Transcript Mesoscopic Electronics with Ultrasound, Diffuse Field Correlations

•
Mesoscopics with Ultrasound
•
We term mesoscopic a diffuse wave field that has mean free paths against absorption or
escape or dephasing that are long compared to mean free paths against elastic scattering.
The corresponding residual coherence in an otherwise fully diffuse wave field can lead to
interesting phenomena often associated with optics and non-interacting mesoscopic
electronics. At MHz frequencies ultrasonic waves in solids have convenient size and time
scales. Ultrasonic systems with high Q's (approaching 10^5 in aluminum) thus lend
themselves to analog investigations in Quantum Chaos and statistical scattering theory.
Phenomena studied include enhanced backscatter, localization, level statistics, S-matrix
statistics, and field-field correlations. Applications appear also in seismology, room acoustics,
and structural acoustics and vibrations.
•
This talk reviews three of our more recent experiments, one on diffuse field correlations with
applications in seismology, one on fidelity decay with relevance to quantum computing, and
one on a reverberant structure with ultrasonic feedback that is an analog for a laser.
Mesoscopics with Ultrasound
classical elastic waves
Richard Weaver
T&AM Illinois
Ole Miss 21 Feb 2006
Typical Experiment
Statistics?
Room Acoustics
With Ultrasound
Object Size ~ 10 cm >> Typical wavelength ~ 6 mm
Signal Duration >> Mean free time between scatterings >> 1/frequency
100 msec
30 msec
2 msec
Theme:
Ultrasonics in High Q Reverberant Bodies
Is an analog for
non-interacting electrons in Q dots
microwave cavities
optical resonators
In which much “mesoscopic” behavior is observable,
and often more accessible.
Experiments have convenient
length scales (wavelengths of mm)
time scales (frequencies of MHz)
Ray-chaotic “cavities”
Multiply scattering bodies
Conventional Ultrasonics looks at
direct signal from one
transducer to another at short times:
0.3
0.2
LL
LLLL
R
0.1
LS?
P
-0.0
-0.1
-0.2
-0.3
0
10
20
30
40
50
60
70
80
90
100
time ( msec)
Observe ray arrivals corresponding to
Surface Rayleigh wave
Surface skimming P wave (L)
Bottom reflected P wave (LL)
P wave reflected twice off bottom(LLLL)
Diffuse Field Ultrasonics looks at the “coda”
What does one measure?
The obvious:
Spectral energy density
Dissipation
Diffusion / Transport
eigenfrequencies
The less obvious:
“Mesoscopics”
1.5
1.0
0.5
273 kHz
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
820 kHz
507 kHz
-3.0
-3.5
-4.0
-4.5
-5.0
-5.5
-6.0
-6.5
-25
0
25
50
75
100
125
150
175
200
225
250
275
time (msec)
Typical Behavior of Spectral Power Density versus time is an
(obvious?) exponential decay, at a rate related to internal friction.
Transport in
media with random
multiple scattering
fits well to a diffusion
equation with dissipation
D 2 E(r,t)  E
  3 (r) (t)  E /t
Spectrum at low frequencies
shows individual modes
Spectrum at high frequencies
is heavily overlapped
cf Ericson oscillations of nuclear cross sections
Mesoscopics (for non-interacting electrons)
• Device size L ~ microns
– So electron transit time, or mean free time before
scattering, is short
• Temperature Low
– So mean free time against inelastic scattering
is long.
e.g. by a phonon, decohering electron wave phase
Wave nature of electron becomes relevant =>
constructive and destructive interferences
The term “ mesoscopic ” has been adopted by
classical wave researchers to describe other types of waves
(e.g. optics, radar, ultrasound, acoustics, and seismic waves)
That are multiply scattered.
But which retain residual coherences
But retains some coherence
Similar physics can arise in
microwave and optical resonators
Mesoscopic Phenomena that we study:
Level statistics - GOE of RandomMatrixTheory
Coherent (enhanced) Backscatter
Anderson Localization (2-d) & (0-d)
S-matrix statistics, and RandomMatrixTheory
Return probabilities / level width statistics
Field-field correlations
*
“Fidelity” decay
*
Random (u/l)aser
*
Many phenomena
that depend on High Q’s and random scattering
* = today’s talk
Field-field correlations
*
Assertion:
A diffuse field’s
field-field correlation
IS
the Green Function of the structure
  (x,t   )  (y,t)   G(x, y,  ) ?

