1 Introduction

Quantum state nonclassicality is a widely studied topic in physics [1]. A quantum state is said to be nonclassical when it cannot be considered a statistical mixture of coherent states. It means that the Glauber-Sudarshan [2, 3] representation of a nonclassical state is either nonpositive (it cannot be interpreted as a classical probability distribution) or includes derivatives of the Dirac delta function. Nonclassical light is essential to producing the entanglement for implementing quantum information protocols with continuous variables [4,5,6].

In general, the \(P\) -function is a generalized function that cannot be directly measured. Hence, various nonclassicality criteria (witness) based on the measurable quantities have been proposed. Moreover, the non-positivity of a variety of well-behaved quasi-probability distribution functions can be exploited for detecting state nonclassicality [7]. A familiar example of such quasi-distributions is the Wigner function [8,9,10,11].

A family of nonclassical states of particular interest is the Schrödinger cat states [12, 13]. They are superpositions of two macroscopically distinct coherent states and are particularly relevant to quantum information and computing [14]. Cat-like superpositions of nearly identical coherent states and their nonclassical properties were first examined in [15, 16]. These states are called near-coherent states and possess semi-classical features similar to coherent states while exhibiting nonclassical behavior. We can show that these states are proportional to the directional derivative of a coherent state with respect to its complex parameter. Therefore, a near-coherent state has an additional characteristic parameter rather than the complex parameter of the original coherent state. Higher-order directional derivatives are also proposed, and their nonclassical properties are studied in [17]. Moreover, quasi-Bell states, as a result of the two-mode superposition of two “Near” coherent states, are introduced in [18], and their entanglement properties are investigated.

In [15], a method for generating near-coherent states has been introduced. This method involves a conditional quantum state transformation, and the corresponding non-unitary operator has been determined [19, 20]. With this approach, any pure or mixed state can be inserted into the input port to obtain the output state. The physical interpretation of these generalized ‘near’ states has also been discussed. This paper focuses on studying the analytical and phase space representation of these generalized ‘near’ states. The remaining sections of this paper are organized as follows:

In Sect. 2, we review the concept of a near-pure state and demonstrate that when the conditional near-state generation operator acts on a coherent state, a near-coherent state is obtained.

Section 3 presents the transformation rule for the analytical representation of pure states.

Section 4 is devoted to determining the transformation rule for the Glauber-Sudarshan representation.

In Sect. 5, we investigate the effect of replacing an input coherent state with a thermal state and analyze thermalization’s impact on the output state’s success probability.

Section 6 focuses on searching for an integral transform that produces the output Wigner functions of near states.

Section 7 contains the conclusion.

Additionally, for completeness, we include an appendix to review the derivation of the conditional transformation operator for an optical setup designed to generate near-coherent states [15].

2 Near-coherent states and beyond

Near-coherent states can be created by superimposing two coherent states with almost identical complex parameters [15]

$$|{\psi }_{1}\rangle \propto \{\left|z+\delta z\rangle -\right|z\rangle \},{\hspace{1em}\hspace{1em}}\delta z=\left|\delta z\right|{e}^{i\phi },{\hspace{1em}\hspace{1em}}\left|\delta z\right|\to 0.$$
(1)

While \(|{\psi }_{1}\rangle\) seems proportional to the derivative of the normalized state \(|z\rangle\) with respect to complex parameter \(z\), this is not the case. Because \({z}^{*}\) is not a differentiable function of \(z\), we can only talk about the directional derivative \(\delta {z}^{*}/\delta z\). Using \(|z\rangle =\text{exp}\left(-z{z}^{*}/2\right){e}^{z{\widehat{a}^{\dagger}}}|0\rangle\) for coherent states and polar representations of \(\delta z=|\delta z|{e}^{i\phi }\), we have

$$\frac{\delta |z\rangle }{\delta z}=-\frac{1}{2}\left({z}^{*}+z{e}^{-i2\phi }\right)|z\rangle +{\widehat{a}^{\dagger}}|z\rangle ,$$
(2)
$$|{\psi }_{1}\rangle =N\delta z\left\{-\frac{1}{2}\left({z}^{*}+z{e}^{-i2\phi }\right)|z\rangle +{\widehat{a}^{\dagger}}|z\rangle \right\},$$
(3)

with \(N\) a normalization factor depending only on the direction of the derivative, \(\phi\), and the phase of the complex parameter \(z\) [15]. Therefore, \(|{\psi }_{1}\rangle\) is proportional to the directional derivative of the coherent state \(|z\rangle\) in the direction \(\phi\) of the complex plane. It is a superposition of the photon-added coherent state \({\widehat{a}^{\dagger}}|z\rangle\) and the coherent state \(|z\rangle\) itself. It must be noted that the direction of \(|{\psi }_{1}\rangle\) depends on phase \(\phi\), leading to an additional control parameter on the state. For example, if \(z=|z|{e}^{i\theta }\) and we chose \(\theta -\phi =\pm \pi /2\), then we have

$${z}^{*}+z{e}^{-i2\phi }=0,{\hspace{1em}}\Rightarrow {\hspace{1em}}\frac{\delta |z\rangle }{\delta z}={\widehat{a}^{\dagger}}|z\rangle ,$$
(4)

and the near-coherent state is just a photon-added state.

