In recent years, considerable theoretical and experimental investigations have been focused on preparing non-Gaussian optical states due to their strongly nonclassical properties, such as entanglement and negativity of the quasi-probability phase-space distributions, which are important for the efficient implementation of quantum information and communication protocols, ^{[1– 5]} quantum estimation tasks, ^{[6, 7]} and quantum cloning.^[8] However, due to higher-order optical nonlinearity limitation, achieving much higher non-Gaussianity of the radiation field is a considerable challenge.

Based on non-Gaussian operations (i.e., photon addition and subtraction), Ourjoumtsev et al. have presented a detailed experimental analysis of kitten state generation by subtracting one photon from a squeezed vacuum beam.^[9] Reference  [10] reported a scheme to generate squeezed superpositions of coherent states. However, highly non-Gaussian states preparation need to subtract more photons and this leads to a rapid decrease of generation fidelity and success probability. Another alternative approach is to take advantage of the effective nonlinearity induced by conditional measurements^[11] and feedback control.^[12] These kinds of measurement-induced control strategies have taught us something about systematic approaches of quantum states generation and the qualitative role of quantum coherence and entanglement between the interested system and measuring equipments.^{[10, 13, 14]} By virtue of conditional measurements, Lvovsky and Mlynek^[15] first introduced a so-called “ quantum optical catalysis” method to prepare coherent superposition states (CSS). This scheme exhibits higher fidelity only for very small amplitude state preparation. Various control strategies have been proposed for the non-Gaussian states preparation, such as photon-added and photon-depleted states, ^[16] photon annihilation-then-creation and photon creation-then-annihilation thermal states, ^[17] photon-added two-mode squeezed coherent states, ^[18] arbitrary superposition of zero- and one-photon, ^[19] odd photon number superposition state, ^[20] photon-subtracted squeezed states, ^[21] and the Schrö dinger cat state.^{[14, 22– 24]}

At present, it is an urgent task to prepare larger-size non-Gaussian states with high fidelity. However, the usual optical resonant cavity is a linear amplifier, which cannot be applied to prepare large-size non-Gaussian states. Noiseless amplification methods^{[25, 26]} can work well only for small-size quantum states preparation. Based on the optical interference, Takahashi et al. have demonstrated a novel way to enhance the size of CSS.^[27] The preparation fidelity would approximate > 90% for | α | ² = 1.2 and 2.0 ideally, while the values are 76% and 60% in the experiment. Very recently, based on quantum optical catalysis mentioned above, Bartley et al. investigated the multi-photon quantum states engineering and generated a weak DCSS (α = 0.9) with > 90% fidelity.^[28]

In this work, by implementing non-Gaussian operations and Bell measurements, we propose a scheme to generate superposition states, the so-called single-photon-added and single-photon-subtracted coherent (SPASC) superposition states. Instead of subtracting more photons, we show that squeezing the input field can enhance non-Gaussianity. In phase space, the quasi-probability distributions of the single-photon-added and -subtracted squeezed coherent (SPASSC) superposition states exhibit a close approximation to that of a displaced coherent superposition states (DCSS). This will provide possibilities to achieve DCSS with higher fidelity. Therefore, we arrange our work as follows. In Section 2, relying on the simple optical elements, we briefly introduce the realization of non-Gaussian operations, namely adding (subtracting) one single photon to (from) Gaussian-like field. Then we propose a conceptual experiment apparatus for preparing SPASC superposition states and investigate the negativity of its Wigner function and non-Gaussian features. In Section 3, in order to enhance quantum non-Gaussianity, we consider a squeezed incident field and demonstrate that squeezing can improve the robustness of quantum non-Gaussianity. As an application, in Section 4 we show that DCSS can be approximately prepared by generating and manipulating SPASSC superposition states, and we then obtain the optimal fidelity. Some conclusive remarks will be made in Section 5.

Figures  3– 5, 6(b), and 6(c) obviously show that non-Gaussianity heavily decreases with rising field intensity. This can be understood from two points: the absence of much stronger non-Gaussian operations and quantum noise. Namely, the added (subtracted) photon number is not greater than 1, and quantum noise covers the effect of non-Gaussian operations. In order to generate much higher amplitude non-Gaussian quantum states, adding (or subtracting) more and more photons or reducing quantum noise is feasible. Here instead of adding (subtracting) more photons, we squeeze the incident coherent field to realize noise reduction. For the case of a squeezed coherent field, we obtain a non-Gaussian single-photon-added and − subtracted squeezed coherent (SPASSC) superposition states and can improve the robustness of non-Gaussianity, while increasing the amplitude.

For a squeezed incident field, generated from Ref.  [42], from Eqs.  (1) and (2) the output is

where | λ , α 〉 ₁ ≡ Ŝ (λ ) D̂ (α ) | 0〉 ₁ denotes a squeezed coherent field. After performing Bell measurements, the conditional states will be determined by the distinguishable measurement results. Therefore, we can generate the single-photon-added and − subtracted squeezed coherent (SPASSC) superposition states, namely,

where | λ , α , − 1〉 ₁ ≡ a| λ , α 〉 ₁ means to subtract a photon from the squeezed coherent state, and | λ , α , 1〉 ₁≡ a^† | λ , α 〉 ₁ means to add a photon. In contrast from this case (Eq.  (7)), the single-photon-subtracted squeezed coherent state | λ , α , − 1〉 ₁ is a non-Gaussian one. Squeezing the field shall enhance the non-Gaussianity of the SPASSC superposition states. Obeying the same derivation procedures (Eqs.  (10)– (13)), we have

in which and κ [ρ _λ , ρ _G] can be derived from

and ρ _λ denotes the density matrix of the SPASSC superposition states in Eq.  (16).