PLAUSIBILITY ARGUMENT number 1
 (x,t)  Re  an un (x)exp{i nt}
Field:
n
Diffuse Field Mode Amplitude Statistics:

 an am*  nm F( n )
The cross correlation of the fields at x and y is then

Compare:
Does
C(x, y,  ) 
1
  (x,t   ) (y,t)  Re  F( )un (x)un (y)exp(in )
2
n

C = dG/dt ?
Almost . . .
They differ by
Factor F, which distorts the spectrum
Support for negative .
Caveats:
Is our field truly diffuse?
?
What is meant by < > ?
Effect of transducer bandwidth?
PLAUSIBILITY ARGUMENT number 2
QuickTi me™ and a
TIFF ( LZW) decompressor
are needed to see thi s pi ctur e.
Experiments
Thermo-elastic surface
excitations from pulsed laser,
~132 different positions
A Q-switched Nd:YAG laser excites elastic waves
in an irregular aluminum block of nominal dimension 15 cm.
The signals from two piezoelectric transducers are
amplified and lo-pass filtered before being digitized
by a PC.
Direct Signal from one transducer to the other:
0.3
0.2
LL
LLLL
R
0.1
LS?
P
-0.0
-0.1
-0.2
-0.3
0
10
20
30
40
50
60
70
80
90
100
time ( msec)
Observe ray arrivals corresponding to
Surface Rayleigh wave
Surface skimming P wave (L)
Bottom reflected P wave (LL)
P wave reflected twice off bottom(LLLL)
Raw Correlation function C:
0.010
0.005
0.000
-0.005
-0.010
-100
-80
-60
-40
-20
0
time msec)
20
40
60
80
100
Comparison of
a) Direct signal between transducers
b) adjusted Correlation
(adjusted according to more complete theory)
0.3
0.2
0.1
-0.0
-0.1
-0.2
-0.3
0
10
20
30
40
50
time msec)
60
70
80
90
100
The best diffuse field is that provided by thermal
fluctuations of elastic waves
A gas of phonons as it were . . . .
The strength of a thermal ultrasonic field at MHz frequencies
1)
2)
3)
Classical Thermal Fluctuation analysis tells us;
Each mode has small energy kT ≈ 4.2 x 10-21 joules
For typical solids,
with mode counts below 1 MHz of ~ 300 modes / cm3
We have energy densities of ~ 10-12 Joules / m3
and rms strain amplitudes of ~ 3 x 10-12
and rms displacement amplitudes of ~ 10-15 meter
< radius of electron!
How difficult is it to detect such weak signals?
We'll see . . . .
Why should we do so?
Answer:
They are perfectly diffuse,
and carry ultrasonic information
Comparison of a
Direct Pulse-Echo
Signal,
and a
Thermal Noise
Correlation
100
15
After
Capturing
320 msec
Of data
10
5
0
(and taking
9 seconds to do so)
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-10
-15
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After
Capturing
320 seconds
Of data
10
5
0
(and taking 2.5
hours to do so)
-5
-10
-15
0.20
50
0.15
. 0.10
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time (microsecondds)
0.05
0.00
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-0.10
-0.15
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time (microseconds)
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Summary
There are two miracles here
1) Diffuse fields carry, in their correlation functions, all the normal ultrasonic information
2) Thermal fluctuations can do the job. Except for their weakness, they are in fact ideal.
Application?
Whenever a source is inconvenient, one might use just receivers
At MHz, one saves a few $k. . . .
there may be no applications
At GHz or higher . . .?
we are looking into this.
At mHz?
Seismological applications are booming
M. Campillo and A. Paul, Science 299 (5606 547-549 (2003).
N.M. Shapiro and M. Campillo Geophys Rsrch Letts 31,(2004)
N M Shapiro, M Campillo, Stehly and Ritzwoller Science, in press (March 11 2005)
M Fehler, K Sabra, P Gerstoft, P Roux and W Kupperman, J Geophys Res, in press(2005)
A map of Surface-wave
Velocity in California
Obtained from correlating
30 days of seismic noise
earthquake
1 year of
correlations
4 one-month
correlations
Fidelity decay
*
Temperature Dependent
Distortion (fidelity)
of Fields
Typical Diffuse field ultrasonic measurement:
Close-up view of signal from two different temperatures
0.8
0.6
0.4
0.2
0.0
-0.2
10.05
Signal has
and
at 30 degrees
at 31 degrees
10.06 10.07 10.08
time (msec)
10.09
a) shifted to the right (dilated actually*)
b) distorted slightly
