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We describe an experimental demonstration of a novel three-photon N00N state generation scheme using a single source of photons based on spontaneous parametric down-conversion (SPDC). The three-photon entangled state is generated when a photon is subtracted from a double pair of photons and detected by a heralding counter. Interference fringes measured with an emulated three-photon detector reveal the three-photon de Broglie wavelength and exhibit visibility > 70% without background subtraction.
©2009 Optical Society of America
1. Introduction
Quantum optical technologies based on multi-photon entangled states can exceed fundamental limits such as the Rayleigh diffraction limit of optical imaging [1,2] or the standard quantum limit of optical phase estimation [3–9]. In particular, the resolution in lithography and metrology can be improved beyond the classical limit by applying what is known as a N00N state, described as |Ψ〉 = (|N,0〉+|0,N〉)/21/2, which embodies photon-number entanglement between two distinct optical modes [1–17]. These N photons comprising a N00N state can collectively behave as a single photon whose wavelength is 1/N of the single-photon wavelength, thereby effectively increasing the imaging resolution and the phase estimation uncertainty by a factor of 1/N and 1/N 1/2, respectively [1–8]. A N00N state generator can also generate few-photon polarization-squeezed light that can be ‘over-squeezed’ [18]. The simplest N00N state with N=2 has been experimentally demonstrated since 1990 by utilizing an SPDC photon pair and a Hong-Ou-Mandel (HOM) interferometer [10–12,19]. The potential benefits of an N=2 state are, however, limited by the relevant de Broglie wavelength [20] being equal to, hence no shorter than the SPDC pump wavelength. Various theoretical schemes for generating N00N states with N>2 have been proposed [21–29], but because of technical difficulties [30,31] only a handful of experiments have succeeded in generating N>2 N00N states [8,15,16,32–35]. Moreover, most of these experiments intrinsically require state projection measurements [8,16,32–35] using coincidence counting of photons in multiple spatial modes to suppress the effect of superfluous states. This method of projecting out unwanted states enables super-resolution interferometry even with classical light sources [35]. However, the original quantum lithography proposal requires the detection of N photons in a single spatial mode [1], which is incompatible with the state projection method. To the best of the authors’ knowledge, only one work in the literature has reported super-resolution interference measurements with an N=3 N00N state with a scheme that does not intrinsically rely on state projection by output photon counters [15].
The major technical difficulty in realizing N00N state generation schemes is the lack of practical near-ideal photon sources. Theoretical proposals typically assume the deterministic generation of a definite number of photons [21–27]. However, practical sources based on SPDC or faint laser pulses are probabilistic in nature and often produce redundant photons that degrade the visibility of N00N state interference fringes [15]. The first demonstration of the three-photon N00N state used two sources, an SPDC source and a weak laser pulse [15]. The probability of producing redundant photons usually increases with the number of independent probabilistic sources, and therefore the super-resolution quality can be further improved by generating the photons from a single probabilistic source. This paper reports the experimental realization of a novel N=3 N00N state generation scheme based on double photon pair emission from a single SPDC source.
2. Scheme
The production of redundant photons by probabilistic photon sources degrades the output purity of a N00N state generation scheme. Here, a photon is said to be redundant if the scheme generates N or more photons and yet the generation of the photon in question is unnecessary to produce a N00N state. Output states with fewer than N photons are less deleterious because the known applications of N00N states such as quantum lithography and quantum metrology employ N-photon detectors that are insensitive to lower-number photon states [1–9]. The effect of redundant photons on the scheme can be quantified in terms of the relative probability to generate the ideal number of photons required by the scheme. A relevant figure of merit is the ratio Pi/Pr, where Pi is the probability per pump pulse to generate just the photons required to build the N00N output state, and Pr is the probability per pump pulse to produce at least one redundant photon. Consider, for example, the N=3 scheme that uses an SPDC photon pair generated with probability γ and a weak laser pulse having a photon with probability α [15]. Pr is obtained by summing over all scenarios giving rise to a redundant photon: SPDC double pair emission, SPDC single pair emission plus two photons from the weak pulse, emission of three photons from the weak pulse, and emissions of higher photon numbers. Since Pi = α⋅γ, we obtain Pi/Pr < [γ/α+α/2+α2/(6γ)]−1 < Pi -1/3 for the scheme of [15]. By comparison, Pi/Pr ≈1/γ = Pi -1/2 in our scheme, where SPDC double pairs and triple pairs are generated with probabilities of γ2 and γ3, respectively. Therefore, the scheme based on SPDC double pair emission can achieve a desired value of Pi/Pr with a higher Pi.
