Super-resolved phase measurements at the shot noise limit by parity measurement L. Cohen, D. Istrati, L. Dovrat, and H. S. Eisenberg∗ Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel ∗ [email protected]
Abstract: Classically, the resolution of optical measurements is limited by the Rayleigh limit and their sensitivity by the shot noise limit. However, non-classical measurements can surpass these limits. Measuring the photon number parity using a photon-number resolving detector, super resolved phase measurements up to 144 better than the Rayleigh limit are presented, with coherent states of up to 4,200 photons on average. An additional measurement that can be implemented with standard single-photon detectors is proposed and demonstrated. With this scheme, super resolution at the shot noise limit is demonstrated with coherent states of up to 200 photons on average. © 2014 Optical Society of America OCIS codes: (120.3180) Interferometry; (120.5050) Phase measurement; (270.5290) Photon statistics.
References and links 1. G. Meyer and N. M. Amer, “Novel optical approach to atomic force microscopy,” Appl. Phys. Lett. 53, 1045– 1047 (1988). 2. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011). 3. J. Aasi,J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, C. Affeldt, O. D. Aguiar, P. Ajith, B. Allen, E. Amador Ceron, D. Amariutei, S. B. Anderson, W. G. Anderson, K. Arai, M. C. Araya, C. Arceneaux, S. Ast, S. M. Aston, D. Atkinson, P. Aufmuth et al., “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photon. 7, 613–619 (2013). 4. J. P. Dowling, “Quantum optical metrology-the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008). 5. C. Kothe, G. Bj¨ork, and M. Bourennane, “Arbitrarily high super-resolving phase measurements at telecommunication wavelengths,” Phys. Rev. A 81, 063836 (2010). 6. B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free Heisenberglimited phase estimation,” Nature 450, 393–396 (2007). 7. C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry-Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A 80, 043822 (2009). 8. G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, “Nonlinear interferometry via Fock-state projection,” Phys. Rev. Lett. 96, 203601 (2006). 9. K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601 (2007). 10. E. Distante, M. Jeˇzek, and U. L. Andersen, “Super resolution with coherent states at the shot noise limit,” Phys. Rev. Lett. 111, 033603 (2013). 11. Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27, A170–A174 (2010). 12. A. Kuzmich and L. Mandel, “Sub-shot-noise interferometric measurements with two-photon states,” Quant. Semiclass. Opt. 10, 493 (1998).
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Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11945
13. M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004). 14. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316, 726–729 (2007). 15. I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010). 16. J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649 (1996). 17. C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811 (2000). 18. C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64, 063814 (2001). 19. C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A 82, 013831 (2010). 20. A. Chiruvelli and H. Lee, “Parity measurements in quantum optical metrology,” J. Mod. Opt. 58, 945–953 (2011). 21. P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: Parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010). 22. W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12, 113025 (2010). 23. L. Pezz´e, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase detection at the quantum limit with multiphoton Mach-Zehnder interferometry,” Phys. Rev. Lett. 99, 223602 (2007). 24. L. Dovrat, M. Bakstein, D. Istrati, A. Shaham, and H. S. Eisenberg, “Measurements of the dependence of the photon-number distribution on the number of modes in parametric down-conversion,” Opt. Express 20, 2266– 2276 (2010). 25. L. Dovrat, M. Bakstein, D. Istrati, and H. S. Eisenberg, “Simulations of photon detection in silicon photomultiplier number-resolving detectors,” Phys. Scr. T147, 014010 (2012). 26. H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004). 27. E. Waks, K. Inoue, W. D. Oliver, E. Diamanti, and Y. Yamamoto, “High-efficiency photon-number detection for quantum information processing,” IEEE J. Sel. Top. Quantum Electron. 9, 1502–1511 (2003). 28. A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008). 29. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994). 30. X. M. Feng, G. R. Jin, and W. Yang, “Quantum interferometry with two outcomes in the presence of phase diffusion,” quant-ph arXiv:1404.1229v2. 31. G. Y. Xiang, H. F. Hofmann, and G. J. Pryde, “Optimal multi-photon phase sensing with a single interference fringe,” Sci. Rep. 3, 2684 (2013).
1.