Near-coherent states can be generated through conditional quantum state preparation [19]. An example of such a preparation method is discussed in [15, 20]. The essential parts of this method are briefly reviewed in an appendix for use in the rest of the paper. This quantum state generation for near-coherent states is fully characterized by a non-unitary transformation operator \(\widehat{Y}\) with the following form

$$\widehat{Y}=\frac{\delta z}{2}{e}^{-\frac{\beta {\beta }^{*}}{2}}{\widehat{a}^{\dagger}}{e}^{\beta {\widehat{a}^{\dagger}}/\sqrt{2}}{\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}},$$
(5)

where \(\beta\) is a controllable complex parameter and

$$\delta z=-i\phi \beta /\sqrt{2},{\hspace{1em}\hspace{1em}}|\phi |\ll 1.$$
(6)

By the transformation operator, we mean that the output state can be obtained from a pure input state \(|{\psi }_{\text{in}}\rangle\) using the following relations.

$$|{\psi }_{\text{out}}\rangle =N\widehat{Y}|{\psi }_{\text{in}}\rangle ,{\hspace{1em}\hspace{1em}}N\equiv \langle {\psi }_{\text{in}}{|{\widehat{Y}^{\dagger}}\widehat{Y}|{\psi }_{\text{in}}\rangle }^{-1/2}.$$
(7)

It is evident that if one ignores all scalar factors in Eq. (5), one can rewrite the transformation operator \(\widehat{Y}\) as the derivative of the operator \(\widehat{Z}(\beta )\), where

$$\widehat{Y}\propto {\widehat{a}^{\dagger}}{e}^{\beta {\widehat{a}^{\dagger}}/\sqrt{2}}{\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}\propto \frac{d\widehat{Z}}{d\beta },{\hspace{1em}\hspace{1em}}\widehat{Z}(\beta )\equiv {e}^{\beta {\widehat{a}^{\dagger}}/\sqrt{2}}{\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}.$$
(8)

When the input state is a coherent state \(|\alpha \rangle ,{\hspace{0.17em}}\alpha \in {\mathbb{C}}\), the output state is a superposition of two other coherent states as follows:

$$|{\psi }_{\text{out}}\rangle =N\widehat{Y}|\alpha \rangle =N\frac{d}{d\beta }\widehat{Z}(\beta )|\alpha \rangle ,$$
(9)

where \(\widehat{Z}(\beta )|\alpha \rangle\) is some un-normalized coherent state. To show this, we use the following operator identity for coherent states:

$${t}^{{\widehat{a}^{\dagger}}\widehat{a}}|u\rangle ={e}^{-\frac{{|u|}^{2}}{2}(1-{t}^{2})}|tu\rangle ,{\hspace{1em}\hspace{1em}}t\in {\mathbb{R}}^{+},$$
(10)

and find

$$|{\psi }_{\text{out}}\rangle =N{e}^{-\frac{1}{4}{|\alpha |}^{2}}\frac{d}{d\beta }{e}^{\beta {\widehat{a}^{\dagger}}/\sqrt{2}}|\frac{\alpha }{\sqrt{2}}\rangle .$$
(11)

Now, we consider coherent states as displaced vacuum states \(|\xi \rangle ={e}^{-\frac{1}{2}{|\xi |}^{2}}{e}^{\xi {\widehat{a}^{\dagger}}}|0\rangle\) to obtain

$$|{\psi }_{\text{out}}\rangle =N{e}^{-\frac{1}{2}{\left|\alpha \right|}^{2}}\frac{d}{d\beta }\left({e}^{\frac{1}{4}{|\alpha +\beta |}^{2}}|\frac{\alpha +\beta }{\sqrt{2}}\rangle \right).$$
(12)

Hence, if one defines two nearly identical complex parameters

$$z\equiv \frac{\alpha +\beta }{\sqrt{2}},{\hspace{1em}\hspace{1em}}z+\delta z\equiv \frac{\alpha +\beta }{\sqrt{2}}+\frac{\delta \beta }{\sqrt{2}},$$
(13)

The output state would be proportional to the superposition \(|{\psi }_{\text{out}}\rangle \propto |z+\delta z\rangle -|z\rangle\), which means that it is near-coherent.

