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A dressed state of an atom and its driving field. Within the rotating-wave approximation, the paired configurations $|e,n⟩$ and $|g,n+1⟩$ mix. A classical-drive rotating-frame Hamiltonian $H=\hslash (-\Delta {\sigma }_z+\Omega {\sigma }_x)/2$ has eigenvalues $\pm \hslash \sqrt{{\Delta }^2+{\Omega }^2}/2$. At resonance its eigenvectors are equal-weight combinations of the two bare configurations. Radiative transitions between neighboring manifolds explain the Mollow triplet.

The rotating-wave approximation to a weak quadrature coupling of a cavity mode to a broadband field retains photon-exchange terms and removes counter-rotating pair creation and annihilation. Flat coupling around resonance produces the displayed interaction in units $\hslash =1$. Its short-time propagator is generated by $\sqrt{\gamma }(ad{B}^{†}-a^{†}dB)$; the Gaussian quantum noise Ito table supplies the quadratic drift correction. The resulting passive cavity has the passive cavity input-output transfer function.

For positive gyromagnetic ratio in static field $B\hat{z}$, transverse components $B_1\cos (\omega t+\phi )$ and $-B_1\sin (\omega t+\phi )$ rotate with the spin precession. Their rotating-frame Hamiltonian operator is time independent, with transverse Rabi angular frequency $\Omega =\gamma B_1$. A linearly polarized field of peak amplitude $B_{\mathrm{peak}}$ contains two oppositely rotating components and gives resonant $\Omega =|\gamma |B_{\mathrm{peak}}/2$ in the rotating-wave approximation. Negative gyromagnetic ratio requires opposite rotation sense.

The first-order response to a cosine drive contains both resonant and counter-rotating terms. The rotating-wave approximation removes the rapidly oscillating contribution only under near-resonant, weak-drive and appropriate time-scale assumptions. It does not remove the separate requirement of weak excitation when the initial amplitude is held fixed.

Choose an isolated pair of energy levels, with ground state $|g⟩=(0,1{)}^T$ and excited state $|e⟩=(1,0{)}^T$. Use the Pauli matrices in the ordered basis $(|e⟩,|g⟩)$, so ${\sigma }_z=\mathrm{diag}(1,-1)$. After subtracting the mean energy, prescribe the laboratory Hamiltonian operatorThe unperturbed energy gap is $\hslash {\omega }_0$, the drive has angular frequency $\omega$, and $\varphi$ fixes its oscillation phase at the chosen time origin. All of ${\omega }_0$, $\omega$, the Rabi angular frequency $\Omega$ and the Rabi detuning $\delta$ are in radians per second; ordinary frequency is angular frequency divided by $2\pi$. In particular, $\Omega$ here is the coefficient multiplying the cosine in $H_{\mathrm{lab}}/\hslash$, not twice that coefficient.

Write $|{\psi }_{\mathrm{lab}}⟩=R(t)|{\psi }_{\mathrm{rot}}⟩$ with $R(t)=\exp (-i\omega t{\sigma }_z/2)$. Transforming the Schrödinger equation gives $H_{\mathrm{rot}}=R^{†}{H}_{\mathrm{lab}}R-i\hslash R^{†}\dot{R}$. The rotating-wave approximation drops terms at angular frequency $2\omega$, leavingThis approximation requires $\Omega ,|\delta |\ll {\omega }_0,\omega$, and an interaction duration over which omitted corrections remain small. It also assumes that other energy levels can be neglected. Resonance alone does not justify either assumption. An exactly co-rotating drive has no discarded counter-rotating component; the linearly oscillating laboratory drive above does.

The Pauli matrix multiplication law gives $K^2={\Omega }_{\mathrm{eff}}^{2}I$. Separating even and odd terms of the matrix exponential therefore proveswith the continuous limit $U=I$ when $\Omega =\delta =0$. Starting from $|g⟩$, the upper probability amplitude is $-i{e}^{-i\varphi }(\Omega /{\Omega }_{\mathrm{eff}})\sin ({\Omega }_{\mathrm{eff}}t/2)$. The Born rule givesThe rotating transformation only changes the quantum phases of these basis states, so their laboratory populations are the same. Rabi detuning tilts the rotation axis of the Bloch vector: it increases the oscillation rate to ${\Omega }_{\mathrm{eff}}$ but caps the transferred population at ${\Omega }^2/({\Omega }^2+{\delta }^2)$. A fixed drive oscillation phase cancels from this single-pulse population when the initial state is $|g⟩$. It still fixes the relative quantum phase of a resulting quantum superposition and matters when another pulse supplies a phase reference.

