Download Oregon Center for Optics

Quantum Optical Measurement:
A Tutorial
Michael G. Raymer
Oregon Center for Optics, University of Oregon
[email protected]
I review experimental work on the measurement of the quantum state of optical fields, and the relevant theoretical
background. The basic technique of optical homodyne tomography is described, with particular attention paid to the
role played by balanced homodyne detection in this process. I discuss some of the original single-mode squeezedstate measurements as well as recent developments. I also discuss applications of state measurement techniques to an
area of scientific and technological importance–the ultrafast sampling of time-resolved photon statistics.
Quantum mechanics is a theory of information, and the state of a system is the means of describing the statistical
information about that system. How can the quantum state of a physical system be completely determined by
measurements? Following Leonhardt, we affirm that “knowing the state means knowing the maximally available
statistical information about all physical quantities of a physical object.” Typically by “maximally available
statistical information” we mean probability distributions. Hence, knowing the state of a system means knowing the
probability distributions corresponding to measurements of any possible observable pertaining to that system. For
multiparticle (or multimode) systems, this means knowing joint probability distributions corresponding to joint
measurement of multiple particles (or modes).
The state of an individual system cannot, even in principle, be measured. However, the state of an ensemble of
identically prepared systems can be measured. Each member of an ensemble of systems is prepared by the same
state-preparation procedure. Each member is measured only once, and then discarded. Thus, multiple measurements
can be performed on systems all in the same state, without worrying about the measurement apparatus disturbing the
system. A mathematical transformation is applied to the data in order to reconstruct, or infer, the state. The relevant
interpretation of the measured state in this case is that it is the state of the ensemble.
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PART 1 :
Quantum Optical Measurement
Michael G. Raymer
Oregon Center for Optics, University of Oregon
[email protected]
----------------------------We review the measurement of optical fields, including direct detection and balanced
homodyne detection. We discuss single-mode squeezed-state measurements as well as
recent developments including: other field states, multimode measurements, and other
new homodyne schemes. We also discuss applications of homodyne measurement
techniques to an area of scientific and technological importance–the ultrafast sampling
of time-resolved photon statistics.
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OUTLINE
PART 1
1. Noise Properties of Photodetectors
2. Quantization of Light
3. Direct Photodetection and Photon Counting
PART 2
4. Balanced Homodyne Detection
5. Ultrafast Photon Number Sampling
PART 3
6. Quantum State Tomography
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REFERENCES CITED
1. RB - RW Boyd, “Radiometry and the Detection of
Radiation”
2. RL - R Loudon, “The Quantum Theory of Light”
3. MW - L Mandel, E Wolf, “Optical Coherence and Quantum
Optics”
4. VR - “Low Noise Techniques in Detectors,” Ann Rev Nucl
Part Sci 38, 217 (1988)
5. MR - M Raymer et al, "Ultrafast measurement of opticalfield statistics by dc-balanced homodyne detection," JOSA B
12, 1801 (1995).
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NOISE PROPERTIES OF DETECTORS
DIRECT PHOTODETECTION (MW Ch9, RB Ch8-11, RL Ch6)
• PHOTOEMISSIVE DETECTORS -- Photoelectric Effect
Assume no detector noise, and independent events
T
light
current
ei(t)
V
t
mean rate = r (constant)
mean current <i>= e r
Number of events in time T is n.
Mean <n>=n=r T
Photoelectron Number Probability=Poisson
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NOISE PROPERTIES OF DETECTORS
mean rate = r (constant)
Number of events in time T is n.
Mean (ensemble average) <n>=n=r T
Photoelectron Number Probability=P(n)
Variance of n: var(n) = <(n -<n>)2> = <n 2> -<n>2
Standard Deviation: std(n)=var(n)1/2
P(n)
std(n)
O
<n >
n
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Photoelectron Number Probability=Poisson
T
n −n
n e
p(n) =
n!
Variance:
var(n) = Δn 2 = n = rT
Standard Deviation: std(n) = Δn = n
t
Shot Noise
Level (SNL)
<n>=0.1
mean rate = r (constant)
p(n)
<n>=1
<n>=5
<n>=10
n
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Photoelectron Number Probability (Semiclassical Theory)
If the mean rate r(t) is not constant.