It must be noted that while optical setup Fig. 2 is designed for the generation of near-coherent states it introduces the transformation operator \(\widehat{Y}\) which can act on various pure and mixed states and generate a new class of “near states”. Moreover, since \(\widehat{Y}\)  is a partial matrix element of the unitary evolution operator \(\widehat{U}\) (relating the input and output ports of the setup) between pre- and post-selected states (a4), i.e.,

$$\widehat{Y}\propto {}_{b}\langle 0|{}_{c}\langle 1|{}_{d}\langle 0|\widehat{U}{|\beta \rangle }_{b}{|1\rangle }_{c}{|0\rangle }_{d},$$
(14)

it linearly depends on the state \({|\beta \rangle }_{b}\) of the input port (b). Therefore, by inserting a superposition state \({c}_{1}{|{\beta }_{1}\rangle }_{b}+{c}_{2}{|{\beta }_{2}\rangle }_{b}\) into this port, one can physically realize a superposition of two transformation operators

$$\widehat{Y}\propto {c}_{1}{\widehat{Y}}_{1}+{c}_{2}{\widehat{Y}}_{2},$$
(15)

where

$${\widehat{Y}}_{k}\propto {}_{b}\langle 0|{}_{c}\langle 1|{}_{d}\langle 0|\widehat{U}{|{\beta }_{k}\rangle }_{b}{|1\rangle }_{c}{|0\rangle }_{d},{\hspace{1em}\hspace{1em}}k=\text{1,2}.$$
(16)

As a result, in cases when \({\beta }_{1}=-{\beta }_{2}\), we have a new method to generate quantum states similar to near-cat states [21]. This issue will be discussed in another article. In the rest of the paper, we will declare the physical content of near states generated by setup Fig. 2 in various representations.

3 Transformation rule for the analytical representation of pure states

Now, let the input state be a pure state \(|{\psi }_{\text{in}}\rangle\). What is the physical interpretation of the output state? Due to the preparation method’s linearity with respect to input states, it is plausible to expand the input state by coherent states [22]. Using the analytical representation of the state we have

$$|{\psi }_{\text{in}}\rangle =\int \frac{{d}^{2}\alpha }{\pi }{\psi }_{\text{in}}({\alpha }^{*}){e}^{-\frac{1}{2}\alpha {\alpha }^{*}}|\alpha \rangle ,{\hspace{1em}\hspace{1em}}{\psi }_{\text{in}}(v)=\langle 0|{e}^{v\widehat{a}}|{\psi }_{\text{in}}\rangle .$$
(17)

In the transformed state, every coherent state \(|\alpha \rangle\) would be replaced by a superposition of two other almost identical coherent states, as given by Eq. (13). The following integral transformation of the input state gives the analytical representation of the output state

$${\psi }_{\text{out}}(z)=N\int \frac{{d}^{2}v}{\pi }{e}^{-{|v|}^{2}}{F}_{Y}(z,{v}^{*}){\psi }_{\text{in}}({v}^{*}),$$
(18)

where \(N\) is a normalization factor and

$${F}_{Y}(u,v)=\sum_{m,n}{Y}_{m,n}\frac{{u}^{m}{v}^{n}}{\sqrt{m!n!}},{\hspace{1em}\hspace{1em}}{Y}_{m,n}=\langle m|\widehat{Y}|n\rangle .$$
(19)

Here, \({Y}_{m,n}\) represents the matrix elements of \(\widehat{Y}\) in the energy basis, so

$${Y}_{m,n}=\frac{1}{\sqrt{2^n}}\langle{m}{\mid}\widehat{a}^{\dagger}{e}^{{\beta{\widehat{a}}^{\dagger}}/{\sqrt{2}}}{\mid}{n}\rangle=\frac{1}{\sqrt{n!2^{n}}}\langle{m}{\mid}{\widehat{a}^{\dagger{n+1}}}{e}^{\beta{\widehat{a}^{\dagger}}/{\sqrt{2}}}{\mid}0\rangle\cdot$$
(20)

By considering the last expression as an inner product of a number state and a coherent state, one can find

$${Y}_{m,n}=\sqrt{2}\frac{1}{\sqrt{m!n!}}{\left(\frac{d}{d\beta }\right)}^{n+1}{\left(\frac{\beta }{\sqrt{2}}\right)}^{m}.$$
(21)

Using this result in (19) we can obtain

$${F}_{Y}(u,v)=\sqrt{2}\sum_{n}{\left(\frac{d}{d\beta }\right)}^{n+1}\frac{{v}^{n}}{n!}{e}^{\frac{u\beta }{\sqrt{2}}}=u{e}^{\frac{u}{\sqrt{2}}\left(\beta +v\right)}.$$
(22)

Finally, Eq. (18) allows us to determine the analytical representation of the output state as follows

$${\psi }_{\text{out}}(z)=N\frac{d}{d\beta }\phi \left(z;\beta \right),{\hspace{1em}\hspace{1em}}\phi \left(z;\beta \right)=\sqrt{2}\int \frac{{d}^{2}v}{\pi }{e}^{-{\left|v\right|}^{2}}{e}^{\frac{z}{\sqrt{2}}\left(\beta +{v}^{*}\right)}{\psi }_{\text{in}}\left({v}^{*}\right).$$
(23)

The discussion above shows that a near- \(|{\psi }_{\text{in}}\rangle\) state can always be interpreted as a superposition of two almost identical states. By ‘almost identical’ we mean two unnormalized states with nearly identical analytical representations, \(\phi (z;\beta )\) and \(\phi (z;\beta +\delta \beta )\), as given by the formula (23).

