Lost photon enhances superresolution A. Mikhalychev,1, ∗ P. Novik,1 I. Karuseichyk,1 D. A. Lyakhov,2 D. L. Michels,2 and D. Mogilevtsev1 1B.I.Stepanov Institute of Physics, NAS of Belarus, Nezavisimosti ave. 68, 220072 Minsk, Belarus 2Computer, Electrical and Mathematical Science and Engineering Division, 4700 King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia (Dated: January 18, 2021) Quantum imaging can beat classical resolution limits, imposed by diffraction of light. In partic- ular, it is known that one can reduce the image blurring and increase the achievable resolution by illuminating an object by entangled light and measuring coincidences of photons. If an n-photon entangled state is used and the pnth-order correlation function is measured, the point-spread func- tion (PSF) effectively becomes n times narrower relatively to classical coherent imaging. Quite surprisingly, measuring n-photon correlations is not the best choice if an n-photon entangled state is available. We show that for measuring (n − 1)-photon coincidences (thus, ignoring one of the available photons), PSF can be made even narrower. This observation paves a way for a strong conditional resolution enhancement by registering one of the photons outside the imaging area. We analyze the conditions necessary for the resolution increase and propose a practical scheme, suitable for observation and exploitation of the effect. Diffraction of light limits the spatial resolution of clas- only n − 1 remaining photons. According to our results, sical optical microscopes [1, 2] and hinders their applica- measurement of (n − 1)th-order correlations effectively bility to life sciences at very small scales. Quite recently, leads to p2(n − 1)=n times narrower PSF than for com- a number of superresolving techniques, suitable for over- monly considered n-photon detection. It is even more coming the classical limit, have been proposed. The ap- strange in view of the notorious entanglement fragility proaches include, for example, stimulated-emission deple- [20]: if even just one of the entangled photons is lost, the tion microscopy [3], superresolving imaging based on fluc- correlations tend to become classical. tuations [4] or antibunched light emission of fluorescence The insight for understanding that seeming paradox markers [5], structured illumination microscopy [6, 7], can be gained from a well-established ghost-imaging tech- and quantum imaging [8{10]. nique [14{16, 21{25] and from a more complicated ap- Quantum entanglement is known to be a powerful tool proach of quantum imaging with undetected photons [26{ for resolution and visibility enhancement in quantum 29]. In our case, detecting n − 1 photons and ignoring imaging and metrology [8{16]. It has been shown that the remaining nth one effectively comprises two possi- using n entangled photons and measuring the n-th order bilities (see the imaging scheme depicted in Fig. 1): the correlations, one can effectivelyp reduce the width of the nth photon can either fly relatively close to the optical point-spread function (PSF) n times [9, 12, 13, 17] and axis of the imaging system towards the detector or go far beat the classical diffraction limit. The increase of the from the optical axis and fail to pass through the aper- effect with the growth of n can naively be explained as ture of the imaging system. In the first case, the photon summing up the \pieces of information" carried by each can be successfully detected and provide us its piece of photon when measuring their correlations. Such logic information. In the second case, it does not bring us the suggests that, being given an n-photon entangled state, information itself, but effectively modifies the state of the the intuitively most winning measurement strategy is to remaining n − 1 photons (as in Refs. [26, 27, 29{31]). It maximally exploit quantumness of the illuminating field effectively produces position-dependent phase shift, thus and to measure the maximal available order of the photon performing wave-function shaping [31] and leading to an correlations (i.e. the nth one). effect similar to structured illumination [6, 7], PSF shap- Surprisingly, it is not always the case. First, it is worth ing [32], or linear interferometry measurement [33], and arXiv:2101.05849v1 [quant-ph] 14 Jan 2021 mentioning that effective narrowing of the PSF and reso- enhancing the resolution. We show that for n > 2 pho- lution enhancement can be achieved with classically cor- tons, the sensitivity-enhancement effect leads to higher related photons [9] or even in complete absence of correla- information gain than just detection of the nth photon, tions between fields emitted by different parts of the im- and measurement of (n−1)-photon correlations surpasses aged object (as it is for the stochastic optical microscopy n-photon detection. [4]). Moreover, the maximal order