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Introduction into QuantumOptomechanics
Part 2 of 2
F.Ya.Khalili
March 20, 2013
1 / 70
1 Standard Quantum Limit
2 Quantum speedmeter
3 Negative optical inertia
4 Some metaphysics
5 Non-gaussian optomechanics
2 / 70
1 Standard Quantum Limit
2 Quantum speedmeter
3 Negative optical inertia
4 Some metaphysics
5 Non-gaussian optomechanics
3 / 70
LENGTHY EQUATIONS AHEAD!
4 / 70
Detection of classical force
Freemass
x(t)
F fl(t)
Meter“x(t)” = x(t) + xfl(t)F sign(t)
5 / 70
Detection of classical force
Freemass
x(t)
F fl(t)
Meter“x(t)” = x(t) + xfl(t)F sign(t)
−mΩ2x(Ω) = F sign(Ω) + F fl(Ω)
“x(Ω)” = x(Ω) + xfl(Ω) =F sign(Ω)
−mΩ2+
the sum noise︷ ︸︸ ︷F fl(Ω)
−mΩ2+ xfl(Ω)
6 / 70
Detection of classical force
Freemass
x(t)
F fl(t)
Meter“x(t)” = x(t) + xfl(t)F sign(t)
−mΩ2x(Ω) = F sign(Ω) + F fl(Ω)
“x(Ω)” = x(Ω) + xfl(Ω) =F sign(Ω)
−mΩ2+
the sum noise︷ ︸︸ ︷F fl(Ω)
−mΩ2+ xfl(Ω)
Ssum(Ω) = Sx +2SxF−mΩ2
+SF
m2Ω4
=~2/4 + S2
xF
SF+
2SxF−mΩ2
+SF
m2Ω4
7 / 70
Detection of classical force
Freemass
x(t)
F fl(t)
Meter“x(t)” = x(t) + xfl(t)F sign(t)
Ssum(Ω) = Sx +2SxF−mΩ2
+SF
m2Ω4
=~2/4 + S2
xF
SF+
2SxF−mΩ2
+SF
m2Ω4
Classical optimization: damn the back action!
8 / 70
Detection of classical force
Freemass
x(t)
F fl(t)
Meter“x(t)” = x(t) + xfl(t)F sign(t)
Ssum(Ω) = Sx +2SxF−mΩ2
+SF
m2Ω4
=~2/4 + S2
xF
SF+
2SxF−mΩ2
+SF
m2Ω4
Classical optimization: damn the back action!
SxF = 0 ⇒ Ssum(Ω) = Sx +SF
m2Ω4=
~2m
(1
Ω2q
+Ω2
q
Ω4
)
Ωq =
(SF
m2Sx
)1/4
∝√〈I 〉m
e2r
9 / 70
Sum noise spectral densityS
(Ω)
Ω
Sx
SFm2Ω4
Ssum(Ω) =~
2m
(1
Ω2q
+Ω2
q
Ω4
)
Ω = Ωq
10 / 70
Sum noise spectral densityS
(Ω)
Ω
Ω (1)q
<
Ω (2)q
<
Ω (3)q
<
Ω (4)q
<
Ω (5)q
11 / 70
Standard Quantum LimitS
(Ω)
Ω
SSQL (Ω) = ~
mΩ 2
12 / 70
Standard Quantum LimitS
(Ω)
Ω
“Quantum”
“Classical”
SSQL (Ω) = ~
mΩ 2
m = 40 kg , Ω = 2π × 100 s−1 ⇒√SSQL ≈ 2.5× 10−21 m/
√Hz
13 / 70
Standard Quantum LimitS
(Ω)
Ω
“Quantum”
“Classical”
SSQL (Ω) = ~
mΩ 2
m = 40 kg , Ω = 2π × 100 s−1 ⇒√SSQL ≈ 2.5× 10−21 m/
√Hz
14 / 70
How to overcome the SQL
Implicit assumptions that have been made:
1 The scheme is stationary (invariant with respect totime shift).
2 The noises are Markovian.
3 The noise are mutually uncorrelated.
4 The test object is a free mass.
Number of schemes based on violation of some of theseassumptions has been proposed.
Only one successful experiment was performed / (fornow).
15 / 70
How to overcome the SQL
Implicit assumptions that have been made:
1 The scheme is stationary (invariant with respect totime shift).
2 The noises are Markovian.
3 The noise are mutually uncorrelated.
4 The test object is a free mass.
Number of schemes based on violation of some of theseassumptions has been proposed.
