It is usually assumed that the energy dependent velocity of a photon moving in a dispersive medium is given by the group velocity at the frequency corresponding to the energy of the photon. This assumption corresponds to the notion that the velocity of a massive particle is determined by its momentum. However, a direct verification of this assumption for a single photon is impossible, since the velocity can only be obtained by measuring the position at two different times, and time-resolved measurements cannot also resolve photon energies. In previous work, I have shown how this limitation can be circumvented for position and momentum, demonstrating that quantum particles do not obey newton's first law in free space. Here, I apply a similar strategy to construct a quantum state in which a non-vanishing percentage of the photons travel a distance x in time t even though the probability of finding any photons with a group velocity of v = x/t is close to zero. Specifically, the suppression of frequencies with group velocities in the vicinity of v = x/t is achieved by destructive interference, while the probabilities of detecting the photons in the initial time window or in the final time window are simultaneously enhanced by constructive interferences between the mutually overlapping wavefunctions. Based on the statistical evidence obtained from separate measurements of single photon arrival times and frequencies, it is then possible to show that the group velocity does not represent the actual velocity at which individual photons propagate through the dispersive medium.
It is usually assumed that the energy dependent velocity of a photon moving in a dispersive medium is given by the group velocity at the frequency corresponding to the energy of the photon. This assumption corresponds to the notion that the velocity of a massive particle is determined by its momentum. However, a direct verification of this assumption for a single photon is impossible, since the velocity can only be obtained by measuring the position at two different times, and time-resolved measurements cannot also resolve photon energies. In previous work, I have shown how this limitation can be circumvented for position and momentum, demonstrating that quantum particles do not obey newton's first law in free space. Here, I apply a similar strategy to construct a quantum state in which a non-vanishing percentage of the photons travel a distance x in time t even though the probability of finding any photons with a group velocity of v = x/t is close to zero. Specifically, the suppression of frequencies with group velocities in the vicinity of v = x/t is achieved by destructive interference, while the probabilities of detecting the photons in the initial time window or in the final time window are simultaneously enhanced by constructive interferences between the mutually overlapping wavefunctions. Based on the statistical evidence obtained from separate measurements of single photon arrival times and frequencies, it is then possible to show that the group velocity does not represent the actual velocity at which individual photons propagate through the dispersive medium.
Since the wavefunction of a photon only describes the probability of photon detection in time and space, it is impossible to derive uniquely defined trajectories describing the path taken by the photon between emission and detection. However, it is possible to test whether a particular set of trajectories is consistent with the statistics observed at different times for photons in the same initial state. Recently, I have shown that quantum interference effects between position and momentum can result in a violation of inequalities associated with motion along straight lines. Here, I present a more detailed analysis on the origin of the effect and its relation with other experimentally observable aspects of quantum statistics such as weak measurements and quantum tomography. It is shown that the interference pattern between a quantum state component of well-defined position and a quantum state component of well-defined momentum describes a modified causality relation between the positions detected at different times. The phase of the interference pattern is identified with the classical action of particle motion and the relation between uncertainty and causality is considered. The specific case of single photon wavefunctions is used to explain the possibilities and limitations of control at the ultimate quantum limit.
Optimal simultaneous control of position and momentum can be achieved by maximizing the probabilities of finding their experimentally observed values within two well-defined intervals. The assumption that particles move along straight lines in free space can then be tested by deriving a lower limit for the probability of finding the particle in a corresponding spatial interval at any intermediate time t. Here, it is shown that this lower limit can be violated by quantum superpositions of states confined within the respective position and momentum intervals. These violations of the particle propagation inequality show that quantum mechanics changes the laws of motion at a fundamental level, providing a different perspective on causality relations and time evolution in quantum mechanics.
We study the propagation of single photons through an atomic vapour cell where group velocity exceeds the velocity of light in vacuum. The mechanism behind this phenomenon is a result of interference of Fourier components due to the dispersive properties of the medium. By considering the distinction between group velocity and information velocity and its implications for special relativity, we show that fast single photons do not violate the principle of relativistic causality. To produce fast single photons, we fine-tune the emission of a single GaAs quantum dot to the 87 Rb D2 atomic transition, where a spectrally narrow region of anomalous dispersion enables a group velocity of vg = 1.21c at room temperature. It is shown that our current configuration does not allow the detection of fast single photons by means of a time resolved measurement of the exciton lifetime. Instead, we propose an alternative detection scheme based on single photon interference, where the peak of the fringe visibility is shifted forward in time by t = -0.4167⌧c (where ⌧c denotes the coherence time of the photon).
