Confinement of hydrodynamic modes on a free surface and their quantum analogs

0.12 MB·English

by  M. Torres

Confinement of hydrodynamic modes on a free surface and their quantum analogs
M. Torres∗,1 J. P. Adrados,1 P. Cobo,2 A. Ferna´ndez,2 C. Guerrero,3 G.
Chiappe,4 E. Louis,4 J.A. Miralles,4 J. A. Verg´es,5 and J.L. Arag´on†6
1Instituto de F´ısica Aplicada, Consejo Superior de Investigaciones Cient´ıficas, Serrano 144, 28006 Madrid, Spain.
2Instituto de Acu´stica, Consejo Superior de Investigaciones Cient´ıficas, Serrano 144, 28006 Madrid, Spain
3Laboratorio de F´ısica de Sistemas Pequen˜os y Nanotecnolog´ıa,
Consejo Superior de Investigaciones Cient´ıficas, Serrano 144, 28006 Madrid, Spain.
5 4Departamento de F´ısica Aplicada and Unidad Asociada del Consejo Superior de Investigaciones Cient´ıficas,
0 Universidad de Alicante, San Vicente del Raspeig, Alicante 03690, Spain.
0 5Departamento de Teor´ıa de la Materia Condensada, Instituto de Ciencia de Materiales de Madrid,
2 Consejo Superior de Investigaciones Cient´ıficas, Cantoblanco, Madrid 28049, Spain.
n 6Centro de F´ısica Aplicada y Tecnolog´ıa Avanzada,
a Universidad Nacional Aut´onoma de M´exico, Apartado Postal 1-1010, Quer´etaro 76000, M´exico.
J (Dated: February 2, 2008)

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Asubtleproceduretoconfinehydrodynamicmodesonthefreesurfaceofafluidispresentedhere.
1
Theexperimentconsistsofasquarevesselwithanimmersedsquarecentralwellvibratingvertically

sothatthesurfacewavesgeneratedbythemeniscusatthevesselboundaryinterferewiththebound
]
n statesofthewell. Thisisaclassicalanalogyofaquantumwellwheresomefundamentalphenomena,
y such as bonding of states and interference between free waves and bound states, can be visualized
d and controlled. The above mentioned interference leads to a novel hydrodynamic transition from
- quasiperiodic to periodic patterns. Tight binding numerical calculations are performed here to
u
show that our results could be transferred to design quantum confinements exhibiting electronic
l
f quasiperiodic surface states and their rational approximantsfor thefirst time.
.
s
c
i The interestin experiments of classicalanalogs,which wasexpectedtoworkasaweakpotentialbindingsurface
s
y accurately model the salient features of some quantum waves upon dependence on the liquid depth. When the
h systems or other fundamental undulatory phenomena, vessel vibrated vertically with such amplitudes that the
p
wasfirstlyraisedbytheacousticexperimentsofMaynard Faraday instability was prevented, such geometry pro-
[
[1], where the analogy between both the acoustic and ducedthreekinds oflinearorweaklynon-linearpatterns

1 Schr¨odinger equations is invoked. In a general overview, on the free surface of the liquid. The first pattern is a
v
a hydrodynamic analogy has also been used to describe sort of bound state restricted to the surface area occu-
1
the flow of electrons in quantum semiconductor devices pied by the immersed well that works as a weak poten-
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0 [2]. Some experiments with liquid surface waves have tialbinding standing planewaves. The secondpatternis
1 beenreportedlaterandtheypresentedabstractconcepts, produced by the meniscus at the vessel walls [11] and it
0 suchasBlochstates,domainwallsandband-gapsinperi- can invade the region of the well depending on the liq-
5
odicsystems[3,4],Bloch-likestatesinquasiperiodicsys- uid depth and the vessel vibration frequency. Finally,
0
tems [5] or novelfindings as the superlensing effect [6] in the last pattern arises from the interference between the
/
s an visually clear manner. On the other hand, the corre- vessel meniscus waves and the bound states of the well.
c
i spondence betweenthe shallowwaterwaveequationand Patterns observed depend on the liquid depth h1, which
s
the acoustic wave equation has also been demonstrated plays the role of an order parameter by controlling the
y
h [4, 7]. Such correspondences could be exploited to in- amplitudesoftheboundstatesinsidethe immersedwell.
p vestigate and address formally similar quantum effects Our main observation is summarized in Fig. 2(a),
v: as those observed in quantum corrals [8], where an opti- which clearly shows the binding of the surface wave
i cal analogy has already been proposed [9], and in grain produced by the drilled well when the vibration am-
X
boundariesorsimplesurfacesteps[10]. Ourmaingoalis plitude is 60 µm. This pattern will be detailed below
ar tobuildupthehydrodynamicanalogyoftheinterference but the physics of its origin can be discussed as fol-
between bound states in a finite quantum well and free lows. The bound states arise from an inertial hydro-
states. Then we realized an experimental set up consist- dynamic instability, balanced by the liquid surface ten-
ing of a square vessel with a single square well drilled at sion [12], that grows over the square well region [5].
its bottom; both squares are concentric and well diag- The boundstate amplitudes increaseonincreasing1/a2;
onals are parallel to the box sides. The immersed well wherea2 =T/ρg,aisthecapillarylength,T istheliquid
surface tension, ρ is the liquid density and g =g αω2
0
±
istheeffectivegravity;g istheaccelerationduetograv-
0
ity, α is the vessel vibration amplitude and ω is the vi-
∗Electronicaddress: [email protected]
brationangularfrequencyofthevessel. Onthecontrary,
†Electronicaddress: [email protected]
themeniscuswaveamplitudedependsonthevariationof
2
the meniscus volume during each vessel oscillation and, liquiddepthandvibrationfrequency. Theinterferenceof
it grows accordingly when a2 increases [11, 12]. In our standing patterns ψ with ψ′ increases the symmetry in
2 2
experiment α is about 60 µm and the meniscus wave the well from square crystalline to octagonal quasicrys-
amplitude reaches a maximum at a vessel vibration fre- talline. Suchpatternmatcheswellwiththeouteronede-
quency of 64 Hz. On the other hand, the vessel vibra- scribed by ψ (k ) = A [exp(ık x)+exp(ık y)]. Accord-
1 1 1 1 1
tion frequency represents the wave state energy. The ing to the dispersion relation for gravity-capillary waves
frequency and the wavelength are related through the describedelsewhere[3,4,5,12],thedifferencebetweenk
1
well known dispersion relation of the gravity-capillary and k is about 2% and the refraction bending of about
2
waves [3, 4, 5, 12]. The present experiment is essentially 1.1o at the boundary of the central window is negligible.
monochromatic as it occurs in the optical analogs [9] of Furthermore,theexternalwavereflectionatthewellstep
quantumcorrals. ToshowthatFig. 1(a)isnotaFaraday [13, 14, 15] is also negligible with such parameters. On
pattern,wepresentasnapshotofthesystemvibratingat the other hand, slender outgoing evanescent waves are
about 70 µm in Fig. 1(b), when the Faraday instability emitted at the boundary of the well and they play an
isreallytriggeredinthesquarewell. Figure1(b)showsa important role in the matching between patterns. It is
higher wavelength that matches with the corresponding important to remark that the parameters of the exper-
period doubling related to a Faraday wave pattern. iment do not allow through the well known dispersion
We chose a square vessel and the orientated square relation [3, 4, 5, 12] that the wave inside the inner well
well configuration to verify that the immersed well con- were a subharmonic Faraday wave and the outer wave
fined wave states. Then we used a square methacrylate were a harmonic meniscus wave. If this were the case,
box with side L of 8 cm where a single square well with then k1 and k2 should be very different.
depth d of 2 mm and side l of 3.5 cm was drilled at its Case III. When h is increased, the potential of the
1
bottom. The bottom of the vessel was covered with a immersedwellisseenincreasinglyweakerbythe system,
shallow ethanol layer of depth h . The liquid depth over and A′ decreases accordingly. Under such conditions,
1 2
the well was then h =h +d. transitional patterns from a quasicrystalline form to a
2 1
As it was already mentioned, the vessel vibration am- crystalline one appear gradually on the hydrodynamic
plitude was60µm, belowthe Faradayinstabilitythresh- window. Figure 2(b) shows a transitional pattern corre-
old set at about 70 µm. The vessel vibrated vertically spondingtoaliquiddepthh1of1.5mmandanexcitation
at a single frequency lying within the range from 35 to frequency of 50 Hz. According to crystallographic tech-
60 Hz. An optimum frequency value was 50 Hz. De- niques of image processing described elsewhere [5], the
pending on the depth h , our experimental results can Fourier transforms of the experimental patterns are cal-
1
be separated into three cases. culated(Figs. 2(a)and2(b))andtheyareusedtodepict
CaseI.Forh1lowerthan1mm,theexperimentshows Fig. 3, where the fast decay of |A′2/A2|2 is shown on
two square lattices rotated 45o between them, namely increasing h1. Figure 4 shows four frames of the numer-
the lattice ofthe immersedsquarewellis separatedfrom ical simulation of the quasicrystal-crystal transition on
the vessel square lattice (Fig. 1a). The external wave the hydrodynamic centralsquarewindow. The complete
amplitudes are lower than the wave amplitudes within movie is available as Auxiliary Material.
the well and, furthermore, the external wave reflection The appearance of a quasiperiodic pattern inside the
at the well step [13, 14, 15] is strong. For very shallow immersed square region, in Case II, confirms that the
liquid layers, waves are only present within the well at squarewell is binding standing plane waves. Such a pat-
the vibration amplitudes of the experiment. tern can only be produced by the interference of two
Case II. At a vessel vibration frequency of 50 Hz, square patterns rotated by 45o, namely the pattern of
whenh is1.2mm,aquasicrystallinestandingwavepat- bound standing waves of the square well is transparent
1
tern appears inside the immersed square region whereas tothepatternofstandingmeniscuswavesofthe vesselif
the outer wave pattern is a square network (Fig. 2(a)). the adequate conditions of the experimental parameters
Theimmersedsquarewellworksasaweakpotentialand are fulfilled. On the other hand, the square pattern of
binds standing plane waves with eigenvectors k u′ and the vessel is only observedbeyond the region of the cen-
2 x
k u′, parallel to the well sides. Nevertheless, it is trans- tral square well. Moreover, there are evanescent waves
2 y
parent for the standing waves of the square box which coming out of the boundary of the square well. The ob-
tunnel the immersed well framework under the experi- servationof such waves also confirms the analogywith a
mental conditions. Inside the square region, the vessel quantum well [16].
eigenstates have eigenvectors k u and k u , parallel to The description of the observed standing wave pat-
2 x 2 y
theouterboxsides. Thus,thestandingstateisfinallyde- ternsgivenin CaseII is the sameas the usedto describe
scribedbyψ (k )+ψ′(k )=A [exp(ık x)+exp(ık y)]+ the stationary states of a particle moving in a potential
2 2 2 2 2 2 2
A′[exp(ık x′)+exp(ık y′)]insidetheregionofthesquare given by a square well. Using these functions, we show
2 2 2
well. Equal phases along x and y, and x′ and y′, respec- a numerical simulation of the observed quasicrystalline
tively, were assumed and A = A′ for the mentioned patterninFig. 3(b). Atrialanderrornumericalmethod
| 2| | 2|
3
was used to fit at the boundaries and a better matching is gratefully acknowledged. This workhasbeen partially
is obtained if slender outgoing evanescent waves emitted supported by the Spanish MCYT (BFM20010202 and
at the boundary of the well are considered. MAT2002-04429), the Mexicans DGAPA-UNAM (IN-
To test the viability of a quantum scenario analogous 108502-3)andCONACyT(D40615-F),theArgentineans
to our experimental results, we numerically studied the UBACYT (x210 and x447) and Fundaci´on Antorchas,
quantum confinement of a double square well by using a and the University of Alicante.
tight-bindingHamiltonianinaL Lclusterofthesquare
×
lattice with a single atomic orbital per lattice site [17],
Hˆ = Xǫm,n|m,nihm,n| [1] S. He and J.D. Maynard, Phys. Rev. Lett. 62, 1888
m,n (1989); J.D. Maynard, Rev.Mod. Phys. 73, 401 (2001).
X tm,n;m′n′ m,n m′,n′ , [2] C.L. Gardner, SIAMJ. Appl.Math. 54, 409 (1994).
− | ih |
<mn;m′n′> [3] M. Torres, J.P. Adrados and F.R. Montero deEspinosa,
Nature (London) 398, 114 (1999)
where m,n represents an atomic orbital at site (m,n), [4] M. Torres, J.P. Adrados, F.R. Montero de Espinosa, D.
andǫ | itsienergy. Inordertosimulatetheinnersquare Garc´ıa-Pablos and J. Fayos, Phys. Rev. E 63, 11204
m,n
(2000).
we have explored several possibilities. Outside the in-
[5] M. Torres, J.P Adrados, J.L. Arag´on, P. Cobo
ner square, the energy of all orbitals are taken equal and S. Tehuacanero, Phys. Rev. Lett. 90,
to zero. Besides, the hopping energies between nearest-
114501 (2003). See also Physical Review Focus:
neighbor sites (the symbol <> denotes that the sum is http://focus.aps.org/story/v11/st11
restricted to nearest-neighbors) were all taken equal to [6] X. Hu, Y. Shen, X. Liu, R. Fu, and J. Zi, Phys. Rev. E
1. Instead, inside the inner square, we either changed 69,030201(R)(2004);M.Peplow,Nature(London)428,
the orbital or the hopping energies. Some illustrative 713 (2004).
[7] T. Chou, J. Fluid Mech. 369, 333 (1998).
results are shown in Fig. 5. Figures 5(a) and (b) cor-
[8] M.F.Crommie,C.P.LutzandD.M.Eigler,Nature(Lon-
respond to wavefunctions close to the band bottom, i.e., don) 363, 524 (1993); M.F. Crommie, C.P. Lutz and
long wavelengths and high linearity. In Fig. 5(a) the D.M. Eigler, Science 262, 218 (1993).
octagonal symmetry within the inner square is clearly [9] G´erardColasdesFrancs,etal.,Phys.Rev.Lett.86,4950
visible. Figure 5(b) illustrates an effect that is purely (2001); C. Chicanne, et al., Phys. Rev. Lett. 88, 097402
quantum,namely,the effectofthe outersquareis visible (2002).
evenonawavefunctionlocalizedintheinnersquare. Al- [10] Y. Hasegawa and P. Avouris, Phys. Rev. Lett. 71, 1071
(1993).
though one cannot expect a one-to-one correspondence
[11] S. Douady,J. Fluid Mech. 221, 383 (1990).
between the experiments discussed here and this simple
[12] L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Perg-
quantumsimulation,theresultsclearlysuggestthatsim-
amon Press, London, 1959).
ilar effects could be observed in a suitable quantum sys- [13] H. Lamb, Hydrodynamics (Cambridge University Press,
tem. Figure5(c)showsapeculiarnullenergystatewhich Cambridge, 1932).
is notlocatedatthe bandbottom. Inthis state the hop- [14] E.F.Bartholomeusz,Proc.CambridgePhil.Soc.54,106
ping energy outside the square is equal to 1, whereas (1958).
[15] J.W. Miles, J. Fluid Mech. 28, 755 (1967).
it is 0.7 inside the square and the general pattern con-
[16] A.Messiah, M`ecanique Quantique (Dunod,Paris,1962).
spicuously resembles our experiment. The Fourier spec-
[17] E. Cuevas, E. Louis, and J. A. Verg´es Phys. Rev. Lett.
trum of the inner square pattern has been performed 77, 1970 (1996).
and it corresponds to a rational approximant of an oc-
tagonal quasicrystal generated by the following vectors:
(2/√5,0), (0,2/√5), 2(2/√5,1/√5), 2(2/√5, 1/√5),
−
2(1/√5,2/√5), 2( 1/√5,2/√5). Fourier spectra of
−
Figs. 5(a) and (b) correspond to quasicrystalline states.
Thissuggeststhatthehoppingenergycouldbeaparam-
etertoinducequasicrystal-crystaltransitionsinconfined
quantum states.
Inconclusion,we havedescribedhere a hydrodynamic
experiment that gives rise to the confinement of wave
states on a free surface. It constitutes a stirring macro-
scopic experimental scenario which models some salient
features of a quantum well andstimulates the study and
visualizationofconfinedquasiperiodicquantumstatesfor
the first time.
TechnicalsupportfromC.ZorrillaandS.Tehuacanero
4
FIG. 1: (a) Snapshot of the system below the Faraday in- FIG. 3: (a) Decay of |A′2/A2|2 when the depth of the liquid
stability onset, vibrating at 35 Hz with h1 = 0.35 mm. Two layer increases; the spots ((cid:7)) on the figure are experimental
clearly separatedwavepatternsareshown. Theamplitudeof points. (b) Simulation of the pattern in Fig. 2(a) by means
themeniscuswaveismuchlowerthanthatofthewell-bound of anumericaltrial and error fittingof thestandingwavesof
states. The reflexion coefficient at the well step is about 0.5. theboxandthosewavesboundbythesquarewell, aswellas
(b)SnapshotofthesystemattheonsetoftheFaradayinsta- theevanescent waves outgoing from thewell boundary.
bility in thesquare well.
FIG. 2: (a) Snapshot of the system vibrating at 50 Hz for FIG. 4: Four frames of the numerical simulation of the
h1 =1.2mm. Thewellistransparentforthemeniscuswave. quasicrystal- crystal transition are shown as theliquid depth
Theinterferencebetweensuchwaveandthewell-boundstates h1increases. ThecorrespondingFouriertransformsareshown
gives rise toa perfect quasicrystalline pattern (Top). Fourier asinsets. AcompletemovieisavailableasAuxiliaryMaterial.
transform of the quasicrystalline wave pattern. Some sec-
ondary rings with a lower intensity appear. Such rings are
due to weak nonlinearities corresponding to the parametri-
cally driven experiment (Bottom). (b) Snapshot of the sys- FIG. 5: Quantum wavefunctions of the tight binding Hamil-
tem vibrating at 50 Hz for h1 = 1.5 mm. The amplitude of tonian described in thetext.
the bound states is lower than that of the meniscus wave.
A quasicrystal-crystal intermediate state appears on thewell
(Top). Fourier transform of the transitional pattern. In the
main octagonal ring thereare four brighterspots (Bottom).
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