Measurement of two-photon position-momentum EPR correlations through single-photon intensity measurements
aa r X i v : . [ qu a n t - ph ] F e b Measurement of two-photon position-momentum EPR correlations throughsingle-photon intensity measurements
Abhinandan Bhattacharjee, Nilakantha Meher and Anand K. Jha ∗ Department of Physics, Indian Institute of Technology Kanpur, Kanpur UP 208016, India (Dated: February 9, 2021)The measurement of the position-momentum EPR correlations of a two-photon state is importantfor many quantum information applications ranging from quantum key distribution to coincidenceimaging. However, all the existing techniques for measuring the position-momentum EPR correla-tions involve coincidence detection and thus suﬀer from issues that result in less accurate measure-ments. In this letter, we propose and demonstrate an experimental scheme that does not require co-incidence detection for measuring the EPR correlations. Our technique works for two-photon statesthat are pure, irrespective of whether the state is separable or entangled. We theoretically showthat if the pure two-photon state satisﬁes a certain set of conditions then the position-momentumEPR correlations can be obtained by doing the intensity measurements on only one of the photons.We experimentally demonstrate this technique for pure two-photon states produced by type-I spon-taneous parametric down-conversion, and to the best of our knowledge, we report the most accuratemeasurement of position-momentum EPR correlations so far.
If two photons are entangled in position and momen-tum variables, the product of the conditional positionand momentum uncertainties of the individual photonsbecomes less than 0 . ~ , the minimum value allowed bythe Heisenberg uncertainty relation. This violation of theconditional Heisenberg uncertainty relation is the signa-ture of position-momentum EPR correlations  and isa witness of the position-momentum entanglement. Fortwo-dimensional variables, such as polarization, entan-glement can be veriﬁed through the violations of Bell’sinequalities  and can be quantiﬁed through measuressuch as concurrence . However, for continuous vari-ables, such as position-momentum, there is no prescrip-tion for quantifying entanglement. One can at best onlyverify entanglement, and the EPR-correlations measure-ments are the primary tool for that. In the past, sev-eral studies have used EPR-correlations measurementsin the position-momentum variables in order to demon-strate position-momentum entanglement [4–11]. EPR-correlation measurements have also been used as wit-nesses of entanglement in many other continuous vari-ables including time-energy [12–14], angle-orbital angularmomentum (OAM) , radial position-radial momen-tum , and quadrature phase-amplitude . More re-cently, even in entangled systems not consisting of pho-tons, EPR-correlations measurements have become im-portant tools for witnessing continuous-variable entan-glement. These include macroscopic objects , BoseEinstein condensate , and cold atoms .The demonstration of position-momentum EPR-correlations is very important for many applications suchas quantum key distribution , quantum informationprocessing , quantum metrology , coincidenceimaging [24, 25], and coincidence holography , sincethe eﬃciency of these applications relies on how accu-rately the EPR correlations could be measured. There-fore, it is very important to have a more accurate tech- nique for measuring EPR-correlations. In the past fewyears, many schemes with increased accuracy have beendemonstrated [4–17]. However, all these methods involvecoincidence detection, implemented either by using twoscanning single-photon detectors , or two scanning slits[5, 6], or array of single-photon detectors [7, 8], or EM-CCD cameras [9–11]. As a result, these measurementmethods suﬀer from either too much loss of light, orstrict alignment requirements, or multiple measurements,which adversely aﬀect the accuracy of measurements.On the other hand, in the context of two-dimensionaltwo-particle state, that is, the two-qubit states, it isknown that if the state is pure, the entanglement quanti-ﬁers [3, 27] can be measured by doing measurements ononly one of the