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1 wwwsciencemagorg/cgi/content/full/scienceaaa3035/dc1 Supplementary Materials for Spatially structured photons that travel in free space slower than the speed of light Daniel Giovannini, Jacquiline Romero, Václav Potoček, Gergely Ferenczi, Fiona Speirits, Stephen M Barnett, Daniele Faccio, Miles J Padgett* This PDF file includes: Materials and Methods Supplementary Text *Corresponding author milespadgett@glasgowacuk Published January 015 on Science Express DOI: 10116/scienceaaa3035

2 Supporting Online Material for: Spatially Structured Photons that Travel in Free Space Slower than the Speed of Light Daniel Giovannini 1, Jacquiline Romero 1, Václav Potoček 1, Gergely Ferenczi 1, Fiona Speirits 1, Stephen M Barnett 1, Daniele Faccio, Miles J Padgett 1 1 School of Physics and Astronomy, SUPA, University of Glasgow, Glasgow, UK School of Engineering and Physical Sciences, SUPA, Heriot-Watt University, Edinburgh, UK These authors contributed equally to this work To whom correspondence should be addressed; milespadgett@glasgowacuk Materials and Methods We use a ultraviolet pump laser with 10 MHz repetition rate, and a pulse length of the order of 15 ps Although our pump laser has a 10 MHz repetition rate, the important photon flux is the rate at which photon pairs are produced through the parametric down-conversion (SPDC) process SPDC is stochastic Our detection rate is of the order of thousands of counts per second, where the time between counts varies according to a Poissonian distribution Unlike a classical pulse, where arrival time is an ambiguous concept which can apply to first light or peak intensity, the arrival time of a single photon is well-defined By operating in the single-photon ultra-low intensity regime, we are not subject to any complications arising from nonlinear detector response In addition, by operating in the single-photon regime, we are able to adopt our Hong-Ou-Mandel approach, in which timing precision can be insured by mechanical stability, and interpretation of our measurements as group velocity is unambiguous 1

3 The pump is incident upon a beta-barium borate (BBO) crystal to produce photon pairs with central wavelength at 710 nm The photons, called signal and idler, pass through an interference filter of spectral bandwidth 10 nm and are collected by polarization-maintaining, single-mode fibers One fiber is mounted on an axial translation stage to control the path length (Fig A) The idler photon goes through polarization-maintaining fibers before being fed to the input port of a fiber-coupled beam splitter (Fig B) (17) Instead of going straight to the other beam splitter input, the signal photon is propagated through a free-space section (Fig C) consisting of fiber-coupling optics to collimate the light and two spatial light modulators (SLMs) SLMs are pixelated, liquid-crystal devices that can be encoded to act as diffractive optical elements implementing axicons, lenses and similar optical components The first SLM can be programmed to act as a simple diffraction grating such that the light remains collimated in the intervening space, or programmed to act as an element to structure the beam (eg axicons or lenses with focal length f) The second SLM, placed at a distance f, reverses this structuring so that the light can be coupled back into the single-mode fiber that feeds to the other input port of the beam splitter The output ports of the fiber-coupled beam splitter are connected to single-photon detectors, which in turn feed a gated counter (Fig D) The advantage of using our Hong-Ou-Mandel approach is that the timing precision is dictated by the mechanical resolution and positional accuracy of our translation stage, which controls the relative path length of the two photons The path delay in the signal arm is adjusted by means of a motorized translation stage, with 007 µm resolution, corresponding to a temporal uncertainty of 0 fs The coincident count rate is recorded as a function of path difference between the signal and idler arms, and the position of the HOM dip is recorded as a function of the spatial shaping of the signal photon For each position of the stage, the coincident count rates for different beam structures (Fig 3(A) and 4) are acquired sequentially The HOM measurements are repeated over a period of several hours, in order to determine the error bars shown A

4 typical experimental data run occurs over a period of about ten minutes The experimental setup for the free-space propagation section is relatively compact (45 45 cm), and the measurements are carried out in a temperature-controlled environment One possible source of systematic error in our results is a local change in the refractive index of the air, and hence group velocity We note that our experimental methods and results preclude this possibility If the local refractive index of the air was changing sufficiently over the time scale of our measurements, this would be manifest as a jitter of the position of the Hong-Ou-Mandel dip, meaning that the dips corresponding to the different cases could not be distinguished No such jitter is observed Our dips remain distinguishable, thereby indicating that no such jitter exists Indeed, the position of our HOM is stable over many hours, ie much longer than the time interval between different measurements, which is typically 4 s or less General Case for Free Propagation The delay that we observe in the experiment was estimated using a geometric, simple ray-optic model Here, we derive a rigorous theory that applies to any arbitrary field It is experimentally established that single photons travel at the group velocity, hence we will concern ourselves only with the group velocity The calculation of the photon travel time is based on the calculation of the group velocity of the photon, which in general depends on the radial profile and position along the propagation axis This is explicitly shown for a Gaussian beam in (18), but we formulate it here for any arbitrary field To take an effective group velocity of the electromagnetic field over a volume in which the velocity is changing, one would need to take a harmonic mean of the group velocity and also weight by the local intensity of the field The group velocity along the propagation of direction z of a beam with phase profile Φ and central frequency ω 0 is given by 3

