Fast imaging of multimode transverse-spectral correlations for twin photons
FFast imaging of multimode transverse-spectral correlations for twin photons
Michał Lipka ∗ and Michał Parniak
1, 2, † Centre for Quantum Optical Technologies, Centre of New Technologies,University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland Niels Bohr Institute, University of Copanhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
Hyperentangled photonic states – exhibiting nonclassical correlations in several degrees of free-dom – offer improved performance of quantum optical communication and computation schemes.Experimentally, a hyperentanglement of transverse-wavevector and spectral modes can be obtainedin a straightforward way with multimode parametric single-photon sources. Nevertheless, experi-mental characterization of such states remains challenging. Not only single-photon detection withhigh spatial resolution – a single-photon camera – is required, but also a suitable mode-converter toobserve the spectral/temporal degree of freedom. We experimentally demonstrate a measurement ofa full 4-dimensional transverse-wavevector–spectral correlations between pairs of photons producedin the non-collinear spontaneous parametric downconversion (SPDC). Utilization of a custom ultra-fast single-photon camera lets us characterize the two-photon state with a high resolution and arelatively short measurement time.
Photonic qubits can be easily created in entangledstates, communicated over many–km distances and ef-ficiently measured [1]. Tremendous effort has been de-voted to improving the success rates of quantum en-hanced protocols and multimode solutions, often accom-panied with active multiplexing, are one of the mostpromising branches of this development [2–11], enablingboth faster transfer and generation of photonic quantumstates. In particular, systems harnessing several degreesof freedom (DoF) offer superior performance [12–14] es-pecially in selected protocols such as superdense coding[15], quantum teleporation [16] or complete Bell-stateanalysis [17]. Utilization of several DoFs brings a qual-itatively new possibility to create hyperentangled statesexhibiting nonclassical correlations in several DoF simul-taneously with a greatly expanded Hilbert space andinformational capacity. Generation of entangled pairsof photons in spectral, temporal, transverse wavevec-tor, spatial, orbital angular momentum (OAM) and withmultiple DoFs has been demonstrated. In particularspontaneous parametric down-conversion (SPDC) can beused to generate hyperentangled states in 4 DoF simul-taneously [18]. Nonetheless, experimental characteri-zation of multidimensional states remains challenging.Single-pixel detectors such as superconducting nanowiresoffer excellent timing resolution [19], as well as spec-tral resolution when combined with dispersive elementssuch as chirped fiber gratings [20] or detector-integrateddiffraction gratings [21]. Such setups provide a wayto implement high-dimensional quantum communication[22], temporal super-resolved imaging [23] or observequantum-interference in time or frequency space [24, 25] -a promising approach for quantum fingerprinting [26, 27].Single-photon-resolving cameras on the other hand nat-urally offer spatial or angular resolution, which can be ∗ [email protected] † [email protected] exploited in super-resolved imaging [28–32], interferom-etry [33], characterization [34, 35] or, similarly as in theprevious case, observation of quantum interference effectssuch as in the Hong-Ou-Mandel-type experiments [36].Recently however, the capability of cameras has been ex-panded by invoking a well-known mode conversion tech-nique, in which Sun et al. simply observed spectral cor-relation with the help of a diffraction grating [37]. It isthus a promising approach to use a camera to observemany DoFs simultaneously.Here, we experimentally demonstrate a measurementof full 4-dimensional quantum correlations between thetransverse and spectral degrees of freedom of a twin-photon state, generated in a non-collinear type I SPDC.An ultra-fast single-photon–sensitive camera, yielding frames per second with × pixels per frame,allows to quickly gather enormous statistic size whilemaintaining high resolution due to a large number ofpixels. In conjunction with recent development inhigh-dimensional entanglement detection [38], our single-photon detection system would enable rapid characteriza-tion of such hyperentangled states. Furthermore, precisecorrelation measurements are vital to fully utilize quan-tum advantage of entangled states e.g. via non-local dis-persion compensation recently demonstrated to improvequantum key distribution rates [39]. Higher-order cor-relation measurements also enable novel superresolvedimaging techniques [29, 30] which particularly benefitfrom fast acquisition rates and high spatial resolution ofemployed single-photon detectors.To generate twin-photon states we have employed aBeta Barium Borate (BBO) non-linear crystal in the typeI SPDC process with noncollinear geometry, as depictedin Fig. 1. For the SPDC pumping we first produce sec-ond harmonic of fs, nm pulses from Ti-Sapphirelaser (Spectra Physics Mai Tai) in a second, similar BBOcrystal with length L = 0 . mm. The nm red pumpis filtered-out with dichroic mirrors and a bandpass filter( nm, nm bandwidth). Blue λ p = 400 nm pumpwith an average power of mW is focused in an L = a r X i v : . [ qu a n t - ph ] J a n BBOPumpBeam stop SlitDichroic mirror Diffraction gratingCustom I-sCMOScamera Bandpass f IdlerSignal f k X λ f Signal Idler (a) (c) (d) (e)(b)
Signal ...