qubits [28, 29], without requiring coinci-dence detection. Furthermore, even in the context of con-tinuous variables, it is known that when the two-photonstate is pure, several two-photon properties such as two-photon angular Schmidt spectrum [30–32], two-photonspatial Schmidt number , and momentum correlations can be measured by doing intensity measurements ononly one of the subsystems. These measurement schemesbased on intensity detection provide much better accu-racy than those based on coincidence detection. In thisletter, utilizing the same physics, we propose a techniquefor measuring the position-momentum EPR-correlationsthat does not require coincidence detection. Our tech-nique is based on measuring the intensities of only oneof the subsystems, and it works for two-photon statesthat are pure, irrespective of whether the state is sepa-rable or entangled. We show that if a pure two-photonstate satisﬁes a certain set of conditions then the position-momentum EPR correlations can be obtained by doingthe intensity measurements on only one of the photons.We experimentally demonstrate our technique with puretwo-photon states produced by type-I spontaneous para-metric down-conversion (SPDC), and we obtain, to thebest of our knowledge, the most accurate measurementof position-momentum EPR-correlations reported so far.A pure state of two photons | Ψ i in the transverse mo-mentum basis can be written as | Ψ i = Z Z d p d p ψ ( p , p ) | p , p i . (1)Here, p ≡ ( p x , p y ) and p ≡ ( p x , p y ) are thetransverse momenta of the ﬁrst and the second pho-ton, respectively, | p , p i is the two-photon state vec-tor, and ψ ( p , p ) represents the two-photon transverse-momentum wavefunction. We note that if the two-photon wavefunction satisﬁes ψ ( p , p ) = ψ ( p ) ψ ( p )then it is separable, otherwise it is non-separable, or inother words, entangled. When the second photon is de-tected with transverse momentum p = 0, then the con-ditional momentum probability distribution P ( p | p =0) of the ﬁrst photon is given by P ( p | p = 0) = | ψ ( p , p = 0) | . (2)Now, the momentum cross-spectral density function ofthe ﬁrst photon can be calculated as W ( p , p ′ ) = h Ψ | E (+) s ( p ) E ( − )1 ( p ′ ) | Ψ i , where E (+)1 ( p ) and E ( − )1 ( p ′ ) are the negative- and positive-frequency partsof the electric ﬁeld operators respectively. For p ′ = − p ,we have W ( p , − p ) = Z Z ψ ∗ ( p , p ) ψ ( − p , p ) d p . (3)Next, we ﬁnd that if the two-photon wavefunction satis-ﬁes the following condition ψ ∗ ( p , p ) ψ ( − p , p ) ∝ | ψ ( p , p = 0) ψ ( p = 0 , p ) | , (4)then using Eqs. (2) and (3), one can show that W ( p , − p ) ∝ P ( p | p = 0) . (5)We note that the condition in Eq. (4) can be satisﬁed byboth separable and inseparable pure two-photon wave-functions. Eq. (5) is the main theoretical result of thisletter. It states that as long as a two-photon state is pure,whether separable or entangled, and satisﬁes the con-dition in Eq. (4), the momentum cross-spectral densityfunction of the ﬁrst photon remains proportional to itsconditional momentum probability distribution function.This implies that the standard deviations of W ( p , − p )and P ( p | p = 0) are equal and that by measuring thestandard deviation of W ( p , − p ), one can obtain thestandard deviation of P ( p | p = 0). We denote the stan-dard deviation of the conditional x -momentum of the ﬁrstphoton by ∆( p x | p x = 0).Now, by writing the two-photon wavefunction ofEq. (1) in the position basis and proceeding in the simi-lar manner, we can show that if the two-photon position wavefunction ψ ( ρ , ρ ) satisﬁes the condition ψ ∗ ( ρ , ρ ) ψ ( − ρ , ρ ) ∝ | ψ ( ρ , ρ = 0) ψ ( ρ = 0 , ρ ) | , (6)where ρ ≡ ( x , y ) and ρ ≡ ( x , y ) are the transverseposition vectors of the ﬁrst and the second photon, thenthe position cross-spectral density function W ( ρ , − ρ )of the ﬁrst photon is proportional to its conditional posi-tion probability distribution function P ( ρ | ρ = 0), thatis, W ( ρ , − ρ ) ∝ P ( ρ | ρ = 0) . (7)Thus, by measuring the standard deviation of W ( ρ , − ρ ), one can obtain the standard deviationof P ( ρ | ρ = 0). We denote the standard deviationof the conditional x -position of the ﬁrst photon by∆( x | x = 0). We note that although the above analysishas been presented with respect to making measure-ments on the ﬁrst photon, we obtain the same resulteven when analysed with the second photon. Now, it isknown that if the two-photon wavefunction is separablethen the product U of the conditional uncertaintiessatisﬁes the Heisenberg uncertainty relation, that is, U ≡ ∆( x | x = 0)∆( p x | p x = 0) > . ~ . (8)However, a violation of this inequality implies that thetwo-photon wavefunction is non-separable and that thetwo photons are entangled having EPR correlations inposition-momentum variables .SPDC is a nonlinear optical process in which a pumpphoton at higher frequency gets down-converted into twophotons of lower frequencies called the signal and idlerphotons. In most experimental situations [4–13, 15–17], one uses a spatially coherent pump ﬁeld for down-conversion. As a result, the joint state of the signal andidler photons produced by SPDC in these experimentalsituations remains pure and very closely resembles thestate given by Eq. (1). Therefore, we experimentallydemonstrate our technique with the two-photon stateproduced by SPDC.For a spatially-coherent Gaussian pump with beamwaist at the crystal plane, the two-photon wavefunctionproduced by SPDC in the momentum basis is given by[8, 9, 22, 36] ψ ( p s , p i ) = Ae − ( p i + p s ) σ p ~ e − | p i − p s | σ − ~ , (9)where p s ≡ ( p sx , p sy ) and p s ≡ ( p ix , p iy ) are the trans-verse momenta at the crystal plane of the signal and idlerphotons, respectively. σ p is the pump beam waist and σ − = p . Lλ p / π , where L is the length of the crys-tal, λ p is the pump wavelength and A is a normaliza-tion constant. By taking the Fourier transform of the experiment (a) (c) (b) f f
50 cm10 cm 40 cm5 cm 15 cm
BBOBBOUV pumpUV pump B.S B.S FT.S 0-240 240 x ~ -320 320 ( ¹m )0 ~~ ~ I ( x , y ) theory ~~ sx sy I ( p , p ) out sx theoryexperiment (d) (e) (f)(h) (i) (j) (g)(k) f f f s s ~ ~ I ( x , y ) s s experiment s -3 3 x ~ ( ¹m ) s -3 3-3 3 000 x ~ ( ¹m ) s x ~ ( ¹m ) s -330 y ~ ( mm ) s p ( ¹m ) -3 3-3 3-3 3 000 ~ sx p ( mm ) ~ sx p ( mm ) ~ sx p ( mm ) -330 p ~ ( mm ) s y I ( x , y ) s s ¢ ~~ sx sy I ( p , p ) P ( x | x = ) i ~~ s P ( p | p = ) ~~ s x i x ~ ~ ¢ ± = ¼ ~~ sx sy I ( p , p ) out ± =0 outoutout ± = ¼ out ± =0 FIG. 1. (a) Lens conﬁguration for measuring position correlation. (b) Lens conﬁguration for measuring momentum correlation.(c) Inversion-based interferometer for measuring position and momentum cross-spectral density functions. B.S: Beam Splitter,T.S: Translational Stage, F: an interference ﬁlter of 10 nm spectral width centered at 810 nm.. (d) and (e) The two interferogramsrecorded at δ c = 0 and δ d = π with the conﬁguration in Fig. 1(a). (f) The diﬀerence intensity image ∆ I (˜ x s , ˜ y s ). (g)Experimental and theoretical conditional probability distribution P (˜ x s | ˜ x i = 0). (h) and (i) The two interferograms recoded at δ c = 0 and δ d = π with the conﬁguration in Fig. 1(b). (j) The diﬀerence intensity image ∆ I (˜ p sx , ˜ p sy ). (k) Experimental andtheoretical conditional probability distribution P (˜ p sx | ˜ p ix = 0). wavefunction given in Eq. (9), we write the two-photonwavefunction in the position basis as ψ ( ρ s , ρ i ) = A ′ e − ( ρ i + ρ s ) σ p e − | ρ i − ρ s | σ − . (10)Here, A ′ is a normalization constant, ρ s ≡ ( x s , y s ) and ρ i ≡ ( x i , y i ) are the transverse position vectors of the sig-nal and idler photons at the crystal plane. We note thatthe above wavefunctions ψ ( p s , p i ) and ψ ( ρ s , ρ i ) satisfythe conditions given in Eqs. (4) and (6), respectively.Next, we present our experiment results demonstrating how the conditional position and momentum uncertain-ties can be obtained by measuring the cross-spectral den-sity functions of just the signal photon. Figures 1(a)-1(c)show the schematics of our experimental setup. An ultra-violet (UV) Gaussian pump beam of wavelength λ p = 405nm and beam waist σ p = 388 µ m is incident on a 2 mmthick β − barium borate (BBO) crystal and produces two-photon state using SPDC with the type-I collinear phase-matching. Figure 1(c) shows an inversion-based interfer-ometer that we use for measuring the cross-spectral den-sity functions [31, 37]. Figures 1(a) and 1(b) show thelens conﬁgurations for imaging, respectively, the crystalplane and the Fourier plane of the crystal onto an EM-CCD camera having 512 ×
