Optical Tweezers

An optical tweezer is a tightly focused dipole trap. Here the particles are individual atoms or molecules: fluorescence identifies their locations, and moving traps can assemble a chosen array from randomly loaded sites. The central tradeoff is between tight localization and light-induced heating or internal-state shifts. Occupancy, survival, motional excitation, and internal coherence describe different aspects of the prepared sample.

The defining physical scale is the diffraction-limited focus. A small waist produces large intensity curvature, so a single trapped particle can have resolved motional levels and can scatter enough fluorescence to be detected site by site. The same high intensity also magnifies differential light shifts, photon scattering, polarization gradients, and sensitivity to optical aberrations.

Required background. Optical dipole traps relates light shifts to confinement; sub-Doppler cooling explains recoil and motional sidebands; POVMs distinguishes an observed outcome from the physical state inferred from it.

A Dipole Trap in a Tight Focus

The numerical aperture of an objective is

$$\mathrm{NA} = n\sin\theta,$$

where $n$ is the refractive index on the object side and $\theta$ is the maximum accepted ray angle. In vacuum ($n=1$), for a paraxial Gaussian beam that underfills an objective,

$$w_0 \simeq \frac{ \lambda_L f }{ \pi w_{\mathrm{in}} } \simeq \frac{ \lambda_L }{ \pi\mathrm{NA}_{\mathrm{eff}} },$$

where $w_0$ is the $1/e^2$ intensity radius, $f$ is the focal length, and $w_{\mathrm{in}}$ is the incident Gaussian radius at the objective.

This is an estimate, not a universal diffraction formula. The actual point spread function depends on pupil filling, aberrations, vector polarization, refractive-index interfaces, and the waist convention. At high NA, a scalar paraxial Gaussian is insufficient: the focused field can contain a longitudinal component and spatially varying ellipticity.

For a vector field with complex peak amplitude $\mathbf E_0$, the leading internal-state light shift is

$$U_a(\mathbf r)=-\frac14\operatorname{Re} \left[\mathbf E_0^*\cdot\boldsymbol\alpha_a(\omega_L)\cdot\mathbf E_0\right].$$

One can write $U_a=-\operatorname{Re}\alpha_a^{\rm eff}I_E/(2\epsilon_0c)$ with $I_E=\epsilon_0c|\mathbf E_0|^2/2$. In a strongly nonparaxial focus, this electric-field intensity is not generally the local axial Poynting flux measured by an intensity camera. Vector diffraction and polarization therefore matter in converting a focus image into an atomic potential.

Mechanical scales

For a red-detuned paraxial Gaussian with depth $U_0$, waist $w_0$, and Rayleigh range $z_R=\pi w_0^2/\lambda_L$, the leading estimates are

$$\omega_r \simeq \sqrt{ \frac{ 4U_0 }{ mw_0^2 } }, \qquad \omega_z \simeq \sqrt{ \frac{ 2U_0 }{ mz_R^2 } }.$$

In a high-NA tweezer, the reliable procedure is to calculate or measure the full potential, locate its equilibrium, and diagonalize its Hessian. The paraxial formulas are useful for scale estimates and consistency checks.

A small waist increases radial curvature as $w_0^{-2}$ at fixed depth. It also reduces $z_R$ and strengthens axial confinement. This produces motional frequencies that can be much larger than in a bulk dipole trap, but it also makes pointing noise and focus drift more consequential.

Quantized motion

Near a stable minimum,

$$H_{\mathrm{mot}} = \sum_{i=x,y,z} \hbar\omega_i \left( a_i^\dagger a_i + \frac12 \right).$$

The ground-state length along principal axis $i$ is

$$x_{0,i} = \sqrt{ \frac{ \hbar }{ 2m\omega_i } }.$$

For a thermal harmonic mode,

$$\bar n_i = \frac{ 1 }{ \exp\left(\hbar\omega_i/k_{\mathrm B}T\right)-1 },$$

and

$$\langle x_i^2\rangle = x_{0,i}^2 \left( 2\bar n_i+1 \right).$$

For separable thermal modes, the three-dimensional motional-ground-state probability is

$$P_{000} = \prod_i \frac{ 1 }{ 1+\bar n_i }.$$

A small temperature in kelvin does not guarantee small $\bar n_i$; the comparison is with $\hbar\omega_i/k_{\mathrm B}$ along each axis.