Idler
FIG. 1. (a) Annular ring of twin-photon emission in the far-field of a BBO crystal mediating non-collinear type-I SPDC.Histogram of single photon positions has been registered over × camera frames. (b) Twin-photon emission in the far-fieldafter passing through the rectangular slit selecting k y ≈ and after diffraction on the grating acting as wavelength-dependentwavevector shift k y → k y + K grating ( λ ) . Distinct fragments correspond to signal and idler photons, respectively. Camera framecoordinates correspond to transverse wavevectors and wavelength of photons. (c),(d) Camera regions corresponding to (c)signal and (d) idler photons. The division into sub-regions with limited wavevectors { k ( s ) x } , { k ( i ) x } and wavelengths { λ s } , { λ i } has been depicted with rectangles. For clarity only wavevector sub-regions (c) or only wavelength sub-regions (d) has beenshown. Signal and idler regions are divided equally. (e) Experimental setup for fast transverse-spectral correlation imaging oftwin-photons. mm BBO with a Gaussian beam width of w = 70 µ mand finally filtered-out with a dichroic mirror. The SPDCemission is far-field imaged with a lens ( f = 60 mm)on an adjustable rectangular slit which selects a rangeof wavevectors [ − ∆ k y / , ∆ k y / around k y = 0 . A sec-ond lens ( f = 300 mm) images the BBO onto a ruleddiffraction grating ( N lines = 1200 lines/mm, resolutionof δλ = 2 λ p /N lines = 0 . nm) mounted vertically in theLittrow configuration and at a small horizontal angle.The grating adds a wavelength-dependent wavevector inthe y direction. A third lens ( f = 100 mm ) far-fieldimages the grating onto a single-photon camera. The ef-fective focal size of the setup from BBO to the camera is f eff = 30 mm.Importantly, while BBO is cut for type I SPDC, byslightly adjusting the angle of the crystal axis with re-spect to the central pump beam wavevector, we canalter the diameter and width of the annular ring thatthe SPDC emission forms in the far-field. The compari-son with theoretical calculations shows that the crystal-axis– z -axis angle (including cutting angle of . ◦ ) is . ◦ ± . ◦ .The single-photon camera consists of a two-stage im-age intensifier (Hamamatsu V7090-D) together with itshigh-voltage supply (Photek FP630) and a kHz gatingmodule (Photek GM10-50B) connected with a custom-built I-sCMOS camera based on a fast CMOS sensor(LUX2100, pixel pitch µ m). Communication with thecamera sensor together with low-level image processingis performed with a programmable logic (FPGA) mod-ule (Xilinx Zynq-7020). The image processing consistsmainly of background subtraction and an algorithm forsingle-photon discrimination (and localization). FPGAmodule is bundled with an ARM-family processor, pro-viding ethernet data transfer to a PC. The CMOS sensor is specifically setup for lowest time-dependent noise atthe cost of lowering the dynamic range. We note thatwith a faster gating module the camera can operate at frames per second with a frame of × px.Single-photon sensitivity is achieved by operating theimage intensifier (II) in the Geiger mode (on/off). A pho-ton striking the II leads to the emission of a photoelectron(with quantum efficiency of which is accelerated( − V) or stopped ( +50
