Laser Cooling

Trap Hamiltonian and Motional States

The atom in the trap is described by a harmonic oscillator:

Physical Meaning of η

When $\eta \ll 1$, the atom's position uncertainty during a photon interaction is much smaller than the photon wavelength. A photon absorption is unlikely to change the motional quantum number ($\Delta n \neq 0$ transitions are suppressed by $\eta$). This suppression is what makes sideband cooling efficient: the carrier transition ($\Delta n = 0$) dominates, and the red sideband ($\Delta n = -1$, cooling) is preferred over the blue sideband ($\Delta n = +1$, heating) by the factor $\eta^2$.

Typical values: $\eta \approx 0.05$–$0.15$ for optical tweezers with $\nu \approx 50$–$200\,\text{kHz}$ and $\lambda \approx 780\,\text{nm}$.

Cooling Cycle

Step 1: Drive the red sideband, a photon at $\omega_0 - \nu$ takes $|g, n\rangle \to |e, n-1\rangle$. Step 2: Spontaneous emission returns the atom to the ground electronic state. In the Lamb-Dicke regime, the emitted photon has equal probability of going to $|g, n-1\rangle$ (the desired path) or $|g, n\rangle$ (recoil back up). The net effect per cycle: one motional quantum removed. The state $|g, 0\rangle$ is dark (no red sideband to drive), so the population accumulates there.

Physical Mechanism

The cooling rate is set by the photon scattering rate on the red sideband: $R_- = \eta^2 \Gamma$. The heating rate from off-resonant blue sideband excitation is $R_+ \propto \eta^2 \Gamma (\Gamma/4\nu)^2$. In steady state, these balance to give $\langle n \rangle_{ss} = ({\Gamma}/{4\nu})^2$.

The formula is striking: the final temperature depends only on the ratio $\Gamma/\nu$, not on the laser intensity or the Lamb-Dicke parameter. Making the trap tighter (larger $\nu$) directly lowers the temperature floor.

The Unresolved Sideband Limit

When the sidebands overlap, the atom cannot distinguish cooling transitions from carrier transitions. The steady-state occupation from Eq. 39 of Phatak et al. is:

The Doppler Temperature

The Doppler limit arises from the competition between two processes: the laser's viscous force damps atomic momentum (cooling), while random recoil from spontaneous emission heats the atom. At the optimal detuning $\Delta = -\Gamma/2$, the minimum achievable temperature is $T_D = \hbar\Gamma/(2k_B)$.

For $^{87}$Rb: $T_D = 146\,\mu\text{K}$. For the ground state in a 100 kHz trap, we need $T \sim \hbar\nu/k_B \approx 5\,\mu\text{K}$. Doppler cooling alone does not reach the ground state for broad-line atoms.

Crossing to Sub-Doppler

The solution is two-photon cooling, the techniques that follow. By using spin-dependent light shifts and coherent dark states, these methods surpass the Doppler limit while still using the same broad transitions. The Lamb-Dicke parameter $\eta$ then controls how efficiently the motional sidebands are addressed.

Position-Dependent Coupling

In a standing wave, the interaction Hamiltonian contains $\sin(k\hat{x})$ or $\cos(k\hat{x})$ instead of $e^{ik\hat{x}}$. Near a node (at $\hat{x} = 0$), expanding to first order:

Enhanced Cooling at the Node

At the node, the $n \to n-1$ coupling scales as $\eta\sqrt{n}$, which goes to zero as $n \to 0$. The state $|n=0\rangle$ is therefore dark at the node, it does not scatter photons. This creates a natural dark state for cooling and allows the motional ground state population to accumulate without being further heated.

Different Trap Frequencies

When the cooling laser and the trap have different length scales (e.g., a red-detuned tweezer and a near-resonant cooling laser), the motional states seen by each can differ. The paper treats this case using a squeezing transformation, showing that an effective Lamb-Dicke parameter

The Sisyphus Mechanism

In the $\sigma^+$-$\sigma^-$ geometry, two beams create a standing wave. The light shift of the $m_F$ sublevels varies sinusoidally in space, and the optical pumping rate between sublevels also varies. An atom in the higher-$m_F$ state climbing a potential hill is pumped by optical pumping to the lower-$m_F$ state at the hill top, like Sisyphus rolling a boulder up forever. Each pump cycle dissipates kinetic energy equal to the light shift $U_0$.

