David S Dean

Antonio Iorio

Enzo Marinari

Gleb Oshanin

Sample-to-sample fluctuations of power spectrum of a random motion in a periodic Sinai model

The Sinai model of a tracer diffusing in a quenched Brownian potential is a much-studied problem exhibiting a logarithmically slow anomalous diffusion due to the growth of energy barriers with the system size. However, if the potential is random but periodic, the regime of anomalous diffusion crosses over to one of normal diffusion once a tracer has diffused over a few periods of the system. Here we consider a system in which the potential is given by a Brownian bridge on a finite interval (0,L) and then periodically repeated over the whole real line and study the power spectrum S(f ) of the diffusive process x(t) in such a potential. We show that for most of realizations of x(t) in a given realization of the potential, the low-frequency behavior is S(f ) ∼ A/f 2 , i.e., the same as for standard Brownian motion, and the amplitude A is a disorder-dependent random variable with a finite support. Focusing on the statistical properties of this random variable, we determine the moments of A of arbitrary, negative, or positive order k and demonstrate that they exhibit a multifractal dependence on k and a rather unusual dependence on the temperature and on the periodicity L, which are supported by atypical realizations of the periodic disorder. We finally show that the distribution of A has a log-normal left tail and exhibits an essential singularity close to the right edge of the support, which is related to the Lifshitz singularity. Our findings are based both on analytic results and on extensive numerical simulations of the process x(t).

I. INTRODUCTION

The statistical classification of time-dependent stochastic processes is often based on the study of their power spectrum

S(f ) = lim →∞ 0 dt e if t x(t) 2 , ( 1
)
where is the observation time and the horizontal bar denotes ensemble averaging with respect to all possible realizations of x(t). Many processes, which are common in nature and are often observed in engineering and technological sciences, are found to exhibit a low-frequency noise spectrum of the universal form [1,2] S(f

) ∼ A f α . ( 2
)
The amplitude A is independent of f , and the exponent α ∈ (1,2), with the extreme cases α = 1 and α = 2 corresponding to the 1/f (flicker) noise and Brownian noise (or noise of the extremes of Brownian noise [3]), respectively. There exist a few physical cases for which the form in (2) with α < 2 extends over many decades in frequency, implying the existence of correlations over surprisingly long times. Relevant examples include electrical signals in vacuum tubes, semiconductor devices and metal films [1,2]. More generally, the form in ( 2) is observed in sequences of earthquakes [4] and weather data [5], in evolution [6], human cognition [7], network traffic [8], and even in the temporal distribution of loudness in musical recordings [9]. Recent experiments have shown the occurrence of such universal spectra in processes taking place in a variety of nanoscale systems. Among them are transport in individual ionic channels [10,11] and electro-chemical signals in nanoscale electrodes [12], bio-recognition processes [13], and intermittent quantum dots [14]. Many other examples, related theoretical concepts, emerging challenges, and unresolved problems have been discussed in [14][15][16][START_REF] Montroll | Proc. Natl. Acad. Sci. USA[END_REF][START_REF] Niemann | [END_REF][19].

An example of a transport process which exhibits the flicker 1/f noise (with logarithmic corrections) was pointed out more than 30 years ago in Refs. [15,16]. This is a paradigmatic example for random motion in a quenched random environment, now known as Sinai diffusion [20], which has been studied in many different contexts [21][22][23][24][25][26][27]. Sinai diffusion is defined as a Brownian motion advected by a quenched drift which is time independent and uncorrelated in space. It can thus be seen as an overdamped Langevin process subject to a quenched force which is uncorrelated in space, so in one dimension it is derived from a Brownian potential V (x). The mean-square displacement of the Sinai diffusion exhibits a remarkably slow logarithmically growth with time t,

E(x 2 (t)) ∼ ln 4 (t) , t → ∞, (3)
where E(•) denotes averaging over realizations of the random potential. The result in (3) is supported by typical realizations of disorder, i.e., it holds for almost all samples with a given potential V (x). Note that despite the slow logarithmic dispersion of the trajectories, the probability currents J L through finite samples of Sinai chains of length L appear to be much larger than the Fickian currents in homogeneous systems [22][23][24][25]; for finite Sinai chains, one has E(J L ) ∼ 1/ √ L, while for homogeneous systems J L ∼ 1/L. Such an anomalous behavior of currents is supported by rare atypical realizations of V (x) which, however, produce the dominant contributions to the average.

1 X V(X) L 2L L 2L FIG. 1. Potential V (x) as a periodically extended Brownian bridge with V (x = 0) = V (x = L) = 0.
In this paper we analyze the power spectrum of random motion in a random quenched potential looking at the problem from a different perspective-we will mainly focus on the amplitude A of the power spectrum, not on the value of the exponent α characterizing the power spectrum. In random environments, this amplitude is itself a random variable fluctuating from realization to realization of the random potential, and this makes the power spectrum itself a random variable. Here we concentrate on a particular model-a periodic Sinai chain [28], in which the potential is a finite Brownian trajectory with constrained end point-the so-called Brownian bridge, defined on the interval (0,L) and then periodically extended in both directions to give an infinite one-dimensional system (see Fig. 1). The origin of the slow logarithmic growth in the original Sinai model (with L = ∞) is due to the unlimited growth of the Brownian potential and the associated energy barriers; however, in our periodic case x(t) ultimately converges to a Brownian motion, on large time and length scales, so the low-frequency spectrum has a form in (2) with α = 2 but the amplitude A-a positive random variable with a finite support (0,A r )-fluctuates from sample to sample. We determine the moments of A and show that the probability distribution function P (A) has a rather nontrivial form characterized by a log-normal left tail (in the vicinity of 0) and a singular right tail (in the vicinity of the right edge A r of the support). In general, A is not self-averaging and its moments are supported by atypical realizations of disorder. These analytic predictions for the periodic Sinai model are confirmed by extensive numerical simulations. An analysis of the distribution of A for the original Sinai model with L ≡ ∞, where the spectrum is described by (2) with α = 1 [15,16] will be presented elsewhere.