V) with an electric potentialcontrolled by the gating module. The accelerated pho-toelectrons are multiplied via avalanche secondary emis-sion in a two-stage microchannel plate (MCP) giving output electrons per photoelectron. The voltage acrossMCP is V. Electrons leaving the MCP are fur-ther accelerated (
V) and strike a phosphor (P46)screen, leaving bright flashes. The phosphor screen isimaged (magnification M = 0 . with a relay lens ontothe CMOS sensor. We note that while we cross-correlatetwo regions for signal and idler photons, respectively, anauto-correlation setup could be employed with a singleregion and simple post-processing [40].The employed gating time was . µ s amounting toan average of ¯ n tot = 0 . photons per frame. Sig-nal and idler photons are observed in × px re-gions each corresponding to rad/mm × . nm (with . rad / ( mm × px ) and . nm / px). The Gaussianmode size σ k -mode was predicted to be . rad/mm andmeasured as . ± . rad / mm (see supplementary ma-terial). The spectral mode size was measured to be σ λ -mode = 4 . ± . nm. The mode sizes correspondto the width of second-order photon number correlations.Using the theoretical prediction of the joint wavefunctionwe numerically get M = 1 / (cid:80) ∞ j =0 λ j ≈ . modes, where λ j are the Schmidt coefficients.While, with a spectrally broad, focused pump beam (a) (b) FIG. 2. Correlation in joint (a) transverse wavevectors and (b) spectrum of signal–idler photon pairs. (a) Color map or (b)left column represents the experimental data (photon number covariance summed over (a) spectral or (b) wavevector regions,see main text). (a) White contours or (b) right column represents theoretical prediction | Ψ { λ s } , { λ i } ( k ( s ) x , k ( i ) x ) | of a two-photonwavefunction component modulus squared, (a) summed over given wavelength ranges of join spectrum { λ s } , { λ i } or (b) summedover transverse wavevector ranges { k ( s ) x } , { k ( i ) x } and normalized to a unity maximum. and sufficiently short crystal, the SPDC emission ishighly multimode both in the spectral and transverseDoF, we shall begin with a single pair of signal –idler modes. A two mode squeezed state | ψ (cid:105) = (cid:80) j =0 χ j/ | j (cid:105) s | j (cid:105) i , generated in SPDC can be approxi-mated to the first order in √ χ as a pair of photons | (cid:105) s | (cid:105) i – one in signal ( s ) mode and one in idler ( i ) mode. Weshall be concerned with the joint wavefunction of the pho-tons in transverse wavevector and spectral coordinates Ψ( k s, ⊥ , λ s ; k i, ⊥ , λ i ) = (cid:104) k s, ⊥ , λ s | (cid:105) s (cid:104) k i, ⊥ , λ i | (cid:105) i . (1)In the experiment we directly measure the x componentof the transverse wavevector, while selecting only thephotons with k y ≈ . Before measurement, a diffractiongrating maps the spectral DoF onto k y ( ω ) . Single-photon camera detects the number of photons with a given trans-verse–spectral coordinate n ( k ( ξ ) x , λ ξ ) ∈ { , } ; ξ ∈ { s, i } separately in signal and idler arms. With a large numberof observed frames, the average over frames (cid:104) n ( k ( ξ ) x , λ ξ ) (cid:105) gives an estimate for the probability of detecting a pho-ton at given coordinates. Hence, the photon number