5 v (z) g = ( ) 1 Φ z ω ω 0 The average velocity over an interval (z 1, z ) becomes ( 1 z z 1 z z 1 ( ) ) 1 [( ) Φ dz = (z z 1 ) Φ z ω ω 0 ω ω 0 ] z z 1 1 which, when the integration is generalized to cover the whole volume between the z 1 and z planes, using the local intensity as a weighting function, simplifies to (z z 1 )/I [ ( ) ] z ψ(z, ω0 ) ψ(z, ω) ω ψ(z, ω 0 ) ψ(z, ω 0 ) ω 0 z 1 Here the function ψ(x, y, z ω), represented in the bra-ket notation as ψ(z, ω), denotes the transverse profile of the spectral component of the beam at frequency ω in a scalar approximation As is the convention, we assume that the phase Φ is the complex argument of the functional value of ψ This only gives positive phase speed (in other words, it describes a wave travelling in the direction of the +z axis) if the notation is chosen so that the kz term has a +i coefficient in the exponent If we use this group velocity to calculate the time it takes a wave packet to cover the distance z z 1, we obtain t = I [ ( ) ] z [ ] z ψ(z, ω0 ) ψ(z, ω) = ω ψ(z, ω 0 ) ψ(z, ω 0 ) ω 0 ω (arg ψ(z, ω 0) ψ(z, ω) ) ω0 z 1 z 1 Thus, the delay of a nonplanar wavefront, compared to a plane wave that travels at speed c (which we approximate with a collimated beam in the experiment) is δz = ct (z z 1 ) = [ ] z k (arg ψ(z, k 0) ψ(z, k) ) k0 z z 1 Our experiment is well within the paraxial approximation, in which case the above expression for δz can be simplified even further The wave function ψ(x, y, z k) evolves from z 1 to z 4

6 according to the paraxial wave equation, where = ( / x) + ( / y) Then, where L = z z 1, and ψ z = i k ψ + ikψ, ( ) il ψ(z, k) = exp k + ikl ψ(z 1, k), ( il ψ(z, k 0 ) ψ(z, k) = ψ(z 1, k 0 ) exp k il ) + i(k k 0 )L ψ(z 1, k), k 0 δz = L Note that can be written as ˆk, where R ψ(z 1, k 0 ) ψ(z 1, k 0 ) ψ(z 1, k 0 ) ψ(z 1, k 0 ) ˆk = i is the quantum mechanical operator corresponding to the transverse wave vector This implies that has a negative real expectation value and the final expression for δz can be established as δz = L ˆk ψ(z1,k 0 ) (1) Interestingly, as a side result we can see that the effective group velocity v g = c 1 + ˆk c ( 1 ˆk is invariant under the free propagation and always smaller than c ) 5

7 Special Cases: Bessel and Gaussian Beams We can now use Eq 1 to calculate the delays for the cases of Bessel and Gaussian beams The former is simple in that there is a single value of k, which is exactly the radial component of the wavevector, denoted k r in the main text Using this we find that in our case (Fig 1A) the delay is δz Bessel = L k r k0 For a Gaussian beam of initial beam waist w 0, focused by a lens of focal length f, the quantity ˆk can be calculated as ˆk =, wf where w f = f/(k 0 w 0 ) is the waist of the focused beam This leads to an expression for the delay in our case (Fig 1B) δz Gauss = w 0 f We underline that these equations are exactly the same as that obtained from a simple geometric model as described in the main text For the cases where we put sharp aperture restrictions (the center and edge stops), a direct application of Eq 1 leads to an infinite slowing down because of the diffraction on the boundaries, which result in a broad-tailed distribution of k However, we can still give an estimate of the delays by recognizing that δz Gauss = w 0 f = ( w 0 ) 1 f = r f, where r is the expectation value of the square of the radius of the beam weighted by the Gaussian intensity distribution For the cases where we place center and edge stops of radius 14 mm, we can renormalize the resulting obstructed and truncated Gaussian distributions and 6

8 calculate r Doing so leads to delay estimates of 116 µm and 1 µm for the center and edge stops, respectively, compared to the collimated case A more careful analysis of the detection part shows that the broad tail of the distribution of k is removed by means of postselection on the principal maximum, naturally implemented in our case by the geometry of the second SLM Taking into account the shape of the aperture at the end of the free-space propagation, it can be shown that the full prediction of δz can actually be expressed as the weak value δz = L R ψ (z 1, k 0 ) ˆk ψ(z 1, k 0 ), ψ (z 1, k 0 ) ψ(z 1, k 0 ) with ψ (z 1, k 0 ) being the effective projection state at z back-evolved to the plane z 1 This reduces to Eq 1 in the case where ψ (z 1, k 0 ) = ψ(z 1, k 0 ), corresponding to detection of the entire wavefront Without going into further detail, we conclude that this description presents a good quantitative agreement with our experimental results and could possibly lead to other interesting cases 7

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