II. THE PERIODIC SINAI MODEL

The precise definition of the model studied is as follows. Consider the Langevin dynamics of a tracer x(t) in a timeindependent potential V (x):

η dx(t) dt = - dV [x(t)] dx(t) + ξ t , ( 4
)
where η is the friction coefficient, ξ t is a Gaussian white noise with zero mean and covariance

ξ t ξ t = 2ηT δ(t -t ) , ( 5
)
and T is the temperature in units of the Boltzmann constant. The potential is periodic, such that V (x + L) = V (x), with L being the periodicity. Furthermore, we assume that the potential V (x) on the interval x ∈ (0,L) is a stochastic, continuous Gaussian process, pinned at both ends so V (0) = V (L) = 0, having zero mean and covariance

E(V (x)V (y)) = 2D V min(x,y) - xy L , 0 x, y L, (6)
where D V = V 2 0 /(2l), V 0 being a characteristic extent of the potential on a small scale of size l. In other words, V (x) on the interval (0,L) is the so-called Brownian bridge (BB) [START_REF] Mörters | Brownian Motion[END_REF] which has the representation

V (x) = W x - x L W L , (7)
where W x is a standard Brownian motion started at W 0 = 0 with correlation function

E(W x W y ) = 2D V min(x,y). ( 8
)
The overall potential on the entire x axis is then given by a periodically repeated realization of the BB (see Fig. 1).

Without loss of generality we set l = 1 in what follows, meaning that we measure L in units of l. We will also skip insignificant numerical factors focusing only on the dependence on the pertinent parameters, such as T , L, and V 0 .

Before we proceed, it is important to emphasize that the dynamics in Eq. ( 4) represents a combination of two paradigmatic situations: random motion in a periodic potential and the Sinai dynamics. Consequently, we expect that x(t) will exhibit two distinct temporal behaviors. At sufficiently short times t, t t c , where t c is a crossover time, the periodicity will not matter and the evolution of x(t) will proceed exactly in the same fashion as in the original Sinai model, Eq. (3). At longer times, t t c , the periodicity of the potential will ensure a transition to a standard diffusive behavior, so x(t) will converge to

x(t) ∼ 2D[V (x)] B t , ( 9
)
where B t is a Brownian trajectory with diffusion coefficient 1 and D[V (x)] is a sample-dependent diffusion coefficient (see, e.g., Refs. [START_REF] Lifson | [END_REF][31][32][33]):

D[V (x)] = D 0 L 0 dx L L 0 dy L exp V (x) -V (y) T , ( 10
)
where

D 0 = T /η is the bare diffusion coefficient in absence of disorder. Note that D[V (x)] D 0 [30] so D[V (x)
] is a random variable with support on (0,D 0 ).

III. NUMERICAL SIMULATIONS

In the main plot of Fig. 2 we show the temporal evolution of E(x 2 (t)) in a periodic BB Sinai model, with L = 64. The numerical evidence for the existence of the two temporal regimes described in (3) (at short times) and in (9) (at large times) is clear. We plot with points the numerical data averaged over 500 000 realizations of the random quenched potential. The dashed line is ln 4 (t) and agrees with the simulated data in the time region (100,1000), while the continuous thin straight line is t and fits perfectly the asymptotically large time region (say, from t min = 10 5 ). An intermediate very slow regime, where both the ln 4 (t) and the t dependence fail to fit the data, also appears clearly. Such a departure from the ln 4 (t) law is not observed for a periodic unconstrained Sinai potential, which we show in the inset, again for L = 64 (here the transition is from a Sinai to a ballistic regime, since for any finite L the potential is biased yielding an constant, but random from sample to sample, force superimposed on a periodic potential). As a matter of fact, this is a surprising feature since one may intuitively expect that in the case of a BB potential the typical barrier which a particle has to overcome should be less, due to stronger correlations, than that for an unconstrained Brownian potential, so for a BB the mean-square displacement E(x 2 (t)) should grow faster with time. This appears not to be the case and an apparent explanation is that for the BB potential the structure of a typical barrier which a particle has to bypass differs from the one for an unconstrained Brownian motion. This may be related to the recent observation [34] that the variance of a maximal positive displacement of a BB on some subinterval (0,L 1 ) with L 1 < L may be greater than the variance of the maximal displacement on the entire interval (0,L).

IV. ANALYTICAL RESULTS

The inset of Fig. 2 shows us that the transition from the Sinai regime at short times to the long time regime is not smeared in time but is sharp and allows us to consistently define a well-defined value of a transition time t c , which we will discuss below. Accounting for the intermediate, subdiffusive regime that appears in the case of the Sinai periodically repeated Brownian bridge, the same procedure allows us to define a transition time also in this case. We may expect that for t t c , the typical behavior of x(t) will be diffusive, so the lowfrequency (f 1/t c ) behavior of the power spectrum (1) will have the form of (2) with α = 2,

A 4D 0 = 1 L 0 dx L L 0 dy L exp V (x)-V (y) T . ( 11
)
Taking into account that for a standard Brownian motion with the diffusion coefficient D the amplitude in ( 2) is A = 4D, we expect A to have support (0,4D 0 ). In what follows we will focus on the statistical properties of A.

