Glauber–Sudarshan P representation
Glauber–Sudarshan P representation
The Glauber–Sudarshan P representation is a suggested way of writing down the phase space distribution of a quantum system in the phase space formulation of quantum mechanics. The P representation is the quasiprobability distribution in which observables are expressed in normal order. In quantum optics, this representation, formally equivalent to several other representations,[1][2] is sometimes championed over alternative representations to describe light in optical phase space, because typical optical observables, such as the particle number operator, are naturally expressed in normal order. It is named after George Sudarshan[3] and Roy J. Glauber,[4] who were working on the topic in 1963. It was the subject of a controversy when Glauber was awarded a share of the 2005 Nobel Prize in Physics for his work in this field and George Sudarshan's contribution was not recognized.[5] Despite many useful applications in laser theory and coherence theory, the Glauber–Sudarshan P representation has the drawback that it is not always positive, and therefore is not a true probability function.
Definition
We also wish to construct the function such that the expectation value of a normally ordered operator satisfies the optical equivalence theorem. This implies that the density matrix should be in anti-normal order so that we can express the density matrix as a power series
Inserting the identity operator
we see that
and thus we formally assign
More useful integral formulas for P are necessary for any practical calculation. One method[6] is to define the characteristic function
and then take the Fourier transform
Another useful integral formula for P is[7]
![P(\alpha )=\sum _{{n}}\sum _{{k}}\langle n|{\hat {\rho }}|k\rangle {\frac {{\sqrt {n!k!}}}{2\pi r(n+k)!}}e^{{r^{2}-i(n-k)\theta }}\left[\left(-{\frac {\partial }{\partial r}}\right)^{{n+k}}\delta (r)\right],](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b2d200196539f2cd30d242df392889691c42dc)
where r and θ are the amplitude and phase of α. Though this is a full formal solution of this possibility, it requires infinitely many derivatives of Dirac delta functions, far beyond the reach of any ordinary tempered distribution theory.
Discussion
Examples
Thermal radiation
From statistical mechanics arguments in the Fock basis, the mean photon number of a mode with wavevector k and polarization state s for a black body at temperature T is known to be
The P representation of the black body is
In other words, every mode of the black body is normally distributed in the basis of coherent states. Since P is positive and bounded, this system is essentially classical. This is actually quite a remarkable result because for thermal equilibrium the density matrix is also diagonal in the Fock basis, but Fock states are non-classical.
Highly singular example
Even very simple-looking states may exhibit highly non-classical behavior. Consider a superposition of two coherent states
where c0 , c1 are constants subject to the normalizing constraint
Despite having infinitely many derivatives of delta functions, P still obeys the optical equivalence theorem. If the expectation value of the number operator, for example, is taken with respect to the state vector or as a phase space average with respect to P, the two expectation values match:
See also
Quasiprobability distribution#Characteristic functions
Nonclassical light
Wigner quasiprobability distribution
Husimi Q representation
Nobel Prize controversies