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Quantum state engineering by a coherent superposition of photon subtraction and addition

We study a coherent superposition of field annihilation and creation operator acting on continuous variable systems and propose its application for quantum state engineering. Specifically, it is investigated how the superposed operation transforms a classical state to a nonclassical one, together with emerging nonclassical effects. We also propose an experimental scheme to implement this elementary coherent operation and discuss its usefulness to produce an arbitrary superposition of number states involving up to two photons.


I. INTRODUCTION
Manipulation of light field at the single-photon level provides a crucial basis for many important applications in quantum information science. In particular, two elementary operations on a single-mode field, i.e. photon subtraction and addition represented by bosonic annihilation and creation operatorsâ andâ † , respectively, can be employed to transform a field state to a desired one [1]. For example, the photon subtraction transforms a Gaussian entangled state (two-mode squeezed state) to a non-Gaussian entangled state for a nonlocality test [2] and entanglement distillation [3]. The photon addition is known to create a nonclassical state from any classical state (e.g. coherent and thermal states) [4], and both of the photonsubtracted [5][6][7][8] and the photon-added squeezed states [9,10] were suggested to improve the fidelity of continuous variable (CV) teleportation. The photon subtraction [11] and the addition [12] are now practically realized in laboratory.
Recently, there appeared a proposal to implement a coherent superpositionââ † ±â †â of two product operations, photon addition followed by subtraction (ââ † ) and photon subtraction followed by addition (â †â ) [13]. Kim et al. particularly suggested an experimental scheme to prove the bosonic commutation relation, [â,â † ] = 1 [13], which was very recently carried out using a thermal state of light field [14]. A cavity-based scheme has also been proposed to prove the bosonic commutation relation, exploiting interaction between three atoms and a cavity field [15]. Based upon the inteferometer setting of [13], Fiurásek suggested an optical scheme to implement an arbitrary polynomial of photon-number operators, e.g. noiseless amplifier [16] and Kerr nonlinearity, via a combination of multiple photon subtraction and addition [17].
In this paper, we consider a coherent superposition of photonic operations at a more elementary level, that is, the superposition of photon subtraction and addition, tâ + râ † , and investigate how it transforms a classical state to a nonclassical one. In general, a coherent superposition of two distinct operations may be created by erasing the "which-path" information relevant to the operations in an interferometer setting. Motivated by the single-photon interferometer in [13], we propose an experimental scheme to implement the coherent operation tâ + râ † in an optical experiment and also show that it can be employed together with displacement operators to engineer an arbitrary quantum state in principle. As an example, we study the case of generating an arbitrary superposition of number states involving up to two photons. The superposition state C 0 |0 +C 1 |1 +C 2 |2 can be used for quantum information processing, e.g., the nonlinear sign-shift(NS) gate (a basic element of the nondeterministic CNOT gate) [18,19], and the optimal estimation of the loss parameter of a bosonic channel [20]. There have been several theoretical proposals to generate the superposition of |0 , |1 , and |2 , e.g., using repeated photon subtractions with squeezing operations [21], repeated photon additions with displacement operations [22], and quantum scissors [23,24]. An experimental realization was recently made conditioned on photodetections using a parametric down-converter with two auxiliary weak coherent states [25]. We study our proposed scheme with experimental imperfections considered and show that it can produce the superposition state with high fidelity. This paper is organized as follows. In Sec. II, we study how the coherent operation tâ + râ † transforms a classical (coherent or thermal) state to a nonclassical one in phase space with the degree of nonclassicality measured by negative volume (area) and by nonclassical depth [26]. In Sec. III, we investigate observable nonclassical effects, squeezing and sub-Poissonian statistics, arising due to the coherent operation, and propose an experimental scheme in Sec. IV to implement the operation tâ + râ † in a single-photon inteferometer setting. In Sec. V, we suggest the coherent operation combined with displacement operation for quantum state engineering and investigate the generation of arbitrary superposition state involving up to two photons with experimental imperfections included. The main results of this paper are summarized in Sec. VI.

