Abstract
Spectroscopy is a method typically used to assess an unknown quantity of energy by means of a frequency measurement. In many problems, resonance techniques^{1,2} enable highprecision measurements, but the observables have generally been restricted to electromagnetic interactions. Here we report the application of resonance spectroscopy to gravity. In contrast to previous resonance methods, the quantum mechanical transition is driven by an oscillating field that does not directly couple an electromagnetic charge or moment to an electromagnetic field. Instead, we observe transitions between gravitational quantum states when the wave packet of an ultracold neutron couples to the modulation of a hard surface as the driving force. The experiments have the potential to test the equivalence principle^{3} and Newton’s gravity law at the micrometre scale^{4,5}.
Main
Generally, a quantum mechanical system that is described by two states can be understood in analogy to a spin1/2 system, where the time development is described by the Bloch equations, assuming two states of a fictitious spin in the multiplet, similar to spinup and spindown states. In magnetic resonance of a standard spin1/2 system, the energy splitting results in the precession of the related magnetic moment in the magnetic field. Transitions between the two states are driven by a transverse magnetic radio frequency field. Similar concepts can be applied to any driven twolevel system, for example in optical transitions with light fields. Variations are inherently connected to highprecision measurements such as atomic clocks^{6}, atom interferometry^{7}, nuclear magnetic resonance^{8}, quantum metrology^{9} and the related spinecho technique^{10}. The sensitivity reached so far^{11} in the search for the electric dipole moment of the neutron is 6.8×10^{−22} eV, or one Bohr rotation every six days.
In this Letter, we demonstrate that energy eigenstates in the gravity potential of the earth can be probed using a new resonancespectroscopy technique, using neutrons bounced off a horizontal mirror. This spectroscopy technique has in common the property that a quantumsystem is coupled to an external resonator. Quantum mechanical transitions with a characteristic energy exchange between the coupling and the energylevels are observed on resonance. A novelty of this work is the fact that the quantum mechanical transition is driven by an oscillating field that does not directly couple an electromagnetic charge or moment to an electromagnetic field. Instead, we observe energy transfer on resonance that is based on gravityquantum states coupled to a modulator. We have named this technique gravity resonance spectroscopy, because the energy difference between these states has a onetoone correspondence to the frequency of the modulator, in analogy to the nuclear magnetic resonance technique, where the energy splitting of a magnetic moment in an outer magnetic field is related to the frequency of a radiofrequency field. This is possible because of the feature of the quantum bouncing ball^{12,13} that the levels are not equidistant in energy. The linear gravity potential leads to measured^{14,15,16} discrete nonequidistant energy eigenstates n〉. A combination of any two states can therefore be treated as a twolevel system, as each transition can be addressed by its unique energy splitting or, in our case, by vibrating the mirror mechanically at the appropriate frequency. It has also been proposed to realize transitions between gravitational quantum states by means of oscillating magnetic gradient fields^{17}. The physics behind these transitions is related to earlier studies of energy transfer where matter waves bounce off a vibrating mirror^{18,19} or a timedependent crystal^{20,21}. In the latter case the transitions are between continuum states, in the quantum bouncer the transitions are between discrete eigenstates. Optical dipole traps of atoms are reviewed in ref. 22.
A measured energy level splitting ΔE can be compared with the prediction in the gravity potential, allowing a test of Newton’s law at short distances. The experiments are linked to current ideas in string theories with large volume compactifications^{4,5} and/or cosmology^{23,24}and can rule out or substantiate forces that are possible via Abelian gauge fields in the bulk^{4,5}, or via a pseudoscalar coupling of axions, which are serious dark matter candidates in the previously experimentally inaccessible astrophysical axion window. Limits from our previous experiments with ultracold neutrons can be found elsewhere^{25,26,27,28}. Owing to the nature of the Schrödinger equation, it also, in principle, provides a new test of the equivalence of inertial and gravitational mass^{3}.
