regions. Their common oscillator supplies the reference quantum phase; the flight between them supplies the interrogation time. The long interval need not contain a driving field. This turns a spatial apparatus into a coherent pulse sequence and makes narrow spectral lines possible without filling the entire flight path with a uniform oscillatory field.

Norman Foster Ramsey, Jr.'s 1950 paper calculates the two-state transition probability and averages it over molecular velocities. It is a theoretical investigation, rather than a report of an atomic clock's measured accuracy.^[ref]

His 1989 lecture contrasts an electric-resonance apparatus photograph with a magnetic-resonance schematic. Coherent drive regions bracket a static-field flight region; state-dependent deflection converts transitions into detector changes, while differing flight times suppress side fringes.^[ref]

The readout infers a change of internal quantum state through state-dependent transmission, as in the Rabi molecular beam magnetic resonance experiment. A useful control is to block one drive region: a narrow fringe requiring both interactions must disappear. Changing the second region's oscillation phase should move the fringe, whereas an incoherent background need not move. These are consequences of the model below, not additional historical measurements. A detector dip alone neither proves perfect preparation nor determines all apparatus-induced frequency shifts; cesium-beam clock procedure adds the controls needed for a practical atomic clock.

The publication sequence begins with Norman Foster Ramsey, Jr.'s 1949 article A New Molecular Beam Resonance Method, Physical Review 76, p. 996, published 1 October; the fuller theoretical paper followed in 1950.^[ref]

NIST-7 supplied a practical realization: optical pumping moved caesium populations into the $F=3$ manifold, without isolating $m_F=0$. Two separated microwave interactions probed the clock transition; fluorescence detected transferred $F=4$ atoms. A synthesized reference, tied to a hydrogen maser, sampled opposite sides of resonance for feedback control. Beam reversal tested cavity quantum phase bias; independent velocity determinations checked motion corrections. The observed central fringe resolved the transition, but power-dependent velocity weighting and light shifts still required evaluation.^[ref]

A time-independent Hamiltonian evolves a state by the unitary operator $U(t)=e^{-iHt/\hslash }$.

For a closed quantum system, unitary time evolution changes a quantum state through a unitary operator, preserving norms and inner products. For a time-independent self-adjoint Hamiltonian operator $H$, choose the initial time as zero:Here $\hslash =h/(2\pi )$ is the reduced Planck constant. The same $U(t)$ acts on every initial quantum state; it depends on the Hamiltonian operator and elapsed time, rather than the chosen preparation.^[ref]

The self-adjointness of $H$ makes the exponential a unitary operator. In finite dimension, conjugating its matrix exponential givesThe two exponents commute because both are multiples of $H$. Hence, for any two initial vectors,This short proof explains why normalization survives, why orthogonality of initial vectors survives, and why the transition probability between two states evolved by the same $U$ is unchanged. It does not require that a fixed measurement in quantum mechanics have time-independent probabilities: its measurement basis need not move with the quantum states.

For an unbounded self-adjoint operator, define the exponential through the spectral theorem, rather than assuming that its power series acts on every vector. It is a unitary operator on the entire Hilbert space; the strong Schrödinger equation holds for initial vectors in the operator domain $D(H)$. For a density operator, unitary time evolution reads $\rho (t)=U(t)\rho (0)U(t{)}^{†}$. It preserves eigenvalues and purity of a density operator, but can change its entries in a fixed basis.

The useful distinction is between conserving total probability and conserving each measurement's probabilities. A fixed energy spectrum supplies different ticking rates for different probability amplitudes. Their relative quantum phases can move the Bloch vector and create quantum interference even while every energy eigenstate population stays fixed. Qubit precession under a constant Hamiltonian makes this visible; Ramsey interferometry converts the accumulated relative quantum phase into a population difference that a simple readout can detect.