T
Define Integrated rate: W = 1 ∫ r(t) dt
(η=QE)
η 0
T
T
t
Probability for Integrated rate to equal W: P(W).
Probability to count n photoelectrons in time T:
p(n) =
Variance:
∫
∞
0
(ηW ) n e−η W
P(W ) dW
n!
var(n) = η 2 var(W ) +
wave noise
(Mandel’s
formula)
n
particle noise
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Photoelectron Number Probability -- Example Classical Blackbody Light in Single Mode (frequency, direction)
T
1
Short-time integration, T<<1/Δω W = ∫ r(t) dt
(η=QE)
η 0
T
filter, Δω
∫ 0 I dt n
T
T
P(W ) = < W >−1 e−W /<W >
t
mean number of counts: n = η W = η r T
Probability to count n photoelectrons in time T:
p(n) =
∫
∞
0
1 ⎛ n ⎞
(ηW ) n e−η W
P(W ) dW =
⎜
⎟
n + 1⎝ n + 1⎠
n!
n
Bose-Einstein Distribution
(derived here semiclassically)
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Photoelectron Number Probability=Bose-Einstein
1 ⎛ n ⎞
p(n) =
⎜
⎟
n + 1⎝ n + 1⎠
Mean:
Variance:
T
n
n =ηrT
var(n) = n 2 + n
t
Thermal-like
Noise
<n>=0.1
mean rate = r (non-constant)
p(n)
<n>=1
<n>=5
<n>=10
n
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Breakdown of the Semiclassical Photodetection Theory
p(n) =
∫
∞
0
(ηW ) n e−η W
P(W ) dW
n!
1
W =
η
Variance: var(n) = η 2 var(W ) +
wave noise
0
r(t) dt
(η=QE)
n
particle noise
There exist light sources for which var(n) <
T
∫
T
n
T
t
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QUANTIZATION OF THE OPTICAL FIELD I
ω j ˆ
(+ )
ˆ
E (r,t) = i ∑
b j u j (r) exp(−iω j t)
2 ε0
j
monochromatic planewave modes:
u j (r) = V −1/ 2 ε j exp(i k j ⋅ r)
photon annilation and
creation operators:
commutator:
†
ˆ
ˆ
[b j , bk ]= δ j k
n-photon state:
(ωj > 0)
polarization
†
bˆ j , bˆk
one-photon state:
†
1ω = bˆω vac
†
nω = (bˆω ) n vac
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QUANTIZATION OF THE OPTICAL FIELD II
Field Uncertainty and Squeezing
ω j ˆ
(+ )
ˆ
A single monochromatic mode: E = i
b u 0 (z)exp(−iω 0 t)
2 ε0
Hermitean operators:
qˆ = (bˆ + bˆ † ) / 21/2
pˆ = (bˆ − bˆ † ) / i21/2
t
Eˆ (+ ) (z,t) ∝ qˆ cos(ω 0 t − k0 z) + pˆ sin(ω 0 t − k0 z)
q, p = quadrature operators
[qˆ, pˆ ] = i
They obey:
p
Uncertainty relation:
std(q) std( p) ≥ 1 / 2
q
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QUANTIZATION OF THE OPTICAL FIELD III
Coherent State - ideal laser output
Eˆ (+ ) (z,t) ∝ qˆ cos(ω 0 t − k0 z) + pˆ sin(ω 0 t − k0 z)
quadrature operators: qˆ = (bˆ + bˆ † ) / 21/2
pˆ = (bˆ − bˆ † ) / i21/2
t
p
Equal Uncertainties:
std(q) = std( p) = 1 / 2
std(p)
std(q)
q
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QUANTIZATION OF THE OPTICAL FIELD IV
Field Uncertainty and Squeezing
Eˆ (+ ) (z,t) ∝ qˆ cos(ω 0 t − k0 z) + pˆ sin(ω 0 t − k0 z)
quadraturesqueezed light:
qˆ = (bˆ + bˆ † ) / 21/2
pˆ = (bˆ − bˆ † ) / i21/2
t
q noise reduced
p noise increased
p
std(p)
std(q)
q
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Coherent and Squeezed States
Eˆ (+ ) (z,t) ∝ qˆ cos(ω 0 t − k0 z) + pˆ sin(ω 0 t − k0 z)
p
p
p0
std(p)
p0
std(p)
std(q)
q0
q
[
std(q)
q0
]
ψ(q) = exp −(q − q0 ) / 2 − i p0 q
2
photon number probability:
Poisson
n n e−n
p(n) = n ψ =
n!
