Abstract:That the speed of light in free space is constant is a cornerstone of modern physics. However, light beams have finite transverse size, which leads to a modification of their wavevectors resulting in a change to their phase and group velocities. We study the group velocity of single photons by measuring a change in their arrival time that results from changing the beam's transverse spatial structure. Using time-correlated photon pairs we show a reduction of the group velocity of photons in both a Bessel beam and photons in a focused Gaussian beam. In both cases, the delay is several micrometers over a propagation distance of the order of 1 m. Our work highlights that, even in free space, the invariance of the speed of light only applies to plane waves. Main textThe speed of light is trivially given as / , where is the speed of light in free space and is the refractive index of the medium. In free space, where = 1, the speed of light is simply . We show that the introduction of transverse structure to the light beam reduces the group velocity by an amount depending upon the aperture of the optical system. The delay corresponding to this reduction in the group velocity can be greater than the optical wavelength and consequently should not be confused with the HÀ Gouy phase shift (1, 2). To emphasize that this effect is both a linear and intrinsic property of light, we measure the delay as a function of the transverse spatial structure of single photons.The slowing down of light that we observe in free space should also not be confused with slow, or indeed fast, light associated with propagation in highly nonlinear or structured materials (3,4). Even in the absence of a medium, the modification of the speed of light has previously been known. For example, within a hollow waveguide, the wavevector along the guide is reduced below the free-space value, leading to a phase velocity greater than . Within the hollow waveguide, the product of the phase and group velocities is given as , = 2 , thereby resulting in a group velocity , along the waveguide less than (5). 2Although this relation for group and phase velocities is derived for the case of a hollow waveguide, the waveguide material properties are irrelevant. It is the transverse spatial confinement of the field that leads to a modification of the axial component of the wavevector, . In general, for light of wavelength , the magnitude of the wavevector, 0 = 2 / , and its Cartesian components { , , } are related through (5)All optical modes of finite , spatial extent require non-zero and , which implies < 0 , giving a corresponding modification of both the phase and group velocities of the light. In this sense, light beams with non-zero k x and k y are naturally dispersive, even in free space. Extending upon the case of a mode within a hollow waveguide, an example of a structured beam is a Bessel beam (Fig. 1A), which is itself the description of a mode within a circular waveguide (1, 6). In free space, Bessel beams can be created using an axicon, or its dif...

The subtraction or addition of a prescribed number of photons to a field mode does not, in general, simply shift the probability distribution by the number of subtracted or added photons. Subtraction of a photon from an initial coherent state, for example, leaves the photon statistics unchanged and the same process applied to an initial thermal state increases the mean photon number. We present a detailed analysis of the effects of the initial photon statistics on those of the state from which the photons have been subtracted or to which they have been added. Our approach is based on two closely-related moment generating functions, one well-established and one that we introduce.

Recently Danan et al. [Phys. Rev. Lett. 111, 240402 (2013)] have performed a much discussed experiment in which which-way information was obtained from the light in a nested Mach-Zehnder interferometer by weak measurement. The presented analysis using the two state vector formalism drew the conclusions that the photons followed disconnected paths. We analyse this experiment using standard quantum optical methods and arrive at analytical expressions that match the experimental results without the need for such disconnected photon paths. We also propose a simple amendment to the experiment capable of displaying new phenomena highlighting the advantages of our description.

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