Echoes from Anharmonic Normal Modes in Model Glasses

1.5 MB·

by  Justin C. Burton

Echoes from Anharmonic Normal Modes in Model Glasses
Justin C. Burton∗
Department of Physics, Emory University
Sidney R. Nagel
James Franck Institute, Enrico Fermi Institute, and Department of Physics, The University of Chicago
(Dated: March 22, 2016)
Glasses display a wide array of nonlinear acoustic phenomena at temperatures T (cid:46) 1 K. This
behavior has traditionally been explained by an ensemble of weakly-coupled, two-level tunneling
states, a theory that is also used to describe the thermodynamic properties of glasses at low tem-
peratures. One of the most striking acoustic signatures in this regime is the existence of phonon
6
echoes, a feature that has been associated with two-level systems with the same formalism as spin
1
echoesinNMR.Herewereporttheexistenceofdistinctlydifferenttypeofacousticechoinclassical
0
modelsofglassymaterials. Oursimulationsconsistoffinite-ranged,repulsivespheresandalsopar-
2
ticles with attractive forces using Lennard-Jones interactions. We show that these echoes are due
r to anharmonic, weakly-coupled vibrational modes, and perhaps provide an alternative explanation
a
M for the phonon echoes observed in glasses at low temperatures.
PACSnumbers: 61.43.Fs,62.25.Jk,62.65.+k,63.20.D-,63.50.Lm
0
2

I. INTRODUCTION (cid:126)ω ≈ k T. In the experiments an acoustic transducer
] B
t attached to the glass sample served to both excite the
f
o Glassy materials behave in a fundamentally different acousticwaveanddetectreflections. Localizedmodes,as
s manner than their crystalline counterparts. Perhaps the postulated in the quantum-mechanical two-level system
.
t moststrikingexampleistheglasstransition;aslow-down model,naturallyleadtoaricharrayofnonlinearacoustic
a
m of kinetic behavior that strongly depends on tempera- behavior such as a temperature-dependent sound veloc-
ture [1, 2]. In contrast to freezing in crystals, there is no ity, saturation of attenuation [9–15], spectral hole burn-
-
d sharp phase transition to a solid phase. Rather the time ing and diffusion [16–19]. However, perhaps the most
n scale for motion of the constituent particles increases dramatic effect observed in these acoustic experiments
o smoothlyasthetemperatureislowered. Atlowtempera- wastheobservationofelectricandphononechoes[17,19–
c tureswhereglassesarerigidsolids,theystillretainmany 23]. Since any two-level quantum system has the same
[
propertiesthataredistinctfromcrystals,yettheseprop- dynamics as an isolated spin, these echoes were thought

2 erties are seemingly universal among disordered solids. tobeanalogousto,andhavethesameformalismas,spin
v Ithasbeenknownsincetheearly1970’sthatthether- echoes studied in magnetic resonance. This observation
7 modynamic properties of dielectric glasses are different was interpreted as evidence for the quantum mechani-
5
fromcrystalsattemperaturesT (cid:46)1K.Atthesetemper- cal nature of the excitations. As we will show here, the
1
5 atures the heat capacity scales approximately linearly in existence of phonon echoes does not necessitate such an
0 T and the thermal conductivity scales as ∼ T2 [3]. For interpretation. Rather, theechoescanbegeneratedbya
. crystals,thisscalingisT3 forbothquantities. Theorigin distinctly different mechanism that is completely classi-
1
ofthesedifferenceshastraditionallybeenattributedtoa cal in origin and is based on the inherent anharmonicity
0
6 diluteensembleoftwo-leveltunnelingstates[4–7]. These of the vibrational modes in disordered solids.
1 states are quantum mechanical in nature and spatially
Inthelastdecade,therehasbeenalargebodyofwork
: localized so that they are weakly coupled to other plane-
v concerning the vibrational modes in jammed, disordered
wave excitations in the solid. Moreover, it has been pre-
i solids which have helped to shed light on the origins of
X supposed that the distribution of two-level energy spac-
the thermodynamic properties of glasses [24, 25]. An ex-
r ings is very broad, leading to an approximately constant
a cessdensityofstatesatlowfrequenciesnaturallyarisesin
density of levels at low temperatures. Phenomenologi-
jammedsystemsandreliesonlyontheexistenceofdisor-
cally, this picture is consistent with most of the exper-
der, not on the details of the particle interaction [26]. In
imental data, yet a fundamental understanding of the
addition, the well-known “Boson peak” has been linked
origin of these localized modes is still lacking [8].
totheonsetofanomalousmodesinjammedsystemsand
Convincingevidencefortheexistenceoftwo-leveltun-
other model glasses [27]. The linear temperature depen-
neling states in glasses comes from acoustic experiments
dence of the thermal conductivity in glasses at inter-
at low temperatures with frequencies ∼ 109 Hz, so that
mediate temperatures requires a constant diffusivity; a
propertythatexistsinjammedsystemsabovetheBoson
peak [28, 29]. Finally, at very low frequencies, jammed
∗ Authortowhomcorrespondenceshouldbeaddressed: systems contain quasi-localized, anharmonic vibrational
[email protected] modes which lie at the heart of mechanical rigidity [30]
2
and indicate the presence of “soft spots” in amorphous (a)
solids [31, 32]. 1.0 excitation
pulses
Unfortunately, inherent difficulties in computing the 0.8
vibrational mode properties in very large systems have max0.6 echo
restricted many studies to higher frequencies and tem- P
/
peratures. The low-temperature regime where two-level P 0.4
tunnelingstatesaresupposedtodominatethethermody-
0.2
namic properties has remained elusive. Our goal is not
to simulate the largest systems and lowest frequencies 0.0
0 500 1000 1500
directly, but rather to illustrate that the observation of
phonon echoes does not require a quantum mechanical
interpretation at all.
