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A good atomic qubit needs two reproducible, orthogonal internal states whose populations and relative quantum phase survive for many gate times. Both should be long lived, with weak spontaneous emission and little decoherence from fluctuating fields or atomic motion. Their transition must nevertheless permit fast, selective, coherent control with little leakage to other states. Reliable initialization and state-selective quantum measurement are also required; optical pumping and fluorescence readout can supply these functions.

For rubidium-87, use the two ground state hyperfine levels in the $5{}^2{S}_{1/2}$ electronic configuration. The electronic angular momentum is $J=1/2$, and addition of angular momentum to the given nuclear spin $I=3/2$ gives $F=|I-J|,\dots ,I+J=1,2$. A suitable Rubidium-87 clock qubit isThese are coupled electron-nucleus angular momentum eigenstates, rather than two bare nuclear-spin projections. Their hyperfine structure supplies a resolvable splitting ${\nu }_{01}\simeq 6.835\mathrm{GHz}$. Both belong to the electronic ground state, avoiding a short optical excited-state lifetime. Since the weak-field Zeeman shift is proportional to $g_F{m}_{F}B$, both have zero linear shift at $B=0$: their rubidium-87 clock transition suppresses first-order magnetic-field dephasing near zero field. A quadratic Zeeman shift of a clock transition and differential optical shifts remain. A small stable bias field can define the quantization axis and help resolve unwanted transitions, at the cost of some residual magnetic sensitivity.^[ref]

Drive the magnetic dipole transition with a resonant microwave field of suitable polarization, here $\Delta m_F=0$, or use two phase-coherent optical fields for a far-detuned stimulated Raman transition with their frequency difference resonant with ${\nu }_{01}$. The optical excited state is then only virtually populated, reducing spontaneous emission. In the rotating frame and rotating-wave approximation, define the drive phase $\varphi$ by the effective HamiltonianHere $X,Y,Z$ are the Pauli matrices on the chosen qubit, and $\Omega$ is the Rabi angular frequency. Pulse duration controls the rotation gate angle, while phase selects the axis in the equatorial plane of the Bloch sphere. A $\pi$ pulse exchanges the populations; a $\pi /2$ pulse makes an equal quantum superposition from either basis state. Controlled Rabi detuning, a differential energy shift, or compositions of these pulses give $z$ rotations. For example, $R_z(\alpha )R_y(\beta )R_z(\gamma )$ realizes any one-qubit unitary operator up to a global phase. Thus the atomic encoding admits arbitrary single-qubit quantum gates.

Let $|g⟩,|e⟩$ be the atomic levels, ${\sigma }_{+}=|e⟩⟨g|$, and $a$ the annihilation operator of the single field mode. Choose the joint ground energy to be zero. The resonant Jaynes-Cummings model isThe first term counts field photons, the second the atomic excitation, and the interaction exchanges one photon for one atomic excitation. A phase redefinition makes $g$ real and positive. Resonance means $\omega ={\omega }_{eg}$; the total excitation $\mathcal{N}=a^{†}a+|e⟩⟨e|$ is conserved. Adding an overall constant gives the equivalent convention using ${\sigma }_z/2$ and a zero-point field term.

The model follows from the electric-dipole interaction and the rotating-wave approximation, retaining the energy-conserving terms while dropping $a{\sigma }_{-}$ and $a^{†}{\sigma }_{+}$. We assume a single quantized monochromatic mode and negligible cavity leakage and atomic relaxation over the time considered. With this convention, the resonant population Rabi angular frequency in the one-excitation subspace is $2g$.

Quantize a single electromagnetic normal mode of angular frequency ${\omega }_c$ and represent one isolated atomic transition as a two-level system with raising operator ${\sigma }_{+}=|e⟩⟨g|$. The bare Hamiltonian operator isThe first main approximation is the electric-dipole approximation: the field varies negligibly across an atom, $k{a}_{\mathrm{atom}}\ll 1$, and the leading coupling is $H_d=-\mathbf{d}\cdot \mathbf{E}({\mathbf{r}}_0)$. For states of definite opposite parity, the relevant dipole part is off-diagonal. After choosing operator phases, this gives $H_d=\hslash g(a+a^{†})({\sigma }_{+}+{\sigma }_{-})$, with $g$ fixed by the transition dipole and the one-photon field wave amplitude.