Lin ⊥ Lin Geometry

In the lin $\perp$ lin configuration, the polarization rotates from linear to circular and back over half a wavelength. The atom's internal state follows this rotation, and energy is dissipated each time the atom climbs a light-shift hill. The cooling is more efficient than $\sigma^+$-$\sigma^-$ because the polarization gradient is steeper.

Temperature Limit

PG cooling does not have a simple closed-form temperature limit, it depends on the spin $F$, the laser intensity, and detuning. The temperature scales approximately as:

Dark States and the $F \to F' = F$ Transition

For a $J \to J' = J$ (or $F \to F' = F$) transition driven with blue-detuned light, coherent superpositions of ground $m_F$ states can be constructed that have zero dipole matrix element to all excited states. These are the "dark states." An atom pumped into a dark state stops scattering. The bright state (orthogonal superposition) has a positive light shift, sitting higher in energy. With blue detuning, bright states are trapped in regions of high intensity and dark states live in low-intensity regions.

The Unified Spin Model (Eq. 72)

After adiabatic elimination and expanding to first order in $\eta$, Phatak et al. derive the effective spin-cooling Hamiltonian:

Cooling Limit (Eq. 72)

The steady-state mean phonon number in the spin model is:

The Λ-Configuration

Two ground states $|g_1\rangle$ and $|g_2\rangle$ (from different hyperfine levels) are each coupled to a common excited state $|e\rangle$ by laser fields $\Omega_1$ and $\Omega_2$ with detunings $\Delta_1$ and $\Delta_2$. The key is to transform to the bright/dark basis:

Why EIT/Λ-GM is Better than GM

In standard GM, the dark state is a single ground-state spin superposition that is dark only at the leading order in the coupling. In the Λ-configuration, the dark state $|g_D\rangle$ is dark to both laser fields simultaneously due to destructive interference, the contribution from $|g_1\rangle$ and $|g_2\rangle$ cancel exactly. This makes $\gamma_- \to 0$ much more effectively, directly reducing the steady-state $\langle n \rangle$.

The same spin-cooling model (Eq. 72) applies, but with significantly smaller $\gamma_-/\gamma_+$, extracted from simulations of $^{87}$Rb: $\gamma_-/\gamma_+ \approx 0.006$ for Λ-GM vs. $\gamma_-/\gamma_+ \approx 0.11$ for standard GM.

EIT Heating Resonance

When $\Delta_1 - \Delta_2 = 2\nu$ (the blue sideband two-photon condition), a heating resonance appears. The optimal cooling occurs at $\Delta_1 = \Delta_2$ when the bright-state light shift equals the trap frequency, $\Omega^2/\Delta = \nu$.

The RSB Cooling Cycle (Fig. 9)

Step 1 (Raman): Two far-detuned counter-propagating lasers at Rabi frequencies $\Omega_1$, $\Omega_2$ and detuning $\Delta$ drive the two-photon transition $|g_1, n\rangle \to |g_2, n-1\rangle$ via stimulated Raman. The Raman coupling:

Step 2 (Optical pumping):

A separate near-resonant beam pumps $|g_2\rangle \to |g_1\rangle$ via spontaneous emission. The key is that angular momentum selection rules ($\Delta m_F = 0, \pm 1$) can be exploited so that $|g_1\rangle$ remains dark to the pump beam. Spontaneous emission in the Lamb-Dicke regime keeps the atom at $n-1$ rather than $n$.

Why RSB Beats EIT/Λ-GM

In EIT and Λ-GM, the counter-propagating lasers that provide cooling also do optical pumping. This means position-dependent collapse operators act on the dark state: the atom in $|g_D\rangle$ scatters photons from the cooling beams at rate $\propto \eta^2 \gamma_h$. This heating rate is fundamental and sets the $\eta^2$ floor in Eq. 72.

In RSB, the optical pump beam is completely separate and designed to be dark to $|g_1\rangle$. The far-detuned Raman beams have negligible scattering ($\Gamma_R \propto \Omega^2/\Delta^2 \approx 0$). So the position-dependent collapse operator from pumping is absent, removing the $\eta^2\gamma_h$ floor entirely.