A schematic of our N00N state generator is shown in Fig. 1 . The SPDC source consists of a non-colinear type-I phase-matched BBO crystal (thickness 1 mm) pumped by a periodic UV pulse train (center wavelength 390 nm; pulse duration 200 fs; repetition rate 76 MHz; average power 500 mW) from a frequency-doubled mode-locked Ti:sapphire laser. The pump beam is focused onto a spot with a full-width at half-maximum (FWHM) diameter of 50 μm in the BBO crystal, where down-conversion produces indistinguishable pairs of horizontally polarized photons that are subsequently collimated with convex lenses (focal length 150 mm).
Fig. 1 Generation and measurement of three-photon N00N states. PBS: polarizing beam splitter; PPBS: partial PBS; HWP: half-wave plate; QWP: quarter-wave plate; IF: interference filter; SPC: single photon counter; SMF: single-mode fiber; FDC: fiber directional coupler.
The preparation stage of the scheme begins with the emission of double pairs of photons by SPDC. The polarization beam splitters (PBSs) transmit (reflect) horizontally (vertically) polarized photons, and wave plate angles given henceforth denote the inclination of the slow axis with respect to the horizontal axis. The two photons along the upper path have their polarization changed to the vertical by a combination of a mirror and a quarter-wave plate (QWP1) aligned along 45°, and merge with the two lower-path photons at PBS2 into a single spatial mode. After these photons pass through a half-wave plate (HWP1) with angle at 22.5°, the four-photon state becomes
where aH (aV) is the annihilation operator for the horizontal (vertical) polarization mode. The ideal partial PBS (PPBS) wholly transmits horizontally polarized photons while transmitting (reflecting) 1/3 (2/3) of the vertically polarized photons [36–38] (the actual custom-coated PPBS made of BK7 glass has measured transmissions of 99% and 31% for horizontal and vertical polarization inputs, respectively). When exactly one photon is reflected by the PPBS, the remaining three photons propagating to QWP2 are in the statewhere ϕ is the magnitude of the birefringence introduced by the PPBS between the horizontal and vertical polarizations. Adjusting the angles of QWP2 and HWP2 to 45° and θ = ϕ/4, respectively, transforms the output from HWP2 into the three-photon N00N state described asAn interference filter (center wavelength 780 nm; FWHM bandwidth 5 nm) is inserted between HWP1 and the PPBS to reduce background noise and improve the degree of temporal/spectral indistinguishability among the four SPDC photons.Barring losses, the probability for the initial double photon pair to yield the above state is 4/27, and the remaining cases produce states that send zero or multiple photons to the heralding single photon counter 1 (SPC1: Perkin-Elmer, SPCM-AQ4C). The avalanche photodiode used as the heralding counter cannot resolve photon number; it detects multiple photons, hence heralds an output of less than three photons with a probability of 4/9 given an SPDC double pair. As mentioned previously, these N<3 output states do not seriously detract from our scheme. Similarly, no degradation of super-resolution results from an output state produced by an SPDC single pair. However, an SPDC triple photon pair emission can degrade the quality of our N=3 scheme because the output state can include redundancy by consisting of three or more photons.