II. WIGNER DISTRIBUTION
We first examine how the coherent operation tâ + râ † transforms a classical state to a nonclassical one in view of the phase-space distribution. For a single-mode state ρ, its Wigner distribution W (α) is generally given by the Fourier transform of the characteristic function C(λ) ≡ Tr{ρD(λ)}, whereD(λ) ≡ e λâ † −λ * â is the displacement operator. That is, W (α) = 1 π 2 d 2 λC(λ)e αλ * −α * λ . (i) Coherent states: Given a coherent state |α 0 as an initial state, the output state after the superposition operation, |Ψ ∼ (tâ + râ † )|α 0 , possesses the Wigner distribution where W 0 (α) ≡ 2 π e −2|α−α0| 2 is the Wigner function of the initial coherent state |α 0 . In Fig. 1 we show the Wigner distribution W (α ≡ x + iy) of the output state (tâ + râ † )|α 0 , where r, t and α 0 are taken as real (|r| 2 + |t| 2 = 1). In general, the negative dip of the Wigner distribution increases with the ratio r of the addition operationâ + in the superposition tâ + râ † . In order to measure the degree of nonclassicality for the output state, we investigate both the negative area and the negative volume in phase space.
Negativity-The negative region with W (α = x+iy) < 0 appears under the condition where C 1 = 1 2 (1 − t r )α 0 . It thus becomes a circle of radius 1/2 regardless of r [ Fig. 1 (d), (e), and (f)], except the case of r = 0 which does not change a coherent state (â|α 0 = α 0 |α 0 ). Therefore, the negative area is independent of the ratio r. However, the depth of the negativity depends on r [ Fig. 1 (a), (b), and (c)], which may be further quantified via the negative volume defined by V N = 1 2 ( d 2 α|W (α, α * )| − 1) [27]. We find that V N generally increases with r, as shown in Fig. 2 (a). For a fixed r, the negative volume V N decreases with the initial amplitude α 0 and the extremal V N thus appears Alternatively, one may quantify the degree of nonclassicality by the nonclasical depth defined by C. T. Lee [26]. The Glauber-P function, P (α), becomes positive when smoothed by a Gaussian of sufficient width, The minimum τ ∈ [0, 1] required for a positive R-function is taken as a nonclassicality measure. Instead of calculating the integration in Eq. (3), the nonclassical depth τ for the output state |Ψ ∼ (tâ + râ † )|α 0 turns out to be unity regardless of r and α 0 by another method: if ρ is orthogonal to a certain coherent state-i.e., β|ρ|β = 0 for some β-the state is maximally nonclassical, i.e. τ = 1 [28]. In our case, there always exists a certain β * = − t r α 0 such that | β|Ψ | 2 ∼ |tα 0 +rβ * | 2 | β|α 0 | 2 = 0. Therefore, the degree of nonclassicality is maximal if measured by the nonclassical depth τ for any r = 0 of the superposed operation tâ + râ † , whereas the nonclassicality increases with r if measured by the negativity volume V N .
(ii) Thermal states: A thermal state ρ th = (1 − e −β )e −βa † a with the average photon numbern = 1 e β −1 being as an initial state, we obtain the Wigner distribu- tion after the coherent operation tâ + râ † as where W th (α) ≡ 2 π(1+2n) e − 2|α| 2 1+2n is the Wigner distribution of the input thermal state. In Fig. 4, we plot the Wigner distribution as a function of r for n = 0.1. On one hand, like the case of coherent-state input, we see that the negative dip of the Wigner distribution increases with the ratio r ofâ † in the coherent operation. On the other hand, the negative region appears under the condition where C 2 = (n + r 2 + 2ntr), C 3 = (n + r 2 − 2ntr), Eq. (5) thus describes an ellipse, as shown in Fig. 4  (d), (e), and (f), with the size of the negative area given by A N = πC4 √ C2C3 . Note that the negativity emerges only with r above a threshold, r > n/(1 + 2n), and the negative area increases with r, unlike the case of coherentstate input. We also find that the negative volume increases with r, as shown in Fig. 2 (b). In comparison, the degree of nonclassicality measured by the nonclassical depth turns out to be τ = (1+n)r 2 n+r 2 using Eq. (3), which give a nonzero value for any r = 0 and also increases with r.

III. OBSERVABLE NONCLASSICAL EFFECTS
In this section, we investigate two observable nonclassical effects, quadrature squeezing and sub-Poissonian statistics, arising from the coherent operation tâ + râ † .

