How to create and detect N-dimensional entangled photons with an active phase hologram
Martin Stuetz, Simon Groeblacher, Thomas Jennewein, Anton Zeilinger
aa r X i v : . [ qu a n t - ph ] J un How to create and detect N-dimensional entangled photons with an active phasehologram
Martin St¨utz, Simon Gr¨oblacher,
1, 2, ∗ Thomas Jennewein, and Anton Zeilinger
1, 2 Faculty of Physics, University of Vienna, Boltzmanngasse 5, A–1090 Vienna, Austria Institute for Quantum Optics and Quantum Information (IQOQI),Austrian Academy of Sciences, Boltzmanngasse 3, A–1090 Vienna, Austria
The experimental realization of multidimensional quantum states may lead to unexplored andinteresting physics, as well as advanced quantum communication protocols. The orbital angular mo-mentum of photons is a well suitable discrete degree of freedom for implementing high-dimensionalquantum systems. The standard method to generate and manipulate such photon modes is to usebulk and fixed optics. Here the authors demonstrate the utilization of a spatial light modulator tomanipulate the orbital angular momentum of entangled photons generated in spontaneous paramet-ric downconversion. They show that their setup allows them to realize photonic entanglement of upto 21 dimensions, which in principle can be extended to even larger dimensions [1].
The ability to manipulate the light modes of single photons is a crucial and necessary tool for any realization ofhigh-dimensional quantum optics experiments based on the orbital angular momentum of photons [2, 3, 4, 5, 6].In addition, many other schemes of quantum optics experiments such as quantum teleportation [7, 8, 9], photonicquantum computing [10, 11, 12] and higher-order photonic entanglement [13, 14] directly rely on photon interference,which requires to overlap photon beams with high quality mode matching. The most common method is to use bulkoptical components such as lenses, mirrors and microfabricated holograms in these setups. However, as such opticalcomponents are static, each of them can only perform a predefined manipulation of the photon beam. Programmablediffractive optics can dramatically extend the capabilities of optical configurations by allowing the generation ofarbitrary and flexible light patterns in real time [15, 16, 17].Our work makes original use of a spatial light modulator (SLM) as an active transformation in a quantum opticsexperiment for the manipulation of photons coming from spontaneous parametric downconversion, which have signif-icantly less temporal and spatial coherence than photons from a laser. The SLM manipulates one photon of a photonpair entangled in its orbital angular momentum (Laguerre-Gaussian functions) of the light mode and hence replacesthe fixed phase-singularity holograms used in preceding experiments [2, 3, 4, 18, 19, 20, 21]. We further demonstratethe possibility of generating 21-dimensional quantum states.The SLM is an array of pixels (in our case 1024 x 768 with a total size of 19.5 x 14.6 mm ) acting as individuallytunable retardation wave plates which can imprint a spatial phase modulation on a light beam. The SLM is madeof nematic liquid crystal pixels whose birefringence is controlled by an external electric field. Our SLM is used inreflection mode, has a refresh rate of 75 Hz and its diffraction efficiency into the first order, which corresponds to thedesired output, is approximately 60%. For technical reasons we were limited by the nonideal modulation depth of theSLM, which has a maximal phase shift of 1 . π for our wavelength of 702 nm [22], a reduced filling ratio of the pixelsof 90% and a reflectivity of 75%.In order to generate the patterns to be applied onto the SLM we implemented an algorithm on a computer in a matlab environment. The function generating the output value for each pixel on the SLM isSLM Pixel (cid:18) x · x max − x min , y · y max − y min (cid:19) == mod h (angle(LG( x − x , y − y , l, z, w , z r )) (1)+ π · λ · f SLM · (cid:0) ast · ( x − x l ) + ( y − y l ) (cid:1) (2)+ x · k x + y · k y ) , π i · π , (3)where λ is the wavelength of the photons impinging on the SLM. This function produces a normalized 8 bits grayvalue (0 – 255) picture over a grid corresponding to the pixel positions of the SLM. The standard range in meters forthe values of x and y are { } and { } , respectively, matching the size of the SLM.Expression (1) of the function SLM Pixel calculates the complex amplitude LG ,l = LG( x − x , y − y , l, z, w , z r )of the desired Laguerre-Gaussian mode with the following parameters: z is the propagation direction of the light field, w is the beam waist, z r is the Rayleigh length and x and y are the distance of the phase singularity from the origin.The function ”angle” is a matlab function which returns the complex phase angle of the evaluated expression. FIG. 1: Experimental setup to demonstrate the manipulation of entangled photons with a spatial light modulator. The opticallynonlinear crystal (BBO) is pumped with an ultra-violet Ar + laser, which generates pairs of photons, entangled in their orbitalangular momentum. In the idler beam the photons are controlled with a spatial light modulator and both signal and idlerbeam are analyzed with fixed phase holograms. The phase hologram of the spatial light modulator is actively controlled witha computer. The index l ∈ [ . . . , − , , +1 , +2 , . . . ] is the azimuthal mode index, with 2 πl being the change in phase of a closedpath around the propagation axis. This phase change gives the Laguerre-Gaussian modes a helical wave front, wherethe angular momentum per photon can only take integer multiples of ¯ h [23]. A phase hologram with l = 0, imprints aphase e − ilθ on the incoming beam, which results in a phase singularity. Such a hologram with one singularity ( l = ± l index by 1 (hologram with l = +1) orlowering it by 1 (hologram with l = − f SLM is the focal length of the lens term in millimeters. Additionally, x l and y l are the distance ofthe lens term from the origin in x and y direction, respectively. The parameter ast is a weighting factor between the x and the y components of the lens term, which is used to compensate for any astigmatism. In the ideal case ast = 1.Furthermore, expression (3) of the function SLM Pixel corresponds to a beam deflection, like a tilted mirror, wherethe