Introduction
One of the most common ways to sense and measure the physical world is by using electromagnetic radiation, and in particular light. Usually, the sensitivity of such measurements is governed by the quality of the measuring device, but ultimately, this sensitivity is bounded by fundamental physical limits on the measurement uncertainty [1]. The resolution of a measurement, defined as the width of its smallest detail, follows in most cases the scale of the light’s wavelength (λ ). Nevertheless, the sensitivity scale is not the resolution. Instead, it is determined by the measurement signal-to-noise ratio and can exceed the resolution by many orders of magnitude. For classical measurement devices, sensitivity is limited by the shot noise limit (SNL) – the discrete division of light energy into photons and their Poissonian statistics. Interestingly, both resolution and sensitivity can be further enhanced. Super-resolved phase measurements, where the smallest detail is smaller than the Rayleigh limit (λ /2), have been demonstrated with many approaches, using either classical or quantum light sources. The SNL is surpassed by super-sensitive methods that incorporate nonclassical states of light and measurements. As states with non-Poissonian statistics can be used, sensitivity is only limited by the Heisenberg uncertainty principle [2]. Most notable is the use of squeezed states for
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Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11946
this purpose [3]. In the last decade there has also been a wide interest in the so-called NOON states [4]. These states are superpositions of N photons in one mode and none in the other, and the opposite arrangement of no photons in the first and N in the second. They exhibit both super-resolution and super-sensitivity. Regrettably, despite their advantages, NOON states are hard to generate, vulnerable to noise, and require photon number resolving detection. Unlike NOON states, optical coherent states are generated by common lasers. They are reduced by absorption to weaker coherent states, thus keeping their qualitative behavior. Enhanced resolution can be achieved by using the photons simultaneously in a few parallel measurements [5], or consecutively by recycling them through a series of measurements [6, 7]. The Fabry-P´erot interferometer is a standard example for the latter. Alternatively, the measuring apparatus can measure properties other than just the average power. Such enhancement has been demonstrated by post-selecting the projection of the output of an interferometer onto a Fock state of 7 photons [8], and onto a NOON state of 6 photons [9]. However, post-selection schemes for super-resolution reduce the sensitivity far from the SNL. Only recently, Distante et al. have reported a deterministic scheme that demonstrated super-resolution at the SNL, using homodyne detection [10]. In this paper we report the experimental results of a recently proposed scheme that demonstrates super-resolution at the shot noise limit [11]. It combines standard interferometry of coherent states with photon-number parity measurements using photon-number resolving detectors. Compared to previous measurements with NOON states [12–15], this scheme demonstrates much higher super-resolution, but can only approach the SNL. Additionally, we show how a similar scheme, that uses only single-photon detectors that cannot resolve the number of photons, achieves super-resolution which is only slightly worse than the original proposal, while keeping the same shot noise limited sensitivity. The rest of the paper is organized as follows: in Sec. 2 we present the theoretical predictions for the resolution and the sensitivity of parity and zero-photon measurements. Section 3 describes the experimental setup. Section 4 describes the experimental results and discusses the deviations from theory due to system imperfections. 2. 2.1.
Theoretical background Parity measurements
ˆ The parity operator is defined as πˆ = (−1)N , where Nˆ is the photon number operator. The parity result of every single measurement is either 1 or -1. Its expectation value, obtained by averaging over many measurement repetitions, is
πˆ =
∞
∑ (−1)n Pn ,
(1)
n=0
where Pn is the probability for n photons in the detected state. The parity expectation value reflects the oddness or evenness of the photon number distribution. Previously, parity measurements were used with trapped ions [16], and later in the context of optical interferometry [17,18]. Apparently, parity measurements are advantageous for many light sources [19,20] and can break the Heisenberg limit if the light source is a two mode squeezed vacuum state [21]. It can be detected using homodyne techniques [22], or by observing the photon-number distribution with a photon-number resolving detector, as will be demonstrated here. Consider a Mach-Zehnder interferometer whose input is a coherent state from one port and the vacuum from the other. The relative phase φ between its arms is the variable we would like to evaluate, while we observe the state of one of the two output ports. The detected output state is a phase-dependent coherent state |α (φ ). The parity expectation value of this state when φ is #206881 - $15.00 USD (C) 2014 OSA
Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11947
close to π is approaching [11] α |πˆ |α = e−2|α | = e−2n¯ cos 2
2( φ ) 2
→ e−
n( ¯ φ −π )2 2
,
(2)
where n¯ = |α (φ = 0)|2 is the maximal detected average photon number, which already includes the effects of photon loss in the interferometer and in the detector. This expectation value is approaching zero as the average number of output photons and its distribution width are increased, because wider distributions include odd and even terms more evenly. On the other hand, when the average photon number is approaching zero, the parity approaches one, as the dominant term as half width at 1/e of the maximum, becomes P0 , an even term. When the resolution is defined ¯ The resolution can be experimentally the phase resolution of the parity measurement is 2/n. verified by comparing parity expectation values at different phases around φ = π . The single-shot sensitivity of the parity measurements saturates the SNL [11, 21]. This sensitivity reflects the level of certainty of estimating φ to be π , when the measured value of the single-shot parity is 1. It is derived using quantum estimation theory to be [8] e4n¯ cos2 ( φ2 ) − 1 (φ − π ) 2 Δπˆ 1 , (3) → √ 1 + (2n¯ + 1) Δφ = dπˆ = 2 2 8 n¯ n¯ sin (φ ) | | dφ
where the parity uncertainty is Δπˆ = 1 − πˆ 2 . The last expression of Eq. (3) is the leading term in the region where φ is close to π , and the parity measurement is approaching the shotnoise limit. Although this limit is achieved only around φ = π , it is possible to achieve the same limit at φ = 0 by monitoring the second output port of the interferometer. Further sensitivity improvement at other phases is also possible, as demonstrated by Ref. [23]. Nevertheless, we suspect that achieving the shot noise limit for every phase value as in that work is not possible in our case, as a two outcome measurement such as parity contains mush less information. 2.2.
