Scalable Spatial Super-Resolution using Entangled Photons
Lee A. Rozema, James D. Bateman, Dylan H. Mahler, Ryo Okamoto, Amir Feizpour, Alex Hayat, Aephraim M. Steinberg
SScalable Spatial Super-Resolution using Entangled Photons
Lee A. Rozema ∗† , James D. Bateman † , Dylan H. Mahler , RyoOkamoto , , , Amir Feizpour , Alex Hayat , , , and Aephraim M. Steinberg , Centre for Quantum Information & Quantum Control and Institute for Optical Sciences,Dept. of Physics, 60 St. George St., University of Toronto, Toronto, Ontario, Canada M5S 1A7 Research Institute for Electronic Science, Hokkaido University, Kita-ku, Sapporo, Japan The Institute of Scientific and Industrial Research,Osaka University, Mihogaoka 8-1, Ibaraki, Osaka, Japan Department of Electrical Engineering, Technion, Haifa 32000, Israel Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada ∗ Correspondence to: [email protected]. † These authors contributed equally to this work. (Dated: October 29, 2018)N00N states – maximally path-entangled states of N photons – exhibit spatial interference pat-terns sharper than any classical interference pattern. This is known as super-resolution. However,even with perfectly efficient number-resolving detectors, the detection efficiency of all previouslydemonstrated methods to measure such interference decreases exponentially with the number ofphotons in the N00N state, often leading to the conclusion that N00N states are unsuitable forspatial measurements. Here, we create spatial super-resolution fringes with two-, three-, and four-photon N00N states, and demonstrate a scalable implementation of the so-called “optical centroidmeasurement” which provides an in-principle perfect detection efficiency. Moreover, we compare theN00N-state interference to the corresponding classical super-resolution interference. Although bothprovide the same increase in spatial frequency, the visibility of the classical fringes decreases expo-nentially with the number of detected photons, while the visibility of our experimentally measuredN00N-state super-resolution fringes remains approximately constant with N. Our implementationof the optical centroid measurement is a scalable method to measure high photon-number quantuminterference, an essential step forward for quantum-enhanced measurements, overcoming what wasbelieved to be a fundamental challenge to quantum metrology.
Many essential techniques in modern science and tech-nology, from precise position sensing to high-resolutionimaging to nanolithography, rely on the creation and de-tection of the finest possible spatial interference fringesusing light. Classically, all such measurements face a fun-damental barrier related to the “diffraction limit,” whichis determined by the wavelength of the light [1], but quan-tum entanglement can be used to surpass this limit bymaking the spatial interference fringes sharper (a resultreferred to as super-resolution) [2, 3]. In particular, theN-photon entangled “N00N” state can display an inter-ference pattern N times finer than that of classical light[4, 5]. However, N00N states suffer from a weakness thathas made their advantage controversial: the probabilityof all N photons arriving at the same place, and thusthe detection efficiency, decreases exponentially with N[6, 7]. Here we implement the optical centroid measure-ment (OCM) proposed by Tsang [8] to completely over-come this problem. A proof-of-principle experiment con-firming the underlying concept of the OCM was recentlyperformed [9], but, being limited to only two photonsand two ‘movable’ detectors, it could not probe the scal-ing properties nor demonstrate the efficiency gain of theOCM. In our experiment, using an array of 11 fixed de-tectors, we measure two-, three-, and four-photon spatialfringes, and find that their visibility does not degradewith the number of entangled photons, clearly displayingthe enhanced efficiency and scalability of the OCM. The visibility of an unentangled OCM, on the other hand, de-cays exponentially. In doing this, we have also achievedthe highest spatial super-resolution to date [10–12].In 2000, Boto et al . pointed out that entangled statesof light offer a way to improve the resolution of interfer-ometers beyond the diffraction limit, which determinesthe smallest spatial features achievable in classical op-tical systems [2]. This limit is set by the fringe spac-ing of the interference pattern created by two beams ofwavelength λ meeting at an angle θ , which is λ/ (2 sin θ ).Under no conditions can a spacing smaller than λ/ et al. circumvented this clas-sical limit by introducing the entangled N00N state, anequal superposition of all N photons in mode (cid:126)k and allN photons in mode (cid:126)k : | ψ N (cid:105) = 1 √ | N, (cid:105) (cid:126)k , (cid:126)k + | , N (cid:105) (cid:126)k , (cid:126)k ) . (1)If (cid:126)k and (cid:126)k are two spatial modes which interfere at adetection plane (figure 1a), the probability of detecting all of the photons at a given position will display spatialfringes with a period of λ/ (2 N sin θ ), where θ is the an-gle between (cid:126)k and (cid:126)k (assuming | (cid:126)k | = | (cid:126)k | = πλ ). Theperiod of the N00N fringes is N times smaller than thatof classical fringes, suggesting that N00N states could beused to increase the resolution of optical systems by afactor of N. This observation has led to much subsequent a r X i v : . [ qu a n t - ph ] D ec FIG. 1.