*because: lower wave speeds at higher temperatures
Theory for dilation
Wave speeds, and sample dimensions, vary with temperature.
In aluminum, the sample dimensions change by:
d ln L / d T = coefficient of thermal expansion = 0.22 x 10-4 per degree K.
More significant is the change in wavespeed with temperature.
S = d ln ( cshear / L )
=
P = d ln ( clongitudinal / L ) =
- 2.9 x 10-4 / °K
- 1.6 x 10-4 / °K
were measured for aluminum alloy polycrystals in the early 80's.
But the diffuse field is a mixture of P and S waves . .
Theory for the dilation, continued
Estimating a typical ray as spending
and
R/1+R = 94.1% of its life as a shear wave,
1/1+R = 5.9% as a longitudinal ray
(where R = 2 cL3 / cS3 = 16 )
We calculate an average dilation of
<>=
(R S + P) / (1+R) = -2.8 x 10-4 / °K
for the total diffuse field.
Experiments
An initially heated sample is allowed to
cool slowly in a vacuum
Temperature is monitored.
Waveforms are taken regularly.
The signals S(t) from different Temperatures are compared by
forming a Normalized Cross Correlation Function X(e) between
two signals,
S1(t) and S2(t)
taken at Temperatures T1 and T2
X(e) 
I
Window at age A
S T (t) S T (t  e}) dt
1
2
•S (t) dt •S (t  e}) dt
2
T1
2
T2
Slopes (after dividing by Temperature change):
200
0.2617 x 10
-3 per degree C
0.2598
104 & 90 degrees
(Fahrenheit)
0.2605
0.2610
0.2602
0.2632
150
102,90
0.2614
100,90
98,90
100
96,90
94,90
50
92, 90
0
0
20
40
60
Age (msec)
80
100
But signals distort as well as dilate . . .
Why?
Because shear and longitudinal wavespeeds dilate differently
with temperature, each ray is dilated a random amount,
The amount of dilation of a ray differs from the mean
by the degree with which that ray . . .
has a different number of P or S segments from the mean.
Theory: Rays strike surfaces at known rate (cA/4V)
Rays mode convert at a rate depending on angle,
and distribution of angle is fully diffuse and
independent of history!
{ lengthy calculation } 
in aluminum, for age less than the Heisenberg time
and assuming good mixing of the rays. :
Distortion, D = -ln fidelity = 3.26 x 10-4
x (Age/msec) (Temperature Difference/°K)2
x (Volume/Surface cm) (frequency/MHz)2
_____ _______ ________
______
for Age beyond tH:
Distortion D ~ Age2
and
Level Velocity Variance
0.25
Distortion D for
0.20
Temperature difference dT = 4 C
at central frequency 300 kHz
(medium block)
0.15
0.10
linear regime
0.05
onset of quadratic regime
at times t > T
Heis enburg
= 88 msec
0.00
0
10
20
30
40
50
60
Age (msec)
Distortion is linear in Age, until tH
70
80
90
100
1.0
Distortion
0.9
versus
Age at
f central = 700 kHz
0.8
Distortion is
Quadratic in
Temperature
Difference.
(medium block)
0.7
0.6
0.5
dT = 5 degrees C
dT = 4
0.4
0.3
dt = 3
0.2
0.1
0.0
0
10
20
30
40
50
Age (msec)
60
70
80
90
100
0.8
Distortion versus frequency at
dT = 4 degrees C
0.7
0.6
Distortion is
Quadratic in
Frequency.
(medium block)
0.5
800 kHz
700kHz
0.4
0.3
f c = 600 kHz
0.2
0.1
0.0
0
10
20
30
40
50
Age (msec)
60
70
80
90
100
Distortion does indeed scale as predicted:
D ~ (Age)(frequency)2(dT)2
But The coefficient of
that dependence
is generally
greater than
predicted
and depends on
shape.
Conclusions:
1) Average dilation is well understood
2) Distortion is not quite as well understood, and has some intriguing features.
2a) It scales as predicted with age, size, frequency, and temperature
2b) Distortion is related to P/S mixing rates
2b) Mixing rates depend on the irregularity of the system.
3) In highly irregular objects, and over a range of a factor of 40 in Volume and 3 in
V/S, distortion is within 15% of theory
4) In slightly irregular objects we see a greater distortion as if the P/S mixing rate is slower.
5) In a regular object, the distortion is MUCH greater,
as if the P/S mixing rate is MUCH slower.
Wave energy gets trapped for excessive times in each type.
A random ultrasonic Uaser
*?
(ultrasonic amplification by self-excited resonance ?)
Ultrasonic
Feedback
Like a Quartz
Clock circuit
H  H o  i  ig | y  x |
Theory for Uaser
S(t)  X(t)  G(x, y,t)  Y (t)  V (t)
V (t)  gS(t)

