Most researches in von Neumann type quantum measurement in recent years has focused on postseleceted weak measurement with sufficiently weak coupling between the measured system and pointer since it is useful to study the nature of the quantum world. This kind of postsselected weak measurement theory is originally proposed by Aharonov, Albert, and Vaidman in 1988,^[1] and considered as a generalized version of standard von Neumann type measurement.^[2] The result of weak coupling postselected weak measurement is called the “weak value”, and generally is a complex number. One of the special feature of the weak value is that it can take the values which lie beyond the normal eigenvalue range of corresponding observable, and this effect will be very clear if the pre- and post-selected states are almost orthogonal. This feature of the weak value is called the signal amplification property of postselected weak measurement and its first weak signal amplification property experimentally demonstrated in 1991.^[3] After that, it has been widely used to elucidate tremendous fundamental problems in quantum mechanics. For details about the weak measurement and its applications in signal amplification processes, we refer the readers to the recent overview of the field.^[4,5] As we mentioned earlier, in postselected weak measurement the interaction strength is weak, and it is enough to consider up to the first-order evolution of a unitary operator for the whole measurement processes. However, if we want to connect the weak and strong measurement, check to clear the measurement feedback of postselected weak measurement and analyze experimental results obtained in nonideal measurements, the full order evolution of a unitary operator will be needed.^[6–8] We call this kind of measurement the postselected von Neumann measurement. We know that in some quantum metrology problems, the precision of the measurement depends on measuring devices and requires to optimize the pointer states. The merits of postselected von Neumann measurement can be seen in pointer optimization schemes.

Recently, the state optimization problem in postselected von Neumann measurement has been presented widely, such as taking the Gaussian states,^[4,9] Hermite–Gaussian or Laguerre–Gaussian states, and non-classical states.^[10,11] The advantages of non-classical pointer states in increasing postselected measurement precision have been examined in recent studies.^[10,12,13] Furthermore, in Ref. [14], the authors studied the effects of postselected measurement characterized by a modular value^[15] to show the properties of semi-classical and non-classical pointer states considering the coherent, coherent squeezed, and Schrödinger cat state as a pointer. Since in their scheme the pointer operator is a projection operator onto one of the states of the basis of the pointer's Hilbert space and interaction strength only has taken one definite value, it cannot completely describe the effects of postselected measurement on the properties of the radiation field. However, to the knowledge of the author, the issue related to the study of the effects of postselected von Neumann measurement considering all interaction strengths to the inherent properties such as photon distribution, photon statistics, and squeezing effects of radiation field have not been handled and need to be investigated.

Furthermore, the above-mentioned properties of quantum radiation fields have many useful technological applications in quantum optics and quantum information processing such as single photon generation and detection,^[16] gravitational wave detection,^[17,18] quantum teleportation,^[19–22] quantum computation,^[23] generation and manipulation of atom–light entanglement,^[24–26] and precision measurements,^[27] etc. We know that the realization of these processes depends on the optimization of the related input quantum states such as coherent state,^[28] squeezed state^[29,30] and Schrödinger cat state.^[31–33] As introduced in previous part, the postselected weak measurement technique has more advantages on signal optimization problem^[34] and precision metrology^[35–43] than traditional measurement theory. Thus, the investigation of the effects of postselected von Neumann measurement on the statistical and squeezing properties of radiation fields is worth studying to optimize the related quantum states to provide more effective methods for the implementation of the above-mentioned technological processes.

In this paper, motivated by the above-listed problems, we describe and examine the effects of postselected von Neumann measurement characterized by postselection and weak value on the properties of single-mode radiation fields. In order to achieve our goal, we choose the three typical states such as coherent state, squeezed vacuum state, and Schrödinger cat state as pointers and consider their polarization degree of freedom as a measured system, respectively. By taking full-order evolution of the unitary operator of our system, we separately study the photon distributions, statistical properties and squeezing effects of radiation fields and compare those with the initial state cases. To give more details about the effects of postselected von Neumann measurement on radiation fields, we plot more figures by considering all parameters that are related to the properties of those radiation fields, respectively. This study shows that the postselected measurement changes the properties of single-mode radiation fields dramatically for strong and weak measurement regimes with specific weak values. Especially the photon statistics and squeezing effect of coherent pointer state is too sensitive for the postselected von Neumann measurement processes.

