Abstract
We describe a method to detect twinbeam multiphoton entanglement based on a beam splitter and weak nonlinearities. For the twinbeam fourphoton entanglement, we explore a symmetry detector. It works not only for collecting twopair entangled states directly from the spontaneous parametric downconversion process, but also for generating them by cascading these symmetry detectors. Surprisingly, by calculating the iterative coefficient and the success probability we show that with a few iterations the desired twopair can be obtained from a class of fourphoton entangled states. We then generalize the symmetry detector to npair emissions and show that it is capable of determining the number of the pairs emitted indistinguishably from the spontaneous parametric downconversion source, which may contribute to explore multipair entanglement with a large number of photons.
Introduction
Since optical quantum systems provide some natural advantages, they are prominent candidates for quantum information processing^{1,2,3}. As a fundamental physical resource, the multiphoton entanglement plays a crucial role in optical quantum computing^{4,5,6}. A standard entangled photon pair is created by means of the nonlinear optical process of spontaneous parametric downconversion (SPDC)^{7}. In SPDC process, one may create photons entangled in various degrees of freedom, for example, polarization entanglement^{8,9,10}, path entanglement^{11,12}, etc.
For creation of polarizationentangled photons, a simplified Hamiltonian^{13,14} of the nonlinear interaction is given by \({H}_{{\rm{SPDC}}}={\rm{i}}\kappa ({\hat{a}}_{H}^{\dagger }{\hat{b}}_{V}^{\dagger }{\hat{a}}_{V}^{\dagger }{\hat{b}}_{H}^{\dagger })+{\rm{H}}{\rm{.c}}{\rm{.}}\), where \({\hat{a}}_{x}^{\dagger }\) and \({\hat{b}}_{x}^{\dagger }\) (with x = H, V) are respectively the creation operators with horizontal (H) or vertical (V) polarization in the spatial modes a and b, and κ is a realvalued coupling constant depended on the nonlinearity of the crystal and the intensity of the pump pulse. In the number state representation, the resulting photon state reads^{14,15,16}
where e.g. \(m{\rangle }_{{a}_{V}}\) means m vertically polarized photons in spatial mode a, and τ = κt/ħ is the interaction parameter with t being interaction time. Each \({\psi }_{n}^{}\rangle \) represents the state of n indistinguishable photon pairs with 〈n〉 = 2sinh^{2} τ. It should be noted that \({\psi }_{n}^{}\rangle \) is different from the general multiphoton entangled state in which each photon represents a qubit. State (2) is usually called the twinbeam multiphoton entangled state.
To avoid multipair emission events, in general, τ is restricted to small enough, such that mainly the firstorder term has been taken into account. For the higherorder terms, these twinbeam multiphoton entangled states have interesting features^{15}, i.e. they are not only entangled in photon number for the spatial modes a and b, but also entangled maximally in polarization degree of freedom. Especially for the secondorder emission, it has been shown that^{17}, depending on the relation between the duration of the pump pulse and the coherence time of the photons, the emitted state is described by two independent pairs or an indistinguishable twinbeam fourphoton entangled state. For the indistinguishable fourphoton entangled state, furthermore, it can be useful for the applications in quantum information processing, for example, testing the quantum formalism against the local realistic theories^{18,19}. Unfortunately, up to now there are only a few reports^{20,21,22} to exploring these analog of a singlet state of two spinn/2 particles.
In this paper, we first focus on the twinbeam fourphoton entangled states and design a quantum circuit of symmetry detector to evolve them by using a beam splitter (BS) and weak nonlinearities. By cascading symmetry detectors, we then propose a scheme of generating the twinbeam twopair entangled state in a near deterministic way. Finally, we generalize the present symmetry detector to highorder emissions.
Results
Symmetry detector based on beam splitter and weak nonlinearities
Throughout the text, for simplicity we write m, n; r, s〉 as an abbreviation for state \(m{\rangle }_{{a}_{H}}\otimes n{\rangle }_{{a}_{V}}\otimes r{\rangle }_{{b}_{H}}\otimes s{\rangle }_{{b}_{V}}\) which means that there are m horizontally polarized photons and n vertically polarized photons in spatial mode a and also there are r horizontally and s vertically polarized photons in spatial mode b.
We first restrict our attention to the fourphoton entanglement and describe a method to explore symmetry detector for the twinbeam entangled states. In general, consider a class of fourphoton entangled states
where c is a constant and normalization factor N satisfies N ^{2} = 1/(2 + c^{2}). Without loss of generality, we may suppose coefficient c to be real.
Consider a lossless 50:50 BS with Hamiltonian \({H}_{{\rm{BS}}}={\rm{i}}\pi ({\hat{a}}^{\dagger }\hat{b}\hat{a}{\hat{b}}^{\dagger })/4\), where \({\hat{a}}^{\dagger }\) (\({\hat{b}}^{\dagger }\)) and \(\hat{a}\) (\(\hat{b}\)) are respectively creation and annihilation operators in the input spatial mode a (b). As shown in Fig. 1, since the interference effect of BS, the input twinbeam state evolves
where (N′)^{2} = 1/(4c ^{2} + 8). An interesting consequence of this evolution is that the input state yields two possible cases, i.e. symmetric state (two photons are in one spatial mode and others are in another spatial mode) and alternatively asymmetric state (the output four photons are in the same spatial mode).