We start by analyzing the typical behavior of A based on an estimate for the typical value of A that we call A typ :

A typ 4D 0 ∼ exp E ln A 4D 0 . ( 12
)
Furthermore,

E ln A 4D 0 = E(ln J + L ) + E(ln J - L ) + 2 ln(L), ( 13
)
where J + L and J - L are stationary currents through a finite sample, of length L, of a Sinai chain,

J + L = 1 L 0 dx exp V (x) T , J - L = 1 L 0 dy exp -V (y) T . ( 14
)
Note that since E(V (x)) = 0, moments of arbitrary order obey

E((J + L ) k ) ≡ E((J - L ) k ) so E(ln J + L ) ≡ E(ln J - L ) (15)
and thus

E ln A 4D 0 = 2E(ln J + L ) + 2 ln(L). (16)
The statistical properties of the currents in finite Sinai chains have been analyzed in Refs. [22][23][24][25] for the case where V (x) is an unconstrained Brownian or an unconstrained fractional Brownian motion. It was shown (see, e.g. Ref. [25] for more details) that for sufficiently large values of L, the behavior of J + L is dominated by the maximum of V (x), V max ≡ max 0 x L V (x). Moreover, any given realization of disorder J + L can be bounded from below and from above by A 1 exp(-V max /T ) and A 2 exp(-V max /T ), where A 1 A 2 are L-independent constants. Consequently, the L dependence (up to an insignificant numerical factor) is captured by the estimate

J + L ∼ exp(-V max /T ).
In principle, this argument can be readily generalized for the case at hand, when V (x) is a BB, and we have merely to use the distribution P BB (V max ) of a maximal positive displacement of a BB on an interval (0,L), instead of the analogous distribution for an unconstrained Brownian motion used in Ref. [25]. This distribution P BB (V max ) is well known from the classical papers [35][36][37] and is given by

P BB (V max ) = 2V max D V L exp - V 2 max D V L , ( 17
)
where D V = V 2 0 /(2l). Using [START_REF] Montroll | Proc. Natl. Acad. Sci. USA[END_REF], we find that, dropping numerical constants,

E ln A 4D 0 ∼ - V 0 T L 1/2 , ( 18
)
so, for arbitrary values of k,

A typ 4D 0 k ∼ exp -k V 0 T L 1/2 . ( 19
)
Therefore, we expect that, for most realizations of the random potential V (x), the amplitude A of the power spectrum will decrease, as a stretched-exponential function exp(-L 1/2 ) of the periodicity L, and will exhibit an Arrhenius dependence on the temperature T .

Next we consider the behavior of the moments E(A k ) of the amplitude with arbitrary (positive or negative) values of k. When V (x) is an unconstrained Brownian motion, a general analysis of the functional in (10) or (11) has been presented in Ref. [28]. The disorder-average value (first moment) of this very functional, which also describes the ground-state energy in a one-dimensional localization problem, was determined in Ref. [38]. It was shown in Refs. [28,38] that the functional of the random potential in (10) or ( 11) can be bounded from below and from above by B 1 exp(-R/T ) and B 2 exp(-R/T ), where B 1 B 2 weakly depend on L and

R ≡ max 0 x L V (x) -min 0 x L V (x) (20)
is the range, or span, of the random potential V (x). Physically, R corresponds to the largest energy barrier that will be encountered by the tracer. Expecting that E(A k ) will show a stronger-than-power-law dependence on L (and we will show in what follows that this is the case), we may drop the constants B 1 and B 2 and write an estimate

A k ∼ exp(-kR/T ), (21)
which should capture the L, k, and T dependence of the moments up to insignificant pre-exponential factors.

To extend this analysis over the case of a BB potential and in order to calculate the moments of A for the case under study, we need to know the distribution of the range of a BB. This distribution was first derived in Ref. [39], in which R was referred to as an adjusted range of Brownian motion, and it is given in series form as

P BB (R) = R d 2 f (R) dR 2 + ∞ n=2 2n(n -1) df [(n -1)R] dR - df (nR) dR + (n -1) 2 R d 2 f [(n -1)R] dR 2 + n 2 R d 2 f (nR) dR 2 , ( 22
)
where, in our notation, f (R) = exp(-R 2 /D V L). For our purposes a slightly different form of P BB (R) will also turn out to be useful. To this end, we exploit here the observation made in Ref. [40] that the range of Brownian bridge and the maximum of Brownian excursion-a Brownian bridge constrained to stay positive-have the same distributions. The distribution of the maximum of a Brownian excursion has been extensively discussed in the literature and several forms of it have been derived (see, for example, Ref. [41]). Choosing a suitable one, we have, in our notation,

P BB (R) = √ 2π 5/2 (2D V L) 3/2 × d dR 1 R 3 ∞ n=1 n 2 exp - π 2 n 2 R 2 D V L . ( 23
)
The two expressions ( 22) and ( 23) coincide.

Now we have all the necessary ingredients to calculate the moments of A. Consider first the moments of negative (not necessarily integer) order. Using the form of P BB (R) in (22), and keeping only the leading exponential dependence on R, we average the estimate in (21) to obtain

E 4D 0 A k ∼ ∞ 0 dR exp k R T - R 2 D V L . ( 24
)
Evaluating this integral via steepest descent, we find that the maximum of the exponential is attained at R ∼ R * = kD V L/2T , and thus

E 4D 0 A k ∼ exp k 2 V 2 0 8T 2 L . ( 25
)
Therefore, the negative moments grow faster than exponentially with k and V 0 , exhibit a super-Arrhenius dependence on the temperature, and grow exponentially with the periodicity L.

The negative moments may also be computed directly by taking the average over the replicated 2k-fold integral to obtain

E 4D 0 A k = 1 0 . . . 1 0 k a=1 du a dw a × exp - D V L 2T 2 a,b |u a -u b | + |w a -w b | -2|u a -w b | + 2 a u a - a w a 2 , ( 26
)
where we have rewritten the integration variables using x a = Lu a and y a = Lw a to obtain the above. The right-hand side of ( 26) has the form of a partition function for k + k interacting particles of two types u and w at inverse temperature β = D V L/2T 2 . The corresponding Hamiltonian is explicitly given by

H = a,b |u a -u b | + |w a -w b | -2|u a -w b | + 2 a u a - a w a 2 . ( 27
)
The particles of type u and w attract particles of the same type with a linear attractive potential, and they repel particles of the other type, again with a linear potential. However, there is an additional interaction which harmonically binds the center of masses of the two particle types.

For k = 1 the integral in ( 26) can be performed exactly to give

E 4D 0 A = T V 0 2π L exp V 2 0 8T 2 L erf V 0 2T L 2 , ( 28
)
where erf(z) is the error function. In the limit L → ∞, the error function converges to 1, so the large-L asymptotic behavior of the latter expression is governed by the exponential function, which has precisely the same form as the one in (25) with k = 1. For larger k, the direct calculation of the integrals in (26) becomes very involved, and we instead resort to an approximate, but physically plausible, argument: In the limit of large L, the partition function is dominated by the ground-state energy. Due to the attraction between the same particle type, we expect that particles of the same type will condense at low temperature about the same point and hence we write u a = U and w a = W for all a. This gives the effective reduced low-temperature Hamiltonian

H 0 = 2k 2 ( 2 -) = 2k 2 - 1 2 2 - k 2 2 , ( 29
)
where = |U -W |. The value = 1/2 minimizes the energy leading to [START_REF] Lifson | [END_REF] in complete agreement with (25).

E 4D 0 A k ∼ exp D V k 2 L 4T 2 = exp k 2 V 2 0 L 8T 2 ,
For positive moments of the amplitude, we use the form of P BB (R) in (23). Keeping only the leading term in L, we find that the leading behavior of A k in ( 21) is given by

E A 4D 0 k ∼ ∞ 0 dR exp - k R T - π 2 D V L R 2 . ( 31
)
Again, we use the steepest descent approach to observe that the dominant contribution to the integral comes from a narrow region around R * = (2π 2 T D V L/k) 1/3 so the overall behavior of the positive moments of the amplitude of (not necessarily integer) order k is given by

E A 4D 0 k ∼ exp - 3 π 2 3 2 k V 0 T 2/3 L 1/3 , (32)
Therefore, the positive moments of the amplitude exhibit a stretched exponential dependence on the order of the moment k and on the characteristic scale of the potential V 0 , a sub-Arrhenius dependence on the temperature, and also decay with the periodicity L as a stretched exponential with the exponent z = 1/3, that is, slower than predicted by the estimate based on the typical realizations of disorder (19). Note, however, that the result in (32) pertains to the asymptotic limit when L → ∞.

For small values of L, we expect that positive moments will exhibit the typical behavior given by (19).

In Fig. 3 we show with symbols our estimates for A 2 obtained from numerical simulations for different values of L. In this case, we are not able, in the limits of our numerical precision, to distinguish a small L regime. The continuous line is our best fit to the form a exp(-L b ), where we obtain the value b = 0.37 ± 0.02. The precision of the numerical data does not support a fit with more parameters (that means that we cannot include subleading corrections). The value b = 0.37 that we find for our estimated exponent (in the sense it is estimated by numerical data in a finite region of values of L) is close to the expected asymptotic value of 1/3 for large L but probably feels the contamination from the low L regime.

Before we proceed to the analysis of the distribution of the amplitude A, two remarks are in order. We first note that there exists another physical system, completely unrelated to the one under study, which exhibits essentially the same behavior. It concerns survival of diffusing particles, with diffusion coefficient D V , in the presence of perfect traps, independently and uniformly distributed on a one-dimensional line. Identifying L as time and 1/T as the density of traps, we see that in one-dimensional systems the behavior of the moments of the probability S L that a particle survives up to time L is identical to the behavior of the moments of A (see, e.g., Ref. [42] and references therein). At sufficiently short times L, S L follows the stretched-exponential form in (19), which is tantamount to the so-called Smoluchowski regime, while for L → ∞, the moments of S L obey the form in (32) as they are supported by the optimal fluctuation R * = (T V 2 0 L/k) 1/3 of a random cavity devoid of traps. This ultimate, late-time, regime has the celebrated fluctuation-induced tails [43,44], which are also intimately related to the so-called Lifshitz singularity in the low-energy spectrum of an electron in a one-dimensional disordered array of scatterers [45]. Below we will show that an analogous essential singularity shows up in the distribution P (A).

Second, we are now in position to estimate the crossover time t c and, hence, to determine the upper bound on the frequency for which the spectrum (2) is characterized by an exponent α = 2. Recalling that our numerical results show a sharp crossover from the Sinai regime (3) to the diffusive behavior in (9), we may estimate t c by simply equating the mean-squared displacement in the Sinai (3) and diffusive regimes (9), i.e., ln 4

(t c ) ∼ E(D[V (x)]) t c , ( 33
)
which gives

t c ∼ 1 E(D[V (x)]) . ( 34
)
Now noticing that D[V (x)] ∼ A, we can expect that t c will display a different dependence on the periodicity L (and the other system parameters) for small and large values of L. For sufficiently small L [but still large enough so the behavior in (3) has enough space to emerge], the typical trajectories of disorder, such that |V (x)| ∼ √ x, will dominate and

t c ∼ exp V 0 T L 1/2 , ( 35
)
which simply tells us that, for sufficiently small L, the crossover time t c to diffusive regime is a time needed for x(t) to travel over a distance L encountering a typical barrier V 0 L 1/2 which x(t) overcomes due to thermal activation. Note the Arrhenius dependence of t c on the temperature T . For larger values of L the behavior of the average amplitude A, given by Eq. ( 32), becomes supported by atypical realizations of disorder with the optimal fluctuation trajectories of |V (x)| ∼ x 1/3 . For such L, we have, by virtue of (32),

t c ∼ exp c V 0 T 2/3 L 1/3 , ( 36
)
where c is a numerical constant; this means that, for larger periodicities, t c exhibits a slower growth with L. Note that in this case t c has a rather unusual sub-Arrhenius dependence on the temperature. In order to discuss this point and to use our numerical data to better understand it, we start by defining a time of exit from the Sinai asymptotic regime. The Sinai regime holds in the first part of the dynamical evolution. We define an exit time from it as the time t (1) c as the minimal time such that

E(|x(t)|) -E(|x(t)|) Sinai > 3σ Sinai (t), (37)
where by the Sinai label we denote an average over the motion in an infinite, unconstrained Sinai potential. In this way we are observing the time where the departure of the motion in the periodic Brownian bridge potential substantially differs from the one in a Sinai infinite potential (σ Sinai is the standard deviation over our numerical estimate for the infinite Sinai motion). On our time scales and sample size this procedure is accurate enough to give a sensible estimate of t (1) c . We assume now that ln t (1) c ∼ a (1) + L b (1) . (38) Since our numerical data are not accurate enough to allow us to disentangle precisely the subleading corrections to this behavior, we analyze our data by defining a size-dependent exponent b (1) (L,2L), computed by using Eq. ( 38) for size L and size 2L. The numerical values computed for t (1) c (L) and the one for t (1) c (2L) are used to disentangle the value of b (1) (L,2L) as estimated from these two values of the lattice size. The limit for large L of b (1) (L,2L) is b (1) .

