Abstract
We propose a fruitful scheme for exploring multiphoton entangled states based on linear optics and weak nonlinearities. Compared with the previous schemes the present method is more feasible because there are only small phase shifts instead of a series of related functions of photon numbers in the process of interaction with Kerr nonlinearities. In the absence of decoherence we analyze the error probabilities induced by homodyne measurement and show that the maximal error probability can be made small enough even when the number of photons is large. This implies that the present scheme is quite tractable and it is possible to produce entangled states involving a large number of photons.
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Introduction
Undoubtedly, entanglement1,2,3,4 is one of the most crucial elements in quantum information processing, e.g. quantum cryptography5, quantum teleportation6, quantum secure direct communication7,8, etc. In recent years, quantum entanglement has been extensively investigated in various candidate physical systems9,10,11,12,13,14,15,16, in particular, one can prepare and manipulate multipartite entanglement in optical systems17,18,19,20,21,22,23,24,25,26,27,28.
Generally, a spontaneous parametric down-conversion (PDC) source29,30 is capable of emitting pairs of strongly time-correlated photons in two spatial modes. As extensions of interest, with linear optics and nonlinear optical materials several schemes for creating multiphoton entangled states have been proposed31,32,33,34,35,36,37. For a large number of photons, however, there are some technological challenges such as probabilistic emission of PDC sources and imperfect detectors. A feasible approach is to use the simple single-photon sources, instead of waiting the successive pairs and quantum nondemolition (QND) measurement38,39,40,41,42,43,44,45 with weak Kerr nonlinearities. Note that the Kerr nonlinearities46,47,48 are extremely weak and the order of magnitude of them is only 10−2 even by using electromagnetically induced transparency49,50. More recently, Shapiro et al.51,52 showed that the causality-induced phase noise will preclude high-fidelity π-radian conditional phase shifts created by the cross-Kerr effect. In these cases, with the increase of the number of photons it is usually more and more difficult to study multiphoton entanglement in the regime of weak nonlinearities.
In this paper, we focus on the exploration of multiphoton entangled states with linear optics and weak nonlinearities. We show a quantum circuit to evolve multimode signal photons fed by a group of arbitrary single-photon states and the coherent probe beam. Particularly, there are only two specified but small phase shifts induced in the process of interaction with weak nonlinearities. This fruitful architecture allows us to explore multiphoton entangled states with a large number of photons but still in the regime of weak nonlinearities.
Kerr nonlinearities
Before describing the proposed scheme, let us first give a brief introduction of the Kerr nonlinearities. The nonlinear Kerr media can be used to induce a cross phase modulation with Hamiltonian of the form 
, where 
is the coupling constant and 
represents the annihilation operator for photons in the signal (probe) mode. If we assume that the signal mode is initially described by the state 
and the coherent probe beam is 
, then after the Kerr interaction the whole system evolves as
where 
with interaction time t. In order to distinguish different cases, one may perform a homodyne measurement39 on the probe beam with quadrature operator 
, where 
is a real constant. Especially for 
, this operation is conventionally referred to as X homodyne measurement; while for 
, it is called P homodyne measurement.
Creation of multiphoton entangled states with linear optics and weak nonlinearities
Let 
represent input ports with respective spatial modes, namely signal modes and 
is a coherent beam in probe mode. The setup of creating multiphoton entangled states is shown in Fig. 1.
The schematic diagram of creating multiphoton entangled states with linear optics and weak nonlinearities.
are input ports and each port is supplied with an arbitrary single-photon state, while 
are the corresponding outputs, respectively. θ and 2θ represent phase shifts in the coherent probe beam 
induced by Kerr interaction between photons. 
is a phase shift gate. Each 
represents a phase shift on the signal photons based on classical feed-forward information.
Without loss of generality we may suppose that each input port is supplied with an arbitrary single-photon state. Then, the total input state reads
where 
, 
, are complex coefficients satisfying the normalization condition 
and 
denotes the sum over all possible permutations of the signal modes, for example,
Each polarizing beam splitter (PBS) is used to transmit 
polarization photons and reflect 
polarization photons. When the signal photons travel to the PBSs, they will be individually split into two spatial modes and then interact with the nonlinear media so that pairs of phase shifts θ and 2θ are induced on the coherent probe beam, respectively. We here introduce a single phase gate 
so as to implement the next X homodyne measurement on the probe beam. 
, 
for even n and 
for odd n, are phase shifts on the signal photons based on the measured values of x via the classical feed-forward information. At last, at the ports 
one may obtain 
output states for even n or 
states for odd n.