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.

For $H=-\gamma B_z{S}_z-\gamma B_x{S}_x\cos (\omega t)$, choose ${\omega }_z=\gamma B_z>0$ and nominal resonance $\omega ={\omega }_z$. Starting from up spin, the rotating-wave approximation gives the normalized Bloch vector in the title, with signed $\Omega =\gamma B_x/2$. It follows by transforming the spinor $(\cos (\Omega t/2),i\sin (\Omega t/2){)}^T$ from the interaction picture back to the Schrödinger picture. Thus weak transverse driving combines slow meridional Rabi oscillations with fast Larmor precession.

This result requires $|B_x|\ll |B_z|$ and cannot be extrapolated as a controlled strong-drive formula. The exact equation is $\dot{\mathbf{s}}=\gamma \mathbf{s}\times (B_x\cos \omega t,0,B_z)$. For $|\gamma B_z|t\ll 1$, neglecting the bias gives the controlled leading solutionThe fixed-axis driving generators commute at all times, making this solution exact when the bias is zero at any prescribed drive frequency. At nominal resonance in a strong-drive experiment, the initial rotation scale is $\Theta \simeq \gamma B_{x}t$, twice the incorrectly extrapolated rotating-wave rate. Bias effects must be retained on longer times.

A motional sideband is a transition that changes internal excitation and a motional quantum number together. Define the internal energy gap as $\hslash {\omega }_0>0$. A carrier drives $|g,n⟩\leftrightarrow |e,n⟩$; the first red sideband drives $|g,n⟩\leftrightarrow |e,n-1⟩$; the first blue sideband drives $|g,n⟩\leftrightarrow |e,n+1⟩$. Their resonant drive angular frequencies follow by subtracting these energy levels. For a Raman transition, the drive is a beam-difference frequency with a sign fixed by the selected internal transition.

Choose a real carrier Rabi angular frequency $\Omega$ defined so that resonant carrier excitation is ${\sin }^2(\Omega t/2)$. In the Lamb-Dicke regime, isolated first-sideband Hamiltonians in a rotating frame can be written, after choosing the pulse phase, asThese formulas use a rotating-wave approximation: neglected carrier and other-sideband terms oscillate rapidly compared with the retained coupling. The drive's effective width and sideband coupling must be small compared with $\nu$; sufficient suppression of off-resonant carrier excitation must also be checked. The red interaction is a Jaynes-Cummings model with a motional quantum replacing a cavity photon. The blue interaction creates internal excitation and a motional quantum together; the external laser supplies energy. The driven-system rotating-frame Hamiltonian alone is not the total laboratory energy.

With $y=x-x_0$ and oscillator length $a_0=\sqrt{\hslash /(m\omega )}$, the first two harmonic oscillator eigenfunctions and energies areThe ground state is even and nodeless; the first excited state is odd with one node of a wavefunction. Their rms widths are $a_0/\sqrt{2}$ and $\sqrt{3/2}{a}_0$. Also, $m{\omega }^2{a}_0^2\sim \hslash \omega$ gives the characteristic extent directly.

For the shaken harmonic trap, expanding about the unshifted Hamiltonian givesThe scalar term gives only a common phase and no transitions. In time-dependent perturbation theory, initially $c_0=1$, $c_{n\ne 0}=0$ and the oscillatory factor for $0\to 1$ is $e^{i(E_1-E_0)t/\hslash }=e^{i\omega t}$. ThusThe odd operator $x$ makes transitions to even $n>0$ vanish by parity. In fact the first-order selection rule for a linearly driven oscillator is stronger: since $x=a_0(a+a^{†})/\sqrt{2}$,All excited $n\ne 1$ amplitudes vanish at first order, including odd $n>1$. This assertion does not apply to the surviving $c_0$ or to higher perturbative orders.

