Experimental simulation of decoherence in photonics qudits
B. Marques, A. A. Matoso, W. M. Pimenta, A. J. Gutiérrez-Esparza, M. F. Santos, S. Pádua
aa r X i v : . [ qu a n t - ph ] N ov Experimental simulation of decoherence in photonics qudits
B. Marques , , ∗ , A. A. Matoso , W. M. Pimenta , A. J. Guti´errez-Esparza , M. F. Santos , and S. P´adua Departamento de F´ısica, Universidade Federal de Minas Gerais,caixa postal 702, 30123-970, Belo Horizonte, MG - Brazil and Department of Physics, Stockholm University, S-10691 Stockholm, Sweden
We experimentally perform the simulation of open quantum dynamics in single-qudit systems.Using a spatial light modulator as a dissipative optical device, we implement dissipative-dynamicalmaps onto qudits encoded in the transverse momentum of spontaneous parametric down-convertedphoton pairs. We show a well-controlled technique to prepare entangled qudits states as well as toimplement dissipative local measurements; the latter realize two specific dynamics: dephasing andamplitude damping. Our work represents a new analogy-dynamical experiment for simulating anopen quantum system.
Introduction
In most cases, when a quantum system interacts with its environment it undergoes decoherence [1], to wit, thesystem-environment interaction “spoils” the state of the system by decreasing its capacity for quantum interference,which is essential for standard quantum information processing. Decoherence is so far one of the major obstaclesfor implementing quantum computation processes in real systems. Despite such nuisance, recent works have shownprocedures to manipulate the system-environment interaction or the information leaked to the environment in suitableways depending on the specific goal: e.g. , estimation of quantum noise [2], protection of coherence and/or entanglement[3], universal quantum computation [4], quantification of entanglement [5–7] and entanglement concentration [8].An interesting study for exploring quantum devices is the experimental simulation of complex dynamics on con-trollable quantum systems of simple implementation [9]. These simulations allow for a better control and, therefore,understanding of the details leading to decoherence as well as the mechanisms underneath the system-environmentexchange of excitation and/or information. Recent works on dynamics simulations have been performed in diversequantum systems such as optical interferometers with polarization-entangled photon pairs generated by spontaneousparametric down-conversion (SPDC) [10–13], spin- nuclear states of carbon atoms accessed by magnetic nuclearresonance [14], and trapped ions [15–18].In particular, the simulation of open system dynamics for qubits (two-level quantum systems) has already beenobserved in many different experiments and has been connected to the observation of phenomena such as entanglementsudden death [10] and Non-Markovian dynamics [19–21], to name a few. The extension of a similar analysis to qudits(d-level quantum systems) presents both a wider range of phenomena to observe and possible applications to exploreranging from full local protection of entanglement to dynamical precursors of entanglement sudden death that are notpresent in pair of qubits. However, the simulation of quantum open systems in qudits is not as easy to implement asin qubits which justifies the rarity of such results in the literature.In this paper we report an experimental technique for simulating decoherence in the the dynamics of a qudit. Ourqudits are encoded in the transverse component of the linear momentum of photon pairs generated by SPDC. Thequantum system is defined in terms of the path entangled photons, namely, down-converted photons propagatedthrough paths outlined by optical diffractive elements (multi-slits) [22]. In particular, we simulate two types ofdecoherence mechanisms namely dephasing [23] and amplitude damping [24], by means of a spatial light modulator(SLM) employed to implement operations on the qudit states. A wide variety of applications extend the use of SLMsfor controlled manipulation of photonic quantum systems encoded, for instance, in polarization [25], in orbital angularmomentum [26, 27] or in transverse momenta of the photons [28–34].This article is organized as follows: in section I we summarize the concepts of open system dynamics; the statepreparation and the dephasing implementation are discussed in sections II and II A, respectively; section II B presentsthe amplitude damping implementation. We summarize and conclude the article in section II B. I. OPEN QUANTUM SYSTEMS