4 Transformation rule for the \(P\)-function of mixed states

A conditional quantum state transformation can be characterized by a non-unitary operator \(\widehat{Y}\) such that [19]

$${\widehat{\rho }}_{\text{out}}=\frac{\widehat{Y}{\widehat{\rho }}_{\text{in}}{\widehat{Y}^{\dagger}}}{{\text{Tr}}\left[\widehat{Y}{\widehat{\rho }}_{\text{in}}{\widehat{Y}^{\dagger}}\right]}.$$
(24)

For the generation of near-coherent states, the operator \(\widehat{Y}\) has been calculated in [20] and is given by the formula (5). A natural way to characterize a conditional quantum state transformation of an input mixed state is to introduce it in terms of quasi-probabilities. Here, we find the Glauber-Sudarshan distribution function for the output states.

Suppose that the input state is a generally mixed state \({\widehat{\rho }}_{\text{in}}\) that can be represented by the Glauber-Sudarshan \(P\) –function \({P}_{\text{in}}\left(\alpha \right)\), so we can write

$${\widehat{\rho }}_{\text{in}}=\int {d}^{2}\alpha {P}_{\text{in}}(\alpha )|\alpha \rangle \langle \alpha |.$$
(25)

Therefore, in view of Eq. (24) and up to a normalization factor, the output state is

$${\widehat{\rho }}_{\text{out}}\propto \int {d}^{2}\alpha {P}_{\text{in}}(\alpha )\widehat{Y}|\alpha \rangle \langle \alpha |{\widehat{Y}^{\dagger}},$$
(26)

which is a similar combination of near-coherent states \(\widehat{Y}|\alpha \rangle \langle \alpha |{\widehat{Y}^{\dagger}}\) with the same weight function \({P}_{\text{in}}(\alpha)\). Now, we determine the normally ordered form of these states. Using identities \(|\xi \rangle ={e}^{-\frac{1}{2}{|\xi |}^{2}}{e}^{\xi {\widehat{a}^{\dagger}}}|0\rangle\) and \(|0\rangle \langle 0|=:{e}^{-{\widehat{a}^{\dagger}}\widehat{a}}:\) for the coherent states, and transformation operator (8) one can write

$$\widehat{Y}|\alpha \rangle \langle \alpha |{\widehat{Y}^{\dagger}}={e}^{-{|\alpha |}^{2}}{\widehat{a}^{\dagger}}{e}^{\beta {\widehat{a}^{\dagger}}/\sqrt{2}} {\hspace{0.17em}}{\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}:{e}^{\alpha {\widehat{a}^{\dagger}}}{e}^{-{\widehat{a}^{\dagger}}\widehat{a}}{e}^{{\alpha }^{*}\widehat{a}}:{\hspace{0.17em}}{\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}{e}^{{\beta }^{*}\widehat{a}/\sqrt{2}}\widehat{a}.$$
(27)

Now, by utilizing following formula, [22] one can find the Glauber-Sudarshan function corresponding to \({\widehat{\rho }}_{\text{out}}\)

$${P}_{\text{out}}\left(z\right)=\frac{{e}^{{\left|z\right|}^{2}}}{\pi }\int \frac{{d}^{2}u}{\pi }{e}^{z{u}^{*}-{z}^{*}u}\left({e}^{+{|u|}^{2}}\langle -u|{\widehat{\rho }}_{\text{out}}|u\rangle \right).$$
(28)

The central part of calculations related to the determination of

$$\begin{array}{c}\langle -u\left|\widehat{Y}\right|\alpha \rangle \langle \alpha \left|{\widehat{Y}^{\dagger}}\right|u\rangle ={e}^{-{\left|\alpha \right|}^{2}}\left(-u{u}^{*}\right){e}^{(\beta {u}^{*}+{\beta }^{*}u)/\sqrt{2}}\langle -u|{\hspace{0.17em}}{\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}:{e}^{\alpha {\widehat{a}^{\dagger}}}{e}^{-{\widehat{a}^{\dagger}}\widehat{a}}{e}^{{\alpha }^{*}\widehat{a}}:{\hspace{0.17em}}{\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}|u\rangle .\end{array}$$
(29)

