1. Introduction

Motivated by the Knill-Laflamme-Milburn scheme for scalable quantum computing with linear optics [1], in recent years the field of quantum interference and entanglement with photonic qubits has grown rapidly [2]. Small numbers of entangled photons have been generated with stimulated parametric down-conversion [3–5 ] and used for implementing various quantum information protocols [6]. More recently, several considerable methods for revealing macroscopic entangled states are reported with linear optics, such as by combining a single photon and a coherent beam on a beam splitter [7], amplifying and deamplifying a two-mode single photon entangled state [8], employing superposition operations to generate photon-number entangled states [9], and so on. For quantum entanglement of a large number of photons [10–13 ], however, whether its fundamental principle or experimental demonstration is still a difficult and subtle task [14–17 ].

A cross-Kerr nonlinear medium is capable of inducing an interaction between the photons [18–21 ], although its strength is very small for all experiments reported to date [22–24 ]. The cross-Kerr nonlinearity has a Hamiltonian of the form $H_{ck}=\overline{h}\chi {\widehat{a}}_s^{†}{\widehat{a}}_s{\widehat{a}}_p^{†}{\widehat{a}}_p$, where ${\widehat{a}}_p^{†}$, â_s ( ${\widehat{a}}_p^{†}$, â_p) are the creation and annihilation operators of the signal (probe) mode, respectively, and χ represents the coupling strength of the nonlinearity. This interaction implies that the probe beam accumulates a phase shift nθ (θ = χt and t is the interaction time) directly proportional to the number of photons n in signal mode. Then, an appropriate homodyne measurement [25] with the operator $\widehat{x}(\varphi )={\widehat{a}}_p{\mathrm{e}}^{1\varphi }+{\widehat{a}}_p^{†}{\mathrm{e}}^{-1\varphi }$ acting on the probe beam allows one to project the signal onto a specific subspace. Specifically, for ϕ = 0 (or ϕ = π/2), this process can be described as the general X (or P) homodyne measurement with α real (|α is the initial coherent probe beam). Based on these weak nonlinearities one can implement photon-number quantum nondemolition measurement [25], entanglement detection [26–28 ], quantum logic gates [29, 30], and multiphoton entanglement [31–34 ]. Since the nonlinearities are extremely weak, it seems natural to improve experimental methods so as to produce large enough nonlinear phase shifts and then follow the previous schemes without bound in the limit of weak nonlinearities. On the other hand, with current technology, it is also important to explore quantum circuit in the regime of weak nonlinearities for quantum information processing [35–38 ].

In this paper, we focus on the exploration of a kind of photon-number entangled states using weak nonlinearities. For each photon number n, we show a quantum circuit to evolve two-mode signal photons, ranging from microscopic to macroscopic systems (i.e., from n = 2 to n ≫ 1). In the regime of weak nonlinearities, more importantly, we consider the maximal phase shift induced in the process of interaction between photons satisfying nθ ≃ 10^{− 2}. Moreover, in the absence of decoherence, we analyze error probability caused by the final homodyne measurement.

2. Exploration of photon-number entangled states

In Fock space, consider an arbitrary two-mode n-photon-number state

Here

Suppose there exist n photons traveling through two spatial modes s ₁ and s ₂, namely signal modes; and we introduce a coherent state $|\alpha 〉=\exp \left(-\frac{1}{2}{|\alpha |}^2\right){\displaystyle {\sum }_{n=0}^{\infty }\frac{{\alpha }^n}{\sqrt{n!}}|n〉}$ in probe mode; let θ and nθ be respective phase shifts on the coherent probe beam according to the two signal modes, as shown in Fig. 1. In order to avoid inducing −θ in the interacting process we herein introduce a single phase gate, i.e., $R_n(\theta )=-\frac{1}{2}n\left(n+1\right)\theta$. After an X homodyne measurement on the probe beam plus appropriate local phase shift operation φ_m (x) on one of the signal modes using classical feed-forward information, at last, the original state can be projected into one of the photon-number entangled states ${|{\psi }_n^m〉}_{s_1{s}_2}$.

 

Fig. 1 The schematic diagram of exploration of photon-number entangled states using weak nonlinearities. Consider n photons traveling through two spatial modes s ₁ and s ₂ (say signal modes). |α⟩ is a coherent state in probe mode. θ and nθ are phase shifts on the coherent probe beam with several weak cross-Kerr nonlinearities respectively. R_n(θ) is a single phase gate used to evolve coherent state. φ_m(x) represents a phase shift on one of the signal modes based on the classical feed-forward information.