co-variance Cov ( k ( s ) x , λ s ; k ( i ) x , λ i ) estimates the probabilityof detecting a non-accidential coincidence – pair of cor-related signal and idler photons in a single camera frame– with given spectral and transverse coordinates, whichis modeled by | Ψ( k s, ⊥ , λ s ; k i, ⊥ , λ i ) | . The covariance isgiven by:Cov ( k ( s ) x , λ s ; k ( i ) x , λ i ) = (cid:104) n ( k ( s ) x , λ s ) n ( k ( i ) x , λ i ) (cid:105) − (cid:104) n ( k ( s ) x , λ s ) (cid:105)(cid:104) n ( k ( i ) x , λ i ) (cid:105) . (2)For visualization, we sum the covariance over selectedsub-regions either in wavelengths { λ s } , { λ i } or wavevec-tors { k ( s ) x } , { k ( i ) x } yielding:Cov { λ s } , { λ i } ( k ( s ) x , k ( i ) x ) = (cid:88) λ s ∈{ λ s } λ i ∈{ λ i } Cov ( k ( s ) x , λ s ; k ( i ) x , λ i ) , (3)andCov { k ( s ) x } , { k ( i ) x } ( λ s , λ i ) = (cid:88) k ( s ) x ∈{ k ( s ) x } k ( i ) x ∈{ k ( i ) x } Cov ( k ( s ) x , λ s ; k ( i ) x , λ i ) , (4)respectively. The selected sub-regions are depicted inFig. 1 (c),(d) on a histogram of signal and idler positionsin wavevector–wavelength space. In the measurement wehave gathered camera frames, each serving as a sep-arate experiment repetition.The joint covariance in transverse wavevector coordi-nates Cov { λ s } , { λ i } ( k ( s ) x , k ( i ) x ) is depicted in Fig. 2 (a)with each panel corresponding to a different pair of wave-length sub-regions { λ s } , { λ i } . Good agreement with thetheoretical prediction (without fitting) can be observed.The details of wavevefunction calculation can be found in Supplementary Material. Similarly, the joint covariancein spectral coordinates Cov { k ( s ) x } , { k ( i ) x } ( λ s , λ i ) is depictedin Fig. 2 (b) for selected pairs of sub-regions in whichthe covariance is non-vanishing.We have demonstrated a capability to measure 4 di-mensional transverse-wavevector–spectral quantum cor-relations between pairs of photons generated in non-collinear SPDC. Due to a custom single-photon cam-era with very fast acquisition rates (an order of mag-nitude improvement) we were able to gather statistics of camera frames (experiment repetitions) in roughlyone day. Large statistics enabled faithful reconstructionof bi-photon wavevefunction in spectral and transverse-wavevector coordinates. For this demonstration we se-lected a single component of the trasnverse wavevectorwhich is far-field imaged (mapped) onto positions on thecamera frame, similarly the spectral part is mapped ontopositions with a diffraction grating. Importantly, our sys-tem is inherently multimode and can be adapted for mea-surements in different mode bases e.g. orbital angularmomentum and for different degrees of freedom. Funding
Ministry of Science and Higher Educa-tion (DI2018 010848); Foundation for Polish Science(MAB/2018/4 “Quantum Optical Technologies”); Officeof Naval Research (N62909-19-1-2127).
Acknowledgements
The "Quantum Optical Technolo-gies” project is carried out within the International Re-search Agendas programme of the Foundation for PolishScience co-financed by the European Union under theEuropean Regional Development Fund. We would liketo thank W. Wasilewski for fruitful discussions and K.Banaszek for the generous support.