No-photon measurements
The major contribution that the P0 component has to parity at its peak suggests that this value is responsible for most of the improvement in resolution and sensitivity. We explore this option as P0 can also be obtained by a standard single-photon detector that cannot discriminate between different photon numbers. Its two outcomes, no photon and any number of photons, corresponds to the measurement of P0 and Pr = ∑∞ n=1 Pn , respectively. One may assume that the difference between these two values is a better approximation of parity, as the deviation is only at the third term P2 . Nevertheless, as P0 − Pr = 2P0 − 1, considering only P0 is equivalent, and much simpler to evaluate. The observable for no detection is the zero photon projector Zˆ = |00|, and its expectation value P0 is n( ¯ φ −π )2 φ ˆ α = e−|α |2 = e−n¯ cos2 ( 2 ) → e− 4 . (4) α |Z| √ This is a very similar result to Eq. (2), with a super-resolved peak which is only wider by 2. The no-photon probability peaks at φ = π , and gets less and less probable as photon numbers are increasing when the φ value deviates from π . In order to find the single-shot sensitivity of this measurement, we follow the same procedure as in Eq. (3) and obtain en¯ cos2 ( φ2 ) − 1 ˆ (φ − π ) 2 1 n¯ ΔZ √ + 1) , (5) → 1 + ( Δφ = = ˆ n¯ 2 2 8 n¯ sin2 (φ ) | dZ | dφ
4
where we used ΔZˆ =
P0 (1 − P0 ). Surprisingly, the P0 measurement is also shot-noise limited.
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Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11948
Fig. 1. The experimental setup. The coherent states are produced by a Ti:Sapphire laser and their average photon number is controlled by a calibrated variable neutral density filter (NDF). The Mach-Zehnder interferometer (MZI) is composed from two polarizers (P) at 45◦ and a phase shifter (PS). One of the output modes from the MZI is filtered spectrally by 3 nm band pass filter (BPF) and spatially by a single mode fiber (SMF). This mode is detected by the silicon photomultiplier (SiPM, Hamamatsu Photonics, S10362-11-100U) detector.
3.
The experimental setup
In order to realize the measurements discussed above, we use a polarization version of the Mach-Zehnder interferometer, where the two spatial modes are replaced by the horizontal and vertical polarizations (see Fig. 1). In order to achieve better visibility, the polarization modes are not coupled by λ /2 wave-plates. Instead, the input and output photons pass through GlanThompson polarizers oriented at 45◦ , which have better performance. The relative phase between the polarization modes is introduced by tilting a calibrated 2 mm thick birefringent calcite crystal. An identical crystal oriented at 90◦ is used to compensate for temporal walk-off. The output is filtered by a 3 nm bandpass filter and directed through a single mode fiber to a silicon photo-multiplier (SiPM) detector, which is an array of many single-photon detecting elements, that has photon-number sensitivity. The input state is generated by a Ti:Sapphire laser with 780 nm wavelength, that produces 150 fs long pulses. The pulses are picked to reduce their rate to 250 kHz. The average number of photons per pulse is controlled by calibrated neutral density filters. There are a few effects that distort the original photon statistics as registered by SiPM detectors [24,25]. The imperfect detection efficiency reduces the average detected photon number, but does not affect its Poissonian statistics. Dark counts are false detections as a result of thermal excitations. Cross-talk can trigger neighboring elements of photon triggered elements, and thus falsely increase the number of detected photons. Finally, there is a chance for more than one photon to hit the same detecting element as a result of their finite number (100 in our case). As around the significant region of our measurement |α | 1, only dark counts affect it, and their effect is important only for relatively small n. ¯ 4.
Results
Figure 2 presents the super-resolved signals obtained for parity (Fig. 2(a)) and P0 (Fig. 2(b)). The data sets are for increasing average photon numbers. A clear narrowing of the signals is observed, where the narrowest peak was measured for parity with about 4000 photons. This peak’s width is λ /(288 ± 3), corresponding to resolution which is 144 times better than what is regularly achieved (presented for comparison √ by a black dashed line). The corresponding widths of the curves for parity and P0 differ by 2, as expected from theory. Another difference between the two results is the weak oscillations that appear only for parity. They are a result
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Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11949
Fig. 2. (a) Parity and (b) P0 dependence on the phase. The presented measurements are for average photon numbers of 4.6 ± 0.2 (purple squares), 25 ± 1 (red circles), 200 ± 8 (blue triangles), 1, 190 ± 50 (pink inverted triangles),and 4, 150 ± 150 (green rhombuses). The dashed black lines represent the classic interference curves, and are presented for comparison reasons. Errors are not shown, as they are smaller than the symbols.
of the truncation of the parity measurement at 26 photons, which becomes more significant as more photons are involved. The degradation of the peak heights as the average photon number is increased, is a result of the imperfect visibility V of the interferometer, and the dark counts, whose average number is nd . The visibility is defined as V = (n¯ − nb )/(n¯ + nb ), where nb is the average number of background counts at φ = π . As dark counts are the major cause for deviations, in our case coh , where Pdc and Pcoh are the probabilities for k clicks in the detector due to Pn = ∑nk=0 Pkdc Pn−k k k dark counts and due to signal photons, respectively. After changing the summation variables, the parity expectation value becomes πˆ =
∞
∞
k=0
n−k=0
∑ (−1)k Pkdc ∑
coh (−1)n−k Pn−k = e−2nd e−2|α | , 2
(6)
where in the last step, Eq. (1) was used twice and we assume the dark counts follow Poisson ¯ b ) cos2 φ2 +nb . To calculate distribution [26]. The imperfect visibility is included as |α |2 = (n−n the effects of dark counts and background on P0 we notice that in this case P0 = P0dc P0coh . Therefore, the corrected result for P0 is P0 = e−nd e−|α |
2
Substitute φ = π , Eqs. (6) and (7) result with the following peak heights h (1 −V ) , h(n) ¯ = exp −β nd + n¯ 1 +V
(7)
(8)
where β = 2 for parity and 1 for P0 . Figure 3 presents the peak height of all measurements with their fits to Eq. (8). The data fits theory very well, where the parity curve slope is about twice as large as that of P0 . From the fit parameters of both curves we extract the dark count rate to #206881 - $15.00 USD (C) 2014 OSA
Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11950
Fig. 3. The effects of finite visibility and dark counts on the signal peak heights for parity (blue squares) and P0 (orange triangles). The peak heights decrease when states with more photons are used, according to Eq. (8).