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We now describe our method in details. For n is even, after the interaction between the photons and the action of the phase gate, the combined system |Ψ_n⟩ ⊗ |α⟩ ≡ |Ψ_n⟩ |α⟩ evolves as

For an X homodyne measurement, as described above, the quadrature operator $\widehat{x}$ is ${\widehat{a}}_p+{\widehat{a}}_p^{†}$. Let |x⟩ be an eigenstate of $\widehat{x}$ with eigenvalue x. Considering the eigenvalue equation for coherent states â_p|α⟩ = α|α⟩, there exists an available result [26, 40] ⟨x|α⟩ = (2π)^{− 1 / 4}exp[−Im(α)² − (x − 2α)²/4]. After the X homodyne measurement on the probe beam, therefore, the signal photons become

At last, we discuss the phase shifts φ_m(x), which are necessary to transform measurement results into the desired states. We take $a_m{\mathrm{e}}^{-\mathrm{i}{\varphi }_m(x)}{|n/2+m,n/2-m〉}_{s_1{s}_2}+b_m{\mathrm{e}}^{\mathrm{i}{\varphi }_m(x)}{|n/2-m,n/2+m〉}_{s_1{s}_2}$ for example, and will transform it into the state $a_m{|n/2+m,n/2-m〉}_{s_1{s}_2}+b_m{|n/2-m,n/2+m〉}_{s_1{s}_2}$ by using a phase shift φ_m(x) on the signal mode s ₂. This process can be described as a mode transformation ${\mathrm{e}}^{\mathrm{i}{\phi }_m(x)}{\widehat{a}}_{s_2}^{†}$ on mode s ₂, and then the initial state will be transformed into $a_m{\mathrm{e}}^{-\mathrm{i}[{\varphi }_m(x)+m{\phi }_m(x)]}{|n/2+m,n/2-m〉}_{s_1{s}_2}+b_m{\mathrm{e}}^{i[{\varphi }_m(x)+m{\phi }_m(x)]}{|n/2-m,n/2+m〉}_{s_1{s}_2}$ up to an unimportant global phase factor. In fact, it is straightforward to observe that if we want to recover it, only φ_m(x) = −ϕ_m(x)/m would be required.

Similarly, for n is odd, the combined system then evolves as

After the measurement on the probe beam, the signal photons become

By now, in the regime of weak Kerr nonlinearities we have described the evolution of n photon combined system. We next discuss some intriguing applications of the exploration of photon-number entangled states.

3. Applications

In many quantum measurements for linear optical quantum computation, one should always attempt to detect the signal photons and then project them into a desired subspace. Surprisingly, for a given photon number n, the setup in our scheme can be used for these processes based on postselection and classical feed-forward (an additional manipulation). So, a direct application of the present scheme leads to an entangling gate for the states $\{{|{\psi }_n^m〉}_{s_1{s}_2},m=0,1,2,\cdots ,[n/2]\}$, and especially, for a_m = b_m it yields one of the maximally entangled number states $({|n-m,m〉}_{s_1{s}_2}+{|m,n-m〉}_{s_1{s}_2})/\sqrt{2}$. Moreover, it is easy to see that the present scheme is suitable for constructing two-qubit polarization parity gate (see Fig. 2 in [29]), discriminating between $|{\psi }_2^0〉$ and $|{\psi }_2^1〉$, nondestructively (see Fig. 1 in [26], also see Fig. 2 in [34]), and so on.

Another important application of the present scheme is as an analyzer for the states $\{{|{\psi }_n^m〉}_{s_1{s}_2}\}$. Consider a state belonging to the set of states $\{{|{\psi }_n^m〉}_{s_1{s}_2}\}$ in signal modes. After the evolution of a series of optical devices followed by an X homodyne measurement on the probe beam, based on the value of the measurement one can infer immediately what the input must have been with a small error probability. Then, a conditional phase shift operation on one of the modes is necessary to restore the output state to that identified. In other words, the suggested analyzer of the states $\{{|{\psi }_n^m〉}_{s_1{s}_2}\}$ is nondestructive and thus the unconsumed signal photons can be recycled for further use.

