Do(es the Influence of) Empty Waves Survive in Configuration Space?

Abstract

The de Broglie–Bohm interpretation is a no-collapse interpretation, which implies that we are in principle surrounded by empty waves generated by all particles of the universe, empty waves that will never collapse. It is common to establish an analogy between these pilot-waves and 3D radio-waves, which are nearly devoided of energy but carry nevertheless information to which we may have access after an amplification process. Here we show that this analogy is limited: if we consider empty waves in configuration space, an effective collapse occurs when a detector clicks and the 3ND empty wave associated to a particle may not influence another particle (even if these two particles are identical, e.g. bosons as in the example considered here).

Appendices

Appendix 1: Testing Phase Coherence Between the Signal and Idler Waves Experimentally

Fig. 5

Simplified scheme for detecting empty waves

Let us consider the scheme presented in Fig. 5. It consists in suppressing the MZ part in the full device in Fig. 2. In other words, we should just keep the HOM device in the green frame of Fig. 2 plus detectors 3 and 4 aimed at generating empty waves. If no 3D empty wave is present, the value of $\Delta \theta$ is irrelevant because it plays the role of a global phase which is like nothing in the standard quantum theory as is well-known. This is obvious if we note that the wave function of the pair of photons, just before entering the beamsplitters connecting to the detectors 3 and 4 then reads

${1\over \sqrt{2}}e^{i\delta \theta }(\Psi ^{idler}(x_1,t)\Psi ^{signal}(y_2,t)+\Psi ^{idler}(x_2,t)\Psi ^{signal}(y_1,t))$.

There is thus no way to estimate the value of $\Delta \theta$ in NCI, and the same results can be shown to hold in 6D CI à la Bohm. If there is a click in detector 3 xor 4, the probability that detector 1 fires is equal to the probability that detector 2 fires, whichever the value of $\Delta \theta$ could be. We predict the same behaviour if we describe this situation in the framework of 3D CI à la de Broglie in absence of phase coherence between the idler and signal pulses. If some coherence (even partial coherence) is present however, the presence of empty waves results into a (partial) interference pattern when $\Delta \theta$ gets varied from 0 to $2\pi$.

In particular, if $\Delta \theta$ is equal to $\pm \pi /2$, in the case of maximal coherence, the surviving photon will always leave the beamsplitter along the same arm, as we noted previously. In this case, it is even not necessary to recombine the two channels at the outputs of the first beamsplitter as done in Fig. 2 in order to reveal the existence of empty waves. As was noted by Croca et al. [1] however, the visibility of the interference pattern is zero whenever the two photon pulses are incoherent in phase, in which case we recover the predictions made in the framework of NCI, and the scheme proposed here is useless. It is however worth doing the experiment outlined in Fig. 5 in order to estimate the degree of coherence between the signal and idler pulses to begin with. If coherence is minimal (full incoherence) it is not conclusive but otherwise this would in itself constitute a crucial experiment. In a sense, 3D empty waves, if they exist and that they exhibit some coherence, would also make it possible then to measure a global phase, which is impossible mission in standard quantum mechanics¹⁴.

Appendix 2: Replacing the PDC Source by a Laser Source

As explained in Sect. 2.3, ... in many situations (where thermal and laser light sources are used for instance) Maxwell’s theory suffices to render account of the observations... This emphasises the role played by the optical source in these experiments. If we send for instance a Fock state $\mid E_N>$ at the input of a beamsplitter then at its outputs we get [44] the state

$\sum _{M=0}^N(\,\!\mid t\!\mid ^2)^{N-M}(\,\!\mid r\!\mid ^2)^{M}\sqrt{{N !\over M!\cdot (N-M) !}}\!\mid E^{transmitted}_{N-M}>\, \!\mid E^{reflected}_{M}>$, where t and r represent the amplitudes of transmission and reflection. Moreover, the N-photons wave function associated to a Fock state can be shown to factorize into products of identical one photon wave functions as shown by us in the Sect. 2.3.