(b)
The first description of echo phenomena was given be
z
Hahn in 1950 by considering the response of an ensem-
ble of nuclear spins to two excitation pulses separated
transverse
by a time τ [33, 34]. Initially, all spins are vertically
excitation
alignedwithanexternalmagneticfieldpointinginthez-
direction. Thefirstpulseactstorotatethespinstowards
y
the x-y plane. In between the pulses, the spins precess
harmonically at their Larmor frequency, and eventually
decohere. The second pulse acts to “time-reverse” the
system, so that the system becomes coherent again at
t=2τ [34]. If we consider an ensemble of spins with dif-
ferent frequencies, the echo manifests as a macroscopic
x
sumofthespinvectors. Themaximumechooccurswhen
thefirstpulserotatesthespinsbyθ =π/2andthesecond
pulse rotates the spins by θ = π. A similar mechanism
FIG.1. (Coloronline)(a)Acousticechogeneratedinjammed
explains photon echoes observed at optical frequencies
systemscomposedof1000particlesatφ=0.70. Theresultis
[35, 36].
an average over the response of 10000 independent systems.
Afteraseriesofexcitationpulsesseparatedbytimeτ,aspon-
Although less well-known, echo phenomena can also
taneous re-phasing of the vibrational modes occurs at a time
be produced by another mechanism [37–41]. One pos-
t = 2τ. (b) Visual representation of one of the jammed sys-
sible mechanism relies on harmonic oscillators that in-
tems used in generating the echo. The red particles are 1.4
teract in a nonlinear way with the excitation pulses, as times larger than the blue particles. The acoustic pulses are
in temperature quench echoes [42, 43]. Another possible excited by a transverse standing wave along the x-direction.
mechanism involves anharmonic oscillators whose reso-
nant frequency shifts with increasing amplitude. This
mechanism is the source of observed echoes in many dif-
ferent systems ranging from cyclotron modes in plasmas II. NUMERICAL MODEL
[37,38,44,45]tothevibrationsofindividualparticlesin
piezoelectric powders [46–48]. Although there are many The majority of our simulations consist of a 3-
similarities between anharmonic echoes and spin echoes, dimensional ensemble of frictionless, spherical particles
therearemanycharacteristicdifferencessuchastherela- with finite-range, repulsive interactions [26]. We use a
tionship between the echo amplitude and pulse spacing, 50-50 binary mixture of particles with two radii, σ and
τ, and the existence of multiple echoes after only two 1.4σ. All particles have a mass m. The pair-potential
pulses for anharmonic echoes. A comprehensive review between any two particles is given by the following:
on both types of echoes can be found in reference [41].

In this paper we use simulations of model glasses to 2(cid:15)(cid:18)1− rij (cid:19)5/2 r <σ +σ ,
showthatclassicalvibrationalmodesindisorderedsolids V(rij)= 5 σi+σj ij i j (1)
can act as weakly-coupled anharmonic oscillators, and 0 r ≥σ +σ ,
ij i j
when excited by a series of pulses, produce echoes simi-
lar to those seen in experiments in glasses at low tem- where r is the distance between the centers of parti-
ij
peratures (Fig. 1a). By varying the pulse amplitude, cles i and j, and (cid:15) is the energy scale of the interaction.
spacing, and number of pulses, we can compare our re- All quantities reported here have lengths measured in
sults directly to experimental data. Our simulations are units of σ, mass in units of m, and frequency in units
(cid:112)
performed with both finite-ranged repulsive spheres and of (cid:15)/mσ2. The 5/2-exponent in the potential is de-
particles with Lennard-Jones interactions. rived from linear, elastic, Hertzian contact mechanics of
3
spherical particles. An important feature of this type of similarresults, yetthetransverseexcitationswerebetter
potential is the natural nonlinearity of inter-particle in- coupled to the anharmonic, low-frequency modes in the
teraction. There will always be a nonlinear correction to systems. Thus the majority of simulations used external
the harmonic approximation for the potential energy, so pulses according to Eq. 3.
thatthefrequencyresponseofindividualmodeswillvary The response of the system can be measured in many
withvibrationalamplitude. Thisfeaturewillturnoutto different ways. We chose perhaps the most natural way,
beessentialforthegenerationofanharmonicechoes,and and measured the response along the same vector that
will be discussed in section III. defined the excitation. That is, the forcing F(cid:126) represents
Individual systems were created by randomly placing avectorwith3N elements,andcanbeexpandedineigen-
N particlesinacubicboxwithperiodicboundarycondi- modes of the system. If the modes do not couple, then
tionsonallsides,whichrepresentsadensegasatT =∞. the total energy in each mode remains constant in time.
Each system is then quenched to T = 0 at the nearest The most convenient way to access the response of the
local potential-energy minimum using the Fast Inertial excitedmodeswastomeasurethepowerP =(cid:126)v·F(cid:126),where
Relaxation Engine (FIRE) algorithm [49]. The resulting (cid:126)visthevelocityvectoroftheparticles. Theresultingsig-
state of the system (i.e., jammed or un-jammed) will de- nal was then normalized by the maximum power (P )
max
pend on the volume fraction φ of particles. For the size during the pulses, as shown in Figure 1a.