Lamb–Dicke parameter

For a transition with wave-vector change $\Delta\mathbf k$, define

$$\eta_i = \left| \Delta\mathbf k \mathbin{\cdot} \hat{\mathbf e}_i \right| x_{0,i}.$$

The Lamb–Dicke regime requires the particle’s spatial extent to be small enough that recoil does not strongly resolve its position. For a thermal mode, a useful condition is

$$\eta_i^2 \left( 2\bar n_i+1 \right) \ll 1.$$

The value of $\eta_i$ depends on beam geometry. A copropagating Raman pair can have a much smaller effective wave-vector difference than a counterpropagating pair. Optical repumping and imaging involve separate single-photon recoil parameters; a small Raman $\eta_i$ does not make those processes recoil free.

High-NA Corrections and State Dependence

Tight focusing introduces effects that may be negligible in a broad dipole trap:

• longitudinal electric-field components;
• polarization gradients across the wave packet;
• vector and tensor light shifts;
• nonseparable radial and axial motion;
• aberration-induced secondary minima;
• surface or window-induced wave-front distortion;
• collection-path and trapping-path misregistration.

The local transition frequency between states $|a\rangle$ and $|b\rangle$ is shifted by

$$\delta\omega_{ba}(\mathbf r) = \frac{ U_b(\mathbf r)-U_a(\mathbf r) }{ \hbar }.$$

Even when the intensity maximum is stable, motion through polarization gradients can create time-dependent vector or tensor shifts. Magic wavelengths, magic polarizations, or magic intensities must be specified for the chosen states and geometry; they are not properties of a wavelength alone.

Loading One Particle

Tweezers are commonly loaded from a MOT, optical molasses, a reservoir dipole trap, an optical lattice, or a preassembled atomic pair. Loading requires both phase-space overlap and a mechanism that prevents uncontrolled multi-particle occupancy.

Collisional blockade

In the simplest neutral-atom loading model, light-assisted collisions eject both particles whenever a second atom enters an occupied tweezer. The site then has only two long-lived states:

$$n \in \{0,1\}.$$

Let $R$ be the arrival rate, assumed independent of occupancy, and $\gamma$ the one-body loss rate. Assume collision removal is fast compared with the interval between arrivals. If a second loading event takes

$$1 \longrightarrow 0,$$

then the occupied probability obeys

$$\frac{ dP_1 }{ dt } = R \left( 1-P_1 \right) - \left( R+\gamma \right) P_1.$$

Equivalently,

$$\frac{ dP_1 }{ dt } = R - \left( 2R+\gamma \right) P_1.$$

The steady filling is

$$P_{1,\mathrm{ss}} = \frac{ R }{ 2R+\gamma } \longrightarrow \frac12$$

when $R\gg\gamma$. The familiar $50\%$ limit is therefore not a fundamental limit of tweezers. It follows from the assumed two-in, zero-out collision outcome.

Enhanced single-particle loading

Light-assisted collisions can be engineered so that an entering pair more often leaves one particle rather than zero. Let $q$ be the conditional probability that one particle remains after a two-particle event. The occupied-to-empty transition rate from loading is then $R(1-q)$, and

$$P_{1,\mathrm{ss}} = \frac{ R }{ R + R(1-q) + \gamma }.$$

The limits are:

$$\begin{aligned} q=0 &: & P_{1,\mathrm{ss}} &\to \frac12, \\ q=1 &: & P_{1,\mathrm{ss}} &\to 1 \qquad (R\gg\gamma). \end{aligned}$$

Blue-detuned light-assisted collisions, controlled energy release, narrow transitions, and dark-state loading can raise site filling, but every scheme must report loss, heating, state preparation, and correlations between neighboring sites.

Loading is species and state dependent

The cooling transition can be strongly shifted at the tweezer center. This may prevent capture, detune imaging light, or change molecular collision potentials. Remedies include:

• magic or near-magic trap conditions;
• alternating trap and cooling light;
• separate reservoir and science traps;
• intensity ramps during loading;
• narrow-line or gray-molasses cooling;
• state-selective association of pre-cooled atom pairs.