q0 + ip0
2
n=α ,α=
2
q
β 2 = (1 / 2)e−2 s
ψ(q) =
[
]
exp −(q − q0 ) / (2β 2 ) − i p0 q
2
2
photon number
probability?
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Quadrature-Squeezed Vacuum States
vacuum
Eˆ (+ ) (z,t) ∝ qˆ cos(ω 0 t − k0 z) + pˆ sin(ω 0 t − k0 z)
p
p
q
ψ(q) = exp[−q 2 / 2
q
β 2 = (1 / 2)e−2 s
]
ψ(q) = exp[−q 2 / (2β 2 )
s=2
p(n)
podd (n) = 0
]
(pair creation)
peven (n) = n ψ
2
=
n
n
⎛
⎞
1 ⎛1
⎞
⎜⎝ n / 2 ⎟⎠ cosh(s) ⎜⎝ 2 tanh(s)⎟⎠
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Two-Mode Squeezed States
by Second-order Nonlinearity:
Optical Parametric Amplification:
pump (ωp) --> signal (ω1) + idler (ω2)
coherent seed fields (amplifier, OPA)
signal(ω1)
signal(ω1)
idler (ω2)
χ(2)
idler (ω2)
pump (ωp)
z=O
z=L
∂
g ˆ ˆ ˆ† ˆ †
ˆ
ˆ
ψ = − i H ψ , H = i (b1 b2 − b1 b2 )
∂z
2
photon difference number ND is a constant of the motion:
†
†
Nˆ = (nˆ − nˆ ) = (bˆ bˆ − bˆ bˆ )
D
1
2
1
[Nˆ D , Hˆ ] = 0
1
2
2
Equal numbers of photons added
to signal and idler beams
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Difference-Number Squeezing:
Classical Particle Model
Case 1. Independent
coherent-state fields
(equal means)
photon
difference
number
n1
n2
(ω1)
(ω2)
N D = n1 - n2
variances add: var(ND)= var(n1)+ var(n2)
=<n1>+<n2>=2<n1>=<nTOT>
SHOT-NOISE LEVEL (SNL)
Case 2. Coherent seeds (OPA)
n1
signal(ω1)
n2
idler (ω2)
χ(2)
pump (ωp)
j1 =n1 +m
j2 =n2 +m
j1
j2
ND= j1 - j2= n1 - n2 (output noise = same as input noise)
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Difference-Number Squeezing:
Classical Particle Model II
Coherent seeds
signal(ω1)
idler (ω2)
n1
n2
(OPA)
j1 =n1 +m
χ(2)
m>> n1 , n2
j1
j2
j2 =n2 +m
ND= j1 - j2= n1 - n2 (output noise = same as input noise)
input: var(ND)=<n1>+<n2> = <nTOT> (input SNL)
output: var(ND)=<n1>+<n2> << <jTOT> (output SNL)
p(N D )
output
input
ND
<jTOT>=
<n1>+<n2> +2m
Gain= <jTOT>/<nTOT>
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300ps, 1064nm
(η=0.8)
KTP
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p(ND), (seeds)
p(ND), amplified
ND
ND
var(ND)
theory
<n1>=
<n2>=106
69%
below SNL
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WIGNER DISTRIBUTION
visualize state of a single mode in (q, p) phase space.
Eˆ (+ ) (z,t) ∝ qˆ cos(ω 0 t − k0 z) + pˆ sin(ω 0 t − k0 z)
projected distributions: Pr(q), Pr(p)
p
p
q
q
Underlying Joint Distribution?