Put $\delta =\omega -\Omega$, $\Sigma =\omega +\Omega$. The exact first-order time integral isNear resonance, the second term oscillates rapidly at approximately $2\omega$ and is bounded by $1/\Sigma$, whereas the first grows as $T/2$ on resonance. Thus $\omega T\gg 1$ justifies the rotating-wave approximation, $\cos \Omega t\mapsto e^{-i\Omega t}/2$. At zeros of the resonant term, the omitted small contribution can dominate the residual; the approximation has no uniform relative-error bound there.

Using the requested order-one approximation $⟨1|x|0⟩\simeq a_0$ and $m{\omega }^2{a}_0/\hslash =\omega /a_0$ gives the square-pulse resonance profileThe exact harmonic oscillator matrix element instead gives half this probability. The finite central maximum is at $\Omega =\omega$; first zeros are at $|\Omega -\omega |=2\pi /T$, with smaller side lobes and a $1/{\delta }^2$ envelope. Increasing $T$ narrows the peak as $1/T$ and increases its height as $T^2$, as long as weak resonant driving keeps $p_1\ll 1$. Small displacement alone does not ensure this for arbitrarily long $T$; near resonance one also needs $\epsilon \omega T/a_0\ll 1$.

Insert the interaction-picture expansion into the Schrödinger equation. The phases multiplying the basis states already solve the unperturbed evolution, leavingHermiticity requires $H_{21}^{\prime }=\hslash ({\omega }^{\prime }{)}^{\ast }$, not $\hslash {\omega }^{\prime }$ unless the coupling is real. Since the diagonal matrix elements vanish and ${\omega }_0={\omega }_2-{\omega }_1$, expanding the cosine givesThe conjugate pair of couplings conserves $|c_1{|}^2+|c_2{|}^2$. Near resonance, the difference frequency is slow while the sum frequency is approximately $2{\omega }_0$. Rapid terms alternate sign before an appreciable slow transfer accumulates. The rotating-wave approximation retains the slow terms when $|{\omega }^{\prime }|,|\omega -{\omega }_0|\ll \omega +{\omega }_0$; resonance alone is insufficient for a strong drive. Counter-rotating corrections include small shifts of the resonance and need not remain negligible at arbitrary strength or arbitrarily long times.

For the spin preparation, use $H=-\mathbf{\mu }\cdot \mathbf{B}=-\gamma \mathbf{s}\cdot \mathbf{B}$ and $\mathbf{s}=\hslash \mathbf{\sigma }/2$. Choose $B>0$ and $\gamma >0$ to fix helicity, with $|\uparrow ⟩=(1,0{)}^T$, $|\downarrow ⟩=(0,1{)}^T$ and ${\omega }_0=\gamma B$. The initial $|\uparrow ⟩$ is the lower state. For negative $\gamma$, reverse the RF rotation sense; pulse areas use $|\gamma |$, and signed Larmor precession determines the corresponding phase adjustment.

A circularly rotating magnetic drive uses two transverse fields in quadrature during $0<t<\tau$:With $U_0(t)=e^{+i{\omega }_{0}t{\sigma }_z/2}$, their interaction Hamiltonian is exactly constant,The minus sign in the second RF component selects co-rotation for the specified positive gyromagnetic ratio. Exponentiating this Pauli matrix givesThe laboratory phase of a resonant spin pulse includes free precession. Returning to the laboratory frame and removing a common phase,Unlike a linearly polarized RF field, this exactly co-rotating field has no counter-rotating component. A single linearly polarized field $B_{x,\mathrm{peak}}\cos ({\omega }_{0}t+\phi )$ gives the same slow rotation in the weak-drive rotating-wave approximation with $\Omega =|\gamma |B_{x,\mathrm{peak}}/2$. For the circular field above, $\Omega =|\gamma |B_1$. Keeping these amplitude conventions distinct avoids a factor-of-two pulse-duration error.