The result should be reported as a per-site probability with uncertainty, not inferred only from the mean number in an array.

Detecting Occupancy

Single-particle fluorescence imaging maps a physical occupancy to camera counts. Let an empty site produce a mean count $\mu_0$, and an occupied site produce

$$\mu_1 = \mu_0 + \eta_{\mathrm{det}} \Gamma_{\mathrm{fl}} \tau,$$

This mean assumes a particle remains bright and present throughout the exposure. Here $\eta_{\mathrm{det}}$ includes collection, transmission, detector quantum efficiency, and analysis aperture; $\Gamma_{\mathrm{fl}}$ is the fluorescence scattering rate; and $\tau$ is the exposure.

If both count distributions were Poisson and a site were called occupied for $n\ge n_\star$, then

$$P_{\mathrm{FP}} = 1 - \sum_{n=0}^{n_\star-1} e^{-\mu_0} \frac{ \mu_0^n }{ n! },$$

and

$$P_{\mathrm{FN}} = \sum_{n=0}^{n_\star-1} e^{-\mu_1} \frac{ \mu_1^n }{ n! }.$$

For equal prior probabilities, the threshold-classification fidelity is

$$F_{\mathrm{class}} = 1 - \frac{ P_{\mathrm{FP}}+P_{\mathrm{FN}} }{ 2 }.$$

Real occupied-site histograms are rarely exact Poisson distributions. Particle loss during the exposure creates a low-count tail; camera noise, stray-light drift, optical pumping, and neighboring point-spread functions create additional structure. Calibration should use empirical conditional distributions.

Fidelity and survival are different

Define:

$$F_{\mathrm{occ}} = P \left( \widehat n=n \right),$$

for inference of the initial occupancy $n=n_{\rm before}$, and

$$S_{\mathrm{img}} = P \left( n_{\mathrm{after}}=1 \mid n_{\mathrm{before}}=1 \right)$$

for imaging survival. High detection fidelity can coexist with poor survival if the particle emits enough photons to be identified and is then lost. Conversely, a nondestructive image can still have poor fidelity if it collects too few photons.

Imaging scatters many photons. The recoil energy added without cooling is of order

$$\Delta E \sim 2N_\gamma E_R.$$

Repeated or long-exposure imaging therefore usually operates with simultaneous laser cooling, sufficiently deep confinement, or both. State detection must also distinguish a dark internal state from physical loss.

(a) A measured occupancy map permits selected atoms to fill an initially empty target; the drawing assumes perfect classification and transport. (b) For independent filling $p=0.6$, the binomial probability of supplying at least ten atoms rises with reservoir size $M$. It is $0.8725$ at $M=20$, whereas ten fixed sites all fill with probability $0.6^{10}\simeq0.00605$. These are supply probabilities before imaging and transport errors.

Generating Tweezer Arrays

Several optical architectures can create multiple focuses.

Method	Main capability	Main limitations
Acousto-optic deflector	fast steering and multi-tone dynamic arrays	frequency-dependent efficiency, RF intermodulation, beat notes
Spatial light modulator	flexible two-dimensional phase holograms	finite refresh rate, phase retrieval, zero order, aberrations
Digital micromirror device	programmable binary amplitude patterns	optical efficiency, diffraction orders, pixelation
Microlens array	passive, parallel, stable focus array	fixed geometry, fabrication nonuniformity, limited rearrangement
Scanned single focus	time-averaged programmable potential	duty cycle, scan-induced micromotion, parametric heating

Hybrid systems often use one element for a static reservoir array and a second mobile tweezer for transport.

Coherent addition and beat notes

If two same-frequency fields overlap, their electric fields add:

$$I \propto \left| \mathbf E_1+\mathbf E_2 \right|^2.$$

A fixed but unknown relative phase creates static interference fringes; phase fluctuations move them and can heat the atom. If the tweezers have frequency difference $\delta\omega$, the cross term beats at $\delta\omega$. A beat much faster than motional response may average out, while one near $\omega_i$ or $2\omega_i$ can drive sloshing or parametric heating.

Orthogonal polarization, frequency offsets, spatial separation, and careful RF-tone design can suppress crosstalk, but each choice can introduce vector light shifts or polarization-dependent trapping. Orthogonal input polarizations need not remain locally orthogonal after high-NA focusing. The relevant interference term uses the actual vector fields at the atom.