Fig. 4. Experimental phase uncertainties for 200 photons on average (blue squares), as a function of the phase for (a) parity and (b) P0 . The theoretical fits to Eqs. (3) and (5) are presented with solid black lines, and to Eq. (9) by orange dashed lines. Errors are shown when larger than their symbols. They represent the standard deviation in the phase uncertainty estimation process.
be 400 ± 100 Hz and the visibility to be 0.99981 ± 10−5 , in agreement with independent direct measurements. The experimental phase uncertainty for parity and P0 were calculated from their data using Eqs. (3) and (5), respectively. Representative results for 200 photons on average are presented in Fig. 4. The uncertainty for P0 is at minimum for a larger range as its peak is also wider. The imperfect visibility is also responsible for the large deviation near φ = π [12]. As the uncertainty in the estimated quantity is a ratio between the uncertainty in the measured quantity
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Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11951
Fig. 5. (a) The resolution and (b) the smallest uncertainty. Parity results are presented by blue squares and P0 results with orange triangles. Solid lines are the theoretical predictions. For the resolution there are no free parameters and the predictions for sensitivity use the λ , 144 times better visibility values from the fits of Fig. 3. The parity resolution reaches 288 than the Rayleigh λ /2 limit (dashed line in (a)). The P0 sensitivity follows the SNL (dashed line in (b)) up to 200 photons. Parity has better resolution but larger deviation from the SNL. Errors were calculated as before and shown when larger than their symbol.
and its slope, it can only be finite at this point as long as both contributions approach zero together. Imperfect visibility results in non-zero observed value and non-zero uncertainty in it. Its slope on the other hand is always zero at this point, where this value peaks, resulting in diverging uncertainty in the estimated value. To modify Eqs. (3) and (5) such that they include the effect of imperfect visibility, Eqs. (6) and (7) are used instead of Eqs. (2) and (4). In this case, the phase uncertainty is 2 V 2 φ 1 +V ¯ 2β n¯ 1+V cos ( 2 ) − 1 h−β (n)e , (9) Δφ = β nV ¯ sin2 (φ ) where h and β are as in Eq. (8). Fits to Eq. (9) are also presented in Fig. 4, with a good agreement with the results. A summary of all the results for the range of 2.5 to 4,200 √ photons on average is presented in Fig. 5. The resolutions of parity and P0 are separated by 2 as expected. They are not affected by the imperfect visibility. The phase uncertainty for P0 is slightly better for small photon numbers, where it follows meticulously the theoretical SNL line. Actually, within experimental errors, the sensitivity of the P0 measurement is shot-noise limited up to the measurement with 200 photons, where the sensitivity is λ /(86 ± 2) and the resolution is λ /(45 ± 1). Although when more photons are used this limit is not followed anymore, phase sensitivity is further improved, until when using 4,200 photons it is λ /(230 ± 15) and the resolution is λ /(202 ± 2). The better sensitivity of P0 is explained by the dependence on a single measured value, compared to parity that depends on many. We should emphasize that the values we present here assume photon detection with perfect quantum efficiency. The detector we have used is far from having a perfect efficiency. Nevertheless, this is a frequently used practice, that demonstrates the limits of the method. Moreover, there are other kinds of photon-number resolving detectors with quantum efficiencies that ap#206881 - $15.00 USD (C) 2014 OSA
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proach unity [27, 28]. Regrettably, they are not available to us. The Fisher information measure is related to the best sensitivity that a measurement using a specific input state can achieve [29]. Recently, it has been shown that single outcome measurements always saturate the Fisher information bound [30]. Therefore, in such measurements, the sensitivity and the Fisher information are directly related. Both Parity and P0 measurements are of this kind. In another work, it has been shown that for a given single outcome measurement the Fisher information can be maximized by choosing the optimal input state [31]. 5.
Conclusions
In conclusion, using a photon-number resolving detector, we have measured the parity and the probability for no-detection of a coherent state that has traveled through a Mach-Zehnder interferometer. Every detected signal is deterministically used for phase evaluation, without post-selection. Super-resolution of these two signals was demonstrated, up to 144 times better than the Rayleigh limit. In addition, these single-shot measurements follow the SNL up to pulses of 200 photons on average, and eventually reach a sensitivity 230 times better than the wavelength. The parity resolution is better than that of P0 , but the P0 sensitivity is slightly better and prevails for larger photon numbers. The main limiting factor for these measurements is the visibility of the interferometer. Acknowledgments The authors would like to thank J. P. Dowling and U. L. Andersen for their insightful remarks about this work.
