HIgh-Noon States with High Flux of Photons Using coherent Beam Stimulated Non-Collinear Parametric Down Conversion

We show how to reach high fidelity NOON states with a high count rate inside optical interferometers. Recently it has been shown that by mixing squeezed and coherent light at a beamsplitter it is possible to generate NOON states of arbitrary N with a fidelity as high as 94%. (Afek I. et al. Science 328, 879 (2010)). The scheme is based on higher order interference between"quantum"down-converted light and"classical"coherent light. However, this requires optimizing the amplitude ratio of classical to quantum light thereby limiting the overall count rate for the interferometric super-resolution signal. We propose using coherent-beam-stimulated non-collinear down converted light as input to the interferometer. Our scheme is based on stimulation of non-collinear parametric down conversion by two-mode coherent light. We have somehow a better flexibility of choosing the amplitude ratio in generating NOON states. This enables super-resolution intensity exceeding the previous scheme by many orders of magnitude. Therefore we hope to improve the magnitude of N-fold super-resolution in quantum interferometry for arbitrary N by using bright light sources. We give some results for N=4 and 5.

ere is a proposal [20] to use electromagnetic fields in NOON states to improve the sensitivity of measurements by a factor of N. In terms of photon number states, a two-mode field can be written as a superposition of two maximally distinguishable N-photon states: Some implementations of this state exist [8][9][10]. e Nphoton coherence is optimally sensitive to small phase shifts between the two modes. In particular, the use of photon pairs in interferometers allows phases to be measured to the precision in the Heisenberg limit, where uncertainty scales as 1/N [21] as compared to the shot noise limit where it scales as 1/ �� N √ . is means that for large number of particles, a dramatic improvement in measurement resolution should be possible. ere are various methods for generating path entangled states with high photon numbers up to N � 5 [8][9][10]. Afek et al. [11] experimentally realized high fidelity NOON states for N � 2, 3, 4, and 5 in a single setup. ey realized the idea introduced by Hoffmann and Ono [22] by mixing coherent state and squeezed vacuum or degenerate parametric down conversion (PDC) (both at single modes) at a 50/50 beam splitter (BS) in two input ports. e idea is based on the indistinguishability of the photons after a BS. e indistinguishable processes always result in quantum interferences.
is means that there is enhancement or suppression of some terms in the superposition. e setup for the observation of path entanglement in the interference of coherent state light and down-converted photon pairs is shown in Figure 1(a). e two beams interfere at the input beam splitter. is method can be interpreted as a cancellation of all output components other than |N0〉 and |0N〉 by destructive quantum interference. As an example, Figures  1(b) and 1(c) shows that there are 2 ways to combine 3 photons at the BS from the sources of coherent light and PDC. ese 2 ways mixes the photons coming from coherent light and PDC. e exact cancellation of unwanted terms cannot be achieved at photon numbers higher than 3 [23], so that the extension to higher photon numbers appears to be difficult. Nevertheless, one can get close to exact path entangled N-photon states with the fidelities of 92% or more for arbitrary N. Particularly, it has been shown in the study by Hoffmann and Ono [22] that the fidelity of 94.1% can be reached for N � 5, and Afek et al. [11] demonstrated this experimentally with visibility of 42%. e NOON state fidelity of the output state can be optimized by tuning the relative strength of average photon numbers between the coherent light and PDC. It appears that the determination of this relative strength depends on the number of ways mixing the two sources. e best results were achieved for the low gain limit of the PDC source. ere is also a proposal [24] to increase the generation probability of NOON states by mixing even/odd coherent states.
In this work, we propose a new idea [25] using stimulated parametric processes along with spontaneous ones to produce path entangled states of arbitrarily high photon number N with fidelity greater than 90% at strong gain regime, meaning that the sources are bright. e stimulated processes enhance the count rate by several orders of magnitude. We use coherent beams at the signal and idler frequencies of a noncollinear PDC source (see Figure 2). We further find that the phases of coherent fields can also be used as tuning knobs to control both the fidelity and the magnitude (repetition rate) of NOON state intensity. It may be borne in mind that the process of noncollinear spontaneous parametric down conversion has been a workhorse for more than two decades in understanding a variety of issues in quantum physics and in applications in the field of imaging and sensing.

