Photon number and optical tomograms for Gaussian states

TL;DR

This paper reviews the probability representation of quantum states in optical and photon number tomography, establishing connections between different measurement approaches and deriving new mathematical relations for Gaussian states.

Contribution

It provides explicit links between photon number and optical tomograms and introduces new integral relations involving Hermite and Laguerre polynomials for Gaussian states.

Findings

Derived explicit connection between photon number and optical tomograms.

Established new integral relations for Hermite and Laguerre polynomials.

Analyzed Gaussian photon states such as squeezed and correlated states.

Abstract

A review of probability representation of quantum states in given for optical and photon number tomography approaches. Explicit connection of photon number tomogram with measurable by homodyne detector optical tomogram is obtained. New integral relations connecting Hermite polynomials of two variables with Laguerre polynomials are found. Examples of generic Gaussian photon states (squeezed and correlated states) are studied in detail.

Equations

$$W(p,q)=d^{-\frac{1}{2}}\exp \left\{-(2d{)}^{-1}\left[{\sigma }_{qq}(p-<p>{)}^2+{\sigma }_{pp}(q-<q>{)}^2-2{\sigma }_{pq}(p-<p>)(q-<q>)\right]\right\}.$$

$$\sigma=\left(\begin{array}[]{cc}\sigma_{pp}&\sigma_{pq}\\ \sigma_{pq}&\sigma_{qq}\end{array}\right).$$

$$d\ge \frac{1}{4},$$

$$d=\det \sigma ={\sigma }_{pp}{\sigma }_{qq}-{\sigma }_{pq}^2.$$

$$T=\text{mbox}Tr\sigma ={\sigma }_{pp}+{\sigma }_{qq}.$$

$${\mathcal{P}}_n=\text{mbox}Tr\hat{\rho }|n><n|,n=0,1,2,...,$$

$$\hat{a}†\hat{a}|n>=n|n>.$$

$${\mathcal{P}}_n={\mathcal{P}}_0\frac{H_{nn}^{\{\mathbf{R}\}}(y_1,y_2)}{n!},$$

$$R_{11}=R_{22}^{\ast }=\frac{2\left({\sigma }_{pp}-{\sigma }_{qq}-2i{\sigma }_{pq}\right)}{1+2T+4d},R_{12}=\frac{1-4d}{1+2T+4d}.$$

$$y_1=y_2^{\ast }=\frac{2\left[(T-1)z^{\ast }+({\sigma }_{pp}-{\sigma }_{qq}+2i{\sigma }_{pq})z\right]}{2T-4d-1}.$$

$$z=2^{-\frac{1}{2}}(<q>+i<p>).$$

$${\mathcal{P}}_0=(d+\frac{1}{2}T+\frac{1}{4}{)}^{-\frac{1}{2}}\exp \left[-\frac{<p{>}^2(2{\sigma }_{qq}+1)+<q{>}^2(2{\sigma }_{pp}+1)-4{\sigma }_{pq}<p><q>}{1+2T+4d}\right].$$

$$\exp \left[-\frac{1}{2}(R_{11}{\beta }^{\ast 2}+R_{22}{\alpha }^2+2R_{12}\alpha {\beta }^{\ast })+R_{11}{\beta }^{\ast }{y}_1+R_{22}\alpha y_2+R_{12}(\alpha y_1+{\beta }^{\ast }{y}_2)\right]$$

$$=\sum _{m,n=0}^{\infty }\frac{H_{mn}^{\{\mathbf{R}\}}(y_1,y_2)}{n!m!}{\alpha }^n{\beta }^{\ast m},$$

$$\hat{X}=\mu \hat{q}+\nu \hat{p},$$

$$w\left(X,\mu ,\nu \right)=\int \exp \left[-ik(X-\mu q-\nu p)\right]W(q,p)\frac{dkdqdp}{(2\pi {)}^2}.$$

$$w_0(X,\theta )=w(X,\cos \theta ,\sin \theta ).$$

$$W(q,p)=\frac{1}{2\pi }\int w\left(X,\mu ,\nu \right)\exp \left[-i\left(\mu q+\nu p-X\right)\right]d\mu d\nu dX.$$

$$\omega (n,\alpha )=⟨n\mid \hat{D}(\alpha )\hat{\rho }{\hat{D}}^{-1}(\alpha )\mid n⟩$$

$$\alpha =\text{mbox}Re\alpha +i\text{mbox}Im\alpha ,$$

$$\hat{D}(\alpha )=\exp (\alpha {\hat{a}}^{†}-{\alpha }^{\ast }\hat{a}).$$

$$\omega(n,\alpha)=\frac{P_{0}(\alpha)H^{\{\mbox{\bf R}\}}_{n\,n}\Big{(}y_{1}(\alpha),y_{2}(\alpha)\Big{)}}{n!},$$