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WIGNER DISTRIBUTION
in (q, p) phase space.
p
W(q,p)
W(q,p)
p
q
1
W (q, p) =
2π
Pr(q) =
∫
∞
−∞
∫
∞
−∞
q
ψ (q + q '/ 2) ψ * (q − q '/ 2) exp(−i q ' p) dq '
W (q, p) dp
,
Pr( p) =
∫
∞
−∞
W (q, p) dq
W(q,p) acts like a joint probability distribution.
But it can be negative.
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WIGNER DISTRIBUTION
for one-photon state | 1 >
ψ(q) = q exp[−q / 2
2
]
p
W(q,p)
q
negative
q
2q 2 + 2 p 2 −1
W (q, p) =
exp(−q 2 − p 2 )
π
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WIGNER DISTRIBUTION
for vacuum state | O >
ψvac (q) = exp[−q / 2
2
]
p
Wvac(q,p)
q
positive
q
1
W (q, p) = exp(−q 2 − p 2 )
π
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Impact of Losses or Detector Inefficiency on the
WIGNER DISTRIBUTION
convolve with a smoothing function:
W after (q, p) =
∫
∞
exp[−(q − x) 2 /ε 2 − ( p − y) 2 /ε 2 ]
πε
−∞
ε2 = 1 / η − 1
2
W (x, y) dx dy
η = Quantum Efficiency
Example: η = 0.5 --> convolve with
vacuum-state Wigner distribution:
W after (q, p) =
∫
∞
−∞
exp[−(q − x) 2 − ( p − y) 2 ]
π
W (x, y) dx dy
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QUANTIZATION OF THE OPTICAL FIELD V
Multimode Fields
Photon-flux amplitude operator:
(+ )
ˆ
Φ (r,t) = i c
∑
bˆ j u j (r) exp(−iω j t)
j
monochromatic planewave modes:
u j (r) = V −1/ 2 ε j exp(i k j ⋅ r)
Photon flux through a plane at z=O:
Iˆ (t) =
∫
(− )
(+)
2
ˆ (x,0,t) ⋅ Φ
ˆ (x,0,t)
d
x
Φ
Det
x
z
Integrated photon number in time T:
Nˆ =
∫
T
0
Iˆ (t) dt
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QUANTIZATION OF THE OPTICAL FIELD VI
Wave-packet Modes
(+ )
ˆ
Φ (r,t) = i c
∑
j
change of
mode basis:
bˆ j u j (r) exp(−iω j t)
u j (r) = V −1/ 2 ε j exp(i k j ⋅ r)
(+ )
ˆ
Φ (r,t) = i c
∑
aˆ k v k (r,t)
k
Non-mon0chromatic Wave-Packet Modes:
Unitary
Transformation
orthogonality:
v k (r,t) = ∑ Ck j u j (r) exp(−iω j t)
j
∫
(temporal
modes)
v k * (r,t) ⋅v j (r,t) d 3 r = δ k j
new annihilation operators: aˆ k = ∑ Ck* m bˆm
m
[aˆ j , aˆ k ] = δ j k
†
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QUANTIZATION OF THE OPTICAL FIELD
Wave-packet Modes
(+ )
ˆ
Φ (r,t) = i c
∑
aˆ k v k (r,t)
k
Non-mon0chromatic (temporal) Wave-Packet Modes:
k=O:
=
+
v 0 (r,t) = ∑ C0 j u j (r) exp(−iω j t)
j
+
+
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QUANTIZATION OF THE OPTICAL FIELD VII
(+ )
ˆ
Φ (r,t) = i c
∑
Non-monchromatic WavePacket Modes
aˆ k v k (r,t)
k
†
aˆ k operator creates one photon in the v k (r,t) wave packet
ˆak † vac = 0, 0,...1k , 0, 0...
v k (r,t)
z
detector
array
one
click
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Summary:
1. Field can be quantized in monochromatic
modes (Dirac), or non-monochromatic wave
packets (Glauber)
2. A single wave packet can be created in onephoton states, coherent states, or squeezed
states.
