Introduction to Nanomechanics
(Spring 2012)
Martino Poggio
Cooling Mechanical Resonators
  k BT
• Achieve ultimate force resolution
• Approach the quantum regime
• Measure mechanical superpositions and
coherences
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Introduction to Nanomechanics
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Superposition & Coherence?
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Introduction to Nanomechanics
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Strategies for Cooling Resonators
  k BT
• “Brute force”: High resonance frequencies &
low reservoir temperatures
• Damping mechanical motion
• Cavity cooling
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Introduction to Nanomechanics
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



1

xrms  x zp 2  2 x zp 2 
 

 k T

B
1 
e
  k BT
xrms  xzp 

2m
  k BT
xrms  xth 
k BT
m 2
xrms (xzp)
T (K)
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“Brute Force”
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6
Real Numbers (T = 1 K)
Top-down doubly clamped
beams (Schwab)
• m = 10-15 kg
•  = 2 x 10 MHz
• xth = 2 x 10-12 m
• xzp= 3 x 10-14 m
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Real Numbers (T = 1 K)
Bottom-up doubly clamped
“clean” nanotubes (Steele/Delft)
• m = 10-21 kg
•  = 2 x 500 MHz
• xth= 4 x 10-11 m
• xzp = 4 x 10-12 m
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Real Numbers (T = 1 K)
Top-down doubly clamped
beams (Schwab)
Bottom-up doubly clamped
“clean” nanotubes (Steele/Delft)
• m = 10-15 kg
•  = 2 x 10 MHz
• m = 10-21 kg
•  = 2 x 500 MHz
• xth = 2 x 10-12 m
• xzp= 3 x 10-14 m
• xth= 4 x 10-11 m
• xzp = 4 x 10-12 m
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Real Numbers (T = 10 mK)
Top-down doubly clamped Si
beams (Schwab)
Bottom-up doubly clamped
“clean” nanotubes (Steele/Delft)
• m = 10-15 kg
•  = 2 x 10 MHz
• m = 10-21 kg
•  = 2 x 500 MHz
• xth = 2 x 10-13 m
• xzp= 3 x 10-14 m
• xth= 4 x 10-12 m
• xzp = 4 x 10-12 m
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Introduction to Nanomechanics
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Technical Challenges
• Resonator Fabrication (high frequency, low
dissipation, low mass)
• Displacement sensing (low measurement
imprecision, i.e. low noise floor)
• Refrigeration (mK temperatures)
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Expectation vs. Reality
Nth
T (K)
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Strategies for Cooling Resonators
  k BT
• “Brute force”: High resonance frequencies &
low reservoir temperatures
• Damping mechanical motion
• Cavity cooling
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14
Usual Cantilever Motion Detection
mx  0 x  kx  Fth
x
cantilever
fiber
interferometer
x
spectrum analyzer
piezo
Simple Electronic Damping
damping
mx  0 x  kx  Fth  g0 x
x
cantilever
x
fiber
interferometer
spectrum analyzer
piezo
d
F fb   g0 x
 g0
dt
Cooling (damping) of a cantilever - T = 4.2K
1000
1000
g=0
Tmode = 3.8 K
100
100
Q0 = 45,660
2
(Å2/Hz)
Sprectral density
Ang /Hz
10
10
11
0.1
0.1
0.01
0.01
1E-3
1E-3
Interferometer
shot noise level
1E-4
1E-4
1E-5
1E-5
3500
3500
3750
3750
4000
4000
Frequency (Hz)(Hz)
Frequency
4250
4250
Cooling (damping) of a cantilever - T = 4.2K
1000
1000
g = 6.8
100
100
2
(Å2/Hz)
Sprectral density
Ang /Hz
10
10
Tmode = 530 mK
Qeff = 5,834
11
0.1
0.1
0.01
0.01
1E-3
1E-3
Interferometer
shot noise level
1E-4
1E-4
1E-5
1E-5
3500
3500
3750
3750
4000
4000
Frequency (Hz)(Hz)
Frequency
4250
4250
Cooling (damping) of a cantilever - T = 4.2K
1000
1000
g = 67
100
100
2
(Å2/Hz)
Sprectral density
Ang /Hz
10
10
11
0.1
0.1
Tmode = 71 mK
Qeff = 674
0.01
0.01
1E-3
1E-3
Interferometer
shot noise level
1E-4
1E-4
1E-5
1E-5
3500
3500
3750
3750
4000
4000
Frequency (Hz)(Hz)
Frequency
4250
4250
Cooling (damping) of a cantilever - T = 4.2K
1000
1000
g = 263
100
100
2
(Å2/Hz)
Sprectral density
Ang /Hz
10
10
11
0.1
0.1
0.01
0.01
1E-3
1E-3
Tmode = 13 mK
Qeff = 173
Interferometer
shot noise level
1E-4
1E-4
1E-5
1E-5
3500
3500
3750
3750
4000
4000
Frequency (Hz)(Hz)
Frequency
4250
4250
Cooling (damping) of a cantilever - T = 4.2K
1000
1000
g = 525
100
100
2
(Å2/Hz)
Sprectral density
Ang /Hz
10
10
11
0.1
0.1
0.01
0.01
1E-3
1E-3
Tmode = 5.3 mK
Qeff = 87
Interferometer
shot noise level
1E-4
1E-4
1E-5
1E-5
3500
3500
3750
3750
4000
4000
Frequency (Hz)(Hz)
Frequency
4250
4250
Cooling (damping) of a cantilever - T = 4.2K
1000
1000
g = 1267
100
100
2
(Å2/Hz)
Sprectral density
Ang /Hz
10
10
11
0.1
0.1
0.01
0.01
1E-3
1E-3
Tmode = 0.62 mK
Q = 36
Interferometer
shot noise level
1E-4
1E-4
1E-5
1E-5
3500
3500
3750
3750
4000
4000
Frequency (Hz)(Hz)
Frequency
4250
4250
Cooling (damping) of a cantilever - T = 4.2K
1000
1000
g = 3043
100
100
2
(Å2/Hz)
Sprectral density
Ang /Hz
10
10
11
0.1
0.1
0.01
0.01
1E-3
1E-3
Interferometer
shot noise level
1E-4
1E-4
Tmode = -0.25 mK
Qeff = 15
1E-5
1E-5
3500
3500
3750
3750
4000
4000
Frequency (Hz)(Hz)
Frequency
4250
4250
Cooling (damping) of a cantilever - T = 4.2K
1000
1000
g = 4565
100
100
2
(Å2/Hz)
Sprectral density
Ang /Hz
10
10
11
Negative
Mechanical
modefeedback
temperature?!
can cancel
photon shot noise!
0.1
0.1
0.01
0.01
1E-3
1E-3
Interferometer
shot noise level
1E-4
1E-4
Tmode = -3.0 mK
Qeff = 10
1E-5
1E-5
3500
3500
3750
3750
4000
4000
Frequency(Hz)
(Hz)
Frequency
4250
4250
Experimental setup
damping
mx  0 x  kx  Fth  g0 ( x  xn )
measurement noise
x
cantilever
fiber
interferometer
x  xn
spectrum analyzer
piezo
d
F fb   g0 ( x  xn )
 g0
dt
Cantilever Noise Temperature with Feedback
mx   0 x  kx  Fth  g  0  x  xn 
Effective Q with feedback:
Measured spectral density:
Qeff
Q0
k


