Atom-Interferometric Free Fall

A light-pulse atom interferometer measures acceleration through a phase assembled from propagation, laser interactions, and the output-port readout. In a dual-species free-fall test, the useful observable is a differential phase whose composition component follows $k_{\rm eff}gT^2$ while vibration, wavefront, recoil, gravity-gradient, and initial-cloud effects follow distinguishable reversal or scaling rules. Calling the entire phase “the gravitational redshift” is model-dependent.

Required background. Equivalence Principles and Gravitational Measurement supplies free fall, proper time, and the distinction between WEP and clock universality. Probability and Statistical Inference supplies likelihoods and nuisance parameters; the local Gaussian phase model below states what is used here.

Helpful background. Weak-Field Gravity supplies the potential and gravity-gradient expansion, while Tests of Gravity connects quantum test masses to other regimes.

Phase combinations that isolate acceleration

For an ideal three-pulse Mach–Zehnder sequence at times $0,T,2T$, the leading vertical phase is

$$\Delta\phi_s=\mathbf k_{{\rm eff},s}\!\cdot\mathbf g_s T_s^2+\phi_{{\rm laser},s}+\phi_{{\rm sys},s},$$

for species $s$, with signed wave-vector projections taken along the same axis. Here $\phi_{{\rm laser},s}$ denotes programmed laser-phase contributions and offsets not already included in the acceleration term; it is not the entire laser phase counted a second time. The $T^2$ factor follows from the sampled position combination. For $z(t)=z_0+v_0t+gt^2/2$,

$$z(0)-2z(T)+z(2T)=gT^2.$$

Thus the initial position and velocity cancel for uniform acceleration and ideal instantaneous pulses. A chirped Raman frequency difference adds a phase $-\dot\omega_L T^2$ in a convention where $\dot\omega_L=k_{\rm eff}g$ follows the falling atoms, giving $(k_{\rm eff}g-\dot\omega_L)T^2$ before other corrections. A gravity gradient or unequal pulse timing spoils the simple cancellation. A scale-factor-matched differential observable is

$$\Delta\phi_d =\Delta\phi_A-\kappa\Delta\phi_B, \qquad \kappa=\frac{k_{{\rm eff},A}T_A^2}{k_{{\rm eff},B}T_B^2}.$$

The acceleration ratio inferred from $\Delta\phi_d$ becomes an Eötvös parameter only after subtracting species-dependent phases and propagating uncertainty in $\kappa$. Common vibration cancels imperfectly when scale factors, pulse timing, or transfer functions differ.

Wave-vector reversal changes the sign of the inertial phase while many even-in-$\mathbf k_{\rm eff}$ shifts do not. Species interchange changes composition labels while leaving apparatus-fixed terms. Varying $T$ identifies the $T^2$ acceleration scaling; changing launch position and velocity diagnoses gravity-gradient and Coriolis terms. Overstreet et al. show how a frequency shift of the mirror pulse suppresses coupling of initial kinematic differences to a gravity gradient Overstreet et al. 2018, main text Eqs. (1)–(4).

From shots to an equivalence result

The dataset must bind pulse times, Raman or Bragg frequencies and angles, effective wave vectors, transition probabilities, initial cloud positions and velocities, vibration reference, magnetic field, wavefront map, and detection calibration. The Asenbaum et al. $^{85}$Rb–$^{87}$Rb experiment reports two seconds of free fall and calibrates beam angles and frequencies to suppress isotope-dependent kinematics Asenbaum et al. 2020, main text and Supplemental Material, PDF. Its article is a primary summary, not a universal machine-readable atom-interferometer release.

For shot vector $\mathbf p$ and phase model $\boldsymbol\phi(\eta,\boldsymbol\nu)$, the ideal output probability is $\mu_i=[1+C_i\cos\phi_i]/2$, where $C_i$ is fringe contrast and the output-port convention fixes the sign. Detected counts can use a binomial likelihood conditional on atom number. In a calibrated Gaussian approximation to the measured probabilities,

$$\mathcal L\propto \exp\!\left[-\frac12(\mathbf p-\boldsymbol\mu)^{\mathsf T} C^{-1}(\mathbf p-\boldsymbol\mu)\right].$$

Priors on wavefront, cloud overlap, recoil, and gravity-gradient corrections must come from calibration data. Selection includes atom-number thresholds, fringe-contrast cuts, pulse-lock failures, vibration range, and accepted launch states; report lost shots and whether thresholds were set blind. Shared laser and vibration channels generate cross-species covariance, so quadrature addition of independent phase errors is generally wrong.