By applying operator identity \({t}^{{\widehat{a}^{\dagger}}\widehat{a}}|u\rangle ={e}^{-\frac{{|u|}^{2}}{2}(1-{t}^{2})}|tu\rangle\) for coherent states, we can find

$$\langle -u|\widehat{Y}|\alpha \rangle \langle \alpha |{\widehat{Y}^{\dagger}}|u\rangle ={e}^{-{\left|u\right|}^{2}}{e}^{-{\left|\alpha \right|}^{2}}\left(-u{u}^{*}\right){e}^{\frac{u{(\alpha +\beta )}^{*}-{u}^{*}(\alpha +\beta )}{\sqrt{2}}}.$$
(30)

Using this result in Eq. (26) and then in Eq. (28), we have

$${P}_{\text{out}}(z)\propto {\partial }_{\beta }{\partial }_{{\beta }^{*}}{e}^{{\left|z\right|}^{2}}\int \frac{{d}^{2}u}{\pi }\frac{{d}^{2}\alpha }{\pi }{\hspace{0.17em}}{e}^{z{u}^{*}-{z}^{*}u}{e}^{-{\left|\alpha \right|}^{2}}{P}_{\text{in}}\left(\alpha \right){e}^{\frac{u{(\alpha +\beta )}^{*}-{u}^{*}\left(\alpha +\beta \right)}{\sqrt{2}}}.$$
(31)

Employing the following formula for integration with respect to \(u\)

$$\int \frac{{d}^{2}u}{\pi }{\hspace{0.17em}}{e}^{\gamma {u}^{*}-{\gamma }^{*}u}=\pi \delta (\gamma ),$$
(32)

the last result can be simplified as follows

$${P}_{\text{out}}(z)=N{e}^{-{\left|z\right|}^{2}}{\partial }_{\beta }{\partial }_{{\beta }^{*}}\left[{e}^{-\beta {\beta }^{*}+\sqrt{2}\left(z{\beta }^{*}+{z}^{*}\beta \right)}{P}_{\text{in}}\left(\sqrt{2}z-\beta \right)\right],$$
(33)

where \(N\) is a normalization constant. Formula (33) is very simple for applications, especially when the Glauber-Sudarshan function is smooth and differentiable. This result shows that if the input state has a well-behaved \(P\)-function, its corresponding near state also has a well-behaved Glauber function. Moreover, due to partial derivatives \({\partial }_{\beta }{\partial }_{{\beta }^{*}}\), classical states can transform into nonclassical ones. For example, if the input state is the thermal state with the mean photon number \(\overline{n }\) and the following \(P\)-function

$${P}_{\text{th}}(\alpha )=\frac{1}{\pi \overline{n}}\text{exp }\left(-\frac{{\left|\alpha \right|}^{2}}{\overline{n} }\right),$$
(34)

we have

$${P}_{\text{out}}(z)=N{e}^{{\left|z\right|}^{2}}\left\{\frac{\overline{n }+1}{\overline{n} }{\left|\sqrt{2}z-\beta \right|}^{2}-1\right\}\text{exp}\left(-\frac{\overline{n }+1}{\overline{n} }{|\sqrt{2}z-\beta |}^{2}\right),$$
(35)

which is consistent with the results of previous calculations [20].

5 Success probability for the near thermal state

Any conditional quantum state generation process randomly results in the desired state. In other words, we can achieve the desired state only when the post-selection is successful. Therefore, experimentally, it is essential to design the generation process so that the probability of success in obtaining the desired state in the output port is high. Moreover, thermal noises can affect input states, which can change the success probability. Now, for the near states’ generation problem, we can study the effect of this thermalization on the success probability.

Given our brief review of the conditional state generation for near states in the appendix section, for a given input state \({\widehat{\rho }}_{\text{in}}\), the success probability of obtaining \({\widehat{\rho }}_{\text{out}}\) is given by [19]

$${P}_{suc.}={\text{Tr}}\left\{{\widehat{\rho }}_{\text{in}}{\widehat{Y}^{\dagger}}\widehat{Y}\right\}.$$
(36)

Scalar factors in Eq. (5) cannot be omitted here to preserve the normalization condition for probabilities. In this section, we study the effect of thermalization on the success probability of generating near states. Let the input state be a thermal state with the density operator

$${\widehat{\rho }}_{\text{in}}=\frac{1}{1+\overline{n} }{\left(\frac{\overline{n} }{1+\overline{n} }\right)}^{{\widehat{a}^{\dagger}}\widehat{a}},$$
(37)

where, \(\overline{n }\) denotes the mean photon number of the state. When \(\overline{n }\to 0\), the input state is the vacuum state, and the output state, according to Eq. (13), will be a superposition of two states near the coherent state \(\left|\beta /\sqrt{2}\right.\rangle\) with yet unknown success probability. Here, we study the effect of increasing the mean photon number on the success probability of obtaining the corresponding near state. Using the proper form of the transformation operator in Eq. (5), we have:

Top of Form

$${P}_{suc.}=\frac{{|\delta z|}^{2}}{2}\frac{1}{1+\overline{n}}{e }^{-{|\beta |}^{2}}{\partial }_{\beta }{\partial }_{{\beta }^{*}}{\text{Tr}}\left\{{\left(\frac{\overline{n} }{1+\overline{n} }\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}{\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}{e}^{\frac{{\beta }^{*}\widehat{a}}{\sqrt{2}}}{e}^{\frac{\beta {\widehat{a}^{\dagger}}}{\sqrt{2}}}{\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}\right\}.$$
(38)

Because of identity \({\text{Tr}}\left(\widehat{A}\widehat{B}\right)={\text{Tr}}\left(\widehat{B}\widehat{A}\right)\) one can write

$${P}_{suc.}=\frac{{|\delta z|}^{2}}{2}\frac{1}{1+\overline{n}}{e }^{-{|\beta |}^{2}}{\partial }_{\beta }{\partial }_{{\beta }^{*}}{\text{Tr}}\left\{{e}^{\frac{\beta {\widehat{a}^{\dagger}}}{\sqrt{2}}}{\left(\frac{\overline{n} }{2\left(1+\overline{n }\right)}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}{e}^{\frac{{\beta }^{*}\widehat{a}}{\sqrt{2}}}\right\}.$$
(39)

Now, the trace can be calculated in the coherent state basis as follows:

Top of Form

$${P}_{suc.}=\frac{{|\delta z|}^{2}}{2}\frac{1}{1+\overline{n}}{e }^{-{|\beta |}^{2}}{\partial }_{\beta }{\partial }_{{\beta }^{*}}\int \frac{{d}^{2}\gamma }{\pi }{e}^{\left(\beta {\gamma }^{*}+{\beta }^{*}\gamma \right)/\sqrt{2}}\langle \gamma |{\left(\frac{\overline{n} }{2\left(1+\overline{n }\right)}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}|\gamma \rangle .$$
(40)

To this end, we use the operator identity \({M}^{{\widehat{a}^{\dagger}}\widehat{a}}=:{e}^{\left(M-1\right){\widehat{a}^{\dagger}}\widehat{a}}:\) for normal ordering and find

$$\langle \gamma |{\left(\frac{\overline{n} }{2\left(1+\overline{n }\right)}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}|\gamma \rangle ={e}^{-\frac{1}{2}\left(\frac{\overline{n }+2}{\overline{n }+1}\right){|\gamma |}^{2}},{\hspace{1em}\hspace{1em}}\Rightarrow$$
(41)
$${P}_{suc.}=\frac{{|\delta z|}^{2}}{2}\frac{1}{1+\overline{n}}{e }^{-{|\beta |}^{2}}{\partial }_{\beta }{\partial }_{{\beta }^{*}}\int \frac{{d}^{2}\gamma }{\pi }{e}^{-\frac{1}{2}\left(\frac{\overline{n }+2}{\overline{n }+1}\right){|\gamma |}^{2}+(\beta {\gamma }^{*}+{\beta }^{*}\gamma )/\sqrt{2}}.$$
(42)

Finally, by evaluating the Gaussian integral and partial derivatives, we can find

$${P}_{suc.}(\overline{n };|\beta |,|\delta z|)={\left|\delta z\right|}^{2}\frac{\overline{n }+1}{{\left(\overline{n }+2\right)}^{2}}\left\{1+\left(\frac{\overline{n }+1}{\overline{n }+2}\right){\left|\beta \right|}^{2}\right\}{e}^{-\left(\frac{1}{\overline{n }+2}\right){|\beta |}^{2}}.$$
(43)

For our problem, Fig. 1 depicts the general behavior of the success probability as a function of \(\overline{n }\). Even for small values of \(0<\overline{n }<1\), the success probability increases considerably, while the output state preserves its nonclassical behavior, as indicated by its calculated \(P\)-function in Eq. (35). Moreover, this probability reaches a maximum value at a specific value \(\overline{n }\) and tends to zero as it tends to infinity.

Fig. 1
figure 1

The success probability plotted against the mean photon number for \(|\beta |=5,{\hspace{1em}}|\delta z|=0.1\)  

Full size image

6 Transformation rule for the Wigner function of a mixed state

Another interesting quasi-probability distribution function, which serves as a witness of nonclassicality, is the Wigner function. The Wigner function of the quantum state \(\widehat{\rho }\) can be defined through the Wigner operator [22]

$${\widehat{U}}_{0}\left(\alpha ,{\alpha }^{*}\right)=2:{e}^{-2\left(\alpha -\widehat{a}\right)\left({\alpha }^{*}-{\widehat{a}^{\dagger}}\right)}:$$
(44)

by the following trace of the product formula

$$W\left(\alpha ,{\alpha }^{*}\right)={\text{Tr}}\left\{\widehat{\rho }{\widehat{U}}_{0}\left(\alpha ,{\alpha }^{*}\right)\right\}.$$
(45)

