Observation of quantum free fall and the consistency with the equivalence principle

TL;DR

This work reports the first measurement of the quantum phase of a freely falling object using a novel quantum Galileo interferometer (QGI) that combines a stationary reference wave-packet with a ballistic one. By deriving and testing the gauge-phase transformation between Einsteinian (free-falling) and Newtonian (lab) frames, the authors predict a characteristic phase Δφ ∝ m g^2 T^3 with a prefactor (m g^2)/(3ħ) and validate it against analytical and numerical models in an atom-chip SG interferometer. The observed agreement supports applying the equivalence principle to quantum wave-packets at low energies and demonstrates a robust interface between quantum mechanics and gravity, with potential extensions to clocks, nanoparticles, and tests of quantum gravity concepts. Overall, the experiment provides a critical stepping stone toward deeper quantum-gravity synthesis and precision tests at the QM-GR boundary.

Abstract

The unification of quantum theory and the general theory of relativity - describing gravity, is one of the most important challenges in science. Einstein's general theory of relativity is based on the principle of equivalence, and has been confirmed to great accuracy for large bodies. However, in the quantum domain the equivalence principle has been predicted to take a unique form involving a gauge phase, equal to the quantum phase of a free-falling object. To measure this phase, we realize a novel cold-atom interferometer in which one wave-packet stays static in the laboratory frame while the other is in free fall. The observed relative-phase of the wave-packets confirms the predicted phase of a free-falling object, and shows that in our low energy regime, the equivalence principle may be applied to the quantum domain. Our observation constitutes a fundamental test of the interface between quantum theory and gravity. The new interferometer also opens the door for further probing of the latter interface, as well as to searches for new physics.

Figures (14)