of correlations is not The discussed sensitivity-enhancement effect can be necessarily the best one [18, 19]. In this contribution, used for increase of resolution in practical imaging we show that, for an entangled n-photon illuminating schemes. One can devise a conditional measurement set- state, it is possible to surpass the measurement of all n- up by placing a bucket detector outside the normal path- photon correlations by loosing a photon and measuring way of the optical beam (e.g. near the lens outside of its aperture) and post-selecting the outcomes when one pho- ton gets to the bucket detector and the remaining n − 1 ∗ [email protected] ones successfully reach the position-sensitive detector, 2 used for the coincidence measurements. We show that E (s) h(s,r) E(r) such post-selection scheme indeed leads to additional in- 0 crease of resolution relatively to (n−1)-photon detection. Resolution enhancement by post-selecting the more infor- mative field configuration is closely related to the spatial s r mode demultiplexing technique [34, 35]. However, in our Source OP Optical system IP case the selection of more informative field part is per- formed by detection of a photon while all the remaining FIG. 1. General scheme of an imaging setup. See the text for photons are detected in the usual way rather than by fil- details. An image of an object placed at the object plane OP tering the beam itself. Also, our technique bears some is formed by the optical system at the image plane IP. resemblance to the multi-photon ghost-imaging [14]. For drawing quantitative conclusions about the resolu- tion enhancement of the proposed technique relatively accomplished by simple coincidence photo-counting. The to traditional measurements of n and (n − 1)-photon detection rate of the n-photon coincidence at a point ~r coincidences, we employ the Fisher information, which is determined by the value of the nth-order correlation has already proved itself as a powerful tool for analysis function (see Methods for details): of quantum imaging problems and for meaningful quan- Z 2 (n) 2 n n tification of resolution [18, 19, 32, 34{38]. Our simu- G (~r) / d ~sA (~s)h (~s;~r) : (2) lations show that for imaging a set of semi-transparent slits (i.e., for multi-parametric estimation problem), one indeed has a considerable increase in the information per The signal, described byp Eq. (2), includes the nth power measurement, and the corresponding resolution enhance- of the PSF, which is n times narrower than the PSF ment. While genuine demonstration of the discussed ef- itself. At least for the object of just twop transparent fects requires at least 3 entangled photons, which can be point-like pinholes, such narrowing yields n times bet- generated by a setup with complex nonlinear processes ter visual resolution of the object than for imaging with (e.g. cascaded spontaneous parametric down-conversion coherent light (see e.g. Refs. [8, 13]). (SPDC) [39], combination of SPDC with up-conversion Alternatively, one may try to ignore one of the photons [40], cascaded four-wave mixing [41], or the third-order and measure correlations of the remaining (n − 1) ones. SPDC [42{44]), a relatively simple biphoton case is still The rate of (n − 1)-photon coincidences is described by suitable for observing resolution enhancement for a spe- the (n − 1)th-order correlation function: cific choice of the region where the nth photon (here, the Z second one) is detected. G(n−1)(~r) / d2~s jA(~s)j2(n−1) jh(~s;~r)j2(n−1) : (3) Here, the 2(n−1)th power of the PSF is present. For n > I. RESULTS p 2, the resolutionp enhancement factor 2(n − 1) is larger than the factor n achievable for n-photon detection. A. Imaging with entangled photons The result obtained looks quite counter-intuitive: each photon carries some information about the illuminated We consider the following common model of a quan- object, while discarding one of the photons leads to ad- tum imaging setup (Fig. 1). An object is described by ditional information gain. This seeming contradiction is a transmission amplitude A(~s), where ~s is the vector of just a consequence of applying classical intuition to quan- transverse position in the object plane. It is illuminated tum dynamics of an entangled system. Due to quantum by linearly polarized light in an n-photon entangled quan- correlations, an entangled photon can affect the state tum state of the remaining ones and increase their \informativity" Z even when it is lost without being detected [26{29]. Here 2~ 2~ (2) ~ ~ jΨni / d k1 ··· d knδ (k1 + ··· + kn) we show that in our imaging scheme such an enhance- ment by loss is indeed taking place. Moreover, an addi- Z n tional resolution increase can be achieved through condi- × a+(~k ) ··· a+(~k )j0i / d2~s a+(~s) j0i; (1) 1 n tioning by detecting the photon outside the aperture of the imaging system (see Fig.