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Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11953
Abstract: Classically, the resolution of optical measurements is limited by the Rayleigh limit and their sensitivity by the shot noise limit. However, non-classical measurements can surpass these limits. Measuring the photon number parity using a photon-number resolving detector, super resolved phase measurements up to 144 better than the Rayleigh limit are presented, with coherent states of up to 4,200 photons on average. An additional measurement that can be implemented with standard single-photon detectors is proposed and demonstrated. With this scheme, super resolution at the shot noise limit is demonstrated with coherent states of up to 200 photons on average. © 2014 Optical Society of America OCIS codes: (120.3180) Interferometry; (120.5050) Phase measurement; (270.5290) Photon statistics.
References and links 1. G. Meyer and N. M. Amer, “Novel optical approach to atomic force microscopy,” Appl. Phys. Lett. 53, 1045– 1047 (1988). 2. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011). 3. J. Aasi,J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, C. Affeldt, O. D. Aguiar, P. Ajith, B. Allen, E. Amador Ceron, D. Amariutei, S. B. Anderson, W. G. Anderson, K. Arai, M. C. Araya, C. Arceneaux, S. Ast, S. M. Aston, D. Atkinson, P. Aufmuth et al., “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photon. 7, 613–619 (2013). 4. J. P. Dowling, “Quantum optical metrology-the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008). 5. C. Kothe, G. Bj¨ork, and M. Bourennane, “Arbitrarily high super-resolving phase measurements at telecommunication wavelengths,” Phys. Rev. A 81, 063836 (2010). 6. B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free Heisenberglimited phase estimation,” Nature 450, 393–396 (2007). 7. C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry-Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A 80, 043822 (2009). 8. G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, “Nonlinear interferometry via Fock-state projection,” Phys. Rev. Lett. 96, 203601 (2006). 9. K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601 (2007). 10. E. Distante, M. Jeˇzek, and U. L. Andersen, “Super resolution with coherent states at the shot noise limit,” Phys. Rev. Lett. 111, 033603 (2013). 11. Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27, A170–A174 (2010). 12. A. Kuzmich and L. Mandel, “Sub-shot-noise interferometric measurements with two-photon states,” Quant. Semiclass. Opt. 10, 493 (1998).
#206881 - $15.00 USD (C) 2014 OSA
Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11945
13. M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004). 14. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316, 726–729 (2007). 15. I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010). 16. J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649 (1996). 17. C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811 (2000). 18. C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64, 063814 (2001). 19. C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A 82, 013831 (2010). 20. A. Chiruvelli and H. Lee, “Parity measurements in quantum optical metrology,” J. Mod. Opt. 58, 945–953 (2011). 21. P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: Parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010). 22. W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12, 113025 (2010). 23. L. Pezz´e, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase detection at the quantum limit with multiphoton Mach-Zehnder interferometry,” Phys. Rev. Lett. 99, 223602 (2007). 24. L. Dovrat, M. Bakstein, D. Istrati, A. Shaham, and H. S. Eisenberg, “Measurements of the dependence of the photon-number distribution on the number of modes in parametric down-conversion,” Opt. Express 20, 2266– 2276 (2010). 25. L. Dovrat, M. Bakstein, D. Istrati, and H. S. Eisenberg, “Simulations of photon detection in silicon photomultiplier number-resolving detectors,” Phys. Scr. T147, 014010 (2012). 26. H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004). 27. E. Waks, K. Inoue, W. D. Oliver, E. Diamanti, and Y. Yamamoto, “High-efficiency photon-number detection for quantum information processing,” IEEE J. Sel. Top. Quantum Electron. 9, 1502–1511 (2003). 28. A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008). 29. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994). 30. X. M. Feng, G. R. Jin, and W. Yang, “Quantum interferometry with two outcomes in the presence of phase diffusion,” quant-ph arXiv:1404.1229v2. 31. G. Y. Xiang, H. F. Hofmann, and G. J. Pryde, “Optimal multi-photon phase sensing with a single interference fringe,” Sci. Rep. 3, 2684 (2013).
1.