Having the Wigner function, we can reconstruct the density operator by the integral

$$\widehat{\rho }=\int \frac{{d}^{2}z}{\pi }W\left(z,{z}^{*}\right){\widehat{U}}_{0}\left(z,{z}^{*}\right).$$
(46)

While one can find the Wigner function of the output state from Eq. (33), we are searching for an integral transform that relates the state’s input and output Wigner functions. Using transformation operator (8) and the relation given by Eq. (46), up to a normalization factor, we have

$${\widehat{\rho }}_{\text{out}}\propto {\partial }_{\beta }{\partial }_{{\beta }^{*}}\int \frac{{d}^{2}z}{\pi }{W}_{\text{in}}\left(z,{z}^{*}\right){e}^{\beta {\widehat{a}^{\dagger}}/\sqrt{2}}{\hspace{0.17em}}{\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}{\widehat{U}}_{0}(z,{z}^{*}){\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}{e}^{{\beta }^{*}\widehat{a}/\sqrt{2}}.$$
(47)

Now, by applying the coherent state ket-bra form of the Wigner operator [23]

$${\widehat{U}}_{0}\left(z,{z}^{*}\right)=2\int \frac{{d}^{2}\gamma }{\pi }{e}^{z{\gamma }^{*}-{z}^{*}\gamma }|z+\gamma \rangle \langle z-\gamma |{\hspace{0.17em}},$$
(48)

and operator identities

$${t}^{{\widehat{a}^{\dagger}}\widehat{a}}|u\rangle =\text{exp}\left[-\frac{{\left|u\right|}^{2}}{2}\left(1-{t}^{2}\right)\right]|tu\rangle ,{\hspace{1em}}t\in {\mathbb{R}}^{+},$$
(49)
$$|\xi \rangle ={e}^{-\frac{1}{2}{|\xi |}^{2}}{e}^{\xi {\widehat{a}^{\dagger}}}|0\rangle ,{\hspace{1em}\hspace{1em}}|0\rangle \langle 0|=:{e}^{-{\widehat{a}^{\dagger}}\widehat{a}}:$$
(50)

for coherent states one can find the following normally ordered Gaussian integral

$${\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}{\widehat{U}}_{0}(z,{z}^{*}){\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}=2{e}^{-{|z|}^{2}}:{e}^{\frac{z{\widehat{a}^{\dagger}}+{z}^{*}\widehat{a}}{\sqrt{2}}-{\widehat{a}^{\dagger}}\widehat{a}}\int \frac{{d}^{2}\gamma }{\pi }{e}^{-{\left|\gamma \right|}^{2}+\left(z-\frac{\widehat{a}}{\sqrt{2}}\right){\gamma }^{*}-\left({z}^{*}-\frac{{\widehat{a}^{\dagger}}}{\sqrt{2}}\right)\gamma }:$$
(51)

By evaluating the integral and simplifying the result we can find

$${\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}{\widehat{U}}_{0}(z,{z}^{*}){\left(\frac{1}{\sqrt{2}}\right)}^{{\widehat{a}^{\dagger}}\widehat{a}}=2{e}^{-2{|z|}^{2}}:{e}^{-\frac{3}{2}{\widehat{a}^{\dagger}}\widehat{a}+\sqrt{2}\left(z{\widehat{a}^{\dagger}}+{z}^{*}\widehat{a}\right)}:$$
(52)

In view of this result, and Eq. (47) one can write

$${\widehat{\rho }}_{\text{out}}\propto {\partial }_{\beta }{\partial }_{{\beta }^{*}}\int \frac{{d}^{2}z}{\pi }{e}^{-2{\left|z\right|}^{2}}{W}_{\text{in}}\left(z,{z}^{*}\right){\hspace{0.17em}}:{e}^{-\frac{3}{2}{\widehat{a}^{\dagger}}\widehat{a}+\left(\sqrt{2}{z}^{*}+\frac{{\beta }^{*}}{\sqrt{2}}\right)\widehat{a}+\left(\sqrt{2}z+\frac{\beta }{\sqrt{2}}\right){\widehat{a}^{\dagger}}}:$$
(53)

