aa r X i v : . [ phy s i c s . op ti c s ] A ug Hyperdiffusion of quantum waves in random photonic lattices
Alexander Iomin
Department of Physics, Technion, Haifa, 32000, IsraelPhys. Rev. E , 022139 (2015) A quantum-mechanical analysis of hyper-fast (faster than ballistic) diffusion of a quantum wavepacket in random optical lattices is presented. The main motivation of the presented analysisis experimental demonstrations of hyper-diffusive spreading of a wave packet in random photoniclattices [L. Levi et al. , Nature Phys. , 912 (2012)]. A rigorous quantum-mechanical calculation ofthe mean probability amplitude is suggested, and it is shown that the power law spreading of themean squared displacement (MSD) is h x ( t ) i ∼ t α , where 2 < α ≤
3. The values of the transportexponent α depend on the correlation properties of the random potential V ( x, t ), which describesrandom inhomogeneities of the medium. In particular, when the random potential is δ correlatedin time, the quantum wave packet spreads according Richardson turbulent diffusion with the MSD ∼ t . Hyper-diffusion with α = 12 / PACS numbers: 42.25.Dd, 05.40.-a, 03.65.-w
I. INTRODUCTION
Recently, it has been demonstrated, experimentallyand numerically [1] that space-time disordered media ac-celerate the transport in a way, when an initial wavepacket spreads at a rate faster than ballistic with themean squared displacement (MSD) h x ( t ) i ∼ t / . Thiseffect has been explained in the framework of classicalmechanical approach due to continuous expansion of thetransverse momentum spectrum in an arbitrary space-time random potential [1–3]. In this paper we suggesta quantum-mechanical explanation of this experimentalobservation of the disorder enhanced transport in pho-tonic lattices [1], which is a more general approach for aquantum wave packet spreading in randomly inhomoge-neous media [1, 4, 5].An investigation of wave spreading in randomly inho-mogeneous media is a long lasting problem, which hasbeen well reviewed already more than thirty years ago[6, 7], where a variety of applications have been consid-ered, and this theory has also a strong impact on statis-tical methods in physics [8] (see also recent review [9]).The main objective of the present research is an esti-mation of the mean squared displacement (MSD) of thewave packet spreading in the transversal direction (whichis the x axis) under its propagation along a wave-guide.Here the main accent is made on the rigorous calculationof the mean probability amplitude. It is known that awave propagation with the wavelength λ in a long range–dependent wave-guide can be described by the parabolicequation in the limit of a small-angle propagation [6, 10].This equation corresponds formally to the Schr¨odingerequation with an effective Planck constant of the orderof λ . Formally, the longitudinal coordinate plays a roleof an effective time t , and the dynamics takes place ina random potential V ( x, t ), which is a space-time de-pendent noise. A rigorous quantum-mechanical consid-eration is suggested for this Langevin-Schr¨odinger equa- tion, and the wave function is obtained as functional of V ( x, t ). We show that the quantum process of spreadingdepends on the time correlation properties of the ran-dom potential. We obtain the hyperfast spreading ofthe quantum wave packet with the MSD h x ( t ) i ∼ t α with the transport exponent 2 < α ≤