Introduction
One of the most common ways to sense and measure the physical world is by using electromagnetic radiation, and in particular light. Usually, the sensitivity of such measurements is governed by the quality of the measuring device, but ultimately, this sensitivity is bounded by fundamental physical limits on the measurement uncertainty [1]. The resolution of a measurement, defined as the width of its smallest detail, follows in most cases the scale of the light’s wavelength (λ ). Nevertheless, the sensitivity scale is not the resolution. Instead, it is determined by the measurement signal-to-noise ratio and can exceed the resolution by many orders of magnitude. For classical measurement devices, sensitivity is limited by the shot noise limit (SNL) – the discrete division of light energy into photons and their Poissonian statistics. Interestingly, both resolution and sensitivity can be further enhanced. Super-resolved phase measurements, where the smallest detail is smaller than the Rayleigh limit (λ /2), have been demonstrated with many approaches, using either classical or quantum light sources. The SNL is surpassed by super-sensitive methods that incorporate nonclassical states of light and measurements. As states with non-Poissonian statistics can be used, sensitivity is only limited by the Heisenberg uncertainty principle [2]. Most notable is the use of squeezed states for
#206881 - $15.00 USD (C) 2014 OSA
Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11946
this purpose [3]. In the last decade there has also been a wide interest in the so-called NOON states [4]. These states are superpositions of N photons in one mode and none in the other, and the opposite arrangement of no photons in the first and N in the second. They exhibit both super-resolution and super-sensitivity. Regrettably, despite their advantages, NOON states are hard to generate, vulnerable to noise, and require photon number resolving detection. Unlike NOON states, optical coherent states are generated by common lasers. They are reduced by absorption to weaker coherent states, thus keeping their qualitative behavior. Enhanced resolution can be achieved by using the photons simultaneously in a few parallel measurements [5], or consecutively by recycling them through a series of measurements [6, 7]. The Fabry-P´erot interferometer is a standard example for the latter. Alternatively, the measuring apparatus can measure properties other than just the average power. Such enhancement has been demonstrated by post-selecting the projection of the output of an interferometer onto a Fock state of 7 photons [8], and onto a NOON state of 6 photons [9]. However, post-selection schemes for super-resolution reduce the sensitivity far from the SNL. Only recently, Distante et al. have reported a deterministic scheme that demonstrated super-resolution at the SNL, using homodyne detection [10]. In this paper we report the experimental results of a recently proposed scheme that demonstrates super-resolution at the shot noise limit [11]. It combines standard interferometry of coherent states with photon-number parity measurements using photon-number resolving detectors. Compared to previous measurements with NOON states [12–15], this scheme demonstrates much higher super-resolution, but can only approach the SNL. Additionally, we show how a similar scheme, that uses only single-photon detectors that cannot resolve the number of photons, achieves super-resolution which is only slightly worse than the original proposal, while keeping the same shot noise limited sensitivity. The rest of the paper is organized as follows: in Sec. 2 we present the theoretical predictions for the resolution and the sensitivity of parity and zero-photon measurements. Section 3 describes the experimental setup. Section 4 describes the experimental results and discusses the deviations from theory due to system imperfections. 2. 2.1.
Theoretical background Parity measurements
ˆ The parity operator is defined as πˆ = (−1)N , where Nˆ is the photon number operator. The parity result of every single measurement is either 1 or -1. Its expectation value, obtained by averaging over many measurement repetitions, is
πˆ =
∞
∑ (−1)n Pn ,
(1)
n=0
where Pn is the probability for n photons in the detected state. The parity expectation value reflects the oddness or evenness of the photon number distribution. Previously, parity measurements were used with trapped ions [16], and later in the context of optical interferometry [17,18]. Apparently, parity measurements are advantageous for many light sources [19,20] and can break the Heisenberg limit if the light source is a two mode squeezed vacuum state [21]. It can be detected using homodyne techniques [22], or by observing the photon-number distribution with a photon-number resolving detector, as will be demonstrated here. Consider a Mach-Zehnder interferometer whose input is a coherent state from one port and the vacuum from the other. The relative phase φ between its arms is the variable we would like to evaluate, while we observe the state of one of the two output ports. The detected output state is a phase-dependent coherent state |α (φ ). The parity expectation value of this state when φ is #206881 - $15.00 USD (C) 2014 OSA
Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11947
close to π is approaching [11] α |πˆ |α = e−2|α | = e−2n¯ cos 2
2( φ ) 2
→ e−
n( ¯ φ −π )2 2
,
(2)
where n¯ = |α (φ = 0)|2 is the maximal detected average photon number, which already includes the effects of photon loss in the interferometer and in the detector. This expectation value is approaching zero as the average number of output photons and its distribution width are increased, because wider distributions include odd and even terms more evenly. On the other hand, when the average photon number is approaching zero, the parity approaches one, as the dominant term as half width at 1/e of the maximum, becomes P0 , an even term. When the resolution is defined ¯ The resolution can be experimentally the phase resolution of the parity measurement is 2/n. verified by comparing parity expectation values at different phases around φ = π . The single-shot sensitivity of the parity measurements saturates the SNL [11, 21]. This sensitivity reflects the level of certainty of estimating φ to be π , when the measured value of the single-shot parity is 1. It is derived using quantum estimation theory to be [8] e4n¯ cos2 ( φ2 ) − 1 (φ − π ) 2 Δπˆ 1 , (3) → √ 1 + (2n¯ + 1) Δφ = dπˆ = 2 2 8 n¯ n¯ sin (φ ) | | dφ
where the parity uncertainty is Δπˆ = 1 − πˆ 2 . The last expression of Eq. (3) is the leading term in the region where φ is close to π , and the parity measurement is approaching the shotnoise limit. Although this limit is achieved only around φ = π , it is possible to achieve the same limit at φ = 0 by monitoring the second output port of the interferometer. Further sensitivity improvement at other phases is also possible, as demonstrated by Ref. [23]. Nevertheless, we suspect that achieving the shot noise limit for every phase value as in that work is not possible in our case, as a two outcome measurement such as parity contains mush less information. 2.2.