Only one successful experiment was performed / (fornow).
16 / 70
Questions?
17 / 70
1 Standard Quantum Limit
2 Quantum speedmeter
3 Negative optical inertia
4 Some metaphysics
5 Non-gaussian optomechanics
18 / 70
SQL: simplified consideration
Position measurement
x(t) = x +pt
m+
1
m
∫ t
0
(t − t ′)F (t ′) dt ′
[x(0), x(t)] =i~tm⇒ ∆x(0) = ∆x(t) >
√~t2m
Ssum >(∆x)2
Ω∼ ~
mΩ2
Momentum measurement
p(t) = p(0) +
∫ t
0
F (t ′) dt ′ ⇒ [p(0), p(t)] = 0
⇒ no SQL
Velocity measurement
We can not measure momentum?Let us measure velocity instead!
19 / 70
SQL: simplified consideration
Position measurement
x(t) = x +pt
m+
1
m
∫ t
0
(t − t ′)F (t ′) dt ′
[x(0), x(t)] =i~tm⇒ ∆x(0) = ∆x(t) >
√~t2m
Ssum >(∆x)2
Ω∼ ~
mΩ2
Momentum measurement
p(t) = p(0) +
∫ t
0
F (t ′) dt ′ ⇒ [p(0), p(t)] = 0
⇒ no SQL
Velocity measurement
We can not measure momentum?Let us measure velocity instead!
20 / 70
SQL: simplified consideration
Position measurement
x(t) = x +pt
m+
1
m
∫ t
0
(t − t ′)F (t ′) dt ′
[x(0), x(t)] =i~tm⇒ ∆x(0) = ∆x(t) >
√~t2m
Ssum >(∆x)2
Ω∼ ~
mΩ2
Momentum measurement
p(t) = p(0) +
∫ t
0
F (t ′) dt ′ ⇒ [p(0), p(t)] = 0
⇒ no SQL
Velocity measurement
We can not measure momentum?Let us measure velocity instead!
21 / 70
The idea
Light interacts with the probe twice:
V.B.Braginsky, F.Ya.Khalili, Phys.Lett.A 147, 251 (1990)22 / 70
The idea
Light interacts with the probe twice:
V.B.Braginsky, F.Ya.Khalili, Phys.Lett.A 147, 251 (1990)23 / 70
Sagnac interferometerDelay lines
aRN
bRN
bLN
aLN
aRE
bRE
aLE
bLEBS
SRM
PRM
z
Laser
p q
PD
ITM ETM
ITM
ETM
Ring cavities
aRN bRN
bLN
aLN
aRE
bRE
aLE
bLE
BS
SRM
PRM
ETM
ETM
z
Laser
p q
PD
Yanbei Chen, Phys.Rev.D 67, 122004 (2003)
24 / 70
Sagnac interferometerDelay lines
aRN
bRN
bLN
aLN
aRE
bRE
aLE
bLEBS
SRM
PRM
z
Laser
p q
PD
ITM ETM
ITM
ETM
Ring cavities
aRN bRN
bLN
aLN
aRE
bRE
aLE
bLE
BS
SRM
PRM
ETM
ETM
z
Laser
p q
PD
δφ(t) ∝ xnorth(t − τ) + xeast(t)
δφ(t) ∝ xeast(t − τ) + xnorth(t)
δφ(t)− δφ(t) ∝ τd
dt[xeast(t)− xnorth(t)]
25 / 70
Overview of 1m speed meter experiment
! 1g mirrors suspended in monolithic fused silica suspensions.
! 1kW of circulating power. Arm cavities with finesse of 10000. 100ppm loss per roundtrip.
! Sophisticated seismic isolation + double pendulums with one vertical stage.
! Large beams to reduce coating noise.
! Armlength = 1m. Target better than 10-18m/sqrt(Hz) at 1kHz.
! No recycling, no squeezing, but plan to use homodyne detection.
! LOTS OF CHALLENGES! (let me know if you want to help…)
S.Hild et al, LSC QNWG telecon, January 30, 201326 / 70
Questions?
27 / 70
1 Standard Quantum Limit
2 Quantum speedmeter
3 Negative optical inertia
4 Some metaphysics
5 Non-gaussian optomechanics
28 / 70
The idea
SSQL =~
mΩ2
It is assumed that minert = mgrav = m. But what if not?