The Centroid measurement: a) Schematic ofthe detection scheme –
Light from two modes, (cid:126)k and (cid:126)k ,is incident on an array of single photon detectors. Correla-tions between all the detectors are measured and recorded. b) Two-photon joint-probability functions – i) and iii) are thetheoretical two-photon joint-probability functions when clas-sical light and N=2 N00N states illuminate the detector array,respectively. A two-photon absorption measurement requires x = x , and as such it only samples the grey diagonal line,drawn in i) and iii), to produce a spatial interference pattern.This results in discarding most of the photon pairs. The cen-troid samples the entire two-photon correlation function by projecting all of the data onto the grey diagonal line, effec-tively detecting all of photon pairs. ii) and iv) are the experi-mental two-photon joint-probability functions. Since there isno way to distinguish photon 1 from photon 2, we could onlymeasure the half of the plot where x > x ; in plots ii) and iv)we have mirrored this data about the diagonal ( x = x ) forcomparison to theory. The diagonal region is dark since twophotons arriving simultaneously at the same SPCM cannotboth be detected. work on N00N states [4, 5, 13–16] and their application intasks such as quantum lithography and quantum imaging[10–12, 17]. However, the individual photons in the N00Nstate cannot be localized to better than λ/
2, regardlessof the narrow spatial scale of the N-photon correlationfringes [6, 7]. This means that the probability of a givenphoton landing within some small region of size r willalways be (cid:46) r/λ , and that the probability for all N pho-tons to arrive at the same region is (cid:46) (2 r/λ ) N . Thus theefficiency of such a detection scheme decreases exponen-tially with N , leading to the conclusion that N00N statesare of little practical use for applications requiring spa-tial interference. The OCM was proposed to address thisproblem, resurrecting the hope of applying such states tohigh-resolution position measurements [8].The OCM displays N-fold super-resolution without re-quiring all the photons to arrive at the same point inspace. Instead, it keeps track of every single N-photonevent, regardless of which combination of detectors fires.By using appropriate post-processing, the OCM never- theless unveils the N-photon quantum interference. Forsimplicity, consider interfering a two-photon N00N stateon an array of photon detectors (figure 1a). Most of thetime, two different detectors fire; occasionally, both pho-tons reach the same detector. The original proposals onlyretained those rare events in which a single detector reg-istered both photons, observing that this rate exhibitedsub-wavelength fringes as a function of detector position.By contrast, the OCM keeps all the events, recordingthe “centroid,” or the average of the detected positionsof the photons – remarkably, sub-wavelength fringes arealso observed as a function of this centroid, obviating theneed to discard the bulk of the events [8].A more visual way to look at the OCM is illustrated infigure 1b, which shows plots of the joint probabilities forphoton 1 to arrive at pixel x and photon 2 to arrive atpixel x in coincidence. In a two-photon–absorber mea-surement, an event is only registered if x = x (bothphotons arrive at the same point). The resulting signalwill be given by the photon correlations along the greydiagonal lines drawn in i (for classical light) and iii ) (fora N00N state). This also means that all of the otherpossible events are discarded. The OCM signal, on theother hand, utilizes all of this data. For two photons, theOCM signal can be visualized as the integral of the joint-probability functions onto the grey diagonal lines (shown i and iii ). In other words, it is a histogram of all of thepoints plotted versus their centroid position, ( x + x ) / .
5, makingit possible to detect super-resolution features which aremuch smaller than the physical pixel size. Similar joint-probability functions can be made for any value of N.In this case, the N-photon centroid is computed from anN-dimensional integral. The N-photon OCM signal atposition X is number (rate) of events whose centroid, X c = ( x + · · · + x N ) /N , is equal to X , where x i is thepixel at which photon i is detected [18]. Unlike the N-photon-absorbing proposal, which keeps only O ( D N ) ofthe N-photon events (if there are D pixels), this methoduses them all.From figure 1b, it can be seen that performing an OCMon a N00N state will result in a signal with the same pe-riodicity as the N-photon absorption detection scheme.It is also evident that performing the OCM on classicallight will result in fringes with this enhanced periodicity.Although the classical OCM indeed results in a signalwith a period of λ/ (2 N sin θ ), its visibility decreases ex-ponentially with N , as V c = N − . (As shown in theSupplemental Material, this is because performing an N-fold OCM on classical light results in signal which is theconvolution of the singles intensity pattern with itself Ntimes.) However, as we show experimentally for two-,three-, and four-photon N00N states, the visibility of theN00N state OCM remains constant, independent of N . FIG. 2.
