Abstract
Matterwave interferometry performed with massive objects elucidates their wave nature and thus tests the quantum superposition principle at large scales. Whereas standard quantum theory places no limit on particle size, alternative, yet untested theories—conceived to explain the apparent quantum to classical transition—forbid macroscopic superpositions. Here we propose an interferometer with a levitated, optically cooled and then freefalling silicon nanoparticle in the mass range of one million atomic mass units, delocalized over >150 nm. The scheme employs the nearfield Talbot effect with a single standingwave laser pulse as a phase grating. Our analysis, which accounts for all relevant sources of decoherence, indicates that this is a viable route towards macroscopic highmass superpositions using available technology.
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Introduction
Matterwave interference with particles of increasing size and mass is a natural and viable method for testing the validity of the quantum superposition principle at unprecedented macroscopic scales^{1,2,3,4}. Macroscopic path separations are nowadays routinely achieved in atom interferometry^{5,6,7}, and technological advances in the control of optomechanical systems^{8} promise that much more massive objects may be delocalized^{9,10,11,12,13}, albeit with spatial separations smaller than a single atom.
Recent proposals put forward nanoparticle interferometry^{14,15} in the mass range of 10^{6}–10^{9} AMU to surpass the mass records currently held by molecule diffraction experiments^{3,16}, while maintaining spatial separations large enough to be resolved by optical means. A first demonstration with molecular clusters^{15} is still far away from the mentioned highmass regime due to difficult experimental challenges, mainly concerning the source and detection. The realization of a proposed doubleslit scheme with silica nanospheres^{14} requires motional groundstate cooling, which is an equally challenging task.
Quite recently, optical feedback cooling has been demonstrated for 100nmsized particles^{17,18}, based on pioneering work that demonstrated the trapping of polystyrene and glass microspheres^{19}, trapping of viruses and bacteria^{20} and even of complete cells^{21} in solutions and high vacuum. Cavity cooling of particles of similar size was proposed^{22} and recently achieved^{23,24} in one dimension, with temperatures in the milliKelvin range. Although this is still far above the ground state of a typical 100kHz trap, we will argue that highmass interference can be realized experimentally with motional temperatures already achieved by optical cooling.
In this paper, we present a nearfield interference scheme for 10^{6} AMU particles. It is based on the singlesource Talbot effect^{25} due to a single optical phase grating, as opposed to the threegrating scenario in Talbot–Lau interference experiments^{3}. Optically trapped silicon nanospheres, feedbackstabilized to a thermal state of about 20 mK, provide a sufficiently coherent source. Individual particles are dropped and diffracted by a standing UV laser wave, such that interference of neighbouring diffraction orders produces a resonant nearfield fringe pattern. In order to record the interferogram, the nanospheres are deposited on a glass slide and their arrival positions are recorded via optical microscopy. We argue that the choice of silicon, due to its specific material characteristics, will yield reliable highmass interference, unaffected by environmental decoherence, in a setup that can be realized with presentday technology.
Results
Proposed experiment
The proposed scheme is sketched in Fig. 1. In the first stage of the experiment, a silicon particle is captured in an optical dipole trap by a lens system of numerical aperture 0.8 focusing a 1,550 nm laser to a waist of 860 nm (ref. 26); the interaction of nanoparticles with light is described further in Supplementary Note 1. The trapping light is collected and used to determine the position of the particle^{18}, which is feedback cooled over many trapping cycles to about T=20 mK of mean translational energy along the horizontal xaxis, implying a momentum uncertainty of about σ_{p}/m=1.2 cm s^{−1}. A laser power of 55 mW results in a trap frequency of ν_{M}=200 kHz and a position uncertainty σ_{x} <10 nm; see Supplementary Note 2. The trap thus serves as a nearly pointlike matterwave source for diffraction.
After feedback cooling, the particle is released from the trap and falls for t_{1}=160 ms before it is illuminated by a frequencytrippled Nd:YAG laser pulse at 355 nm with a pulse length of 10 ns and an energy E_{G}≤500 μJ. The pulse is retroreflected by a mirror to form a standingwave phase grating with period d=λ_{G}/2, which diffracts the particle by modulating the matterwave phase through the dipole interaction. The Talbot time, which sets the scale for nearfield interference^{3}, is thus given by t_{T}=md^{2}/h≈80 ms. The laser beam must be expanded such that the waist is larger than the uncertainty in position σ_{p}t_{1}/m≈2 mm accrued during free flight. Moreover, the orientation of the grating must be angularly stable to less than microradians to avoid blurring of the interferogram due to acceleration of the particle under gravity, and positionally stable to within 30 nm relative to the initial particle position; see Supplementary Note 3.
