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research34 min studyLesson 483 in Physics

Magnetic and optical dipole traps

Compare state-selective magnetic confinement with conservative light shifts and calibrate Gaussian trap frequencies while accounting for scattering and geometry.

Before you begin
On this page

The question for today

What you’ll understand

  • Identify which magnetic moment sign is attracted to a field minimum.
  • Relate dipole potential sign and scattering to laser detuning.
  • Derive harmonic trap frequencies from a Gaussian optical potential.

After laser cooling, atoms are often transferred to a trap that does not rely on continuous resonant scattering. Magnetic traps use internal-state energy in an inhomogeneous magnetic field, while optical dipole traps use the light-induced energy shift. These conservative approximations can hold a cold cloud with much less recoil heating than a MOT. Their depths, curvatures, and state dependence are different quantities, so calibrating one does not automatically determine the others.

A magnetic minimum selects internal states

For a chosen adiabatic Zeeman state, the magnetic potential follows its energy as field magnitude changes. A state with positive derivative of energy with respect to field seeks lower field magnitude and can be confined near a minimum. A high-field-seeking state cannot be trapped by the same static field minimum. The effective moment may vary with field, so a constant magnetic moment is a local approximation rather than a universal label across an entire hyperfine manifold.

Near a true field zero, slow atoms can fail to follow the changing field direction adiabatically and flip into untrapped states. A nonzero bias minimum, time-averaged configuration, or optical plug can reduce this Majorana loss. Magnetic confinement is therefore not automatically spin independent. State preparation, field calibration, and the local adiabaticity condition belong in the trap specification, especially when comparing spin mixtures or tuning interactions with a large bias field.

Um(B)≃Um(B0)+μeff(B−B0),μeff=∂Eint∂BU_m(B)\simeq U_m(B_0)+\mu_{\mathrm{eff}}(B-B_0),\qquad \mu_{\mathrm{eff}}=\frac{\partial E_{\mathrm{int}}}{\partial B}
B is magnetic-field magnitude in teslas, Eint is internal-state energy in joules, and μeff is in J/T. A positive μeff is low-field seeking. The linear expression assumes one adiabatically followed state over a small enough field range.↔ Scroll wide equations horizontally.

Light shifts come with residual photon scattering

An off-resonant optical field induces a dipole whose real response gives an energy shift and whose absorptive response gives photon scattering. For a positive real polarizability, high intensity lowers potential energy and attracts atoms toward a beam focus. In a simple two-level description this corresponds to red detuning. Blue detuning can instead confine atoms near intensity minima. Real multilevel atoms require the appropriate scalar, vector, and tensor light shifts rather than one universal two-level polarizability.

Within the far-detuned rotating-wave two-level limit, depth scales as intensity divided by detuning magnitude, while scattering scales as intensity divided by detuning squared. Increasing detuning and intensity together can therefore preserve depth while reducing scattering. This scaling is not valid without limit across additional resonances or into the quasi-static optical regime. Technical intensity and pointing noise can also heat a cloud even when spontaneous scattering is weak.

Udip=−Re⁡α2ϵ0cI,Γsc≃Γ∣Δ∣U0ℏU_{\mathrm{dip}}=-\frac{\operatorname{Re}\alpha}{2\epsilon_0c}I,\qquad \Gamma_{\mathrm{sc}}\simeq\frac\Gamma{|\Delta|}\frac{U_0}{\hbar}
I is intensity in W/m², α is SI polarizability, and U₀ > 0 is the magnitude of a red trap’s central depth. The scattering relation assumes Γ ≪ |Δ| ≪ optical resonance frequency, weak saturation, and an effectively isolated two-level transition.↔ Scroll wide equations horizontally.

Curvature determines frequency, not depth alone

A red Gaussian beam has its most negative potential at the focus. Expanding its radial Gaussian and axial diffraction envelope around that point gives a harmonic well. Radial curvature depends on waist squared, while axial curvature depends on Rayleigh range squared. Two beams can have identical central depth but different confinement because their waists differ. The beam's Rayleigh range is linked to waist and wavelength, so an arbitrary axial frequency cannot be chosen independently of optical geometry.