ratio and particle interactions studied here, jamming oc-
For systems quenched from initially random positions
curs at φ = φ ≈ 0.64 [26]. All simulations reported
c (T = ∞), we found that some modes often went unsta-
here are for a volume fraction φ=0.70, so that the sys-
ble during excitation by an acoustic pulse. This is likely
tems are well into the jammed regime. In addition, at
due to the crossing of a significant energy barrier in the
thisvolumefraction,approximately0.3%oftheparticles
system, and was followed by a ≈ 10−20% drop in the
havenooverlapsaftertheinitialquench. Theseparticles
potential energy of the system. Upon re-quenching the
contributetrivialzero-frequencymodestothesystem,so
systemfollowingsuchaninstability,theminimumpoten-
theyareremovedpriortoacousticexcitation. A3-Drep-
tial energy at T =0 also decreased by ≈10−20%. The
resentation of a quenched 1000-particle system is shown
excess potential energy is likely due to the preparation
in Figure 1b.
of the system by quenching from T = ∞, without any
In addition to finite-ranged, repulsive interactions, we
annealingsteps. Byrepeatedlypulsingeachsystemwith
also simulated Kob-Andersen binary Lennard-Jones sys-
acousticpulsesofdecreasingF ,witheachpulsefollowed
0
tems [50]. Each system consists of 800 A and 200 B par-
byaquench,wefoundthatthestabilityofthesystemin-
ticleswithequalmassminteractinginthreedimensions.
creased dramatically. Thus all data reported here comes
The pair-potential between particles is given by:
from systems which have been prepared using this an-
(cid:15) (cid:34)(cid:18)σ (cid:19)12 (cid:18)σ (cid:19)6(cid:35) nealing protocol.
V(r )= ij ij − ij , (2)
ij 72 r r
ij ij
III. CHARACTERIZATION OF MODES IN
where (cid:15) =1.5(cid:15) , (cid:15) =0.5(cid:15) , σ =0.8σ , and
AB AA BB AA AB AA JAMMED SYSTEMS
σ = 0.88σ [27]. The potential is cut off at r =
BB AA ij
2.5σ and the potential is shifted so that V(2.5r ) =
ij ij Oneofthedefiningcharacteristicsofcrystallineelastic
0. We also add an additional linear correction so that
solids is that at sufficiently low frequencies, all vibra-
V(cid:48)(2.5r ) = 0. All Lennard-Jones systems were created
ij tional modes are plane-wave acoustic modes. In stark
at a density ρ = 1.2, then quenched using the FIRE
contrast, glasses display an excess of anomalous modes
algorithm.
at low frequencies, some of which are spatially localized.
Once a system is quenched, we excited the vibrational
Jammed systems of soft, frictionless spheres contain a
modes using external pulses. Each pulse consisted of ap-
very large number of low-frequency modes as well [25].
plying a transverse, spatially-varying sinusoidal force to
Theamountofexcessmodeswilldependonthedistance
the particles. The force on the ith particle is given by:
from the critical volume fraction, ∆φ = φ−φ , where
c
F(cid:126) =F sin(k x −ω t)yˆ, (3) φc is the volume fraction when the system first begins to
i 0 x i 0
jam. The peak in the excess number of modes (which is
wherek =2π/Listhewavevector,andListheboxsize. knownastheBosonpeakintheglassliterature)occursat
x
We used the longest wavelength that could fit along one a characteristic frequency ω∗, which tends towards zero
boundaryofthedomain(asshowninFigure1b). Inorder asφ isapproached: ω∗ ∝∆φ3/4 forthepotentialchosen
c
to maximize the coupling of the pulse to a narrow band in equation 2 [25].
of vibrational modes, the pulse frequency was restricted Figure 2a shows the average density of states for sys-
sothatω /k ≈v ,wherev isthespeedofsoundinthe temsofN =1000particles. Theapproximatelocationof
0 x s s
system. The pulse amplitude F , and duration t were ω∗ is shown by the red point. The inset shows the same
0 p
adjustableparameters,althoughtypicalrangesoft were dataonalogscale. Themodesbelowω∗ consistofamix
p
10-45cycles,wheretheperiodof1cycle=2π/ω . Longi- of extended and quasi-localized vibrational modes [51].
0
tudinalpolarizationswerealsostudiedwithqualitatively Since each system only contains 1000 particles, the low-
4
3.0 100
(a)
2.5 10-2 (a)
s
of state 12..50 100 1100--46 slo pe = 2 freq uency
ty 1.0 10-2
si 10-8
n
de 0.5 10-140-3 10-2 10-1 100 10-10
0.0
0.0 0.1 0.2 0.3 0.4 0.5 10-7 10-6 10-5 10-4 10-3 10-2 10-1
frequency mode amplitude A
11..00
(b) 11..00 106
0.8 0.8 (b)
o
ti 104
ra 0.6
n 0.6
o
ati 0.4 102
p
ci 0.4 0.2
ti
par 0.0 100
0.2
10-2 10-1 100
0.0
0.0 0.1 0.2 0.3 0.4 0.5
frequency
FIG. 3. (Color online) (a) Normalized frequency shift ver-
sus amplitude of 3 modes in a single 1000-particle system.
FIG. 2. (Color online) (a) Density of states for systems of
Overabroadrangeofamplitudes,equation7isvalidathigh
1000bi-disperseparticlesatφ=0.70. Thesolidblacklinesare
(red), intermediate (blue), and low (black) frequencies. De-
theaverageof1000independentsystems. Theinsetshowsthe
viations at higher amplitudes are due to coupling between
same data on a log scale. The dashed line represents an ω2
nearbymodes. (b)Coefficientoffrequencyshiftξversusmode
behavior, consistent with Debye theory. The red dot shows
frequency for 500 modes in 5 different 1000-particle systems.
the approximate position of ω∗, and the blue dot shows the
ForagivenamplitudeA,modesatlowerfrequencywillexpe-
frequencyatwhichmostofthesimulationsareperformed. (b)
rience a larger frequency shift.
Participationratioofallmodesin1000independentsystems.