In this section we present a brief review of the theory of open quantum systems in order to outline this work. Asystem ( S ) interacting with an environment ( E ) is described by the Hamiltonian [11] H = H S ⊗ I E + I S ⊗ H E + H int , (1)where H S and H E are the system and environment Hamiltonian operators, respectively; H int is the system-environment coupling Hamiltonian, and I S ( I E ) is the system (environment) identity operator. We can describeonly the system evolution by the equation of motion given by˙ ρ S = − i ~ Tr E [ H, ρ SE ] , (2)where ρ SE is the total ( S + E ) density operator.The system evolution for the system-environment coupling can always be expressed as an unitary evolution andthe total density matrix can be written as ρ SE = U SE ( t ) ρ SE (0) U † SE ( t ) , (3)where U SE ( t ) = exp ( − iHt/ ~ ) and ρ SE (0) is the density matrix of the initial state. In the particular case where weconsider the initial state as a product state between the system and the environment, ρ SE (0) = ρ S (0) ⊗ | i E h | E ,the effective evolution of the system is given by: ρ S ( t ) = T r E h U SE ( t ) ρ SE (0) U † SE ( t ) i , (4)= X ǫ ( E h ǫ | U SE ( t ) | i E ) ρ SE ( E h | U † SE ( t ) | ǫ i E ) , where {| ǫ i} form an orthonormal basis for the environment. This evolution can be expressed only in terms of operatorsacting on S in the following form ρ S ( t ) = X ǫ K ǫ ( t ) ρ S (0) K † ǫ ( t ) , (5)where the operators K ǫ ( t ) = E h ǫ | U SE ( t ) | i E (6)are the so-called Kraus operators [35] and define a trace preserving positive map: K † ǫ ( t ) K ǫ ( t ) > P ǫ K † ǫ ( t ) K ǫ ( t ) = 1, T r S ( ρ S ( t )) = 1. Note that the Kraus operators are not uniquely defined because there are many bases for describingthe environment. As a consequence, we deal with the equivalent operators from different sets, which originate differentdecompositions of the same resulting density matrix.Under certain well established circumstances known as the Born-Markov approximations, the evolution of thesystem can also be expressed in terms of a time continuous Master equation, given by [36]˙ ρ S = − i ~ [ H S , ρ S ] + N − X j =1 γ j (cid:18) A j ρ S A † j − ρ S A † j A j − A † j A j ρ S (cid:19) , (7)where ρ S is the system density operator, A j are the so-called Lindblad operators and γ j is a non-negative quantitywhich has dimensions of the inverse of time if A j is dimensionless. The first term on the right side of the masterequation represents the unitary part of the dynamics generated by the Hamiltonian H S . In this case, an intuitive setof Kraus operators is given by K = 1 − iH S dt − P j γ j dt A † j A j and K j = p γ j dtA j where γ j dt ≪ dt are ignored. This so-called unravelling of the Master equation is associated to the Quantum Trajectoriesmethod where K and K j are respectively known as the No-Jump and Jump operators. This method is connectedboth to an alternative way to calculated the evolution of the system on average as well as a direct way to infer itsevolution at any single realization where a sequential measurement of the state of the environment is performed.In this work, as mentioned above, we focus on two types of open quantum system evolution: dephasing andamplitude-damping. In the following sub-sections we outline such evolutions. A. Dephasing
In a dephasing dynamics the system loses coherence due to the system-environment interaction without any popu-lation exchange. This occurs when a noisy environment couples to a system [23]. We can describe this dynamics fora system with dimension d using the Kraus operators K d = I d , (8) K j = d − X i =0 e iπδ ij | i i h i | , (9)where 0 ≤ j ≤ d − I d is the identity operator for a system of dimension d . The system evolution in a dephasingdynamics can be obtained from ρ S ( { p i } ) = d X i =0 p i K i ρ S K † i , (10)where { p i } is the set of time dependent parameters that represent the weight of each Kraus operator. Writing thesystem density operator in a matrix form such that h i | ρ | j i = ρ ij , the dynamics for each matrix density element canbe obtained. The diagonal elements are constant, ρ ′ ii = ρ ii , ∀ i, (11)and the off-diagonals elements evolve as ρ ′ ij = (1 − p i − p j ) ρ ij ∀ i = j, (12)showing the system decoherence. The experimental implementation can be simplified if we consider the particularcase p i = p/ ≤ j ≤ D −
1, inducing a single-parameter dependence in the system evolution. Thus, the offdiagonals elements are ρ ′ ij = (1 − p ) ρ ij ∀ i = j. (13)Dephasing dynamics implementation is presented in section II A. B. Amplitude Damping
Damping dynamics represents the dissipative interaction between the system and its environment. A commonexample is the loss of photons from a cavity into a zero-temperature environment of electromagnetic-field modes [37].This dynamics can be described using the master equation˙ ρ = 2 γaρa † − γρa † a − γa † aρ, (14)where a ( a † ) is the operator annihilation (creation) of a photon inside the cavity and γ is the decay rate inside thecavity.One approach to describe the system dynamics is based on the theory of quantum trajectories [38–40], which consistin monitoring the system’s environment. Environmental monitoring during a time interval { t, t + δt } indicates whetheror not a loss of excitation (a quantum jump) can occur. If no loss of excitation occurs, the system evolves without aquantum jump; thus, this evolution is given by ρ S ( t + δt ) = e − i ~ H eff δt ρ S ( t ) e i ~ H eff δt , (15)where H eff = i ~ γa † a/