Figure  7 shows the non-Gaussianity of the SPASC (λ = 0) and SPASSC (λ = 0.2) superposition states for four different values of k. When the squeezing parameter value λ = 0.2, we find that its non-Gaussianity is robust against the amplitude rising. For various values of k, the sensibility of non-Gaussianity is different. According to the predictive mentioned in Section  2.4, a squeezing field can convert single-photon-subtraction coherent state to a non-Gaussian one. In order to interpret this point, we can utilize

to describe this process. The corresponding non-Gaussianity depends on the probability distribution of the photon-addition-coherent state, i.e.,

which is related with the squeezing parameter λ and ratio k. From its partial differential function , we can confirm that variation range of its non-Gaussianity has a lower dependence on the larger or smaller initial value of k (e.g., k = 0.2, 0.4 in Fig.  6(b) and k = 2.5, 3.0 in Fig.  6(c)).

Fig.  7.

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	Fig.  7. Non-Gaussianity of the SPASC and SPASSC superposition states. Hollow points: the squeezing parameter λ = 0, i.e., the SPASC superposition states. Solid points: the SPASSC superposition states with the squeezing parameter λ = 0.2. The circle, star, triangle, and rectangle points correspond to k = 0.5, 1.0, 1.5, and 3.0, respectively.

How to accurately and steadily generate target quantum states is a greatly important and indispensable step on the road to realizing quantum information processing. As a particular kind of non-Gaussian state, generating a coherent superposition states (CSS) has attracted some special attention due to their remarkable usefulness for universal quantum computation^[43] and fundamental tests of quantum theories.^{[44– 46]} DCSS can be defined as

where β is the displaced parameter, z denotes the amplitude of CSS, and the normalization constant N is

In phase space, its Wigner distribution function is shown in Fig.  8(a). Comparing Fig.  8(a) with Fig.  8(b), we can observe that the quasi-probability distribution features of DCSS are very similar to those of the SPASSC superposition states. In other words, the SPASSC superposition states can be controlled to approximately generate DCSS. Fidelity, which describes a distance measurement between an interested quantum state ρ and a given reference quantum state σ , can be defined as^{[47, 48]}

Fig.  8.

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	Fig.  8. (a) Wigner functions of the displaced CSS with z = 0.9 and β = 0.7. (b) Wigner functions of the SPASSC superposition states with α = 0.4, λ = 0.8, k = 0.76 in Eq.  (16).

For the case of pure state (e.g., ρ = | Ψ 〉 〈 Ψ | , σ = | Φ 〉 〈 Φ | ), this definition can be simplified to be F(ρ ; σ ) = | 〈 Ψ | Φ 〉 | ², which can be also called “ transition probability” . Suppose that DCSS is a reference quantum state with its amplitude z and displaced parameter β , we calculate the overlap squared between the states in Eqs.  (16) and (17). Thus, we have

where M_λ is the normalization constant of ρ _λ shown in Eq.  (16).

For | α | = 1.5 (here the phase is 1.05), the displaced parameter β = 0.9, by employing the above analytical expression shown in Eq.  (19), we plot the behavior of the optimal fidelity of generating the target state for a range of coherent amplitude z (as shown in Fig.  9). It should be pointed out that, when taking λ = 0, equation  (19) shows the fidelity of the SPASC superposition states with respect to the target DCSS. In the case of λ = 0, taking k = 1.045, the optimal fidelity is labelled by dotted line (red) with F = 0.703 for z = 1.1. Its tendency will rapidly decrease with increasing size of amplitude z. When taking λ = 0.307 and k = 1.050, we achieve the maximum fidelity F = 0.981 with z = 1.53 detailed by solid line (green). Then the fidelity will also decrease with increasing amplitude of DCSS. For λ = 0.559, k = 1.050, dashed line (blue) shows that the fidelity reduces to F = 0.906 with z = 2.36.

Fig.  9.

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	Fig.  9. Fidelity of generating DCSS. Dotted line (red) denotes the fidelity with λ = 0, k = 1.045, solid line (green) for λ = 0.307, k = 1.050, dashed line (blue) for λ = 0.559, k = 1.050.

Employing the strategy of subtracting (adding) single photon from (to) the radiation field, we have proposed a scheme to generate the single-photon-added and -subtracted coherent (SPASC) superposition states by implementing Bell measurements, and investigated its nonclassical features by adjusting the reflectivity of the high transmission beamsplitter (HT-BS) and the gain of the parametric down-conversion (PDC) process. Instead of the original coherent light field, we utilize a squeezed field to reduce quantum noise and achieve the enhancement of quantum non-Gaussianity (e.g., the SPASSC superposition states), and then further demonstrated a better strategy to approximately generate the displaced coherent superposition states (DCSS) with higher fidelity by controlling the relative proportion parameter and the squeezing parameter. The experimental implementation of the distinguishable Bell measurements may be quite challenging, but our scheme does not need to subtract and add more photons. Actually, for a low-efficiency multiplexing avalanche photodiode (APD), this scheme remains useful. Its larger size and high fidelity indicates its possible applications to study quantum information processing and test the fundamentals of quantum theory.