For a single harmonic pulse detuned by $\Delta ={\omega }_0-\omega$, the effective rotating-frame field tilts out of the transverse plane. Its rotation rate becomes $\sqrt{{\Omega }^2+{\Delta }^2}$, and its maximum population transfer is ${\Omega }^2/({\Omega }^2+{\Delta }^2)$. Exact inversion to $-z$ is impossible with one fixed nonzero-detuning pulse. Half transfer, needed for a transverse eigenstate, gives the equatorial reachability of a detuned spin pulse criterion $|\Delta |\le \Omega$. With that condition, changing the duration and RF phase can still prepare $+x$ or $+y$ at the chosen laboratory stopping time. The mere statement $\omega \approx {\omega }_0$ does not ensure this condition, because the drive can be much weaker than either frequency. If $|\Delta |>\Omega$, neither transverse target is reached by that single pulse. These restrictions concern fixed-frequency, fixed-amplitude single-pulse control; retuning, chirping or phase-controlled composite pulses change the available controls and can remove the restriction. No dissipative spin relaxation is included in this coherent two-state model.

Use the isolated nuclear spin Zeeman effect implied by the supplied nuclear magnetic moment, with hyperfine structure neglected. The proton gyromagnetic ratio isThe two spin levels are separated by $g_p{\mu }_N{B}_0$. Resonant nuclear magnetic resonance therefore requires the Larmor frequency

The duration is a Rabi pi pulse: in a frame rotating with the resonant transverse magnetic field, the spin must rotate through $\pi$. If $B_1$ denotes the magnitude of a circularly rotating field with the correct sense of rotation, the Rabi angular frequency is ${\Omega }_R={\gamma }_p{B}_1$. The shortest pulse is then

There is an amplitude convention to specify. If the applied transverse magnetic field is instead the linearly oscillating $B_1\cos (\omega t)\hat{\mathbf{x}}$, its resonant circular component has magnitude $B_1/2$. The rotating-wave approximation gives ${\Omega }_R={\gamma }_p{B}_1/2$ and henceThe wording does not specify this polarization convention, so the two pulse durations should be distinguished. Odd multiples of either shortest Rabi pi pulse also invert the spin within the corresponding ideal model.

Write ${\omega }_z=\gamma B_z$ and use the ordered spin basis $(|\uparrow ⟩,|\downarrow ⟩)$. The free Hamiltonian operator is $H_0=-\hslash {\omega }_z{\sigma }_z/2$, soResolving the cosine into its two complex exponentials gives the exact upper off-diagonal factorwith the complex conjugate in the lower entry. Choose the field orientation with ${\omega }_z>0$, as implicit in the printed near-resonance condition, and put $\delta =\omega -{\omega }_z$. The rotating-wave approximation discards the terms at $\omega +{\omega }_z$, retaining the slow terms at $\delta$. The resulting interaction picture equations areThe factor $1/4$ contains $1/2$ from $S_x=\hslash {\sigma }_x/2$ and $1/2$ from the cosine decomposition. This is a Driven two-level Rabi model, with signed on-resonance coupling $\Omega =\gamma B_x/2$ and generalized Rabi angular frequency ${\omega }_R=\sqrt{{\delta }^2+{\Omega }^2}$. Here the Rabi detuning uses the paper's convention, drive minus free frequency.

Small detuning alone is insufficient to derive the printed equations. One also requires $|\gamma B_x|,|\delta |\ll \omega +{\omega }_z$, and an evolution time over which neglected terms stay small. At resonance this is the weak transverse-field regime $|B_x|\ll |B_z|$. The supplied solution is thus an approximate solution for the linearly oscillating field, not an exact one at arbitrary drive strength. If $\gamma B_z<0$, the other exponential is resonant and the physical positive resonance frequency is $|\gamma B_z|$.

With $\Omega =\gamma B_x/2$, the Born rule applied to the supplied down-spin probability amplitude givesFor nonzero transverse field, the maxima occur atThe first maximum is reached at $\pi /{\omega }_R$; on resonance this is a Rabi pi pulse. The resonance envelope of a Rabi transition is a Lorentzian line shape, centred on ${\omega }_{\mathrm{res}}={\omega }_z$ under the orientation convention above. It has half its peak value at $|\omega -{\omega }_z|=|\Omega |$, so its full width at half maximum isIf resonance width is instead defined as the half-width, the ratio differs by a factor of two but has the same requested proportionality. The quality factor is large precisely in the regime where the rotating-wave approximation is justified. The envelope is obtained by choosing the best time separately at each drive frequency; a measurement at one fixed pulse duration has a different line profile.