2. On the other hand, if the system loses an excitation, the system evolves with a quantumjump ρ s ( t + δt ) = δtδp aρ s ( t ) a † , (16)where δp = δtγ Tr[ aρ s ( t ) a † ] is the probability that a quantum jump occurs within the time interval. Note that, thehigher the excitation in the cavity, the greater the chance of a quantum jump occurs, δp = δtγ Tr[ a | N i h N | a † ] = δtγN ,where | N i is the photon number state inside the cavity.In Section II B we demonstrate a partial dynamics of this open system: the no-jump trajectories. Such interestingdynamics occurs when the evolution exhibits no jump at all. II. EXPERIMENTAL SETUP
In this section we describe the setup for simulating experimentally dephasing and damping dynamics on qudits.The experimental setup is illustrated in Figure 1. A 100 mW solid state laser operating at λ = 355 nm pumps a 5 mmthick type I BiBO crystal (BiB O ) and creates degenerate non-collinear photon pairs with horizontal polarization.A dichroic mirror placed after the crystal reflects the pump beam out of the setup and transmits the photon pairs.Signal ( s ) and idler ( i ) photons ( λ s,i = 710 nm) are transmitted through a polarizing beam splitter (PBS) beforecrossing a multi-slit array placed perpendicularly to the propagation axis of the pump beam at a distance of 250 mmfrom the crystal. Taking the pump beam direction as the z longitudinal axis, the multi-slit plane lies in the x − y transverse plane. The slits are 0 . .
25 mm. A 300 mm focallength lens L p placed 50 mm before the crystal is used for focusing the pump beam at the multi-slit array plane.When the beam waist at this plane is smaller than the separation between the slits, the spatial part of the two-photonstate after crossing the aperture will be given by [22, 30, 41–46] | ψ i = 1 √ d ℓ d X ℓ = − ℓ d | ℓ i s |− ℓ i i , (17)where ℓ d = ( d − / D is the number of slits, and | ℓ i s ( | ℓ i i ) is the so-called signal (idler) slit state or photon pathstate. At this point, a maximally path entangled state is prepared. FIG. 1: Experimental setup for implementing dephasing and damping dynamics for qudits encoded in transverse path statesof the twin photons. The lens L p focuses the pumping beam at the multi-slit plane, generating the state given by equation17 after the multiple slit [41]. The SLM1, together with a PBS and the multi-slit can generate partially entangled states. Toperform the dephasing dynamics, a spherical lens L c is placed in the configuration 2 f − f , creating a multi-slit image on theSLM2 plane. A beam splitter (BS) depicted in dotted line is not used for this implementation. On the amplitude dampingdynamics, a cylindrical lens is used to project the image at an infinite distance, instead of the lens L c . In order to realize thedamping operations, we place the BS (dotted line) to construct a Sagnac interferometer. The lenses L i and L s are used tocreate an image or interference pattern according to their positions at the focal plane. We use a half wave plate (HWP) torotate the polarization of signal photons because the SLM2 modules only horizontal polarization. Now, we describe how we prepare a path state with any degree of entanglement. A Holoeye LC-R 2500 spatial lightmodulator, depicted as SLM1 in Figure 1, is positioned just behind the multi-slit array, at a distance of ∼ A. Dephasing implementation
We characterize a ququart photonic path state under the dephasing dynamics. Ququart states are prepared byplacing a four-slit in front of SLM1, perpendicular to the photon pair path (Figure 1). In this particular dynamics,the SLM1 does not change the initial ququart state which is given by equation 17. On the idler arm, the idler photonpasses through the lens L i , with a focal length of 200 mm, that projects the interference pattern at the detector 2plane. On the signal arm, the signal photon passes through L c , with a focal length of 125 mm, which is positionedin the configuration 2 f − f with the SLM2 plane. This configuration allows to create the multi-slit image on theSLM2 screen. The SML2 model is a Hamamatsu LCOS-SLM X10468 which is used to perform the dephasing. TheSLM2 display is addressed with four rectangular regions (see upper inset of Figure 1), each one matching a given slit ℓ from the four-slit array. Each gray level region modulates a phase φ ℓ and performs the dephasing operations K i independently. Under this implementation, the BS is not present. The lens L s , with a focal length of 200 mm projectsthe interference pattern on the detector 1 plane. Any operation required to implement the dephasing dynamics isperformed by the SLM2. FIG. 2: The graphic shows how the films with the SLM patterns are formed to produce the dephasing dynamic. The differentshades of gray represent different Kraus operators. The Kraus operators are implemented by the SLM2, which is divided infour rectangular regions corresponding to the four paths available to the signal photon in a ququart state.