This paper is organized as follows: In Section 2, we describe the model of our theory by brief reviewing the postselected von Neumann measurement and outline the tasks we want to investigate. In Section 3, we separately calculate exact analytical expressions of photon distribution, second-order correlation function, Mandel factor and squeezing parameter of those three-pointers by taking into account the full-order evaluation of unitary operator under final states given after postselected measurement finished and present our main results. Finally, we summarize our findings of this study in Section 4.

To build up our model, we begin to review some basic concepts of the von Neumann measurement theory. The standard measurement consists of three elements; measured system, pointer (measurement device or meter) and environment, which induce the interaction between the system and the pointer. The description of the measurement process can be written in terms of Hamiltonian, and the total Hamiltonian of a measurement composed of three parts:^[44]

$$ \begin{eqnarray}H={H}_{{\rm{s}}}+{H}_{{\rm{p}}({\rm{m}})}+{H}_{{\rm{int}}},\end{eqnarray}$$ (1)

$$ \begin{eqnarray}{H}_{{\rm{int}}}=g(t)\hat{A}\hat{P},\,\,\,\displaystyle {\int }_{{t}_{0}}^{t}g(t){\rm{d}}t=g\delta (t-{t}_{0}),\end{eqnarray}$$ (2)

One can express the position X^ and momentum operator P^ of the pointer using the annihilation a^ and creation operator a^† (satisfying [a^,a^†]=1) as^[45]

$$ \begin{eqnarray}\begin{array}{lll}\hat{X} & = & \sigma ({\hat{a}}^{\dagger }+\hat{a}),\end{array}\end{eqnarray}$$ (3)

$$ \begin{eqnarray}\begin{array}{lll}\hat{P} & = & \frac{{\rm{i}}}{2\sigma }({\hat{a}}^{\dagger }-\hat{a}),\end{array}\end{eqnarray}$$ (4)

Assume that the system initially prepared in the state |ψ_i〉 and the initial pointer state is |ϕ〉, then after interaction we project the state e−i∫t0tHintdτ|ψi〉|ϕ〉 onto the postselected system state |ψ_f〉, we can obtain the final state of the pointer. Furthermore, the normalized final state of the pointer can be written as

$$ \begin{eqnarray}\begin{array}{lll}|\varPhi \rangle & = & {\mathscr{N}}\left[\frac{1}{2}(\hat{I}+{\langle A\rangle }_{w})\otimes D\left(\frac{s}{2}\right)\right.\\ & & \left.+\frac{1}{2}(\hat{I}-{\langle A\rangle }_{w})\otimes D\left(-\frac{s}{2}\right)\right]|\phi \rangle.\end{array}\end{eqnarray}$$ (6)

In this study, we take the transverse spatial degree of freedom of single-mode radiation field as pointer and its Hamiltonian H_m(p) is related to the generating processes of single mode radiation fields such as coherent state, squeezed vacuum state and Schrödinger cat state, etc. The polarization degree of freedom of single mode radiation field is taken to be the measured system, and its Hamiltonian H_s is related to the polarization of the beam. Based on the basic requirements of quantum measurement theory,^[44] the H_s and H_m(p) do not affect our measurement results. Thus, in our current research, it is not necessary to give the explicit expressions of the system and meter in hereafter. We suppose that the operator to be observed is the spin x component of a spin-1/2 particle, i.e.,

$$ \begin{eqnarray}A={\sigma }_{x}=|{\uparrow }_{z}\rangle \langle {\downarrow }_{z}|+|{\downarrow }_{z}\rangle \langle {\uparrow }_{z}|,\end{eqnarray}$$ (8)