3. When thenoise is a Markov ( δ -correlated) process, the quantumwave packet dynamics corresponds to Richardson diffu-sion [11] with the MSD of the order of t . This classicalturbulent diffusion is obtained here by the rigorous quan-tum mechanical treatment. A phenomenological statis-tical approach dated back to works by Kolmogorov andObukhov [12, 13] suggested this turbulent accelerationby means of a Gaussian δ -correlated noise [14], added tothe dynamical system ¨ x + V ( t ) = 0. In this case, dueto the noise term V ( t ), Richardson diffusion takes placewith the MSD h x ( t ) i ∼ t , which is due to the diffu-sive spread of the velocity profile h ˙ x ( t ) i ∼ t . In quan-tum mechanics, the Kolmogorov-Obukhov approach wasfirst applied in Ref. [15] to study a non-diffusive motion,where a Gaussian δ correlated in time random processwas treated in the framework of the Furutsu-Novikov for-mula [16, 17] for the mean probability amplitudes [6, 9].Recently, it was applied to described a tracer behavior[18] for an explanation of a limiting case of experimen-tal realization of quantum-mechanical superdiffusion ofultra cold atoms [19].However, in real experimental realizations of the dis-order in photonic lattices [4, 5], the random potentialdoes not possess this Markov property due to the finitesize of the optical wavelength λ . Therefore, the quantumdynamics is considered in an arbitrary random potential V ( x, t ), which is correlated in both time and space. Inthis case a rigorous quantum-mechanical analysis cannotbe performed, and a suitable approximation is suggestedto treat this random quantum dynamics.It is well known that the quantum dynamics can bedescribed by a complex Gaussian kernel in functional in-tegration [20, 21]. When V ( x, t ) is δ correlated in time,it does not affect the quantum Gaussian paths in thefunctional integration that makes it possible to treat thequantum mechanics rigorously, like in the Richardson dif-fusion case. The situation changes essentially, when therandom process is strongly correlated. Then the quan-tum paths are affected by the inhomogeneities of themedia. The rigorous analysis is impossible, and terms,which are responsible for this “intertwine”, are treatedapproximately by averaging this part of the quantumpaths. The suggested averaging procedure is performedin self-consistency with the quantum spreading, and as aresult of this, we obtain hyperdiffusion of the quantumpacket spreading, when the MSD is of the order of t / .which coincides with a result obtained in Refs. [2, 3] inthe ray dynamics limit.Therefore, as the result of the parabolic equation ap-proximation of the wave process, the wave spreading inrandomly inhomogeneous media is investigated in theframework of quantum mechanics with a random poten-tial, which is the Langevin Schr¨odinger equation. Animportant motivation for this analysis is experimentalinvestigations of quantum wave packet spreading in ran-dom optical lattices [1, 4, 5]. Another interesting motiva-tion of the present analysis is investigation (experimentaland theoretical) of sound waves spreading in underwateracoustics in the presence of random environments (seee.g. recent results in Refs. [22, 23]). A. Parabolic equation approximation
The method of parabolic equation approximation wasfirst applied by Leontovich in studying radio-wavesspreading [24] and later it has been developed in de-tail by Khohlov [25] (see also [10]). Parabolic equationfor monochromatic light propagation in two dimensionalrandomly inhomogeneous media reads [1, 4, 6] i∂ z Ψ = [ − k ∂ x Ψ − kn ∆ n ( z, x )]Ψ . (1)Here ∆ n ( x, z ) is local fluctuations of refractive index n = n + ∆ n , and z is the propagation direction of thewave with the wave index k = 2 πn /λ , therefore an ef-fective semiclassical parameter is of the order of 1 /k . Inwhat follows it is convenient to work with dimensionlessvariables and parameters. Taking into account that Eq.(1) has a form of a Schr¨odinger equation, one defines thedimensionless effective time t = z/λ and the dimension-less effective Planck constant ˜ h = kλ = πn , then, the This presentation of the quantum dynamics by means of an aux-iliary Markov field in the framework of the Feynman-Kac formuladoes not suppose any Markovian property of quantum mechan-ics. dimensionless quantum momentum is λkλ ∂ x = ˜ h∂ x , where x/λ → x . Note that the wavelength in the experimentalsetup is λ ∼ . µ m and ∆ n/n ∼ − ≪ II. QUANTUM LANGEVIN EQUATION
Formally, the wave function Ψ( x, t ) describes the dy-namics of a quantum wave packet (particle) in randomtime-dependent optical potential V ( x, t ), and it is gov-erned by the Schr¨odinger equation, which reads ∂ t Ψ( x, t ) = [ i ˜ h∂ x / iV ( x, t ) / ˜ h ]Ψ( x, t ) (2)with the initial conditionΨ( x, t = 0) = Ψ ( x ) . (3)Considering the optical random potential V ( x, t ) as anexpansion of a quasiperiodic function, one has [1, 3] V ( x, t ) = 1 √ N N X m = − N A m exp( ik m x − iω m t ) + c.c. , (4)where the coefficients of the expansion A m are randomcomplex values, while k m and ω m are independent ran-dom real values. Denoting averaging over the Gaussianensemble by h . . . i V , we obtain that A m are controlled bya Gaussian distribution with the averaging property h A m i V = h A m A n i V = 0 , h A ∗ m A n i V = σ δ m,n . (5)From this property, one obtains for the 1D space-timedependent potential V ( x, t ) h| V ( x.t ) | i V = 2 σ . (6)Note that this formulation of the random potential isgeneral and corresponds to the experimental setup [1].Since V ( x, t ) is a random function, the Schr¨odinger Eq.(2) is a Langevin equation with a multiplicative noisepotential V ( x, t ).Following Ref. [6], this equation can be solve exactly.The solution of Eq. (2) can be presented in the formof a functional integration over an auxiliary Gaussianfield λ ( t ). The details of the calculation can be foundin Ref. [9]. However, here we present an alternating wayof the solution, which is more suitable for the quantum-mechanical consideration. A. Solution of Langevin equation
A formal integration of the Schr¨odinger Eq. (2) yields a T ordered (time ordered) form of the evolution operator,which acts on the initial wave functionΨ( x, t ) = ˆ T exp h i ˜ h Z t ∂ x dτ + i ˜ h Z t V ( x, τ ) dτ i . (7)Under the sign of the time ordering operator ˆ T , all valuesare commuted, and the kinetic and potential exponentialscan stay separate. Therefore, for the kinetic term, oneapplies the Hubbard-Sratonovich transformation [26, 27]exp h i ˜ h Z t ∂ x dτ i = Z Y τ dλ ( τ ) p π ˜ hi × exp h i h Z t dτ λ ( τ ) i · exp h Z t dτ λ ( τ ) ∂ x i . (8)Taking into account that the last exponential acts as ashift operator, one obtains the solutionΨ( x, t ) = Z Y τ dλ ( τ ) p π ˜ hi exp (cid:20) i h Z t dτ λ ( τ ) (cid:21) × Ψ (cid:16) x + Z t dτ λ (cid:17) exp (cid:20) i ˜ h Z t dτ V (cid:16) x + Z tτ dτ ′ λ , τ (cid:17)(cid:21) . (9)Therefore, the quantum-mechanical estimation of theMSD h x ( t ) i leads to two standard procedures of aver-aging. First one obtains a mean probability amplitude h| Ψ( x, t ) | i V by averaging of the obtained result in Eq.(9) over all realizations of the random field V ( x, t ), andthen performs a standard quantum-mechanical calcula-tion of the MSD. Therefore the MSD reads h x ( t ) i = Z x h| Ψ( x, t | i V dx . (10) III. MEAN PROBABILITY AMPLITUDE