29 / 70
The idea
SSQL =~
mΩ2
It is assumed that minert = mgrav = m. But what if not?
SSQL(Ω) =~minert
m2gravΩ4
30 / 70
The idea
SSQL =~
mΩ2
It is assumed that minert = mgrav = m. But what if not?
SSQL(Ω) =~minert
m2gravΩ4
minert
mgrav→ 0 ⇒ SSQL(Ω)→ 0
31 / 70
The idea
SSQL =~
mΩ2
It is assumed that minert = mgrav = m. But what if not?
SSQL(Ω) =~minert
m2gravΩ4
minert
mgrav→ 0 ⇒ SSQL(Ω)→ 0
Impossible? But...
F.Khalili et al, Phys. Rev. D 83, 062003 (2011)
32 / 70
Optical springs to help
Mechanical responce function:
χ−1(Ω) = −mΩ2 + K
33 / 70
Optical springs to help
Mechanical responce function:
χ−1(Ω) = −mΩ2 + K
Two optical springs (γ δ, just for simplicity):
K ≈ 4ωp〈I1〉cL
δ1
δ21 − Ω2
+4ωp〈I2〉cL
δ2
δ22 − Ω2
34 / 70
Optical springs to help
Mechanical responce function:
χ−1(Ω) = −mΩ2 + K
Two optical springs (γ δ, just for simplicity):
K ≈ 4ωp〈I1〉cL
δ1
δ21 − Ω2
+4ωp〈I2〉cL
δ2
δ22 − Ω2
〈I 〉1δ1
+〈I 〉2δ2
= 0 , Ω δ1,2 ⇒ K ≈ −µΩ2
µ =4ωp
cL
〈I1〉δ1 + 〈I2〉δ2
δ21 + δ2
2
: the optical inertia!
35 / 70
Optical springs to help
Mechanical responce function:
χ−1(Ω) = −mΩ2 + K
Two optical springs (γ δ, just for simplicity):
K ≈ 4ωp〈I1〉cL
δ1
δ21 − Ω2
+4ωp〈I2〉cL
δ2
δ22 − Ω2
〈I 〉1δ1
+〈I 〉2δ2
= 0 , Ω δ1,2 ⇒ K ≈ −µΩ2
µ =4ωp
cL
〈I1〉δ1 + 〈I2〉δ2
δ21 + δ2
2
: the optical inertia!
mgrav = m , minert = m + µ
µ < 0 ⇒ minert < mgrav36 / 70
Questions?
37 / 70
1 Standard Quantum Limit
2 Quantum speedmeter
3 Negative optical inertia
4 Some metaphysics
5 Non-gaussian optomechanics
38 / 70
~F = m~a
(deterministic!)
Quantum-classical cut
i~d |ψ〉dt
= H |ψ〉
(deterministic!)
39 / 70
~F = m~a (deterministic!)
Quantum-classical cut
i~d |ψ〉dt
= H |ψ〉
(deterministic!)
40 / 70
~F = m~a (deterministic!)
Quantum-classical cut
i~d |ψ〉dt
= H |ψ〉 (deterministic!)
41 / 70
~F = m~a (deterministic!)
Quantum reduction: THE randomness!
i~d |ψ〉dt
= H |ψ〉 (deterministic!)
42 / 70
The root of all evil:the (in)famous Reduction Postulate:
|ψn〉 =En|ψ〉√
Pn
Pn = 〈ψ|En|ψ〉
which cuts off randomly brunchesof the World Wave Function.
43 / 70
Where and when the reduction REALLY happens?
• Copenhagen interpretation: We do not know and donot want to know; shut up and calculate theprobabilities.
• David Bohm: Reduction is a human invention. Thereal world is classical (but non local!)
• Eugene Wigner: Reduction happens in the humanbrain (which does not obey the ordinary laws ofphysics)
• Hugh Everett: Reduction is a human invention. Thereal world is completely quantum.
• Most of the physicists (in their subconsciousness):Reduction happens somewhere in meso-scale (andtherefore open systems blah blah blah...)
44 / 70
Where and when the reduction REALLY happens?
• Copenhagen interpretation: We do not know and donot want to know; shut up and calculate theprobabilities.
• David Bohm: Reduction is a human invention. Thereal world is classical (but non local!)
• Eugene Wigner: Reduction happens in the humanbrain (which does not obey the ordinary laws ofphysics)
• Hugh Everett: Reduction is a human invention. Thereal world is completely quantum.
