Feshbach Resonance and Interactions in Ultracold Quantum Gases

Table of Contents

1. Introduction
2. Overview of Ultracold Atomic Interactions
3. Scattering Length and Low-Energy Collisions
4. Magnetic Tuning of Interactions
5. Feshbach Resonance: Basic Principle
6. Open and Closed Channels
7. Resonance Width and Strength
8. Effective Scattering Length Formula
9. Experimental Realization of Feshbach Resonances
10. Species with Accessible Feshbach Resonances
11. Applications in Bose Gases
12. Control of Collapse and Explosion in BECs
13. Feshbach Molecules and Binding Energies
14. Applications in Fermi Gases
15. BEC–BCS Crossover Physics
16. Efimov States and Universal Few-Body Physics
17. Optical and Radio-Frequency Feshbach Resonances
18. Multichannel Quantum Defect Theory (MQDT)
19. Limitations and Technical Challenges
20. Conclusion

1. Introduction

Feshbach resonances provide a powerful method for tuning atomic interactions in ultracold gases. By varying an external magnetic field, researchers can precisely control the scattering length, enabling exploration of quantum many-body physics from weak to strong coupling regimes.

2. Overview of Ultracold Atomic Interactions

At ultralow temperatures, atomic interactions are dominated by s-wave scattering. The interaction potential can be characterized by a single parameter: the scattering length \( a \).

3. Scattering Length and Low-Energy Collisions

The s-wave scattering length \( a \) governs the low-energy behavior of the scattering amplitude:
\[
f(k) pprox -rac{a}{1 + ika}
\]
Positive \( a \): repulsive; negative \( a \): attractive; \( a = 0 \): non-interacting limit.

4. Magnetic Tuning of Interactions

A magnetic field shifts the energy of internal atomic states via the Zeeman effect. Near a resonance, the energy of a bound molecular state (closed channel) aligns with a scattering state (open channel), modifying \( a \).

5. Feshbach Resonance: Basic Principle

A Feshbach resonance occurs when the bound state of a closed channel becomes degenerate with the collisional energy of the open channel. This causes a resonant enhancement in the scattering amplitude.

6. Open and Closed Channels

• Open channel: free atom pair in an entrance channel
• Closed channel: bound molecular state in a different hyperfine configuration
  Coupling between channels modifies scattering properties near resonance.

7. Resonance Width and Strength

Resonances are characterized as:

• Broad: strong coupling, large range of tunability
• Narrow: weak coupling, sensitive to field stability
  The resonance strength impacts thermalization and loss rates.

8. Effective Scattering Length Formula

The scattering length near a magnetic Feshbach resonance is given by:
\[
a(B) = a_{ ext{bg}} \left( 1 – rac{\Delta}{B – B_0}
ight)
\]
where \( a_{ ext{bg}} \) is the background scattering length, \( B_0 \) is the resonance position, and \( \Delta \) is the width.

9. Experimental Realization of Feshbach Resonances

Feshbach resonances were first observed in \(^{85} ext{Rb}\) and \(^{23} ext{Na}\) using magnetic field sweeps. They are now routinely used in experiments on bosons and fermions alike.

10. Species with Accessible Feshbach Resonances

• Bosons: \(^{85} ext{Rb}\), \(^{133} ext{Cs}\), \(^{39} ext{K}\)
• Fermions: \(^{6} ext{Li}\), \(^{40} ext{K}\)
  These species feature rich and well-characterized resonance spectra.

11. Applications in Bose Gases

• Tuning interaction strength to observe collapse (attractive \( a < 0 \)) and stable condensates (repulsive \( a > 0 \))
• Engineering solitons, droplets, and quantum turbulence

12. Control of Collapse and Explosion in BECs

Adjusting \( a \) can induce:

• BEC collapse at large negative \( a \)
• Controlled “Bosenova” explosions
• Stabilization via three-body loss or dipolar interactions

13. Feshbach Molecules and Binding Energies

On the attractive side of a resonance, weakly bound diatomic molecules form. These molecules serve as precursors to deeper bound states or BECs of molecules.

14. Applications in Fermi Gases

Feshbach tuning enables crossover from:

• Weakly bound Cooper pairs (BCS side, \( a < 0 \))
• Tightly bound bosonic molecules (BEC side, \( a > 0 \))
• Unitary regime (\( |a| o \infty \)): strongly interacting quantum fluid

15. BEC–BCS Crossover Physics

Ultracold fermions across a Feshbach resonance allow exploration of superfluidity, pairing gaps, vortex lattices, and pseudogap behavior in the crossover regime.

16. Efimov States and Universal Few-Body Physics

Near a Feshbach resonance, three-body systems support Efimov states—universal trimers whose energy levels follow a geometric scaling:
\[
E_n \propto e^{-2\pi n/s_0}
\]
These have been observed in three-body loss spectra.

17. Optical and Radio-Frequency Feshbach Resonances

Optical Feshbach resonances use laser-induced coupling to modify scattering length. RF fields can also induce transitions between channels, offering time-dependent interaction control.

18. Multichannel Quantum Defect Theory (MQDT)

MQDT provides a framework for modeling multi-channel scattering, allowing predictive understanding of resonance positions, widths, and universal properties.

19. Limitations and Technical Challenges

• Magnetic field noise and resolution
• Three-body loss near resonance
• Heating and stability of BEC/DFG
• Need for species-dependent calibration

20. Conclusion

Feshbach resonances have revolutionized the study of ultracold matter by enabling tunable interactions. From quantum phase transitions to universal few-body physics, they provide a key to exploring and engineering new quantum states.