$$ \begin{eqnarray}|{\psi }_{{\rm{i}}}\rangle =\cos \frac{\theta }{2}|{\uparrow }_{z}\rangle +{{\rm{e}}}^{{\rm{i}}\phi }\sin \frac{\theta }{2}|{\downarrow }_{z}\rangle,\end{eqnarray}$$ (9)

$$ \begin{eqnarray}|{\psi }_{{\rm{f}}}\rangle =|{\uparrow }_{z}\rangle,\end{eqnarray}$$ (10)

$$ \begin{eqnarray}{\langle {\sigma }_{x}\rangle }_{{\rm{w}}}={{\rm{e}}}^{{\rm{i}}\phi }\tan \frac{\theta }{2},\end{eqnarray}$$ (11)

Furthermore, we take the initial pointer state |ϕ〉 as a coherent state, squeezed vacuum state and Schrödinger cat state to study the effects of postselected measurement on the properties of those pointers, respectively. To achieve our goal: (1)We study the conditional probability of finding n photons after postselected measurement. For the state |Φ〉, the conditional probability of finding n photons can be calculated by 12Ppost(n)=|〈n|Φ〉|2, and we compare it with the probability P(n) = |〈 n|ϕ〉|² of initial pointer state |ϕ〉.(2)We investigate the second-order correlation function g⁽²⁾(0) and Mandel factor Q_m for the |Φ〉 state. The second-order correlation function of a single-mode radiation field is defined as 13g(2)(0)=〈a†a†aa〉〈a†a〉2, and the Mandel factor Q_m of the radiation field can be expressed in terms of g⁽²⁾(0) as 14Qm=〈n〉[g(2)(0)−1]. If 0 ≤ g⁽²⁾(0) < 1 and –1 ≤ Q_m < 0 simultaneously, the corresponding radiation field has sub-Poissonian statistics and more nonclassical. We have mention that the quantity Q_m can never be smaller than –1 for any radiation field, and negative Q_m values, which are equivalent to sub-poissonian statistics, cannot be produced by any classical field.^[46](3)We check the squeezing parameter of single-mode radiation field for the |Φ〉 state. The squeezing parameter of the single-mode radiation field reads^[47]15Sϕ=〈:Xϕ2:〉−〈Xϕ〉2, where 16Xϕ=12(ae−iϕ+a†eiϕ),[Xϕ,Xϕ+π2]=i is the quadrature operator of the field, and :: stands for the normal ordering of the operator defined by :aa^†: = a^†a, whereas aa^† = a^†a + 1. We can see that S_θ is related to the variance of X_θ, i.e., 17Sϕ=(ΔXϕ)2−12, where ΔXϕ=〈Xϕ2〉−〈Xϕ〉2. The minimum value of S_ϕ is –1/2 and for a nonclassical state S_ϕ∈[–1/2,0).

In the next section, we will study the above properties of three typical radiation fields by taking into the postselected measurement with arbitrary interaction strengths and weak values, respectively.

In this section, we study the statistical properties and squeezing effects of typical single-mode radiation fields such as a coherent state, squeezed vacuum state and Schrödinger cat state, respectively, and compare the results with the corresponding initial state’s case.

The coherent state is typical semi-classical, quadrature minimum-uncertainty state for all mean photon numbers. The coherent state was originally introduced by Schrödinger in 1926 as a Gaussian beam to describe the evolution of a harmonic oscillator,^[48] and the mathematical formulation of coherent state (also called Glauber state) has been introduced by Glauber in 1963.^[28] Coherent states play an important role in representing quantum dynamics, especially when the quantum evolution is close to classical.^[49,50] Here we take the coherent state as initial pointer state^[51]

$$ \begin{eqnarray}|\alpha \rangle =D(\alpha )|0\rangle,\end{eqnarray}$$ (18)

$$ \begin{eqnarray}\begin{array}{lll}|\varPsi \rangle & = & \frac{\lambda }{\sqrt{2}}\left[\left(1+{\langle A\rangle }_{{\rm{w}}}\right){{\rm{e}}}^{-{\rm{i}}\frac{s}{2}{\rm{Im}}(\alpha )}|\alpha +\frac{s}{2}\rangle\right. \\ & & \left.+(1-{\langle A\rangle }_{{\rm{w}}}){{\rm{e}}}^{{\rm{i}}\frac{s}{2}{\rm{Im}}(\alpha )}|\alpha -\frac{s}{2}\rangle \right],\end{array}\end{eqnarray}$$ (19)