Schematic of the experimental apparatus: a)N00N state preparation –
Laser light and light from type-Icollinear down conversion are combined to create polarizationN00N states for a range of different values of N. b) CentroidMeasurement –
The polarization N00N states are convertedinto path-entangled N00N states, and interfered on a multi-mode fiber ribbon connected to 11 single-photon countingmodules (SPCM). c) Locking Measurement –
A small amountof down-converted and laser light is sent into the “lockingport”, and used to measure any phase drift between the laserand down-conversion paths so that it can be corrected.
Experimentally, we produce N00N states in polariza-tion using a relatively new technique which, given brightdown-conversion sources, can produce N00N states ofarbitrary N [15, 16]. Our source (figure 2a) createsN00N states by combining laser light with light froma Type-I collinear down-conversion (DC) source. TheDC light is generated by pumping a 1mm BBO crys-tal, cut for type-I phase matching, with 600mW averagepower of 404nm light. The 404nm pump is generated byfrequency-doubling light from a femtosecond Ti:Sapphlaser (running at 1.4W average power and 76 MHz rep-etition rate, and centered at 808nm) using a 2mm longcrystal of BBO. We measure approximately 12,000 two-photon counts/s from the DC source, with a couplingefficiency (pairs/singles) of 9 .
5% when single-photon de-tectors are placed directly after the in-fiber polarizingbeamsplitter (PBS) of panel b (including a diversion of ≈
3% of the photon pairs to our locking measurement,to be discussed shortly). The DC light is spatially over-lapped with light from the Ti:Sapph laser at a PBS,passed through a half waveplate at 22 . ◦ , and coupledinto polarization-maintaining single-mode fiber. To gen-erate N00N states, the relative phase between the twoarms must be set to zero, and the relative amplitudesbalanced. The source is optimized for N00N states ofdifferent N by simply changing the relative amplitude be-tween the laser and DC light [15]. Ideally, two- and three-photon N00N states can be made perfectly by setting thetwo-photon rate from the laser equal to that from the DClight (which we will refer to as configuration 1). Althoughour source cannot make perfect four-photon N00N states,it can, in principle, make a state which has 93% fidelitywith the ideal four-photon N00N state (the visibility of an OCM using this state should also be 93%). Makingthis four-photon state requires that the laser two-photonrate be three times larger than the DC two-photon rate.However, even if the laser rate is further increased, in-creasing the four-photon count rate, the fidelity of thefour-photon state will not be significantly degraded [19].To make four-photon N00N states, we choose to makethe laser two-photon rate 8.5 times larger than the DCrate (configuration 2), which leads to a fidelity with theideal state of 85%, a theoretical OCM visibility of 85%,and increases the four-photon rate by about a factor of 10(compared to the configuration in which the four-photonfidelity is optimized).The phase between the two arms must also be stabi-lized in order to make N00N states. We accomplishedthis by using a piezoelectric-driven trombone arm in thelaser path. To generate a feedback signal, small amountsof DC and laser light are sent through the other port ofthe state-preparation PBS, to the “locking measurement”(figure 2c). We measure 500 down-converted two-photoncounts/s ( ≈
3% of the detected down-converted light) atthe locking measurement and 5000 two-photon counts/sfrom the laser, creating a low-fidelity two-photon N00Nstate. This state will be phase shifted if the phase be-tween the two arms drifts, and can therefore be used totrack the phase drift. After acquiring counts for 5s, thephase of the state is measured, and then any phase driftbetween the two arms is corrected. Using this lockingmechanism, we are able to keep the N00N state sourcestable for days.Once polarization N00N states are collected intopolarization-maintaining single-mode fiber from thesource, they are sent to the OCM apparatus (figure2b). There, they are converted into path-entangled N00Nstates using an in-fiber PBS. Once the output polariza-tions are matched, the two modes are spatially interfered,by overlapping them at a 50 : 50 beamsplitter, at an angleof θ = 0 .