In the interaction picture, $a{\sigma }_{+}$ and $a^{†}{\sigma }_{-}$ oscillate at the difference ${\omega }_a-{\omega }_c$, whereas $a{\sigma }_{-}$ and $a^{†}{\sigma }_{+}$ oscillate at the sum ${\omega }_a+{\omega }_c$. The second main approximation is the rotating-wave approximation: near resonance and for $g\sqrt{n+1}\ll {\omega }_a+{\omega }_c$, the latter terms average away on the interaction time scale. The resulting Jaynes-Cummings model hasThe two-level system and single-normal mode truncations are additional assumptions: other atomic transitions and optical normal modes must be sufficiently far off resonance or weakly coupled. Dissipation is neglected over the interaction time. The retained terms exchange one atomic excitation for one photon, conserving $a^{†}a+{\sigma }_{+}{\sigma }_{-}$.

Use $t_0=0$ as in the displayed carrier-phase convention. The drift is diagonal, and conjugation givesMultiplying by $A(t)\cos (\omega t+\varphi )$ and expanding the cosine into exponentials yieldsThe rotating-wave approximation averages away the components oscillating at $2\omega$, retainingThe envelope is slowly varying and the effective coupling is weak compared with the carrier frequency. The pulse phase selects the transverse rotation axis on the Bloch sphere; its pulse area determines the angle. Counter-rotating corrections are neglected under this approximation, not identically zero.

Work with real positive pulse envelopes and define $\Omega =\sqrt{{\Omega }_{12}^2+{\Omega }_{23}^2}>0$. Then $\cos \theta ={\Omega }_{12}/\Omega$ and $\sin \theta ={\Omega }_{23}/\Omega$. In the ordered basis $|1⟩,|2⟩,|3⟩$, the candidate dark state of a driven Hamiltonian has coordinatesMultiplication by the given rotating-wave approximation matrix givesThusIts absence of an excited-state component is the destructive interference that makes it dark. The orthogonal bright ground-state combination is $|B⟩=\cos \theta |1⟩+\sin \theta |3⟩$. Since $H|B⟩=\Omega |2⟩$ and $H|2⟩=\Omega |B⟩$, the two remaining eigenstates are $(|B⟩\pm |2⟩)/\sqrt{2}$ with eigenvalues $\pm \Omega$.

For signed envelopes, use a continuous two-argument angle with $\cos \theta ={\Omega }_{12}/\Omega$ and $\sin \theta ={\Omega }_{23}/\Omega$, rather than an ambiguous arctangent branch. When both fields vanish, every state has zero eigenvalue and this formula does not select a unique dark vector.

Use units $\hslash =1$. A Hermitian quadrature coupling of a cavity mode to a continuum of external modes can be writtenwith real coupling amplitude $\kappa$. The free Quantum Hamiltonian is $H_0={\omega }_c{a}^{†}a+\int \omega b_{\omega }^{†}{b}_{\omega }d\omega$. In the interaction picture, the two photon-exchange products $a{b}_{\omega }^{†}$ and $a^{†}{b}_{\omega }$ acquire phases with frequency $\omega -{\omega }_c$, whereas $a^{†}{b}_{\omega }^{†}$ and $a{b}_{\omega }$ oscillate at $\omega +{\omega }_c$.

For weak coupling near resonance, the rotating-wave approximation drops those rapidly oscillating pair-creation and pair-annihilation terms and retainsFor a broadband flat coupling $\kappa =\sqrt{\gamma /(2\pi )}$, define the envelope annihilation operator $b(t)=(2\pi {)}^{-1/2}\int b_{\omega }{e}^{-i(\omega -{\omega }_c)t}d\omega$. The Markov approximation extends the detuning integral over the full real line and gives $[b(t),b^{†}(s)]=\delta (t-s)$. Then the cavity-reservoir interaction isThe phases and normalization of the external field fix the sign convention. The creation operators and annihilation operators exchange one excitation between cavity and field, rather than creating or destroying a pair.