Disclosures
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Bi-photon amplitude and photon number covariance
Let us begin with the positive part of the blue pumpclassical electric field: E (+) p ( r , t ) = E p (cid:90) d k p, ⊥ d ω p A p ( k p, ⊥ , ω p ) exp[ i ( k p · r − ω p t )] , (5)where E p denotes the pump pulse amplitude, k p, ⊥ itstransverse wavevector and A p corresponds to the nor-malized slowly varying envelope of the pulse. From thewavefunction of the state generated in SPDC we shallconsider only the biphoton part [41], which can be de-noted as: | Ψ (cid:105) = (cid:90) d k s, ⊥ d k i, ⊥ d ω s d ω i Ψ( k s, ⊥ , ω s ; k i, ⊥ , ω i ) × ˆ a † ( k s, ⊥ , ω s )ˆ a † ( k i, ⊥ , ω i ) | vac (cid:105) , (6)where s, i indices correspond to signal and idler photons,respectively. For a crystal with length L and assumingthe z axis points along the pump beam’s central wavevec-tor, the biphoton amplitude Ψ( k s, ⊥ , ω s ; k i, ⊥ , ω i ) is givenby: Ψ( k s, ⊥ , ω s ; k i, ⊥ , ω i ) = N (cid:90) L/ − L/ d z (cid:8) A p ( k s, ⊥ + k i, ⊥ , ω s + ω i ) exp[ i ∆ k z ( k s, ⊥ , ω s ; k i, ⊥ , ω i ) z ] (cid:9) , (7)where the phase mistmatch ∆ k z ( k s, ⊥ , ω s ; k i, ⊥ , ω i ) is de-termined by the z components of the wavevectors: ∆ k z ( k s, ⊥ , ω s ; k i, ⊥ , ω i ) = k p,z ( k s, ⊥ + k i, ⊥ , ω s + ω i ) − k s,z ( k s, ⊥ , ω s ) − k i,z ( k i, ⊥ , ω i ) . (8)Integrating along z we get: Ψ( k s, ⊥ , ω s ; k i, ⊥ , ω i ) = N A p ( k s, ⊥ + k i, ⊥ , ω s + ω i ) sinc [ L ∆ k z ( k s, ⊥ , ω s ; k i, ⊥ , ω i )2 ] . (9)In our experiment we select a narrow range of k y around k y = 0 . Hence, we shall simplify the notation k α, ⊥ → k ( α ) x ; α ∈ { s, i } . Importantly, | Ψ( k ( s ) x , ω s ; k ( i ) x , ω i ) | isproportional to the probability of simultaneously gen-erating a signal photon with transverse wavevector k ( s ) x and wavelength λ s = 2 πc/ω s and an idler photon with k ( i ) x and λ i = 2 πc/ω i . Clearly, if the bi-photon termvanishes | Ψ( k ( s ) x , ω s ; k ( i ) x , ω i ) | = 0 for some coordi-nates ( k ( s ) x , ω s , k ( i ) x , ω i ) , the probability of observing a s + i p [nm]250255075 k ( s ) x + k ( i ) x [r a d / mm ] (b) = 10.02 ± 0.46[rad/mm] FIG. 3. (a) Second order photon number correlation func-tion g (2) ( k ( s ) x + k ( i ) x , λ i + λ s ) between signal and idler photons,in sum coordinates of spectral and temporal DoF. (b) Crosssection for λ i + λ s = 2 λ p . Data (blue crosses) is depictedagainst a gaussian fit (red curve) with Gaussian width pa-rameter σ = 10 . ± .