512 pixels and 60 second ac-quisition time.For measuring the position cross-spectral density func-tion of the signal photon, we use the conﬁguration ofFig. 1(a) with f = 10 cm and f = 40 cm and imagethe crystal onto the EMCCD plane, kept at 40 cm from f , with a magniﬁcation M = 4. We take ( x s , y s ) and(˜ x s , ˜ y s ) to be the position coordinates at the crystal planeand at the EMCCD plane, respectively. The two sets ofcoordinates are related as ˜ x s = M x s and ˜ y s = M y s . Theintensity I δ out (˜ x s , ˜ y s ) of the output interferogram at theEMCCD plane is given by I δ out (˜ x s , ˜ y s ) = k I (˜ x s , ˜ y s ) + k I ( − ˜ x s , ˜ y s ) + 2 √ k k W (˜ x s , ˜ y s , − ˜ x s , ˜ y s ) cos δ . Here, k and k are the scaling constants, while k I (˜ x s , ˜ y s )and k I ( − ˜ x s , ˜ y s ) are the intensities at the EMCCDplane coming through the two arms of the interferom-eter. The quantity δ is the phase diﬀerence between thetwo interferometric arms. If we take two interferograms I δ c out (˜ x s , ˜ y s ) and I δ d out (˜ x s , ˜ y s ) at δ = δ c and δ = δ d , re-spectively, then it can be shown that the diﬀerence in-tensity ∆ I out (˜ x s , ˜ y s ) = I δ c out (˜ x s , ˜ y s ) − I δ d out (˜ x s , ˜ y s ) is pro-portional to the position cross-spectral density, that is,∆ I out (˜ x s , ˜ y s ) ∝ W (˜ x s , ˜ y s , − ˜ x s , ˜ y s ) [31, 37]. Figures 1(d)and 1(e) show the two experimentally measured inter-ferograms at δ = δ c ≈ δ = δ d ≈ π , respectively,and Fig. 1(f) shows the diﬀerence intensity ∆ I out (˜ x s , ˜ y s ).From Eq. (7), we have that W (˜ x s , ˜ y s , − ˜ x s , − ˜ y s ), is pro-portional to the conditional position probability distribu-tion function of the signal photon, that is, P (˜ x s , ˜ y s | ˜ x i =0 , ˜ y i = 0) ∝ W (˜ x s , ˜ y s , − ˜ x s , − ˜ y s ). Therefore, we ob-tain the one-dimensional conditional position probabil-ity distribution function P (˜ x s | ˜ x i = 0) by averaging∆ I out (˜ x s , ˜ y s ) over the ˜ y s -direction and plotting it inFig. 1(g). Using Eq. (10) and the relevant experimen-tal parameters, we calculate the theoretical P (˜ x s | ˜ x i = 0)and plot it in Fig. 1(g) (solid curve). We scale the P (˜ x s | ˜ x i = 0) plots in Fig. 1(g) such that the maximumvalue is one. We ﬁt the experimental P (˜ x s | ˜ x i = 0) witha Gaussian function and ﬁnd the standard deviation tobe 26 . µ m. The standard deviation of the theoreticalplot is 30 . µ m. Now, we use ˜ x s = M x s and obtain theexperimental and theoretical values of ∆( x s | x i = 0) tobe 6 . µ m and 7 . µ m, respectively.For measuring the momentum cross-spectral densityfunction of the signal photon, we use the conﬁgura-tion of Fig. 1(b) with f = 5 cm, f = 10 cm, and f = 30 cm. The eﬀective focal length of this combi-nation is f e = 15 cm. The EMCCD is kept at 30 cmfrom f , which is the Fourier plane of this conﬁguration.We take ( p sx , p sy ) to be the transverse momentum atthe crystal plane and (˜ p sx , ˜ p sy ) is the position coordi-nate at the EMCCD plane. These coordinates can beshown to be related as p sx = k ~ f e ˜ p sx and p sy = k ~ f e ˜ p sy , where k = πλ . The intensity I δ out (˜ p sx , ˜ p sy ) ofthe output interferogram at the EMCCD plane in thiscase can be written as I δ out (˜ p sx , ˜ p sy ) = k I (˜ p sx , ˜ p sy ) + k I ( − ˜ p sx , ˜ p sy ) + 2 √ k k W (˜ p sx , ˜ p sy , − ˜ p sx , ˜ p sy ) cos δ .Here, k I (˜ p sx , ˜ p sy ), and k I ( − ˜ p sx , ˜ p sy ) are the intensitiesat the EMCCD plane coming through the two interfer-ometric arms. Just as discussed above, the diﬀerenceintensity ∆ I out (˜ p sx , ˜ p sy ) = I δ c out (˜ p sx , ˜ p sy ) − I δ d out (˜ p sx , ˜ p sy )is proportional to the cross-spectral density function W (˜ p sx , ˜ p sy , − ˜ p sx , ˜ p sy ) [31, 37]. Figures 1(h) and 1(i)show the two experimentally measured interferogramsat δ = δ c ≈ δ = δ d ≈ π , respectively, andFig. 