Write $\cos {\omega }_{1}t=(e^{i{\omega }_{1}t}+e^{-i{\omega }_{1}t})/2$ and $\Delta \omega ={\omega }_2-{\omega }_1$. Multiplication of the matrix from part (e) givesThese are exactly the constant, difference-frequency, and sum-frequency terms in the PDF, with each Hermitian pair visible. For a weak, slowly varying envelope, the terms at $2{\omega }_1$ and ${\omega }_1+{\omega }_2$ average out by the rotating-wave approximation. To discard the other transition as well, its detuning must be nonzero and large compared with the drive coupling and envelope bandwidth. Sufficient scale conditions arewhere $B$ is a characteristic envelope bandwidth; turn-on transients must satisfy the same spectral restrictions. ThenSmall coupling compared with the optical frequencies alone is not enough for this selective approximation: if $|\Delta \omega |$ is comparable with the pulse bandwidth or Rabi scale, the spectator transition remains driven. A long selective gate also accumulates a small off-resonant Stark phase of order $(A_1{d}_2{)}^2/\Delta \omega$, so that error must be acceptable or compensated. These assumptions distinguish resonant quantum control from an exact decoupling.

An atom-field dressed state is an eigenstate of the coupled atom and drive, rather than an eigenstate of the isolated atom. A strong near-resonant field mixes the bare two-level states and changes the transition spectrum. In a frame rotating at the drive frequency, the rotating-wave approximation gives, for one convenient convention,Here $\Omega$ is the resonant Rabi frequency. The two eigenvalues areAt resonance, the dressed states are equal-weight symmetric and antisymmetric combinations of the bare atom–field configurations. For a quantized drive, the appropriate paired configurations are $|e,n⟩$ and $|g,n+1⟩$; their coupling scales as $\sqrt{n+1}$. A strong coherent drive gives the familiar approximately classical Rabi frequency over its occupied photon-number range.

Radiative transitions connect neighboring dressed manifolds. Same-branch transitions have frequency near ${\omega }_L$, while opposite-branch transitions give sidebands near ${\omega }_L\pm {\Omega }_R$. Under sufficiently strong resonant driving this produces the Mollow triplet in resonance fluorescence, with sidebands separated from the center by $\Omega$. Radiation damping determines widths and weights; detuning changes the relative intensities. Sidebands must be sufficiently separated compared with the decay scale to be resolved.^[ref]

A fluorescence experiment drives the transition with a narrow-band laser, collects scattered light away from the intense drive beam, and resolves its spectrum. Changing drive intensity changes $\Omega$ and hence the splitting. This measures transitions of the interacting system, not merely a population oscillation in time. The emitted spectrum can include an elastic coherent contribution as well as the inelastic fluorescence peaks, so the observed line shape needs the specified drive, damping and detection model.

At first order in time-dependent perturbation theory set $c_1\simeq 1$ and use the parity result $V_{22}=0$. Decomposing the cosine into its two exponentials givesBoth exponentials contribute to the exact first-order result:Each apparent zero denominator has a continuous limit. In particular the near-resonant fraction tends to $it$ as $\omega \to {\omega }_0$, while the counter-rotating fraction remains bounded on the optical timescale. One may equivalently write the expression asThis form makes $c_2(0)=0$ explicit and avoids cancellation at resonance. At extremely short time both sinc functions tend to one, so $c_2\simeq -i\Omega t$, agreeing with the initial derivative of the original cosine-driven equation. No rotating-wave approximation has yet been made. The separate assumption of weak excitation remains necessary: the first-order expression must not be extended until it predicts probabilities comparable to or larger than one.