We plot this estimated exponent as a function of L in Fig. 4. In this case, the crossover we have derived analytically clearly emerges from the numerical data, which give an estimated exponent close to 1/2 for small L values and close to 1/3 for larger values of the size L.

We finally turn to the analysis of the distribution P (A) of the amplitude of the low-frequency power spectrum (see Fig. 5). Examining first the negative moments of A, we observe that they are growing functions of L and k, which hints that such a behavior of A is derived from the left tail of the distribution P (A), i.e., when A is close to 0. Furthermore, the quadratic dependence of the moments on the order of the moment k in the exponential is a fingerprint of the log-normal distribution, which suggest that the left tail of P (A) has the form: Note that this distribution is unimodal, with the most probable value of A mp ∼ exp(-V 2 0 L/4T 2 ), which is, for sufficiently large L, much smaller and closer to 0 than the typical value in (19). Further, positive moments in (32) are, for large L, much larger than those expected from the typical realizations of disorder (19). This means, in turn, that the behavior in (32) stems apparently from the right tail of the distribution P (A) when A is close to the right edge of the support, i.e., A ≈ A r = 4D 0 . Let us formally write 4D 0 0 A k dA P (A) ∼ A k r exp -

P (A) ∼ 1 A exp - 2T 2 ln 2 (A) V 2 0 L . ( 39
kV 0 √ L T 2/3 , ( 40
)
where, for simplicity of notation, any numerical constant in the exponential of the right-hand side is included in V 0 . We assume that the major contribution to the integral on the left-hand side of (40) comes from a narrow region close to the right edge of 39) is shown for small A as a solid green curve while the prediction of Eq. (47) for the right tail is shown for large values of A by the dashed blue line. the support. Changing the integration variable as

z = T V 0 √ L ln A r A , (41)
we cast (40) into the form

4D 0 V 0 √ L T ∞ 0 dz exp - kV 0 √ L T z × exp - V 0 √ Lz T P (z) ∼ exp - kV 0 √ L T 2/3
. (42) Using then the formal definition of the Laplace transform of one-sided stable Lévy distribution L ν (z) with index ν (see, e.g., Ref. [46]), ∞ 0 exp(-pz)L ν (z) ≡ exp(-p ν ), (43) we immediately infer that

P (A) ∼ T V 0 √ LA L 2/3 T V 0 √ L ln A r A . ( 44
)
Note that the result in ( 44) is expected to hold only in the vicinity of the right edge of the support, and we consider its asymptotic form in this domain. For A ≈ A r , the argument z in the one-sided Lévy distribution L 2/3 (z) is close to zero, so its asymptotic behavior is given by

L 2/3 (z) ∼ z -2 exp - b z 2 , ( 45
)
where b is a computable constant. For A ≈ A r , we have that

z ≈ T V 0 √ L 1 - A A r , ( 46
)
so eventually we find the following asymptotic representation of the distribution P (A) close to the right edge of the support:

P (A) ∼ 4D 0 V 0 √ L (4D 0 -A) 2 exp - 4D 0 V 0 √ bL T (4D 0 -A) 2 .
(47)

Note that the distribution in (47) exhibits an essential singularity in the vicinity of A r , which is related to the Lifshitz singularity. In Fig. 5 we plot the empirical probability distribution obtained in numerical simulations, together with the best fits to the asymptotic forms (39) and (47): The agreement is remarkable.

FIG. 3 .

 3 FIG. 3. The second moment of the fitted amplitudes of the power spectrum in a BB potential. Numerical results are shown as circles along with the fit of the data by a exp(-L b ), shown as a solid line, yielding the fitted value b = 0.37.

)FIG. 4 .

 4 FIG.4. The exponent b(1) (L,2L) in Eq. (38) as a function of L.

FIG. 5 .

 5 FIG. 5. Distribution P (A) of the amplitudesA for a BB potential, plotted with circles (numerical results). In the main figure L = 64 while the inset shows the results for L = 32. The log-normal fit corresponding to Eq. (39) is shown for small A as a solid green curve while the prediction of Eq. (47) for the right tail is shown for large values of A by the dashed blue line.

  E(x 2 (t)) in a periodic Sinai model (with a periodic BB), numerical data shown as points. Also shown by the dashed line is the fit c 1 ln 4 (t) for the short-time Sinai regime along with the solid line late-time fit c 2 t. Inset: As in the main figure but for an unconstrained periodic Sinai potential.

	10 6 10 6 10 6
			10 6 10 6 10 6
	10 4 10 5 E(x 2 (t)) 10 4 10 5 E(x 2 (t)) E(x 2 10 4 10 5 (t))	E(x 2 (t)) E(x 2 (t)) E(x 2 (t))	10 2 10 2 10 2 10 2 10 2 10 2 10 3 10 4 10 5 10 3 10 4 10 5 10 3 10 4 10 5	10 3 10 3 10 3	10 4 10 4 10 4	10 5 10 5 10 5
					t t t
	10 3 10 3 10 3
	10 2 10 2 10 2
	10 2 10 2 10 2		10 3 10 3 10 3		10 4 10 4 10 4	10 5 10 5 10 5	10 6 10 6 10 6	10 7 10 7 10 7
						t t t
	FIG. 2. Main:

ACKNOWLEDGMENTS E.M. and G.O. thank the Kavli Institute for Theoretical Physics of the University of California at Santa Barbara for warm hospitality during their participation at the program Active Matter: Cytoskeleton, Cells, Tissues and Flocks, where this work was initiated. D.S.D. and G.O. acknowledge partial support from the Office of Naval Research Global Grant No. N62909-15-1-C076 and also thank the Institute for Mathematical Sciences of the National University of Singapore for warm hospitality and financial support. E.M. used funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (Grant No. 694925).


Mushegh Rafayelyan

Etienne Brasselet
email: etienne.brasselet@ubordeaux.fr

Bragg-Berry mirrors: reflective broadband q-plates

We report on the experimental realization of flat mirrors enabling the broadband generation of optical vortices upon reflection. The effect is based on the geometric Berry phase associated with the circular Bragg reflection phenomenon from chiral uniaxial media. We show the reflective optical vortex generation from both diffractive and nondiffractive paraxial light beams using spatially patterned chiral liquid crystal films. The intrinsic spectrally broadband character of spin-orbit generation of optical phase singularities is demonstrated over the full visible domain. Our results do not rely on any birefringent retardation requirement and, consequently, foster the development of a novel generation of robust optical elements for spin-orbit photonic technologies. Tailoring intensity, phase, and polarization of electromagnetic fields is a cornerstone of today's information and communica tion technologies. In particular, the ever increasing demand for a higher information rate has led photonic technologies to play a paramount role, especially through the use of the orbital an gular momentum that light may carry, which can be associated with an infinite dimensional basis. The most common illustra tion corresponds to the paraxial description of electromagnetic fields by means of the so called Laguerre Gauss modes. Indeed, each of these modes is characterized by an optical phase singu larity with an integer topological charge l that is described by a complex field amplitude proportional to expilϕ (ϕ is the usual azimuthal angle in the transverse plane) and associated with orbital angular momentum lℏ per photon [1] (ℏ is the reduced Planck constant). Therefore, the production of a singular phase mask having an amplitude transmittance proportional to expilϕ is a basic issue for which several tech nologies have been developed [2], including powerful ones based on helicity dependent geometric phases [3,4]. In particu lar, the use of azimuthally varying anisotropic structures for helicity controlled generation of optical vortex beams emerged in the early 2000s [5]. Namely, a circularly polarized paraxial light beam with helicity σ 1 passing through a half wave plate having an optical axis orientation angle of the form ψ qϕ with a q half integer (in short, a "q plate") experiences a change of its orbital state l by an amount Δl 2σq. Retrospectively, its significance started to be fully appreciated after nematic liquid crystal q plates working in the visible do main were realized in 2006 [6] and the coupling between the spin and orbital angular momentum of light was pointed out. Indeed, since then, q plates eased and made possible numerous fundamental and applied realizations in optics, both in the classical and quantum regimes, as reviewed in [7].

Importantly, the latter q plates are transmissive spin orbit optical elements that require π, a birefringent phase retardation condition. In this context, we recently unveiled the geometric Berry phase associated with the circular Bragg reflection phenomenon in chiral anisotropic optical media [8], which brings a novel paradigm to achieve wavelength independent pure spin orbit topological shaping of light from two and three dimensional "Bragg Berry" mirrors (i) in the reflection mode and (ii) without need for any birefringent retardation requirement. Existence of a geometric phase at reflection from cholesteric liquid crystals planar slabs is also discussed in another recent independent report [9]. From the application point of view, we also note a closely related recent work intro ducing the control of reflected wavefronts from planar optics with patterned cholesteric liquid crystals [10] with the realiza tion of reflective lenses and deflectors from Bragg Berry mirrors.

Here we propose, realize, and experimentally demonstrate the broadband reflective optical vortex generation from flat Bragg Berry mirrors for both diffractive and nondiffractive par axial light beams. This is done by using either Gaussian or Bessel incident beams whose reflection off a chiral Bragg mirror endowed with suitable surface orientational boundary condi tions led to the production of Laguerre Gauss and higher order Bessel like beams, respectively. From the spectral point of view, we report on optical vortex generation over the full visible range, at least from 450 to 650 nm wavelengths. By doing so, our results substantially extend the reach of previous works [8 10] and set the basis for the development of a novel generation of spin orbit optical elements whose demonstrated robustness against polychromaticity and fabrication constraints offer a valuable alternative to existing q plate limitations. Still, broadband transmissive liquid crystal q plates of arbitrary order can be realized using chiral multilayer structures [START_REF] Li | Proc. SPIE[END_REF]. Solid state strategies realizing effective q plates endowed with broadband capabilities have also been reported [START_REF] Bouchard | [END_REF]13]. However, the latter approaches are limited to q 1 and the poor spatial resolution of the effective optical axis patterning (several millimeters) limits their application potential.

In practice, cholesteric liquid crystals are prime choice ma terials to realize chiral Bragg mirrors. In fact they combine the strong optical anisotropy, Δn, of nematic liquid crystals with a helical supramolecular ordering of the liquid crystal molecules characterized by a pitch, p, which is the distance over which the average molecular axis rotates by 2π. As such, thick enough cholesteric films (a few pitches are enough) may selectively re flect one of two circular polarization states via the circular Bragg reflection phenomenon [14]. Considering light propagation along the helical axis, the circular Bragg reflection corresponds to the existence of a photonic bandgap centered on the wave length λ np (n is the average refractive index of the choles teric) and characterized by a spectral width Δλ pΔn [START_REF] Oswald | Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments[END_REF]. The Bragg reflected photons are those having the helicity σ χ, where χ 1 is the handedness of the supramolecu lar cholesteric helix. As shown in [8], it is the preservation of the photon helicity at circular Bragg reflection, associated with the molecular orientation at the surface of the mirror, that leads to the production of a geometric Berry phase for the reflected light. More precisely, previous works established that a cholesteric Bragg mirror characterized by a surface molecular orientation angle ψ generates a phase factor 2σψ for the Bragg reflected field [8 10]. We thus expect that Bragg Berry mirrors, having a patterned surface molecular orientation of the form ψ qϕ, q half integer, produce a change of the incident orbital state l by an amount Δl 2χq.

