We study two families of four-photon superpositions of the Fock states: even vacuum squeezed states (EVSS) and orthogonal-even coherent states (OECS). These families are distinguished due to several properties: for certain values of parameters, they give the fourth-order uncertainty products close to the known minimal value (which is lower than for the Gaussian states); they have equal dimensionless values of the second- and fouth-order moments of the coordinate and momentum for all values of parameters; they possess zero covariances for all values of parameters. Since these states are obviously non-Gaussian, we consider them as good candidates to compare several different measures of non-Gaussianity proposed by different authors for the past fifteen years. The reference Gaussian states in all examples are thermal states dependent on a single parameter (an effective temperature or the coordinate variance). We analyze the measures based on the normalized Hilbert–Schmidt distance and the relative entropy (introduced by Genoni–Paris–Banaszek), the fidelity measure (Ghiu–Marian–Marian) and its logarithmic analog (Baek–Nha), as well as the Mandilara–Karpov–Cerf “Gaussianity parameter”. These measures are compared with the kurtosis of the coordinate probability density and with the non-Gaussian behavior of the Wigner function.

Non–Gaussian states play an important role in quantum optics. It is enough to remember that the basic Fock states are strongly non-Gaussian. Many families of such states were studied during recent decades: see, e.g., the review [

It is well known that the following equality holds for any Gaussian state (either pure or mixed):

Therefore, one may be tempted to consider the normalized kurtosis as a simple measure of non-Gaussianity

For this reason, the main trend is to use the statistical operator

Several authors proposed simple measures containing the only positive parameter

The measure

The relative entropy

Our interest in the problem of non-Gaussianity originated from studies of various generalizations of uncertainty relations. It is well known that all

Several generalizations of Inequality (

Looking at inequality Equation (

Another idea is to replace zero in the right-hand side of Equation (

This definition holds for mixed quantum states as well as for pure ones. Obviously,

Various families of quantum states were analyzed in connection with the non-Gaussianity measures described above. We give a brief and certainly incomplete list of publications mentioning pure quantum states only. Single Fock states were considered in [

In this paper, we add two interesting families of pure non-Gaussian states to this list. From all known measures, we choose five examples in which all ingredients can be analytically calculated rather easily: Equations (

The first example, considered in detail in

Using the well-known methods of the

Consequently,

Similar superpositions with

The second family of states, considered in

These kinds of states were studied for a long time from different points of view. They were named “four-photon states” [

The four-photon superpositions

These are consequences of the relations between the quadratures and annihilation/creation operators,

(hereafter, we use dimensionless variables with

Respectively,

The normalized thermal statistical operator of the harmonic oscillator can also be written in terms of the annihilation/creation operators as

Consequently, the operator

The relative entropy measure of non-Gaussianity for pure quantum states can be written as follows,

Now, let us consider an arbitrary operator

Formula (

In view of Formula (

Using formula 5.12.1.4 from [

Consequently, the wave function

The probability density has the form

After the integration, using identities Equations (

Calculating the Fourier transform of function

Consequently,

Simple calculations show that the second-order moment also depends on the only parameter

Note that

The fourth-order moment depends on

The minimum of

Numerical calculations yield

For small values of

The Wigner function of state Equation (

This function is invariant with respect to the rotation by 90 degrees in the phase plane:

Remember that we use dimensionless variables, assuming formally

Therefore, the non-Gaussianity of

Note that the section

The sum Equation (

Plots of measures Equations (

The behavior of the normalized kurtosis is more interesting. For

In the case of OECS Equation (

On the other hand,

The minimum

The normalized wave function of OECS in the coordinate representation has the following form:

The behavior of this function strongly depends on the phase

If

The Wigner function is the sum of 16 exponential terms with

Plots of the sections

In the case involved, the trace

Plots of measures Equations (

The evolution of the kurtosis depends on the phase

It is interesting to compare the degrees of non-Gaussianity of two families of states: EVSS and OECS. For this purpose, these degrees must be calculated for equal values of some parameter. A relevant physical parameter could be the state energy, or, equivalently for the four-photon superpositions, the coordinate variance

We have compared five different measures of non-Gaussianity for two interesting four-photon superpositions of the Fock states: even vacuum squeezed states (EVSS) and orthogonal-even coherent states (OECS). Four measures show a monotonous growth when parameters characterizing the “size” of superpositions are increased. However, it is difficult to choose any of these measures as the “best” one: all of them seem more or less equivalent, at least for the pure quantum states studied in this paper. The behavior of the “Gaussianity” measure Equation (

M.C.d.F.: analytical and numerical calculations, plotting figures; V.V.D.: conceptualization, methodology, analytical calculations, and writing the paper. All authors have read and agreed to the published version of the manuscript.

This research received no external funding.

The authors thank A.C. Pedroza and A.E. Santana for the interest in the work and useful remarks. V.V.D. acknowledges the partial support of the Brazilian funding agency Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

The authors declare no conflict of interest.

The probability density Equation (

The section

The behavior of the non-Gaussianity measures (

Real wave functions of OECS for different values of parameter

The coordinate probability density of OECS for different values of parameter

The sections of function

The sections of function

The behavior of the non-Gaussianity measures (

Non-Gaussianity measures as functions of the inverse fidelity