After the grating, the particle undergoes free fall for t_{2}=126 ms, forming an interference pattern when it arrives on the glass slide. The arrival position can be detected by absorption imaging with visible light. Fitting to the known pointspread function of the imaging system permits 100 nm positional accuracy^{27}; see Supplementary Note 4. The density pattern depicted in Fig. 2a is predicted to appear after many runs of the experiment. In the following, we discuss the theoretical description of the interference effect and the experimental constraints.
Theoretical model
Our starting point for evaluating the interference effect is the trapped thermal state of motion, a Gaussian mixture with standard deviations and . The particle will be illuminated by a uniform standingwave pulse oriented along the horizontal xaxis (see Fig. 1), so that the y and zmotion can be ignored.
The nearfield diffraction effect including all relevant decoherence mechanisms is best captured in a quantum phasespace description^{28}. For the present purposes it is most useful to work with the characteristic function representation χ(s, q), that is, the Fourier transform of the Wigner function^{29} of a given quantum state ρ. Here, we summarize the detailed derivation given in Supplementary Methods. The initial Gaussian state,
first evolves freely for a time t_{1}, χ_{1}(s, q)=χ_{0}(s−qt_{1}/m, q), before it is illuminated by the optical grating pulse of period d. Given an almost pointlike initial spread σ_{x}/d≪1, the matter waves must evolve for at least the Talbot time t_{T} to ensure that they are delocalized over adjacent grating nodes in order to be able to interfere. The initial momentum, on the other hand, is spread over many grating momenta, σ_{p}d/h>>1, so that the timeevolved state extends over many grating periods. That is, if particles are only detected in a finite detection window around the centre of the distribution in the end, we can neglect the Gaussian density profile by writing
The particle interacts with the standingwave pulse through its optical polarizability , determined by the particle radius R and its complex refractive index n_{Si} at the grating wavelength λ_{G}=2d. In the limit of short pulse durations τ, this imprints the phase φ(x)=φ_{0} cos^{2}(πx/d) on the matterwave state^{15}, where φ_{0}=2Re(α)E_{G}/ħcε_{0}a_{G} depends on the energy E_{G} and spot area a_{G} of the pulse. The characteristic function transforms as , where the B_{n} are Talbot coefficients, given in terms of Bessel functions^{30},
Incoherent effects due to absorption or scattering of laser photons are negligible for the nanoparticles considered here (Supplementary Methods); nevertheless, our numerical simulations include both effects.
The final density distribution w(x)=‹xρx›, that is, the probability to find the particle at position x after another free time evolution by t_{2}, then takes the form
It describes a periodic fringe pattern oscillating at the geometrically magnified grating period D=d(t_{1}+t_{2})/t_{1} (ref. 25). The fringe amplitudes, given by the Talbot coefficients (3), are diminished the larger the spread σ_{x} of the initial state (1).
An exemplary density pattern (4) is plotted in Fig. 2a for varying time t_{2}. The simulation was performed for 10^{6} AMU silicon particles, assuming realistic experimental parameters and including the influence of environmental decoherence. It shows pronounced interference fringes with visibilities of up to 75%.
The pattern in Fig. 2b is the result of a classical simulation assuming that the particles are moving on ballistic trajectories. A lensing effect due to the strong dipole forces exerted by the standingwave field is here responsible for the density modulation. This classical result is obtained simply by replacing sin πξ by πξ in expression (3) for the grating coefficients^{30}.
The clear difference between the quantum and the classical pattern is captured by the sinusoidal fringe visibility, the ratio between the amplitude and the offset of a sine curve of period D fitted to the density pattern (4):
As shown in Fig. 3, the classical and the quantum prediction differ significantly: the classical theory predicts many regions of low contrast as a function of φ_{0}, whereas the quantum prediction exhibits a slow φ_{0} dependence. The highest quantum visibility amounts to 83% at φ_{0}=1.4π.
Accounting for decoherence
A realistic assessment of the proposed scheme must also include the influence of collisional and thermal decoherence^{28}. This is incorporated into (4) by multiplying each Fourier component with a reduction factor of the form
where Γ gives the rate and f(x) determines the spatial resolution of decoherence events of a certain class. In our simulation we accounted for collisions with residual gas particles, scattering and absorption of blackbody photons, and thermal emission of radiation using a realistic microscopic description. Each process contributes another factor R_{n} listed in the Supplementary Methods; the rate of thermal emission depends on time since the particle loses internal energy and cools during flight.
Experimental constraints
As a major concern for the successful implementation of the experiment, environmental decoherence must be kept sufficiently low. According to our simulations, collisional decoherence can be essentially avoided at ultrahigh vacuum pressures of 10^{−10} mbar.
Radiative decoherence is suppressed by choosing silicon spheres because they are essentially transparent at typical wavelengths of room temperature blackbody radiation. The thermal emission of photons is determined by the internal temperature of the nanospheres, which is set in the trapping stage of the experiment. A trapping intensity of 90 mW μm^{−2} leads to an initial heating rate ∂_{t}T_{int}=200 K s^{−1} and an equilibrium temperature of 1,600 K. This high value is a consequence of the low blackbody emissivity of silicon^{31}, implying that the particle does not lose heat efficiently while in the trap. Nevertheless, due to the high refractive index n_{Si}=3.48 of silicon, the particle may be trapped for well in excess of a second before the temperature rises that high. This time corresponds to about 10^{5} trap oscillations, a sufficient period to perform parametric feedback cooling of the motion to T=20 mK; see Supplementary Note 5. The low emissivity of silicon is the essential advantage compared to other materials such as silica, for which much work in this field has been done^{10,11,23}. We find that to perform this experiment with silica would require cryogenic cooling of both apparatus and nanoparticle to 100 K, whereas thermal decoherence of silicon becomes important only at internal temperatures in excess of 1,000 K. Moreover, the high refractive index of silicon compared to the value n_{SiO2}=1.44 of silica means that less optical power is required to trap the sphere and to monitor its position^{24}.
As an additional advantage, silicon absorbs strongly at optical frequencies, which simplifies the detection of the interferogram. In principle, this would also affect the interaction with the grating laser, since a particle at the antinode of the grating absorbs on average n_{0}=0.12φ_{0} photons. For a grating laser waist of 30 mm we anticipate a phase modulation of φ_{0}/E_{G}=50 rad mJ^{−1} and hence we can access φ_{0}≤4π. The finite absorption of grating photons, which is included in the simulations, disturbs the interferogram little.
Discussion
We presented a viable scheme for highmass nanoparticle interferometry, which employs only a single optical diffraction element and requires only moderate motional cooling. The setup would operate in ultrahigh vacuum at room temperature. It is limited to masses up to 10^{6} AMU mainly by the growing Talbot time and freefall distance^{32}. Interferometry in a microgravity environment could pave the way to even higher masses^{33}.
Remarkably, with path separations of up to 150 nm and interrogation times of 300 ms, the presented scheme is already sensitive to alternative theories beyond the Schrödinger equation. The renowned collapse model of continuous spontaneous localization (CSL)^{34} could be probed in its current formulation^{4}. In fact, a successful demonstration of interference with a visibility exceeding 42% would bound the localization rate to λ_{CSL}<1.4 × 10^{−11} Hz, a value at the lower end of recent estimates for this parameter^{35,36}; see Supplementary Discussion. Such a superposition experiment can be associated with a macroscopicity value of μ=18 (ref. 37), substantially exceeding that of every presentday matterwave experiment and comparing well with the most ambitious micromirror superposition proposals^{9}.
Additional information
How to cite this article: Bateman, J. et al. Nearfield interferometry of a freefalling nanoparticle from a pointlike source. Nat. Commun. 5:4788 doi: 10.1038/ncomms5788 (2014).
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Acknowledgements
Funding by the EPSRC (EP/J014664/1), the Foundational Questions Institute (FQXi) through a Large Grant, and by the John F Templeton foundation (grant 39530) is gratefully acknowledged. This work was also partially supported by the European Commission within NANOQUESTFIT (No. 304886).
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H.U. conceived the interferometer and initiated the research. J.B. and S.N. designed the scheme and performed calculations and simulations. K.H. advised on the theory. All authors discussed the results and wrote the manuscript.
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Supplementary Figures 15, Supplementary Notes 15, Supplementary Discussion, Supplementary Methods and Supplementary References (PDF 935 kb)
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Bateman, J., Nimmrichter, S., Hornberger, K. et al. Nearfield interferometry of a freefalling nanoparticle from a pointlike source. Nat Commun 5, 4788 (2014). https://doi.org/10.1038/ncomms5788
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DOI: https://doi.org/10.1038/ncomms5788
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