Let us consider the state ρ S that describes an ensemble with N components. When the dephasing occurs, theconstituent N p i evolves according to the operator K i (see equation 10), with i = 0 , , , ,
4. In order to implementthis dynamics in our experimental setup, we explore a way to divide the ensemble described by ρ S . This noveltechnique makes partitions over the acquisition time where different operators K i acts on each time division. Insteadof using a single static image on the SLM2, we use a film, which are kinetic images related to the operations K i .During a certain time interval which corresponds to a single kinetic image, the four rectangular regions at SLM2 willhave different gray levels but constant at this time division. The gray levels are chosen such that the path phasesadded by the SLM2 implement the Kraus operators K i (eq. 8 and 9). A sequence of 32 of these gray levels patternschanging at each time division constitutes what we call a film. The time duration of a film is equal to the acquisitiontime. Because the SPDC process generates randomly photon pairs, on average the same number of down-convertedphotons are generated at equal intervals. So, our ensemble is formed by twin-photons generated over equal timeintervals.Therefore the acquisition time is partitioned into 32 equal time intervals and the parameter p is implemented over awhole acquisition time. Also, the parameter p varies according to each film. In Figure 2 illustrates the implementationof the dephasing dynamics for distinct values of parameter p . For parameter p = 0 . K (identity). For parameter p = 0 . K ), followed by four consecutive images related to operators K i ,with i = 0 , , ,
3, respectively. For p = 1 . K , K , K and K exposed at equal time intervals, while the image related to operator K isnot included in the slides sequence. Repeating the formerly procedure, a dephasing dynamics was performed varyingthe parameter p in steps of 0 .
125 within the interval [0 , p . The conditional pattern on the detection plane [47]is described by the equation below P ( x i , x s ) = Asinc (cid:18) kax i f (cid:19) sinc (cid:18) kax s f β (cid:19) × " ℓ d X ℓ>m = ℓ d (1 − p ) | h ℓ, − ℓ | ρ (0) | m, − m i | sinc (cid:18) ( ℓ − m ) kdbf (cid:19) sinc (cid:18) ( ℓ − m ) kdbf β (cid:19) × cos (cid:18) ( ℓ − m ) kdx i f − ( ℓ − m ) kdx s f β + arg( h ℓ, − ℓ | ρ (0) | m, − m i ) (cid:19)(cid:21) , (18)where k is twin-photon wave number, 2 a = 0 . mm is the slit width, d = 0 . mm is the distance between the slits, f = 200 mm is the focal length of a convergent lens placed at a distance f from the detectors plane, 2 b = 0 . mm isthe detectors width, A is a normalization constant and ρ (0) = | ψ i h ψ | is the initial state given by equation 17. Thescale factor β has a dimensionless value of 0 .
62 and appears due to the combination of the lens L c and L s . TABLE I: Experimental and predicted values of p for dephasing dynamics on ququarts. Experimental parameter p x =0 ( p x = x π )is obtained by fitting the interference pattern curves measured by detecting the photons in coincidence, keeping fixed the signaldetector (det. 1) at x = 0 ( x = x π ) and scanning the idler detector (det. 2). Predicted values p predicted are the parameters weattempt to implement. p x =0 p x = x π p predicted . ± .
063 0 . ± .
044 0 . . ± .
066 0 . ± .
055 0 . . ± .
071 0 . ± .
054 0 . . ± .
071 0 . ± .
052 0 . . ± .
066 0 . ± .
053 0 . . ± .
073 0 . ± .
058 0 . . ± .
062 0 . ± .
051 0 . . ± .
050 0 . ± .
061 0 . . ± .
069 0 . ± .
