Bonato et al. Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B A175
Phase control of a path-entangled photon state bya deformable membrane mirror
Cristian Bonato,1,2 Stefano Bonora,1 Andrea Chiuri,3 Paolo Mataloni,3,4 Giorgio Milani,3
Giuseppe Vallone,5,3,* and Paolo Villoresi1
1CNR-INFM LUXOR, Department of Information Engineering, University of Padova, Padova, Italy2Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, the Netherlands
3Dipartimento di Fisica, Università Sapienza di Roma, Piazzale Aldo Moro 5, Roma 00185, Italy4Istituto Nazionale di Ottica Applicata (INOA-CNR), L.go E. Fermi 6, 50125 Florence, Italy
5Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, Via Panisperna 89/A,Compendio del Viminale, Roma 00184, Italy
. INTRODUCTIONdaptive optimization of experimental parameters is anxtremely powerful tool for researchers. In optics, higher-rder material dispersion, broadband phase matchingonditions in inhomogeneous media, as well as the non-inear and thermal deformation to the pulse wavefront,ose difficulties in the achievement of the optimal condi-ions for the processes under study. The adaptive ap-roach, i.e., the use of suitable devices that may circum-ent the limits of conventional components by adaptingheir shape, has paved the way to several breakthroughsn the generation of quantum states via nonlinear opticalhenomena, as well as in the transformation in time andrequency of laser pulses up to the single optical cycle re-ime.
Adaptive optics was developed with the idea to act onortions of an optical beam to correct aberrations. The ac-ion is driven by the direct measure of the alterations, asxplored over a century ago in the case of astronomical in-trumentation by Hartmann [1]. Indeed, the initial appli-ations of adaptive optical devices were in astronomy be-ause of the possibility of correcting the wavefront of aeam gathered by a telescope, thus compensating the deg-adation due to atmospheric turbulence [2]. From thesenitial applications, adaptive optics has spread into differ-nt fields, like laser physics [3,4], biomedical imaging,nd vision [5]. Deformable mirrors and, in particular,embrane mirrors appear to be particularly interesting
ecause of their low loss, insensitivity to chromatism, andarge dynamics. Furthermore, these mirrors are cheapnd are characterized by a low power consumption.
More recently, deformable mirrors have been used in aew seminal experiments in quantum optics. In a first ex-eriment [6], a segmented michroelectromechanical mi-romirror was used to demonstrate that a coherent imagef a pure phase object can be obtained using the inter-eam coherence of a pair of spatially incoherent en-angled photon beams. In a second experiment [7], aembrane deformable mirror was used to demonstrate
he even-order aberration cancellation effect in quantumnterferometry. The adaptive mirror allowed a precise andlean implementation of selected optical aberrations, sohat it was possible to show experimentally that theecond-order correlation function for a pair of entangledhotons is sensitive only to odd-order aberrations. In bothxperiments, however, the deformable mirror was used as
static device, where a specific shape was dialed andhen kept fixed for the duration of the experiment.
In the present work we give a paradigmatic example ofhe potentialities of a membrane mirror for classicallosed-loop control of a two-qubit entangled optical state.n particular, we have used it to stabilize the phase of awo-photon state entangled in two optical paths per pho-on [8] subject to random fluctuations. The experimentas been realized by adopting a simplified version of thepparatus recently introduced to demonstrate the en-anglement of two photons in many spatial optical modesmultipath entanglement [9]). A stream of path-entangledhoton pairs propagates through an interferometric opti-al system in which random optical path length instabili-ies result in fluctuations of the relative phase of theuantum superposition state. We show that the simple
010 Optical Society of America
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A176 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 Bonato et al.
se of a deformable mirror in a closed-loop configurationllows us to reduce the state noise deriving from phase in-tabilities.
The compensation of phase fluctuations in the case ofuantum interferometry shows some differences from theorresponding classical cases. The typical emission ratesf the entangled photon sources are of the order of 105
airs per second, and currently used single-photon detec-ors saturate at �3·106 counts per second. Therefore in-egration times of the order of a second are typicallyeeded to reach a sufficient signal-to-noise ratio. This setsome limits on the time scale of the phase fluctuationshat affect the system.