parameters k x and k y define an inclined plane and are used to change the diffraction direction of the light beam,with k x ≈ πθ x /λ in the approximation of a small diffraction angle θ x (and analogous for k y ).Effectively the SLM represents three adjustable diffractive optical elements: a phase-singularity, a tunable lens anda tunable mirror, which are readily usable in experiments for achieving mode matching and beam pointing of a lightbeam.The experimental versatility of the SLM is demonstrated with the manipulation of entangled photon pairs created byfocusing an Ar + laser with a wavelength of 351 nm and a power of about 95 mW into a β -barium-borate crystal (BBO)with type-I mode matching (see Fig. 1 for the experimental setup). Due to spontaneous parametric downconversion(SPDC), pairs of photons entangled in their orbital angular momentum are produced [2, 24, 25], called the signaland the idler beam, both at a central wavelength of 702 nm and a bandwidth of 2 nm full width at half maximum.The photons in the signal beam pass a lens L ( f = 250 mm) to guarantee that they effectively couple into the finalfibers. In contrast, the idler beam is widened using lenses L ( f = −
30 mm) and L ( f = 100 mm) to optimallyilluminate the SLM. With another lens L ( f = 750 mm) the beam is focused again after being reflected from theSLM. In order to discriminate between the +1, -1 and 0 modes we use a probabilistic mode analyzer (for a detaileddescription see Ref. [2] and [3]).In order to achieve a high pair coupling efficiency of both the photons from the SPDC into the optical fibers, whichis required for maximal coincidence rates as well as perfect correlation in the orbital angular momentum, the signaland idler beams must have well matched spatial mode functions. A striking advantage of the SLM over fixed optics FIG. 2: Images (a) show how the astigmatism of the lens term is compensated with the parameter ast. For ast = 1 .
029 an LG , is obtained (on the lower right), whereas for all other values general superpositions of Hermite-Gaussian modes are produced.(b) Pictures of the resulting mode for different values of x l and y l , which determine the position of the lens term on the SLM.The camera is kept at the same position throughout the measurements. is the fact that now the idler beam can be easily matched to the mode of the signal beam via tuning the lens termon the SLM. The dependence of the spot size of the idler beam at the crystal on the focal length f SLM of the SLM ismeasured by sending a laser beam in reverse through the idler path, and measuring the mode diameter with a chargecoupled device (CCD) camera. The focal length required to achieve the same spot size for the signal and idler beamsis found to be f SLM = 940 mm.Since the beam is reflected not perfectly orthogonal from the SLM, the lens term applied to the SLM shows someastigmatism. Hence the resulting mode of the reflected light beam will no longer be a pure first order LG mode buta general superposition of Hermite-Gaussian modes. This is compensated by tuning the parameter ast which putsan astigmatism on the lens term, which also allows to perform experiments of photons involving Hermite Gaussianmodes [26]. In our setup the ideal value to obtain clean LG modes was ast = 1 .
029 throughout the experiment [seeFig. 2(a)].A further, important parameter is the position of the displacement of the hologram in the x direction x and inthe y direction y with respect to the beam center. Tuning this parameter can be used to obtain superpositions ofdifferent LG modes, as was done in order to violate a Bell-type inequality [3] and to obtain a secret key in a quantumcryptography scheme [20]. Figure 2(b) shows the resulting mode for different values of x and y .The most important aspect of our experiment is to show that entangled photons from downconversion can easily bemanipulated with the SLM device [27]. We therefore place a CCD camera with a long-pass filter (cutoff at 800 nm)in the idler beam and apply transformations between LG modes with different l indices on the SLM. As the majorityof the downconversion photons are produced in the LG , state, the intensity profiles on the CCD correspond to the l index of the transformation on the SLM. Figure 3(a) shows images of the idler beam recorded for various values of l ( f SLM = 131 mm and k y = 2 × rad/m). The mode with the highest index shown is l = 10, which can in principlebe used for quantum communication and information protocols with dimension of up to 21. The transformations andtherefore the accessible Hilbert spaces for quantum optics experiments are in principle not limited.As a test of the SLM we analyze the perfect correlations of the entangled photon pairs in their orbital angularmomentum. The signal beam is transformed with conventional quartz holograms and the idler beam with the SLMscheme. These correlations are the crucial ingredients for any quantum information processing scheme such as quantumcryptography. The expected correlations for a maximally entangled qutrit (a trinary quantum system; l=-1,0,+1)state for four different couplers on the signal and idler side are shown in Fig 3(b). Various phase holograms calculatedfor the desired transformations were displayed on the SLM and the experimentally obtained coincidences for thedifferent couplers are also shown in Fig. 3(b). The entangled character of the produced pairs can be clearly seenand the ability of the spatial light modulator to manipulate the pairs on the single photon level is proven. Thereason for the reduced visibility of the observed correlations in the trinary quantum systems is mainly due to residualmismatching of the two SPDC modes, as well as the nonperfect initial correlation of the photon pairs produced in thedownconversion process.We have shown a scheme of manipulating entangled photons produced by spontaneous parametric downconversion FIG. 3: (a) Pictures of the downconverted light, transformed into higher-order modes by corresponding phase hologramson the SLM. (b) Coincidence counts of the downconverted beams per 10 s. With the SLM a transformation is performedand the correlations are observed via probabilistic mode analyzers. The individual settings are coupler 2, l = 0; coupler 3, l = −
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