Insert the expansion with amplitudes $c_1,c_2$ and the two unperturbed dynamical phases into $i\hslash \dot{\Psi }=(H_0+V)\Psi$. Differentiation of the explicit phases cancels $H_0$, leavingEquivalently, with $\Omega =(\omega -{\omega }_0)/2$,These interaction-picture equations conserve $|c_1{|}^2+|c_2{|}^2$. They are exact for the circularly rotating magnetic drive; a rotating-wave approximation has not been used.

Let $\Delta ={\Omega }_0-\omega$ and $R(t)=e^{-i\omega t{S}_z/\hslash }$. Differentiating $|\Psi ⟩=R|{\Psi }_R⟩$ gives the rotating-frame Hamiltonian operator $R^{†}HR-i\hslash R^{†}\dot{R}$. The angular momentum commutator yields $R^{†}{S}_{\pm }R=e^{\pm i\omega t}{S}_{\pm }$, while $-i\hslash R^{†}\dot{R}=-\omega S_z$. Thus the rotating-frame Rabi Hamiltonian isThis transformation is exact for the circular drive; no rotating-wave approximation has been used.

A near-resonant drive selects a transition and, in an appropriate rotating frame and rotating-wave approximation, controls a two-level effective Quantum Hamiltonian. If $H_{\mathrm{eff}}=\Omega (t){\sigma }_x/2$ with fixed axis, then $U(T)=\exp [-i{\sigma }_x{\int }_0^T\Omega (t)dt/2]$. Pulse area fixes the rotation angle; drive phase chooses a transverse axis. Selectivity and neglect of rapidly oscillating terms require the appropriate detuning and drive-strength regime.

A circularly driven spin becomes time-independent after a rotation $R=e^{-i\omega t{S}_z/\hslash }$. The transformed Hamiltonian operator is $R^{†}HR-i\hslash R^{†}\dot{R}$, so the longitudinal frequency is shifted by $-\omega$. This is an exact change of frame for a circular drive. The rotating-wave approximation is a separate approximation needed when a drive also contains a counterrotating component.

Consider a two-level system with a real coupling energy $g$ and constant angular drive frequency $\Omega$,The opposite phases and real $g$ make this Hamiltonian Hermitian. Write the state amplitudes as $a=U(t)b$, with the unitary matrixSubstitution into the Schrödinger equation gives the constant quantum rotating frame HamiltonianConsequently $b(t)=[I\cos (\Lambda t/\hslash )-i{H}_{\mathrm{eff}}\sin (\Lambda t/\hslash )/\Lambda ]b(0)$. Starting in the first basis state gives transition probabilityThe diagonal transformation changes phases but not populations. This is an exact solution of the specified rotating-coupling model, without a rotating-wave approximation. If the input phase is instead written $e^{i\omega t/\hslash }$, then $\omega =\hslash \Omega$ has units of energy, rather than angular frequency.

A constant-amplitude drive lasting $T$ gives the Fourier transform in the title, where $\mathrm{sinc}x=\sin x/x$ and its value at zero is one. The transition probability in weak driving has a finite resonant maximum, first zeros at $\delta =\pm 2\pi /T$, and decreasing side lobes. Longer duration increases the peak as $T^2$ and narrows its width as $1/T$. The rotating-wave approximation discards a rapidly oscillating counterrotating term; it can have a small absolute error but a large relative error at the resonant term's zeros.

The Tavis-Cummings model extends the Jaynes-Cummings model to several two-level systems exchanging excitations with a common quantized normal mode under the rotating-wave approximation. It conserves total excitation number. Equal coupling preserves permutation-symmetric atomic sectors, allowing finite-dimensional reductions for a fixed initial excitation number.

For two coherent square pulses of equal duration $\tau$, pulse-start separation $T\ge \tau$, resonant Rabi angular frequency $\Omega$, and Rabi detuning $\delta$, the initially lower-state transition probability has the displayed leading weak-excitation form when $|\Omega |\tau \ll 1$. Integrating the first-order probability amplitude over both windows gives a sum of two equal Fourier transforms, with relative quantum phase $\delta T$. The factor four from coherent addition makes the central value $(\Omega \tau {)}^2$. The rotating-wave approximation additionally requires many carrier cycles per pulse and near-resonant weak driving.