$$ \begin{eqnarray}\begin{array}{lll}{\lambda }^{-2} & = & 1+|{\langle A\rangle }_{{\rm{w}}}{|}^{2}+{{\rm{e}}}^{-\frac{1}{2}{s}^{2}}(1+|{\langle A\rangle }_{{\rm{w}}}{|}^{2}\\ & & +2{\rm{Re}}{\langle A\rangle }_{{\rm{w}}})\cos (2s{\rm{Im}}(\alpha )),\end{array}\end{eqnarray}$$ (20)

The conditional probability of finding n photons after taking a postselected measurement for state |Ψ〉 is obtained from Eq. (11) by changing |Φ〉 to a specific normalized state |Ψ〉. We plot the conditional probability of finding n photons under the normalized state |Ψ〉 and the analytical results are shown in Fig. 1. As indicated in Fig. 1, the black solid line represents the photon distribution probability for the initial state of coherent pointer state before postselected measurement, and it has a Poisson probability distribution. However, as shown in Fig. 1 the postselected measurement can change the photons Poisson distribution with increasing the weak value and changing the interaction strength s. Furthermore, from Fig. 1(a) we can know that P_coh(n) is larger, in the postslected case for some small photon number regions rather than the no interaction case.

For coherent state |α〉 (Eq. (17)) the second-order correlation function g⁽²⁾(0) is equal to one, i.e., g⁽²⁾(0) = 1 and the Mandel factor is equal to zero. Now we will calculate the gcoh(2)(0) and Q_m,coh considering the final pointer state |Ψ〉 for the coherent pointer state. The expectation value of the photon number operator n^=a†a under the state |Ψ〉 is

$$ \begin{eqnarray}\begin{array}{lll}{\langle n\rangle }_{\varPsi } & = & \frac{|\lambda {|}^{2}}{4}\left\{{\left|1+{\langle A\rangle }_{{\rm{w}}}\right|}^{2}{\left|\alpha +\frac{s}{2}\right|}^{2}+{\left|1-{\langle A\rangle }_{{\rm{w}}}\right|}^{2}{\left|\alpha -\frac{s}{2}\right|}^{2}\right.\\ & & +2{{\rm{e}}}^{-\frac{{s}^{2}}{2}}{\rm{Re}}\left[{{\rm{e}}}^{2{\rm{i}}s{\rm{Im}}(\alpha )}(1-|{\langle A\rangle }_{{\rm{w}}}{|}^{2}-2{\rm{i}}\,{\rm{Im}}{\langle A\rangle }_{{\rm{w}}}))\right.\\ & & \left.\left.\times {\left(\alpha +\frac{s}{2}\right)}^{\ast }(\alpha -\frac{s}{2})\right]\right\}.\end{array}\end{eqnarray}$$ (21)

From Fig. 2 we can deduce that in postselected weak measurement region with large weak values (see the curves in Figs. 2(b) and 2(c)), the final state of the pointer state possesses sub-Poisson statistics (0 < g⁽²⁾(0) < 1 and –1 < Q_m < 0). From Fig. 2(d), we can see that in strong postselected measurement, the system has super Poisson statistics (g⁽²⁾(0) > 1 and Q_m > 0). Thus, the results summarized in Fig. 2 confirm that the postselected measurement can change the statistical properties of the coherent state significantly.