Entangling two photons in different optical paths is anfficient way to create quantum states of light spanningigh-dimension Hilbert spaces, namely, qu-dit �d�2� andyper-entangled states. It has already been demonstratedhat multidimensional entangled states enable the real-zation of important quantum information tasks, such asell state analysis [10–12], superdense coding [13], secureuantum key distribution [14,15], and high fidelity one-ay quantum computation [16–19]. Besides other tech-iques adopted to generate path entanglement, the opti-al setup used in the present experiment uses photonairs emitted over the light cone of a nonlinear paramet-ic crystal pumped by a laser source. This technique rep-esents a useful resource for an efficient generation andistribution of entangled photon states, since it allowsne to maximize the emission of photon pairs for a givenalue of the pump power.
. EXPERIMENTn the experiment, entangled photons are generated withorizontal polarization by spontaneous parametric downonversion (SPDC) by a �-borate (BBO) Type-I nonlinearrystal pumped by a continuous wave (cw) ultravioletUV) laser beam (wavelength �p=266 nm). Couples ofhotons are emitted at degenerate wavelength �=2�p532 nm and selected by two interference filters withandwidth ��p=5 nm. By virtue of momentum conserva-ion, the two photons are emitted with uniform probabil-ty distribution along the external surface of a cone, asoted, with photon A �B� emitted along the up (down)ide. A positive lens LP is then used to transform the coni-al emission into a cylindrical one. In the present experi-ent (see Fig. 1) two pairs of opposite correlated direc-
ions are collected by four integrated systems, each giveny a GRaded INdex (GRIN) lens glued to a single modeber [9,20]. Then the four integrated systems, pre-alignedo maximize photon coincidences, are glued to a four-holecreen, building in this way a single compact device whichan be used to study the effects of photon entanglement.
The entangled state deriving from the selection of twoairs of SPDC modes is expressed as
������ =1
�2����A�r�B − ei��r�A���B�, �1�
here ��� ��r�� refers to the left (right) mode of the corre-ponding photon [9]. The radiation coupled by each GRINens and travelling through the corresponding optical fi-
er is injected by a collimator into the interferometric ap-aratus shown in Fig. 1, where the left and right modeswith waist �1 mm) belonging to the A and B modes are
ixed on a common beam splitter (BS). This configura-ion allows us to overcome the mechanical instabilities ofhe apparatus, since any mirror or BS fluctuation affectsoth photons in the same way and does not influence theelative phase of the quantum state. On the other handhe phase � is strongly affected by the intrinsic thermalnstabilities of the optical fibers. Since the two events��A�r�B and �r�A���B assume the same phase of the laseream through the SPDC process, their relative phase isue to the differences in fiber and bulk optical paths, Spe-ifically �=�+2� /��rA+�B−�A−rB�, where �A ��B� and rArB� are, respectively, the left and right paths of the pho-on A �B�.
The BS action on the input modes ��� and �r� for bothhotons can be written as
����j →1
�2�����j + i�r��j�
�r�j →1
�2��r��j + i����j�� j = A,B, �2�
here ���� and �r�� are the output modes.The state corresponding to the BS output is
������� =1
�21 + ei�
2�����A�r��B − �r��A����B�
+ i1 − ei�
2�����A����B + �r��A�r��B� , �3�
here � depends on the path length difference betweenr�A and �r�B. Each photon travels through a Glan–Taylorolarizer that selects the horizontal polarization and is
ig. 1. (Color online) Experimental setup. The SPDC sourceonsists of a BBO Type I crystal pumped by a UV laser beam.he parametric radiation, given by four k optical modes, is col-
ected by a corresponding number of integrated systems of GRINenses and single mode fibers and injected into a two-arm inter-erometer. Polarization restoration of the photons is performedy proper � /4 and � /2 wave plate sets after fiber transmission.or each photon, the right ��r�� mode is spatially matched on theS with the left ����� mode. A translation stage allows fine adjust-ent of the left optical paths �x1 to obtain temporal indistin-
uishability (and thus interference) between the modes. The de-ormable mirror is placed on the right mode side and allowshanging the state phase. Two single photon detectors are placedfter two horizontal (Glan–Taylor) polarizers at the output portsf the BS, one on the A and the other on the B side.
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Bonato et al. Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B A177
hen focused on a large-area single photon detector100 �m100 �m wide sensitive area) by a lens (focalength 5 cm). The use of a large-area detector allows us toeglect any beam wandering effect caused by the defor-ation of the adaptive mirror. The rate of the coincidence
vents at ports ����A and �r��B is
C��� =N0
4�1 + cos �� =
N0
2cos2
�
2, �4�
here N0 represents the generated pairs per second and he integration time.