Let V(t) = exp{-ilt}
Then
G˜ (x, y, l )g  1
Is the condition for free vibration

G˜ (x, y, l )g  1
 If g = 0,
Solutions l are in general complex
G˜  
i.e, a pole, ie. l = r-igr
 Fastest growing solution is that l with largest Imaginary part
-
could be predicted from measured passive G’s.
 If g is slowly increased from zero, then the first instability is at
G˜ (x, y, ) 1/ g
Prediction:
First instability takes place at such that

G˜ (x, y, )g˜ () 1 0
3
2
1
0
~700
kHz
-1
-2
Behavior (in time domain) after gain ~ 50 dB is suddenly turned on
-3
-2.00
-1.75
-1.50
-1.25
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
time (seconds)
3
2
~700
kHz
1
0
-1
-2
Behavior (in time domain) after gain ~51 dB is turned on
-3
-2.00
-1.75
-1.50
-1.25
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
time (seconds)
We observe exponential growth, at a rate that depends on gain,
until amplifiers saturate.
Response of system to impulse . . . at gain = 0,
| G˜ (x, y,  ) |

Transducers have
strongest response at
~700kHz
and at ~1500 kHz
6000
5000
4000
Uaser spectrum
3000
2000
1000
0
0
500
1000
frequency (kHz)
1500
2000
2500
700
600
500
400
"Pitch-Catch" Spectrum
Uaser Spectrum
(single line at 1465.05)
300
200
100
0
1464.0
1464.5
1465.0
1465.5
1466.0
frequency (kHz)
Uaser line occurs at/near a maximum of | G() |
Line drifts over minutes ( as system temperature fluctuates )
100
80
system spectrum without gain
Uaser Spectrum
60
40
20
0
761.0
761.2
761.4
761.6
frequency (kHz)
Sometimes the single line is near 750 kHz
761.8
762.0
Sometimes, the Uaser has several lines . . . .
250000
200000
Uaser Spectrum
150000
100000
50000
0
500
1000
1500
frequency (kHz)
2000
2500
160
140
120
Uaser spectrum
Passive system spectrum
100
80
60
40
20
0
721.0
721.1
721.2
721.3
721.4
721.5
721.6
721.7
721.8
frequency (kHz)
A rare occurrence (triple line at 721.25 kHz) . . .
721.9
722.0
Test theory, that
G˜ (x, y, ) 1/ g˜ ();
g complex in general
Gives  at which first instability occurs
as we slowly increase the gain

0.15
0.10
0.05
0.00
190
192
194
196
198
200
Alternative design for a Uaser:
(ultrasonic amplification by stimulated emission of radiation)
A “single atom” uaser,
A piezoelectric transducer
in an auto-oscillating
Wein Bridge
100
75
50
25
0
132.0
132.5
133.0
frequency (kHz)
133.5
134.0
Other positions, other bridge parameters . . .
250
200
150
100
50
0
661
662
663
664
665
666
667
frequency (kHz)
668
669
670
671
200
150
100
50
0
125
130
frequency (kHz)
135
140
More recently . . .
Other designs:
A Van-der Pol
oscillator
We observe:
Single uaser chooses a frequency dictated by acoustic medium
Single uaser synchronizes to an applied CW field.
and does so with a phase that corresponds to
stimulated emission
Two or three uasers synchronize to each other,
Power output proportional to
Square of number of uasers !
But what is the Uaser good for?
?
Analog to optical lasers ? permit experiments and measurements not convenient in optics
Direct applications in acoustics?
In Summary
Ultrasonic reverberant bodies offer a venue for study of
mesoscopic wave phenomena,
Anderson localization
Enhanced backscatter
Ericson fluctuations
Ray chaos
Level statistics
Width statistics/return probability
Field and intensity correlations *
Fidelity decay
*
Ultrasonic u(l)asers
*