Motional sideband (${\omega }_{\mathrm{drive}}={\omega }_0,{\omega }_0-\nu ,{\omega }_0+\nu$)

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.

A motional carrier transition changes an atom's internal state without changing its oscillator quantum number. Its matrix element from the motional ground state contains the factor $e^{-{\eta }^2/2}$. A first blue sideband is smaller in amplitude by $\eta$ and changes the motional energy by one trap quantum. Conditional energy changes must be distinguished from averages over all motional outcomes.

Consider an original ideal calculation with no heating, decoherence, detuning or spectator modes. In the ordered basis $(|g,n⟩,|e,n-1⟩)$ for $n\ge 1$, the red motional sideband has $H_r=\hslash \eta \Omega \sqrt{n}{\sigma }_x/2$. Here ${\sigma }_x$ is the Pauli matrix exchanging the two basis states. Since ${\sigma }_x^2=I$, its unitary time evolution isThus $U_n^{†}{U}_n=I$, and excitation from $|g,n⟩$ has probability $P_r(n,t)={\sin }^2(\theta \sqrt{n}/2)$. For $n=0$ it vanishes. The blue motional sideband similarly gives $P_b(n,t)={\sin }^2(\theta \sqrt{n+1}/2)$. A red Rabi pi pulse calibrated for $n=1$ has $\theta =\pi$; it is not a Rabi pi pulse for every occupied $n$.

Choose $\eta =0.08$, $\Omega /(2\pi )=100\mathrm{kHz}$ and $\nu /(2\pi )=2\mathrm{MHz}$. The $n=1$ red Rabi pi pulse lasts $t_{\pi }=\pi /(\eta \Omega )=62.5\mu \mathrm{s}$. Let motion have the Boltzmann distribution of a quantum harmonic oscillator, with chosen mean occupation $\bar{n}=0.25$:Here $\sum p_n=1$, $\sum n{p}_n=\bar{n}$ and $p_0=0.8$. Reindex the red sum using $n=m+1$ and $p_{m+1}=r{p}_m$: $P_r(t)=r{P}_b(t)$ for equal resonant pulse parameters, even outside the short-pulse limit. At times with $P_b>0$, the ratio $R=P_r/P_b$ therefore gives $\bar{n}=R/(1-R)$. A nonthermal distribution, unequal drive strengths, detuning or state-detection bias can invalidate that thermometer. Ground-state population cannot generally be deduced from the mean alone without specifying the distribution.

The chosen pulse gives $P_r\simeq 0.181393$ and $P_b\simeq 0.906967$. These are internal excitation probabilities, not quantum gate fidelities or cooling-cycle efficiencies. An independent matrix exponentiation of each two-state block agrees with the sine formula and preserves normalization. Summing $n=0,\dots ,79$ omits probability $r^{80}<1.21\times 10^{-56}$; because every sine squared lies between zero and one, this also bounds each omitted response. The thermal relation supplies a separate dimensional check:Here $k_B$ is the Boltzmann constant. Substitution recovers $\bar{n}=0.25$; the thermal root-mean-square expansion parameter is $\eta \sqrt{2\bar{n}+1}\simeq 0.098$. This checks the populated part of the Lamb-Dicke regime, while exponentially rare large $n$ still exceed any fixed small-parameter criterion. The pulse is a leading-order model; it does not bound corrections from the full coupling or experimental noise.