As noted, in the measurement setup shown in Fig. 1ny temperature variation modifies the optical length ofhe fibers, resulting in time dependent phase fluctuations�t� between the two events ���A�r�B and �r�A���B. In our ex-eriment, as in most quantum interference experiments,he signal is given by the average number of coincidencevents integrated over a time :
C�t� =N0
2 �t−/2
t+/2
dt� cos2��t��
2 . �5�
veraging over random phase fluctuations, the visibilityf the interference fringes is reduced [21]. For example, inhe case of a Gaussian distribution of the phase noisewith variance �2), the average coincidence rate is
C = C0�1 + cos �e−�2/2 . �6�
ntegration over time has a further consequence, actings a low-pass filter:
C =N0
2 � d�C̃���H,t���, �7�
here C̃��� is the Fourier-transform of the time-ependent coincidences C���t� and H,t���ei�tSinc�� /2�. Therefore, higher-frequency fluctuationsre averaged and decrease the visibility of the interfero-etric fringes, while low-frequency fluctuations show up
n the temporal evolution of the output coincidence rate.xperimental data for a 2 s integration time are shown
n Fig. 2(left). In the present experiment the interferom-ter was located within a thermally isolated polystyreneox to reduce the fast temperature fluctuations. This op-ration then cuts off fast noises. The coincidence count
(a)
ig. 2. (Color online) (left) Coincidence counts in interference cosolated box and when no phase control is activated. Each experimoincidence rate shows temperature fluctuations within a time scn condition of no interference. Since the counts follow a Poissonump power instabilities.
ate shows fluctuations over time, with a characteristicime scale of the order of tens of seconds. The typical vis-bility of the interference fringes is 75%.
Let us suppose we want to apply a general phase shift� to the interferometer in order to create a particularuantum state. The state phase will be expressed as0�t�+��, where �0�t� is a stochastic function describinghe intrinsic fluctuations of the interferometer. In order toerform a measurement, such �0�t� should be compen-ated to zero. In other words, first we need to take the sys-em to a coincidence maximum ��0=0� and stabilize it,hen we can apply the required phase-shift ��. To takehe coincidence rate to a maximum and keep the quantumtate stable over time, we compensated phase fluctuationsith a deformable mirror placed in one arm of the inter-
erometer and controlling the length difference betweenhe �r�A and �r�B optical paths. An optical path-length dif-erence corresponds to a phase shift �→�+2� /�,here � is the wavelength of the two photons.
. DEFORMABLE MIRRORcustom deformable mirror was used for phase compen-
ation (see Fig. 3). It consists of an aluminized nitrocellu-ose membrane that is deflected by the electrostatic pres-ure applied through a series of pads placed 100 �melow the membrane [22]. The electrodes were controlledy a high voltage �0–265 V� driver that can indepen-ently address the actuators. Such deformable mirrorsre usually used for aberration compensation [23] or, inome cases, for the compression of ultrafast pulses [4].he mirror design was such that two square areas (size.4 mm by 1.4 mm) of the membrane behave like flat, par-llel mirrors with controllable relative displacement dsee Fig. 3). This allowed us to control the relative phasehift of the two photon beams, which were spatially a fewillimeters apart (beam diameter � 1 mm).The best membrane shape for carrying on this task is
ectangular, because the membrane boundaries are paral-el to the planes. Because of the strong crosstalk betweenhe deformation caused by the single electrodes we had toompute the square area position that would allow a largenough d displacement while keeping the flatness andarallelism of the planes suitable for the experiment. Areliminary study of the deformation M�x ,y� was carriedut solving the Poisson equation for membranes under anpplied voltage [24]:
(b)
ns detected by the interferometric setup enclosed in a thermallypoint represents the number of detected coincidences in 2 s. Thethe order of hundreds of seconds. (right) Coincidences measuredtistic, the variation shown in the left picture cannot arise from
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A178 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 Bonato et al.