1(j) shows the diﬀerence intensity ∆ I out (˜ p sx , ˜ p sy ).From Eq. (6), we have that P (˜ p sx , ˜ p sy | ˜ p ix = 0 , ˜ p iy = 0) ∝ W (˜ p sx , ˜ p sy , − ˜ p sx , − ˜ p sy ). Therefore, we obtain the one-dimensional conditional probability distribution function P (˜ p sx | ˜ p ix = 0) by averaging ∆ I out (˜ p sx , ˜ p sy ), over ˜ p sy -direction and plotting it in Fig. 1(k). Using Eq. (9) andthe relevant experimental parameters, we calculate thetheoretical P (˜ p sx | ˜ p ix = 0) at the EMCCD plane and plotit in Fig. 1(k) (solid curve). We scale the P (˜ p sx | ˜ p ix = 0)plots in Fig. 1(k) such that the maximum value is one.We ﬁt the experimental P (˜ p sx | ˜ p ix = 0) with a Gaussianfunction and ﬁnd the standard deviation to be 49 . µ m.The standard deviation of the theoretical plot is 49 . µ m.Using p sx = k ~ f e ˜ p sx , we obtain the experimental and the-oretical values of ∆( p sx | p ix = 0) to be 2 . × − ~ µ m − and 2 . × − ~ µ m − , respectively.As deﬁned in Eq. (8), the experimentally measuredvalue of the conditional uncertainty product U ex is 1 . × − ~ . This is much smaller than 0 . ~ and thus impliesa strong EPR correlations between the two entangledphotons. We ﬁnd the theoretical conditional uncertaintyproduct U th to be 1 . × − ~ , and we thus ﬁnd a goodmatch between the theory and experiments. Now, in or-der to quantify the accuracy of our measurement scheme,we use the quantity F ≡ | U th − U ex | U th × F implies better accuracy of EPR cor-relation measurements. For our experimental results, weobtain F = 14 . F obtained throughour measurement scheme is much smaller than the pre-viously reported values of 27 .
1% in Ref. , 43 .
7% inRef. , 66% in Ref. , 190% in Ref. , and 376%in Ref. . There is another quantity that is quite of-ten used for quantifying the EPR-correlations measure-ments. It is called the degree of violation and is deﬁnedas D = (0 . ~ /U exp ) . The degree of violation D does notquantify the measurement accuracy but gives an estimateof the degree with which the Heisenberg bound of 0 . ~ isviolated. For our experimental measurements, D = 896,whereas the degree of violations reported earlier include576 in Ref. , 380 in Ref. , 42 in Ref. , 25 inRef. , and 4 in Ref. . Thus we report not only themost accurate EPR-correlations measurements but alsothe highest degree of violation. We note that since weare using collinear phase-matching, what gets recordedby the EMCCD camera is the sum of the interferogramsproduced by the signal and idler ﬁelds. However, sincesignal and idler photons are identical in their spatial de-gree of freedom, functional form of the sum interferogramis same as that of the individual interferograms producedby signal and idler photons.In conclusion, we have demonstrated a scheme formeasuring two-photon position-momentum EPR corre-lations that does not require coincidence detection. Ourscheme works for any two-photon state that is pure, ir-respective of whether the state is separable or entan-gled. We have experimentally demonstrated this tech-nique with pure two-photon states produced by type-ISPDC and have obtained the most accurate measure-ment of position-momentum EPR correlations reportedso far. Our scheme can be extended for measuring EPR-correlations in other continuous variables such as time-energy [12–14] and angle-OAM . It is also appli-cable to pure two-particle states produced using otherprocesses such as spontaneous four-wave mixing .Thus, we expect our work to have wide practical im-plications for continuous-variable quantum informationapplications.We thank Siddharth Ramachandran for useful discus-sions and acknowledge ﬁnancial support through the re-search grant no. EMR/2015/001931 from the Scienceand Engineering Research Board (SERB), Department ofScience and Technology, Government of India, and the re-search grant no. DST/ICPS/QuST/Theme-1/2019 fromthe Department of Science and Technology, Governmentof India. NM acknowledges post-doctoral fellowship fromIndian Institute of Technology Kanpur. ∗ [email protected] A. Einstein, B. Podolsky, and N. Rosen, Physical review , 777 (1935). J. S. Bell, Physics , 195 (1964). W. K. Wootters, Phys. Rev. Lett. , 2245 (1998). M. D’Angelo, Y.-H. Kim, S. P. Kulik, and Y. Shih, Phys-ical review letters , 233601 (2004). J. C. Howell, R. S. Bennink, S. J. 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