16 mrad. These overlapped modes are focusedonto a “fiber-ribbon” (which serves as our fixed-detectorarray) using cylindrical lenses. The visibility of the spa-tial interference pattern formed across the detector arrayis measured to be 90% using classical light. The phaseof this spatial interference signal is actively locked by ob-serving the phase on the detector array, and feeding thisback to a piezoelectric actuator which moves one of thefiber collimators.Our array of single-photon detectors consists of 12multi-mode fibers, mounted in a standard MTP12 fiberribbon. The fiber cores are 62.5 µ m in diameter, and aremounted linearly with a 250 µ m separation between fibercenters. This results in a fill factor (and therefore maxi-mum coupling efficiency) of ≈ FIG. 3.
Experimentally measured centroids for one tofour photons:
The first column is the result of a centroidmeasurement performed on classical light, and the second col-umn the result for N00N states. The circles are the measureddata, and the solid curves are fits from which the visibility isextracted. The total number of counts is plotted. The two-,and three-photon N00N-state data were acquired for 66 min-utes, and the four-photon N00N-state data were acquired for57 hours. To take the classical data, the down-converted lightwas blocked and the laser intensity was turned up. Then theone-, two-, and three-photon classical centroid measurementswere made in 10 minutes, and the four-photon measurementsin 45 minutes. The error bars are calculated assuming Poiso-nian counting statistics, and are not shown for the one- andtwo-photon data as they are much smaller than the circles.The vertical dashed lines indicate the period of the classicalinterference pattern. combinations. Using classical light, we measure an over-all coupling efficiency of 20% into the 11 fibers makingup our detector array. The additional losses in the OCMapparatus further reduces the final coupling efficiency ofthe DC photons (pairs/singles) from 9 .
5% to 0 .
2% (mea-sured in the fiber ribbon by summing the two-photoncounts, and dividing by the sum of the singles).We run the N00N-state source in the two configura-tions (described above) to take data. In configuration 1,we observed 150 two-photon N00N states per second and328 three-photon N00N states in 66 minutes, using thefiber ribbon. In configuration 2, we measured 284 four-photon N00N states in 57 hours. The resulting OCMs areplotted in the second column of figure 3 (circles). Thethree panels, from top to bottom, are the two-, three-, and four-photon OCMs. A sinusoid with a Gaussianenvelope is fitted to the data (solid lines), and the vis-ibility extracted from the sinusoidal part of the fit. Inthe fits to the two- to four-photon data, the period ofthe N-photon centroid is constrained to be the single-photon period divided by N. The resulting visibilities are49 ± ±
5% and 41 ± FIG. 4.
Scaling of visibility with photon number:
Aplot of the visibility of the centroid pattern versus photonnumber. The blue curve is the visibility which is predictedfor the classical centroid measurement, the blue triangles arethe experientially measured classical visibilities, and the redcircles are the measured visibilities of the quantum opticalcentroid measurement. its of 25% and 12 . ± ±
6% and59 ± ± ±
4% and 14 ± N .To study the scalability of our experiment, we plotthe various visibilities versus photon number in figure4. The solid curve is the theoretical prediction for thevisibility of the classical OCM signal. The experimen-tally measured classical visibilities (triangles) agree wellwith this curve, demonstrating a clear exponential decay.The visibility of the OCM signal performed on a N00Nstate should be constant (in principle it should be 1 . [1] L. Rayleigh, Philosophical Magazine Series 5 , 261(1879).[2] A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P.Williams, and J. P. Dowling, Physical Review Letters ,2733 (2000).[3] P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L.Braunstein, and J. P. Dowling, Physical Review A ,063407 (2001).[4] M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg,Nature , 161 (2004).[5] P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gas-paroni, and A. Zeilinger, Nature , 158 (2004).[6] O. Steuernagel, Journal of Optics B: Quantum and Semi-classical Optics , S606 (2004).