Flat Bragg Berry mirrors are prepared from a 10 μm thick right handed cholesteric Bragg mirror made of liquid crystal material MDA 02 3211 (from Merck), which is characterized by pitch p 347 nm, birefringence Δn 0.195, and average refractive index n 1.604 at 589.3 nm wavelength and temperature 20°C. The cholesteric slab is sandwiched between two glass substrates coated by an azobenzene based surface alignment layer with submicron thickness (from Beam Co.), as depicted in Fig. 1(a). Two alignment layers are optically treated to ensure azimuthally varying orientation of the liquid crystal molecules at both ends of the liquid crystal layer of the form ψ qϕ with q half integer. In this work, without loss of generality, we restrict our demonstration of principle to the case q 1. This is illustrated in Fig. 1(c), which shows the spatial distribution of the orientational boundary conditions obtained by polarization microscopy (CRi Abrio Micro imaging system) of a coated glass substrate alone. The residual birefringent phase retardation of the alignment layer is on the order of 0.01π, and its contribution to the reflected field is weak enough to be safely discarded.

Experimental demonstration of optical vortex generation is done at a 532 nm wavelength (that falls in the circular photonic bandgap of our material) by using a continuous laser beam following the setup shown in Fig. 1(d). First, the Bragg Berry mirror is illuminated at normal incidence by a circularly polarized Gaussian beam with helicity σ χ with a waist ra dius in the plane of the mirror w ∼ 0.25 mm [see Fig. 2(a)].

The far field reflected light is collected by camera CCD 2 by placing the lens L in f f configuration with respect to the patterned chiral Bragg mirror and the camera [see Fig. 2(b)]. A doughnut intensity profile characteristic of the presence of the expected on axis optical phase singularity of topological charge 2χ is observed. Theoretically, this is described by a Fourier transform, which expresses in the polar coordinate k; θ as Ek; θ ∝ R 2π 0 R ∞ 0 ΦϕGr exp ikr cosθ ϕ rdrdϕ, where Gr exp r 2 ∕w 2 is the Gaussian beam incident amplitude on the mirror and Φϕ exp 2iχϕ is the reflective phase mask of the Bragg Berry mirror. Integration over the azimuthal coordinate gives Ek; θ ∝ exp 2iχθ R ∞ 0 GrJ -2χ krrdr, where J n is the nth order Bessel function of the first kind, which unveils the optical phase singularity. The corresponding intensity profile is axisymmetric and is given by

Ik ∝ 1 exp k 2 w 2 ∕41 k 2 w 2 ∕4∕k 2 .
The result of simulations is shown in Fig. 2(c), which displays both the intensity (luminance) and phase (colormap) profiles, which give fair agreement.

Further support of singular beam shaping dictated by a change Δl 2χq of the incident orbital state is made by an alyzing the phase profile of the field just after it reflects off the sample. This is shown in Fig. 3(a), which displays the intensity pattern collected by camera CCD 2 by placing the lens L in 2f 2f configuration with respect to the sample and the cam era. Obviously, the observed intensity pattern corresponds to a Gaussian envelope since diffraction has not yet taken place. However, an optical phase singularity is already imprinted into the field, which is revealed by interfering with the field with a reference Gaussian beam. The resulting interference pattern is shown in Fig. 3(b), which exhibits a fork like intensity pattern with two teeth whose intensity modulation contrast is opti mized by placing a polarizer before CCD 2 [see Fig. 1(d)]. This demonstrates experimentally that the mirror behaves as an effective reflective singular phase mask with topological charge 2 as indicated above while describing the reflected far field. The latter "near field" observations are confronted with simulations in Fig. 3(c), which displays the intensity pattern jΦG G ref j 2 where G ref refers to the referenced Gaussian beam whose propagation direction is slightly tilted with respect to that of the probed field.

Then we formally extend above topological shaping of light to diffraction free optical fields. This is done by preparing the incident field as a circularly polarized fundamental Bessel beam with helicity σ χ and placing a glass axicon in the course of the above incident Gaussian beam [START_REF] Mcgloin | [END_REF], between beamsplitters BS 1 and BS 2 [see Fig. 1(d)]. The incident intensity pattern collected by camera CCD 1 is shown in Fig. 4(a), which corre sponds to an axicon with a full apex cone angle of 178°. The reflected field after a few centimeters of propagation is imaged on CCD 2 and shown in Fig. 4(b). The corresponding intensity pattern is representative of the expected higher order Bessel beam as expected from the reflective phase mask ρϕ, whose order two is unveiled by interferometry as shown in Fig. 4(c), where a tilted Gaussian reference arm is used. Indeed, by count ing the fringes around the central part of the pattern, one can retrieve the characteristic 4π exploration of the phase along the circle.

Finally, we explore the polychromatic behavior of the Bragg Berry mirror in the visible domain by replacing the monochro matic laser beam with a supercontinuum whose spectrum is pre sented in Fig. 5(a). It corresponds to the image of the reflected beam from a usual broadband mirror using a dispersion prism. Replacing the latter mirror with the Bragg Berry mirror, one gets the Bragg reflection spectrum shown in Fig. 5(b), which exhibits the characteristic high reflectance spectral bandwidth Δλ of a cholesteric mirror. However, the singular nature of a Bragg reflected field goes beyond the spectral region of the circular photonic bandgap. This is illustrated in the bottom part of Fig. 5 that shows the intensity patterns of the Bragg reflected polychromatic vortex beam with topological charge 2 for a set of five wavelengths, from 450 to 650 nm by steps of 50 nm, fol lowing the monochromatic approach used to obtain Fig. 2(b) by using a set spectral filter with 10 nm full width at half maxi mum placed before the camera CCD 2 . As one can see from Figs. 5(c) to 5(g), the quality of the generated optical beam is decreased outside the photonic bandgap, where the reflected sig nal is low and several factors can alter topological shaping purity, such as the imperfect refractive index matching conditions be tween glass substrate and the average refractive index of the aniso tropic medium [14] and the presence of photoalignment layers. In practice, this can be easily circumvented by using either exter nal field (thermal, electrical, magnetic) effects or oblique inci dence, which are well known to drastically shift the bandgap [START_REF] Oswald | Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments[END_REF].