10 1 . To obtain a single value of parameter p , we measure two interference patterns by varying the idler detector’sposition along the x-direction while the signal detector is positioned at x s = 0 and x s = x π ( kdx π /f β = π ). Figure 3shows the interference patterns and their fits, which are performed by fixing all parameters, except p . As expected,the interference pattern visibilities decrease when the parameter p increases. In Table I, we report the experimental(obtained from conditional interference patterns) and the predicted (calculated from time duration of slides at thefilm) values of p . B. Amplitude damping implementation
The damping dynamics is performed in spatial qutrit states using a three-slit for defining the photon paths. Atthe SLM1, each slit region is modulated differently to get a specific amplitude and, as we mentioned above at thebeginning of section II, we prepare a different two-qutrit state with the state coefficients modified. In this way, weprepare a partial entangled qutrit state of the form | ψ (0) i = a |− s , i i + b | s , i i + c | s , − i i , (19)with a + b + c = 1. This qutrit system behaves like a truncated harmonic oscillator, where the relation between thestates {| i , | i , | i} and the slits are shown in Figure 4. The no-jump trajectory of this amplitude damping dynamic p=0.000 p=1.000 p=0.875p=0.750 p=0.625p=0.500 p=0.375p=0.250 p=0.125 -0,9 -0,6 -0,3 0,0 0,3 0,6 0,9100200300400500 C o i n c i den c e Position (mm) -0,9 -0,6 -0,3 0,0 0,3 0,6 0,90100200300400500600 C o i n c i den c e Position (mm) -0,9 -0,6 -0,3 0,0 0,3 0,6 0,90200400600 C o i n c i den c e Position (mm) -0,9 -0,6 -0,3 0,0 0,3 0,6 0,90200400600800 C o i n c i den c e Position (mm)-0,9 -0,6 -0,3 0,0 0,3 0,6 0,902004006008001000 C o i n c i den c e Position (mm) -0,9 -0,6 -0,3 0,0 0,3 0,6 0,90200400600800 C o i n c i den c e Position (mm) -0,9 -0,6 -0,3 0,0 0,3 0,6 0,902004006008001000 C o i n c i den c e Position (mm) -0,9 -0,6 -0,3 0,0 0,3 0,6 0,902004006008001000 C o i n c i den c e Position (mm) -0,9 -0,6 -0,3 0,0 0,3 0,6 0,9020040060080010001200 C o i n c i den c e Position (mm)
FIG. 3: Ququart interference pattern when the idler detector is scanned and the signal photon is fixed in x s = 0 (closedsquares) and x s = x π (open circles) for all p values measured. To find the p value for each pattern, we fitted the graphs usingthe equation 18 with D = 4. is implemented only in the path states of the signal photon and the two-qutrit state evolution is described by | ψ ( t ) i = e − i ~ ( H eff ⊗ I ) t | ψ (0) i (20)= 1 N ( t ) (cid:0) a | s i i + be − γt | s i i + ce − γt | s i i (cid:1) , where N ( t ) is a normalization factor and γ is the analogous constant decay rate present in the treatment of cavityloss. FIG. 4: Schematic representation of the correspondence between the photon path states defined by slits and the harmonicoscillator.
To implement the no-jump operations, we placed a cylindrical L c lens for projecting the image of the three-slitarray at infinity. This image is propagated along the signal arm to detector 1, passing into a Sagnac interferometerwhose input and output ports are defined by a 50 /
50 beam-splitter (dotted BS showed in Figure 1). Inside the Sagnacinterferometer there is a HWP which changes vertical polarization into horizontal and vice-versa. This interferometerintroduces phase differences ( φ ℓ ) between transmitted and reflected paths inside the Sagnac interferometer since theSLM2 modulates phases only for horizontally polarized photons. So, this modified Sagnac interferometer performsmodulation in each slit state described by the unitary operation U sag | ℓ s i = sin (cid:18) φ ℓ (cid:19) | ℓ s i . (21)Comparing the above Sagnac operation expression with the state evolution (equation 21), it is possible to find acorrespondence between to analogous physical systems: the photon decay rate inside a cavity and the phase modula-tion given by the interferometer (figure 1). Therefore, the amplitude damping (no-jump) dynamics can be performedby introducing the following phase differences between the reflected and transmitted photon path states at the inter-ferometer φ ( t ) = 0 ,φ ( t ) = arcsin (cid:0) e − γt (cid:1) , (22) φ ( t ) = arcsin (cid:0) e − γt (cid:1) . State evolution is obtained by detecting coincidence counts of the three-slit aperture at the image plane. On the idlerarm, a