Atomic & Optical Physics

Nobel Prize Physics: Prof. Ketterle won the 2001 Nobel Prize for achieving Bose-Einstein condensation. Quantum mechanics meets experiment!

Atomic & Optical Physics

⚛️ Quantum Mechanics Meets Experiment

This course bridges the gap between theoretical quantum mechanics and cutting-edge experimental physics. You'll learn how abstract quantum concepts like superposition, entanglement, and quantum statistics manifest in real laboratory systems.

🔬 Laboratory Physics

  • • Laser cooling to nanokelvin temperatures
  • • Trapping single atoms and ions
  • • Creating Bose-Einstein condensates
  • • Measuring quantum states of light

🎯 Applications

  • • Quantum computing with trapped ions
  • • Precision metrology beyond Heisenberg limit
  • • Quantum simulation of many-body systems
  • • Fundamental tests of quantum mechanics

🏆 Nobel Prize-Winning Research

Prof. Wolfgang Ketterle won the 2001 Nobel Prize in Physics (shared with Eric Cornell and Carl Wieman) for achieving Bose-Einstein condensation in dilute gases of alkali atoms. This was the first realization of a quantum state of matter predicted by Einstein in 1925!

"At ultralow temperatures, a gas of bosonic atoms can condense into the same quantum ground state, forming a macroscopic quantum object where all atoms move in lockstep. This allows us to see quantum mechanics with the naked eye."

📚 Course Content

Part I: Quantum Optics (Topics 1-6)

  • • Classical vs quantum description of light
  • • Non-classical light states: squeezing, single photons
  • • Entanglement and EPR paradox
  • • Quantum metrology and Heisenberg limit
  • • Coherent states and photon statistics

Part II: Light-Atom Interactions (Topics 7-12)

  • • Two-level atoms in electromagnetic fields
  • • Optical Bloch equations and Rabi oscillations
  • • Light forces: dipole force and radiation pressure
  • • Dressed atom picture and AC Stark shift
  • • Laser cooling mechanisms (Doppler, Sisyphus)
  • • Achieving sub-microkelvin temperatures

Part III: Ultracold Atoms & Quantum Gases (Topics 13-21)

  • • Evaporative cooling to nanokelvin regime
  • • Bose-Einstein condensation: theory and experiment
  • • Weakly interacting Bose gases (Gross-Pitaevskii equation)
  • • Degenerate Fermi gases and Fermi surfaces
  • • BEC-BCS crossover with Feshbach resonances
  • • Ion trapping and quantum gates for quantum computing

🔗 Connections to Other Courses

Prerequisites:

Leads to:

  • • Quantum Field Theory: Quantized EM field
  • • Quantum information and computing
  • • Condensed matter physics (superfluidity, superconductivity)

🔬 Fundamental Equations & Concepts

Two-Level Atom Hamiltonian

The simplest model for light-atom interaction uses a two-level atom (ground state |g⟩ and excited state |e⟩) in a classical electromagnetic field:

$$\hat{H} = \frac{\hbar\omega_0}{2}\hat{\sigma}_z + \hbar\Omega(t)\cos(\omega t)\hat{\sigma}_x$$

where $\omega_0$ is the atomic transition frequency, $\Omega(t)$ is the Rabi frequency proportional to the electric field, and $\hat{\sigma}_x$, $\hat{\sigma}_z$ are Pauli matrices representing transitions between levels.

Rabi Oscillations

When a resonant laser field ($\omega = \omega_0$) drives the atom, the population oscillates between ground and excited states:

$$P_e(t) = \sin^2\left(\frac{\Omega t}{2}\right)$$

The Rabi frequency $\Omega = -\vec{d}\cdot\vec{E}_0/\hbar$ depends on the atomic dipole moment $\vec{d}$ and laser field amplitude $\vec{E}_0$. Typical values: $\Omega \sim 2\pi \times (1\text{-}100\text{ MHz})$ for optical transitions.