Within the framework of the ‘Pairwise Interference Model’ of light–matter interaction, it is postulated that light particles follow definite trajectories, and thus, for example, in the double-slit experiment, a particle can pass through only one of the two slits. On this premise, the interference of two photons is examined in greater detail. It is hypothesized that the interference of a photon pair requires the involvement of matter and, initially, that both molecular excitations and photons oscillate at the same frequency and that the continuity of the oscillation is not disrupted by absorption and emission. The molecular excitations in the light source are supposed to oscillate synchronously. Both photons arrive at the same point on the detector but may travel along different paths and therefore arrive at different times. Under these conditions, their phase difference, φ(2) – φ(1), is always zero. Consequently, no interference pattern can form on the detector screen and therefore the assumptions made are incorrect. Alternatively, one may assume that only the photon oscillates, whereas the excitation stores the phase, φ, possessed by the photon at the instant of absorption. In this case, φ(2) – φ(1) corresponds to the difference in travel times modulo the oscillation period, and an interference pattern can form. However, this assumption is problematic for some reasons: the nature of the matter-based memory remains unclear; according to the special theory of relativity, no time elapses for a particle travelling at the speed of light, meaning that it cannot oscillate; and the assumption contradicts Bohr’s correspondence principle. In a classical wave, the phase information propagates away from the source at the wave velocity without changing. This leads to the alternative possibility that the excitations oscillate, whereas the photons do not. Instead, the photons transport information about the phase of the excitation at the instant of emission. As can be shown, in this case, φ(2) – φ(1) likewise corresponds to the difference in travel times modulo the oscillation period. However, this hypothesis contradicts the de Broglie postulate, according to which all particles exhibit wave properties, and the relation ε = hν must therefore be interpreted differently. It also raises the question of how the dimensions of a laser, specifically the mirror separation, can influence the wavelength of the amplified light if a photon has no wavelength. Using replicator dynamics and the ‘Pairwise Interference Model’ of light–matter interaction, a simulation demonstrates that, in a simplified laser model, assuming that photons do not oscillate whereas excitations do, self-organisation gives rise to a phase pattern in the sequence of photons within the emitted light beam that corresponds to the pattern obtained by discretising a classical wave. Thus, coherent light can exhibit wave properties even if photons themselves do not. Finally, the possibility of how incoherent light acquires wave properties is briefly discussed.
Understanding the nature of single-photon propagation and any differences to the classical multi-photon regime is essential to better use the escalating number of single-photon technologies. Novel techniques and results are reported that demonstrate a consistent ability to measure the classical group velocity in an optical fiber to an accuracy in the time-offlight around ±0.2 ns in 30,000 ns. Identical techniques are then applied to single photons to determine their velocity and assess if there is any evidence that Schrödinger frequencies (N+1/2)ω alter the ω-k dispersion diagram. The evidence suggests that photons in a circular fiber show no additional dispersion but travel at the classical group velocity.
The speed of light is nominally given by c/n, where n is the refractive index of the medium in which the light is traveling. Since the refractive index of free space is 1, it is natural to expect that light travels at c in free space. We show that this is not the case if one considers real beams that have finite transverse extent. The modification of the wavevectors comprising the beam result in a change in the phase and group velocities. We study the group velocity of single photons, whose arrival times are unambiguously established by a click in a single-photon detector.We generate time-correlated photon pairs via spontaneous parametric down-conversion. One photon is sent through a fiber delay of a fixed length; the arrival time of this photon is the reference time to which we compare the arrival time of the other photon, which we can choose to be either unstructured or structured. The structuring is done by spatial light modulators that can be switched on and off, hence allowing photons to travel through exactly the same optical path, much like in a race. We consider photons in a Bessel mode and a focused Gaussian mode, and show that in both cases, the reduction in group velocity results in a delay of several micrometers over a propagation distance of 1 m or ~30 femtoseconds in terms of arrival time. The effect is well-understood from a geometric argument and from a rigorous calculation of the harmonic average of the group velocity, and hence, is not anomalous at all. Our work highlights that, even in free space, the invariance of the speed of light only applies to plane waves.