For the random quantum process, the physical char-acteristics are described by the mean probability ampli-tude (MPA), or distribution function ρ ( x, t ), obtainedfrom the random wave function (9) by averaging overthe Gaussian distribution ρ ( x, t ) = h| Ψ( x, t ) | i V . (11)Obviously, this value is normalized R dxρ ( x, t ) = 1. Fol-lowing [28–30], let us obtain this normalization condition.The initial condition can be presented by means of theFourier integrationΨ ( x ) = 12 π Z ∞−∞ ¯Ψ ( k ) e − ipx dp . (12) Note that an important information about the random processis carried by the correlation function of V ( x, t ), which can beobtained by the ensemble averaging. Substituting this expression in Eq. (9), one obtains forthe MPA ρ ( x, t ) = Z Y τ dλ ( τ ) dλ ( τ )2 π ˜ h exp (cid:20) i h Z t ( λ − λ ) dτ (cid:21) × Z dp dp π ¯Ψ ( p ) ¯Ψ ∗ ( p ) exp[ − ix ( p − p )] × exp (cid:26) − i Z t [ p λ ( τ ) − p λ ( τ )] dτ (cid:27) × D exp h i ˜ h Z t (cid:16) V (˜ x , τ ) − V ∗ (˜ x , τ ) (cid:17) dτ iE V , (13)where ˜ x j = x + R tτ λ j ( τ ′ ) dτ ′ and j = 1 , A. Integration over the Gaussian distribution
Now one can treat the random potential term by in-tegration over the 2 N + 1 dimensional Gaussian packet,and this procedure coincides with integration over many-dimensional coherent states [31] d h P (cid:16) { A ∗ m , A m } (cid:17)i = exp (cid:16) − X m | A m | /σ (cid:17) Y m d A m πσ , (14)where d A m = d [ Re ( A m )] d [ Im ( A m )]. Therefore, aftertaking into account Eq. (4), the ensemble averaging pro-cedure corresponds to the following 2 N + 1 dimensionalintegration h . . . i V = Y m Z d A m πσ exp h − X | A m | /σ i × exp h X m (cid:16) A m α ∗ m − A ∗ m α m (cid:17)i , (15)where α m is the following complex function α m = i ˜ h √ N Z t dτ h e − ik m ˜ x ( τ )+ iω m τ − e − ik m ˜ x ( τ )+ iω m τ i . (16)Using the property of integration of coherent states [31],namely Z d βπ e −| β | e α ∗ β f ( β ∗ ) = f ( α ∗ ) , (17)one obtains from the integration in Eq. (15) h . . . i V = exp[ − σ X m | α m | ] ≡ F h λ ( τ ) , λ ( τ ) i . (18)The next step of the quantum analysis is functional in-tegration over the auxiliary Gaussian fields λ and λ .However, the exact quantum-mechanical treatment ispossible only for the δ correlated in time random po-tential V ( x, t ) h V ∗ ( x, t ) V ( x ′ , t ′ ) i V = C ( x, t ; x ′ , t ′ ) = C ( x, x ′ ) δ ( t − t ′ ) , (19)where C ( x, x ) = 2 σ ( cf . Eq. (6)). First, we considerthis case, noting that the restriction of δ correlation cor-responds also to the Obukhov mechanism of Richardsondiffusion [14]. IV. RICHARDSON DIFFUSION
Richardson diffusion [11] was the first phenomenolog-ical observation of developed turbulence [32], and thisphenomenon has been discussed in a variety of experi-mental and numerical studies, see reviews [12, 32] and asadmitted in [32, 33], it still lacks sufficient experimentalconfidence.Let us define the property of V ( x, t ) by means ofthe spectral density S ( k, ω ) of the correlation function C ( x, t ; x ′ , t ′ ) with the δ correlated constraint (19). Fol-lowing Refs. [2, 3], we present the correlation functionin the following translational invariant in space and timeform C ( x, x ′ ) δ ( t − t ′ ) = σ N X m h e ik m ( x − x ′ ) − iω m ( t − t ′ ) + c.c. i = σ Z dk Z dω ˜ S ( k, ω ) h e i [ k ( x − x ′ ) − ω ( t − t ′ )] + c.c. i = σ Z dkS ( k ) cos[ k ( x − x ′ )] δ ( t − t ′ ) , (20)where S ( k ) = 4 π ˜ S ( k, ω ).Using this delta correlated property, one can describethe dynamics of | α m ( t ) | in Eq. (18) by means of thespectral density S ( k ). Substituting Eq. (16) in Eq. (18)and taking into account Eq. (20), one obtains F [ λ ( τ ) , λ ( τ )] = exp h − σ X m | α m | i = exp n − σ ˜ h N Z t dτ Z t dτ N X m = − N × h e ik m ˜ x ( τ ) − iω m τ − e ik m ˜ x ( τ ) − iω m τ i × h e − ik m ˜ x ( τ )+ iω m τ − e − ik m ˜ x ( τ )+ iω m τ io = exp (cid:26) − σ ˜ h Z t dτ Z ∞−∞ dkS ( k ) × n − cos h k