$$y_1(\alpha )=y_2^{\ast }(\alpha )$$

$$P_0(\alpha )$$

$$\omega (n,\alpha )=\int w(X,\mu ,\nu )K(X,\mu ,\nu ,n,\alpha )dXd\mu d\nu .$$

$$K(X,\mu ,\nu ,n,\alpha )=\frac{1}{2\pi }⟨n\mid {\hat{D}}^{+}(\alpha )e^{i(X-\mu \hat{q}-\nu \hat{p})}\hat{D}(\alpha )\mid n⟩.$$

$$K(X,\mu ,\nu ,n,\alpha )=\frac{1}{2\pi }\exp \left[iX+\frac{\nu -i\mu }{\sqrt{2}}{\alpha }^{\ast }-\frac{\nu +i\mu }{\sqrt{2}}\alpha \right]⟨n\mid \hat{D}\left(\frac{\nu -i\mu }{\sqrt{2}}\right)n⟩.$$

$$D_{nn}(\gamma )=⟨n\mid \hat{D}(\gamma )\mid n⟩=e^{-\frac{\mid \gamma {\mid }^2}{2}}{L}_n(\mid \gamma {\mid }^2),$$

$$\gamma =\frac{\nu +i\mu }{\sqrt{2}}$$

$$K(X,\mu ,\nu ,n,\alpha )=\frac{1}{2\pi }\exp \left[iX+\frac{\nu -i\mu }{\sqrt{2}}{\alpha }^{\ast }-\frac{\nu +i\mu }{\sqrt{2}}\alpha \right]L_n\left(\frac{{\nu }^2+{\mu }^2}{2}\right).$$

Full text

1 Introduction

In probability representation [1] of quantum mechanics the system states are associated with a probability distribution function. This probability distribution is connected with wave function or density matrix by an integral transform like, e.g. Radon transform [2]. There are different kinds of probabilities (called tomographic probabilities or tomograms) describing quantum states. For example symplectic tomograms were introduced in [3]. They, for particular choice of parameters, provide optical tomograms [4, 5]. The photon number tomography [6] uses probability distributions considered in [7, 8] related to density matrices by some integral kernels. Photon number tomogram is probability distribution for discrete random variable $n=0,1,2\ldots$. The probability distribution depends also on a complex amplitude parameter $\alpha$. The photon number tomograms were discussed in [9]-[14].

The aim of our paper is to connect the photon number tomogram with the symplectic and optical tomograms and to find the corresponding integral kernels in explicit form. As concrete example we will consider the quantum states of photon for Gaussian Wigner function.

2 Photon distributions in the states described by the Gaussian

Wigner functions

The most general mixed squeezed state of the one-mode light is described by the Wigner function $W(p,q)$ of the generic Gaussian form with five real parameters (see, e.g., [15]),

[TABLE]

Here $<p>$ and $<q>$ are the mean values of the “momentum” and “position” operators (we assume $\hbar=\omega=1$). The other three parameters are matrix elements of the real symmetric variance matrix,

[TABLE]

These matrix elements cannot be quite arbitrary, since they must satisfy the Schrödinger-Robertson uncertainty relation [16, 17]

[TABLE]

where d is the invariant of matrix $\sigma$,

[TABLE]

Another invariant parameter which will be used below is

[TABLE]

The photon distribution function ${\cal P}_{n}$, i.e. the probability to have $n$ photons in the state described by the density operator $\hat{\rho}$,corresponding to the Wigner function (1) is given by the formula

[TABLE]

where $|n>$ is the eigenstate of the number operator $\hat{a}{\dagger}\hat{a}$ ($\hat{a}$ and $\hat{a}^{\dagger}$ are photon annihilation and creation operators)

[TABLE]

Evidently, ${\cal P}_{n}$ is nothing but the the diagonal matrix element $\rho_{nn}$ of the density operator $\hat{\rho}$ in the Fock basis.

In [18] the photon distribution function ${\cal P}_{n}$ for the generic Gaussian state (1) is obtained in the form

[TABLE]

where $H_{nn}^{\{{\bf R}\}}(y_{1},y_{2})$ are Hermite polynomials of two variables. Elements of the symmetric matrix

{\bf R}=\left(\begin{array}[]{cc}R_{11}&R_{12}\\ R_{12}&R_{22}\end{array}\right)

determining the Hermite polynomial are given by formula

[TABLE]

The arguments of Hermite polynomials are of the form

[TABLE]

The complex parameter $z$ is given by the relation

[TABLE]

The probability to have no photons ${\cal P}_{0}$ is given by the formula

[TABLE]

The generating function for the Hermite polynomials of two variables is [19, 20]

[TABLE]

[TABLE]

3 Notion of Quantum State in symplectic and optical tomography approaches

It was shown [3] that for the generic linear combination of quadratures which is a measurable observable $\left(\hbar=1\right)$

[TABLE]

where $\hat{q}$ and $\hat{p}$ are the position and momentum, respectively, the symplectic tomogram $w\,(X,\,\mu,\,\nu)$ (normalized with respect to the variable $X$), depending on the two extra real parameters $\mu$ and $\nu,$ is related to the state of the quantum system expressed in terms of its Wigner function $W(q,\,p)$ as follows:

[TABLE]