The color scale represents the density of the points. Red is
themaximumdensityofstatesD ,andpurpleisnearzero.
max
larger amplitude induces nonlinear effects in the vibra-
tion at smaller energies. Low-frequency plane waves in
estfrequencyplane-wavemodewouldoccuratω ≈0.025. crystallinesolidsarespatiallyextendedandarethemost
Thedegreeoflocalizationofthemodesisillustratedin harmonic modes in the system due to the small relative
Figure 2b, which shows the participation ratio for each displacements between neighboring particles. However,
mode. The participation ratio measures the fraction of in jammed solids, the lowest frequency modes are the
particles participating in a given vibrational mode: most anharmonic [30].
(cid:16) (cid:17)2
(cid:80) |eˆ |2
l m,l
p(ω )= , (4)
m N(cid:80) |eˆ |4
l m,l
A. Anharmonic Frequency Shifts
where eˆ is the lth component of the unit eigenvector
m,l
corresponding to the mth eigenmode. At low frequen-
An important consequence of nonlinearity is that the
cies, thejammedsystemscontainabroaddistributionof
fundamental frequency of the mode will shift with am-
participationratios,asshownbythelargespreadinpur-
plitude. Let us consider the simplest model of an anhar-
ple data points. Many of these modes are quite localized
monicoscillatorwithmassmandfundamentalfrequency
(low participation ratio). It has also been shown that
ω with a cubic perturbation to the potential V:
these modes exist independent of ∆φ [30].
Onenaturalconsequenceofspatiallocalizationinavi-
brationalmodeisthatforagivenamountofenergy,fewer mω2 mω2
V = x2+ x3. (5)
particlesareundergoingalargeramplitudemotion. This 2 3x
0
5
The equation of motion of this oscillator is thus ξA2). We have also ignored any other higher harmonics
1
in the solution stemming from the nonlinearity of the
(cid:18) (cid:19)
x
mx¨=−mω2x 1− . (6) oscillator, and only consider the frequency shift to the
x0 fundamental mode.
At time t = τ, we apply a second pulse, which adds
To second order, it can be shown that the frequency of
amplitude A to the position of the oscillator:
the oscillator depends on the square of the amplitude of 2
vibration [52, 53]:
x(τ)=A2+A1eiωτ(1−ξA21). (9)
∆ω
=−ξA2, (7)
For simplicity, we will assume that A and A are real,
ω 1 2
although the same analysis can be done in the case that
where ξ =5/12x20. they are complex. The new amplitude of oscillation is:
We can measure the anharmonicity of the modes in
jammed systems by applying an initial amplitude to the |x(τ)|2 =A2+A2+2A A cos(cid:0)ωt(cid:0)1−ξA2(cid:1)(cid:1). (10)
1 2 1 2 1
modes at t=0 with all particles at rest, then letting the
system evolve in time. Specifically, this is accomplished The evolution of the oscillator after the second pulse de-
by adding a vector Aeˆ to the initial position vector of pends on its amplitude, so that
m
all of the particles, where A is the amplitude and eˆ is
m
theeigenvectorassociatedwiththemthmode. After200 x(t≥τ)= (11)
ficytctloesa, tshineurseosidualtlinfugnmctoiotniontooofbtthaeinptahretifcrleeqsuaelnoncygoeˆfmthies (cid:16)A2+A1eiωτ(1−ξA21)(cid:17)eiω(t−τ)(1−ξ|x(τ)|2).
mode.
Figure 3a shows the normalized frequency shift as a Combiningequation10and11,weobtainthefullsolution
function of amplitude for three modes at high, inter- of the oscillator when t≥τ:
mediate, and low frequencies. The frequency shift is
x(t≥τ)= (12)
quadratic in amplitude, and is larger at low frequen-
(cid:16)(cid:16) (cid:17) (cid:17)
cies. Athigheramplitudes,theinteractionamongnearby A2+A1eiωτ(1−ξA21) eiω(t−τ)(1−ξ(A21+A22)) ×
modesbecomespronouncedandenergyistransferredbe-
tweenmodes,leadingtoadampingofthevibrationsand e−i2ξω(t−τ)A1A2cos(ωτ(1−ξA21)).
other forms of nonlinearities. Eventually, at very large
amplitudes,particlerearrangementsoccurandtheeigen- The complexity here is due to the fact that there is a
modes have changed, so our analysis is no longer valid. cosine function in the argument of the exponential. We
Figure 3b shows how ξ depends on mode frequency. can simplify this part by use of the Jacobi-Anger expan-
Localized modes at high are anharmonic, but the most sion [54]:
anharmonicmodeslieatlowfrequencieswherethemodes
are quasi-localized. The broad distribution of ξ at lower ∞
(cid:88)
frequenciesisrelatedtothebroaddistributioninthepar- eizcosθ = inJn(z)einθ, (13)
ticipation ratio. n=−∞
whereJ isaBesselfunctionofthe1stkind. Thenequa-
n
tion 12 becomes:
IV. ECHOES FROM ANHARMONICITY
x(t≥τ)= (14)
Inordertounderstandhowanensembleofanharmonic
(cid:16)(cid:16) (cid:17) (cid:17)
oscillators can give rise to an echo, let us first consider A2+A1eiωτ(1−ξA21) eiω(t−τ)(1−ξ(A21+A22)) ×
theresponseofasingleoscillatortotwoexcitationpulses.
∞
Our derivation is similar to previous derivations [38, 45, (cid:88) inJn(2ξω(τ −t)A1A2)e−inωτξA21einωτ.