In order to distinguish between the symmetric state and the asymmetric state, we here consider quantum nondemolition detection^{23,24} by using weak nonlinearities. As an important nonlinear component for alloptical quantum computing, Kerr medium^{25,26,27} is capable of evolving photons in signal and probe modes with the interaction Hamiltonian \({H}_{{\rm{Kerr}}}=\hslash \chi {\hat{n}}_{s}{\hat{n}}_{p}\), where χ is the coupling strength of the nonlinearity and \({\hat{n}}_{s}\) (\({\hat{n}}_{p}\)) represents the number operator for the signal (probe) mode. As a result, if there are n photons in the signal mode, then it yields nθ in the probe mode, where θ = χt is a phase shift on the coherent probe beam induced by the interaction via Kerr media and t represents the interaction time. In this way, these Kerr media are used mainly in creating and manipulating multiphoton entanglement^{28,29,30,31,32,33,34,35,36,37,38,39,40,41,42}. Since the Kerr nonlinearities are extremely weak^{27,31}, we here only take small but available phase shifts into account.
As shown in Fig. 1, after an overall interaction between the photons with Kerr media, the combined system \({{\rm{\Phi }}}_{{\rm{BS}}}\rangle \otimes \alpha \rangle \) then evolves as
where c _{1} = 2/(1 + c) is the derived coefficient connected with the original c, P = 1/{1 + (1 − c)^{2}/[2 + (1 + c)^{2}]} and \({N}_{1}^{2}=1/\mathrm{(2}+{c}_{1}^{2})\) are respectively the probability and normalization factor for the symmetric state.
We next turn to the question of how to project the signal photons into the symmetric state or the asymmetric state. For a real coherent state, generally, one may perform an X homodyne measurement^{43,44,45,46} with the quadrature operator \(\hat{x}=\hat{a}+{\hat{a}}^{\dagger }\). In terms of the result^{47} 〈xα〉 = (2π)^{−1/4} exp[−(Im(α))^{2} − (x − 2α)^{2}/4], after the X homodyne measurement on the probe beam, for x > α(1 + cos θ), one can obtain the symmetric state
Alternatively, for x < α(1 + cosθ), we get the asymmetric state
up to a phase shift φ _{ x } = −α sin θ(x − 2α cos θ)/2 mod 2π on the spatial mode b _{2} according to the value of the measurement.
That is, if one permits the phase shift with feedback from the value of the measurement, then the twinbeam fourphoton asymmetric state \({{\rm{\Phi }}}_{x}^{0}\rangle \) can be prepared. Also, for each symmetric state, it is interesting to note that the output state is analogous to the input state, up to a correlation coefficient (relative amplitude). Especially, for c = 1, i.e.
it is the twinbeam entangled state emitted by the SPDC source with the duration of the pump pulse is much shorter than the coherence time of the photons, and we here refer to this pair of the indistinguishable fourphoton entangled states as twopair, for simplicity. Obviously, for this twopair, one can immediately obtain the result that the output state is the same as the input.
Twopair generation by cascading symmetry detectors
For a fourphoton entangled state created in SPDC process^{17}, besides the above twopair, the four photons can be emitted as two independent pairs in the opposite limit, or any intermediate situation depended on the ratio between the duration of the pump pulse and the coherence time of the created photons. Since the relative phase relation and the equal weight of the terms, as a singlet spin1 state, the twopair satisfies rotational symmetry. Similar to the rotationally symmetric Bell state, it may be useful for quantum information processing and quantum computation in the future.
As an important application of the symmetry detectors, we now present a scalable scheme for generating the twopair with the general fourphoton entangled states. For this purpose, we construct a quantum circuit diagram by cascading these symmetry detectors, as shown in Fig. 2, where the input/output modes correspond to the signal photons. In each symmetry detector, we here simplify the initial model straightforwardly by discarding the result of the asymmetric state.
Then, after the i th cascading, one can obtain the symmetric state
where c _{ i } = 2/(1 + c _{ i−1}), \({N}_{i}^{2}=1/\mathrm{(2}+{c}_{i}^{2})\). The total success probability reads
As a result, it is not difficult to find that such a cascading symmetrydetector is capable of generating twopair from the mentioned fourphoton entangled states.
Clearly, we here take c = 2 for example. We calculate the iterative coefficient c _{ i } and the probability P _{ i } and plot the relationships of the correlation coefficients and the success probabilities versus the number of iterations (10 times), as shown in Fig. 3. The result shows that with a few iterations the correlation coefficient approaches 1 and the success probability gets close to 1. Furthermore, since we take only those events into account that yield the required results via postselection, the present scheme of the twopair generation is near deterministic.
Symmetry detector for the higherorder emissions
So far, we have addressed the symmetry detector and its interesting application, involving the secondorder emission of the SPDC process. In order to enable us to explore multiphoton entanglement with a large number of photons from SPDC source, we now describe a method to generalize the symmetry detector from the secondorder to the higherorder emissions of the SPDC source.