Materials and Methods
We now describe the idea and the results of preliminary calculations that support the above assertion. Consider the scheme shown in Figure 2. Here a 1 and b 1 are the signal and idler modes driven by the coherent fields. e usual case of spontaneous parametric down conversion is recovered by setting α 0 � β 0 � 0. e ψ is the phase introduced by the object or by an interferometer. For down conversion of type II, the signal and idler would be two photons in two different states of polarization. e input state before the first beam splitter is defined by where |0〉 is the two-mode vacuum state. e two-mode down conversion operator is characterized by a real gain parameter r and phase ϕ that determines the phase of the down-converted photons. e two-mode displacement operator is a product of displacement operators for each mode. e input state in the Schrödinger picture can be written as the following superposition where |Ψ N 〉 is the N-photon component given by where C(m, N − m) are the coefficients of N-photon states in the input state |α 0 , β 0 〉 (r,ϕ) of the interferometer shown in Figure 2. For α 0 � β 0 � 0, the coefficients are given by C(m, n) � δ mn ((− e iϕ tanh(r)) n /cosh(r)). It is easy to produce two-photon NOON state using Hong-Ou-Mandel scheme [26]. However, for N > 2, we need some extra parameters to cancel the unwanted terms inside the interferometer. is can be done using coherent states as seeds, initially in the vacuum modes of the noncollinear down conversion (see Figure 2). When the stimulation is on, the coefficients C(m, n) become functions of coherent field amplitude, and it can be expressed in a closed form [27] as where We take α 0 � β 0 � |α 0 |e iθ and ϕ � 0 for simplicity. We use the phase of coherent fields, θ, for controlling the interference between coherent states and down-converted photons. is interference results from the indistinguishability between coherent state photons and down-converted photons [25] because their polarization, frequency, and momentum states are completely the same. e creation of an ideal NOON state would require suppression of all the non-NOON components after the first beam splitter. By tuning the parameters available in the scheme, we can reach this with a high fidelity using multiphoton interference. e 2 International Journal of Optics fidelity of the output state's normalized N-photon component with a NOON state is where U BS is the 50 : 50 beam splitter unitary transformation and |Ψ norm N 〉 is given by equation (6) with a normalization constant. ere are more than one way to represent U BS , and here, we adapt the choice given in the study by Dunningham and Kim [28]. It is easy to calculate the fidelity given by Pittman et al. (12) by the inner product of the states U BS |NOON〉 and |Ψ norm N 〉. e beam splitter transformation of the NOON state is As an example, let us consider the calculation of F 4 . e beam splitter transforms |4004〉 to be and equations (6)- (14) would lead to upto a normalization. e interferometric phase measurement is done by the photon number resolving detection (see Figure 2). For example, for four-fold resolution enhancement, we use an array of four single-photon coincidence counting modules in 2 by 2 arrangement as proposed by Kolkiran and Agarwal [19] (it means two of the detectors are in the upper exit port and two of them in the lower exit port) or 3 by 1 scheme as proposed by Kolkiran and Agarwal [29]. ere are two parameters to optimize the fidelity once we fix the phase and the flux of PDC photons; the phase of the coherent fields, θ, and the pair amplitude ratio of the coherent state and PDC state is given by for weak fields. For the strong fields, this ratio should be replaced by because the magnitude of pair flux from PDC is dominated by the term sinh 2 (r) [25]. is optimization shows a different character for weak and strong-field regimes. In the weak-field regime, the maximum fidelity is very sensitive to c, and it is a limiting factor for the total strength of the total signal. On the other hand, in the limit of high gain, c takes larger values together with flexibility in the optimization. is makes possible the super-resolving phase    for weak fields. e horizontal axis is the phase of the coherent field. e phase of PDC is chosen to be zero. Here, the gain parameter of the PDC is r � 0.1 and the amplitude ratio is optimized at c � |α| 2 /r � 2.26 for the maximum fidelity of F 4 � 93.3%. e maximum fidelity is reached at phases of 0 and π. is is much better than the case (with down-converted photons only) with a fidelity of 75% and the case (coherent fields only) with a fidelity of 50%. (b) e fidelity profile in the limit of high gain with r � 4.5 (this gain has been reported in [30]). e black, red, and blue curves are with fidelities of 92%, 90%, and 81%, respectively. It is clear from the plots that we have much more flexibility in pair amplitude ratio. By choosing a larger amplitude ratio, we can reach a total flux of coherent state pair photons having approximately 3 orders of magnitude higher than PDC pair photons. measurements with high-NOON states at much brighter light resources.
We would like to explain the physics behind the advantage of using our scheme compared with the idea of Hoffmann and Ono [22]. Suppose that we want to construct N � 3-NOON state, in our scheme, there are six distinct ways to mix three photons coming from the laser light (coherent states) and PDC (see Figure 3). is is three times larger than the scheme used in other studies [11,22]. For N � 4 and N � 5, in our scheme, there are 9 and 12 ways, respectively, whereas there are 2 and 3 ways in other studies [11,22]. e number of photon mixing ways is getting even larger and larger for higher NOON states. As a result, our scheme had higher-order interference effects leading to higher count rates with similar fidelity. We give detailed results in the next section.

