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Experimental Test of Quantum Theory Using Atom Optics

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Experimental Test of Quantum Theory Using Atom Optics 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.
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