• Most of the physicists (in their subconsciousness):Reduction happens somewhere in meso-scale (andtherefore open systems blah blah blah...)
45 / 70
Where and when the reduction REALLY happens?
• Copenhagen interpretation: We do not know and donot want to know; shut up and calculate theprobabilities.
• David Bohm: Reduction is a human invention. Thereal world is classical (but non local!)
• Eugene Wigner: Reduction happens in the humanbrain (which does not obey the ordinary laws ofphysics)
• Hugh Everett: Reduction is a human invention. Thereal world is completely quantum.
• Most of the physicists (in their subconsciousness):Reduction happens somewhere in meso-scale (andtherefore open systems blah blah blah...)
46 / 70
Where and when the reduction REALLY happens?
• Copenhagen interpretation: We do not know and donot want to know; shut up and calculate theprobabilities.
• David Bohm: Reduction is a human invention. Thereal world is classical (but non local!)
• Eugene Wigner: Reduction happens in the humanbrain (which does not obey the ordinary laws ofphysics)
• Hugh Everett: Reduction is a human invention. Thereal world is completely quantum.
• Most of the physicists (in their subconsciousness):Reduction happens somewhere in meso-scale (andtherefore open systems blah blah blah...)
But what about cats?47 / 70
Where and when the reduction REALLY happens?
• Copenhagen interpretation: We do not know and donot want to know; shut up and calculate theprobabilities.
• David Bohm: Reduction is a human invention. Thereal world is classical (but non local!)
• Eugene Wigner: Reduction happens in the humanbrain (which does not obey the ordinary laws ofphysics)
• Hugh Everett: Reduction is a human invention. Thereal world is completely quantum.
• Most of the physicists (in their subconsciousness):Reduction happens somewhere in meso-scale (andtherefore open systems blah blah blah...)
48 / 70
Where and when the reduction REALLY happens?
• Copenhagen interpretation: We do not know and donot want to know; shut up and calculate theprobabilities.
• David Bohm: Reduction is a human invention. Thereal world is classical (but non local!)
• Eugene Wigner: Reduction happens in the humanbrain (which does not obey the ordinary laws ofphysics)
• Hugh Everett: Reduction is a human invention. Thereal world is completely quantum.
• Most of the physicists (in their subconsciousness):Reduction happens somewhere in meso-scale (andtherefore open systems blah blah blah...)
49 / 70
The most consistent options
• Copenhagen interpretation: We do not know and donot want to know; shut up and calculate theprobabilities.
• David Bohm: Reduction is a human invention.The real world is classical (but non local!)
• Eugene Wigner: Reduction happens in the humanbrain (which does not obey the ordinary law ofphysics)
• Hugh Everett: Reduction is a human invention.The real world is completely quantum.
• Most of the physicists (in their subconsciousness):Reduction happens somewhere in meso-scale(and therefore open systems blah blah blah...)
50 / 70
The options which can be tested
• Copenhagen interpretation: We do not know and donot want to know; shut up and calculate theprobabilities.
• David Bohm: Reduction is a human invention.The real world is classical (but non local!)
• Eugene Wigner: Reduction happens in the humanbrain (which does not obey the ordinary law ofphysics)
• Hugh Everett: Reduction is a human invention.The real world is completely quantum.
• Most of the physicists (in their subconsciousness):Reduction happens somewhere in meso-scale(and therefore open systems blah blah blah...)
51 / 70
The paper to read tonight
Lev Vaidman, “On schizophrenic experiences of theneutron or why we should believe in the many-worldsinterpretation of quantum theory”,arXiv:quant-ph/9609006
52 / 70
The Gaussianity problem
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
0.2
0.4
0.6
0.8
1.0
Quantum Wignerfunction W (x , p) of
|ψ〉 = |0〉.
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
0.2
0.4
0.6
0.8
1.0
Classical probabilitydistribution W (x , p) for
T =~Ω0
2kB
Find any difference!
53 / 70
The Gaussianity problem
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
0.2
0.4
0.6
0.8
1.0
Quantum Wignerfunction W (x , p) of
|ψ〉 = |0〉.
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
0.2
0.4
0.6
0.8
1.0
Classical probabilitydistribution W (x , p) for
T =~Ω0
2kBFind any difference!
54 / 70
An always positive Wigner function can serve as thehidden-variable probability distribution with respect tomeasurements corresponding to any linear combinationof x and p.