Since the coherent state is a minimum uncertainty state, there is no squeezing effect for coherent state |α〉, i.e., S_ϕ = 0. Next, we investigate the squeezing parameter Sϕcoh for |Ψ〉 of the coherent pointer state. The expectation value of X_{ϕ, coh} under the state |Ψ〉 is given by

$$ \begin{eqnarray}\begin{array}{lll}{\langle {X}_{\phi,{\rm{coh}}}\rangle }_{\varPsi } & = & \frac{|\lambda {|}^{2}}{\sqrt{2}}\left\{(1+|{\langle A\rangle }_{{\rm{w}}}{|}^{2})|\alpha |\cos (\phi -\theta )+s\cos \phi \,{\rm{Re}}[{\langle A\rangle }_{{\rm{w}}}]+\frac{1}{2}{{\rm{e}}}^{-\frac{1}{2}{s}^{2}}{\rm{Re}}[{{\rm{e}}}^{2{\rm{i}}s{\rm{Im}}(\alpha )}(1-{\langle A\rangle }_{{\rm{w}}})(1+{\langle A\rangle }_{{\rm{w}}}^{\ast })\right.\\ & & \times (2r\cos (\theta -\phi )+{\rm{i}}s\sin \phi )].\end{array}\end{eqnarray}$$ (23)

If s = 0, then the above expression will reduce to the expectation value of X_ϕ under the initial coherent pointer state |α〉, i.e., 〈 X_{ϕ, oh}〉_{s = 0} = Re [α e^{−i ϕ}]. The expectation value of Xϕ,coh2 with |Ψ〉 is given by

$$ \begin{eqnarray}\begin{array}{lll}{\langle {X}_{\phi,{\rm{coh}}}^{2}\rangle }_{\varPsi } & = & \frac{1}{2}\left\langle (a{{\rm{e}}}^{-{\rm{i}}\phi }+{a}^{\dagger }{{\rm{e}}}^{{\rm{i}}\phi })(a{{\rm{e}}}^{-{\rm{i}}\phi }+{a}^{\dagger }{{\rm{e}}}^{{\rm{i}}\phi })\right\rangle \\ & = & \frac{1}{2}\left[\langle {a}^{2}\rangle {{\rm{e}}}^{-2{\rm{i}}\phi }+\langle {a}^{\dagger 2}\rangle {{\rm{e}}}^{2{\rm{i}}\phi }+2\langle {a}^{\dagger }a\rangle +1\right]\\ & = & \frac{1}{2}\left[2{\rm{Re}}[\langle {a}^{2}\rangle {{\rm{e}}}^{-2{\rm{i}}\phi }]+2\langle n\rangle +1\right],\end{array}\end{eqnarray}$$ (24)

$$ \begin{eqnarray}\begin{array}{lll}\langle {a}^{2}\rangle & = & \frac{|\lambda {|}^{2}}{4}\left\{|1+{\langle A\rangle }_{{\rm{w}}}{|}^{2}{\left(\alpha +\frac{s}{2}\right)}^{2}\right.\\ & & +{\left|1-{\langle A\rangle }_{{\rm{w}}}\right|}^{2}{\left(\alpha -\frac{s}{2}\right)}^{2}\\ & & +{{\rm{e}}}^{2{\rm{i}}s{\rm{Im}}(\alpha )}{{\rm{e}}}^{-\frac{{s}^{2}}{2}}(1-{\langle A\rangle }_{{\rm{w}}})(1+{\langle A\rangle }_{{\rm{w}}}^{\ast }){(\alpha -\frac{s}{2})}^{2}\\ & & +\left.{{\rm{e}}}^{-2{\rm{i}}s{\rm{Im}}(\alpha )}{{\rm{e}}}^{-\frac{{s}^{2}}{2}}(1+{\langle A\rangle }_{{\rm{w}}})(1-{\langle A\rangle }_{{\rm{w}}}^{\ast }){\left(\alpha +\frac{s}{2}\right)}^{2}\right\}.\end{array}\end{eqnarray}$$ (25)

We can observe from Fig. 3 that after postselected measurement the squeezing parameter of the initial coherent state changes dramatically, and we can see the phase-dependent squeezing effect. The X quadrature (ϕ = 0) of the final state is not squeezed (see Fig. 3(c)), but P quadrature (ϕ=π2) of the final state has a squeezing effect for moderate interaction strengths (see Figs. 3(a), 3(b), and 3(d)) with any weak values.

