[Paper Review] Delocalised Excitation and Conditional Negative Delays of Single Photons

Daniela Angulo|arXiv (Cornell University)|Sep 5, 2024

Quantum Mechanics and Applications4 citations

TL;DR

This paper experimentally demonstrates that a single photon can spend a negative amount of time as an atomic excitation in a resonant medium, using weak measurement via the cross-Kerr effect to probe atomic excitation time. The measured mean excitation time—ranging from −0.82±0.31τ₀ to +0.54±0.28τ₀—matches the group delay, proving that negative group delay has physical significance as the actual time atoms spend in excited states during photon transmission.

ABSTRACT

This paper presents the concept of Stochastic Zero, a mathematical framework designed to model time as a stochastic variable in physical systems. This approach is especially relevant for high-volatility experiments, where temporal uncertainties and phenomena like 'negative time' are observed. By complementing traditional methods such as Itô calculus, Stochastic Zero provides a balance of accuracy and flexibility in capturing temporal fluctuations. The simulations highlight how this framework offers deeper insights compared to conventional models, particularly in quantum systems characterized by significant temporal variability.

Motivation & Objective

• To determine whether the group delay experienced by a transmitted photon corresponds to the actual time atoms spend in excited states during light-matter interaction.
• To test the hypothesis that negative group delays correspond to physically meaningful negative excitation times in atomic media.
• To experimentally verify the theoretical prediction that the time-integrated atomic excitation due to a transmitted photon equals the group delay.
• To resolve the long-standing debate on whether group delay reflects actual time spent in excitation or is merely a mathematical artifact.

Proposed method

• Using a weak, off-resonant probe beam to measure the phase shift induced on the probe via the cross-Kerr effect, which is proportional to the instantaneous atomic excitation probability.
• Measuring the time-dependent phase shift on the probe beam to infer the time-integrated excitation of atoms due to a transmitted single photon.
• Employing quantum trajectory theory and weak value formalism to theoretically predict the atomic excitation time as equal to the group delay.
• Applying post-selection to only transmitted signal photons to isolate the effect of transmission on atomic excitation.
• Using the relation φ_T = −τ_g ∂ω₀/∂m to link the measured probe phase shift to the group delay τ_g and the AC Stark shift ∂ω₀/∂m.
• Integrating the probe phase shift over a broad time window to improve reliability despite increased statistical error, informed by theoretical insights into the shape of φ_T(t).

Figure 1: Schematics of experimental setup. (a) Atomic level scheme. (b) Conceptual diagram of the experimental apparatus: a resonant pulsed beam (signal) and off-resonant continuous-wave beam (probe) counter-propagate through a cloud of cold 85 Rb atoms, detected at opposite sides of the apparatus.

Experimental results

Research questions

• RQ1 Does the group delay of a transmitted photon correspond to the actual time atoms spend in excited states?
• RQ2 Can negative group delays be interpreted as physically meaningful negative excitation times?
• RQ3 Is the time-integrated atomic excitation caused by a transmitted photon quantitatively equal to the group delay?
• RQ4 How does post-selection on transmitted photons affect the measured excitation time of atoms?
• RQ5 What is the physical origin of the phase shift observed on a probe beam due to a single transmitted photon?

Key findings

• The mean atomic excitation time due to a transmitted photon was measured to be (−0.82±0.31)τ₀ for the narrowest pulse, confirming a negative excitation time.
• For the broadest pulse, the excitation time was measured as (+0.54±0.28)τ₀, showing positive excitation even with broadband excitation.
• The measured excitation times are consistent with the theoretical prediction that they equal the group delay of the transmitted pulse.
• The results validate that negative group delays are not just mathematical artifacts but represent actual physical times when atoms are excited.
• The probe phase shift was found to be proportional to the group delay, with the scaling factor determined by the AC Stark shift of the atomic levels.
• The integration of the phase shift over a broad time window, though increasing statistical error, yielded more reliable results than peak-based analysis.

Figure 2: Phase shift in a shot (time window containing one pulse) acquired by the probe over time. Blue circles represent the phase shift due to a transmitted photon, with the orange solid line indicating the theoretical expectation. Green squares show the phase shift due to an average incident pho