3. Measurement techniques are needed that can
determine the properties of these wave-packet
states.
next sections:
A. importance of high quantum efficiency (Q.E.)
and temporal selectivity
B. properties of detectors
C. balanced homodyne detection
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1. Importance of high quantum efficiency (Q.E.)
and fast time response
for quantum-state characterization
model Q.E. as a loss, such as from a beam splitter:
example: n-photon number state |n>
ideal detector
m counts
|n>
TBS=
transmission
loss
j
probability(m) = binomial
distribution
n!
m
pr(m) =
TBS (1− TBS ) n−m
m!(n − m)!
quantum state is changed by losses, or by low Q.E.
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2. Importance of temporal selectivity:
Multimode Fields
(+ )
ˆ
Φ (r,t) = i c
∑
aˆ k v k (r,t)
k
Non-mon0chromatic
Wave-Packet Modes:
=
+
Want to measure the
light in just one of
these packets.
+
How to select?
+
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PROPERTIES OF DETECTORS
1. Photo-Cathodes (photomultiplier, PMT)
sensitivity single photon
quantum efficiency 10-20%
gain noise 10-20%
dark current 10-14 amp
time response 10ps-10ns
Good
Bad
Bad
Good
Good
gain ~ 106
light
e-
current
i(t)
t
V
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2. Avalanche Semiconductor (Silicon) Photodiode
sensitivity quantum efficiency
gain noise dark current time response -
single photon
80%
100%
large
10ps
Good
Good
Bad
Bad
Fair
electron-hole pair created
n-type
+
p-type
--> avalanche
gain~106
current meter
large reverse bias
V
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3. Linear -Response Semiconductor (Silicon) Photodiode
quantum efficiency
read-out noise gain noise dark current time response -
99%
100-300 photons
small
10-9 amp
10ns
Excellent
Bad
Good
Bad
Bad
electron-hole pair created
n-type
+
p-type
(no gain)
current meter
small reverse bias
V
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Origin of Read-out Noise in Linear -Response Semiconductor
(Silicon) Photodiode
Integrate current for a time T =RcCf= 2 µs
1. Dark Current iD=10-9A --> mean number of counts = iDT/e
std(nD)=(iDT/e)1/2 =110 counts
Rc
Cf
V
Rf
Cd
2. Read-out Noise:
FET
A. Thermal Noise - mean counts = O
std(nTh) = std( iTh)T/e ;
var( iTh)=2 kB Temp/ (T Rf)
std(nTh) = 80 counts (room temp)
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2. Read-out Noise (cont’d):
B. Series resistance Noise - mean counts = O
Tf= carrier transit time through FET channel = 10-8s
Cd = detector capacitance = 10 pF
var(nS)=(4 kB Temp/e2)(Tf/T) Cd
std(nS) = 250 counts (room temp)
Cf
V
Rc
Rf
Cd
FET
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Summary: Linear -Response Silicon Photodiode
var(nDARK)~ var(nTHERMAL)~ T
;
var(nSERIES)~1/ T
Optimum Integration Time, T ~ 1-10 µs
Electronic Noise ~ 200 electrons per pulse
Can detect ~ 300 photoelectrons per pulse using a linearresponse photodiode.
Quantum Efficiency is high ~ 99%
Example: <n> = 4 106 --> SNL = 2 103 = 10 x Electronic Noise
Smithey, Beck, Belsley, M.R., Phys. Rev. Lett. 69, 2650 (1992)
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1. Would like to use the Linear-Response
Silicon Photodiode’s
high Quantum Efficiency
2. Would like to find a technique to select a
single wave-packet mode
3. Would like to measure quadrature
amplitudes of selected single wave-packet
mode
Eˆ (+ ) (z,t) ∝ qˆ cos(ω 0 t − k0 z) + pˆ sin(ω 0 t − k0 z)
qˆ = ( aˆ + aˆ † ) / 21/2 pˆ = (aˆ − aˆ † ) / i21/2
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end of PART 1
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