Uniformity

For fixed geometry,

$$\omega_{i,j} \propto \sqrt{U_j}.$$

Small site-to-site depth variations therefore produce

$$\frac{ \delta\omega_i }{ \omega_i } \simeq \frac12 \frac{ \delta U }{ U }.$$

The same $\delta U$ can create differential transition shifts, unequal Raman sidebands, position-dependent loading, and nonuniform gate detunings. Uniform camera brightness is not a sufficient proxy for uniform depth: fluorescence depends on cooling detuning and collection as well as occupancy.

Useful site-resolved calibrations include:

• AC Stark spectroscopy;
• parametric or sloshing frequencies;
• survival under a calibrated depth ramp;
• sideband spectra;
• beam-power measurements corrected for optical transfer;
• loading and imaging response maps.

Why Rearrangement Is Needed

Suppose each of $N$ target sites loads independently with probability $p$. The probability of a defect-free array without rearrangement is

$$P_{\mathrm{full}} = p^N.$$

Even good local filling becomes poor global filling as $N$ grows. For $p=0.55$ and $N=50$,

$$P_{\mathrm{full}} \simeq 1.0\times10^{-13}.$$

Rearrangement changes the question. Load $M>N$ reservoir sites, image them, and ask whether at least $N$ particles are available. Under the independent Bernoulli model,

$$P_{\mathrm{supply}} = \sum_{k=N}^{M} \binom{M}{k} p^k \left( 1-p \right)^{M-k}.$$

This can be close to unity even when $p^N$ is tiny. Rearrangement does not make stochastic loading deterministic; it converts excess spatial resources plus measurement and control into a high-probability target.

Assignment and path planning

After imaging, the controller must:

1. classify the occupancy of every reservoir and target site;
2. select which particles to keep;
3. assign loaded sites to target sites;
4. choose collision-free trajectories;
5. transport particles without spilling or excessive excitation;
6. re-image or otherwise verify the result.

A minimum-cost matching can reduce total distance, but distance is not the only cost. One may penalize crossings, weak traps, transfers between traps, long holds, or motion through occupied sites. Path planning and assignment should therefore be separated from the optical actuation model.

If every move survives independently with probability $s$ and $L$ moves are needed, the move-survival factor is

$$P_{\mathrm{move}} \simeq s^L.$$

This simple product is useful for budgeting but can fail when errors share a common cause such as laser-power drift or a bad spatial region.

Quantum Transport in a Moving Tweezer

Consider one harmonic axis with a moving center $x_c(t)$:

$$H(t) = \frac{ p^2 }{ 2m } + \frac12m\omega^2 \left[ x-x_c(t) \right]^2.$$

Assume a constant positive $\omega$, that the particle starts in the motional ground state, and that the trap begins and ends at rest. The coherent excitation relative to the final stationary trap, using $a=\sqrt{m\omega/(2\hbar)}(x-x_c)+ip/\sqrt{2m\hbar\omega}$, is

$$\alpha(T) = - \sqrt{ \frac{ m\omega }{ 2\hbar } } e^{-i\omega T} \int_0^T \dot x_c(t) e^{i\omega t} dt.$$

The added mean occupation is

$$\Delta\bar n = |\alpha(T)|^2.$$

The result follows from $\dot\alpha=-i\omega\alpha- \sqrt{m\omega/(2\hbar)}\dot x_c$ with $\alpha(0)=0$. The final ground-state overlap is $e^{-|\alpha|^2}$; for an initially thermal mode, the same displacement adds $|\alpha|^2$ to the mean occupation.

Transport is excitation-free in this model when the velocity waveform has zero Fourier component at $\omega$. Slow smooth motion accomplishes this adiabatically, but carefully shaped faster trajectories can also cancel the resonant component. Merely specifying the average speed is insufficient.

Acceleration tilts the trap in its comoving frame. A finite-depth tweezer can spill the particle even when the harmonic excitation formula predicts a small $\Delta\bar n$. The maximum acceleration must therefore be compared with the full restoring-force curve, and transfers between stationary and mobile traps must preserve both depth and phase-space overlap.