47], except that here we explicitly deal with mechanical
n=−∞
oscillators for arbitrary amplitudes. We will then sum
the contribution of many single oscillators with different At this point it is helpful to define a characteristic fre-
natural frequencies. For simplicity, we will only consider quency shift Ω = ωξA2 and pulse amplitude ratio α =
1
δ-function pulses which add a finite amount of energy to A /A . Withthesesubstitutionsandsomealgebraicma-
2 1
the oscillator in a short period of time. nipulations, equation 14 becomes:
At time t = 0, the first pulse excites the oscillator so
that it begins with amplitude A , then evolves in time: x(t≥τ)= (15)
1
∞
x(0<t<τ)=A1eiωt(1−ξA21). (8) (cid:88) eiω(t−nτ)i3nA1eiΩ(nτ−t)eiΩα2(τ−t)×
n=−∞
Here the frequency is slightly less than the fundamental
[J (2αΩ(t−τ))+iαJ (2αΩ(t−τ))].
frequencyduetothefiniteamplitude,sothatω =ω(1− n n−1
1
6
Wemaynowidentifytermsinthesolutionthatvaryon
different time scales. Oscillatory terms containing “ωt”
in their argument vary rapidly in time, whereas terms
with“Ωt”willvarymuchmoreslowlysincethefrequency
shift is much smaller than the fundamental frequency
(Ω/ω (cid:28)1). Thus we can write
∞
(cid:88)
x(t≥τ)= G(Ωt)eiω(t−nτ) (16)
n=−∞
where
G(Ωt)=i3nA eiΩ(nτ−t)eiΩα2(τ−t)× (17)
1
[J (2αΩ(t−τ))+iαJ (2αΩ(t−τ))].
n n−1
Equations 16 and 17 apply to only a single oscillator FIG. 4. (Color online) Proxy for echo amplitude |G2τ|, nor-
malized by the first pulse amplitude A , as a function of
which is excited by two delta function pulses. An echo 1
Ωτα (see equation 19). Over a broad range of values for
involves the coherent sum of many oscillators at a given
theratioofpulseamplitudesα,themaximumechooccursfor
point in time. Each oscillator may have a different fun-
1<Ωτα<2.
damentalfrequency,ω. Thustheechoamplitude,X,will
be given by
∞
(cid:88) (cid:88) excitation pulses, then its effect on the dynamics will
X(t≥τ)= G(Ω t)eiωm(t−nτ), (18)
m be reduced. The appearance of multiple echoes and the
m n=−∞ dependence on pulse spacing will be discussed in more
detail in section V.
where ω is the fundamental frequency of the mth os-
m
cillator, and Ω = ω ξA2. When performing the sum Figure 4 shows shows how equation 19 depends on the
m m 1
overm,theexponentialtermeiωm(t−nτ) willvaryrapidly quantity Ωτα. Of particular importance is where the
maximum echo is located. When the second pulse am-
with time and sum to zero since ω is different for every
plitude is comparable to the first pulse amplitude (i.e
oscillator, i.e., the total signal will be decoherent. How-
α≈1),thenthemaximumechoisachievedwhenΩτ ≈1.
ever, if t = nτ, the exponential term will be near unity
This means that the characteristic frequency shift of the
andtheoscillatorswillbecoherent. Theechoamplitude,
oscillatorsshouldbetheinverseofthepulsespacing. The
X will then depend only on G(Ω t), which varies slowly
m
requirements to achieve the maximum echo amplitude
with time since Ω (cid:28) ω . One important consequence
m m
are technically different in spin echoes, where the sec-
of equation 18 is that not only do we expect an echo at
ond pulse (π-pulse) should be twice as large as the first
t=2τ,butalsomultipleechoesatt=3τ,4τ,etc. Thisis
(π/2-pulse), given they are the same duration.
adistinguishingfeatureofclassicalechoesinanharmonic
oscillators. The simplest description of a quantum me-
chanical spin echo only contains features at t = 2τ. At
V. ECHOES IN MODEL GLASSES
later times the precessing spins become incoherent.
Let us assume that the excitation pulse excites a nar-
row band of oscillators with similar anharmonicity, so Thediscussionuptothispointhasonlyconsiderediso-
that Ω is approximately constant, and we may drop lated,independent,anharmonicoscillators. Wenowturn
m
the subscript m. If we only consider the first echo, i.e. our attention to echoes in model glasses. We emphasize
t=2τ and n=2, then G becomes: here that each “oscillator” is a normal vibrational mode
of the disordered solid. The echo signal is the sum of
G =−A e−iΩτα2(iαJ (2Ωτα)+J (2Ωτα)). (19) the vibrational motion of all of the excited modes. For
2τ 1 1 2
very small amplitudes, each mode is linearly indepen-
so that the echo amplitude is approximately given by dent. However,forlargeramplitudestheywillnecessarily
couple energy between different modes, invalidating our
(cid:112)
X ∝|G |=A α2J (2Ωτα)2+J (2Ωτα)2. (20) analysisin theprevious section. Asillustrated byFigure
2τ 1 1 2
3, the normal modes are naturally anharmonic, so that
In this form it is apparent that the echo amplitude de- an echo should be observable so long as the amplitude
pends on the pulse spacing, in contrast to spin echoes. of each mode is not too large, and they remain linearly
Specifically, the echo amplitude tends to zero at small independent.
pulse spacings. This can be seen by considering the fre- WhenexcitedbyanacousticpulsenearT =0,asingle
quency shift of the oscillators as a slowly-varying phase. vibrationalmodewillincreaseinamplitude,andthefinal
If the phase does not have time to evolve between the amplitudeofvibrationwilldependondifferencebetween
7
(a) 0.25
0.8 0.20
x
0.4 ma0.15
x P
a
m /
P 0.0 P 0.10
/
P
0.8 0.05
0.00
0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (x 10-5)
pulse amplitude F
0
0.0
0 500 1000 1500 2000 2500 3000
FIG. 6. Echo amplitude versus pulse amplitude F . Each
0
(b) datapointistheaverageof10,000independentsystems,each
composedof1000particles. Thepulsewidthist =45cycles,
p
0.25 and the pulse frequency is ω = 0.025, which is below ω∗
0
as shown in Figure 2a. The decay at long times is due to
0.2 nonlinear coupling between the modes which causes energy
to spread to other modes in the system.