Our second pointer state is the squeezed vacuum state. Squeezed states of the radiation field are generated by degenerate parametric down-conversion in an optical cavity.^[52] The squeezed state have important applications in many quantum information processing tasks, including gravitational wave detection,^[18,17] quantum teleportation.^[19,20] Suppose that a measuring device can be initially prepared in the squeezed vacuum state^[51] defined by

$$ \begin{eqnarray}|\xi \rangle =S(\xi )|0\rangle,\end{eqnarray}$$ (26)

The probability amplitude of finding n photons in a squeezed coherent state is given by

$$ \begin{eqnarray}\begin{array}{l}&\langle n|\pm \frac{s}{2},\xi \rangle \\ =&\frac{1}{\sqrt{\cosh \eta }}\exp \left[-\frac{1}{2}{\left|\pm \frac{s}{2}\right|}^{2}-\frac{1}{2}{\left(\pm \frac{s}{2}\right)}^{\ast 2}{{\rm{e}}}^{{\rm{i}}\delta }\tanh \eta \right]\\ &\times \frac{{\left(\frac{1}{2}{{\rm{e}}}^{{\rm{i}}\delta }\tanh \eta \right)}^{\frac{n}{2}}}{\sqrt{n!}}{H}_{n}\left[\chi {({{\rm{e}}}^{{\rm{i}}\delta }\sinh (2r))}^{-\frac{1}{2}}\right]\end{array}\end{eqnarray}$$ (29)

The second-order correlation function g⁽²⁾(0) and the Mandel factor Q_m of the initial squeezed vacuum pointer state |ξ〉 is given by

$$ \begin{eqnarray}{g}^{(2)}(0)=3+\frac{1}{{\sinh }^{2}\eta }\end{eqnarray}$$ (30)

$$ \begin{eqnarray}{Q}_{{\rm{m}}}=1+2{\sinh }^{2}\eta,\end{eqnarray}$$ (31)

The analytical results for gsq(2)(0) and Q_m,sq under the state |φ〉 are presented in Fig. 5. It can be observed from Figs. 5(a) and 5(c) that both gsq(2)(0) and Q_m,sq can take the values, in which only sub-Poisson radiation field can possess the region s ≳ 0.5 for large weak values. For definite measurement strength s and larger weak value, the value gsq(2)(0) is lower than one and the value of Q_m,sq is less than zero when the squeezed state parameter η is less than one (see Figs. 5(b) and 5(d)). Thus, it is apparent that the postselected measurement dramatically changes the photon statistical properties of the initial squeezed pointer state |ξ〉.

The squeezing parameter S_ϕ of the initial squeezed vacuum state |ξ〉 can be calculated using Eqs. (14)–(16) and (33), and written as

$$ \begin{eqnarray}{S}_{\phi }=\frac{1}{2}\left[{\cosh }^{2}\eta -\sinh (2\eta )\cos (2\phi -\delta )+{\sinh }^{2}\eta \right]{\rm{}}-\frac{1}{2}.\end{eqnarray}$$ (32)

The squeezing parameter S_ϕ,sq of the final normalized state |φ〉 after postselected von-Nuemann measurement can be calculated as the same processes of the initial state |ξ〉, and the analytical results are shown in Fig. 6. According to Figs. 6(a) and 6(b), the postselected measurement would have a negative effect on the squeezing effect of the squeezed vacuum state since S_ϕ,sq gradually becomes larger with increasing the measurement strength s. However, as indicated in Fig. 6(c), in weak measurement regime where 0 < s < 1, the postselected measurement still has the effects on squeezing of the squeezed vacuum pointer state. As mentioned above the squeezing effect of the squeezed vacuum state is phase-dependent, and the postselected measurement has no positive effect on the squeezing of Xϕ=π2 quadrature (see Fig. 6(d)).

In the above two subsections, we have already confirmed that postselected measurement characterized by a weak value can really change the statistical and squeezing effect of single-mode radiation fields. To further increase the reliability of our conclusions, in this subsection, we check the same phenomena by taking the Schrödinger cat state as a pointer. Schrödinger’s cat is a gedanken experiment in quantum physics, proposed by Schrödinger in 1935,^[53] and its corresponding state is called the Schrödinger cat state. The Schrödinger cat state plays significant roles not only in fundamental tests of quantum theory,^[54–57] but also in many quantum information processing such as quantum computation,^[23] quantum teleportation,^[21,22] and precision measurements.^[27] The Schrödinger cat state is composed by the superposition of two coherent correlated states moving in the opposite directions, and defined as^[47]

$$ \begin{eqnarray}|\varTheta \rangle =K\left(|\alpha \rangle +{{\rm{e}}}^{{\rm{i}}\omega }|-\alpha \rangle \right),\end{eqnarray}$$ (33)

$$ \begin{eqnarray}K={\left[2+2{{\rm{e}}}^{-2|\alpha {|}^{2}}\cos \omega \right]}^{-\frac{1}{2}}\end{eqnarray}$$ (34)

$$ \begin{eqnarray}\begin{array}{lll}|\chi \rangle & = & \frac{\kappa }{2}\left[(1+{\langle A\rangle }_{{\rm{w}}})D(\frac{s}{2})\right.\\ & & \left.+(1-{\langle A\rangle }_{{\rm{w}}})D(-\frac{s}{2})\right]|\varTheta \rangle \end{array}\end{eqnarray}$$ (35)

$$ \begin{eqnarray}\begin{array}{lll}{\kappa }^{-2} & = & \frac{1}{2}(1+|{\langle A\rangle }_{{\rm{w}}}{|}^{2})+{K}^{2}(1-|{\langle A\rangle }_{{\rm{w}}}{|}^{2})\cos (2s{\rm{Im}}[\alpha ]){{\rm{e}}}^{-\frac{{s}^{2}}{2}}\\ & & +\frac{{K}^{2}}{2}{\rm{Re}}[(1-{\langle A\rangle }_{{\rm{w}}})(1+{\langle A\rangle }_{{\rm{w}}}^{\ast })\\ & & \times ({{\rm{e}}}^{{\rm{i}}\omega }{{\rm{e}}}^{-\frac{1}{2}|2\alpha +s{|}^{2}}+{{\rm{e}}}^{-{\rm{i}}\omega }{{\rm{e}}}^{-\frac{1}{2}|2\alpha -s{|}^{2}})].\end{array}\end{eqnarray}$$ (36)

Here we must mention that ω ∈ [0,2π], and when ω = 0 (ω = π) it is called the even (odd) Schrödinger cat state. Similar to the previous two cases, the conditional probability P_sh(n) of the Schrödinger cat state can be obtained by replacing the |ϕ〉 in Eq. (5) to |χ〉, and the analytical results of the even Schrödinger cat state is presented in Fig. 7. As shown in Fig. 7, the postselected measurement can change photon distribution of the field, and in fewer photon regions with large weak values, P_sh(n) is larger than the initial photon distribution of the Schrödinger even cat state. By comparing Fig. 7 and Fig. 4 we can also obtain the same common sense that the even Schrödinger cat state has similar properties as a squeezed state.

The Mandel factor and second-order correlation function for the Schrödinger cat state are given by

$$ \begin{eqnarray}\begin{array}{ll}{Q}_{{\rm{m}}} & =\frac{4|\alpha {|}^{2}{{\rm{e}}}^{-2|\alpha {|}^{2}}\cos \omega }{1-{{\rm{e}}}^{-4|\alpha {|}^{2}}{\cos }^{2}\omega }\end{array}\end{eqnarray}$$ (37)

$$ \begin{eqnarray}{g}^{(2)}(0)=1+\frac{4{{\rm{e}}}^{-2|\alpha {|}^{2}}\cos \omega }{{(1-{{\rm{e}}}^{-2|\alpha {|}^{2}}\cos \omega )}^{2}},\end{eqnarray}$$ (38)

The gsh(2)(0) and Q_m,sh of the Schrödinger cat state can be calculated using Eqs. (13) and (14) with the normalized final pointer state |χ〉. As indicated in Fig. 8, after postselected measurement, the statistical property of the Schrödinger cat state changes from sub-Poisson to super-Poisson gradually with increasing the interaction strength and weak value. However, as we can see from Figs. 8(b) and 8(d), in the postselected weak measurement regime, where 0 < s < 1, we can achieve the best performance for nonclassicality of the odd Schrödinger cat state with small coherent state modulus r.