1  g c 1  g   0

c4 / k 2

S x  xn ( ) 
  2   2 2   2 2 / Q 2
c
c
eff

Actual cantilever spectral density:
S x ( ) 
Cantilever mode temperature:
Tmod e 
Tmod e


  2   2 2   2 2 / Q 2
c
c
0
 SF  
 th   2   2 2   2 2 / Q 2
c
c
eff


c4 / k 2
2


2 2
c
 c2 2 / Qeff2


g 2 k 2 2
S

S
 Fth
xn 
2 2

Q
c 0


k x2
kB
 g2  1
T
1 k


c 
 S xn
1  g 4 k B  1  g  Q0

 Sx
 n

Cantilever Noise Temperature with Feedback
mx  0 x  kx  Fth  g0  x  xn 
Effective Q with feedback:
Measured spectral density:
Actual cantilever spectral density:
Cantilever mode temperature:
For optimum
feedback gain
Qeff 
Q0
k

1  g c 1  g  0

4
2

/
k
c
S x  xn ( )  
  2   2 2   2 2 / Q 2
c
c
eff


  2   2 2   2 2 / Q 2
c
c
0
 SF  
 th   2   2 2   2 2 / Q 2
c
c
eff




g 2 k 2 2
S x ( ) 
S

S
 Fth
xn 
2 2
2
2 2
2 2
2

Q
c 0

  c   c  / Qeff 
c4 / k 2
Tmod e 
k x2
kB
Tmode,min 
k c T
S xn
k B Q0

 Sx
 n

T = 4.2 K
Cooling (damping) of a cantilever - T = 4.2K → 4.6mK
1000
1000
T = 4.2 K
Tmode = 5.3 K
100
100
2
2
Spectral density
Ang (Å
/Hz/Hz)
10
10
Tmode = 530 mK
11
0.1
0.1
Tmode = 73 mK
0.01
0.01
Tmode = 16 mK
Tmode = 8.3 mK
Tmode = 4.6 mK
1E-3
1E-3
1E-4
1E-4
Tmode = 5.3 mK
Tmode = 9.3 mK
1E-5
1E-5
3500
3500
3750
3750
4000
4000
Frequency (Hz)
Frequency
(Hz)
4250
4250
Cooling (damping) of a cantilever – model and experiment
10000
10000
T = 4.2 K
Q0 = 45,660
100
100
Tmode, min = 4.6 mK
Qeff = 36
mode
T
(K)
T mode(mK)
1000
1000
10
10
Theoretical Limit
11
0.1
0.1
0
1000
1000
2000
2000
3000
3000
gg
4000
4000
5000
5000
6000
6000
Cooling (damping) of a cantilever – model and experiment
102
100
Tmode (K)
Teff (K)
1
10
10
1001
T = 295 K
-1
10
0.1
Tmode = 2.9 mK
10-2
0.01
T = 4.2 K
T = 2.2 K
Theoretical Limit
10-3
1E-3
0
2000
2000
4000
4000
g
g
6000
6000