If a coherent state $\mid \alpha >_{coh.}\equiv e^{-\mid \alpha \vert ^2/2}\sum _{N=0}^\infty {\alpha ^N\over \sqrt{N!}}\mid E_N>$ enters the beamsplitter, then the output state obeys

$$\begin{aligned}{} & {} \mid \Psi>^{out}=e^{-\!\mid \alpha \!\mid ^2/2}\sum _{N=0}^\infty {\alpha ^N\over \sqrt{N!}}\sum _{M=0}^N(\mid t\!\mid ^2)^{N-M}( \mid r\!\mid ^2)^{M}\\{} & {} \qquad \sqrt{{N !\over M!\cdot (N-M) !}}\!\mid E^{transmitted}_{N-M}>\,\!\mid E^{reflected}_{M}>\\{} & {} \quad =e^{-\mid \alpha \mid ^2(\mid t\mid ^2+\mid r\mid ^2)/2}\sum _{N=0}^\infty {\alpha ^M\alpha ^{N-M}\over \sqrt{N!}}\sum _{M=0}^N(\mid t\!\mid ^2)^{N-M}(\mid r\!\mid ^2)^{M}\\{} & {} \qquad \sqrt{{N !\over M!\cdot (N-M) !}}\mid E^{transmitted}_{N-M}>\, \mid E^{reflected}_{M}>\\{} & {} \quad =\left( e^{-\mid \alpha \mid ^2\mid t\mid ^2/2}\sum _{N-M=0}^\infty (\alpha \mid t\mid ^2)^{N-M}\sqrt{{1\over (N-M) !}}\mid E^{transmitted}_{M-N}>\right) \\{} & {} \qquad \left( e^{-\mid \alpha \mid ^2\mid r\mid ^2/2}\sum ^\infty _{M=0}(\alpha \mid r\mid ^2)^{M}\sqrt{{1\over M!}}\mid E^{reflected}_{M}>\right) \\{} & {} \quad =\ \mid \alpha ^{in}\cdot t>_{coh.}^{Transmitted}\cdot \mid \alpha ^{in}\cdot r>_{coh.}^{Reflected}, \end{aligned}$$

which is the product of a coherent transmitted state with a coherent reflected state.

If we use a laser source coupled to a beamsplitter (as plotted in Fig. 6) instead of a source of two equivalent photons (e.g. a crystal where a single photon of the pump gets replaced, during a Parametric Down Conversion process, by two equivalent photons, the signal and idler photons), the wave function $\ \Psi ^{out}$ thus obeys, instead of equation (19), the following equation:

$$\begin{aligned} \Psi ^{out}=\mid \alpha ^1>_{coh.}\mid \alpha ^2>_{coh.}\mid \alpha ^3>_{coh.}\mid \alpha ^4>_{coh.} \end{aligned}$$

(21)

where the source produces a coherent state

$$\begin{aligned} \mid \alpha>_{coh.}\equiv e^{-\mid \alpha \mid ^2/2}\sum _{N=0}^\infty {\alpha ^N\over \sqrt{N!}}\vert N>, \end{aligned}$$

while the complex amplitudes assigned to the coherent states present in outcome channels 1,2,3,4 are equal (up to a global phase) to

$$\begin{aligned}{} & {} \alpha _1={i\alpha \over 4}(-1+e^{i\Delta \theta }+e^{i\Delta \phi }+e^{i(\Delta \theta +\Delta \phi }) \text { (detector 1)},\\{} & {} \alpha _2={i\alpha \over 4}(-1+e^{i\Delta \theta }-e^{i\Delta \phi }-e^{i(\Delta \theta +\Delta \phi }) \text { (detector 2)},\\{} & {} \alpha _3=i\alpha /2 \text { (detector 3)},\\{} & {} \alpha _4=\alpha /2 \text { (detector 4)}. \end{aligned}$$

Fig. 6

Replacing the PDC source by a laser source

This distribution is the same that would be obtained in the framework of Maxwell’s theory, and in the framework of NCI à la de Broglie and à la Bohm as well. It is also fully consitent with a CI because the state of light is factorisable (it is a product of coherent states localized at the level of each of the 4 outcome channels/detectors). Therefore the clicks at the levels of detectors 1,2,3,4 are uncorrelated (statistically independent). This explains why no crucial experiment aimed at discriminating between these various interpretational schemes is conceivable when the source is a laser state. These results could appear to be useful however in case they are used to calibrate the device described in Fig. 2.