“The word ‘classical’ means only one thing in science: it's wrong!” We have previously emphasized the fact that all states of light are quantum mechanical and are thus nonclassical, deriving some quantum features from the discreteness of the photons. Of course, in practice, the nonclassical features of light are difficult to observe. (We shall use “quantum mechanical” and “nonclassical” more or less interchangeably here.) Already we have discussed what must certainly be the most nonclassical of all nonclassical states of light – the single-photon state. Yet, as we shall see, it is possible to have nonclassical states involving a very large number of photons. But we need a criterion for nonclassicality. Recall that in Chapter 5 we discussed such a criterion in terms of the quasi-probability distribution known as the P function, P (α). States for which P (α) is positive everywhere or no more singular than a delta function, are classical whereas those for which P (α) is negative or more singular than a delta function are nonclassical. We have shown, in fact, that P (α) for a coherent state is a delta function, and Hillery has shown that all other pure states of the field will have functions P (α) that are negative in some regions of phase space and are more singular than a delta. It is evident that the variety of possible nonclassical states of the field is quite large.
In 1963 Glauber /Phys. Rev. 130, 2529 and 131, 2766/ published a basic theoretical study that revealed the broad variety of possible photon states. Later on a quantum-statistical analysis showed that states with classical analog (this light Is termed classical light) as well as states without classical analog (so-called nonclassical light) can occur. Following the definition given by Teich, Saleh, and Perina /JOSAB2 (1985) 275) nonclassical light exhibits one or more of the attributes squeezed, antibunched, sub-Poissonian, what is connected with a negative value of the field-fluctuation excess, the bunching excess, or the Poisson excess. These negative values mean quantum fluctuations below the classical standard quantum limits of ideal laser light; thus, nonclassical light may yield an essential improvement of the optical measurement accuracy, in particular of the signal-to-noise ratio. There exists a common property of the three attributes concerning the Glauber-Sudarshan representation P(c ), which is the real weight function belonging to the incoherent mixture (density operators) of Glauber states with different complex amplitudes. In the case of classical light P(oç ) is a positive definite function, whereas nonclassical light requires a nonpositive definite function P(o.), which cannot be interpreted as a proper probability distribution - as it is possible in the case of light with a classical analog /M. Schubert, Ann. d. Phys. 4k (1987) 53/. In addition to its possible important applications nonclassical light yields a deep insight into the quantum nature of the radiation field.
Nonclassical states of light are necessary resources for quantum technologies such as cryptography, computation and the definition of metrological standards. Distinguishing nonclassical states from those that can be described without invoking quantum mechanics is both a necessary practical task and of fundamental interest. In general, signatures of nonclassicality arise from apparently pathological behaviour of the quasi-probability distributions used to describe the states. While tomographic reconstruction of the state is therefore sufficient to show nonclassicality, it is often more convenient to probe photon-counting statistics of on-off detectors directly. This is formalised by the Mandel Q-parameter QM= 〈Δn〉 / 〈n〉 - 1, which is negative for sub-Poissonian statistics arising from nonclassical states. However, the Mandel Q-parameter can fail when using a multiplexed array of on-off detectors to measure states with several photons; splitting in this manner can corrupt its original statistics. A new parameter, QB= N 〈(Δ c) 2 〉 / 〈 c 〉 (N - 〈 c 〉) can be used to witness nonclassicality directly from such click statistics. For QB<; 0, such statistics are thus deemed sub-binomial and therefore arise from a nonclassical state.
There is a common understanding in quantum optics that nonclassical states of light are states that do not have a positive semidefinite and sufficiently regular Glauber-Sudarshan
P function. Almost all known nonclassical states have
P functions that are highly irregular, which makes working with them difficult and direct experimental reconstruction impossible. Here we introduce classes of nonclassical states with regular, non-positive-definite
P functions. They are constructed by ``puncturing'' regular smooth positive
P functions with negative Dirac-$\ensuremath{\delta}$ peaks or other sufficiently narrow smooth negative functions. We determine the parameter ranges for which such punctures are possible without losing the positivity of the state, the regimes yielding antibunching of light, and the expressions of the Wigner functions for all investigated punctured states. Finally, we propose some possible experimental realizations of such states.
The definition of nonclassical states in quantum optics by the nonpositivity of their Glauber–Sudarshan quasiprobability is investigated and it is shown that it hides some serious problems. It leads to a subdivision of squeezed thermal states into classical and nonclassical states which is difficult to interpret physically by some qualitatively different behaviour of the states. Nonclassical states are found in arbitrarily small neighbourhoods of every classical state that is illustrated by a very artificial modified thermal state. The observability of the criterion in comparison to that for nonclassicality of states determined by the nearest Hilbert–Schmidt distance to a class of reference states is discussed. The behaviour of the nonclassicality of states in models of phase-insensitive processes of damping and amplification is investigated and it is found that every nonclassical state eventually makes a transition to a classical state. However, this is not specific for the negativities or singularities of the Glauber–Sudarshan quasiprobability and is found in similar form for other quasiprobabilities, for example, for the Wigner quasiprobability. We discuss in quite general form some defects of the Glauber–Sudarshan quasiprobability if compared with classical distribution functions over the phase space, in particular the failure of an earlier advertised superposition formula.
A bosonic state is commonly considered nonclassical (or quantum) if its Glauber-Sudarshan
P function is not a classical probability density, which implies that only coherent states and their statistical mixtures are classical. We quantify the nonclassicality of a single qubit, defined by the vacuum and single-photon states, by applying the following four well-known measures of nonclassicality: (1) the nonclassical depth, $\ensuremath{\tau}$, related to the minimal amount of Gaussian noise which changes a nonpositive
P function into a positive one; (2) the nonclassical distance
D, defined as the Bures distance of a given state to the closest classical state, which is the vacuum for the single-qubit Hilbert space; together with (3) the negativity potential (NP), and (4) concurrence potential, which are the nonclassicality measures corresponding to the entanglement measures (i.e., the negativity and concurrence, respectively) for the state generated by mixing a single-qubit state with the vacuum on a balanced beam splitter. We show that complete statistical mixtures of the vacuum and single-photon states are the most nonclassical single-qubit states regarding the distance
D for a fixed value of both the depth $\ensuremath{\tau}$ and NP in the whole range
[0,1] of their values, as well as the NP for a given value of $\ensuremath{\tau}$ such that $\ensuremath{\tau}>0.315\phantom{\rule{0.16em}{0ex}}4$. Conversely, pure states are the most nonclassical single-qubit states with respect to $\ensuremath{\tau}$ for a given
D, NP versus
D, and $\ensuremath{\tau}$ versus NP. We also show the ``relativity'' of these nonclassicality measures by comparing pairs of single-qubit states: if a state is less nonclassical than another state according to some measure then it might be more nonclassical according to another measure. Moreover, we find that the concurrence potential is equal to the nonclassical distance for single-qubit states. This implies an operational interpretation of the nonclassical distance as the potential for the entanglement of formation.
Nonclassical quantum effects gradually reach domains of physics of large systems previously considered as purely classical. We derive a hierarchy of operational criteria suitable for a reliable detection of nonclassicality of light from an arbitrarily large ensemble of independent single-photon emitters. We show, that such large ensemble can always emit nonclassical light without any phase reference and under realistic experimental conditions including incoherent background noise. The nonclassical light from the large ensemble of the emitters can be witnessed much better than light coming from a single or a few emitters.