x0.15
a
m
P
/ 0.1
P
shows the averaged amplitude at two different values of
0.05
pulseseparationτ. Bothpulseshaveidenticalamplitudes
(F ≈ 5×10−5), and identical pulse durations: t = 45
0.00 0 p
0 500 1000 1500 2000 2500 3000 cycles(thisvaluefortp waschosenbecauseitwascloseto
thevalueusedintheoriginalexperimentswhichobserved
phonon echoes at low temperatures in glasses [22]). In
both plots, the echo is apparent at t ≈ 2τ. Taking into
FIG. 5. (Color online) (a) Echo amplitude, normalized by
account the finite pulse width (t ), the exact position of
P , vs. time for two different values of pulse spacing τ. A p
max theechois2τ+2t ,sinceτ ismeasuredfromthecenterof
third example, with an intermediate value of τ, is shown in p
each excitation pulse and the first pulse begins at t=0.
Figure 1a. Each signal is the average of 10,000 independent
Figure 5b shows the echo amplitude (normalized by
systems, each composed of 1000 particles. The pulse width
is tp = 45 cycles, and the pulse frequency is ω0 = 0.025. (b) Pmax)asτ isvaried. Thisdependencecanbeunderstood
Normalized echo amplitude vs. τ. The error bars represent using equation 20. We fit the data to the form:
the size of the noise between the second pulse and the echo.
(cid:112)
The red line is a fit to the data using equation 21. |A |=K J (K τ)2+J (K τ)2, (21)
2τ 1 1 2 2 2
where K and K are fitting parameters. The best fit
1 2
is shown by the red line in Figure 5. This is essentially
the frequency of the oscillator and the frequency of the the same curve as the red line in Figure 4. The error
exciting pulse, in addition to the spatial coupling to the bars represent the average noise in the amplitude in the
polarization of the excitation (equation 3). The number region between the second pulse and the echo. Although
of modes excited by a given pulse is inversely propor- equation 21 is derived from the dynamics of a single os-
tional to the duration of the pulse. For long pulses, only cillator, both K and K represent an average over the
1 2
modeswithfrequenciesneartheexcitationfrequencywill different modes excited by the pulses. Since K ∝ A ,
1 1
be driven to large amplitudes, whereas for short pulses, and A represents the initial amplitude, its value will
1
many modes of different frequencies will be excited (e.g., varyconsiderablyfrommodetomode. However, K will
2
adelta-functionpulsewillexcitemodesofallfrequencies be more uniform since it represents the frequency shift,
equally). Ω, and only modes that satisfy Ωτ ≈1 will contribute to
A second pulse at a later time can either increase or the echo.
decrease the amplitude of an individual mode, depend- ThereasonableagreementinFigure4isduetothefact
ing on the phase difference between the mode and the that the condition Ωτ ≈ 1 can be achieved by increas-
excitation. An echo will be formed by the average of an ingτ ratherthantheamplitude, sothatthenonlinearity
ensemble of vibrational modes, which become coherent remains a perturbation to the system and their is little
at a later point in time. For systems composed of 1000 cross-talk between adjacent modes. However, if we vary
particles, we found that averaging over 10,000 indepen- thepulseamplitudeinsteadofτ,thenweinjectmoreen-
dent systems was necessary in order to achieve a suffi- ergy into each mode at higher amplitudes and the echo
cient echo signal above the background noise. Figure 5a amplitudeisreducedduetononlinearcouplingsbetween
8
1.0 1.0
0.8 0.8
x
ax0.6 ma0.6
m P
P
/
/0.4 P 0.4
P
0.2 0.2
0.0 0.0
0 500 1000 1500 2000 2500 0 1000 2000 3000 4000
FIG. 7. Multiple echoes are not observable in the average FIG.8. (Coloronline)Echosignalfrom10,000averagedsys-
response of 10,000 systems due to noise limitations. Each tems, as in Figure 7, with the addition of a third pulse after
systemhas1000particles. Thepulseamplitudeandfrequency the first echo. The three-pulse echo sequence produces four
are the same as in Figure 1a and Figure 5a. The red arrow total echoes, with the positions indicated by the red arrows.
indicates the predicted position of the echo at t=3τ. This is a characteristic of both parametric (spin) echoes and
anharmonic echoes.
modes, which is not accounted for in the model. Fig-
1.0
ure 6 shows a peak in the echo amplitude as the pulse
amplitude is varied. This is expected from equation 20, 0.8
and illustrated in Figure 4. However, the data in Fig- max0.6
ure 6 decays much more rapidly, which is likely due to P
/ 0.4
the coupling between modes for larger pulse amplitudes, P
whereenergyisbeingredistributedtoothermodesinthe 0.2
system.
0.0
When compared to spin echoes, a defining character- 0 500 1000 1500
istic of anharmonic echoes is the occurrence of multiple
echoesafterjusttwopulses(equation16). Figure7shows
the average response of 10,000 systems of 1000 particles FIG. 9. Echoes in systems with Lennard-Jones interactions.
each, identical to Figure 1a and Figure 5a, except ex- Thesignalistheaverageof10,000independentsystems,each
tended to longer times. Multiple echoes are clearly not composedof1000particles. Thepulsewidthistp=45cycles.
visible. This is likely due to the signal noise in this re-
gion. One possible remedy is to average over many more
√
systems, since the noise decreases as N, although this
was computationally prohibitive. One may reasonably Although jammed systems of frictionless spheres pro-
expect the 3τ echo to be reduced in amplitude by the vide the simplest example of model glasses, we have also
same factor as the 2τ echo is with respect to the pulse studiedechoeswithmorerealistictwo-particlepotentials.
amplitude. If this is true, then it is not surprising that Figure 9 shows the average response of 10,000 systems,
the 3τ echo is not visible since it would clearly lie below each composed of 1000 particles with Lennard-Jones in-
the noise. teractions. Specifically, we use a Kob-Andersen binary
If one applies a third pulse to the system, then there system, as described in section II. The density of states
will be a total of four echoes that can be observed. The of these systems looks somewhat different from jammed
positions of these echoes are τ , τ , τ + τ , and τ − systems [27]. The excitation frequency was chosen to be
1 2 1 2 1
τ , where the times are referenced with respect to the approximately 5% of the maximum frequency in the sys-
2
positionofthethirdpulse. Thepulsespacingτ refersto tem, and consistent with the longest-wavelength plane
1
thefirstandsecondpulse,andτ referstothesecondand wave which could fit inside the simulation boundaries
2
thirdpulse. Thisistrueforbothtwo-levelsystemechoes (Figure 1b). The echo looks nearly identical to those
(i.e. spin echoes), as well as the classical anharmonic in Figure 1a and 5a, where the particles interact via
echoesthatwearetreatinghere. Figure8showsathree- Hertzian potentials.
pulse echo sequence in jammed systems. The signal is All of the data that we have reported here so far has
theaverageof10,000systemscomposedof1000particles been taken on systems with 1000 particles, then aver-
each, and is identical to Figure 1a, with the addition of aged over many configurations. This is partially due to
a third pulse at a later time. If the third pulse is placed the fact that the calculation and characterization of the
prior to the first echo, then τ −τ is positive, and all dynamical matrix and vibrational modes is straightfor-
1 2
four echoes occur after the three pulses. ward. We can also average over fewer configurations of
9
systemswithalargernumberofparticlesandobtainsim-
1.0
ilarresults. Figure10showsanechosignalresultingfrom
averaging the response of 5000 systems, each composed 0.8
x
of 8000 particles. Each cubic system of particles is twice ma0.6
as long on one side as a 1000 particle system, so the fre- P
/ 0.4
quency of excitation was smaller by a factor of two, and P
the wavelength was longer by a factor of two (cf. Figure 0.2
1b). 0.0
However, for systems with more than 8000 particles, 0 500 1000 1500
the computational requirements to observe an echo be-
come expensive. This is mostly due to the fact that our
simulations require simulating many thousands of cycles FIG.10. Amplitudeversustimeshowinganechoinsystemsof
of low-frequency oscillations. However, we can estimate 8000particles. Thesignalistheaverageof5000independent
the conditions necessary to observe an echo in only one systems. The pulse frequency is half of that used in 1000-
system, rather than averaging over many systems. In particle systems. The pulse width is tp = 45 cycles.
order to observe an echo clearly, there must be a suffi-
cient number of excited modes, N , which will average
e
to zero in regions between the pulses and the echo. The
constraint, we suggest that we only apply frequencies in
number of excited modes is proportional to the number
the region of the quasi-localized modes.
of systems, N , the density of vibrational states at the
s
excitationfrequency,D(ω),andinverselyproportionalto The amount of coupling depends on the frequency dif-
the pulse width, tp: ference between two modes: ω1−ω2. As we increase the
number of particles, the density of states also increases,
N D(ω) so the frequency spacing between the modes decreases.
N ∝ s . (22)
e t This is unavoidable. Also, the amount of coupling de-
p
pends on the spatial overlap between the modes. Plane
For simplicity, let us assume that the density of states wavesareextendedmodesthatwillinevitablysharepar-
obey’s a Debye-like behavior, so that D(ω) ∝ N ω2, ticlevibrations. However,twolocalizedorquasi-localized
p
where N is the number of particles in the system. The modes, if sufficiently far away from each other, will have
p
pulse width used in our simulations is t = 45 cycles very little coupling, regardless of the frequency.
p
=45×2π/ω. Then equation 22 reduces to Thus,givenafixedexcitationfrequencyω,asN →∞,
p
the energy flow between modes will eventually destroy
(cid:18) ω (cid:19)3 the coherence of the echo. This can only be remedied if
N ∝N N . (23)
e s p ω mostoftheexcitedmodesarequasi-localizedsothatthey
D
can still behave as independent anharmonic oscillators.
The frequency in equation 23 has been normalized by
Wesuspectthatechoescouldbeobservedinasingle,very
theDebyefrequencyinordertoeasilycomparetoexper-
large system provided that the density of plane waves is
iments. We are interested in comparing our simulations
much smaller than the density of quasi-localized modes.
with experiments from low temperature glasses where
possible. For that reason we chose t = 45 cycles, and
p
ω/ω ≈0.0002,whicharetypicalvaluesusedintheorig-
D
VI. CONCLUSIONS
inalexperimentswhichobservedphononechoesinglasses
at low temperatures [22]. Using these values, N ∼ 640.
e
We would need the same number of excited modes in a These results illustrate how the anharmonic vibra-
single system to see the echo. If we assume we have one tionalmodesinajammedsystemofparticlescangiverise
system (N = 1), then we would need N ≈ 8×1013 tophononechoes,similartothosemeasuredinglassesat
s p
particles to observe an echo at such low frequencies. We low temperatures. The mechanism of echo generation is
have also assumed that the density of states is quadratic distinctly different from echoes produced by quantum-
infrequency. AttemperaturesbelowT =100K,theden- mechanical two-level systems [41]. In our simulations,
sity of states in glasses is known to decrease faster that echoes are produced by a purely classical mechanism
ω2 [55–57]. This would only strengthen the dependence caused by the frequency shift of the anharmonic vibra-
of N on ω, and necessitate even larger systems in order tional modes. This shift acts as a slowly-varying phase
e
to observe an echo, thus our estimate constitutes a lower which evolves in the time between the pulses, resulting
bound on the system size. in a non-zero average of the ensemble.
