The concept of mean photon number is commonly used in quantum optics to describe the average number of photons present in a particular quantum state. It quantifies the intensity or brightness of the electromagnetic radiation associated with that state.
In classical optics, light is often described in terms of a continuous wave, and the intensity of the light can take any positive value. However, in quantum mechanics, light is quantized into discrete particles called photons. The number of photons in a particular quantum state can be a non-integer value, including fractions or even zero.
The mean photon number (often denoted as ȡ or n) is the average value of the number of photons in a given quantum state. It is obtained by calculating the expectation value of the photon number operator for that state. Mathematically, it is expressed as:
ȡ = ⟨n⟩ = ∑(n P(n)),
where n represents the number of photons in the state, P(n) is the probability of having n photons, and the summation is taken over all possible values of n.
Now, regarding your question about the mean signal photon number being less than one, this situation can arise in certain quantum states, such as squeezed states or sub-Poissonian states. In these states, the photon number distribution is non-classical, and the mean photon number can be reduced below unity.
To understand this, it's important to note that the mean photon number represents an average value over a large number of measurements or observations. In quantum mechanics, the measurement outcomes can exhibit statistical fluctuations. In certain quantum states, these fluctuations can lead to a mean photon number that is less than one.
In classical optics, an average value less than one might seem counterintuitive because we usually think of intensity as a positive quantity. However, in quantum optics, the probabilistic nature of quantum states allows for sub-Poissonian statistics, where the fluctuations in photon number can be suppressed. This leads to a mean photon number that can be less than one, indicating a quantum state with reduced average photon occupancy.
It's worth noting that mean photon number values less than one do not imply a lack of photons or complete darkness. Rather, they indicate the peculiar statistical behavior of light in certain quantum states, where the average number of photons is reduced compared to classical expectations.