Introduction to Weak Coherent States (WCSs)
A key idea in quantum optics is weak coherent states (WCSs), which are simply light fields produced by a weak laser. WCSs differ from ideal single-photon sources in that they are probabilistic, which means that they cannot be relied upon to emit a single photon at a given time. Rather, they depict a linear superposition of states with photon numbers. Since the emission of both multi-photon and vacuum pulses needs to be precisely controlled, this probabilistic feature makes it impossible to produce a single-photon pulse with complete certainty.
Features and attributes of WCSs
In a weak coherent state, the photon number is distributed according to the Poisson distribution. The average number of photons per pulse is kept much below one in the “weak regime,” which is essential for many applications. Multi-photon contributions can frequently be disregarded for very low mean photon numbers, such as 0.01; the likelihood of detecting one photon is much higher than that of detecting two or three photons, with ratios of 200 and 300, respectively.
The ability of WCSs to essentially resemble an ideal single photon in terms of both photon count and undetermined phase is a crucial feature. These superposed weak coherent states exhibit basic single-photon properties such as Hong-Ou-Mandel interference and antibunching, and can attain high fidelity with single-photon states. By varying factors like the mean photon number and the number of phases, it is possible to directly observe whether certain single-photon features are present or absent. One characteristic of single photons that WCSs assist to clarify is the intrinsic ambiguity between photon number and phase.
Applications of WCSs
In quantum cryptography systems, weak coherent states are frequently used, especially for quantum key distribution (QKD). Avoiding multi-photon pulses is crucial in QKD because eavesdroppers can use a photon-number splitting attack to take advantage of them. Usually, to combat this, the light source is greatly attenuated to keep the average number of photons per pulse well below one.
Although this raises the likelihood of vacuum emission, methods such as decoy states can enable a larger average photon number without jeopardizing the QKD system’s security. In quantum key distribution applications, it has been demonstrated that superposed weak coherent states perform better than conventional methods utilizing phase-randomized weak coherent states. The practical application of measurement-device-independent quantum key distribution methods has also been greatly aided by WCSs.
Two-Photon Interference with WCSs
One of the fundamental concepts of quantum physics and experimental quantum optics is multiphoton interference, namely two-photon interference (TPI) and the Hong-Ou-Mandel (HOM) phenomenon. TPI studies have been widely carried out at the single-photon level using traditional light sources, such as WCSs, although they are frequently explored using strongly correlated photon pairs.
In interference studies involving two coherent states, it is frequently observed that the interference visibility is limited to 50% for two spatial modes. This restriction is typically ascribed to the potential for multi-photon emission that is intrinsic to WCSs. It is true that the maximally visible visibility in phase-randomized HOM-type TPI experiments with WCSs is restricted to 0.5. The reason for this is that a path-correlated two-photon state’s phase-sensitive interference is randomized, producing a continuous coincidence that doesn’t add to the interference fringe.
Experimental Demonstrations of TPI with WCSs
TPI has been investigated in experiments with different WCS configurations:
- Phase-Randomized WCSs: In these studies, an interferometer (such as a Michelson or Mach-Zehnder interferometer) actively varies the relative phase between the two pathways. A phase-insensitive HOM-type TPI fringe, which is frequently seen as a “dip” in coincidence counts, is all that remains after the phase-sensitive interference vanishes.
- Temporally Separated WCSs: HOM-type TPI can be seen by employing electrical delay and coincidence time windows for post-selection, as well as by adding an optical delay in between successive weak coherent pulses. The HOM fringe may show up as a dip or a peak, depending on how distinguishable the photons are (e.g., in terms of polarization).
- Spatially Separated WCSs: In this configuration, two interferometers are spatially separated, and WCSs follow different routes. A transition between peak and dip fringes based on the relative phase of the two two-photon amplitudes is observed in the HOM-type TPI effect with synchronized phase randomization and path-length modifications.
Theoretical Interpretation and Characterization
A more thorough understanding indicates that the underlying physics should be interpreted as interference between two-photon states at the single-photon level within the interferometer, even though the observed interference results with WCSs, such as fringe visibility and shape, can frequently be explained by classical intensity correlation. To properly understand the interference effect, particularly when weak coherent pulses are taken into account at the single-photon level, this quantum framework is necessary.
Using two-photon interference in a Hong-Ou-Mandel interferometer, a technique for spectrum characterization of WCS sources has been established. Two feeble laser sources with different frequencies are fed into the interferometer, and the interference pattern that results is fitted using a theoretical model. The equivalence of both methods for spectral characterization can then be demonstrated by comparing the fitting parameters, such as frequency mismatch and coherence length, with findings from classical optical beat measurements using bright copies of the sources.
Limitations and Challenges
There may be fundamental and practical restrictions on the characterization of WCSs. Accuracy may be constrained by hardware characteristics, such as the HOM interferometer’s delay generators’ resolution and step-size. More importantly, at greater frequency mismatches, the interference pattern’s visibility may diminish due to the sources’ mutual coherence and spectral separation. To prevent averaging out the interference pattern and losing information, the gate’s temporal breadth of single-photon detectors must also be less than the beat note’s oscillation.
On a more abstract level, weak coherent states are also problematic in theoretical physics, especially when it comes to representations of path integrals. Weak coherent states do not provide a resolution of unity expressed as a local integral, in contrast to normal coherent states. This implies that path integrals cannot be directly constructed using the conventional time-slicing approach. Instead, using a continuous-time regularization strategy, researchers use a well-defined path integral using Wiener measure.
Weak coherent states are essentially like a feeble, shimmering light, where each glimmer is a mixture of potentialities rather than a single photon. However, we can infer characteristics that are very comparable to those of a single, elusive quantum particle by looking at how these tiny twinkles interact.
