A Proposed Experimental Test of Quantum Theory using the Techniques of Atom Optics
Peter J. Riggs Department of Quantum Science, Australian National University, Canberra, ACT 2601, Australia
It may be possible to empirically discriminate between the predictions of Orthodox Quantum Theory and the deBroglie-Bohm Theory of Quantum Mechanics. An experiment using the measurement methods of Atom Optics is suggested in which an atom trap having evanescent light ‗mirrors‘ in front of each of its walls is used to determine whether trapped atoms are in motion inside the trap. An absence of detected motion would be contrary to the prediction of Orthodox Quantum Theory and would support the deBroglie-Bohm Theory.
1. Introduction conduct of such an experiment has had to wait until a sufficient level of advancement The interpretation of the quantum in measurement science has been reached – mechanical formalism has a controversial this has now been achieved (e.g. see: [12– history dating back to the theory‘s initial 19]). formulation and remains unresolved to this day. In Orthodox Quantum Theory (OQT), 2. Different Predictions no physical reality is attributed to matter waves (as described by wavefunctions) [1– In order to devise an experiment that 3]. Yet, convincing evidence for the would empirically discriminate these two objective existence of matter waves has theories, there must be at least one been steadily mounting over the last two situation where they make different decades (e.g. [4–9]). In contrast, the predictions. Contrary to a common belief, deBroglie-Bohm (deBB) Theory of it is not correct that deBB Theory produces Quantum Mechanics (also known as the exactly the same theoretical predictions to Causal Theory of Quantum Mechanics) OQT in every conceivable circumstance. postulates the existence of matter waves The ‗infinite‘ potential well is one case whilst making the same statistical where predictions differ in the two predictions as does OQT [10, 11]. Given theories. Consider a neutral, spinless the accumulating evidence in favour of the particle trapped in a rectangular box of reality of matter waves, it would be a side lengths Lx, Ly, Lz, with zero potential highly significant breakthrough in quantum inside and ‗infinite‘ potential outside (i.e. a physics if an experimental test could be three-dimensional ‗infinite‘ well). Taking conducted that discriminates between OQT one corner of this well as the origin of a and deBB Theory. A proposal is presented Cartesian coordinate system we find that a here for such an experimental test particle of mass m inside the well has the employing Atom Optics. Similar to the following stationary state wavefunction: case of the testing of Bell‘s Theorem, the
(8 L L L )1/2 sin (n x L ) sin (n y L ) sin (n z L ) exp (–iEt/ℏ) ... (1) nxnynz / x y z x / x y / y z / z where n , n , n = 1, 2, 3, ... , E = E = [(n 2 L 2) + (n 2 L 2) + (n 2 L 2)] (2ℏ2 2m) x y z nxnynz x / x y / y z / z / is the particle‘s energy, t is time, and the integrating the momentum probability other symbols have their usual meanings. density (p)2 over a specified range in According to OQT, the particle inside momentum space, i.e. P(p) = the well is in motion as the ground state (p) 2 dp dp dp where p , p , p are (kinetic) energy is non-zero. If the particle x y z x y z were not in motion then its momentum the Cartesian components of the would be zero, contrary to the Uncertainty momentum p, and h (p) is the Fourier Principle (as understood in OQT) [20–23]. integral transform of with h being Since the particle in the well is not in an Planck‘s constant. The deBB Theory eigenstate of momentum [24], OQT allows for the possibility where quantum requires measurements of the particle‘s equilibrium has not been established, i.e. momentum to result in values that occur where the Born Statistical Postulate (and with probabilities given by applying the therefore the standard quantum probability Born Statistical Postulate. Then the actual density) does not hold [25, 26]. momentum values found on measurement In the three-dimensional well described will occur with a frequency that closely above, the momentum probability density approaches the probability P(p) of finding is [27]: the particle with its momentum in a given range. This probability is calculated by
2 2 n 2 2 2 (p) = {(2nx Lx/ℏ) [1 – (–1) x cos (px Lx/ℏ)] / [(nx) – (px Lx/ℏ) ] } 2 ny 2 2 2 {(2ny Ly/ℏ) [1 – (–1) cos (py Ly/ℏ)] / [(ny) – (py Ly/ℏ) ] } 2 nz 2 2 2 {(2nz Lz/ℏ) [1 – (–1) cos (pz Lz/ℏ)] / [(nz) – (pz Lz/ℏ) ] } ... (2)
OQT predicts that various values of momentum will be measured with for a zero spin quantum particle [28, 29]. probabilities calculated by integrating Inside the well, the particle is at rest Eq. (2) but the particle cannot be found to because all of the particle‘s energy has have zero momentum. become stored in the standing matter wave In deBB Theory, the configuration [30, 31]. The particle being at rest can be wavefunction of a quantum system seen by applying Eq. (3) to the S function provides a mathematical description of its in Eq. (1), then p = S = (– Et) = 0, i.e. (physically real) matter wave. The the particle has zero momentum for all wavefunction is expressed in polar form: values of nx, ny, and nz. This result can also R exp (iS/ℏ), which is a natural hold when the well contains many expression representing the amplitude and particles. Since deBB Theory makes a phase of the matter wave, where R and S different prediction to OQT for the are real-valued functions of the space and ‗infinite‘ potential well, the possibility of time coordinates. The momentum p of a conducting an empirical test is opened up. single particle is given by:
p = S … (3)
3. Reflection of Atoms from Evanescent linearly polarized laser light is incident (at Wave ‘Mirrors’ greater than the critical angle) on an interface between transparent mediums of This section will briefly review the higher to lower refractive indices, the laser relevant aspects of atom optics. When beam undergoes total internal reflection. 2
An evanescent light wave is then generated is the value of the potential at the surface, on the lower refractive index side of the and (1/) is its decay length. In the case interface surface [32], which decays of a glass-vacuum interface, ½ exponentially with distance from the = (2π/λ) (n2 sin2θ − 1) with λ being the surface. The potential of the electric field incident laser wavelength, n is the index of of this evanescent wave has the form refraction of the glass, and θ is the angle of Uev(z) = Uo exp (−2z, where z is the incidence [33, 34] (as shown in Figure 1). direction perpendicular to the surface, Uo
Figure 1. An evanescent wave is formed on the vacuum side of an interface surface when θ > critical angle.
The decay length depends on the intensity low energy entering the evanescent wave of the incident laser beam (IL) and the will be repelled without them reaching the angle θ [35, 36]. If the incident laser is interface surface [43–45]. The matter wave transverse electric (TE) polarized then the itself will extend to the surface. If an generated evanescent wave will have atom‘s speed is sufficiently low, it will be intensity I given by [37, 38]: reflected elastically [46–49] and, for ev normal incidence, its (vector) momentum
will become oppositely directed to what it I = (4n cos2θ) I (n2 − 1) ev L was before reflection. Under these circumstances, the evanescent wave acts as Suppose we have atoms that are two- an effective ‗infinite‘ potential barrier to level systems with resonance frequency ωA the atoms [50, 51]. The reflection of ultra- (determined by an atom‘s energy level cold neutral atoms by this method is spacing), natural linewidth of atomic experimentally well established [52, 53]. transition Γ, and saturation intensity Isat Evanescent wave ‗mirrors‘ may be used to construct a practical version of a three- (= 2 Γc , where c is the speed of light in vacuum) [39]. Also assume a regime of dimensional ‗infinite‘ potential well which coherent atom optics in which spontaneous will have desirable measurement emission is negligible (achieved when the advantages. detuning = (ωL − ωA) is large, i.e. when 4. Proposed Test || ≫ Γ) [40–42], where ωL is the frequency of the incident laser beam. In this section, an outline of an When > 0 (denoted ‗blue detuning‘), the experimental proposal to test OQT is interaction energy is positive and atoms of presented which was originally suggested 2
some years ago [54] and is now feasible. (In practice, the bottom of a super-polished The proposal is to make measurements prism would form each of the walls.) inside a specially constructed atom trap (an Consider first the case where the trap effective three-dimensional ‗infinite‘ contains many atoms. Imagine that a dilute potential well). The application of standard ‗cloud‘ of neutral, spinless atoms is placed measurement methods inside particle traps, in the centre of such a trap with the state e.g. removing part of a trap‘s wall to preparation being an energy eigenstate. facilitate measurement, introduce Prior to being placed in the trap, the atoms disturbances to the quantum system within would need to be ultra-cooled to put them [55–58]. Such disturbances limit the into their (preferably) lowest energy state, information on momentum that can be to have a deBroglie wavelength of the obtained to the distribution given by the order of the trap‘s dimensions (so that Born Statistical Postulate. These wave effects dominate) [62], and to have a restrictions have previously ensured that low enough speed to guarantee reflection OQT and deBB Theory were empirically by the evanescent light waves. When the equivalent. The proposed experiment will atoms are in position, the lasers that avoid the limitations inherent in standard generate each evanescent wave are measurement methods by using atom activated. If the atoms are in motion once optics techniques. set inside the trap, they will be reflected Information may be gained without when approaching close to any of its walls. introducing disturbances by reflecting The presence of atoms near to the trap‘s atoms from evanescent wave ‗mirrors‘ interior surfaces will result in small phase [59]. Following a suggestion by Cook & shifts in the reflected laser beams caused Hill and by Dowling & Gea-Banacloche, a by slight changes in the refractive index of suitable rectangular atom trap might be regions adjacent to the trap‘s walls [63, constructed [60, 61]. Laser light incident 64]. on the external walls of the trap totally Aspect et al. calculated that if TE internally reflects and generates evanescent polarized lasers are used then the phase waves on the inside of each of the walls. shift (φ) of a reflected laser beam due to atoms ‗rebounding‘ is given by [65]:
2 2 12 n cos p Isat
( ) 2 ( )( )( )in … (4) (n 1) M Iev where p is the magnitude of the maximum Phase shifts in the reflected laser beams momentum of the atoms, M is the atomic could be established by interferring each mass, ρin is the incident atomic density (i.e. beam with a reference laser [70, 71]. The the density when not close to the detection of phase shifts in one or more of evanescent waves), and the other symbols the reflected laser beams over the lifetime are as previously defined. The of the trap would indicate that the OQT measurement of a phase shift in any of the prediction that the atoms are in motion, is reflected laser beams constitutes, in the verified. The absence of phase shifts would first instance, a measurement of the atomic indicate that the deBB Theory prediction is density of the ‗cloud‘ of atoms being correct. Aspect et al. have suggested that reflected from the relevant trap wall [66]. such measurements would be feasible [72] Importantly, it is a quantum non- (see also the discussion on phase shift demolition measurement [67, 68] and no enhancement in Section 5). extra energy is imparted to the atoms [69]. The absence of any phase shifts is dependent on the standing matter wave 2
being undisturbed after the atoms are these values of momenta with their placed in the well, as disturbances to the different probabilities would also be stationary state would result in the atoms acceptable as evidence against OQT and in being accelerated [73]. However, favour of the deBB Theory. determining whether the system has been disturbed or not may not be practical for a 5. Experimental Issues ‗cloud‘ of atoms. The possibility that a disturbance to the stationary state of a The proposed experiment would be many-atom system may be unavoidable in technically challenging. The use of neutral practice suggests that the experiment atoms is necessitated by the requirements would be better conducted with just a of having spinless objects that do not single atom if measurements can be made interact with each other. Below are briefly sufficiently sensitive. mentioned a few of the practical The use of only a single atom would experimental issues: avoid some complications, such as random (i) Effect of Gravity collisions or the natural tendency for the The most obvious problem is the presence atomic momentum distribution to move of the gravitational force. Although gravity 2 over time to (p) (i.e. moving to will not affect the atom‘s motion along the quantum equilibrium) [74]. An isolated, length of an atom trap, it will affect trapped, one-particle quantum system is vertical motion. This would result in a non- achievable in practice [75–81]. Discussion standing matter wave. Unless some means of the conditions for detection of a single is devised to incorporate the effects of atom may be found in Courtois et al. gravity without affecting the stationary (1995) [82]. The absence of any phase shift state or otherwise prejudicing the would indicate a lack of detection of the measurements made, this experiment atom in motion, contrary to the OQT would need to be conducted in a free-fall prediction. environment. The single atom case also offers the (ii) Vibrations of the Trap Walls possibility of making individual Vibrations of the walls of the atom trap momentum measurements which would would likely affect the standing matter permit a determination of whether the wave and might possibly heat the atoms if system is undisturbed or not. Since Eq. (4) they approach too near to the walls. shows that the phase shift depends on the Vibrations may be avoided by acoustically, atomic momentum, measurement of the mechanically, and thermally isolating the phase shift would allow the momentum to apparatus and ultra-cooling it. The be determined from the experimental data. apparatus should also be shielded from If the OQT prediction is correct, then for external light or other radiation. multiple rebounds of a single atom, (iii) The van der Waal Interaction successive measurements of the phase shift The van der Waal interaction arises when in the reflected laser light should yield atoms closely approach a surface (< 100 values of momentum with probabilities nm). This attractive potential falls off with given by integrating Eq. (2) [83]. If the cube of the distance from the surface measurements of the phase shift and reduces the height of the potential consistently showed values of momentum barrier in front of each wall [84]. The van but with probabilities that differ from those der Waal interaction would need to be predicted by OQT, this would indicate that incorporated into a resultant potential [85, a disturbance to the standing matter wave 86] which is the sum of the evanescent proved unavoidable in practice and the wave and van der Waal potentials. The atom was thereby accelerated. However, resultant potential is still repulsive for 3
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