�M�x,y� = −1
Tp�x,y�, �8�
here p is the electrostatic pressure,
p�x,y� =�0
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h �2
, �9�
is the mechanical tension of the membrane, and h theembrane-to-electrode distance. The simulations were
arried out through the finite element method, which al-owed us to design the deformable mirror. In order to ad-ress such a membrane deformation we used two pairs oflectrodes with the size of 1.4 mm by 15 mm spaced.4 mm apart. The electrodes were empty in the centralart in order to completely remove the transverse radiusf curvature [25]. In order to compute the voltages neces-ary for controlling the membrane deformation we mea-ured the shape obtained applying the maximum voltageo each actuator (influence function) using an interfero-etric technique. Under the hypothesis of linearity, valid
or small membrane deformations, we combined the influ-nce functions to determine the voltages that create thewo parallel planes with a controllable displacement dinimizing the root-mean-square (rms) flatness error. To
ompute the position of the planes we used the followingrocedure: the position of plane 1 was kept fixed and theosition of plane 2 was increased until the rms deviationrom a plane parallel to a reference was smaller than ahreshold value of 30 nm rms. Then we repeated this al-orithm, changing the position of plane 1. Following thistrategy in the first instance we determined the optimallane distance, which was 11.2 mm. Then we character-zed the performance of the deformable mirror as phasehifter. The maximum displacement achievable was �00 nm with 8 bit control resolution. We measured overhe whole displacement range the average rms deviationrom the reference plane, which was of 26.0±1.5 nm forhe fist plane and 23.8±1.5 nm for the second. Moreover,he average parallelism of the two planes was measuredo be 18±4 �rad.
The deformation of the membrane for different appliedoltages is illustrated in Fig. 4. The thin portion of theurves show the shape of the membrane, whereas the twohick segments in each curve are used for the relativehift of the two beams.
Fig. 3. (Color online) Scheme of the deformable mirror.
. MEASUREMENTSefore applying a control algorithm, the system was char-cterized scanning with the deformable mirror over aime scale much shorter than the instability time. Start-ng from a coincidence maximum, was increased byteps of � /4 (30 steps in total, over a 60 s time). The co-ncidence count rates exhibit a sinusoidal behavior as aunction of the externally applied phase (see Fig. 5), withmaximum of 800 and a minimum of 100 coincidences ins (visibility 78±2%). It is worth nothing that the phase
hift introduced by the mirror deformation does not de-rade the quality of the beam as seen by the constantalue of the visibility for several optical cycles.
In order to select a proper phase � for the quantumtate we assume to take the coincidence rate to the maxi-um, which we set as �0=0, and then apply the needed
hase-shift �. Therefore, our problem can be reduced tohat of taking the interferometer into the �0=0 state,aximizing the number of coincidences. From the pre-
iminary characterization of the system parameters, wessume to know the expected value of coincidences perecond of the maximum Cmax.
In a typical closed-loop experiment, a measurement iserformed on the system at each step, and the measure-ent result is used as a parameter to drive the controller.
ig. 4. (Color online) Measurements of the plot of the cross sec-ion X�-X� of the membrane for four different values of relativeisplacement. The flat portions of the membrane used for phasehifting of the two beams are shown with thick lines.The mea-urement was carried out with an interferometric technique.
ig. 5. (Color online) Mirror calibration. Measurement of coin-idence counts as a function of mirror deformation. This allowss to calibrate the deformation in terms of the state phase. In theraph each step corresponds to a � /4 phase shift.
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Bonato et al. Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B A179
n our case, we estimate the phase of the quantum statey measuring the coincidence rate and comparing it to itsaximum value. To remove the phase ambiguity (differ-
nt phase values giving the same coincidence rate) weompare this value with the coincidence rate obtained in-reasing the phase by a small amount. In other words, westimate the phase � of the quantum state by measuringhe coincidence rate C��� and its derivative with respect tohase dC��� /d�. Then we use this estimate of � to guesshe phase we need to apply in order to take the system tohe maximum. We take a threshold value T as the valuebove which the coincidence rate can be assumed to be athe maximum. For example, we can take the threshold toe one standard deviation below the expected maximumalue: T=Cmax−�Cmax. In details, the maximization algo-ithm we propose works as follows:
1. (i) if the coincidence rate C is above T /2 and below T,hen we can apply exactly the phase shift we need to geto the closest maximum: ��=2 arc cos�C /T. Knowingnly C we cannot determine the sign of ��, since we coulde either in the ascending or descending side of the maxi-um. Therefore we use a double-step procedure: to deter-ine on which side of the maximum we are, we apply a
mall phase shift. If the number of coincidences increases,e are on the ascending side and we apply +��. If theumber of coincidences decreases, then we apply −��.2. (ii) if the coincidence rate is below T /2, then we take
he system into case (i), shifting the phase by �.3. (iii) above the threshold T the maximization proce-
ure is successful and the mirror remains in a fixedosition.