In this formula, the right-hand side is normally ordered, so to obtain the Wigner function of the output state, one can use the ket-bra form of the Wigner operator, as given in Eq. (48), and find

$$\begin{array}{c}{W}_{\text{out}}(\alpha ,{\alpha }^{*})\propto {\partial }_{\beta }{\partial }_{{\beta }^{*}}\\ \int \frac{{d}^{2}z}{\pi }\frac{{d}^{2}\gamma }{\pi }{e}^{-2{\left|z\right|}^{2}}{e}^{-2{\left|\gamma \right|}^{2}}{W}_{\text{in}}\left(z,{z}^{*}\right){\hspace{0.17em}}{e}^{-\frac{3}{2}{\left(\alpha -\gamma \right)}^{*}\left(\alpha +\gamma \right)+\left(\sqrt{2}z+\frac{\beta }{\sqrt{2}}\right){\left(\alpha -\gamma \right)}^{*}+\left(\sqrt{2}{z}^{*}+\frac{{\beta }^{*}}{\sqrt{2}}\right)\left(\alpha +\gamma \right)}.\end{array}$$
(54)

Integration with respect to \(\gamma\) can be evaluated separately

$$\begin{array}{c}{W}_{\text{out}}\left(\alpha ,{\alpha }^{*}\right)\propto {e}^{-\frac{3}{2}{|\alpha |}^{2}}{\partial }_{\beta }{\partial }_{{\beta }^{*}}\\ {e}^{\frac{\alpha {\beta }^{*}+{\alpha }^{*}\beta }{\sqrt{2}}}\int \frac{{d}^{2}z}{\pi }{e}^{-2{\left|z\right|}^{2}+\sqrt{2}\left(\alpha {z}^{*}+{\alpha }^{*}z\right)}{W}_{\text{in}}\left(z,{z}^{*}\right)\int \frac{{d}^{2}\gamma }{\pi }{e}^{-\frac{1}{2}{\left|\gamma \right|}^{2}+\left(\sqrt{2}{z}^{*}+\frac{{\beta }^{*}}{\sqrt{2}}-\frac{3}{2}{\alpha }^{*}\right)\gamma -\left(\sqrt{2}z+\frac{\beta }{\sqrt{2}}-\frac{3}{2}\alpha \right){\gamma }^{*}},\end{array}$$
(55)

and, after simplifying the result, we can find the final result as follows

$$\begin{array}{c}{W}_{\text{out}}(\alpha ,{\alpha }^{*})\propto {e}^{-6{|\alpha |}^{2}}{\partial }_{\beta }{\partial }_{{\beta }^{*}}\\ {e}^{-\beta {\beta }^{*}+\frac{4}{\sqrt{2}}\left(\alpha {\beta }^{*}+{\alpha }^{*}\beta \right)}\int \frac{{d}^{2}z}{\pi }{e}^{-6{\left|z\right|}^{2}+2\left[\left(2\sqrt{2}{\alpha }^{*}-{\beta }^{*}\right)z+\left(2\sqrt{2}\alpha -\beta \right){z}^{*}\right]}{W}_{\text{in}}\left(z,{z}^{*}\right).\end{array}$$
(56)

While this result is more complicated than Eq. (33), unlike Glauber’s function, the Wigner function always exists, and this formula provides the Wigner function of the generated near state in output (a) of setup Fig. 2. Specifically, formula (56) can be easily applied to Gaussian states. For example, when the input state is a vacuum state, up to a normalization factor \(N\), one can find

$${W}_{\text{in}}\left(z,{z}^{*}\right)=2{e}^{-2{|z|}^{2}}{\hspace{1em}}\Rightarrow {\hspace{1em}}{W}_{\text{out}}\left(\alpha ,{\alpha }^{*}\right)=N{e}^{-2{\left|\alpha \right|}^{2}}{\partial }_{\beta }{\partial }_{{\beta }^{*}}{e}^{-\frac{1}{2}\beta {\beta }^{*}+\sqrt{2}\left(\alpha {\beta }^{*}+{\alpha }^{*}\beta \right)},$$
(57)

that after some algebra can be simplified as follows

Fig. 2
figure 2

An optical setup for the conditional generation of near-coherent states and more

Full size image
$${W}_{\text{out}}\left(\alpha ,{\alpha }^{*}\right)=N{e}^{-\frac{1}{2}{\left|\beta \right|}^{2}}{e}^{-2{\left|\alpha \right|}^{2}+2\sqrt{2}\text{Re}\left(\alpha {\beta }^{*}\right)}\left({\left|\sqrt{2}\alpha -\frac{\beta }{2}\right|}^{2}-\frac{1}{2}\right).$$
(58)

This Wigner function takes negative values in the region \(\left|\alpha -\frac{\beta }{2\sqrt{2}}\right|<\frac{1}{2}\) of the complex plane, which is evidence for the output state’s nonclassicality.

7 Conclusion

In this article, we studied the generation of near states in both analytical (for pure states) and phase space representations (for generally mixed states). We derived relations for the transformation of the analytical and phase space representations, which are closely related to the state nonclassicality, such as the Glauber-Sudarshan and Wigner functions. Additionally, we investigated the effect of thermalization of the input state on the success probability in the conditional generation of the near states. We demonstrated that a minor thermalization of the input state effectively increases the success probability without destroying the nonclassical properties of the state.