[7] M. Tsang, Physical Review A , 043813 (2007).[8] M. Tsang, Physical Review Letters , 253601 (2009).[9] H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd,Physical Review Letters , 083603 (2011).[10] M. D’Angelo, M. V. Chekhova, and Y. Shih, PhysicalReview Letters , 013602 (2001).[11] Y. Kawabe, H. Fujiwara, R. Okamoto, K. Sasaki, andS. Takeuchi, Optics Express , 14244 (2007).[12] Y.-S. Kim, O. Kwon, S. M. Lee, J.-C. Lee, H. Kim, S.-K.Choi, H. S. Park, and Y.-H. Kim, Optics Express , 24957 (2011).[13] T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, andS. Takeuchi, Science , 726 (2007).[14] L. K. Shalm, R. B. A. Adamson, and A. M. Steinberg,Nature , 67 (2009).[15] H. F. Hofmann and T. Ono, Physical Review A ,031806 (2007).[16] I. Afek, O. Ambar, and Y. Silberberg, Science , 879(2010).[17] M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C.Teich, Physical Review Letters , 083601 (2003).[18] Q. Gulfam and J. Evers, Physical Review A , 023804(2013).[19] S. Rosen, I. Afek, Y. Israel, O. Ambar, and Y. Silberberg,Physical Review Letters , 103602 (2012).[20] D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam,Physical Review A , 061803 (2005).[21] A. E. Lita, A. J. Miller, and S. W. Nam, Optics Express , 3032 (2008).[22] M. Ghioni, A. Gulinatti, I. Rech, F. Zappa, and S. Cova,IEEE Journal of Selected Topics in Quantum Electronics , 852 (2007).[23] S. Ramelow, A. Mech, M. Giustina, S. Groblacher,W. Wieczorek, J. Beyer, A. Lita, B. Calkins, T. Gerrits,S. W. Nam, A. Zeilinger, and R. Ursin, Optics Express , 6707 (2013).[24] M. Giustina, A. Mech, S. Ramelow, B. Wittmann,J. Kofler, J. Beyer, A. Lita, B. Calkins, T. Gerrits, S. W.Nam, R. Ursin, and A. Zeilinger, Nature , 227 (2013).[25] B. G. Christensen, K. T. McCusker, J. B. Altepeter,B. Calkins, T. Gerrits, A. E. Lita, A. Miller, L. K.Shalm, Y. Zhang, S. W. Nam, N. Brunner, C. C. W.Lim, N. Gisin, and P. G. Kwiat, Physical Review Let-ters , 130406 (2013). SUPPLEMENTAL INFORMATIONVisibility of Classical Centroid
When performing the optical centroid measurement(OCM), the resulting signal, C ( X ) (where X the centroidcoordinate), can be written in terms of the N-photonprobability distribution, P ( x , . . . , x N ) (the probabilityof the N-photons arriving at positions x to x N ), as: C ( X ) = (cid:90) dx . . . (cid:90) dx N P ( x , . . . , x N ) δ ( x + · · · + x N N − X ) . (2)If the incident light is classical, P ( x , . . . , x N ) factorizesinto P ( x ) × · · · × P ( x N ). In this case, C ( X ) becomesthe N-fold convolution of N identical single-photon prob-ability distributions: C cl ( X ) = ( P ( x ) ∗ · · · ∗ P ( x N ))( N X ) . (3)The single-photon probability distribution for classicallight (of wavelength λ ) interfering at an angle of θ is P ( x ) ∝ f x ) (where f = (4 π sin θ ) /λ ). TheFourier transform of this single-photon probability dis-tribution is ˜ P ( ω ) ∝ δ ( ω + f ) + δ ( ω ) + δ ( ω − f ).Due to the Convolution Theorem (the Fourier trans-form of the convolution is the product of the Fouriertransforms of the individual functions) the Fourier trans-form of the classical centroid signal, C cl ( X ), is ˜ C cl ( ω ) ∝ N δ ( ω + f ) + δ ( ω ) + N δ ( ω − f ). So the C cl ( X ) becomes C cl ( X ) ∝ N − cos( N f X ) . (4)The visibility of C cl ( X ) is N − . Thus, although thecentroid signal will exhibit fringes with a frequency times N larger than the classical signal, the visibility of thosefringes decreases exponentially. Subtraction of Accidentals
In our N00N-state source, we combine classical lightwith down-converted light, and then post-select on de-tecting N photons. Assuming perfect detectors, when Nphotons are detected they will be in a state which hashigh fidelity with the ideal N00N state. In reality, ourdetectors have approximately 60% efficiency and photonsmay be lost before they even impinge on the detectors. Inthese conditions, unwanted “accidental” counts can oc-cur. To understand this, consider making a three-photonN00N state. In this case, the N00N state should arise dueto interference between the possibility of detecting three laser photons and no down-converted photons and thatof detecting one laser photon and two down-convertedphotons. Any other detection event will lower the fi-delity with the N00N state. Since the detectors and the
FIG. 5.
Scaling of visibility with photon number: