Entropy, purity and optical hysteresis in markovian optical modes

P. Martinez Vara; J. C. Atenco Cuautle; G. Arenas Muñoz; E. Saldivia Gomez; A. Garcia Guzman; I. I. Cazares Aguilar; G. Martinez Niconoff

doi:10.1364/OE.411977

1. Introduction

In the present manuscript we implement randomly non-stationary processes to generate optical fields, whose properties are studied trough the evolution of the probabilistic parameters. We focus our attention on the synthesis of markovian modes generated by a random linear combination of Bessel modes in a tandem array. The linear combination associated to the amplitude function is changing according to a stochastic process of Markov-chain type Ehrenfest [1,2]. These markovian processes has associated a matrix representation known as transition matrix TM which transforms an initial random vector into another random vector generating a probability redistribution. The stochastic process is generated applying recursively the TM on an initial random vector, generating a sequence of random vectors. The $n$-power of the TM generates the $n$-step matrix which carries the information for the complete evolution process [3,4]. From the $n$-step matrix structure is possible to identify when the process becomes stable. This is obtained when all the terms of the $n$-step matrix become different to zero and the set of random vectors describe a convergent sequence tending towards a final vector which acts as an attractor for the stochastic process. This final vector has associated the maximum entropy value for the process. A natural way to generate the markovian mode is using Bessel modes of integer order whose amplitude value is linked to the sequence of random vectors i.e, the random vector allows to associate a random linear combination of Bessel modes which evolves towards a final and stable configuration that is associated to a maximum entanglement [5]. This stable mode has the characteristic to be indistinguishability [6], thus it does not exist a unique inverse process that can lead to the initial random vector. This effect constitutes the optical hysteresis, as a consequence of this last statement the energy to reach the stable mode depends on the trajectory determined by the probabilistic parameters. This feature can be measure from the $n$-value in the $n$-step matrix when it reaches a stable configuration. In this way, the Markov modes correspond to a non-conservative optical field. A very important consequence of this proposal consists in the possibility to control interference effects, this is because during the transitory times or “unstable time” interference does not exist and it only appears when the process reaches a stable configuration. We remark that the optical field associated to markovian modes cannot be analyzed with the traditional model of partial coherent theory. Mostly because the markovian modes do not fulfill the hypothesis of stationarity, which is a required in the partial coherence model [7]. However, a great variety of optical fields do not satisfy stationary properties [8]. One example is in the statistical analysis of processes with few photons whose behavior display a stochastic process similar to queuing process [9] which are a type of Markov-chain process. We focused our study on a markovian process of Ehrenfest type which was designed and implemented to model the thermodynamic equilibrium. These allow to stablish an analogy between optical equilibrium with a thermodynamic process. In the present manuscript, we describe an ensemble of Bessel optical modes, whose amplitude function is described by a stochastic succession of random vectors that follows a Markov-chain process. The amplitude fluctuations are characterized with the concept of purity [10]. This property allows to stablish a similitude between the markovian mode with a dominant Bessel mode of integer order. Also, its physical description is related with the entanglement characterized by the transition probabilities associated to the $n$-step matrix. Finally, when the number of steps $n$ is large enough, the stochastic process reaches a stable configuration that allows the implementation of interference effects. The resulting model offers interesting applications. For instance, in cryptographic information [11], dynamic holography [12], tunable spectroscopy [13] and time-depending self-healing processes [14]. The stochastic matrix elements are related to the transition probabilities among possible states. The dynamics of the process is analyzed by interpreting the matrix as a transformation applied to a initial random vector, that represents the initial state of the chain. The evolution of the initial state is obtained by sequentially applying the TM, which generates a final $n$-step stochastic matrix which carries information of the probability of initial states that reach a final state in $n$-steps. A fundamental point consists on the analysis of the stability of the process. In order to do so, we studied processes whose $n$-step stochastic matrix acquires a regular form, this means that all the matrix elements are different from zero and then no forbidden states exists, assuring then the stability of the markovian process. The issues to be addressed are the following: for a given process Markov-chain type and assuming an initial state which has associated an initial random vector, it is necessary to identify how this vector evolves to generate entanglement among the elements of the resulting random vector, allowing to generate interference effects. The final state is described by calculating the entropy values implicit in the $n$-step stochastic matrix [15,16]. From this analysis we are able to determinate the time evolution of the purity of the markovian optical field. To maintain a geometrical point of view, we take advantage on the fact that each Markov-chain type process is associated to an oriented graph named digraph [17], where each node represents a “physical state” that for our case corresponds to a Bessel mode. The evolution of the process generates connectivity among the Bessel modes which describe to the entanglement of the process. Our manuscript is organized in the following form. We start with the Ehrenfest process with four states and two examples with different probabilistic parameters are presented, showing that in both cases the $n$-step matrix converges to the same matrix differing in the number of the $n$-steps for each matrix which constitutes the hysteresis features. The process is implemented in the optical context transforming an initial random vector with the $n$-step matrix and the resulting vector is interpreted as a linear combination of Bessel modes of integer order which constituting the Ehrenfest mode. To identify the preferent Bessel mode we calculate the purity value. This purity value is obtained of the final entropy from the resulting mode. Computer simulation are presented.

2. Description of markovian chains

A stochastic process describes the evolution of a random variable time parametrized $X(t), i=0,1,2, . ., n$. This process is completely determined when the $n$-th correlation function is known which does not occur in most of the cases [7]. The correlation function is given by $P(x_{0}, x_{1}, \ldots , x_{n})$ with the suffix refers to time. With the Bayes theorem, the $n$-th correlation function acquires the form

(1)$$\begin{aligned} P\left(x_{0}, x_{1}, \ldots, x_{n}\right) & = P\left(x_{n} | x_{n-1}, \ldots, x_{1}\right) P\left(x_{n-1}, x_{n-2}, \ldots, x_{1}, x_{0}\right)\\ &= P\left(x_{n} | x_{n-1}, \ldots, x_{1}\right) P\left(x_{n-1} | x_{n-2}, \ldots, x_{1}\right) P\left(x_{n-2}, \ldots, x_{1}\right) \ldots P\left(x_{1} | x_{0}\right), \end{aligned}$$

where $P(x_{0}|x_{n-1}, x_{n-2}, \ldots )$ are the conditional probability that represents the probability of occurrence of $x_{n}$ given the occurrence of an event defined by $x_{n-1}, x_{n-2}, \ldots$. When the process depends on its recent history, which is known as the markovian hypothesis, Eq. (1) acquires a simplified form given by

(2)$$P\left(x_{0}, x_{1}, \ldots, x_{n}\right)=P\left(x_{n} | x_{n-1}\right) P\left(x_{n-1} | x_{n-2}\right) \ldots P\left(x_{1} | x_{0}\right) P\left(x_{0}\right).$$

The previous expression defines the Markov chain and the term $P(x_{i}|x_{i-1})$ is known as the transition probability. To analyze the evolution of the stochastic process, is convenient to associate a matrix representation, which is obtained from Table 1

Table 1. Array to generate the transition matrix.

View Table

Where $P_{ij}$ is the transition matrix TM and each term represents the probability for the stochastic process goes from $i$-state to the $j$-state. Then, all the process is characterized the vector transformed

(3)$$\vec{\upsilon_{0}} \mathbf{P}= \begin{matrix} {a_{0}} & {\cdots} & {a_{n}}\end{matrix} \left(\begin{matrix} {P_{00}} & {P_{01}} & {\cdots} & {P_{0n}} \\ {P_{10}} & {P_{11}} & {\cdots} & {P_{1n}} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ {P_{n0}} & {P_{n1}} & {\cdots} & {P_{nn}} \end{matrix}\right ) =\vec{\upsilon_{1}},$$

where $\vec {\upsilon _{0}}=\left (a_{0}, \ldots , a_{n}\right )$ is the random initial vector. An important property is that the probability dispersion on each file in the TM must satisfy

(4)$$\sum_{j} P_{i j}=1, \quad i=0,1,2, \ldots, n.$$

In this way, the TM is an stochastic matrix. The global evolution of the markovian process is obtained by applying recursively the matrix to the initial random vector given by

(5)$$\vec{\upsilon}_{1}=\vec{\upsilon}_{0} \mathrm{P}, \quad \vec{\upsilon}_{2}=\vec{\upsilon}_{1} \mathrm{P}=\vec{\upsilon}_{0} \mathrm{P}^{2}, \quad \cdots \quad \vec{\upsilon}_{N}=\vec{\upsilon}_{0} \mathrm{P}^{n},$$

where stochastic matrix $P^{n}$ is known as the $n$-step transition matrix. Very important properties of the $n$-step transition matrix can be highlighted: Firstly, when the transition matrix is applied to a given initial state, forbidden states can appear meaning that this state cannot be reached from this initial state. This effect is closely related to bifurcation properties [18]. Secondly, another effect consists on the generation of chaos, which consists on small changes on the probability transitions values can transform the initial state after $n$-steps toward probability states completely different. An important property of the $n$-step matrix is that from its structure is possible to identify equilibrium states. These occur when the file entropy reaches a non-variable random vector which acts as an attractor for all the process. All of the properties previously mentioned depend on the type of markovian chain. For this reason, in the following section we focus our attention on stochastical process which has associated a final “stable configuration” which is defined below. The evolution of entropy values and the connection among the nodes can be identified by associating a digraph to the $n$-step stochastic matrix. The recursive application of the $n$-step matrix is matched to the evolution of the digraph. It can be easily identified from the digraph when the $n$-step stochastic matrix acquires a regular. This means that non-forbidden states appear and then the process must be reached for $n$ large enough a stable configuration. A consequence of this fact is all the elements in the $n$-step stochastic matrix must be different from zero, this property is completely fulfilled for a Markov-chain type Ehrenfest process. This type of process was designed to model the thermodynamic equilibrium, then this allows related it with the stability of optical processes.

3. Markov-chain type Ehrenfest process

We define a markovian mode type Eherenfest, as the optical field generated by means of a tandem array of Bessel modes of integer order whose amplitude values are changing according a Markov chain-type Ehrenfest. The structure of the chain and its convergence can be easily understood using a box model as follows. Suppose two boxes labeled as $A$ and $B$. Box $A$ contains $M$ balls, and box $B$ contains $N-M$ balls. The balls are labeled from $1$ to $N$ and they are randomly distributed in each box. The Ehrenfest process consists of selecting one ball and transferring it to the other box with probability $\alpha$, or letting it remain in the same box with probability $1-\alpha$. Keeping this idea in mind, we can easily show that the transition matrix when the process has four states is given by a $5\times 5$ matrix, given by

(6)$$E= \left( \begin{matrix} {1-\alpha} & {\alpha} & {0} & {0} & {0} \\ {\frac{1}{4}\alpha} & {1-\alpha} & {\frac{3}{4} \alpha} & {0} & {0} \\ {0} & {\frac{1} {2}\alpha} & {1-\alpha} & {\frac{1}{2} \alpha} & {0} \\ {0} & {0} & {\frac{3}{4}\alpha} & {1-\alpha} & {\frac{1}{4}\alpha} \\ {0} & {0} & {0} & {\alpha} & {1-\alpha} \end{matrix}\right ).$$

To obtain a better understanding of the Ehrenfest process evolution, we present two examples considering the probabilistic values $\alpha =1/2$ and $\alpha =1/4$. In spite that our analysis was made considering two specific values, the generalization is easily understood for arbitrary alfa-values. The main issues discussed herein is how the convergence of each matrix depends on the alfa value and how the final state can be obtained from the corresponding $n$-step matrices which allow us to identify the tandem array with maximum entropy corresponding to the most probable. The expression for the transition matrix and its corresponding $n$-step matrix are given by

(7)$$\begin{array}{c}E=\left( \begin{array}{@{}ccccc@{}} {0.5} & {0.5} & {0} & {0} & {0} \\ {0.125} & {0.5} & {0.375} & {0} & {0} \\ {0} & {0.25} & {0.5} & {0.25} & {0} \\ {0} & {0} & {0.375} & {0.5} & {0.125}\\ {0} & {0} & {0} & {0.5} & {0.5} \end{array}\right ),\; E^{2}=\left( \begin{array}{@{}ccccc@{}} {0.3125} & {0.5} & {0.1875} & {0} & {0} \\ {0.125} & {0.4062} & {0.375} & {0.0937} & {0} \\ {0.0312} & {0.25} & {0.4375} & {0.25} & {0.0312} \\ {0} & {0.0937} & {0.375} & {0.4062} & {0.125}\\ {0} & {0} & {0.1875} & {0.5} & {0.3125} \end{array}\right ), \ldots ,\\ \; E^\textbf{65}=\left(\begin{array}{@{}ccccc@{}} {0.0625} & {0.25} & {0.375} & {0.25} & {0.0625}\\ {0.0625} & {0.25} & {0.375} & {0.25} & {0.0625}\\ {0.0625} & {0.25} & {0.375} & {0.25} & {0.0625}\\ {0.0625} & {0.25} & {0.375} & {0.25} & {0.0625}\\ {0.0625} & {0.25} & {0.375} & {0.25} & {0.0625} \end{array}\right ).\end{array} $$

The matrices considering $\alpha =1/4$ are

(8)$$\begin{array}{c}E=\begin{pmatrix} {0.75} & {0.25} & {0} & {0} & {0} \\ {0.0625} & {0.75} & {0.1875} & {0} & {0} \\ {0} & {0.125} & {0.75} & {0.125} & {0} \\ {0} & {0} & {0.1875} & {0.75} & {0.0625} \\ {0} & {0} & {0} & {0.25} & {0.75} \\ \end{pmatrix},\; E^{2}=\begin{pmatrix} {0.5781} & {0.375} & {0.0468} & {0} & {0} \\ {0.0937} & {0.6015} & {0.2812} & {0.0234} & {0} \\ {0.0078} & {0.1875} & {0.6093} & {0.1875} & {0.0078} \\ {0} & {0.0234} & {0.2812} & {0.6015} & {0.0937} \\ {0} & {0} & {0.0468} & {0.375} & {0.5781} \end{pmatrix}, \cdots ,\\ \; E^{\bf 138}=\begin{pmatrix} {0.0625} & {0.25} & {0.375} & {0.25} & {0.0625}\\ {0.0625} & {0.25} & {0.375} & {0.25} & {0.0625}\\ {0.0625} & {0.25} & {0.375} & {0.25} & {0.0625}\\ {0.0625} & {0.25} & {0.375} & {0.25} & {0.0625}\\ {0.0625} & {0.25} & {0.375} & {0.25} & {0.0625} \end{pmatrix}.\end{array} $$

An interesting result can be deduced by comparing Eq. (7) and Eq. (8) that represent two different transition matrices which evolves toward the same $n$-step stochastic matrix. From this representation we can to identify the most probable state. We remark that the $n=n(\alpha )$ describes the number of steps to reach the most probable state. For the examples presented $n(1/2)\neq n(1/4)$. Once that the $n$-equilibrium configuration has been obtained, all the elements in each column have the same value. The structure of Eq. (7) shows that the product of any arbitrary random vector generates the same row of the $n$-step stochastic matrix. This corresponds to the equilibrium of the process that is non-dependent on the initial random vector. From this equilibrium condition, we can deduce some generic features, particularly the maximum entanglement related to the succession of the digraphs shown in Fig. 1. The entanglement features correspond to the evolution of the different states of the probability values among the nodes. This analysis implies the evolution of the entropy values. It must be noted that the changes in the assigned probability values of the initial stochastic matrix, implies the modification of the connectivity between states. The optical states are $0$ which mean dark field and $J_{0},\; J_{1},\; J_{2},\; J_{3}$ are the Bessel modes of integer order.

Fig. 1. Depicture of the stochastic matrix process corresponding to Eq. (7). a) Digraph associated to the initial stochastic matrix. b) Digraph for $E^{2}$. c) Digraph for $E^{65}$ when the process reaches a stablish configuration.

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4. Generation of Ehrenfest optical modes

In this section, we implement the Ehrenfest process in the optical context. To perform this, we conveniently start to define the basis for the markovian mode given by the Bessel beams of integer order, given by

(9)$$\left\{e^{i \beta z} J_{n}(2 \pi r d) e^{i n \theta}\right\} \quad n=0, \pm 1, \pm 2, \dots ,$$

where $\beta$ is a constant that describes the phase along the z-coordinate, $r$ and $\theta$ are the polar coordinates. To generate the markovian modes, we use the random vectors generated with the $N$-step matrix. Using this representation, we propose to define the a stochastic mode as a sequence of modes whose structure follows a stochastic process and locally presents diffraction free features, a particular case occurs when integer order Bessel modes are selected following a Markov chain type process.

Equation (9) can be matched with the box model of the Ehrenfest process described in the previous section. By replacing the label in each ball by $J_{0}, J_{1}, \ldots , J_{n}$ we describe the evolution of initial state $x_{0}=\left (a_{0}, \ldots , a_{n}\right )$. This vector corresponds to the coordinates that represent the appearance of the mode with the following interpretation: Assuming that the process has time duration $T$, divided by $n$ subintervals of length $\Delta T$. In each subinterval, a Bessel mode of integer order is selected. Thus, the optical field consists of a succession of mode-type chains where the occurrence of the ith Bessel mode is $n \alpha _{i}$. With this interpretation, the optical field assumes the structure shown in Fig. 2. A liquid crystal display (LCD) is implemented to generate the boundary condition that consists of an annular slit angularly modulated for synthetizing the corresponding Bessel mode [19,20]. The structure of the chain corresponds to the Ehrenfest mode.

Fig. 2. Experimental set up to generate the markovian mode in a tandem array of Bessel modes. The LCD contains an annular slit with time-dependent angular modulation and it is illuminated with a coherent plane wave.

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For a better understanding of the evolution of the markovian mode, it is convenient to describe a tree diagram, shown in Fig. 3. The bold line represents a posible sequence of Bessel modes.

Fig. 3. Tree diagram to describe the entanglement between Bessel modes.

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The markovian mode consists of a sequence of Bessel modes of integer order where each sequence appears according to a certain probability value. The state starts with a zero order Bessel beam, whose irradiance is shown in Fig. 4(a). This state evolves following the chain and after the stability is reach, acquires a stable configuration, this is shown in Fig. 4(b). When the experiment is performed $n$-times, the number of occurrence of each element of the basis can be obtained from the transformed random vector. We analyzed the global optical fields once the equilibrium is reached. The mean irradiance distribution presents diffraction free features. The later is easily understood because the irradiance associated to each mode is non- depending on the $z$-coordinate. The optical field consist in a sequence of time changing blocks that do not follow a stationary process. For this reason, the mode is characterized using the entropy models, this is performed in the following section.

Fig. 4. a)Bessel mode of zero order, corresponding to the initial state. b) Mean irradiance associated to the markovian mode. c) Irradiance associated to the coherent interaction between modes. d) Mean irradiance for the interference between markovian modes. The geometrical structure is the same as shown in b), however they have different irradiance scales as it is shown in the color bar.

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5. Entropy, purity and interference between markovian modes

From the fact that a markovian process has associated a stochastic matrix, we can identify the generic features through entropy calculation. This allows to describe the mode’s structural properties. We proposed the calculus of the Von Neumann entropy [21] in order to obtain the entropy value from the $n$-step stochastic matrix. The entropy is calculated from the principal diagonal elements. The resulting value acts as a reference value for the entropy measurement obtained from the elements of the secondary diagonal. This value contains information about the correlation among the constitutive modes. By comparing these entropy values, we can deduce how the correlation function evolves, which allows to understand the irradiance distribution as a function of $n$, that represents the number of applications of the initial stochastic matrix. The Von Neumann entropy is defined as

(10)$$S_{v}=-\operatorname{Tr}\left(E^{n} \ln E^{n}\right),$$

where $\operatorname {Tr}$ denotes the trace of stochastic matrix $E^{n} \ln \left (E^{n}\right )$. It must be noted that this type of entropy contains information of the irradiance distribution. However, we need to describe the entanglement among the elements of the basis. This description can be obtained by proposing the correlation entropy calculus as an ansatz using the elements in the secondary diagonal. This entropy takes the following form:

(11)$$S_{c}=-\operatorname{Tr} D\left(E^{n} \ln E^{n}\right),$$

where $\operatorname {TrD}$ denotes the trace of the secondary diagonal. Then, a good method of describing the mode structure is applied by calculating the difference of the entropy values, which is expressed as

(12)$$\Delta S=S_{v}-S_{c}.$$

We propose this definition, since it can be easily proven that the correlation entropy is always upper-bounded by the Von Neumann entropy $S_{v}\geq S_{c}$. To compare the entropy values, each diagonal needs to satisfy the normalization condition. From Eq. (12), certain interesting cases can be identified. The limit case occurs when $\Delta S=0$. Its physical meaning is that all irradiance events involved during the process participate in the global irradiance distribution. The other case occurs when $\Delta S=S_{v}$. There is no interaction among the elements of the basis; thus, they are statistically independent. However, this case is not permitted in the Markov-chain type Ehrenfest process. Finally, more entropy measurements can be obtained from the $q$th-row elements, expressed as

(13)$$S_{q}=-\sum_{i=1}^{N+1} \alpha_{i q} \ln \left(\alpha_{i q}\right),$$

where $\alpha _{i q}$ denotes the elements in the $q$th row and satisfies $\sum _{i} \alpha _{i q}=1$ for $q=1,2,\ldots ,N+1$. From this entropy row, an order relationship is easily identified, e.g.,

(14)$$S_{q}< S_{3}=S_{5}<S_{6}<\dots,$$

this allows to describe how the irradiance is redistributed between the rows. From this order relationship, we can associate a purity measurement to the Ehrenfest mode as follows:

(15)$$p_{q}=1-\frac{S_{q}}{\sum_{i=0}^{n} S_{i}},$$

which determines the similitude of the markovian mode with the $J_{q}$ mode because $\sum _{q=1}^{N+1} p_{q}=1$. For an Ehrenfest mode, once the equilibrium is reached, all of the rows have the same entropy value as shown below.

The concept of purity can be implemented to describe the superposition between two modes defining in this way the “markovian interference”. The analysis is performed by means of the $n$-step correlation matrix. Let two processes where its corresponding transition matrix have different probability parameters. For example, for the Ehrenfest matrix given by $\bf E$ in Eqs. (7–8) with $\alpha _{1}=1/2$ and $\alpha _{2}=1/4$ the resulting vectors after $s$ and $k$ steps are given by

(16)$$^{(1)}e_{s}=^{(1)}e_{0}^{(1)} E^{s} ; \quad ^{(2)}e_{k}=^{(2)}e_{0}^{(2)} E^{k}.$$

Where the left upper indexes $(1-2)$ referes to the process with different probabilistic parameters. Hence, the corresponding ensemble of modes has different statistical parameters. To obtain the correlation between these modes we calculate the moduli of the sum of the modes given by

(17)$$\left| ^{(1)}e_{s}-\, ^{(2)}e_{k}\right|^{2} =|^{(1)}e_{s}|^{2}+|^{(2)}e_{k}|^{2}-\left[^{(1)}e_{0}\left({ }^{(1)} E^{s}\; {^{(2)}E^{k\dagger}}\right)\, ^{(2)}e_{0}^{\dagger}+{ }^{(2)}e_{0}\left({ }^{(2)} E^{k} {^{1}E^{s\dagger}}\right)\, ^{(1)}e_{0}^{\dagger} \right].$$

The expression between the square braquets represents the interference term. As particular case we consider the evalution of the processes when both of them have the same initial vector $^{(1)}e_{0}=\, ^{(2)}e_{0}$, the intereference term acquires the form

(18)$$^{(1)}e_{0}\left[\left({ }^{(1)} E^{s}\, {^{(2)}E^{k \dagger}}\right)+\left({ }^{(2)} E^{k}\, {^{(1)}E^{s \dagger}}\right)\right]\, ^{(1)}e_{0}^{\dagger}.$$

For the case of the markovian process, the $n$-step matrices has associated a regular form as can be observed from matrices given by Eq. (7,8). Surprisingly, the markovian interference will acquire a maximum value for $s$ and $k$ large because the $n$-step matrices converges to the same final vector which is non-dependent on the initial vector. With this analysis, the expression for the markovian mode is

(19)$$\varphi(r, \theta, z)=\exp i \beta z\left( A_{0}(0)+ A_{1} J_{0}(r)+A_{2} J_{1}(r) e^{i \theta}+A_{3} J_{2}(r) e^{i 2 \theta}+A_{4} J_{3}(r) e^{i 3 \theta}\right),$$

where $A_{i}$ are the compounds of a random vector that satisfies $<A_{0}>=0.06$, $<A_{1}>=0.25,$ $<A_{2}>=0.37,$ $<A_{3}>=0.25$ and $<A_{4}>=0.06$. To describe the interference between the two markovian modes, it is necessary to describe a measure of superposition among them. This can be obtained from the correlation matrix implicit in the irradiance expression, given by

(20)$$I(r, \theta)=|\varphi(r,\theta, z)|^{2} =\left(\; 0,\; J_{0}(r),\; J_{1}(r) e^{i \theta},\; J_{2}(r) e^{i 2 \theta},\; J_{3}(r) e^{i 3 \theta}\; \right)\left(\begin{array}{ccccc} c_{00} & c_{01} & c_{02} & c_{03} & c_{04} \\ c_{10} & c_{11} & c_{12} & c_{13} & c_{14} \\ c_{20} & c_{21} & c_{22} & c_{23} & c_{24} \\ c_{30} & c_{31} & c_{32} & c_{33} & c_{34} \\ c_{40} & c_{41} & c_{42} & c_{43} & c_{44} \end{array}\right)\left(\begin{array}{c} 0\\ J_{0}(r) \\ J_{1}(r) e^{-i \theta} \\ J_{2}(r) e^{i 2 \theta} \\ J_{3}(r) e^{i 3 \theta} \end{array}\right),$$

where the matrix $C_{ij}$ is the correlation matrix and term is obtained by the product between the amplitude terms of the linear combination given by Eq. (19) and a local purity term given by

(21)$$c_{ij}=A_{i}A_{j}p_{ij},$$

where the local purity satisfies

(22)$$p_{ij}=1-\frac{S_{ij}}{\sum_{lm} S_{lm} },$$

$S_{ij}$ is the local entropy between the amplitude terms. This correlation purity is a measure of the interaction between the corresponding modes. It must be noted that this definition constitutes a deep difference with partially coherent modes. In the last case, the coherence degree modulates all the interference term, however in the markovian modes the correlation purity modulates the interaction between constitutive modes. Therefore, it is not possible to define a global coherence degree. The local purity is a measure of the spreading of the row entropy. In Fig. 4(a) shows the irradiance associated to the markovian mode once the stable configuration has been reached. To have a reference point, Fig. 4(b) shows the irradiance associated to a completely coherent optical field whose amplitude values satisfies Eq. (19). Finally, Fig 4(c) shows the interference between two markovian modes where the irradiance distribution carry on the information of the correlation purity. In these last cases, the angular dependence observed in Figs. 4(b-c) is a manifestation of the phase term associated to the dominant mode.

