**The superluminal group velocity difference of neutrinos that was NOT measured by Opera**

I take advantage of the Physics Nobel Prize 2015 rewarding the discovery of neutrino oscillations to shine a different light on a famous experiment that made big headlines in september 2011 (and gave me opportunity to start this blog!)

we discuss the possibility that the apparent superluminality is a quantum interference effect, that can be interpreted as a weak measurement[2, 3, 4, 5]. Although the available numbers strongly indicate that this explanation is not correct, we consider the idea worth exploring and reporting – also because it might suggest interesting experiments, for example on electron neutrinos, about which relatively little is known. Similar suggestions, though not interpreted as a weak measurement [6, 7] or not accompanied by numerical estimates [6, 8], have been proposed independently.

The idea, following analogous theory and experiment [9] involving light in a birefringent optical fibre, is based on the fact that the vacuum is birefringent for neutrinos. We consider the initial choice of neutrino flavour as a preselected polarization state, together with a spatially localized initial wavepacket. Since a given flavour is a superposition of mass eigenstates, which travel at different speeds, the polarization state will change during propagation, evolving into a superposition of flavours. The detection procedure postselects a polarization state, and this distorts the wavepacket and can shift its centre of mass from that expected from the mean of the neutrino velocities corresponding to the different masses. This shift can be large enough to correspond to an apparent superluminal velocity(though not one that violates relativistic causality: it cannot be employed to send signals). Large shifts, corresponding to states arriving at the detector that are nearly orthogonal to the polarization being detected, are precisely of the type considered in weak measurement theory.

It seems that only muon and tau neutrino flavours are involved in the experiment... The initial beam, with ultrarelativistic central momentum p, is almost pure muon, which can be represented as a superposition, with mixing angle θ, of mass states |+> and |- >, with m_{+ }> m_{- }...The two mass states evolve with different phases and group velocities neglecting the spreading and distortion [10] of the individual packets– both negligible in the present case. E_{±}and v_{±}are the energies and group velocities of the two mass states, and we write E_{±}=E±1/2ΔE, v_{±}=v±1/2Δv, x=vt+ξ... in which the new coordinate ξ measures deviation from the centre of the wavepacket expected by assuming it travels with the mean velocity.In the experiment, the detector postselects the muon flavour [1]... thus the shift in the measured final position of the wavepacket [can be interpreted]... as an effective velocity shift, that is

[where the prefactor, tΔv is the relative shift of the two mass wavepackets, expected from the difference of their group velocities (it is small compared with the width of the packet ... in the neutrino case). The main factor represents the influence of the measurement-that is of the pre- and postselection and the evolution]...The possibility of superluminal velocity measurement arises because the amplification factor in (8) can be arbitrarily large if sin.^{2}2θ and sin^{2}(tΔE/2ℏ) are close to unity, corresponding to near-orthogonality of |pre> and |post>

For neutrinos with momentum p, ... the group velocity [difference Δv is given by -ΔE/p]. Thus Δv<0, so, in order for the apparent velocity to be superluminal, Δveff in (8) must be positive; this can be accommodated by making cos2θ negative.

Note also that v_{+}and v_{--}are less than c if both neutrino masses are nonzero, so the individual mass eigenstate wavepackets move with subluminal group velocities; any superluminal velocity arising from (8) is a consequence of pulse distortion ... associated with the postselection, i.e. considering only arriving muon neutrinos. In the more conventional superluminal wave scenario [10], group velocities faster than light, and the pulse distortions that enable them to occur, are associated with propagation of frequencies near resonance, for which there is absorption, i.e. non-unitary propagation. That is also true in the optical polarization experiments [9] and in the neutrino situation considered here, with the difference that the nonunitarity, which gives rise to the superluminal velocity, is not continuous during propagation but arises from the sudden projection onto the postselected state.

In the [Opera] experiment, the energies of the neutrinos varied over a wide range, with an average of cp = 28.1GeV. For the difference in the squared masses,with electron neutrinos neglected andm_{+}andm_{-}identified with the standard m_{2}and m_{3}, a measured value [13] is m_{+}^{2}c^{4}-m_{-}^{2}c^{4}≃2.43×10^{-3}eV^{2}.This gives

Δv/c=-1.5×10^{-24}. (16)

The apparent velocity measured in the experiment [1] was (1+2.5×10.^{-5})cComparison with the quantum velocity shift Δv_{eff}in (8) would require knowlege ofm_{+}andm_{-}, not just their squared difference, and the individual masses arenot known. But even on the most optimistic assumption, that m._{-}=0, it is immediately clear that it is unrealistic to imagine that the quantum amplification factor in (8) can bridge the gap of 19 orders of magnitude between (16) and the measured superluminal velocity

(Submitted on 13 Oct 2011 (v1), last revised 14 Nov 2011 (this version, v2))

Remark: for the anecdote the abstract of this article by the distinguished mathematical physicist Michael Berry and his collaborators might be the shortest one ever written since it answered laconically to the question asked in the title : "probably not". And time has proved that it was right...

**A superluminal group velocity of photons that was effectively measured**

While the theoretical prediction from the last paragraph has not been tested by the Opera experiment and will stay quite hard to test empirically given the smallness of the effect, the physics behind it is pretty sound and falsifiable in other contexts. I think the article below is a nice illustration:

The physics of light propagation is a very timely topic because of its relevance for both classical [1] and quantum [2] communication.Two kind of velocities are usually introduced to describe the propagation of a wave in a medium with dispersion ω( k): the phase velocity v_{ph}=ωk and the group velocity v_{g}=∂ω/∂k . Both of these velocities can exceed the speed of light in vacuum c in suitable cases [3]; hence, neither can describe the speed at which the information carried by a pulse propagates in the medium. Indeed, since the seminal work of Sommerfeld, extended and completed by Brillouin [4], it is known that information travels at the signal velocity, defined as the speed of the front of a square pulse. This velocity cannot exceed c [5]. The fact that no modification of the group velocity can increase the speed at which information is transmitted has been directly demonstrated in a recent experiment [6].Superluminal (or even negative) and, on the other extreme, exceedingly small group velocities, have been observed in several media [7]. In this letter we report observation of both superluminal and delayed pulse propagation in a tabletop experiment that involves only a highly birefringent optical fiber and other standard telecom devices.

Before describing our setup, it is useful to understand in some more detail the mechanism through which anomalous group velocities can be obtained. For a light pulse sharply peaked in frequency, the speed of the center-of-mass is the group velocityv_{g}of the medium for the central frequency [3]. In the absence of anomalous light propagation, the local refractive index of the medium isn_{f}, supposed independent on frequency for the region of interest. The free propagation simply yieldsv_{g}=L/t_{f}where L is the length of the medium andt_{f}=n_{f }L/c is the free propagation time. One way to allow fast- and slow-light amounts to modify the properties of the medium in such a way that it becomes opaque for all but the fastest (slowest) frequency components. The center-of-mass of the outgoing pulse appears then at a time t = t_{f}+<t>, with<t>the mean time of arrival once the free propagation has been subtracted; obviously<t><0 for fast-light,<t>>0 for slow-light. If the deformation of the pulse is weak, the group velocity is still the speed of the center-of-mass, now given by

v_{g}=Lt_{f}+<t>. (1)

This can become either very large and even negative (<t>→−∞) or very small (<t>→∞) — although in these limiting situations the pulse is usually strongly distorted, so that our reasoning breaks down.

(Submitted on 20 Jul 2004 (v1), last revised 10 Jan 2005 (this version, v2))