lens L i is used to project the three-slit image at the plane of detector 2. Both detectors 1 and 2, considered aspoint-like detectors Figure 1, are fixed at the positions corresponding to the slit image l s and m i (with l, m = − , , {| l i s | m i i } . The experimental resultsfrom the measurements described above are shown in Table II and Figure 5. Such measurements alone can not fullycharacterize the state, however they do show the population change which is the main effect generated by the dampingdynamics.Furthermore, another interesting feature of this dynamics is the increase of entanglement in the system accordingto the initial state. In this implementation, we generate a initial state with the constraint ( a < b < c ). In Figure6 it is shown the entanglement dynamics of the system. To quantify the entanglement we used the normalizedI-concurrence [48] defined as C ( ψ ( t )) = 1Ω q − T r ( ρ i )] = 1Ω p − T r ( ρ s )] , (23)where ρ i ( ρ s ) is the reduced state of idler (signal) and Ω = p d − /d . Note that the experimental error of theI-concurrence calculated from the experimental measurements increases over time. This happens due to the decreasein coincidence counts, as shown in Table II. From the above results we can infer from the decrease of the totalcoincidence counts that when the system evolves fewer ensemble components have no jump trajectories, as expected. FIG. 5: System population measurement over time. As expected, the slit populations related to higher energy level decreaseand the lower energy levels increase.TABLE II: Coincidence counts between photons transmitted by different slit states for different values of γt . The I-concurrence C e ( | ψ ( γt ) i ) is calculated from the measured states and compared with the predicted I-concurrence C p ( | ψ ( γt ) i ). γt = 0 . γt = 0 . γt = 0 . γt = 0 . γt = 0 . γt = 1 . γt = 1 . γt = 1 . γt = 1 . | , i ± ± ± ± ± ± ± ± ± | , i ± ± ± ± ± ± ± ± ± | , i ±
16 220 ±
15 262 ±
16 252 ±
16 273 ±
17 232 ±
16 248 ±
16 245 ±
16 252 ± | , i ± ± ± ± ± ± ± ± ± | , i ±
31 814 ±
29 775 ±
28 517 ±
23 465 ±
22 339 ±
18 345 ±
19 227 ±
15 125 ± | , i ± ± ± ± ± ± ± ± ± | , i ±
45 1490 ±
39 1222 ±
35 635 ±
23 476 ±
22 290 ±
18 265 ±
16 98 ±
10 120 ± | , i ± ± ± ± ± ± ± ± ± | , i ± ± ± ± ± ± ± ± ± C e ( | ψ ( γt ) i ) 0 . ± .
007 0 . ± .
008 0 . ± .
009 0 . ± .
012 0 . ± .
014 0 . ± .
016 0 . ± .
016 0 . ± .
021 0 . ± . C p ( | ψ ( γt ) i ) 0 .
864 0 .
881 0 .
914 0 .
942 0 .
960 0 .
972 0 .
962 0 .
945 0 . Summary
In this work we experimentally demonstrated simulations of dissipative dynamics on quantum systems in a simpleimplementation. Our quantum systems are spatial qudits encoded in the transverse paths of photons pairs generatedby SPDC. The dissipative operators for simulating dephasing and amplitude damping dynamics were realized by meansof a spatial light modulator. Dephasing dynamics was performed completely and amplitude damping dynamics wasimplemented partially, in which we performed only the no-jump trajectory. In the dephasing dynamics we measuredthe interference patterns to calculate the parameter p , related to coherence loss. Besides, in the amplitude dampingdynamics we measured the population which allows us to identify population changes from higher to lower levels. Toidentify the implementation success we used a parameter that represents the principal characteristic of the evolution:coherence loss for dephasing and population changes for amplitude damping. Spatial photonic states are interestingsystems to make quantum computation using qudits. The implemented dissipative dynamics are general and do notdepend on the initial state and can also be extended to different system dimensions. Moreover, the experimentaltechnique using films instead of images can be manipulated to implement other types of operations, increasing the0 FIG. 6: Entanglement dynamics for no-jump trajectory of amplitude damping acting on the qutrit system. I-concurrence iscalculated from experimental data (red dots) showed in the table II and the theoretical prediction (black line).
SLM uses.
Acknowledgments
This work is part of Brazilian National Institute for Science and Technology for Quantum Information and wassupported by the Brazilian agencies CNPq, CAPES, and FAPEMIG. We acknowledge the EnLight group for veryuseful discussions.
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