Results and Discussion
We now give our results of the theoretical fidelity for the generated NOON states in Figure 4 and 5. Figure 4(a) shows the fidelity of interferometer state's N photon component with NOON state with N � 4 for weak fields. e horizontal axis is the phase of the coherent field. e phase of PDC is chosen to be zero. Here, the gain parameter of the PDC is r � 0.1, and the amplitude ratio of the coherent state and PDC given by c � |α| 2 /r is optimized at 2.26 for the maximum fidelity of F 4 � 93.3%. e maximum fidelity is reached at phases of 0 and π. is is much better than the case in which α � 0 (with down-converted photons only) with a fidelity of 75% and the case in which r � 0 (coherent fields only) with a fidelity of 50%. Figure 4(b) shows the fidelity profile in the limit of high gain with r � 4.5 (this gain value has been reported experimentally in [30]). e black, red, and blue curves are for c � 10, 50 and 150 with fidelities of 92%, 90%, and 81%, respectively. It is clear from the plots that we have much more flexibility in the amplitude ratio. By choosing a larger amplitude ratio, we can reach a total flux of coherent state photons having approximately two orders of magnitude higher than PDC photons. For example, for c � 50, when the mean number of photons in PDC sinh 2 (4.5) ≈ 2 × 10 3 , we have ≈ 10 5 coherent state photons. In physical terms, as proposed by [10], c 2 is the two-photon probability of the classical source divided by that of the quantum source. In this case, we have c 2 � 2500, meaning that for every pair of PDC photons, 2500 pairs of laser photons are used. Although the fidelity is down only by 3% as proposed by Afek et al. [11] with c 2 � 3, we reach three orders of magnitude in signal repetition rate.
is would increase the coincidence count rate tremendously when the method is used in experimental quantum interferometry. We note here that a slight decrease in the fidelity would only decrease the visibility slightly. As noted in the study by Hofmann and Ono [22], the phase sensitivity of an N-photon fringe with visibility V is given by δψ � 1/VN, we can expect a phase sensitivity of 1/0.90 N in the high photon number limit, a result that is only slightly lower than the Heisenberg limit of 1/N, achieved by maximal path entanglement. In the strong-field regime, one can notice the strange dependence of fidelity on the coherent state phase θ. is is unlike the weak-field regime, which shows a nice periodic dependence. In the strong-field regime, as c gets larger, the bandwidth of phase dependence for the maximum fidelity gets narrower in some sense. e physics of this behaviour needs to be investigated further. Figure 5 shows the results for N � 5. Figure 5(a) shows the optimized fidelity (maximum at 91%) profile at the low gain limit, r � 0.1 and c � 0.6 . Figure 5(b) shows the fidelity profile in the limit of high gain with r � 4.5. e black, red, and blue curves are the fidelity profiles for c � 10, 50 , and 150, respectively. e respective maximum fidelities are approximately 91%, 88%, and 84%, respectively. Here, the fidelity of 91% is being reached when c 2 � 100 meaning that for every pair of PDC photons, 100 pairs of laser photons are used to see five-photon interference pattern.

Conclusions
In conclusion, we have shown that using stimulating coherent fields in the noncollinear PDC setup generates not International Journal of Optics only high-NOON states with large N but also with arbitrary intensity for N � 4 and 5. e realistic application of NOON states in quantum metrology requires high-intensity flux of photons. e theoretical improvement using coherent field stimulated noncollinear PDC photons over the method of mixing squeezed light with coherent state [11,22],which implies a fundamental connection between nonlocality of the source and creation of NOON states. e number of ways mixing nonclassical photons with the classical ones in the proposed scheme is multiple times larger, which leads to higher-order interference effects. is is why the scheme is using more classical photons over nonclassical ones (three orders of magnitude larger) to reach Heisenberg limited sensitivity. e ongoing development of high gain parametric down conversion together with efficient detectors shows promise for realizing the scheme proposed in this paper. We also state that the scheme can be applied to utilize NOON states of higher N number. Further study to this end is required.

Data Availability
e analytic data used to support the findings of this study are included within the article. e plots in Figures 4 and 5 can be obtained by plotting the closed forms given by equations (6)-(15) using an appropriate "function plotting software," such as mathematica or matlab.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.