The physical meaning of the parameters $\mu$ and $\nu$ is that they describe an ensemble of rotated and scaled reference frames in which the position $X$ is measured. For $\mu=\cos\,\theta$ and $\nu=\sin\,\theta,$ the symplectic tomogram (14) is the distribution for the homodyne-output variable used in optical tomography [5] and named optical tomogram

[TABLE]

Formula (14) can be inverted and the Wigner function of the state can be expressed in terms of the symplectic tomogram [3] :

[TABLE]

Since the Wigner function determines completely the quantum state of a system and, on the other hand, this function itself is completely determined by the symplectic tomogram (or its partial case optical tomogram), one can understand the notion of the quantum state in terms of the classical marginal distribution for squeezed and rotated quadrature. So, we say that the quantum state is given if the position probability distribution $w\left(X,\,\mu,\,\nu\right)$ in an ensemble of rotated and scaled reference frames in classical phase space is given. It is worth noting, that the information contained in the symplectic tomogram $w\left(X,\,\mu,\,\nu\right)$ is overcomplete. To determine the quantum state completely, it is sufficient to give the function for arguments with the constraints $\left(\mu^{2}+\nu^{2}=1\right)$ which corresponds to the optical tomography scheme [5, 21, 22], i.e., $\mu=\cos\theta$ and the rotation angle $\theta$ labels the reference frame in classical phase space.

4 Photon number tomography

The photon-number tomogram defined by the relation

[TABLE]

is the function of integer photon number $n$ and complex number

[TABLE]

where $\hat{\rho}$ is the state density operator and $\hat{D}(\alpha)$ is the Weyl displacement operator

[TABLE]

The photon-number tomogram associated with one-mode mixed light state,described by the Wigner function of generic Gaussian form (1) was obtained explicitly in terms of the Hermite polynomials of two variables in [9] and is of the form

[TABLE]

where the elements of the matrix R, which determines the Hermite polynomial, are described by formulae (8). The arguments of the Hermite polynomial are

[TABLE]

where $d$ the determinant of real symmetric quadrature variance matrix $\sigma$, and $T$ is its trace are determined by formulas (4,5). The probability to have no photons $P_{0}(\alpha)$ reads

[TABLE]

where $L=1+2T+4d$. Thus we obtained the photon number tomogram of the one-mode Gaussian state in explicit form expressed in terms of Hermite polynomial of two variables.

5 The relation of optical tomogram and photon number tomogram

In the section we find the relations among optical, symplectic and photon numbers tomograms of quantum states. The photon number tomogram can be expressed in terms of the symplectic tomogram by using the integral transform

[TABLE]

Here the kernel of the integral transform is expressed in terms of matrix elements of the displacement operator

[TABLE]

The explicit dependence of the kernel on the real parameters of the symplectic transform is given by the expression

[TABLE]

Using the known formula for the diagonal elements of the displacement operator

[TABLE]

where

[TABLE]

one has the kernel expressed in terms of Laguerre polynomial

[TABLE]

In view of this the photon number tomogram is expressed in terms of optical tomogram as follows

[TABLE]

Here $\alpha$ is the complex number

[TABLE]

Let us introduce the characteristic function

[TABLE]

Then the photon number tomogram reads

[TABLE]

Here we introduce momenta of optical tomogram

[TABLE]

For example, one can check for the ground oscillator state we have

[TABLE]

Let us consider an example of excited oscillator state

[TABLE]

One can show that the photon number tomogram of the state reads

[TABLE]

and

[TABLE]

The symplectic tomogram of this state reads

[TABLE]

Applying general relation (21) we get for $m\leq n$

[TABLE]

which provides a new integral relation of Hermite and Laguerre polynomials. For Gaussian states with the optical tomogram

[TABLE]

we have

[TABLE]

Comparing this formula with (18) we see that the Hermite polynomials of two variables are expressed in terms of Laguerre polynomials via the integral transform. For ground state with optical tomogram

[TABLE]

one has the Poissonian photon number tomogram

[TABLE]

6 Conclusion

We point out the main results of our work. We expressed measurable optical tomogram [21, 22] in terms of photon number tomogram by invertable map. It means that one can measure photon number tomogram by the homodyne detector. The entropies associated with photon number tomograms like tomographic Shannon entropy [23] also can be measured by homodyne detecting photon qunatum states. We will study this aspect of photon number tomography in future papers.

The obtained relation of optical tomogram and photon number tomogram provides possibility of measuring photon statistics by means of homodyne detector. In fact, the primary experimental data in homodyne measuring the photon quantum state are given in the form of optical tomogram $w_{0}(X,\theta)$. All the other quantum characteristics can be expressed in terms of the optical tomogram. Thus tomogram (26) with the $\alpha_{1}=\alpha_{2}=0$ provides the photon distribution function in terms of the optical tomogram. This relation can be also used to check accuracy of measuring quantum state by means of measuring photon statistics using both measuring homodyne quadrature and counting the photons. The results of such measurements must be compatible and the compatibility condition is given by (26).

7 Acknoledgements

This study was supported by the Russian Foundation for Basic Research under the project Nos.07-02-00598-a, 09-02-00142-a.