Measured oscillation frequencies probe local curvature and can calibrate intensity or waist when the other quantity is known. Thermal clouds exploring the wings see anharmonicity and can produce amplitude-dependent frequencies. Gravity lowers the effective escape barrier and displaces the minimum, while interactions can shift collective modes relative to single-particle motion. A sudden depth change can excite cloud motion; a static harmonic estimate does not by itself predict the temperature after that quench.

ωr2=4U0mw02,ωz2=2U0mzR2,zR=πw02λ\omega_r^2=\frac{4U_0}{mw_0^2},\qquad \omega_z^2=\frac{2U_0}{mz_R^2},\qquad z_R=\frac{\pi w_0^2}{\lambda}
m is atomic mass in kg, waist w₀ and Rayleigh range zR in metres, λ the vacuum wavelength, and ωr,ωz angular frequencies in rad/s. These formulas use the local expansion of a single ideal red Gaussian beam, neglecting gravity, other fields, and interactions.↔ Scroll wide equations horizontally.

Work it through

Calculate the curvature of a long focus

An ideal Gaussian optical trap has U₀/m = 0.010 m²/s², waist 100 μm, and Rayleigh range 0.030 m. Find radial and axial angular frequencies and compare their sizes.

  1. The radial squared frequency is 4(0.010)/(10−4)2=4.0×106 s−24(0.010)/(10^{-4})^2=4.0\times10^6\,\mathrm{s^{-2}}, giving ωr = 2000 rad/s.

  2. The axial squared frequency is 2(0.010)/(0.030)2=200/9 s−22(0.010)/(0.030)^2=200/9\,\mathrm{s^{-2}}, giving ωz=200/3≃4.71 rad/s\omega_z=\sqrt{200}/3\simeq4.71\,\mathrm{rad/s}.

  3. The frequency ratio is approximately 424. The strong anisotropy follows from the long Rayleigh range relative to the waist, even though both directions share the same central trap depth.

The result

The local angular frequencies are 2000 rad/s radially and 4.71 rad/s axially. The corresponding ordinary frequencies require division by 2π.

The living notebook

Explore the model

Interactive lab

Interactive lab · function

Inspect a local quadratic curvature

Reset and set BOTH Coefficient a to 0.5 and Position x to 1; predict y = 0.5 and slope 1. Then Reset and set BOTH a to 1 and x to 1; predict y = 1 and slope 2. The exact parabola illustrates how changing curvature changes restoring force. It does not model a Gaussian beam, escape depth, gravity, scattering, or a physical trap-frequency calibration.

Inspect a local quadratic curvatureThis lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Current readings: y at chosen x: 0.608; Tangent slope · 2ax: 1.35; Signed area · ax³/3: 0.182. All parameters are adjustable with the labeled controls below.−2.4−1.201.22.4−9−4.504.59xy(x, ax²)
y = ax²TangentSigned integral
y at chosen x
0.608
Tangent slope · 2ax
1.35
Signed area · ax³/3
0.182
t = 0 s

This lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forLocal curvature and global depth are distinct aspects of a trapping potential.

Pause & reason

Check your understanding

In the stated far-detuned two-level scaling, double |Δ| and intensity together. What happens to depth and scattering rate?

Your turn

Try it yourself

A magnetic state has μeff = +2 × 10⁻²³ J/T over a local field range. Moving outward from the minimum raises field magnitude by 5 × 10⁻⁴ T. Find its potential-energy change. What would the sign imply for a state with the opposite moment?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Use the energy derivative with respect to field magnitude, not a force formula without a gradient.

Compare with the solution

The energy rise is ΔU=μeffΔB=(2×10−23)(5×10−4)=10−26 J\Delta U=\mu_{\mathrm{eff}}\Delta B=(2\times10^{-23})(5\times10^{-4})=10^{-26}\,\mathrm J. The positive-moment state pays energy to move outward and is attracted toward the local field minimum. Reversing μeff gives an energy decrease of the same magnitude, so that state would seek larger field and would not be confined by this minimum alone. Adiabatic following remains an assumption.

Keep the idea

Take it with you

  • A magnetic trap’s confinement depends on the prepared state and adiabatic following.
  • Optical depth and photon-scattering heating obey different detuning scalings within a specified regime.
  • Trap frequencies measure local curvature and require geometry and anharmonicity checks.

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