Optical Bloch Equations

Including spontaneous emission (rate $\Gamma$) and detuning $\delta = \omega - \omega_0$, the density matrix evolves according to:

$$\frac{d}{dt}\rho_{ee} = -\Gamma\rho_{ee} + i\frac{\Omega}{2}(\rho_{ge} - \rho_{eg})$$
$$\frac{d}{dt}\rho_{ge} = -(i\delta + \frac{\Gamma}{2})\rho_{ge} + i\frac{\Omega}{2}(\rho_{ee} - \rho_{gg})$$

These equations describe the complete dynamics of a two-level atom including coherence decay and population transfer. They are central to understanding laser cooling, optical pumping, and spectroscopy.

❄️ Laser Cooling Mechanisms

Doppler Cooling

An atom moving with velocity $v$ towards a red-detuned laser ($\delta < 0$) sees the light Doppler-shifted closer to resonance, absorbing more photons from the opposing beam than from the co-propagating beam. This creates a velocity-dependent friction force:

$$F = -\beta v \quad \text{where} \quad \beta = \frac{8\hbar k^2\delta\Omega^2}{(\delta^2 + \frac{\Gamma^2}{4} + 2\Omega^2)^2}$$

The Doppler cooling limit arises from the random kicks during spontaneous emission:

$$T_{Doppler} = \frac{\hbar\Gamma}{2k_B}$$

For sodium ($\Gamma = 2\pi \times 10$ MHz), $T_{\text{Doppler}} \approx 240$ μK. This was considered the fundamental limit until sub-Doppler mechanisms were discovered.

Sisyphus Cooling (Sub-Doppler)

In optical molasses with polarization gradients, atoms climb potential hills created by the AC Stark shift, losing kinetic energy before being optically pumped to a lower state at the hill top. This process can cool atoms below the Doppler limit to temperatures:

$$T_{recoil} = \frac{(\hbar k)^2}{2m k_B}$$

For sodium, $T_{\text{recoil}} \approx 2.4$ μK. Sisyphus cooling routinely achieves temperatures of 1-10 μK, about 100 times colder than the Doppler limit!

Evaporative Cooling

To reach nanokelvin temperatures for BEC, we use evaporative cooling: atoms in a magnetic trap have different energies. By selectively removing the hottest atoms (lowering the trap depth), the remaining atoms rethermalize at a lower temperature. The cooling efficiency is characterized by:

$$\frac{dT}{T} = \frac{1}{3}\left(\frac{\eta - 5}{\eta - 3}\right)\frac{dN}{N}$$

where $\eta = E_{\text{cut}}/k_BT$ is the ratio of the cut energy to thermal energy. For $\eta \approx 7\text{-}10$, each factor of 10 reduction in atom number yields a factor of 3-5 reduction in temperature. This allows cooling from $\sim10$ μK to $\sim10$ nK!

🌌 Bose-Einstein Condensation

Quantum Statistics & Phase Transition

For a gas of $N$ non-interacting bosons in a harmonic trap with frequencies $\omega_x, \omega_y, \omega_z$, the critical temperature for BEC is:

$$T_c = \frac{\hbar\bar{\omega}}{k_B}\left(\frac{N}{\zeta(3)}\right)^{1/3} \quad \text{where} \quad \bar{\omega} = (\omega_x\omega_y\omega_z)^{1/3}$$

and $\zeta(3) \approx 1.202$ is the Riemann zeta function. For $N = 10^6$ atoms and $\bar{\omega} = 2\pi \times 100$ Hz, we get $T_c \approx 300$ nK.

Below $T_c$, a macroscopic fraction of atoms occupy the ground state: $N_0/N = 1 - (T/T_c)^3$. At $T = 0.5T_c$, already 87.5% of atoms are in the condensate!