IV. EXPERIMENTAL SCHEME
In this section, we propose how the superposed operation tâ + râ † can be implemented in experiment. A key idea here is to erase which-path information on whether the implemented operation refers to the photon subtractionâ or the photon additionâ † , as described below.
First, note that the success of both the operations,â andâ † , can be heralded by a detection of a single photon in an optical scheme. When an arbitrary initial state |Ψ is injected into a beam-splitter with the other input in a vacuum state, the detection of a single-photon at one output port heralds that a photon is subtracted from the initial state, due to the conservation of photon number. This corresponds to the actionâ|Ψ , which holds well particularly when the transmissivity of the beam splitter is large [11]. On the other hand, if the state |Ψ is injected to a signal mode of a nondegenerate parametric amplifier (NDPA) with the idler mode in a vacuum state, the detection of a single photon at the output idler mode heralds that one photon is added to the input state, due to the pairwise photon-creation and -annihilation process in the NDPA. This corresponds to the action a † |Ψ , which holds well particularly when the interaction strength in the NDPA is small [12]. With these two schemes combined, if the which-path information on the detected single-photon is erased by using an additional beam splitter (BS 2 ) with transmissivity t 2 [ Fig. 7], the coherent superposition tâ + râ † can be conditionally implemented.
In Fig. 7, an arbitrary state |Ψ is injected into the parametric down-converter with small coupling strength s ≪ 1, which acts as Next, the BS1 (transmissivity: t 1 ≈ 1) acts on the state [13] asB Finally, the beam splitter BS2 (transmissivity: With the detection of single-photon at PD1 (PD2) and no photon at PD2 (PD1), we see from Eq. (13) that the state collapses to |ψ out ∼ (tâ + râ † )|ψ a , where t ∼ r * 1 t1 t 2 ( r * 1 t1 r * 2 ) and r ∼ sr 2 ( −st * 2 ). Experimental imperfections, e.g. nonideal photo-detection, will be further considered in the next section to investigate the implemented operation particularly on generating an arbitrary superposition state.
As a remark, we note the identityŜ †âŜ =â cosh s + a † e iφ sinh s andŜ †â †Ŝ =â † cosh s +âe −iφ sinh s, wherê S ≡ e 1 2 (ξâ †2 −ξ * â2 ) is the squeezing operator (ξ ≡ se iφ ). Thus, one may alternatively implement the coherent operation tâ + râ † by a sequence of squeezing operation, photon subtraction (addition), and the inverse squeezing,Ŝ †âŜ (Ŝ †â †Ŝ ), for the case of |t| > |r| (|t| < |r|). However, this scheme seems to be more demanding than the above proposal based on a single-photon interferometer, as the number of required nonlinear resources is increased. Moreover, a very large squeezing s → ∞ is needed to implement tâ + râ † when the subtraction and the addition parts are comparable to each other, i.e. |t| ≈ |r|. Even assuming that the squeezing operations can be performed perfectly, we have calculated the output fidelity for superposition states C 0 |0 + C 1 |1 + C 2 |2 similar to Fig. 8 and found that this alternative scheme does not make improvement. Therefore, in the next section, we will focus on the interferometric setting described above to implement the coherent operation.

V. GENERATION OF A SUPERPOSITION OF
|0 , |1 AND |2 In this section, we show that the coherent operation tâ + râ † together with a displacement operatorD(β) = e βâ † −β * â can be employed to generate an arbitrary field state. In particular, we investigate the generation of a superposition state involving up to two photons in detail.

VI. CONCLUSION
In this paper, we have investigated a coherent superposition tâ+ râ † acting on continuous variable systems that can be used for quantum state engineering. It has been shown that the operation creates nonclassicality out of a classical state with observable effects such as squeezing and sub-Poissonian statistics. For the coherent-state inputs, the degree of nonclassicality measured by negative volume in phase space increases with the ratio r, while that measured by nonclassical depth due to C. T. Lee is maximal regardless of r. In particular, given an input coherent amplitude α 0 , the squeezing effect occurs optimally for r < 1, which implies that the coherent operation generally achieves better squeezing than the bare photon additionâ † . For the thermal-state inputs, the degree of nonclassicality increases with r measured by both the negative volume and the nonclassical depth.
We have also proposed an optical experimental scheme to implement the coherent operation tâ + râ † in a singlephoton intereference setting. Furthermore, it has been shown that the coherent operation combined with the displacement operations can be employed to generate an arbitrary superposition state and that a high fidelity particularly in producing a superposition state involving up to two photons is achievable against experimental imperfections using the currently available techniques.