To date, although optics-based quantum information processing is an attractive possibility, however, many fundamental challenges remain for practical application, and the most difficult one is quantum noise (sometimes called decoherence). Especially, a recent study [41] showed that a causal, non-instantaneous model precludes the possibility of high-fidelity π-radian conditional phase shifts created by the cross-Kerr effect in optical fiber. Even for the ideal case of no loss, no dispersion, and no self-phase modulation, because of the non-commuting nature of the phase-noise operators this continuous-time multi-mode model [42] will not lead to a favorable application to fiber-based cross-phase modulation, just as performing the parity gate proposed by Munro et al. [20]. Nevertheless, notice that a cavitylike system may support a high-fidelity quantum gate operation [43]. Also, a very recent study on quantum noise effects involved in quantum nonlinear dynamics of coupled gain-loss waveguides shows an interesting result of the enhancement of the Kerr nonlinearity due to much strengthened light intensity in the symmetry-breaking regime [44]. Consequently, we here only show a novel method to explore photon-number entangled states with weak nonlinearities, and this flexible method may be suited to an available system with future modifications, such as a cavity.

4. Discussion and summary

Note that the cross-Kerr nonlinearities are extremely weak and the order of magnitude of them is only 10^{− 2} even by using electromagnetically induced transparency [22, 23]. In the present scheme, let nθ = 1.0×10^{− 2} and then obtain ${\epsilon }_{\max }=\text{erfc}\left[{\left(1-1/n\right)}^2\alpha \times {10}^{-4}/2\sqrt{2}\right]/2$. Therefore, by applying an appropriate coherent probe beam the present scheme can has a small enough error probability and then be realized in a nearly deterministic manner. Given ${\left(1-1/n\right)}^2\alpha =4\sqrt{2}\times {10}^4$ with n = 2,3,…, for example, then we have ε _max ≃ 0.003. Clearly, let n = 2 and mean photon numbers of coherent state n_α = |α|² = 5.12 × 10¹⁰, then the above value of error probability holds. Also, when n ≫ 1 and letting n_α = 3.2×10⁹ we can also have the given error probability.

In addition, we discuss an issue arising from the necessary feed-forward phase shift. Without loss of generality, we take even n for example, thus φ_m(x) = − (α/m)sin[m(n−1)θ]{x − 2α cos[m(n − 1)θ]} mod 2π. In particular, for m(n − 1)θ ≪ 1, we have |φ_m(x)| ~ |α|(n−1)θ|x−2α cos[m(n − 1)θ] mod 2π. If we further assume that |x − 2α cos[m(n − 1)θ]|~1, with nθ ~ 10⁻² and |α| ~ 10⁵, then |φ_m(x)| ~ 10³ mod 2π. It means that the coherent state amplitude α might need to be supplied rather accurately in order to implement the feed-forward phase shift. Alternatively, we consider |x−2α cos[m(n−1)θ] ≪ 1 and the uncertainty in the homodyne measurement is much less than 2πm/{|α| sin[m(n − 1)θ]}. Then, the desired feed-forward phase shift may also be achieved with this available homodyne measurement [26, 45]. In any event, an appropriate feed-forward phase shift will need to be known with sufficient accuracy in order to undo the unwanted phase factor.

In summary, we show an architecture of exploration of photon-number entangled states using weak nonlinearities. Also, we suggest some interesting applications of the present scheme and analyze its error probabilities. Notice that the two-mode photon-number entangled states we studied are different from the general photon-number entangled states [9], or twin beams [46], which are two separate beams of light that bear almost identical intensity. Compared with the previous reports [46–48 ], we not only suggest a method to create these particular two-mode photon-number entangled states, but also discuss how one might detect them in the regime of weak nonlinearities. There are several remarkable advantages in the present scheme. First, our scheme is feasible with the current experimental technology, because there is no large phase shift (−θ with θ ≤ 10^{− 2}, for example) in the interacting process with weak Kerr nonlinearities and then the strength of the nonlinearities we required are orders of magnitude in current practice. Second, in the absence of decoherence, by analyzing the error probability we show that our scheme works in a nearly deterministic way. At last, the present model allows one to explore photon-number entangled states, both creating and detecting, even for a huge photon-number. This, along with the source consists mainly of two phase shifts in the interacting process and a final homodyne measurement, makes our architecture relatively simple but extremely fruitful.