For the higherorder emissions, consider an npair multiphoton entangled state \({\psi }_{n}^{}\rangle \). When the photons passing through the 50:50 BS, the transformation between the incoming modes (a _{1} and b _{1}) and the outgoing modes (a _{2} and b _{2}) is
Then, as the multiphoton interference effect at the symmetric BS, the input npair entangled state \({\psi }_{n}^{}\rangle \) will be transformed into
This result implies that when the photons passed through a symmetric BS the npair entangled state remains unchanged.
In the process of the nonlinear interactions, for clearer statement, we here rewrite the Kerr phase shifts θ/3 and 2θ/3 in spatial modes a _{2} and b _{2}, and the original phase shift −5θ is accordingly replaced by −θ. On the basis of the methods of exploring multiphoton entanglement via weak nonlinearities^{44,45}, for arbitrary mpair, 1 ≤ m ≤ n, the total phase shift in the probe mode is (m − 1)θ. Then after an X homodyne measurement, one may obtain the npair with the value x < α{cos[(n − 1)θ] + cos[(n − 2)θ]}, (n − 1)pair with α{cos[(n − 1)θ] + cos[(n − 2)θ]} < x < α{cos[(n − 2)θ] + cos[(n − 3)θ]}, \(\cdots \), mpair with the value α{cos(mθ) + cos[(m − 1)θ]} < x < α{cos[(m − 1)θ] + cos[(m − 2)θ]}, \(\cdots \), twopair with α[cos θ + cos 2θ] < x < α[1 + cos θ] or onepair (the singlet state) with x > α[1 + cos θ]. Obviously, as a particular case of the npair emissions, a onepair emission is simple but instructive. Since these multipair structures are robust against losing of photons they are maybe contribute to explore multiphoton entanglement from microscopic to macroscopic systems.
Discussion
By now, we have concentrated on the means to explore symmetry detectors for twinbeam multiphoton entanglement. A realistic SPDC source with the higherorder emissions, however, inevitably emits onepair, twopair or npair entangled photons, spontaneously. A surprising result of the present symmetry detector is that the number of the pairs emitted from the SPDC source can be determined exactly and then be collected. Indeed, after the X homodyne measurement on the probe beam one can immediately infer the number of the pairs by means of the value of measurement. Also, the signal photons are specifically projected onto a particular multipair entangled state.
In recent years, one of the most intriguing developments of quantum theory, both theoretical and experimental, is opticsbased quantum information processing^{3,5,6,48,49,50}. However, many fundamental challenges remain for practical application, for example, how to obtain (approximately) πradian phase shifts with crossKerr nonlinearities. In 2003, theoretically, Hofmann et al.^{51} showed that a nonlinear phase shift of π can be obtained by using a single twolevel atom in a onesided cavity. However, it exists a challenge to experimental realization due to additional reflections of mismatched pump light. A recent study^{52} showed that it is still unsatisfactory for creating highfidelity πradian conditional phase shifts by the crossKerr effect in optical fiber. In view of this fact, we here only take weak nonlinearities into account, i.e. θ ≤ 10^{−2}. More concretely, if we take α = 2.0 × 10^{6} and θ = 2.0 × 10^{−3}, then the error probability of our symmetry detector is \(\varepsilon ={\rm{erfc}}\,({x}_{d}/2\sqrt{2})/2\simeq 3\times {10}^{5}\), where x _{ d } is the distance between two peaks of Gaussian curves. Furthermore, it is exactly the maximal value of the error probabilities in the process of determining the multipair emissions. Therefore, it is in this sense that the present scheme can be realized in a nearly deterministic manner.
In conclusion, we explore an efficient symmetry detector for detecting the twinbeam multiphoton entanglement based on BS and weak nonlinearities. Especially, as a typical application we suggest a scalable scheme of twopair generation with a class of fourphoton entangled states by cascading these symmetry detectors. Note that such twopair entangled states may be very useful for multiphoton quantum information processing in the future. In the present architectures, there are several remarkable advantages. First, for the higherorder emissions, it is capable of determining the number of the pairs emitted indistinguishably from the SPDC source. Second, since we here only use a symmetric BS and two small Kerr nonlinearities, our symmetry detector is simple and novel. At last, it is possible to extend our means to general circuits constructed from linear elements, SPDC sources, and detectors. We hope that our scheme will stimulate investigations on the applications of higherorder emissions from the SPDC source.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant Nos: 11475054, 11371005, 11547169, Hebei Natural Science Foundation of China under Grant No: A2016205145, Fundamental Research Funds for the Central Universities of Ministry of Education of China under Grant No: 3142017069, Foundation for Highlevel Talents of Chengde Medical University under Grant No: 201701, the Research Project of Science and Technology in Higher Education of Hebei Province of China under Grant No: Z2015188.
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Y.Q.H., D.D., F.L.Y. and T.G. contributed equally to this work. All authors wrote the manuscript text and reviewed the manuscript.
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Correspondence to Dong Ding or FengLi Yan or Ting Gao.
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On fourphoton entanglement from parametric downconversion process
Quantum Information Processing (2018)
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