• Figure 1: The QGI experiment to measure the phase accumulation of a free-falling particle. (A) Space-time diagram illustrating the classical trajectories of the WPs in the interferometer. A beam splitter creates a coherent superposition of two WPs with different spin states, launching one WP into a freely-falling ballistic trajectory for a duration of $2T$, while the second WP is held stationary, serving as a reference. A second beam splitter recombines the two WPs at the end of the ballistic trajectory. State-dependent detection (not shown) measures the population in each state, revealing the phase difference between the two trajectories. (B) A schematic of our experimental realization: In the longitudinal (1D) QGI interferometer, the stationary WP is held against gravity using magnetic gradients produced by currents in microfabricated wires on the atom chip. This WP, associated with state $|1\rangle \equiv|F=2,m_F=1\rangle$ of a $^{87}$Rb atom, is positioned $\approx 113\,\rm\mu m$ below the chip surface. The ballistic WP, associated with state $|0\rangle \equiv|F=1,m_F=0\rangle$, travels upwards reaching, in this example, a splitting of about $7.5\,\rm\mu m$ [about seven times the WP size (SM)]. Interchanging currents in the three central wires (shown in the figure) create a 2D quadrupole to realize a strong magnetic gradient with a small magnetic field, which minimizes phase noise. Finally, in order to measure the population in each spin state, another (reversed) magnetic gradient separates the two states in position so that they can be imaged by a CCD. See the Methods section for more details.
• Figure 2: Population in outport 1 vs. the free-fall duration of the ballistic path ${2T}$. (A) The relative spin population in outport 1 (state $|0\rangle$) oscillates with a distinct chirp as a function of interferometer time. The blue points represent the experimental data, with error bars derived from the standard error of the mean (SEM). The red line is a fit to the data of the form $P = P_{\text{mean}}(2T) + \frac{1}{2}V(2T)\cos\left[\phi(2T)\right]$, where $V$ is the visibility of the oscillations, $\phi$ is the phase of the oscillations and $P_{\text{mean}}$ is the mean value of the population. To obtain the fit, we first identify the upper and lower envelopes and fit them to a polynomial to get $P_{\text{mean}}$ and $V(2T)$. We then fit the data to the model where the phase $\phi(2T)$ is a third-order polynomial. During the fitting we exclude the first and last oscillation as they suffer from larger uncertainty in the values of the envelope. The data shows 13 complete oscillations, with low phase noise, resulting in an average population error (SEM) of $1.8 \pm 1.5$ [%] and good visibility, starting at $V=80\,\%$ and decreasing to $V=20\,\%$ at $2T = 2000 \, \mu\mathrm{s}$. The decrease in visibility is mainly due to a varying overlap between the WPs due to differences in shape evolution under the influence of curved gradients as well as imperfection in the path recombination. The figure includes $633$ experimental cycles, each $30$ s, so the graph was taken over a period of $5.3$ hours. The short-term and long-term stability of the interferometer are good due to the high stability of the chip trap, and current source. (B) Atomic distribution (heat map of optical density) adapted from CCD images (on-resonance absorption imaging), of the atoms in the two interferometer outports [see Fig. 1(B)], showing two oscillations. The top cloud is outport 1 (state $|0\rangle$), and the bottom cloud is outport 2 (state $|1\rangle$, which is magnetically sensitive). The cloud in outport 2 is focused by the final splitting pulse, increasing its optical density, and making it appear slightly denser even when the two clouds have the same population. The vertical scale of the image is 1.1 mm.
• Figure 3: Phase and its derivative vs. the free-fall duration of the ballistic path $2T$. (A) The phase difference between the two interferometer arms as a function of the free-fall duration of the upper arm. The red line represents the experimental data, where the phase is extracted from Fig. 2. To estimate the uncertainty bounds, we repeat the experiment with an increase (decrease) of the magnetic-pulse current $I_{kick}$ by $0.5\,\%$ (which is our estimated experimental precision) and use the result to plot the upper (lower) uncertainty bounds. The green points are the result of a numerical simulation based on tools we developed for calculating WP evolution japha2021, which include the effects of shape and rotation of the WP. The blue dashed line is the result of the analytical approximation of the experiment given in Eq. \ref{['eq:phase_diff']}. The analytical model assumes a homogeneous magnetic gradient, and a set of square pulses of current so that the acceleration of each arm is a piecewise constant function. The black dashed line depicts the phase of the Gedanken experiment. The inset shows the derivative of the phase with respect to $2T$. To obtain the derivative of the phase for the numeric simulation, we fit a third-order polynomial to the set of points in the main figure and take its derivative. The derivative of the analytical approximation and the Gedanken experiment are calculated by direct differentiation. The yellow dashed lines represent 5% deviations from the $T^3$ phase accumulation model, by modifying the analytical prediction to a power law of $T^{3\pm0.15}$. The thin blue dashed lines represent the analytical approximation with 5% deviations from the theoretical prefactor (note that the prefactor in our experiment is $mg^2/3\hbar$, as explained in Eq. \ref{['eq:ballistic']}). This is meant to show that the data excludes such deviations from the cubic phase or the predicted prefactor. (B) The residuals between the phase of the data and the numerical simulation, where the latter is a "blind" simulation with no free parameters or fine tuning. A difference of 2 rad over 13 oscillations spanning a phase of $\approx 80$ rad, constitutes a deviation of about 2.5%. The error bars represent the statistical noise of the phase in the experiment, which shows a relative noise of 1.3% of the phase. The red band represents the systematic uncertainties [which also appear in (A)]. The blue line shows the result of the numeric simulation after fine-tuning of the currents and the magnetic field along the z-axis. The kick current is increased by 0.15% , the idle current is changed from 0.55 mA to 0.5 mA, and $B_z$ from 0.33 G to 0.35 G, all within the experimental uncertainties. (C) The residuals between the phase of the data and the analytical approximation. The 5% lines allow one to put an upper limit of a few percent on the possible deviation from the predicted prefactor.
• Figure 4: The experimental scheme and trajectories of the WPs in the experiment. (A) The complete experimental scheme of our longitudinal (1D) QGI, for the case of $2T = 2.4\,\rm ms$. The x-axis is time, where we set $t=0$ at the moment of trap release. At this time the atoms are in the state $|2\rangle \equiv|F=2,m_F=2\rangle$. We apply Delta-kick cooling (DKC) at $t=1.1\,\rm ms$, which collimates the WP's expansion and launches the atoms into a ballistic trajectory upwards. Before the start of the interferometer, we transfer the atoms to the state $|F=2,m_F=1\rangle$ by applying an on-resonance RF $\pi$ pulse (grey wavy line). During the interferometer, we control the spin state of the atoms with four MW pulses (orange wavy lines) and the momentum of the atoms with three magnetic gradient pulses (green areas). The MW pulses are on resonance with the transition between the $|1\rangle \equiv|F=2,m_F=1\rangle$ and $|0\rangle \equiv|F=1,m_F=0\rangle$ states, where the $|0\rangle$ state is magnetically non-sensitive in the Zeeman first order. The first $\pi/2$ pulse puts the atoms in an equal superposition of $|1\rangle + |0\rangle$. At the end of the interferometer, the trajectories of these two states are joined to a single trajectory with two spin states, after which the second $\pi/2$ pulse is applied, creating four overlapping WPs which interfere, two WPs per each spin state, imprinting the phase difference into a population difference (for simplicity of graphics, we have not differentiated the four WPs in the plot). After the second $\pi/2$ pulse, we detect the relative population in each state. In the detection scheme, we first spatially separate the states by applying a fourth magnetic gradient and, finally, image the atoms using on-resonance absorption imaging. The symmetric temporal positioning of the $\pi/2$ and $\pi$ pulses also serves as a dynamic-decoupling scheme, increasing the coherence time, canceling the linear phase accumulation due to possible detuning of the MW and reducing the shot-to-shot phase fluctuations. (B) Zoom-in on the interferometer sequence. The first gradient (kick) pulse applies a maximal acceleration $a_{kick}$ for a total duration of $T_{kick}$, which launches the top arm into a ballistic trajectory. $50\,\rm \mu s$ after the end of the gradient pulse, we apply a $\pi$ pulse (with a duration of $16\,\rm \mu s$) that inverts the spin state of the arms so that the bottom arm is now in the magnetically-sensitive state $|1\rangle$. $5\rm\,\mu s$ after the $\pi$ pulse, we start the holding pulse (duration of $T_h$ and rise time $\tau_h = 12\,\rm \mu s$), which applies an acceleration opposite and equal to gravity, to hold the bottom arm stationary in the laboratory frame. $50\,\rm \mu s$ after the holding pulse, we again apply a $\pi$ pulse and $5\rm\,\mu s$ later we recombine the trajectories of the two WPs by applying a magnetic gradient pulse with a maximal acceleration $a_{kick}$ for a total duration of $T_{kick}$, as in the first pulse.
• Figure 5: The trajectories of the WPs for different holding durations. The trajectories of the two arms of the interferometer, calculated by the numerical simulation for holding durations $T_h = 232, 742, 1252, 1762, 2272 \,\rm \mu$s. Dashed blue lines represent the ballistic trajectory, and solid red lines the static WP. At the end of the interferometer, a small spatial splitting can be seen, on the order of $0.1 \,\rm \mu m$. For $T_h = 2272 \,\rm \mu$s, the spatial splitting between the two arms halfway through the interferometer reaches $7.5 \,\rm \mu$m. The trajectories are plotted until the time in which the final (second) $\pi/2$ is applied.