We describe our method in details. For n is even, after the interaction between the photons with Kerr media and followed by the action of the phase gate, the combined system evolves as
In order to create the desired multiphoton entangled states, we here perform an X homodyne measurement38,45 on the probe beam. If the value x of the X homodyne measurement is obtained, then the signal photons become
where
, are respectively Gaussian curves which are associated with the probability amplitudes of the outputs, and
, are respectively phase factors based on the values of the X homodyne measurement. Note that the peaks of these Gaussian distributions locate at 
. Thus, the midpoints between two neighboring peaks 
and the distances of two nearby peaks 
with 
. Obviously, with these 
midpoint values 
there exist 
intervals and each interval corresponds to an output state.
We now consider the phase shifts 
. The signal photon evolves as 
, 
. A straightforward calculation shows that
After these feed-forward phase shifts have been implemented and the signal photons pass through the PBSs, one can obtain the desired states as follows. Clearly, for 
we have
for 
, we obtain the states
and for 
we obtain the state
Similarly, for odd n, we have
Also,
Here, the functions 
, phase shifts 
and 
are approximately the same as those described for even n, except for 
and then the similar results hold for the midpoints 
and the distances 
with 
. Of course, with 
midpoint values 
there may be 
output states; that is, one can obtain the states 
for
for
and
As an example of the applications of interest for the present scheme, we introduce a class of remarkable multipartite entangled states
where
and
are two orthonormal states, namely Dicke states53,54. In view of its “catness”, the state 
can be referred to as cat-like state and especially for 
it can be expressed as the canonical n-partite Greenberger-Horne-Zeilinger (GHZ) state. In the present scheme, obviously, for 
, 
, we can obtain these cat-like states with 
for even n and 
for odd n, where the qubits are encoded with the polarization modes 
and 
. Of course, more generally, we may project out a group of multiphoton entangled states involving generalized Dicke states.
Discussion
There are two models commonly employed in the process of Kerr interaction, single-mode model and continuous-time multi-mode model51. The former implies that one may ignore the temporal behavior of the optical pulses but the latter is causal, non-instantaneous model involving phase noise. It has been shown that52 this causality-induced phase noise will preclude the possibility of high-fidelity CPHASE gates created by the cross-Kerr effect in optical fiber. To solve this problem, one may need to find an optimum response function for the available medium, or to exploit more favorable systems, such as cavitylike systems55. After all, the ultimate possible performance of Kerr interaction with a larger system is an interesting open issue. More recently, we note that Feizpour et al.56 showed the first direct measurement of the cross-phase shift due to single photons. It may be possible to open a door for future studies of nonlinear optics in quantum information processing. In the present scheme, we restrict ourselves to ignoring the phase noise and concentrate mainly on showing a method for exploring multiphoton entangled states in the regime of weak cross-Kerr nonlinearities, i.e. 
.
It is worth noting that, there are only small phase shifts θ and 2θ instead of a series of related functions of the number of photons in the process of interaction with Kerr nonlinearities. This implies that the present scheme is quite tractable especially for creating entangled states with a larger number of photons. In addition, the error probabilities 
are 
, which come from small overlaps between two neighboring curves. Considering the distances of two nearby peaks 
with 
for even n and 
for odd n, the maximal error probability 
, which is exactly the result described by Nemoto and Munro in40. Obviously, the error probabilities in our scheme are no more than that one even when the number of photons is large. Therefore, by choosing an appropriate coherent probe beam the error probability can be reduced to as low a level as desired and then the present scheme may be realized in a nearly deterministic manner.
In summary, based on linear optics and weak nonlinearities we have shown a fruitful method for exploring a class of multiphoton entangled states, the generalized cat-like states. Evidently, three aspects are noteworthy in the present framework. First, since there are no large phase shifts in the interacting process with weak Kerr nonlinearities, our scheme is more feasible compared with the previous schemes. Second, the system is measured only once with a small error probability and it means that the present scheme might be realized near deterministically. Finally, the fruitful architecture allows us to explore a group of multiphoton entangled states involving a large number of photons, i.e., to produce entangled states approaching the macroscopic domain.
Additional Information
How to cite this article: He, Y.-Q. et al. Exploration of multiphoton entangled states by using weak nonlinearities. Sci. Rep. 6, 19116; doi: 10.1038/srep19116 (2016).
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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: A2014205060, the Research Project of Science and Technology in Higher Education of Hebei Province of China under Grant No: Z2015188, the Graduate Innovation Project of Hebei Province of China under Grant No: sj2015004, Langfang Key Technology Research and Development Program of China under Grant No: 2014011002.
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Y.Q.H., D.D., F.L.Y. and T.G. contributed equally to this work. All authors wrote the main manuscript text and reviewed the manuscript.
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He, YQ., Ding, D., Yan, FL. et al. Exploration of multiphoton entangled states by using weak nonlinearities. Sci Rep 6, 19116 (2016). https://doi.org/10.1038/srep19116
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