In general, the gravityresonancespectroscopy method consists of the three steps shown in Fig. 1a. In region one the initial state p〉 is prepared by a state selector. In region two a socalled π pulse induces a transition into a second state q〉. In region three another state selector selects the state finally measured by the detector. In our experiments, all three regions are realized in a single device consisting of a polished mirror on the bottom and a rough scatterer on top (Figs 1b and 2). The scatterer suppresses the population of higher levels by scattering and absorption. It allows only the ground state to pass and prepares the state p〉. Transitions to level q〉 are induced by vibrating the whole system in the vertical direction. The population of level q〉 is again destroyed by the state selector, leaving the remaining population of p〉 to be detected. The lifetimelimiting interactions of the scatterer are described phenomenologically by adding decay terms to the equations of motion termed as damped oscillation (see Methods).
The mirror is made of polished optical glass and is mounted on a piezodriven vibration system. The experiment itself is mounted on a polished plane granite slab with active and passive antivibration tables underneath. The slab is levelled with a precision better than 1 μrad. A mumetal shield suppresses the coupling of residual fluctuations of the magnetic field to the magnetic moment of the neutron adequately. As in previous measurements^{14,15,16}, the neutrons are taken from the ultracold neutron installation PF2 at Institut LaueLangevin (ILL). We restrict the horizontal velocity to 5.7 m s^{−1}<v<7 m s^{−1}.
To test our state selector, we analysed the spatial neutron density distribution with a spatial resolution detector directly after the mirror and for a scatterer system without vibrations. The measurement agrees well with the quantum mechanical prediction^{16} for our geometric parameters. 57% of the neutrons were found in the ground state, 37% in the second state, and 6% in higher states. To optimize the contrast in later measurements, we chose levels 1〉 and 3〉 as states p〉 and q〉 respectively.
Within the qBounce^{12,13} experiment, we performed two resonancespectroscopy measurements with different geometric parameters (mirror length L and gap height h), resulting in different resonance frequencies and widths, as outlined in Fig. 2. In general, the oscillator frequency at resonance for a transition between states with energies E_{p} and E_{q} is:
Here, one has to take into account that because of the additional potential of the state selector, the energy eigenvalues depend on the gap height h, see Fig. 2. Although the first energy level stays practically unchanged, higher states are more strongly affected.
The transition probability p〉 to q〉 is given in the Methods section. The transfer is referred to as Rabi transition. In our experiments we chose states 1〉 and 3〉 as a twolevel system. In resonance, if the modulation frequency ω equals the transition frequency ω_{13}=ω_{3}−ω_{1}, the system is driven into a coherent superposition of state 1〉 and 3〉 and we can choose amplitude a in such a way that we have complete reversal of the state occupation.
The observable is the measured transmission left after the transition 1〉 to 3〉 as a function of the modulation frequency ω and amplitude a, see Fig. 3. During the time the neutron spends on the mirror, the acceleration on the mirror is practically constant and does not change significantly. We measure a with a noise and vibration analyser attached to the neutron mirror system. In addition, the positiondependent mirror vibrations were measured using a laserbased vibration analysis system. The piezosystem by itself does not show significant selfresonant behaviour, which might influence the neutron transmission in the frequency range considered.
For the first experiment, Fig. 3a shows the measured count rate as a function of ω. Blue (brown) data points correspond to measurements with moderate (high) vibration strength 1.5≤a≤4.0 m s^{−2}(4.9≤a≤7.7 m s^{−2}). The corresponding Rabi resonance curve was calculated using their mean vibration strength of 2.95 m s^{−2} (5.87 m s^{−2}). The black data point sums up all of the measurements at zero vibration. The grey band represents the one sigma uncertainty of all offresonant data points. The brown line is the quantum expectation as a function of oscillator frequency ω for Rabi transitions between state 1〉 and state 3〉 within an average time of flight τ=L/v=23 m s. The normalization for transmission T, frequency at resonance ω_{13} and global parameter f are the only fitting parameters. f is a multiplicator for the vibration amplitude and takes into account that the amplitude is not uniform over the whole surface of the mirror. The averaged effective amplitude acting on the neutron is somewhat smaller than the measured one and was found to be f=0.56±0.16. It is a global parameter, independent of the vibration frequency, and can therefore easily be corrected for. A sharp resonance was found at frequency ω_{13}=2π×(705±6) Hz, which is close to the frequency prediction of ω_{13}=2π×671 Hz, remembering that the gap height measurement has an uncertainty due to the roughness of the state selector. The fullwidth at halfmaximum is the prediction made on the basis of the time the neutrons spend in the modulator. The prediction was convoluted by the time of flight distribution function. The significance for 1〉→3〉 excitations is 3.5 standard deviations.