For a stored internal superposition, motion also accumulates a differential phase:

$$\phi_{ba} = \int_0^T \delta\omega_{ba} \left[ \mathbf r(t),t \right] dt.$$

With this definition, the coherence evolves as $\rho_{ba}(T)=e^{-i\phi_{ba}}\rho_{ba}(0)$ when only the differential energy acts. Coherent transport requires reproducible phase or an echo/calibration protocol, not only particle survival.

Cooling and State Preparation

Loading and rearrangement generally leave motional excitation. Subsequent cooling can improve localization, suppress Doppler dephasing, reduce interaction fluctuations, and prepare resolved motional states.

Sideband cooling

In a resolved-sideband protocol, a coherent transition removes one motional quantum and optical pumping resets the internal state. A minimal cycle is

$$|g,n\rangle \longrightarrow |e,n-1\rangle \longrightarrow |g,n-1\rangle.$$

The cooling transition linewidth must be narrow enough to distinguish motional sidebands in the relevant effective model. The red-to-blue first-sideband-area ratio for a harmonic mode is

$$\frac{ A_{\mathrm{red}} }{ A_{\mathrm{blue}} } = \frac{ \bar n }{ \bar n+1 },$$

provided the probe is weak, sidebands are resolved, and the Lamb–Dicke first-sideband matrix elements apply. The ratio determines the mean for any diagonal motional distribution; converting it to temperature or ground-state probability requires further distributional assumptions. The reset photons must satisfy their own recoil-heating constraints.

Three-dimensional cooling requires coupling to all principal axes. High-NA polarization structure and unfavorable Raman-beam geometry can leave a weakly cooled mode.

Internal-state preparation

Optical pumping, microwave or Raman pulses, and coherent clock transitions can prepare hyperfine, Zeeman, nuclear-spin, electronic, rotational, or vibrational states. The preparation error budget should include:

• residual population outside the computational manifold;
• position-dependent Rabi frequency and detuning;
• differential light shifts;
• motional sidebands and Doppler phase;
• Raman scattering from the tweezer;
• magnetic-field and polarization drift.

State-preparation fidelity and occupancy must be measured separately. An empty site cannot be silently counted as successful preparation of a dark state.

Rydberg excitation in a tweezer

Rydberg excitation can switch on interactions between particles at chosen sites; Rydberg blockade explains the resulting collective dynamics. The tweezer itself matters because the ground and Rydberg states can experience different, even oppositely signed, light shifts. Some protocols switch the trap off during excitation, which trades differential shifts for free motion and recapture loss.

Position fluctuations change the interaction energy, and velocity causes Doppler and optical-phase fluctuations. Thus particle survival after a pulse neither establishes internal coherence nor identifies a Rydberg population without a calibrated loss model. See neutral-atom qubits for the complete information-processing protocol.

Molecules, Alkaline-Earth Atoms, and Clocks

Tweezers are not restricted to alkali atoms.

Molecules

Single molecules can be loaded by direct laser cooling or assembled from individually trapped atoms. Their additional rotational, vibrational, and hyperfine structure introduces anisotropic polarizability, dense Raman channels, and collision chemistry. A molecule can remain physically trapped while leaving the intended internal manifold, so state-resolved detection is essential.

Merging two tweezers permits controlled two-particle collisions or association. The relative motional ground state, species-dependent trap centers, differential sag, and state-dependent light shifts determine the short-range wave-function overlap.

Alkaline-earth-like atoms

Two-valence-electron atoms offer narrow intercombination and clock transitions alongside strong imaging transitions. This supports narrow-line cooling, nuclear-spin qubits in isotopes with nonzero nuclear spin, and magic trapping, but also requires careful control of metastable-state loss and state-dependent polarizability.

Metrology

Tweezer clocks combine single-particle readout with optical-clock spectroscopy. Their uncertainty budget includes site-resolved light shifts, collisional effects, motional sampling, dead time, atom-number fluctuations, and selection bias from postselection or rearrangement. Precision Spectroscopy explains how these effects enter line fitting and uncertainty estimates.