The squeezing parameter of the initial Schrödinger cat state |Θ〉 reads^[47]

$$ \begin{eqnarray}\begin{array}{lll}{S}_{\phi }^{{\rm{i}}n} & = & \frac{|\alpha {|}^{2}{{\rm{e}}}^{-4|\alpha {|}^{2}}}{{(1+{{\rm{e}}}^{-2|\alpha {|}^{2}}\cos \omega )}^{2}}\\ & & \times [1+(\cos 2\phi {({{\rm{e}}}^{2|\alpha {|}^{2}}+\cos \omega )}^{2}\\ & & +{\sin }^{2}\omega {\cos }^{2}\phi -1)].\end{array}\end{eqnarray}$$ (39)

The analytic expression of S_ϕ,sch for the Schrödinger cat state after postselected measurement can be achieved using the final normalized state |χ〉, and the analytical results are summarized in Fig. 9. As indicated in Fig. 9, after postselected measurement, the squeezing effect of the Schrödinger cat state still depends on phase ϕ, and the squeezing effect of both the even and odd Schrödinger cat states are increased with increasing the interaction strength s and for small weak values (see Figs. 9(a), 9(b), and 9(d)). However, the odd Schrödinger cat pointer state has no squeezing effect (see Figs. 9(c) and 9(d)).

In summary, we have investigated the effects of postselected measurement characterized by postselection and weak value on the statistical properties and squeezing effects of single-mode radiation fields. To achieve our goal, we take the coherent state, squeezed vacuum state and Schrödinger cat state as a pointer, and their polarization degrees of freedom as the measurement system, respectively. We have derived analytical expressions of the pointer state’s normalized final state after postselected measurement considering all interaction strengths between the pointer and the measured system. We separately present the exact expressions of photon distributions, second-order correlation functions, Mandel factors and squeezing parameters of the above three typical single-mode radiation fields for the corresponding final pointer states, and plot the figures to analyze the results.

We find that the photon distributions of those three pointer states change significantly after postelected measurement, especially the coherent pointer state. It is shown that postselected measurement changes the photon statistics and squeezing effect of coherent state dramatically, and it is noticed that the amplification effect of weak value plays a major role in this process. We also show that the postselected measurement changes the photon statistics of the squeezed vacuum state from super-Poisson to sub-Poisson for large weak values, and moderate interaction strengths. However, the photon statistics of the Schrödinger cat state changes from sub-Poisson to super-Poisson with increasing the interaction strength. In accordance with previous findings, the squeezing effects of squeezed vacuum and Schrödinger cat pointer states are still phase-dependent and the squeezing effect of squeezed vacuum pointer is decreased with increasing the interaction strength. On the contrary, the squeezing effect of the even Schrödinger cat state is increased with increasing the interaction strength for small weak values compared with the initial pointer state case.

This work belongs to the state optimization using postselected von Neumann measurement. Those properties of radiation fields we investigated are directly effect on the implementations of the important quantum information processing mentioned in the introduction section. Thus, we anticipate that the results shown in this research can be useful to provide other effective methods for studying the related quantum information processing.

In our current research, we only consider the three typical single-mode radiation fields to investigate the effects of postselected von Neuman measurement on their inherent properties, but real light beams are in fact the time-dependent and the representation of time dependence requires the use of two or more modes of the optical systems. Thus, it would be interesting to check the effects of postselected von Neumann measurement on other useful radiation fields in quantum optics and quantum information processing such as single photon-added^[58,59] and photon-subtracted^[60] states, pair-coherent state (two-mode field),^[61–64] and other multimode radiation fields.^[65]