46 [ rad/mm ] . photon pair in ( k ( s ) x , ω s , k ( i ) x , ω i ) is equal to the prod-uct of marginal probabilities of observing a signal pho-ton at ( k ( s ) x , ω s ) and of observing an idler photon at ( k ( i ) x , ω i ) . Hence, | Ψ( k ( s ) x , ω s ; k ( i ) x , ω i ) | is proportionalto photon number covariance between signal and idlermodes Cov ( k ( s ) x , λ s ; k ( i ) x , λ i ) .For direct comparison with measured covariances wesum the modulus squared amplitudes in the selectedwavelength and wavevectors regions, respectively: | Ψ { λ s } , { λ i } ( k ( s ) x , k ( i ) s ) | = (cid:88) λ s ∈{ λ s } ,λ i ∈{ λ i } | Ψ( k ( s ) x , πc/λ s ; k ( i ) x , πc/λ i ) | , (10) | Ψ { k ( s ) x } , { k ( i ) x } ( λ s , λ i ) | = (cid:88) k ( s ) x ∈{ k ( s ) x } ,k ( i ) x ∈{ k ( i ) x } | Ψ( k ( s ) x , πc/λ s ; k ( i ) x , πc/λ i ) | . (11) Non-classical correlations
To quantify the non-classical character of signal–idler correlations we employthe second-order photon number correlation function de-fined for a single pair of signal–idler modes as: g (2) ( k ( s ) x , λ s ; k ( i ) x , λ i ) = (cid:104) n ( k ( s ) x , λ s ) n ( k ( i ) x , λ i ) (cid:105)(cid:104) n ( k ( s ) x , λ s ) (cid:105)(cid:104) n ( k ( i ) x , λ i ) (cid:105) . (12)For visualization we sum the coincidences and the nor-malizing factor over the uncorrelated directions to get: g (2) ( k + , λ + ) = (cid:88) k − ,λ − (cid:104) n ( k ( s ) x , λ s ) n ( k ( i ) x , λ i ) (cid:105) ( k + , k − ; λ + , λ − ) / (cid:8) (cid:88) k − ,λ − (cid:104) n ( k ( s ) x , λ s ) (cid:105) ( k + , k − ; λ + , λ − ) × (cid:104) n ( k ( i ) x , λ i ) (cid:105) ( k + , k − ; λ + , λ − ) (cid:9) (13)with k ± = k ( s ) x ± k ( i ) i , λ ± = λ s ± λ i and where we im-plicitly transformed the mean photon numbers to ± co-ordinates. The cross-section for degenerate wavelength λ i + λ s = 2 λ p has a Gaussian shape with width σ =10 . ± .
46 [ rad/mm ] . This width corresponds to thethe transverse wavevector mode size of SPDC emissionas σ k -mode = σ/ √ , where the √ factor comes from theJacobian of ( k ( s ) x , k ( i ) x ) → ( k ( s ) x + k ( i ) x , k ( s ) x − k ( i ) x ) trans-formation. Similarly, the Gaussian fit for k ( s ) x = − k ( i ) x cross-section gives σ λ -mode = 4 . ± . nm. If we traceover the idler (signal) mode, the remaining signal (idler)has a thermal photon count statistics with autocorrela- tion g (2) idler,auto = g (2) signal,auto = g (2) therm,auto ≤ ; hence, ac-cording to Cauchy-Shwarz inequality the upper classicalbound on the second order cross-correlation function is g (2) classical ≤ (cid:113) g (2) signal,auto g (2) idler,auto = 2 . Number of modes
We estimate the number of modes by consideringthe thoretical prediction for the biphoton wavefunction,given by Eq. (9). We numerically compute the wavefunc-tion in a range of experimentally observed wavelengthsand transverse wavevectors. The obtained tensor is re-shaped into a two dimensional matrix with a single di-mension corresponding to the spectral and transverse co-ordinates of a single photon. Singular value decomposi-tion of the resulting matrix yields the Schmidt coefficients { λ j } . The total number of modes is given by: M = ( ∞ (cid:88) j =0 λ j ) − ≈ ..
We estimate the number of modes by consideringthe thoretical prediction for the biphoton wavefunction,given by Eq. (9). We numerically compute the wavefunc-tion in a range of experimentally observed wavelengthsand transverse wavevectors. The obtained tensor is re-shaped into a two dimensional matrix with a single di-mension corresponding to the spectral and transverse co-ordinates of a single photon. Singular value decomposi-tion of the resulting matrix yields the Schmidt coefficients { λ j } . The total number of modes is given by: M = ( ∞ (cid:88) j =0 λ j ) − ≈ .. ..