6. Optical hysteresis

The Ehrenfest mode acquires a final equilibrium configuration which is independent of the probabilistic parameters. However, the number $n$ of steps to reach these configuration depends on the parameter $\alpha$ implicit in the transition matrix, Eq. (6), i.e. $n=n(\alpha )$. Once that the $n$-step matrix reaches its final configuration, the markovian mode acquires an indistinguishable configuration, and it is not possible to define a unique inverse process that can lead to the initial vector. In this inverse process is implicit the optical hysteresis for a markovian optical mode type Ehrenfest. The hysteresis can be defined through the difference of irradiance for modes whose transition matrices has different parameters $\alpha$ as given in Eqs. (7,8). Then the hysteresis has the form

(23)$$H=<I_{1}>n(\alpha_{1})-<I_{2}>n(\alpha_{2}),$$

where $<I>$ describe the irradiance mean transition between each step, in this way $<I>n(\alpha )$ describes the total irradiance used in each process. Because in general, the optical hysteresis depends on the trajectory. This means that the set of markovian type Ehrenfest modes $E(\alpha _{i})$ is a non-conservative field. An interesting mathematical problem to be solved consists in the calculus of the minimum number of steps $n(\alpha )$ to allow the process to reach reach its stable configuration. This stable configuration corresponds to the maximum entropy and with the strongest entanglement among the elements of the basis.

7. Conclusions

We proposed a theoretical model to generate stochastic optical modes using a set of Bessel modes in tandem array. The array is changing according a stochastic Markov-chain type process. The model was implemented using as prototype a Ehrenfest process. The evolution of the array was obtained by applying recursively the transition matrix to an initial random vector. In this way the final vector has associated a $n$-step stochastic matrix, which represents the probability that, for a given initial state it reaches a final state in $n$-steps. This $n$-step matrix also takes information of the process evolution, obtained through the entropy evolution calculated from the rows which have a random vector structure. The set of entropy values allowed us to associate a purity degree for the resulting mode giving information about the entanglement of the basic elements represented by the Bessel modes on integer order. Computational simulations were performed with MATLAB software for a Markov-chain type Ehrenfest process. This type of process was implemented because it describes the conditions under which a thermodynamic process reaches equilibrium. The optical field evolves towards an stabilized optical mode. The set of modes in tandem array describe a markovian chain that is not a conservative optical field, which presents effects of optical hysteresis. Once that the markovian mode is stabilized, interference effects can be implemented. This is possible by counting the number of coincidences among the basis elements. The model offers applications to generate tunable holography, and process cryptographic to transfer information, self-healing analysis and tunable optical tweezers.

Disclosures

The authors declare no conflicts of interest.

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	$P_{0}$	$P_{1}$	$P_{2}$	$P_{3}$	$\dots$
$P_{0}$	$P_{00}$	$P_{01}$	$P_{02}$	$P_{03}$	$\dots$
$P_{1}$	$P_{10}$	$P_{111}$	$P_{12}$	$P_{13}$
$P_{2}$	$P_{20}$	$P_{21}$	$P_{22}$	$P_{23}$
$P_{3}$	$P_{30}$	$P_{31}$	$P_{32}$	$P_{33}$
$⋮$	$⋮$				$⋱$

Entropy, purity and optical hysteresis in markovian optical modes

Abstract

1. Introduction

2. Description of markovian chains

3. Markov-chain type Ehrenfest process

4. Generation of Ehrenfest optical modes

5. Entropy, purity and interference between markovian modes

6. Optical hysteresis

7. Conclusions

Disclosures

References

Cited By

Figures (4)

Tables (1)

Equations (23)

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