In the second measurement, the reduced length reduces the average flight time to τ=15 m s. The gap height h is 1.6 μm larger, thus changing the predicted resonant frequency to ω_{13}=615 Hz. We observe a resonance frequency ω_{13}=2π×(592±11) Hz, close to the prediction, and f=0.99±0.29. Figure 3b shows the combined result for measurements with both mirror length L=10 cm and L=15 cm. The transmission in units of the unperturbed system is displayed as a function of detuning.
In total, the significance for gravity resonance spectroscopy between state 1〉 and 3〉 at ω_{13} is 4.9 standard deviations. The left and right hand data points combine all offresonant measurements with (ω−ω_{13})/Δω≥1.5, where Δω is the fullwidth at halfmaximum. The statistical sensitivity of the measured energy difference between the gravity level 1〉 and 3〉 is 7.6×10^{−3}, corresponding to ΔE=2×10^{−14} eV. The limiting systematic uncertainty in this experiment is the roughness of the state selector (0.4 μm).
The new method profits from small systematic effects in such systems, mainly owing to the fact that, in contrast to atoms, the electrical polarizability of neutrons is extremely low. Neutrons are not disturbed by short range electric forces such as van der Waals or Casimir forces. Together with its neutrality, this provides the key to a sensitivity of several orders of magnitude below the strength of electromagnetism^{29}. The long term plan is to apply Ramsey’s method of separated oscillating fields^{30} with an expansion of the current setup, which works without any influence of the state selector on the energy analysis.
Methods
Rabi resonance spectroscopy^{1} measures the energy difference between a twolevel system p〉 and q〉 with a coupled oscillating field ω and damping γ. The wave function of the two level system is:
with the timevarying coefficients C_{p}(t) and C_{q}(t).
With the frequency difference ω_{pq} between the two states, the frequency ω of the driving field, the detuning δ ω=ω_{pq}−ω, the Rabi frequency Ω_{R} and the time t, the coupling between the timevarying coefficients is given by:
with a transformation into the rotating frame of reference:
Ω_{R} is a measure of the strength of the coupling between the two levels and is related to the vibration strength a via^{30}:
The probability of being found in the excited state as a function of time is:
where the effective Rabi frequency is:
Such oscillations are damped out and the damping rate depends on how strongly the system is coupled to the environment. In our case, the damping is caused by the state selector at height h above a mirror. A key point for the demonstration of this method is that it allows the detection of resonant transitions at a frequency that is tuned by the height of the state selector. The additional mirror potential shifts the energy of state 3〉 as a function of height, see Fig. 2. The absorption is described phenomenologically by adding decay terms γ_{p} and γ_{q} to the equations of motion:
The solution of this equation is given by:
The fit used in Fig. 3a contains three parameters, the resonant frequency ω_{pq}, the transmission normalization N, and vibration strength parameter f, which is multiplied by the measured acceleration. The damping as a function of Ω_{R} was measured separately, see Fig. 3c, as well as the width of the Rabi oscillation and background (0.0050±0.0002 s^{−1}).
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Acknowledgements
We gratefully acknowledge support from the Austrian Science Fund (FWF) under Contract No. I529N20 and the German Research Foundation (DFG) as part of the Priority Programme (SPP) 1491 ‘Precision experiments in particle and astrophysics with cold and ultracold neutrons’, the DFG Excellence Initiative ‘Origin of the Universe’, and DFG support under Contract No. Ab128/21. The neutron mirrors were characterized by SDH Sputterdünnschichttechnik, Heidelberg.
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Jenke, T., Geltenbort, P., Lemmel, H. et al. Realization of a gravityresonancespectroscopy technique. Nature Phys 7, 468–472 (2011). https://doi.org/10.1038/nphys1970
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DOI: https://doi.org/10.1038/nphys1970
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