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics,Cambridge Univ. Press, Cambridge, 1987
S. L. Braunstein, P. van Loock, RMP 77, 513 (2005)
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
The real tests ofquantumness requirenon-linear measurementsor non-Gaussian states.
55 / 70
An always positive Wigner function can serve as thehidden-variable probability distribution with respect tomeasurements corresponding to any linear combinationof x and p.
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics,Cambridge Univ. Press, Cambridge, 1987
S. L. Braunstein, P. van Loock, RMP 77, 513 (2005)
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
The real tests ofquantumness requirenon-linear measurementsor non-Gaussian states.
56 / 70
1 Standard Quantum Limit
2 Quantum speedmeter
3 Negative optical inertia
4 Some metaphysics
5 Non-gaussian optomechanics
57 / 70
Generic scheme of experiment
• Prepare some really macroscopic mechanical objectin a really non-classical (non-Gaussian) state.
• Show that this state can survive for considerabletime (no spontaneous reduction!)
• Unprepare the mechanical object back to its initialstate (crucial for Everett’s point of view)
It would be good to include into the last step also theguy who do this experiment (resolving thus the Wigner’sfriend paradox).
58 / 70
Generic scheme of experiment
• Prepare some really macroscopic mechanical objectin a really non-classical (non-Gaussian) state.
• Show that this state can survive for considerabletime (no spontaneous reduction!)
• Unprepare the mechanical object back to its initialstate (crucial for Everett’s point of view)
It would be good to include into the last step also theguy who do this experiment (resolving thus the Wigner’sfriend paradox).
59 / 70
Generic scheme of experiment
• Prepare some really macroscopic mechanical objectin a really non-classical (non-Gaussian) state.
• Show that this state can survive for considerabletime (no spontaneous reduction!)
• Unprepare the mechanical object back to its initialstate (crucial for Everett’s point of view)
It would be good to include into the last step also theguy who do this experiment (resolving thus the Wigner’sfriend paradox).
60 / 70
Generic scheme of experiment
• Prepare some really macroscopic mechanical objectin a really non-classical (non-Gaussian) state.
• Show that this state can survive for considerabletime (no spontaneous reduction!)
• Unprepare the mechanical object back to its initialstate (crucial for Everett’s point of view)
It would be good to include into the last step also theguy who do this experiment (resolving thus the Wigner’sfriend paradox).
61 / 70
Mechanical non-Gaussian preparation
Two ways:
• Direct use the mechanical non-linearity.
Impossible at the current technological level (?):mechanical systems are too linear and too noisy.
• Injection of an optical non-Gaussian state into themechanical resonator.
The way to go?
62 / 70
Mechanical non-Gaussian preparation
Two ways:
• Direct use the mechanical non-linearity.Impossible at the current technological level (?):mechanical systems are too linear and too noisy.
• Injection of an optical non-Gaussian state into themechanical resonator. The way to go?
63 / 70
Direct linear interaction: simple and effective
Ωmech ωe.m.
Λ
Hint = −~K (a + a†)x
Ωmech = ωe.m.
|φ〉|ψ〉 ⇐⇒t=π/Λ
|ψ〉|φ〉
(feasible in microwave band)64 / 70
Up-converting: effective as well
Ωmech ωe.m.− ωp
Λ
Hint = −2~G (Ae−iωpt a + A∗e iωpt a†)x
ωp = ωe.m. − Ωmech
|φ〉|ψ〉 ⇐⇒t=π/Λ
|ψ〉|φ〉
(feasible in both microwave and optical bands)65 / 70
The idea
The setup is identical tothe optical cooling in theresolved sideband regime.
The only difference:the input field ina non-Gaussian state,instead of the ground one.
J.Zhang, K.Peng, and S.L.Braunstein, PRA 68, 013808 (2003)
66 / 70
The idea
The setup is identical tothe optical cooling in theresolved sideband regime.
The only difference:the input field ina non-Gaussian state,instead of the ground one.
J.Zhang, K.Peng, and S.L.Braunstein, PRA 68, 013808 (2003)
67 / 70
(A bit) more real-world proposal
Adaption to LIGO and table-top interferometers.
F.Khalili et al, PRL 105, 070403 (2010)68 / 70
Quantum memory?
Storage times of:
optical cavities: . 10−3 s
MW resonators: . 1 s
mechanical pendulums: up to 108 s!
69 / 70
Shrodinger cat in AdvLIGO?
M = 40 kg70 / 70