The heralding photon reflected from the PPBS is coupled into a single-mode fiber (SMF: mode field diameter 5.6 μm; numerical aperture 0.12) and detected by SPC1. The interference between the horizontal and vertical polarization components of the output state from HWP2 is measured by means of QWP3, HWP3, PBS3, and two SMF directional couplers connected to three SMF-coupled SPCs (SPC2~SPC4). QWP3 set at 45° converts horizontal and vertical polarizations into left- and right-circular polarizations, respectively, while HWP3 swaps the two circular polarizations and applies an angle-dependent phase difference between them. The combined action of QWP3 and HWP3 can be expressed as follows:
where θ is the HWP3 angle. This transformation is equivalent to 50/50 coupling with a relative phase shift of 4θ between the horizontal and vertical polarization modes. Interference between the two circular polarization components produces horizontally polarized photons that are transmitted by PBS3 and coupled into the SMF using an aspheric lens (focal length 8.0 mm). All three photons of states |3H,0V〉HWP2 and |0H,3V〉HWP2 pass through PBS3 with a probability of 1/8, while all three photons of states |2H,1V〉HWP2 and |1H,2V〉HWP2 pass through PBS3 with a probability of 3/8 [39]. Both SMF directional couplers have a beam-splitting ratio of 43:57, hence the three photons entering the SMF divide equally among the three SPCs with a probability of 21%. The coincidence counting by these SPCs detects three-photon state in the single spatial/polarization mode of the SMF before FDC1. This measurement emulates a three-photon absorber for quantum lithography in which a single photographic element changes its atomic level when three identical photons are absorbed simultaneously.3. Experimental results
The first step in preparing the scheme for N00N state generation is the adjustment of the position of the back-reflecting mirror with HOM interferometry. For this purpose only, the average pump power for SPDC is lowered to 100 mW to reduce the relative probability of double pair emission that reduces the visibility of the HOM dip. Coincidence counting of SPC1 and SPC2 with the angles of QWP2, HWP2, QWP3, and HWP3 set uniformly along 0° projects the state entering PPBS to |1H,1V〉. Spatio-temporal mode overlap between the upper and lower path photons is achieved by obtaining a HOM dip with a visibility of 0.97, as shown in Fig. 2(a) . After restoring the average pump power to 500 mW, the HWP2 angle is tuned to maximize four-photon coincidence counts with QWP2, QWP3, and HWP3 aligned along 45°, 0°, and 0°, respectively. These coincidence counts are proportional to the probability that all three photons are horizontally polarized before entering QWP3. As shown in Fig. 2(b), the optimal HWP2 angle is a multiple of 45°, which implies that the PPBS does not introduce significant birefringence.
Fig. 2 Preparation of the three-photon N00N state generation scheme: (a) HOM interferometry to match the two optical path lengths: two-fold coincidence counts per 1 s as a function of the time delay between the photons entering PBS2 through the upper and the lower paths. The solid curves are Gaussian least-square fits. (b) Adjustment of the HWP2 angle: four-fold coincidence counts of SPC1~SPC4 per 300 s. The dashed line is a sinusoidal fit.
The interference between the horizontal and vertical polarization modes measured with respect to the N=3 N00N state is shown in Fig. 3 . The QWP3 angle has been reset to 45°, and the interference fringes are obtained by varying the HWP3 angle. Figure 3(a) shows the two-fold coincidence counts of SPC1 and SPC2. Here, the interference fringes are due to single-photon interference between the horizontal and vertical polarization components of the right-circularly polarized photons entering QWP3. Figure 3(b) shows four-fold coincidence counts of SPC1~SPC4, which reveal super-resolution interference with fringes that oscillate three times faster compared to single-photon interference. The solid line is a sinusoidal fit to the experimental data, and the error bars represent the standard deviations that correspond to the square roots of the measured counts. The visibility of these heralded three-photon interference fringes is 0.72±0.03 without background subtraction, which clearly surpasses the level of visibility (10% for N=3) achievable with a proposed technique based on classical nonlinear optics [40]. The unheralded three-fold coincidence counts of SPC2~SPC4 shown in Fig. 3(c) are, on average, two orders greater than in Fig. 3(b), and mostly attributable to double pairs of SPDC photons transmitting through the PPBS with no photon going to SPC1. In this case, all four photons incident on HWP3 are horizontally polarized and produce a sharp peak when the HWP3 angle is a multiple of 90°. The single photon detection probability at SPC1 is measured to be (1.92±0.02)×10−3, and this multiplied by the unheralded three-photon coincidence probability approximately equals the four-photon coincidence probability due to triple pair production by the SPDC source. Subtracting our estimated triple-pair background from the interference pattern of Fig. 3(b) yields Fig. 3(d), which exhibits three-photon interference with a visibility of 0.91±0.03. The values for the visibility we have obtained indicate that the fidelity 〈Ψideal|ρexp|Ψideal〉 between the generated ρexp and the ideal N00N state |Ψideal〉 is greater than 0.68±0.03 (0.90±0.03 with background subtraction) [41].