Gross-Pitaevskii Equation

When atoms interact via s-wave scattering (characterized by scattering length $a$), the condensate wavefunction$\psi(\vec{r},t)$ obeys a nonlinear Schrödinger equation:

$$i\hbar\frac{\partial\psi}{\partial t} = \left[-\frac{\hbar^2}{2m}\nabla^2 + V_{ext}(\mathbf{r}) + g|\psi|^2\right]\psi$$

where $g = 4\pi\hbar^2 a/m$ is the interaction strength. This equation successfully describes:

  • • Collective excitations (sound waves in the condensate)
  • • Expansion dynamics when the trap is turned off
  • • Quantized vortices in rotating BECs
  • • Solitons and matter-wave interference

Experimental Signatures

BEC is typically detected via absorption imaging after releasing the atoms from the trap. Key signatures:

  • Bimodal distribution: Below $T_c$, you see a narrow peak (condensate) on top of a broad thermal cloud
  • Anisotropic expansion: The condensate expands faster along the tightly confined direction (inverted aspect ratio) due to interaction energy converting to kinetic energy
  • Coherence length: Matter-wave interference between two overlapping condensates shows high-contrast fringes, proving macroscopic phase coherence

💡 Quantum Optics: Light as Quanta

Coherent States

Laser light is well described by coherent states |α⟩, which are eigenstates of the annihilation operator:

$$\hat{a}|\alpha\rangle = \alpha|\alpha\rangle \quad \text{where} \quad |\alpha\rangle = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle$$

Coherent states have Poissonian photon statistics with $\langle n \rangle = |\alpha|^2$ and $\Delta n = \sqrt{\langle n \rangle}$. They minimize the Heisenberg uncertainty relation $\Delta X \cdot \Delta P = \hbar/2$ with equal uncertainty in both quadratures.

Squeezed States

Non-classical squeezed states reduce noise in one quadrature below the standard quantum limit at the expense of increased noise in the conjugate quadrature. The squeezing operator is:

$$\hat{S}(r) = \exp\left[\frac{r}{2}(\hat{a}^{\dagger 2} - \hat{a}^2)\right]$$

Applications include gravitational wave detection (LIGO uses squeezed light!), precision spectroscopy, and quantum communication. Squeezing of 15 dB (factor of ~5.6 noise reduction) has been demonstrated.

Photon Antibunching & g⁽²⁾(τ)

The second-order coherence function distinguishes classical from quantum light:

$$g^{(2)}(\tau) = \frac{\langle\hat{a}^\dagger(t)\hat{a}^\dagger(t+\tau)\hat{a}(t+\tau)\hat{a}(t)\rangle}{\langle\hat{a}^\dagger(t)\hat{a}(t)\rangle^2}$$
  • Coherent light: $g^{(2)}(0) = 1$ (Poissonian, no correlation)
  • Thermal light: $g^{(2)}(0) = 2$ (bunched, photons tend to arrive together)
  • Single photon source: $g^{(2)}(0) = 0$ (antibunched, cannot have two photons simultaneously)

Antibunching ($g^{(2)}(0) < 1$) is a purely quantum effect with no classical analog and proves the particulate nature of light.

⚡ Light Forces & Trapping

Dipole Force (Gradient Force)

An atom in a spatially varying laser field experiences a conservative force due to the AC Stark shift:

$$\mathbf{F}_{dip} = -\nabla U_{dip} \quad \text{where} \quad U_{dip} = \frac{\hbar\Omega^2}{4\delta}$$

For red-detuned light ($\delta < 0$), atoms are attracted to intensity maxima (optical traps). For blue-detuned light ($\delta > 0$), atoms are repelled from intensity maxima (optical barriers, hollow-beam traps).

Typical trap depths: $U/k_B \sim 1$ mK. Far-detuned dipole traps ($|\delta| \gg \Gamma$) have negligible heating from spontaneous emission, enabling very long atom storage times.

Radiation Pressure (Scattering Force)

Each absorbed photon transfers momentum $\hbar k$ to the atom. The scattering force in the low-intensity regime is:

$$F_{scatt} = \hbar k \cdot \Gamma_{scatt} = \hbar k\Gamma\frac{I/I_{sat}}{1 + I/I_{sat} + 4\delta^2/\Gamma^2}$$

where $I_{\text{sat}} = \pi hc\Gamma/(3\lambda^3)$ is the saturation intensity. Maximum force $F_{\text{max}} = \hbar k\Gamma/2$ occurs on resonance at saturation. For sodium, $F_{\text{max}} \approx 10^{-20}$ N, producing accelerations of $\sim 10^5$ g!