Although the reflected intensity is obviously wavelength dependent outside the circular photonic bandgap, the purity of the generated vortex is wavelength independent. This is illus trated in Figs. 6(a) and 6(b) where the reflectance and phase spectra of the helicity preserved reflected field for the incident Bragg polarized light is calculated from the Berreman approach [17] as a function of the orientation, say ψ 0 , of the liquid crystal director at z 0 (see Fig. 1). Therefore, our approach does not require the use of post polarization filtering as is the case of the usual transmissive q plate. In our case, low reflectance outside the bandgap implies the need to use larger acquisition time, hence lower signal to noise ratio, as one can see in Figs. 5(c) and 5(g). Moreover, the robustness of the pure broadband optical vortex generation versus the angle of incidence is illus trated in Figs. 6(c) and 6(d) for a 30°external incidence angle in glass substrate. Note that the two main effects of oblique in cidence are (i) to shift the bandgap to the blue spectral region and (ii) to break axisymmetry even more than the incidence angle is increased, as one can see from the modulated reflectance versus ψ 0 in Fig. 6(c). This emphasizes the large acceptance angle of Bragg Berry mirrors.

Summarizing, we have demonstrated that flat spin orbit reflective optical elements enable complex phase engineering of light fields, whatever their diffractive or nondiffractive nature. Noticeably, existing strategies to go beyond the present limitations associated with a helicity dependent, high reflectance regime over a spectral window of width of a few tens of nanometers which had been originally developed for non patterned cholesteric mirrors as reviewed in [18] could formally be extended to the case of patterned surface anchoring conditions. In general, the use of anisotropic chiral metamate rials may offer exotic photonic bandgaps [19]. Moreover, the sensitivity of liquid crystals to external fields should bring tun able or self induced operating conditions. On the other hand, state of the art point by point photoalignment techniques allow for considering arbitrary singular or non singular patterns [20,21]. Finally, recalling that Bragg Berry mirrors have no re strictions in terms of cell thickness and that curvature may lead to additional topological shaping features [8], this work should foster the development of spin orbit photonic technologies.

Note: During the revision stage of this Letter, a closely related work dealing with polychromatic optical vortex gener ation from patterned cholesteric liquid crystal slabs was published, see [22].

  OCIS codes: (160.1585) Chiral media; (160.3710) Liquid crystals; (050.4865) Optical vortices.

Fig. 1 .

 1 Fig. 1. (a) Sketch of the Bragg Berry flat mirror that consists of a space variant cholesteric liquid crystal flat mirror sandwiched between two planar glass substrates provided with photoalignment layers that ensure radial ordering of the liquid crystal molecules at both ends of the liquid crystal slab. (b) Illustration of the cholesteric liquid crystal supramolecular right handed (χ 1) helix with pitch p that is ori ented along the z axis. The radial surface anchoring condition is visu alized by the "molecule" at z 0 in magenta color. (c) Experimental map of the space variant orientation ψ of the alignment layers in the x; y plane. (d) Experimental setup for optical vortex generation from an incident beam impinging at normal incidence on the Bragg Berry mirror. M , mirror; BS 1;2;3 , beamsplitters; L, lens; P, polarizer; and CCD 1;2 , imaging devices.

Fig. 2 .

 2 Fig. 2. (a) Experimental incident Gaussian intensity profile on the sample collected by camera CCD 1 . (b) Far field intensity profile collected by camera CCD 2 owing to the lens L placed in f f con figuration. (c) Calculated intensity and phase that correspond to panel (b). The luminance refers to the intensity and the colormap refers to the phase from 0 to 2π.

Fig. 3 .

 3 Fig. 3. (a) Experimental intensity profile just after a normally incident Gaussian beam is reflected off the Bragg Berry mirror, that is collected by camera CCD 2 placing the lens L in 2f 2f configura tion. The speckle like pattern is reminiscent of residual imperfections of the alignment layer. (b) Interference pattern between the field that corresponds to panel (a) and a reference Gaussian beam. A fork like interference pattern of order two is observed. (c) Simulation of the observed singular interference pattern shown in panel (b).

Fig. 4 .

 4 Fig. 4. (a) Experimental incident fundamental Bessel intensity pro file on the sample collected by camera CCD 1 . (b) Reflected field in tensity profile collected by camera CCD 2 . (c) Interference pattern between the field that corresponds to panel (b) and a reference Gaussian beam. A fork like interference pattern of order two is ob served. Dashed segments and fringes numbering are given as a guide.

Fig. 5 .

 5 Fig. 5. (a) White light laser source spectrum of the reflected beam off a usual broadband dielectric mirror using a dispersion prism. (b) Same as panel (a) in the case of a Bragg Berry mirror with q 1. (c) (g) Selection of far field vortex beam intensity profiles for a set of wavelengths.

Funding.

  Erasmus Mundus Action 2 BMU MID (MID20121729).

Fig. 6 .

 6 Fig. 6. (a), (b) Reflectance and phase spectra of the helicity pre served light field component of the reflected field for Bragg polarized incident field as a function of ψ 0 . (c), (d) Same as in (a) and (b) for 30°i ncident angle in the glass substrate. Calculations are made using the refractive indices parallel and perpendicular to the director of our liquid crystal mixture, namely n ∥ 1.7013 and n ⊥ 1.5064 at 589.3 nm wavelength and temperature 20°C and glass substrate refrac tive index n glass 1.52.