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
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 . 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 . Most notable is the use of squeezed states for this purpose . In the last decade there has also been a wide interest in the so-called NOON states . 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 , or consecutively by recycling them through a series of measurements [6, 7]. The Fabry-Pérot 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 , and onto a NOON state of 6 photons . 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 .
In this paper we report the experimental results of a recently proposed scheme that demonstrates super-resolution at the shot noise limit . 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. Theoretical background
2.1. Parity measurements
The parity operator is defined as π̂ = (−1)N̂, where N̂ 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, is16], 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 . It can be detected using homodyne techniques , 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 close to π is approaching 
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 Eq. (3) is the leading term in the region where ϕ is close to π, and the parity measurement is approaching the shot-noise 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. . 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 , 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 Ẑ = |0〉 〈0|, and its expectation value P0 isEq. (2), with a super-resolved peak which is only wider by . 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
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 Glan-Thompson 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 n̄.
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 , as expected from theory. Another difference between the two results is the weak oscillations that appear only for parity. They are a result 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 = (n̄ − nb)/(n̄ + 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 , where and are the probabilities for k clicks in the detector due to dark counts and due to signal photons, respectively. After changing the summation variables, the parity expectation value becomesEq. (1) was used twice and we assume the dark counts follow Poisson distribution . The imperfect visibility is included as . To calculate the effects of dark counts and background on P0 we notice that in this case . Therefore, the corrected result for P0 is Eqs. (6) and (7) result with the following peak heights h 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 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 . As the uncertainty in the estimated quantity is a ratio between the uncertainty in the measured quantity 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.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 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 approach 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 . Recently, it has been shown that single outcome measurements always saturate the Fisher information bound . 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 .
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.
The authors would like to thank J. P. Dowling and U. L. Andersen for their insightful remarks about this work.
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