Z tτ (cid:16) λ ( τ ′ ) − λ ( τ ′ ) (cid:17) dτ ′ io(cid:27) . (21)To take the functional integrals over the auxiliary fields λ ( τ ) and λ ( τ ), one performs the following linear changeof the fields [28] λ ( τ ) = 2 µ ( τ ) + ˜ hν ( τ ) / λ ( τ ) = 2 µ ( τ ) − ˜ hν ( τ ) / , (22) where the Jakobian of the transformation is ˜ h for eachvalue of τ . Then the functional part of the integrand inEq. (13) reads Y τ dµ ( τ ) dν ( τ )2 π exp h i Z t µ ( τ ) ν ( τ ) dτ i × exp h − i ( p − p ) Z t µ ( τ ) dτ − i ˜ h p + p ) Z t ν ( τ ) dτ i F [ ν ( τ )] , (23)where we use the fact that F [ λ ( τ ) , λ ( τ )] = F [ λ ( τ ) − λ ( τ )], which follows from Eq. (21). Taking integrationover x in Eq. (13) one obtains the δ function δ ( p − p ).Then, functional integration over µ ( τ ) yields the deltafunctions Q τ δ ( ν ( τ )), since the rest of the integrand doesnot depend on µ ( τ ). Finally, after integration over ν ( τ )one obtains that the MPA is normalized to 1 Z ∞−∞ ρ ( x, t ) dx = 1 . (24) A. Mean squared displacement
Handling the exact expression of the MPA, we arriveat the main objective of the work and can evaluate therate of the wave packet spreading by calculation of theMSD h x ( t ) i in the (transversal) x direction. Taking intoaccount Eqs. (13) and (21), one obtains for the MSD h x ( t ) i = Z ∞−∞ ρ ( x, t ) x dx = Z dp dp π ¯Ψ ( p ) ¯Ψ ∗ ( p ) δ (2) ( p − p ) × Z Y τ dµ ( τ ) dν ( τ )2 π exp h i Z t µ ( τ ) ν ( τ ) dτ i × exp h − i ( p − p ) Z t µ ( τ ) dτ − i ˜ h p + p ) Z t ν ( τ ) dτ i F [ ν ( τ )] , (25)where we use the following definition of the second deriva-tive of the delta function δ (2) ( p − p ) ≡ ∂ p ∂ p δ ( p − p ).Now, we can repeat the previous calculations of Eqs.(13), (21), and (23). Functional integration over µ ( τ )yields Q τ δ [ ν ( τ ) − p − p )]. Therefore, functional in-tegration over ν ( τ ) is rigorous, as well. Performing in-tegration with δ (2) ( p − p ), one obtains finally for theMSD h x ( t ) i = P t + D t . (26)Here the first term ∼ t describes a well known wavepacket spreading in homogeneous media with the meansquared momentum P = ˜ h π Z ∞−∞ p | ˆΨ ( p ) | dp . The second term, which is obtained by the rigorous quan-tum mechanical calculations, is of a pure classical na-ture and corresponds to Richardson diffusion [11]. How-ever, its contribution in the quantum process of the wavepacket spreading is dominant ∼ D t , where the gener-alized diffusion coefficient is D = σ Z ∞−∞ k S ( k ) dk . (27) V. HYPER-DIFFUSION
It should be stressed that the experimental realizationof photonic lattices with the δ -correlated random poten-tial is technically impossible [1, 4, 5]. Therefore, the esti-mation of the MSD for the realistic arbitrary correlatedrandom potential V ( x, t ) leads to essential complicationof the analysis. Let us return to Eqs. (20) and (21)in a general form of the spectral density S ( k, ω ). Thecorrelation function reads C ( x − x ′ ; t − t ′ ) = σ N X m h e ik m ( x − x ′ ) − iω m ( t − t ′ ) + c.c. i = σ Z dk Z dω ˜ S ( k, ω ) h e i [ k ( x − x ′ ) − ω ( t − t ′ )] + c.c. i . (28)In this case, the functional action I [ λ ( τ ) , λ ( τ )] = − σ P m | α m | in F [ λ ( τ ) , λ ( τ )] = exp {I [ λ ( τ ) , λ ( τ )] } in Eq. (21) is a more complicated expression, which is nottreatable rigorously. After some algebraic manipulations,this reads I [ λ ( τ ) , λ ( τ )] = − σ ˜ h Z t dτ Z t dτ Z ∞−∞ dk Z ∞−∞ dω × S ( k, ω ) h e ik R τ τ λ dτ ′ − ω ( τ − τ ) (cid:16) − e ik R tτ ( λ − λ ) dτ ′ (cid:17) + e ik R τ τ λ dτ ′ − ω ( τ − τ ) (cid:16) − e − ik R tτ ( λ − λ ) dτ ′ (cid:17)i . (29)Problematic terms here are the exponentialsexp h ik R τ τ λ j dτ ′ i , where j = 1 ,