At resonance, retain the signed coupling $\Omega =\gamma B_x/2$. In the interaction picture, the given solution reduces to $c_1=\cos (\Omega t/2)$ and $c_2=i\sin (\Omega t/2)$. Returning to the Schrödinger picture givesFor a normalized spinor $(a,b{)}^T$, the Pauli matrices give $⟨\mathbf{\sigma }⟩=(2\mathrm{Re}{a}^{\ast }b,2\mathrm{Im}{a}^{\ast }b,|a{|}^2-|b{|}^2)$. Therefore the spin expectation value within the rotating-wave approximation isIts magnitude is $\hslash /2$: the Bloch vector stays on the unit sphere. In a frame rotating about $z$ at angular velocity $-{\omega }_z$, it makes a uniform rotation in the $yz$ plane. For $|B_x|\ll |B_z|$, this gives slow Rabi oscillations between the poles, accompanied by rapid laboratory Larmor precession about $z$.

If the same formula is formally used for $|B_z|\ll |B_x|$, it predicts fast meridional rotations whose plane turns slowly about $z$. However, the original PDF really does ask for this opposite limit, and it lies outside the assumption used in part (b). That formal description is not a controlled prediction for the stated linearly driven Hamiltonian operator.

A valid strong-drive description starts from the exact equation for the Bloch vector $\mathbf{s}=2⟨\mathbf{S}⟩/\hslash$:For times $|\gamma B_z|t\ll 1$, neglecting the weak bias gives a controlled leading approximation, since the time integral of the omitted Hamiltonian operator has small norm. The remaining field has a fixed $x$ axis, so its generators commute at different times andAt the nominal resonance $\omega =\gamma B_z$ and on the early spin-rotation scale $t\sim 1/|\gamma B_x|$, this becomes $\Theta \simeq \gamma B_{x}t$. The initial rotation rate is consequently $\gamma B_x$, twice the rate predicted by extrapolating the rotating-wave approximation. For longer times the exact differential equation must be retained. The following original calculation illustrates both regimes rather than asserting an exact strong-drive Rabi formula.

Expand the wavefunction in unperturbed energy eigenstates,The $c_j$ are interaction picture probability amplitudes, so $|c_j{|}^2$ are level populations and $|c_1{|}^2+|c_2{|}^2=1$ in this closed two-level system. A classical field ${\mathbf{E}}_0\cos \omega t$ supplies the electric-dipole interaction $-\mathbf{d}\cdot {\mathbf{E}}_0\cos \omega t$. Choosing the relative phase of the energy eigenstates makes $\Omega =-⟨1|\mathbf{d}\cdot {\mathbf{E}}_0|2⟩/\hslash$ real. This is the resonant Rabi angular frequency.

Combining the field's two exponentials with the unperturbed quantum phases produces terms oscillating at $\omega -{\omega }_0=\delta$ and at $\omega +{\omega }_0$. The rotating-wave approximation drops the latter, leaving exactly the supplied equations with coupling $\Omega /2$. It requires $|\Omega |,|\delta |\ll {\omega }_0$ and a field envelope slow enough on the carrier timescale. Spontaneous emission and dephasing must be negligible during the sequence. For a ground-state hyperfine level transition, the analogous direct drive is a magnetic dipole transition; the same equations hold with the magnetic matrix element replacing the electric one.

Assume an initially unexcited chemical atom, two nonoverlapping square pulses starting at $0,T$, and a common coherent field phase. In the weak-excitation limit $|\Omega |{\tau }_p\ll 1$, substitute $c_1=1+O({\Omega }^2)$ into the second probability amplitude equation. After the second pulse,where the sinc function means $\mathrm{sinc}u=\sin u/u$, with value one at zero. Squaring givesThe printed prefactor is a factor of four too small for the supplied equations and pulse durations. A direct counterexample is $\delta =0$: the exact resonant Rabi oscillation gives $P_2={\sin }^2(\Omega {\tau }_p)$ for two pulses of duration ${\tau }_p$, because the intervening resonant rotating-frame evolution is trivial. Its small-area limit is $(\Omega {\tau }_p{)}^2$, rather than $(\Omega {\tau }_p/2{)}^2$. The latter is the single-pulse result before coherent addition.

