Quantum Fluctuations and Van der Waals Forces Between Neutral Atoms

Summary

MIT OpenCourseWare's lecture explores how neutral atoms interact through quantum fluctuations, producing van der Waals and Casimir-Polder forces. It connects classical intuition with quantum electrodynamics, using perturbation theory and photon exchange to show how short-range and long-range forces emerge between atoms.

Overview

The lecture analyzes the force between two neutral atoms A and B separated by a distance R, arguing that the Coulomb term cancels for neutral, localized objects and that the leading interaction arises from dipole fluctuations. In classical physics the expectation value of the dipole moment for each atom vanishes due to isotropy, and there is no spontaneous attraction between neutral atoms. Quantum mechanically, however, zero point fluctuations make the dipole moments nonzero on average, and when these fluctuations are correlated between the atoms they produce a nonzero attraction. The teacher emphasizes that the quantum Wanderwaals (van der Waals) force is fundamentally a fluctuation phenomenon, and that a purely classical treatment cannot capture it.

Three complementary perspectives are introduced to understand the van der Waals interaction. The first is a perturbative quantum-mechanical approach in which the leading interaction for two ground-state atoms arises at second order, yielding a 1/R^6 London dispersion potential. In first order, the interaction has a dipole-dipole character that scales as 1/R^3, but this contribution cancels for isotropic ground states when both atoms are in their ground state, so the observable leading order is the second-order London term. The second perspective uses a mechanical analogy with two coupled harmonic oscillators or LC circuits. Each oscillator has a zero point energy, and coupling through a fluctuating dipole field produces a shifted energy that also scales as 1/R^6 at short range. The third perspective reinterprets the effect in terms of vacuum fluctuations of the electromagnetic field. The vacuum contains zero point fluctuations in each mode, and the correlated dipole moments induced by these field fluctuations give rise to a Casimir-Polder type interaction with a 1/R^7 long-range tail when propagation effects are included. The density of field modes plays a crucial role in determining which wavelengths dominate the interaction, and long wavelengths can become important when retardation effects are included.

Quantum Fluctuations and the Vacuum

The instructor then highlights how the underlying physics depends on two kinds of zero point fluctuations: those of the atomic oscillators themselves and those of the quantized electromagnetic field. In the atomic picture, the dipole moment fluctuations of each atom couple to the field, inducing a second dipole in the other atom, and this dipole-dipole interaction gives the short-range 1/R^6 potential. In the field picture, a photon exchange process mediates the interaction. When one considers the field as a bath of harmonic oscillators with their own zero point energy, the interaction emerges from the exchange of virtual photons rather than real radiation, particularly in the regime where no real photons are emitted.

Long Range Versus Short Range: Retardation and Casimir-Polder

A central theme is the difference between short-range and long-range behavior. In the short-range regime, propagation delays are negligible, and a instantaneous dipole-dipole interaction leads to a 1/R^6 potential. In the retarded or long-range regime, the finite speed of light and the structure of the electromagnetic field become important. Photons can propagate between the atoms, and the energy denominators in perturbation theory pick up contributions from the photon frequency relative to the atomic transition frequency. This retardation changes the distance dependence to a slower, additional power, yielding the retarded Casimir-Polder potential, typically scaling as 1/R^7 at large separations. The transition between these regimes is governed by the wavelengths of the resonant radiation of the atoms and the density of electromagnetic modes available at those frequencies.

Diagrammatic View and Fourth-Order Perturbation Theory

The lecture then introduces a diagrammatic, quantum-electrodynamic view. Because there is no direct coulomb interaction between neutral atoms in the QED formulation, the interaction must arise from the exchange of quanta of the quantized radiation field. In ground-state atom interactions, the leading contribution comes from fourth-order perturbation theory, corresponding to two-photon exchange processes. Two representative diagrams illustrate how intermediate states with either two atomic excitations or two photons contribute, and how the energy denominators in these diagrams determine the short-range and long-range scaling. The analysis shows that even though some diagrams look similar, their dominant contributions differ between short and long range, culminating in the 1/R^6 short-range London dispersion and the 1/R^7 retarded Casimir-Polder form for large separations.

Interplay of Fields and Atoms

To connect the different pictures, the lecturer emphasizes the need for a consistent description in which both the atomic system and the electromagnetic field are quantized. He notes that while one can describe aspects of the effect by focusing solely on atomic oscillators or solely on a single mode of the field, a complete understanding requires considering the full spectrum of field modes and their zero point energies. In a pedagogical detour, the two coupled LC circuit model shows how coupling shifts the normal modes and introduces a zero-point energy correction that mirrors the quantum mechanical van der Waals interaction. This dual viewpoint clarifies how fluctuations in one system induce correlated fluctuations in another, producing an attractive force that is inherently quantum mechanical and not present in a purely classical framework.

Practical Takeaways and Subtleties

The lecture concludes with several important caveats. First, the van der Waals and Casimir-Polder forces are manifestations of quantum fluctuations, and energy shifts arise from virtual processes rather than real photon emission. Second, the long-range behavior depends crucially on retardation and the density of electromagnetic modes, which favor long-wavelength fluctuations at certain distances. Third, a fully consistent treatment requires quantization of both the atoms and the electromagnetic field; treating one as a classical background can miss essential physics. The instructor points to additional resources such as Spruch's papers for a more detailed methodological treatment. Finally, the discussion ties the atomic physics of dispersion forces to broader concepts in quantum electrodynamics and intends to present a unifying diachromatic approach that blends short-range and retarded effects in one framework.