2. Let us simplifythese terms by introducing an average momentumfunction ¯ p j = Z τ τ λ j ( τ ′ ) dτ ′ τ − τ . (30)Obviously, ¯ p = ¯ p = ¯ p ( t ), where we stressed that the av-eraged momentum function is a function of time. Chang-ing the integration from times ( τ , τ ) to τ = τ and s = τ − τ , one recasts Eq. (29) in the form I [ λ ( τ ) , λ ( τ )] = − σ ˜ h Z t dτ Z t − t ds Z ∞−∞ dk Z ∞−∞ dω × S ( k, ω ) e i ( k ¯ p − ω ) s h − cos (cid:16) k Z tτ ( λ − λ ) dτ ′ (cid:17)i . (31)Integration over s can be approximated by a δ function.Namely, this integration yields Z t − t e i ( k ¯ p − ω ) s ds → Z ∞−∞ e i ( k ¯ p − ω ) s ds = 2 πδ ( ω − k ¯ p ) . Now integration over the frequency ω can be performedthat yields the action function I [ λ ( τ ) , λ ( τ )] = I [ λ ( τ ) − λ ( τ )]= − σ ˜ h Z t dτ Z ∞−∞ dkS ( k, k ¯ p ) × h − cos (cid:16) k Z tτ ( λ − λ ) dτ ′ (cid:17)i . (32)Finally, one obtains F [ λ ( τ ) , λ ( τ )] = exp {I [ λ ( τ ) − λ ( τ )] } , which is analogous to the expression obtained forRichardson diffusion. Performing again the variablechange of Eq. (22), we obtain an expression for the MSDanalogous to Eq. (25). The MSD reads h x ( t ) i = Z ∞−∞ ρ ( x, t ) x dx = Z dp dp π ¯Ψ ( p ) ¯Ψ ∗ ( p ) δ (2) ( p − p ) × Z Y τ dµ ( τ ) dν ( τ )2 π exp h i Z t µ ( τ ) ν ( τ ) dτ i × exp h − i ( p − p ) Z t µ ( τ ) dτ − i ˜ h p + p ) Z t ν ( τ ) dτ i F [ ν ( τ )] . (33)The essential difference between Eqs. (33) and (25) is thespectral density, which now is a two dimensional function S ( k, k ¯ p ). Integration over the fields µ and ν and differ-entiation over p and p yields h x ( t ) i = πσ Z t dτ ( t − τ ) Z ∞−∞ S ( k, k ¯ p ) k dk (34)We obtain the asymptotic behavior of Eq. (34) for largevalues of ¯ p , following a similar procedure presented inRefs. [2, 3, 34]. Therefore, by rescaling the variables, k ′ = k ¯ p , one obtains h x ( t ) i = πσ Z t dτ ( t − τ ) ¯ p ( τ ) Z ∞−∞ S ( k ′ ¯ p , k ′ ) k ′ dk ′ ≈ D Z t ( t − τ ) ¯ p ( τ ) dτ . (35)Here it was reasonable to suppose that S ( k ¯ p , k ) is a slowfunction of k/ ¯ p . For ¯ p = const the MSD correspondsto Richardson diffusion ∼ t . Such behavior supposesfor the averaged momentum function to be an increasingfunction of time. Moreover, it has been suggested in Ref.[3] that for large ¯ p , one obtains S ( k/ ¯ p, k ) ≈ S (0 , k ) thatyields nonzero generalized diffusion coefficient D = πσ Z k S (0 , k ) dk . (36)It is also supposes a physical meaning of ¯ p ( t ), which be-haves as a velocity-velocity correlation function. There-fore, one suggests a self-consistent procedure, presentedin Appendix A, to find this function. This yields for ¯ p ( t )¯ p ( t ) = (5 D / / t / . (37)Taking this behavior into account, one obtains h x ( t ) i ∼ ¯ Dt / , (38)which corresponds to hyperdiffusion, observed experi-mentally [1]. Here ¯ D = (2 / / D / VI. CONCLUSION
An enhanced spreading of a quantum wave packetin randomly inhomogeneous media is considered. Thisquantum process is realized in an arbitrary space-timedependent potential V ( x, t ). A rigorous quantum-mechanical calculation of the mean probability amplitude(MPA) is suggested that makes it possible to calculatethe mean squared displacement (MSD) of the spreadingwave packet. The obtained result establishes the powerlaw spreading of the MSD, which is h x ( t ) i ∼ t α , where2 < α ≤