he results shown in Fig. 6 demonstrate that the deform-ble mirror can compensate the slow temperature fluc-uations, causing the coincidence variation given in Fig.. The maximization procedure rapidly converges to a co-ncidence value above the threshold. The standard devia-ion of the stabilized data is �=36. This demonstrateshat the active control induces only a modest increase ofhe error of the average coincidence with respect to aoissonian distribution.In Fig. 7 the discrete Fourier transform of the data
ith and without adaptive compensation is plotted. Thentensity of the frequency components below 1 mHz, dueo the slow phase fluctuations shown in Fig. 2, is clearlyeduced.
In our approach we stabilized the phase at �=0 thatepresents the position with the maximum signal/noise
ig. 6. (Color online) Coincidence counts with activated deform-ble mirror. The first point represents the initial random phase.he optimization algorithm rapidly controls the phase state toaximize the coincidences and to keep the phase constant to �0.
atio. The mirror can now be used to fix an arbitraryhase state. According to the calibration we can obtainhe precise phase state variation in terms of the mirroreformation. Thus, after a preliminary stabilization with=0 the mirror can be used to deterministically generatestate with arbitrary phase �1. The typical time scale of
he stability of the new phase �1 is given by nearly 200 sas shown in Fig. 2(left)].
It is worth noting that the control scheme we propose isifferent from quantum feedback control. In quantumeedback control, a single quantum system is subject toontrol to force its dynamics according to some require-ent. Measurements used to provide input information
or the control are quantum: the result is probabilisticnd the measurement process itself has a back-action onhe state, in the sense that it projects the state to one ofhe eigenvectors of the measurement operator. In our casehe measurements we are performing on the system aren a sense “classical,” since we repeat the experiment onifferent copies of the same input quantum state travel-ng through the optical interferometric system, having ac-ess to the mean value of the result. We do not have aingle quantum state that evolves under the influence ofhe external environment. We have a source that emits atream of quantum states and that evolves in an unpre-ictable way over time, and we want to keep it stable.oreover, in photon-counting experiments, the currently
vailable photodetectors absorb photons, so that eachingle quantum state produced is destroyed in the mea-urement process and cannot be used for further opera-ions.
Another important aspect of our experiment regardshe involved time scales. In photon counting experimentshe number of counts is a Poissonian process, and a suffi-ient number of counts N needs to be collected in order toave a good signal-to-noise ratio (which scales like 1/�N).ntangled photon sources based on SPDC typically pro-ide a few thousand pairs per second, which means thato reach a signal-to-noise ratio around 1% one needs toollect counts for a certain amount of time (which isypically of the order of 1 s). As we described in Section 2,his temporal integration sets the time scale for the ex-eriment: fluctuations with time scale faster than areveraged and result in a reduced visibility of the fringes,
ig. 7. (Color online) Discrete Fourier transform of the datahown in Figs. 2 and 6. The Fourier components are normalizeduch that the sum of their squares is equal to 1. Except for theonstant component at 0 Hz frequency, the frequency compo-ents below 2 mHz (see inset) are filtered out by the action of theeformable mirror.
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A180 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 Bonato et al.
hile slower oscillations manifest themselves in the fluc-uation of the coincidence rate over time. The compensa-ion technique we report here can get rid of these sloweructuations, resulting in a more precise and reliable set-ing of the interferometer phase. As noted, the fast fluc-uations are erased by the thermal box.
It is worth noting that in order to stabilize the phase its not possible to replace the deformable mirror with aigid mirror moved by a piezoelectrical stage. In fact, inhe configuration used in the present experiment allowingvery high mechanical stability, two modes (�r�A and �r�B)
mpinge on the same mirror. By moving the piezo, nohase would be changed since the length of both modes isaried. Then, only a device able to independently vary theengths of the paths �r�A and �r�B can be used in this case:he deformable mirror is precisely this device.
. CONCLUSIONSn conclusion, we experimentally demonstrated a feed-ack control of the states generated by a source of en-angled photon pairs by means of a custom-design de-ormable mirror. We believe this technique can bextremely beneficial in quantum interference experi-ents, since it decouples the quantum state produced by
he source from the random phase fluctuation induced byhe environment.
CKNOWLEDGMENThis work was partially supported by Strategic-Research-roject QUANTUMFUTURE of the University of Padova.
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