No-photon measurements
The major contribution that the P0 component has to parity at its peak suggests that this value is responsible for most of the improvement in resolution and sensitivity. We explore this option as P0 can also be obtained by a standard single-photon detector that cannot discriminate between different photon numbers. Its two outcomes, no photon and any number of photons, corresponds to the measurement of P0 and Pr = ∑∞ n=1 Pn , respectively. One may assume that the difference between these two values is a better approximation of parity, as the deviation is only at the third term P2 . Nevertheless, as P0 − Pr = 2P0 − 1, considering only P0 is equivalent, and much simpler to evaluate. The observable for no detection is the zero photon projector Zˆ = |00|, and its expectation value P0 is n( ¯ φ −π )2 φ ˆ α = e−|α |2 = e−n¯ cos2 ( 2 ) → e− 4 . (4) α |Z| √ This is a very similar result to Eq. (2), with a super-resolved peak which is only wider by 2. The no-photon probability peaks at φ = π , and gets less and less probable as photon numbers are increasing when the φ value deviates from π . In order to find the single-shot sensitivity of this measurement, we follow the same procedure as in Eq. (3) and obtain en¯ cos2 ( φ2 ) − 1 ˆ (φ − π ) 2 1 n¯ ΔZ √ + 1) , (5) → 1 + ( Δφ = = ˆ n¯ 2 2 8 n¯ sin2 (φ ) | dZ | dφ
4
where we used ΔZˆ =
P0 (1 − P0 ). Surprisingly, the P0 measurement is also shot-noise limited.
#206881 - $15.00 USD (C) 2014 OSA
Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11948
Fig. 1. The experimental setup. The coherent states are produced by a Ti:Sapphire laser and their average photon number is controlled by a calibrated variable neutral density filter (NDF). The Mach-Zehnder interferometer (MZI) is composed from two polarizers (P) at 45◦ and a phase shifter (PS). One of the output modes from the MZI is filtered spectrally by 3 nm band pass filter (BPF) and spatially by a single mode fiber (SMF). This mode is detected by the silicon photomultiplier (SiPM, Hamamatsu Photonics, S10362-11-100U) detector.
3.
The experimental setup
In order to realize the measurements discussed above, we use a polarization version of the Mach-Zehnder interferometer, where the two spatial modes are replaced by the horizontal and vertical polarizations (see Fig. 1). In order to achieve better visibility, the polarization modes are not coupled by λ /2 wave-plates. Instead, the input and output photons pass through GlanThompson polarizers oriented at 45◦ , which have better performance. The relative phase between the polarization modes is introduced by tilting a calibrated 2 mm thick birefringent calcite crystal. An identical crystal oriented at 90◦ is used to compensate for temporal walk-off. The output is filtered by a 3 nm bandpass filter and directed through a single mode fiber to a silicon photo-multiplier (SiPM) detector, which is an array of many single-photon detecting elements, that has photon-number sensitivity. The input state is generated by a Ti:Sapphire laser with 780 nm wavelength, that produces 150 fs long pulses. The pulses are picked to reduce their rate to 250 kHz. The average number of photons per pulse is controlled by calibrated neutral density filters. There are a few effects that distort the original photon statistics as registered by SiPM detectors [24,25]. The imperfect detection efficiency reduces the average detected photon number, but does not affect its Poissonian statistics. Dark counts are false detections as a result of thermal excitations. Cross-talk can trigger neighboring elements of photon triggered elements, and thus falsely increase the number of detected photons. Finally, there is a chance for more than one photon to hit the same detecting element as a result of their finite number (100 in our case). As around the significant region of our measurement |α | 1, only dark counts affect it, and their effect is important only for relatively small n. ¯ 4.
Results
Figure 2 presents the super-resolved signals obtained for parity (Fig. 2(a)) and P0 (Fig. 2(b)). The data sets are for increasing average photon numbers. A clear narrowing of the signals is observed, where the narrowest peak was measured for parity with about 4000 photons. This peak’s width is λ /(288 ± 3), corresponding to resolution which is 144 times better than what is regularly achieved (presented for comparison √ by a black dashed line). The corresponding widths of the curves for parity and P0 differ by 2, as expected from theory. Another difference between the two results is the weak oscillations that appear only for parity. They are a result
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Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11949
Fig. 2. (a) Parity and (b) P0 dependence on the phase. The presented measurements are for average photon numbers of 4.6 ± 0.2 (purple squares), 25 ± 1 (red circles), 200 ± 8 (blue triangles), 1, 190 ± 50 (pink inverted triangles),and 4, 150 ± 150 (green rhombuses). The dashed black lines represent the classic interference curves, and are presented for comparison reasons. Errors are not shown, as they are smaller than the symbols.
of the truncation of the parity measurement at 26 photons, which becomes more significant as more photons are involved. The degradation of the peak heights as the average photon number is increased, is a result of the imperfect visibility V of the interferometer, and the dark counts, whose average number is nd . The visibility is defined as V = (n¯ − nb )/(n¯ + nb ), where nb is the average number of background counts at φ = π . As dark counts are the major cause for deviations, in our case coh , where Pdc and Pcoh are the probabilities for k clicks in the detector due to Pn = ∑nk=0 Pkdc Pn−k k k dark counts and due to signal photons, respectively. After changing the summation variables, the parity expectation value becomes πˆ =
∞
∞
k=0
n−k=0
∑ (−1)k Pkdc ∑
coh (−1)n−k Pn−k = e−2nd e−2|α | , 2
(6)
where in the last step, Eq. (1) was used twice and we assume the dark counts follow Poisson ¯ b ) cos2 φ2 +nb . To calculate distribution [26]. The imperfect visibility is included as |α |2 = (n−n the effects of dark counts and background on P0 we notice that in this case P0 = P0dc P0coh . Therefore, the corrected result for P0 is P0 = e−nd e−|α |
2
Substitute φ = π , Eqs. (6) and (7) result with the following peak heights h (1 −V ) , h(n) ¯ = exp −β nd + n¯ 1 +V
(7)
(8)
where β = 2 for parity and 1 for P0 . Figure 3 presents the peak height of all measurements with their fits to Eq. (8). The data fits theory very well, where the parity curve slope is about twice as large as that of P0 . From the fit parameters of both curves we extract the dark count rate to #206881 - $15.00 USD (C) 2014 OSA
Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11950
Fig. 3. The effects of finite visibility and dark counts on the signal peak heights for parity (blue squares) and P0 (orange triangles). The peak heights decrease when states with more photons are used, according to Eq. (8).