3, and the values of the transport exponent α depend on the correlation properties of the random po-tential V ( x, t ). The main motivation of the presentedanalysis is experimental demonstrations on wave packetspreading in random photonic lattices [1, 4, 5]. Anotherpossible application of the presented analysis can be re-lated to a sound waves monitoring in underwater acous-tics [22], at the conditions when the parabolic equationapproximation is valid and the refractive index has ran-dom local fluctuations ∆ n ( x, z ), which leads to a domi-nant random potential as in Eqs. (1) and (2).The rigorous formal expression for the wave functionis obtained in a form of paths integration, such that thewave function (9) is a functional of the random poten-tial V ( x, t ). When V ( x, t ) is δ correlated in time as inEq. (20), the MSD is rigorously calculated in the frame-work of quantum-mechanical consideration. The domi-nant term in the MSD of the order of t is due to turbu-lent Richardson diffusion [11]. Another important resultof Eq (26) is that the quantum homogeneous spread ∼ t stays separate from the dominant classical one ∼ t . One can understand this property from the structure of thewave function (9)Ψ( x, t ) = Z D [ λ ( τ )] exp (cid:20) i h Z t dτ λ ( τ ) (cid:21) × e i ˜ h R t dτV (cid:16) x + R tτ dτ ′ λ ,τ (cid:17) Ψ (cid:16) x + Z t dτ λ (cid:17) , (39)where D [ λ ( τ )] = Q τ dλ ( τ ) √ π ˜ hi . This is a kind of Feynman-Kac formula [20, 21], obtained by means of the auxiliaryMarkov process with the Gaussian distribution in thepotential V ( x, t ). However, since V ( x, t ) is random itself,the details of the potential are not important, and themain information, and contribution to the MPA is dueto the correlation function C ( x − x ′ , t − t ′ ), or the spec-tral density S ( k, ω ), correspondingly. When the randompotential is δ correlated in time, the auxiliary field λ doesnot intertwine with the potential V ( x, t ). This is reflectedin the solution for the MPA ρ ( x, t ), where the averagedevolution kernel F depends only on the quantum part ofthe auxiliary fields, namely F = F [ λ ( τ ) − λ ( τ )]. As aresult of this, rigorous integration over λ and λ is per-formed. Therefore, each Markov process contributes sep-arately to the MSD in Eq. (26). The quantum mechan-ics leads to the ballistic ∼ t spread of the initial wavepacket, while the classical Obukhov mechanism of tur-bulent diffusion reveals itself in pure quantum mechanicswith the dominant ∼ t spread of the wave packet.The situation changes dramatically, when the randompotential is correlated in both space and time. In thiscase the auxiliary λ fields and the random potential areintertwined due to the nonlocal terms R t t λ ( τ ) dτ in theMPA. To make the problem treatable, this nonlocal termis presented in form of an averaged quantum momentumfunction ( t − t )¯ p , where ¯ p is related to a velocity-velocitycorrelation function of random quantum paths. After thisapproximation, the integration over the λ s is performedrigorously again. Now the quantum ballistic spread isaccompanied by hyperdiffusion ∼ t α . Assuming that thespectral function after rescaling S ( k ¯ p , k ) is a slow functionof k/ ¯ p , like in Eq. (36), it is obtained that α = 12 /
5. Asalready admitted, this result coincides with one obtainedin Refs. [2, 3] in the classical limit of the ray dynamics.However, contrary to Refs. [2, 3], in the present analy-sis we did not suppose any restriction conditions for therandom potential V ( x, t ).In the general case, one obtains that 2 < α <
3. Thisresult follows from Eq. (35), where R ∞−∞ S ( k ¯ p , k ) k dk isa slow varying function, which approaches to the trans-port constant D for the asymptotic large times t → ∞ . Note that quantum mechanics itself is not the Markovian dy-namics
Therefore, α = 12 / Acknowledgments
I thank Professor S. Fishman for helpful and infor-mative discussions, and comments to the text. This re-search was supported by the Israel Science Foundation(ISF-1028).
