Gamow Vectors Explain the Shock "Batman" Profile

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by  Maria Chiara Braidotti

Gamow Vectors Explain the Shock ”Batman” Profile
Maria Chiara Braidotti,1,2 Silvia Gentilini,1 and Claudio Conti1,3
1Institute for Complex Systems, National Research Council (ISC-CNR), Via dei Taurini 19, 00185 Rome (IT).
2Department of Physical and Chemical Sciences,
University of L’Aquila, Via Vetoio 10, I-67010 L’Aquila (IT).∗
3Department of Physics, University Sapienza, Piazzale Aldo Moro 5, 00185 Rome (IT).†
(Dated: January 25, 2016)
Thedescriptionofshockwavesbeyondtheshockpointisachallengeinnonlinearphysics. Finding
solutions to the global dynamics of dispersive shock waves is not always possible due to the lack
of integrability. Here we propose a new method based on the eigenstates (Gamow vectors) of a
reversed harmonic oscillator in a rigged Hilbert space. These vectors allow analytical formulation
6 forthedevelopmentof undularboresof shock wavesin anonlinearnonlocal medium. Experiments
1 by a photothermal induced nonlinearity confirm theoretical predictions: as the undulation period
0 as a function of power and the characteristic quantized decays of Gamow vectors. Our results
2 demonstratethatGamowvectorareanovelandeffectiveparadigmfordescribingextremenonlinear
phenomena.
n
a
J
In physics, shock waves emerge in a wide variety of Recently, unnormalizable wavefunctions named nonli-
1
fields ranging from fluidodynamic to astrophysics, to near Gamow vectors (GVs) proved to be fruitful: the
2
dispersive gas dynamics [1] and plasma physics [2, 3], to shock waves are described by the eigensolutions of the

] granular systems [4], to Bose-Einstein condensation and so called reversed harmonic oscillator (RHO). [24, 25]
S
polaritons [5]. [6–13] The shock waves ubiquity arises Here we show that this approachcan be the key to solve
P
from the universal properties of hyperbolic systems of analyticallyshockwavepropagationinthe far field. Our
.
n partialdifferentialequations,which aretypically present analysis allows to describe and analyze the development
i in various contexts. However the exact description of of the characteristic undular bores during the shock
l
n the shockwaveprofilesis rarelyavailableandtechniques formation, and provides a complete description of the
[ like the Whitham approach are limited to integrable ”Batman ears”.

1 or nearly integrable systems. [14, 15] Very simplified
v hydrodynamic models like the Hopf equation are used We consider a light beam with amplitude A (I = A2
| |
6 in most of the cases. [15–19] These methods allow the is the intensity), wavelength λ, and propagating in a
9 descriptionofthe wavepropagationsince the occurrence mediumwithrefractiveindexn . LetZ andX thepropa-
0
7
of the shock point, but do not provide a global and gation and polarization directions respectively, then the
5
comprehensive analysis of the phenomenon. Issues paraxial propagationequation reads as
0
that are often overlooked include the field decay after
1. the shock, and the long term evolution (far field). In 2ik∂A + ∂2A +2k2∆n[|A|2](X)A=0, (1)
0 ∂Z ∂X2 n
addition, peculiar features like the oscillation period of 0
6
1 the so called undular bores (fast oscillation observed with PMKS = IdX the beam power per unit length in
v: ians trheseulwtsavoef pnruomfileericinaldciasplceurlsaitvieonssy.stOemnse)sopnelcyifircemopaeinn tbheer.trIannasvneorsneloYRcadlimreecdtiiounmanthdekre=fr2aπctniv0/eλintdheexwvaavreinatuimon-
i
X question concerns the appearance of the fast oscillations ∆n can be written as
in the internal part of a Gaussian beam undergoing a
r
a shock in a nonlocal medium. This is in contrast with ∆n[I](X)=n G(X X′)I(X′)dX′. (2)
2
−
the hydrodynamical formulation, which predicts that Z
the bores are expected to appear in the external part, ThefunctionGisproportionaltotheGreenfunctionand
i.e., at the beam edges. During the shock formation the it is normalized such that GdX = 1. We observe that
beam displays a characteristic double peaked M-shape Eq. (2) originates from the solution of the Fourier heat
R
(”Batman ears”) that is also a riddle in the field of equation for thermal nonlocal nonlinearity.[19]
normal dispersion mode-locked laser [20–23]. No ana- We write Eq. (1) in terms of the normalized variables
lytical solution is available to describe the phenomenon. x=X/W and z =Z/Z with Z =kW2
0 d d 0
The challenge is finding new ideas and paradigms for
∂ψ 1∂2ψ
describing wave propagation beyond the breaking point i + PK(x) ψ(x)2ψ =0, (3)
and hence cover the knowledge gap in predicting this ∂z 2∂x2 − ∗| |
universal shock features. where ψ = AW /√P , with ψ ψ = 1 denoting
0 MKS
h | i
Hilbert space scalar product. P is set as P /P ,
MKS REF
with P = λ2/4π2n n . n is the nonlinear opti-
REF 0 2 2
| |
∗ [email protected] cal coefficient and K(x)=W0G(xW0) is the nonlocality
† http://www.complexlight.org kernel function.