Fig. 4. Experimental phase uncertainties for 200 photons on average (blue squares), as a function of the phase for (a) parity and (b) P0 . The theoretical fits to Eqs. (3) and (5) are presented with solid black lines, and to Eq. (9) by orange dashed lines. Errors are shown when larger than their symbols. They represent the standard deviation in the phase uncertainty estimation process.
be 400 ± 100 Hz and the visibility to be 0.99981 ± 10−5 , in agreement with independent direct measurements. The experimental phase uncertainty for parity and P0 were calculated from their data using Eqs. (3) and (5), respectively. Representative results for 200 photons on average are presented in Fig. 4. The uncertainty for P0 is at minimum for a larger range as its peak is also wider. The imperfect visibility is also responsible for the large deviation near φ = π [12]. As the uncertainty in the estimated quantity is a ratio between the uncertainty in the measured quantity
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Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11951
Fig. 5. (a) The resolution and (b) the smallest uncertainty. Parity results are presented by blue squares and P0 results with orange triangles. Solid lines are the theoretical predictions. For the resolution there are no free parameters and the predictions for sensitivity use the λ , 144 times better visibility values from the fits of Fig. 3. The parity resolution reaches 288 than the Rayleigh λ /2 limit (dashed line in (a)). The P0 sensitivity follows the SNL (dashed line in (b)) up to 200 photons. Parity has better resolution but larger deviation from the SNL. Errors were calculated as before and shown when larger than their symbol.
and its slope, it can only be finite at this point as long as both contributions approach zero together. Imperfect visibility results in non-zero observed value and non-zero uncertainty in it. Its slope on the other hand is always zero at this point, where this value peaks, resulting in diverging uncertainty in the estimated value. To modify Eqs. (3) and (5) such that they include the effect of imperfect visibility, Eqs. (6) and (7) are used instead of Eqs. (2) and (4). In this case, the phase uncertainty is 2 V 2 φ 1 +V ¯ 2β n¯ 1+V cos ( 2 ) − 1 h−β (n)e , (9) Δφ = β nV ¯ sin2 (φ ) where h and β are as in Eq. (8). Fits to Eq. (9) are also presented in Fig. 4, with a good agreement with the results. A summary of all the results for the range of 2.5 to 4,200 √ photons on average is presented in Fig. 5. The resolutions of parity and P0 are separated by 2 as expected. They are not affected by the imperfect visibility. The phase uncertainty for P0 is slightly better for small photon numbers, where it follows meticulously the theoretical SNL line. Actually, within experimental errors, the sensitivity of the P0 measurement is shot-noise limited up to the measurement with 200 photons, where the sensitivity is λ /(86 ± 2) and the resolution is λ /(45 ± 1). Although when more photons are used this limit is not followed anymore, phase sensitivity is further improved, until when using 4,200 photons it is λ /(230 ± 15) and the resolution is λ /(202 ± 2). The better sensitivity of P0 is explained by the dependence on a single measured value, compared to parity that depends on many. We should emphasize that the values we present here assume photon detection with perfect quantum efficiency. The detector we have used is far from having a perfect efficiency. Nevertheless, this is a frequently used practice, that demonstrates the limits of the method. Moreover, there are other kinds of photon-number resolving detectors with quantum efficiencies that ap#206881 - $15.00 USD (C) 2014 OSA
Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11952
proach unity [27, 28]. Regrettably, they are not available to us. The Fisher information measure is related to the best sensitivity that a measurement using a specific input state can achieve [29]. Recently, it has been shown that single outcome measurements always saturate the Fisher information bound [30]. Therefore, in such measurements, the sensitivity and the Fisher information are directly related. Both Parity and P0 measurements are of this kind. In another work, it has been shown that for a given single outcome measurement the Fisher information can be maximized by choosing the optimal input state [31]. 5.
Conclusions
In conclusion, using a photon-number resolving detector, we have measured the parity and the probability for no-detection of a coherent state that has traveled through a Mach-Zehnder interferometer. Every detected signal is deterministically used for phase evaluation, without post-selection. Super-resolution of these two signals was demonstrated, up to 144 times better than the Rayleigh limit. In addition, these single-shot measurements follow the SNL up to pulses of 200 photons on average, and eventually reach a sensitivity 230 times better than the wavelength. The parity resolution is better than that of P0 , but the P0 sensitivity is slightly better and prevails for larger photon numbers. The main limiting factor for these measurements is the visibility of the interferometer. Acknowledgments The authors would like to thank J. P. Dowling and U. L. Andersen for their insightful remarks about this work.
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Received 24 Feb 2014; revised 24 Apr 2014; accepted 25 Apr 2014; published 9 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011945 | OPTICS EXPRESS 11953