Motional scales of a rubidium tweezer

Consider an $^{87}\mathrm{Rb}$ atom in a red-detuned tweezer with assumed depth

$$\frac{ U_0 }{ k_{\mathrm B} } = 1.0\,\mathrm{mK},$$

measured $1/e^2$ waist

$$w_0 = 0.90\,\mu\mathrm m,$$

and wavelength

$$\lambda_L = 850\,\mathrm{nm}.$$

Using the paraxial Gaussian formulas as estimates, the Rayleigh range is

$$z_R = \frac{ \pi w_0^2 }{ \lambda_L } \simeq 2.99\,\mu\mathrm m.$$

For

$$m = 1.443\times10^{-25}\,\mathrm{kg},$$

the trap frequencies are

$$\frac{ \omega_r }{ 2\pi } \simeq 109\,\mathrm{kHz}, \qquad \frac{ \omega_z }{ 2\pi } \simeq 23.3\,\mathrm{kHz}.$$

The ground-state lengths are

$$x_{0,r} \simeq 23\,\mathrm{nm}, \qquad x_{0,z} \simeq 50\,\mathrm{nm}.$$

For a single 780 nm photon projected along a radial principal axis,

$$\eta_r \simeq \frac{ 2\pi }{ 780\,\mathrm{nm} } x_{0,r} \simeq 0.19.$$

A counterpropagating Raman pair can have nearly twice this wave-vector change, while a copropagating pair can have much less. The radial level spacing is

$$\frac{ \hbar\omega_r }{ k_{\mathrm B} } \simeq 5.25\,\mu\mathrm K.$$

Thus a temperature of $20\,\mu\mathrm K$ is cold in everyday units but not near the radial motional ground state without further cooling. For three uncorrelated thermal modes at that temperature, $\bar n_r\simeq3.3$ and $\bar n_z\simeq17.4$, giving $P_{000}\simeq0.003$. Cooling only the tight radial directions leaves the weaker axial mode excited. The example also illustrates why a measured vector point-spread function and Hessian should replace the paraxial estimate in precision work.

Exercises

1. A bright classification with a low occupancy prior

Use the classifier of Exercise 5, but suppose only $p=0.01$ of sites are initially occupied. Given a positive classification, what is the probability that the site was occupied? Compare it with the equal-prior fidelity.

Solution

Bayes’ rule gives

$$P(n=1\mid\widehat n=1)= \frac{p(1-P_{\rm FN})} {p(1-P_{\rm FN})+(1-p)P_{\rm FP}}\simeq0.733.$$

Although the equal-prior classification fidelity is $0.9944$, most tested sites are empty, so their false positives appreciably contaminate selected positives. For these Poisson models, equal error costs, and prior $p$, the optimal decision is occupied when

$$n\ln(\mu_1/\mu_0)\ge\mu_1-\mu_0+\ln[(1-p)/p].$$

It shifts the integer threshold from $5$ at equal priors to $7$ at $p=0.01$. This inference concerns initial occupancy, not survival through the image.

2. The collisional-blockade limit

Solve

$$\dot P_1 = R - \left( 2R+\gamma \right) P_1$$

for an initially empty site. Find the steady filling and loading time constant.

Solution

The linear equation has solution

$$P_1(t) = \frac{ R }{ 2R+\gamma } \left[ 1 - e^{-(2R+\gamma)t} \right].$$

Therefore

$$P_{1,\mathrm{ss}} = \frac{ R }{ 2R+\gamma }, \qquad \tau_{\mathrm{load}} = \frac{ 1 }{ 2R+\gamma }.$$

When $R\gg\gamma$, the steady filling approaches $1/2$.

3. One-particle-retaining collisions

Let a two-particle event leave one particle with probability $q=0.80$. Assume $R=20\,\mathrm{s^{-1}}$ and $\gamma=0.50\,\mathrm{s^{-1}}$. Find the steady single-particle filling.

Solution

The occupied-to-empty rate is

$$R(1-q)+\gamma = (20)(0.20)+0.50 = 4.50\,\mathrm{s^{-1}}.$$

Hence

$$P_{1,\mathrm{ss}} = \frac{ R }{ R+R(1-q)+\gamma } = \frac{ 20 }{ 24.5 } \simeq 0.816.$$

The improvement over $1/2$ comes from changing the collision branch, not from violating collisional blockade.