2
Asshownin[24],inahighlynonlocalmedium,i.e. when writing ψ =e−iPz/2σφ, Eq. (3) reads as (see [24])
the nonlocality function width is much wider than the
+∞
field intensity and the highly nonlocal approximation
(HNA) is valid [K(x) ψ 2 κ(x)], the solution of i∂zφ=Hˆrhoφ= En− f−n f+n φ. (6)
Eq. (3) is strongly linke∗d|to| th≃e eigenstates of a RHO. nX=0 (cid:12) (cid:11)(cid:10) (cid:12)
(cid:12) (cid:12)
These states are the so called Gamow vectors, firstly in- TheeigenfunctionsφcanbeexpandedintermsofGamow
troduced by Gamow in 1920sin nuclear physics in order eigenvectors as
todescribeparticledecaysandresonances. [26]Itcanbe
shown that GVs can be obtained by extending the har- N
φ(x)= Γ f− f+ ψ(x,0) . (7)
monic oscillatoreigenfunctions in the complex plane:[27] n n n
n=0
Xp (cid:10) (cid:12) (cid:11)
f±n(x)=e±iπ/8 2√nn±!√iγπ 1/2e∓iγ2x2Hn( ±iγx), (4) wγ(h2enre+ψ1)(xa,r0e)thise GthVesiqnuitaianltizpehdysdi(cid:12)eccaalysrtaattees.and Γn =
(cid:18) (cid:19) p RHOGamoweigenstateshavethepeculiarcharacteristic
of being the eigenvectors of Fourier transform operator.
where H (x) are the n-order Hermite polynomials. The
n
f± arediscretestates belongingto ariggedHilbertspace Indeed,onecanobservethatthereversedoscillatoreigen-
n
(RHS) ×,whichisanextensionofthestandardHilbert valueequationhasthesameformasitsFouriertransform
space H. In × the Khalfin theorem [28] does not hold withinaphasefactor. ConsideringtheRHOHamiltonian
H H in the position basis (xˆ x and pˆ i∂ ) we have:
true and exponentially decaying wavefunctions are ad- x
→ →−
mitted. Indeed,theeigenvaluesoftheRHOHamiltonian
pˆ2 1
Hˆ (pˆ,i∂ )= + γ2∂2 = Hˆ ( i∂ ,xˆ). (8)
pˆ2 1 rho p 2 2 p − rho − x
Hˆ (pˆ,xˆ)= γ2xˆ2 (5)
rho 2 − 2 To describe the far field with this formalism, we cannot
neglect that GVs have an infinite support. Hence, to
arepurelyimaginarynumbersE± = iγ(n+1/2),where account for the spatial confinement of the experiment,
n ±
γ is the decaying coefficient of the associated classical we introduce the windowed Gamow vectors:
system. For z > 0 the eigenfunction f+ is exponentially
n N
increasing while f−n is decreasing. For this reason we φW(x)= Γ f− f+ ψ(x,0) rect (x), (9)
choose the latter to describe exponential decaying dy- G n n n W
n=0
namics when z grows. Figure 1 shows the square modu- Xp (cid:10) (cid:12) (cid:11)
(cid:12)
lus of f− and their tilt, calculated as the x derivative of where rect (x) = 0 for x > W and rect (x) = 1 for
n W | | W
the phase ϕ of f−, ∂ ϕ, for even n. Notice the resem- x < W, which is the range of the spatial confinement.
n x | |
blanceofthesefunctionswiththestandardintensityand During evolution each Gamow component in Eq. (7) ex-
phase profile observed during numerical simulations and ponential decay with rate Γ : the ground state has the
n
experiments in shock waves.[19] lowestdecayrateΓ0 =γ andhigherorderGamowstates
decay faster than the fundamental one. This allows to
consider only the fundamental GV in the far field. We
compute the Fourier transform of the fundamental
(a) 0.4 (b) F
state of Eq. (9):
−2|f(x)|n00..24 n=8 n=0 ∂φx−00..022 nnnn====0246 ψ˜(kx)=F(cid:0)f−0(x)(cid:1)×=((cid:18)−41E+rf4(cid:20)i(cid:19)(21e−−ik22ixγ)2(√(k−γxi−γWπW)1γ/4)×(cid:21)+
−0.4 n=8 (1 i)(k +Wγ)
−010 0 10 −40 −20 0 20 40 +Erf 2 − 2 x .
x x (cid:20) √γ (cid:21))
(10)
Equation (10) gives an analytical expression of the far
FIG. 1. (color online) First five even reversed oscillator field, which is compared below with the experiments
Gamow eigenstates square modulus (a) and their tilt (b), (Fig. 2). Equation (10) predicts in closed form the typi-
calculated as ∂xϕ, for the fundamental state and first four cal ”Batman profile”, the fact that undular bores are
excited states (n=0, 2, 4, 6 and 8).
internalin the beam profile,andthe correctscaling with
respect to the power of the undulation period.
It is worthwhile to notice that we can analyze the beam We validate this analysis by experiments in a nonlocal
evolution during wave breaking in a nonlocal nonlinear optothermal medium. The experimental set-up is illu-
medium using GVs of RHO. When HNA holds true and strated in Fig. 2a. A continuous wave (CW) laser beam
3
y (a)(a) (b)
x 2
m]
m
L2 Y [4
L1 61W 2W 3W 4W
2 2.5 3 2 2.5 3 2 2.5 3 2 2.5 3
X [mm] X [mm] X [mm] X [mm]
.4
(d) (e)
.2
5
.8
.6
.4
0 0.05 0.1 0.15 0.2 0.25
Power [arb.units]
FIG. 2. (color online) (a) Experimental setup scheme used to collect images of the laser beam transmitted by a RhB sample.
L1isthecylindricallensusedtomakethebeamelliptical andtofocusitinthesample. L2isthesphericallensusedtocollect
the beam with a CCD camera. (b) CCD images of the light beam at different laser powers (PMKS = 1,2,3 and 4W). (c)
Eq. (10) square modulus for two different power values (P1 >P2). (d) Experimental normalized intensity profile for different
powervalues: PMKS =2W(dashedline)and4W(continuousline). (e)Intensitydecaysasafunctionofpower. Thecoefficients
of the straight lines describe the Gamow vectors decaying rates (γ1 = 8.0 and γ2 = 1.6). Their quantized ratio is 5 as
− −
expected from theory (see [24]).
at 532nm wavelength is focused through a cylindrical shownin Fig.2d. We alsoobservethatthe experimental
lens (L1) with focal length f = 20cm in order to mimic dataexhibitareductioninthecentralpartoftheprofile.
a nearly one-dimensional propagation. Letting Z the This is mostly causedby the presenceof nonlinearlosses
propagationdirection,thelensfocusesthebeamintheX (not included in the model): the thermal effect induces
direction. The light is collected by a spherical lens (L2) Rhodaminediffusionoutofthehighestintensity regions,
andaChargedCoupledDevice(CCD)camera. Thespot which, in turn, are hence subject to a reduced absorp-
dimension is 1.0mm in the Y direction and 35µm in the tion. [19]
X direction. These geometrical features make the uni- Exponential decays are the major signature of Gamow
dimensional approximation valid and allow to compare states.[24, 27, 29] An important aspect of our analysis
experimental results with the theoretical model. The is that the elliptical beam has an intensity that varies
diffraction length in the X direction is L = 3.0mm. GaussianlyalongY. Thisimpliesthat,observingaCCD
diff
A solution of Rhodamine B (RhB) and water at 0.1mM image,intensity profiles atdifferent Y correspondto dif-
acts as a nonlocal optical medium and is placed in a cu- ferent powersin the one-dimensionalapproximation;the
vette 1mm thick in the propagation direction. RhB is a link between the Y position and the power follow the
dye with a high nonlinear index of refraction n , its ab- Gaussian profile (P exp Y2). Correspondingly
2 MKS
∝ −
sorption length is L =1.0mm. [19] theexpectedexponentialtrendwithrespecttopowercan
abs
We collect CCD images of the beam for different powers be extractedfroma singlepicture by lookingatdifferent
(see Fig.2b). The transverseX sectionbroadensandwe Y positions. This analysis is carried out by considering
observe intensity peaks (”Batman ears”) on the lateral a region in Fig. 2b at P = 4W; the resulting pro-
MKS
edges of the beam, resembling the shape of Gamow vec- file versus power is shown in Fig. 2e: two exponential
tors(seeFig.2cand2d). Theundularboresoftheshock trends are clearly evident and the two straight lines cor-
appear between the lateral peaks, in the internal part responding to different decay coefficients are drawn (the
of the Gaussian beam. For low power (P 1W) conversion from Y to P correspond to a logarith-
MKS MKS
≤
thebeamprofileisGaussian. Figure2cshowsthesquare mic scale in which exponentials are replaced by straight
modulusofthefarfieldanalyticallyexpressedinEq.(10) lines). The extracted ratio of the two decay coefficients
for two different power values (P > P ) and we stress is5andhence inagreementwiththe expectedquantized
1 2
the remarkable agreement with the experimental results
4
theoretical value [24]. as a function of optical power which matches the theo-
retical expectation as shown in Fig. 3b.
The control of extreme nonlinear phenomena is at the
basis of the future developments of nonlinear physics,
but requires novel theoretical tools and paradigms. In
10
(a) (b) this article we propose a novel approach to describe
4.0W 12
8 the occurrence of undular bores and the M-shape
ntensity [arb.units] 246 323...550WWW Period [arb.units]1680 (eeab”vqnyBoudnlaauwetttwimeioonaptn.enrco”Trhv)uhnildieienqedstutaeernbsogsynlfiortgybothnmapelorndironlerifiosnelcnveerleaiorpdrcsitutiaibyrolilnnpengroqoofunadtaluihnhncetieeugawshmrsalhyvSmoehncebrkoc¨ornhewdlaaiiannnkviegicenaessgrr.,
I 0 4 Our experiments quantitatively confirm the new theo-
2.0W retical scenario. We believe that this approach is not
−2 2
0 0.1 0.2 0.3 2.0 3.0 4.0 only limited to the spatial case considered here, but has
y [µm] Power [W] an impact in temporal pulse dynamics (as for example
modelocked lasers in the normal dispersion regime) and
also,moreingeneral,inthevastnumberoffieldsdealing
FIG.3. (coloronline) (a)Characteristic intensityoscillations
with shock waves. Understanding the way GVs occurs
for different power values. (b) Measured undulation period
T as a function of power. Continuous line is the fit function during extreme nonlinear phenomena may lead to the
T √4P as expected by thetheory. control of these processes in fields like nonlinear optics,
∝ polaritonsandultracoldphysics,andtothedevelopment
of novel devices including supercontinuum and X-ray
We analyze the undular bores of shock waves (see generation.
Fig. 2b). Equation (10) predicts that the field inten-
sity undulation period T grows like T √4P. Figure 3a We acknowledgesupport the ERC projectVANGUARD
∝
shows the oscillatory behavior of the far field as visua- (grant number 664782), the Templeton Foundation
lized on the CCD camera when removing the collecting (grantnumber58277)andtheERCprojectCOMPLEX-
lens L2. Through spectral analysis we obtain the period LIGHT (grant number 201766).
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