Appendix A: Inferring of the averaged momentum ¯ p Let us obtain analytical expression (37) for the aver-aged momentum ¯ p ( t ) in the framework of a self-containedprocedure, where we take into account that ¯ p ( t ) is a cor-relation function. First, it is worth noting that Eq. (30)is a definition of ¯ p . However it does not determine thelatter, since the averaging of the random auxiliary field λ ( t ) over the time interval s = t − t is not well de-fined. Second, we admit that the integral R τ τ λ ( t ) dt isnot zero. Moreover, we replace this integral by a quan-tum path. One can reasonably suppose that the MSDof these quantum paths is determined by the velocity-velocity correlation function ¯ p ( t ) ≈ h ˙ x ( t ) i λ = h ˙ x ( t ) i .Therefore, we relate the averaged momentum function tothe real quantum path x ( t ), and to estimate its tempo-ral behavior, we consider its classical random dynamics.Integrating the dynamical equation¨ x = − d V ( x, t ) d x , one obtains from the definition of the random potentialin Eq. (4)˙ x ( t ) = Z t √ N X m ( − i ) k m A m e ik m x ( t ′ ) − iω m t ′ dt ′ + c.c. . (A1)Therefore, the self-correlation function reads h ˙ x ( t ) i = 2 σ N X m Z t dt ′ Z t dt ′′ × cos[ k m ( x ′ − x ′′ ) − ω ( t ′ − t ′′ )] k m = 2 σ Z t dt ′ Z t dt ′′ Z dk Z dωk S ( k, ω ) × cos[ k ( x ′ − x ′′ ) − ω ( t ′ − t ′′ )] . (A2) Note that according the property (5), h ˙ x ( t ) i = 0. Afterchanging integration times τ = t ′ and s = t ′ − t ′′ , wearrived at the same expression as in Eq. (31).Following the solution of the wave function in Eq. (9),the evolution of the coordinates x ( t ) is due to the shiftoperator x ( t ) = x + R t λ ( τ ) dτ . Therefore, accordingEq. (30), the difference x ( t ′ ) − x ( t ′′ ) yields ¯ p ( τ ) = ( x ′ − x ′′ ) / ( t ′ − t ′′ ) = ˙ x . Taking into account Eq. (35), oneobtains approximately from Eq. (A2)¯ p ( t ) ≈ D Z t dτ ¯ p ( τ ) . (A3)Differentiating Eq. (A3) over time, one obtains¯ p ( t ) d ¯ p ( t ) d t = D . (A4)This equation can be also obtained by using the wellknown expression (see e.g. , [35]) h x ( t ) i λ = 2 Z t ( t − τ ) h ˙ x ( τ ) ˙ x (0) i λ dτ (A5)and consider that h x ( t ) i = h x ( t ) i λ . Now we put for-ward the physical meaning of the momentum function ¯ p by substituting it in Eq. (A5) h x ( t ) i λ = 2 Z t ( t − τ )¯ p ( τ ) dτ . (A6)Differentiating twice Eqs. (A6) and (35) over t and com-paring the obtained results, one obtains¯ p ≈ D Z t dτ ¯ p ( τ ) . Differentiating this over time again, one obtains Eq.(A4). Solving this equation, one obtains¯ p ( t ) = (5 D / / t / . (A7) [1] L. Levi, Y. Krivolapov, S. Fishman and M. Segev, NaturePhys. , 912 (2012).[2] E. Arvedson, M. Wilkinson, B. Mehlig, and K. Naka-mura, Phys. Rev. Lett. , 030601 (2006).[3] Y. Krivolapov, L. Levi, S. Fishman, M. Segev and M. Wilkinson, New J. Phys. , 043047 (2012).[4] ) L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O.Manela and M. Segev, Science , 1541 (2011).[5] M. Rechtsman, L. Levi, B. Freedman, T. Schwartz, O.Manela and M. Segev, Optics and Photonics News (Spe- cial Issue: Optics in 2011) (12), (2011).[6] V.I. Kliatskin, Stochastic equations and waves in ran-domly inhomogeneous media (Nauka, Moscow, 1980) (inRussian).[7] R. Dashen J. Math. Phys.
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