Quasi-probability Distribution

Introduction

In the most general form, the dynamics of a quantum-mechanical system are determined by a master equation in Hilbert space: an equation of motion for the density operator (usually written ) of the system. The density operator is defined with respect to a complete orthonormal basis. Although it is possible to directly integrate this equation for very small systems (i.e., systems with few particles or degrees of freedom), this quickly becomes intractable for larger systems. However, it is possible to prove that the density can always be written in a diagonal form, provided that it is with respect to an overcomplete basis. When the density operator is represented in such an overcomplete basis, then it can be written in a way more like an ordinary function, at the expense that the function has the features of a quasiprobability distribution. The evolution of the system is then completely determined by the evolution of the quasiprobability distribution function.

The coherent states, i.e. right eigenstates of the annihilation operator serve as the overcomplete basis in the construction described above. By definition, the coherent states have the following property:

$\begin{align}\hat{a}|\alpha\rangle&=\alpha|\alpha\rangle \\ \langle\alpha|\hat{a}^{\dagger}&=\langle\alpha|\alpha^*. \end{align}$

They also have some additional interesting properties. For example, no two coherent states are orthogonal. In fact, if and are a pair of coherent states, then

Note that these states are, however, correctly normalized with . Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof.

Proof of the overcompleteness of the coherent states
Integration over the complex plane can be written in terms of polar coordinates with . Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function: $\begin{align}\int \|\alpha\rangle\langle\alpha\| \, d^2\alpha &= \int \sum_{n=0}^{\infty}\sum_{k=0}^{\infty} e^{-{\|\alpha\|^2}} \cdot \frac{\alpha^n (\alpha^*)^k}{\sqrt{n!k!}} \|n\rangle \langle k\| \, d^2\alpha \\ &= \int_0^{\infty} \int_0^{2\pi} \sum_{n=0}^{\infty}\sum_{k=0}^{\infty} e^{-{r^2}} \cdot \frac{r^{n+k+1}e^{i(n-k)\theta}}{\sqrt{n!k!}} \|n\rangle \langle k\| \, d\theta dr \\ &= \sum_{n=0}^{\infty} \int_0^{\infty} \sum_{k=0}^{\infty} \int_0^{2\pi} e^{-{r^2}} \cdot \frac{r^{n+k+1}e^{i(n-k)\theta}}{\sqrt{n!k!}} \|n\rangle \langle k\| \, d\theta dr \\ &= 2\pi \sum_{n=0}^{\infty} \int_0^{\infty} \sum_{k=0}^{\infty} e^{-{r^2}} \cdot \frac{r^{n+k+1}\delta(n-k)}{\sqrt{n!k!}} \|n\rangle \langle k\| \, dr \\ &= 2\pi \sum_{n=0}^{\infty} \int e^{-{r^2}} \cdot \frac{r^{2n+1}}{n!} \|n\rangle \langle n\| \, dr \\ &= \pi \sum_{n=0}^{\infty} \int e^{-u} \cdot \frac{u^n}{n!} \|n\rangle \langle n\| \, du \\ &= \pi \sum_{n=0}^{\infty} \|n\rangle \langle n\| \\ &= \pi \hat{I}.\end{align}$ Clearly we can span the Hilbert space by writing a state as On the other hand, despite correct normalization of the states, the factor of π>1 proves that this basis is overcomplete.

Proof of the overcompleteness of the coherent states

Integration over the complex plane can be written in terms of polar coordinates with . Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function:

$\begin{align}\int |\alpha\rangle\langle\alpha| \, d^2\alpha &= \int \sum_{n=0}^{\infty}\sum_{k=0}^{\infty} e^{-{|\alpha|^2}} \cdot \frac{\alpha^n (\alpha^*)^k}{\sqrt{n!k!}} |n\rangle \langle k| \, d^2\alpha \\ &= \int_0^{\infty} \int_0^{2\pi} \sum_{n=0}^{\infty}\sum_{k=0}^{\infty} e^{-{r^2}} \cdot \frac{r^{n+k+1}e^{i(n-k)\theta}}{\sqrt{n!k!}} |n\rangle \langle k| \, d\theta dr \\ &= \sum_{n=0}^{\infty} \int_0^{\infty} \sum_{k=0}^{\infty} \int_0^{2\pi} e^{-{r^2}} \cdot \frac{r^{n+k+1}e^{i(n-k)\theta}}{\sqrt{n!k!}} |n\rangle \langle k| \, d\theta dr \\ &= 2\pi \sum_{n=0}^{\infty} \int_0^{\infty} \sum_{k=0}^{\infty} e^{-{r^2}} \cdot \frac{r^{n+k+1}\delta(n-k)}{\sqrt{n!k!}} |n\rangle \langle k| \, dr \\ &= 2\pi \sum_{n=0}^{\infty} \int e^{-{r^2}} \cdot \frac{r^{2n+1}}{n!} |n\rangle \langle n| \, dr \\ &= \pi \sum_{n=0}^{\infty} \int e^{-u} \cdot \frac{u^n}{n!} |n\rangle \langle n| \, du \\ &= \pi \sum_{n=0}^{\infty} |n\rangle \langle n| \\ &= \pi \hat{I}.\end{align}$

Clearly we can span the Hilbert space by writing a state as

On the other hand, despite correct normalization of the states, the factor of π>1 proves that this basis is overcomplete.

In the coherent states basis, however, it is always possible to express the density operator in the diagonal form

where f is a representation of the phase space distribution. This function f is considered a quasiprobability density because it has the following properties:

(normalization)
If is an operator that can be expressed as a power series of the creation and annihilation operators in an ordering Ω, then its expectation value is

(optical equivalence theorem).

The function f is not unique. There exists a family of different representations, each connected to a different ordering Ω. The most popular in the general physics literature and historically first of these is the Wigner quasiprobability distribution, which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:

If the quantum system has a classical analog, e.g. a coherent state or thermal radiation, then P is non-negative everywhere like an ordinary probability distribution. If, however, the quantum system has no classical analog, e.g. an incoherent Fock state or entangled system, then P is negative somewhere or more singular than a delta function.

This sweeping statement is unavailable in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog.

In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti-normal order. More recently, the positive P representation and a wider class of generalized P representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.

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Quasi-probability Distribution - Introduction

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