An independent check injects blinded acceleration and wavefront terms into real sampling or a realistic simulation, then applies wave-vector reversal, species interchange, and at least three interrogation times. The recovered gravity term must follow $k_{\rm eff}gT^2$, while gravity-gradient and wavefront injections follow their separately predicted changes. Null ports and deliberately mismatched scale factors test vibration leakage.

Evidence and interpretation at 2026-08-19

Position or test	Evidence/status at 2026-08-19	Limitation or counterevidence
Dual-species inertial WEP test	Published simultaneous isotope interferometers and the Asenbaum et al. null comparison support a differential-acceleration interpretation	Composition contrast is narrow; wavefront, initial kinematics, recoil, and shared-laser covariance limit generalization
Single-internal-state “Compton clock” redshift reading	Some EEP-violating frameworks map atom and conventional-clock observables to common coefficients	In broad quadratic-Lagrangian treatments the propagation action difference cancels, leaving a WEP-sensitive laser phase; there is no independent clock oscillation at the Compton frequency
Quantum-clock interferometry with distinct internal states	Controlled proposals and developing experiments can compare proper-time-sensitive internal phases	Sensitivity depends on state preparation, closure, recoil, and theory mapping; a general experimental verdict remains open

The apparent dispute is resolved by stating the model. Wolf et al. demonstrate the cancellation of the propagation action difference for a closed, symmetric light-pulse interferometer described consistently by one quadratic Lagrangian, with matched pulse intervals and recoil conditions. In that setting the gravimeter measures WEP-sensitive acceleration rather than an independent universal clock rate Wolf et al. 2011, Eqs. (4)–(13), PDF. The closure and internal-state conditions matter; “quadratic” alone does not characterize every atom interferometer. Hohensee et al. show that within a specified gravitational Standard-Model Extension mapping, matter-wave, clock, and torsion-balance observables constrain related EEP-violating coefficients Hohensee et al. 2011, Eqs. (1)–(8). These are conditional statements, not equal evidence for incompatible descriptions of one model.

No published null establishes that every quantum system obeys every sense of equivalence. The strongest conservative conclusion is a constraint on composition-dependent free fall for the tested states and apparatus. An alternative interpretation is warranted only after its action, laser coupling, internal-state clock, and parameter map are specified.

Independent discriminator. Operate an interferometer with a resolved internal-state clock superposition, close both output paths, reverse $\mathbf k_{\rm eff}$, and vary $T$. A phase that follows internal energy and proper-time separation after the ordinary acceleration phase is removed would distinguish clock universality from a pure center-of-mass WEP measurement.

What could change the conclusion. The interpretation could change after a public calibrated shot-level release, a dual-species result with materially different composition charge, a closed quantum-clock interferometer measurement, or a theory calculation that changes the phase decomposition while passing wave-vector, species, and interrogation-time controls.

Exercise

Two species have phases $\phi_A=k_A(g_A+a_v)T^2$ and $\phi_B=k_B(g_B+a_v)T^2$, where the same mirror acceleration $a_v$ drives both interferometers but $k_A\ne k_B$. Why does simply subtracting phases leave vibration, and what combination removes it in this idealization?

Solution

Direct subtraction gives $[k_Ag_A-k_Bg_B+(k_A-k_B)a_v]T^2$, so unequal scale factors leave a vibration term. Normalize the second phase by $\kappa=k_A/k_B$ and form $\phi_A-\kappa\phi_B$. The result is $k_AT^2(g_A-g_B)$: the common $a_v$ cancels in this idealization because it has the same transfer function apart from the scale factor. Real pulse timing and transfer-function differences can spoil that cancellation, which is why deliberate scale-factor mismatch is a useful control.

References

• Asenbaum, Peter, Chris Overstreet, Minjeong Kim, Joseph Curti, and Mark A. Kasevich. “Atom-Interferometric Test of the Equivalence Principle at the $10^{-12}$ Level.” Physical Review Letters 125 (2020): 191101. DOI; Open PDF.
• Hohensee, Michael A., Steven Chu, Achim Peters, and Holger Müller. “Equivalence Principle and Gravitational Redshift.” Physical Review Letters 106 (2011): 151102. DOI.
• Overstreet, Chris, et al. “Effective Inertial Frame in an Atom Interferometric Test of the Equivalence Principle.” Physical Review Letters 120 (2018): 183604. DOI.
• Wolf, Peter, Luc Blanchet, Christian J. Bordé, Serge Reynaud, Christophe Salomon, and Claude Cohen-Tannoudji. “Atom Interferometry and the Einstein Equivalence Principle.” In Experimental Gravity and Gravitational Waves, proceedings of the 46th Rencontres de Moriond and GPhyS Colloquium. Thé Gioi (2011): 327–331. arXiv:1109.6247; Open PDF.