Magneto-Optical Trap (MOT)

The MOT combines optical molasses with a quadrupole magnetic field to create a position-dependent restoring force. The spatially varying Zeeman shift makes atoms at position r⃗ more resonant with one of the counter-propagating beams, creating both damping and confinement.

  • Typical parameters: $10^8 - 10^{10}$ atoms, $T \sim 100$ μK, density $\sim 10^{10} - 10^{11}$ cm⁻³
  • Loading time: 1-10 seconds from background vapor
  • Trap lifetime: 1-100 seconds (limited by collisions with background gas)

The MOT is the workhorse of atomic physics, serving as the starting point for nearly all ultracold atom experiments.

🔮 Advanced Topics & Applications

Feshbach Resonances & BEC-BCS Crossover

By tuning an external magnetic field near a Feshbach resonance, the s-wave scattering length $a$ can be varied from $-\infty$ to $+\infty$, allowing exploration of the crossover from BEC (tightly bound molecules, $a > 0$) to BCS superfluidity (loosely bound Cooper pairs, $a < 0$):

$$a(B) = a_{bg}\left(1 - \frac{\Delta B}{B - B_0}\right)$$

At unitarity ($a \to \pm\infty$), the system is scale-invariant and interactions are as strong as quantum mechanics allows. This regime has been used to study universal thermodynamics and quantum hydrodynamics.

Quantum Simulation

Ultracold atoms in optical lattices (periodic potentials created by interfering laser beams) can simulate condensed matter Hamiltonians:

$$\hat{H} = -J\sum_{\langle i,j\rangle}(\hat{a}_i^\dagger\hat{a}_j + \text{h.c.}) + \frac{U}{2}\sum_i\hat{n}_i(\hat{n}_i-1)$$

This is the Bose-Hubbard model with hopping $J$ and on-site interaction $U$. It exhibits a quantum phase transition from superfluid to Mott insulator as $U/J$ increases.

  • • Quantum magnetism and frustrated systems
  • • Topological states (synthetic dimensions, gauge fields)
  • • Many-body localization
  • • Real-time dynamics beyond classical computation

Ion Traps & Quantum Computing

Single ions trapped in RF Paul traps or Penning traps provide pristine two-level systems (qubits) for quantum information processing. Key advantages:

  • Long coherence times: $T_2 > 10$ minutes for hyperfine qubits
  • High-fidelity gates: Single-qubit gates $> 99.99\%$, two-qubit gates $> 99.5\%$
  • Programmable connectivity: Phonon-mediated gates between distant ions
  • High-fidelity readout: State-dependent fluorescence with $> 99.99\%$ accuracy

Current ion-trap quantum computers (IonQ, Honeywell) have achieved quantum volumes $> 10^6$ and are approaching the threshold for quantum error correction.

Precision Metrology & Atomic Clocks

Ultracold atoms enable the most precise measurements in physics. Optical lattice clocks using Sr or Yb achieve fractional frequency uncertainty below $10^{-18}$, measuring time to 1 second in the age of the universe!

The Allan deviation for these clocks:

$$\sigma_y(\tau) = \frac{1}{2\pi\nu_0\sqrt{N}}\frac{1}{\sqrt{\tau}}$$

Applications include geodesy (measuring gravitational redshift to probe Earth's gravitational field), tests of general relativity, searches for dark matter, and potential redefinition of the second.

🎓 Video Lectures

This course features 21 topics (27 video segments) from Prof. Wolfgang Ketterle's MIT course on Atomic and Optical Physics. The lectures cover both theoretical foundations and experimental techniques used in modern quantum optics laboratories.

Watch MIT Ketterle Lectures →