4. Supply probability for rearrangement

Twenty reservoir sites load independently with $p=0.60$. What is the probability that at least ten particles are available for a ten-site target? Compare this with the probability that ten fixed sites load without defects.

Solution

For rearrangement,

$$P_{\mathrm{supply}} = \sum_{k=10}^{20} \binom{20}{k} (0.60)^k (0.40)^{20-k} \simeq 0.872.$$

Without rearrangement, ten specified sites are all filled with probability

$$P_{\mathrm{fixed}} = (0.60)^{10} \simeq 6.05\times10^{-3}.$$

The reservoir converts extra sites into a much larger supply probability, but move and imaging errors still reduce the final success.

5. A photon-count threshold

Assume Poisson counts with $\mu_0=1$ for an empty site and $\mu_1=12$ for an occupied site. Classify the site as occupied for $n\ge5$. Calculate $P_{\mathrm{FP}}$, $P_{\mathrm{FN}}$, and the equal-prior classification fidelity.

Solution

The false-positive probability is

$$P_{\mathrm{FP}} = 1 - \sum_{n=0}^{4} e^{-1} \frac{ 1^n }{ n! } \simeq 3.66\times10^{-3}.$$

The false-negative probability is

$$P_{\mathrm{FN}} = \sum_{n=0}^{4} e^{-12} \frac{ 12^n }{ n! } \simeq 7.60\times10^{-3}.$$

Thus

$$F_{\mathrm{class}} = 1 - \frac{ P_{\mathrm{FP}}+P_{\mathrm{FN}} }{ 2 } \simeq 0.9944.$$

This number does not include particle loss during imaging and therefore is not an imaging-survival probability.

6. Excitation from a constant-velocity move

A harmonic tweezer moves by distance $d$ at constant velocity for time $T$, with instantaneous starts and stops:

$$\dot x_c(t) = \frac{d}{T}, \qquad 0<t<T.$$

Use the transport formula to find the added mean occupation.

Solution

The resonant Fourier integral is

$$\int_0^T \dot x_c(t) e^{i\omega t} dt = \frac{ d }{ T } \frac{ e^{i\omega T}-1 }{ i\omega }.$$

Its magnitude is

$$d \left| \frac{ \sin(\omega T/2) }{ \omega T/2 } \right|.$$

Therefore

$$\Delta\bar n = \frac{ m\omega d^2 }{ 2\hbar } \left[ \frac{ \sin(\omega T/2) }{ \omega T/2 } \right]^2.$$

The excitation vanishes in the ideal harmonic model when $\omega T=2\pi\ell$ for nonzero integer $\ell$. The instantaneous changes in velocity are experimentally unrealistic and can excite anharmonic modes, so smooth endpoint conditions are still desirable.

7. Depth uniformity from frequency data

Two nominally identical tweezer sites have radial frequencies that differ by $1.5\%$. Assuming their geometry is identical, estimate their fractional depth difference.

Solution

Because $\omega_r\propto\sqrt U$,

$$\frac{ \delta\omega_r }{ \omega_r } \simeq \frac12 \frac{ \delta U }{ U }.$$

Hence

$$\frac{ \delta U }{ U } \simeq 2 \left( 1.5\% \right) = 3.0\%.$$

This inference assumes equal waists and principal-axis geometry. A waist error can change frequency without the same proportional change in depth.

8. Common loading fluctuations

On half the experimental repetitions, all twenty reservoir sites load independently with $p=0.4$; on the other half they load independently with $p=0.8$. Find the probability of having at least ten atoms. Does substituting the mean filling $p=0.6$ in a single binomial distribution work?

Solution

Condition on the shared loading environment first. If $B(p)=\sum_{k=10}^{20}\binom{20}{k}p^k(1-p)^{20-k}$, then

$$P_{\rm supply}=\tfrac12[B(0.4)+B(0.8)] \simeq\tfrac12(0.24466+0.99944)=0.62205.$$

Using $B(0.6)$ instead gives $0.87248$, a substantial overestimate. The unconditional sites are correlated even though they are independent within each environment. Likewise ten fixed target sites are all occupied with probability $(0.4^{10}+0.8^{10})/2\simeq0.05374$, not $0.6^{10}$. Mean filling alone does not determine either global success probability.

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