Fig. 3 Measurement results of single- and three-photon interference. Data points are coincidence counts per 300 s, and error bars represent standard deviations calculated as (counts)1/2. (a) Single-photon interference: two-fold coincidence counts of SPC1 and SPC2; (b) heralded three-photon interference: four-fold coincidence counts of SPC1~SPC4; (c) unheralded three-photon interference: three-fold coincidence counts of SPC2~SPC4; (d) heralded three-photon interference after subtracting the background coincidences due to the triple-pair generation by SPDC.
From the measurement results in Fig. 3 and nominal detection efficiency (64%) of SPCs, the relative probabilities of photon-number states in the output can be estimated. When SPC1 clicks, the output is in vacuum, single-, two-, and three-photon states with probabilities of 92%, 8%, 0.06%, and 0.005%, respectively, which means that the heralded output state is mostly a vacuum state or a single-photon state that originate from single pair emission by SPDC. If the total photon flux impinging on a sample is a major concern, an ideal double photon pair source and a high-efficiency photon-number-resolving heralding detector are required to suppress these unnecessary lower-number states. Since the success probability of the initial four photons to yield a N00N state is 4/27, our scheme does not satisfy the criterion for phase sensitivity to beat the standard quantum limit ηV 2 N > 1, where η is the success probability, V is the visibility, and N is the number of photons used [35]. However, the raw visibility of 0.72 in Fig. 3(b) satisfies the criterion expressed as V 2>1/3 if the source and the heralding detector are ideal and only photons passing through the phase shifter are counted.
4. Conclusion
We have demonstrated the generation of three-photon N00N states using a novel scheme based on linear optics and a single SPDC source that emits two pairs of photons. Three-photon number entanglement between two orthogonal polarization modes has been verified by observing interference fringes with a de Broglie wavelength three times smaller than the single-photon wavelength. The super-resolution interference has been measured in a single spatial mode without the aid of the state projection method, hence our scheme is compatible with the original proposal for quantum lithography [1]. To our knowledge, this is the first quantitative demonstration of quantum-lithography-compatible N=3 super-resolution with a visibility that exceeds the present expectation of classical nonlinear optics. Our scheme can be extended to generate N>3 N00N states by utilizing multiple SPDC photon pairs and heralding counters although the probability of successful N00N state generation decreases with increasing N. For example, an N=4 N00N state can be generated from a triple photon-pair state |3H,3V〉 by replacing the PPBS in our scheme with two PPBSs having reflectances of 1/3 and 0 for the vertical and horizontal polarizations, respectively, and one HWP at 45° in between. Simultaneous detection of single photons reflected by each PPBS, which occurs with a probability of 16/243 for a given |3H,3V〉, heralds the generation of an N=4 N00N state after HWP2. The structure of this N=4 N00N state generation scheme turns out to be topologically equivalent to the proposal by Lee et al. [21]. The method we have outlined is essentially a procedure for subtracting heralding photons from an initial product state so that the remaining photons are projected to a desired quantum state [42]. Although the generation efficiency decreases as N increases, our extendable scheme is currently an experimentally feasible way to realize higher N00N states and other kinds of nonclassical states as well.
Acknowledgements
This work was supported by the KRISS project ‘Single-Quantum-Based Metrology in Nanoscale.’
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