The condition $\omega {\tau }_p\ll 1$ printed in the source also does not justify the derivation from a physical carrier field. A consistent rotating-wave approximation pulse regime isIf the supplied equations are treated as the mathematical model regardless, their weak-excitation integral remains valid for any ${\tau }_p$; subcycle pulses of the original real field require the discarded counter-rotating term and depend on carrier phase. $T$ here is the separation of pulse starts, as specified, rather than the dark interval $T-{\tau }_p$.

Choose the relative phases of the two energy eigenstates so that $d=⟨{\psi }_2|{\hat{D}}_0|{\psi }_1⟩$ is real and nonnegative. Define the resonant Rabi angular frequency $\Omega =d/\hslash$, the Rabi detuning $\delta ={\omega }_0-\omega$, and $\Sigma ={\omega }_0+\omega$. Substitution into the Schrödinger equation, followed by projection onto the two eigenstates, gives the exact interaction picture equationsKeep both probability amplitudes dynamical: replacing $c_1$ by one would incorrectly impose weak excitation and miss complete population transfer.

For $|\delta |,\Omega \ll {\omega }_0$, apply the rotating-wave approximation to discard the rapidly oscillating terms at $\Sigma$. Differentiating the second remaining equation and using the first givesThe initial conditions are $c_2(0)=0$ and ${\dot{c}}_2(0)=-i\Omega /2$. With ${\Omega }_R=\sqrt{{\Omega }^2+{\delta }^2}$, their solution isConsequently,This result permits order-one excitation, including full population transfer on resonance, even though the drive is small compared with the optical angular frequency.

For an average from switch-on to time $\tau$, integrate the squared sine:On resonance this limiting average is $1/2$, although the instantaneous probability repeatedly reaches one. The zero-coupling, zero-Rabi detuning case is understood separately as $P_2=0$.

Near resonance alone does not make this an exact result for arbitrarily strong driving. The stated sinusoidal Hamiltonian also needs the rotating-wave approximation. A concrete counterexample is $\omega ={\omega }_0$ and $\Omega ={\omega }_0$: the exact equations give $P_2(t)={\Omega }^2{t}^2+O(t^4)$ at switch-on, while the claimed rotating-wave sinusoid gives ${\Omega }^2{t}^2/4+O(t^4)$. Thus “without assuming weak radiation” can mean without assuming small excitation, but cannot remove the drive-to-carrier scale separation. The infinite-time average above belongs to the rotating-wave model; omitted corrections can accumulate over sufficiently long times.

Rabi oscillations require a coherent drive with a stable phase over several population cycles and an isolated, appreciably coupled pair of quantum states. For large-amplitude oscillations, the Rabi detuning must be comparable to or smaller than the Rabi angular frequency. The population lifetime, optical coherence time, transit time and drive coherence time should all exceed $2\pi /{\Omega }_R$ by enough to resolve the cycles. An ensemble also needs sufficiently uniform Rabi frequencies and Rabi detunings that phase averaging does not wash the oscillations out. The usual sinusoidal formula additionally assumes the rotating-wave approximation.

The Einstein coefficients describe incoherent transition rates, rather than the evolving optical probability amplitudes. Their simplest use is appropriate for broadband, randomly phased radiation, with field correlation time short relative to population evolution, or for rapidly dephased chemical atoms whose optical coherences can be eliminated. The radiation energy density per unit frequency must be smooth over the relevant spectral line. Stimulated upward and downward transitions and spontaneous emission then enter rate equations, without predicting coherent population return. A short observation time can give a small excitation probability even for a coherent field; that condition alone does not justify an Einstein coefficients rate description. Coherent laser irradiation can also admit steady rate equations after optical coherence has relaxed, with the appropriate saturation response.

