Trapped-ion quantum control

Trapped-ion quantum control prepares an ion's internal quantum state, couples it to quantized motion with laser pulses, and measures the result through state-dependent fluorescence. Its appeal is access to both a controllable two-level system and a quantum harmonic oscillator in one apparatus.

Confinement, preparation, control and readout solve different problems. The Paul trap's RF electric field confines the ion through its RF-averaged trapping potential; it still produces Paul trap micromotion. Doppler cooling and resolved sideband cooling reduce motional excitation. Optical pumping selects an internal quantum state. A coherent pulse then changes amplitudes, while electron shelving converts an internal population into a detectable stream of photons. A bright/dark count by itself does not reveal a quantum superposition's relative phase; an additional controlled rotation before detection can make that phase affect the measured population.

An internal qubit uses two distinguishable electronic or hyperfine structure quantum states, denoted $|g⟩,|e⟩$. The ion's centre-of-mass motion has a different Hilbert space. In one approximately harmonic direction, a number state $|n⟩$ has energy $E_n=\hslash \nu (n+1/2)$, where $\nu$ denotes a motional angular frequency, not ordinary frequency. The tensor product of quantum systems gives $|g,n⟩$. Cooling towards $n=0$ prepares motion; it does not automatically prepare the internal qubit. Nor does the ground state mean a stationary point particle: zero-point energy and position spread remain.

For several ions, Coulomb forces couple their displacements into collective normal modes. Each selected normal mode has its own quantum harmonic oscillator number states, while each ion retains internal qubit states. A pulse addressing one ion can change a shared mode. This supplies a route to correlations between distant internal quantum states; it also makes residual motional excitation and unwanted excitation of other normal modes relevant. A single-ion pulse calculation alone establishes no experimental multi-ion quantum gate performance.

The Lamb-Dicke regime means the wave-induced phase change across the occupied motional wavepacket is small. For one normal mode, let $x_0=\sqrt{\hslash /(2m\nu )}$ be the ground state's position standard deviation, and write $x=x_0(a+a^{†})$ using creation and annihilation operators. Here $m$ is mass and $\hslash$ the Planck constant divided by $2\pi$. The effective wave vector component $k_{\mathrm{eff}}$ is the single beam's component for a direct transition, or the difference of two beam components for a Raman transition. With dimensionless $\eta =|k_{\mathrm{eff}}|x_0$, the useful condition is $\eta \sqrt{2n+1}\ll 1$ over the populated number states, rather than merely $\eta \ll 1$.

Absorb the sign of $k_{\mathrm{eff}}$ into the pulse phase. Expanding the coupling factor givesThe constant term preserves $n$; the annihilation operator gives $a|n⟩=\sqrt{n}|n-1⟩$; the creation operator gives $a^{†}|n⟩=\sqrt{n+1}|n+1⟩$. The different square roots explain why removing a motional quantum becomes impossible at $n=0$, while adding one remains possible. Smallness of $\eta$ controls this spatial expansion; resolving the resulting transitions is a separate frequency requirement.

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.

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.

Resolved sideband cooling removes motional quanta by selectively driving a red motional sideband and then resetting the internal quantum state through spontaneous emission or optical pumping. Spectral resolution requires the effective cooling-transition width smaller than the motional angular frequency in the same units. A narrow Raman transition can meet that requirement even when the optical transition used for Doppler cooling is broad.

The reset is essential: the coherent red motional sideband merely exchanges internal and motional excitation reversibly. To see the distinction, an ideal pulse sends $|g,1⟩$ to $-i|e,0⟩$; the same interaction can send it back. An ideal dissipative reset instead takes $|e,0⟩$ to $|g,0⟩$, with information and energy carried into the environment. Repeating this cycle accumulates population near the motional ground state. Recoil during the reset can add motion, and off-resonant excitation can disturb $|g,0⟩$; neither is included in the ideal pulse below.

In 1995, Christopher Monroe and collaborators cooled one ${}^9{\mathrm{Be}}^{+}$ ion in a ring/endcap Paul trap. They used $313\mathrm{nm}$ Doppler cooling and optical pumping into $|F,m_F⟩=|2,2⟩$. Red-sideband Raman transitions transferred population to $|1,1⟩$, followed by dissipative recycling. Repeating across three directions yielded a reported three-dimensional ground state population of about $92\%$.^[ref]

A variable-frequency Raman transition probe, an exchange pulse and a cycling detection beam converted state populations into fluorescence collected through $f/2$ optics onto a position-sensitive photomultiplier. Red/blue asymmetry inferred motional occupation; varying probe duration tested the thermal assumption. Off-resonant processes and independently observed heating limited cooling.^[ref]

Electron shelving distinguishes an internal quantum state outside a cycling optical transition from one that scatters many photons. In the mercury-ion example in David J. Wineland's lecture, $S\leftrightarrow P$ excitation produces $194\mathrm{nm}$ fluorescence; transfer to a metastable atomic level $D$ suppresses it. A photomultiplier records bright/dark intervals. Coherent $282\mathrm{nm}$ interrogation precedes detection to avoid perturbation by the detection beam.^[ref]

“Dark” means weak scattering under a particular detection setting. The motional ground state is dark to an ideal red motional sideband for a different reason: there is no lower number state. It can still fluoresce strongly on a cycling detection transition. Shelving's population readout also need not preserve a quantum superposition's quantum coherence; scattered photons carry information about the internal state, and their recoil can disturb motion.

For an original illustrative readout model, take detected photon count $K$ in a fixed exposure to have a Poisson distribution, with stipulated means ${\lambda }_B=12$ for a bright state and ${\lambda }_D=0.2$ for a dark state. Classify $K\ge 3$ as bright. Direct summation givesWith equally likely preparations, the average misclassification probability is $0.0008354$. These numbers describe only the chosen count model. Actual fluorescence includes short-time photon correlations, detector effects and possible state changes; it need not follow a fixed-rate Poisson process. Longer exposure can improve separation of count distributions while increasing unwanted optical pumping, metastable decay or recoil.

If the prepared bright population is $p$, let $\alpha$ and $\beta$ denote the two conditional errors above. The observed bright-label probability is $q=(1-\alpha )p+\beta (1-p)$, hence $p=(q-\beta )/(1-\alpha -\beta )$ when the denominator is nonzero. Known-state controls must estimate $\alpha ,\beta$ in the actual apparatus; the formula cannot correct unknown drift. For $M$ independent preparations, the raw bright fraction has variance $q(1-q)/M$. Counting uncertainty, preparation error, pulse error and state-detection bias therefore enter different parts of the inference. Repeating a biased measurement reduces its sampling uncertainty without removing its bias.