Matter-Wave Interferometry

Matter-wave interferometry compares amplitudes for different center-of-mass histories of a particle or composite object. Its foundational value lies in measuring spatial coherence while controlling ordinary sources of decoherence. A large mass is useful only together with a known path geometry, coherence time, and reliable distinction between quantum fringes and classical density modulation.

Helpful background. Interference with Matter introduces the experimental evidence for matter waves.

Wavelength, geometry, and coherence

For a nonrelativistic object of mass $m$ and speed $v$,

$$\lambda_{\rm dB}=\frac{h}{mv}.$$

The center-of-mass wavelength can be much smaller than the object’s physical size. Interference concerns the center-of-mass amplitude, not a requirement that the object fit inside one wavelength. Internal excitation is compatible with center-of-mass coherence if internal or environmental states do not acquire distinguishable path information.

For a grating period $d$, a conventional Talbot length is $L_T=d^2/\lambda_{\rm dB}$; fractional and doubled recurrence conventions must be specified when comparing apparatuses. With $m=10^4$ atomic mass units, $v=100\ {\rm m\,s}^{-1}$, and $d=100$ nm, $\lambda_{\rm dB}\simeq4.0\times10^{-13}$ m and $L_T\simeq2.5$ cm. Near-field methods exploit such propagation scales rather than resolving the subpicometre wavelength directly.

From a measured pattern to a coherence claim

A scanned grating or position-sensitive detector measures a count pattern. The quantum prediction depends on velocity distribution, grating transmission or phase, source coherence, forces, and detector response. A classical moiré shadow can also be periodic; agreement with the predicted dependence on velocity, grating interaction, and propagation distance is more discriminating than periodicity alone.

Fein and collaborators reported molecular interference beyond $25$ kDa in a Talbot–Lau apparatus in 2019. This is a dated example of the method, not a statement of the current mass record. Their comparison with the modeled quantum visibility illustrates why the apparatus transfer function matters.

Decoherence and localization sensitivity

For random momentum kicks with rate $R$ and normalized distribution $\mu(q)$, a simple one-dimensional localization model gives

$$\Gamma(\Delta x)=R\left[1-\int dq\,\mu(q)e^{iq\Delta x/\hbar}\right].$$

For a symmetric kick distribution this is real. Small path separation can be unresolved by the kicks, while large separation suppresses the Fourier term and approaches a rate $R$. Along an interferometer, the relevant factor is approximately

$$\exp\!\left[-\int_0^t\Gamma(\Delta x(s))\,ds\right].$$

Thus a quoted maximum separation alone is insufficient; the full separation history matters. Collisions, thermal radiation, vibrations, and fluctuating fields can reduce visibility through different mechanisms. Model exclusion requires treating those known effects and their uncertainty, rather than identifying every missing fringe with fundamental collapse.

Exercise

For isotropically emitted photons of a fixed wave-number magnitude $k$, show that the direction-averaged distinguishability factor for separation $\Delta x$ is $\sin(k\Delta x)/(k\Delta x)$.

Solution

Choose the polar axis along the separation. The angular average is $\frac12\int_{-1}^{1}e^{ik\Delta x u}du=\sin(k\Delta x)/(k\Delta x)$. It tends to one for $k\Delta x\ll1$, so a single long-wavelength photon poorly resolves the paths. It tends to zero after averaging at large separation, where an emission can efficiently decohere them.

References

• Fein, Y. Y., et al., “Quantum superposition of molecules beyond 25 kDa,” Nature Physics 15, 1242–1245 (2019), doi:10.1038/s41567-019-0663-9 — a molecular-interference realization.
• Nimmrichter, S., and Hornberger, K., “Macroscopicity of mechanical quantum superposition states,” Physical Review Letters 110, 160403 (2013), arXiv:1205.3447 — connecting coherence tests to modified dynamics.