Photon momentum in a medium with negative group velocity

Peter Peter, Milonni

Holger Friedrich Hofmann

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.

B. Macke, B. S�gard

Nikolai S. Bukhman

It is shown that there are no dispersing media with «subluminal only» or «superluminal only» real group velocity – if in some medium at some frequencies the real group velocity is subluminal, then in the same medium there are necessarily frequencies at which the real group velocity is superluminal. Conversely, if a medium has frequencies with a superluminal group velocity, then it necessarily has frequencies with a subluminal group velocity. In other words, there is a necessary connection between subluminal and superluminal group velocity – if any medium is capable of generating subluminal group velocity, then the same medium is capable of generating superluminal group velocity. Similarly, there are no media with «only positive» or «only negative» imaginary part of the complex group velocity – if at some frequencies the imaginary part of the group velocity is positive, then at other frequencies in the same medium it is negative and vice versa. This means that if there is a carrier frequency in the medium at which some linearly frequency-modulated signal propagates at a speed lower than the group speed, then in the same medium there is necessarily a carrier frequency at which the same signal will propagate at a speed higher than the group speed and vice versa.

L. J. Wang, Arthur Dogariu, A. Kuzmich

Dexin Ye, Guoan Zheng, Jingyu Wang +4 more

Scientific community has well recognized that a Lorentzian medium exhibits anomalous dispersion behavior in its resonance absorption region. To satisfy the Krammers-Kronig relation, such an anomalous region has to be accompanied with significant loss, and thus, experimental observations of negative group velocity in this region generally require a gain-assisted approach. In this letter, we demonstrate that the negative group velocity can also be observed in the absence of absorption resonance. We show that the k-surface of a passive uniaxial Lorentzian medium undergoes a distortion near the plasma frequency. This process yields an anomalous dispersion bandwidth that is far away from the absorption resonance region, and enables the observation of negative group velocity at the plasma frequency band. Introducing anomalous dispersion in a well-controlled manner would greatly benefit the research of ultrafast photonics and find potential applications in optical delay lines, optical data storage and devices for quantum information processing.