We would of course like to observe an echo in a single The anharmonicity of the vibrational modes can be
system, but this is computationally unfeasible. Not only studied at T = 0. At low frequencies, model glasses
does it require very large systems, but it also requires based on jammed sphere packings have quasi-localized
thattheanharmonicoscillatorsdonotcoupletheenergy modesthatcontributetothedensityofstates[30]. Ithas
inbetweenthemstrongly. Inordertominimizethislatter been argued that jammed systems are marginally stable
10
cated. Qualitatively,wenotethattheappearanceofmul-
tipleechoesaftertwoexcitationpulsesintheechoexper-
y
g iments[22]isanaturalanduniquefeatureinanharmonic
r
ne echoes,anddoesnotdependonthedetailsofthesystem.
e
At very low temperatures, a quantum mechanical pic-
ture of the dynamics is certainly necessary. The tradi-
double-well anharmonic tional explanation for the excess excitations in glasses
potential potential
at very low temperatures relies on two-level tunneling
systems created by the splitting of the ground state en-
FIG. 11. (Color online) Quantum mechanical picture of en-
ergy in a double-well potential (Fig. 11). The distribu-
ergy levels corresponding to the low-frequency excitations in
tion of the energy barriers are assumed to be broad, and
glasses. The double-well is the traditional view of the ori-
the states are spatially localized. We offer an alterna-
ginoftwo-levelsystems,wherethelowestenergylevelissplit
in two by the presence of the barrier. Alternatively, a wide, tive picture based on localized, anharmonic vibrational
anharmonicpotentialcanproducelow-frequencymodes,non- modesthatcanbeunderstoodbothclassicallyandquan-
uniformlevelspacing,andnonlinearacousticphenomenasuch tum mechanically. A wide and shallow anharmonic po-
as echoes. tential (Fig. 11), characterized by low-frequency, nearly-
unstable modes, will have energy levels which are not
equally spaced. These modes arise naturally due to the
amorphous nature of the solid, and do not depend on
andinherentlyclosetoaninstabilitywherethestructure
specific particle interactions.
will rearrange [58–60]. The contribution of such incipi-
Ourresultshavefocusedsolelyontheoriginofphonon
ent instabilities to the density of states is currently be-
echoes in glasses, and have not addressed many other
ing evaluated [61]. The anharmonicity that is important
well-known nonlinear acoustic properties in glasses, such
for echoes is generated by the frequency shift of a mode
as saturation of attenuation and hole-burning. However,
with increasing amplitude; this is due to expansion non-
the existence of these phenomena in glasses may not be
linearity [62, 63], and is measurable at small amplitudes.
restricted to models that require two-level systems. Past
If the amplitude becomes too large, then there will be
theoretical results suggest that some universal thermo-
energy transfer between modes and the modes will lose
dynamic properties in glasses need only modest assump-
coherence. At even higher amplitude in a system with
tions about the nature of the low-temperature modes
only finite-ranged interactions, the contacts can break
[65]. More recent results show that universal features
and reform [64]. This would destroy echo formation.
of acoustic attenuation in glasses can be explained by
Theechoesobservedinoursimulationshavemanyfea-
generic,elastically-coupledresonantmodes,andthatthe
tureswhichareconsistentwithparametric,two-levelsys-
details and origins of the resonant modes are less impor-
tem echoes, such as the three-pulse echo sequence (Fig.
tant [15].
8). However, many features are quite different. First,
Finally, we note that the simulations presented here
anharmonicechoesdonothaveasimple, intuitivecondi-
were performed on small systems. The detailed prop-
tion for maximizing the echo signal, such as a π/2 pulse
erties of the anharmonic modes at very low-frequencies
followed by a π pulse. In fact, Figure 4 shows that the
have yet to be investigated due to computational lim-
maximum echo amplitude is a complicated function of
itations on system size. One benefit of using jammed
pulsespacingandamplitude. However,herewenotethat
spheres with finite-ranged repulsions as a model glass is
for small amplitudes, equation 20 reduces to:
that there are two limiting regimes to investigate [62].
Although the limit N → ∞ is inaccessible, we can take
|G |=A α2Ωτ =A A2ξωτ, (24)
2τ 1 1 2 the limit ∆φ → 0, bringing the system on the verge of
instability. In this regime it is well known that jammed
which is in agreement with previous authors [45], and
systems develop an enormous increase in the density of
has the same dependence on A and A as the small-
1 2
states. The fate of these excess modes as the the system
amplitude result for spin echoes [22].
is compressed above ∆φ = 0 provides a starting point
Graebner and Golding [22] measured this small-
for our understanding of the low-frequency, anharmonic
amplitude dependence in silica glass, and also showed
modes.
that the maximum echo intensity does not precisely oc-
cur when A = A /2, among other discrepancies with a
1 2
model of echoes based on two-level systems. In addition,
at small amplitudes, Graebner and Golding observed a VII. ACKNOWLEDGEMENTS
smallincreaseinechoamplitudewithpulsespacing. This
featureischaracteristictoanharmonicechoesandisseen We are grateful to Brent Busby for assistance with
in our simulations (Fig. 5b). However, the experiments computational servers. We thank Efi Efrati, Carl
probe≈1000timeslowerfrequenciesthanwecanaccess Goodrich, Dustin Kleckner, Andrea Liu, and Ning Xu
in the simulations, and also involve additive reflections for important discussions. We acknowledge support
of pulses, so a more quantitative comparison is compli- from NSF MRSEC DMR-1420709, NSF DMR-1455086