Fix the qubit polarization convention by $\rho =(I+\vec{P}\cdot \vec{\sigma })/2$, where the Pauli matrices are ordered in the basis $(|1⟩,|2⟩)$ and ${\sigma }_z=|1⟩⟨1|-|2⟩⟨2|$. Thus the ground state is the north pole of the Bloch sphere. In a rotating frame and within the rotating-wave approximation, choose the drive phase and detuning sign so thatFor example, with this ground-state convention one may take $\Delta ={\omega }_L-{\omega }_0$ and choose the transverse coupling positive. The Bloch vector precesses around the effective field $(\Omega ,0,\Delta )$ at angular frequency ${\Omega }_R=\sqrt{{\Omega }^2+{\Delta }^2}$, tracing a circle at fixed projection on that axis. Its length is conserved under coherent unitary time evolution. On resonance the axis is transverse and the Rabi oscillation can fully invert an initially ground-state atom. Away from resonance the axis tilts, andThere is no relaxation towards an equilibrium Bloch vector in this ideal model. An inhomogeneous ensemble can nevertheless lose its averaged qubit polarization through dephasing even without spontaneous emission; the circle description assumes a common drive and detuning.

For $|a{|}^2+|b{|}^2=1$, the stated quantum state givesThe diagonal elements are the two state populations. The off-diagonal elements are quantum coherences: they retain the relative quantum phase and determine interference under a subsequent pulse. This is a pure state, even if both populations are nonzero: ${\rho }^2=\rho$, $\mathrm{Tr}{\rho }^2=1$, and $|\vec{P}|=1$. A mixed state averages projectors over preparation probabilities and generally has $\mathrm{Tr}{\rho }^2<1$. Equal populations alone do not distinguish an equal coherent superposition from an unpolarized mixture.

For use in all three cases, multiplication of the gates gives the Hadamard-phase-Hadamard qubit rotationThe global phase cancels in ${\rho }^{\prime }=U\rho U^{†}$. Thus the Bloch vector map is

The final unlabelled Ramsey spectroscopy comparison applies to the three cases as follows. In an atomic clock, an initial coherent pulse prepares a superposition, a free interval $T$ accumulates a relative quantum phase proportional to detuning times $T$, and a final coherent pulse converts that phase to a population. Here $\alpha$ plays the role of that accumulated phase, with sign and fringe offset set by pulse conventions. In (a), the first Hadamard gate creates the needed superposition and the final gate reveals full-contrast Ramsey interferometry. In (b), the first Hadamard gate instead puts the state in $|1⟩$, so the intervening phase gate has no relative quantum phase to imprint: the output is independent of $\alpha$. In (c), the unpolarized density matrix remains unchanged under every gate, so the ensemble has no Ramsey interferometry contrast. A coherent unitary gate alone cannot initialize a maximally mixed ensemble into a pure clock state.

The Ramsey interferometry analogy concerns preparation, phase accumulation, and readout; a Hadamard gate is not literally a transverse $\pi /2$ rotation on every input. In this basis $H=R_y(\pi /2){\sigma }_z$, and its Bloch sphere action is a $\pi$ rotation about $(\hat{x}+\hat{z})/\sqrt{2}$, up to a global phase. The phase conventions of physical Ramsey spectroscopy pulses must be specified to compare their outputs for arbitrary initial states.

Treat $|a⟩=|0⟩$ and $|b⟩=|1⟩$ as an isolated ground-state hyperfine qubit. Their transition angular frequency is ${\omega }_0=2\pi \times 320.7\mathrm{MHz}$. The radiofrequency magnetic field couples them through an allowed magnetic dipole transition. After choosing its phase reference, take the two-level laboratory Hamiltonian operator to bewhere $X,Z$ are Pauli operators. Here $\Omega$ is defined by the on-resonance rotating-frame off-diagonal matrix element $\hslash \Omega /2$.

Use a quantum rotating frame and the rotating-wave approximation, valid for $\Omega ,|\delta |\ll {\omega }_0$, with Rabi detuning $\delta ={\omega }_0-\omega$. Ignoring off-resonant coupling to other levels givesAt resonance $\delta =0$, the unitary time evolution isThus the Born rule gives coherent Rabi oscillations,The stated amplitudes are in the rotating frame; the laboratory carrier phases are tracked by the drive reference.

These logical states are orthogonal, individually resolvable hyperfine levels in the same electronic ground state. They avoid fast optical decay, can be coherently coupled by radiofrequency control, and have a state-selective optical readout through $|c⟩$. The excited level $|c⟩$ is useful for fluorescence detection rather than for the storage superposition in this model. Long lifetimes alone do not guarantee a useful qubit: control isolation and adequate quantum coherence are also required.