Quantum optical measurement

Quantum Optical Measurement:
A Tutorial
Michael G. Raymer
Oregon Center for Optics, University of Oregon
raymer@uoregon.edu
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 squeezed-
state 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.
M.G.Raymer_TTRL2a_V2_2005
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PART 1 :
Quantum Optical Measurement
Michael G. Raymer
raymer@uoregon.edu
-----------------------------
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
3 of 42

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 optical-
field 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
mean rate = r (constant)
mean current <i>= e r
e-
light current
i(t)
V
t
Number of events in time T is n.
Mean <n>=n=r T
Photoelectron Number Probability=Poisson
T
5 of 42

NOISE PROPERTIES OF DETECTORS
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
O <n> n
std(n)P(n)
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Photoelectron Number Probability=Poisson
p(n) =
n n
e−n
n!
Variance:
Standard Deviation:
var(n) = Δn2
= n = rT
std(n) = Δn = n
t
T
Shot Noise
Level (SNL)
<n>=0.1
<n>=1
<n>=5 <n>=10
p(n)
n
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Photoelectron Number Probability (Semiclassical Theory)
If the mean rate r(t) is not constant.
Define Integrated rate:
p(n) =
(ηW )n
e−ηW
n!0
∞
∫ P(W ) dW
Variance: var(n) = η 2
var(W ) + n
T
t
T
W =
1
η
r(t) dt
0
T
∫
Probability for Integrated rate to equal W: P(W).
Probability to count n photoelectrons in time T:
(η=QE)
wave noise particle noise
(Mandel’s
formula)
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Photoelectron Number Probability -- Example -
Classical Blackbody Light in Single Mode (frequency, direction)
Short-time integration, T<<1/Δω
p(n) =
(ηW )n
e−ηW
n!0
∞
∫ P(W ) dW =
1
n +1
n
n +1
⎛
⎝
⎜
⎞
⎠
⎟
n
T
t
T
W =
1
η
r(t) dt
0
T
∫
Probability to count n photoelectrons in time T:
(η=QE)
I dt
0
T
∫ n
filter, Δω
P(W ) = < W >−1
e−W /<W >
n = ηW = ηr Tmean number of counts:
Bose-Einstein Distribution
(derived here semiclassically)
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Photoelectron Number Probability=Bose-Einstein
Mean:
Variance: var(n) = n2
+ n
t
T
Thermal-like
Noise
<n>=0.1
<n>=1
<n>=5 <n>=10
p(n)
n
mean rate = r (non-constant)
p(n) =
1
n +1
n
n +1
⎛
⎝
⎜
⎞
⎠
⎟
n
n = ηr T
10 of 42

Breakdown of the Semiclassical Photodetection Theory
p(n) =
(ηW )n
e−ηW
n!0
∞
∫ P(W ) dW
Variance: var(n) = η 2
var(W ) + n
W =
1
η
r(t) dt0
T
∫
There exist light sources for which
(η=QE)
wave noise particle noise
var(n) < n
T
t
T
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QUANTIZATION OF THE OPTICAL FIELD I
ˆE
(+)
(r,t) = i
ω j
2ε0j
∑ ˆbj u j (r) exp(−iω j t)
u j (r) = V−1/2
ε j exp(ik j ⋅ r)
[ˆbj , ˆbk
†
]=δ j k
monochromatic plane-
wave modes:
commutator:
ˆbj, ˆbk
†
photon annilation and
creation operators:
polarization
(ωj >0)
one-photon state:
1ω = ˆbω
†
vac
n-photon state: nω = ( ˆbω
†
)n
vac
12 of 42

QUANTIZATION OF THE OPTICAL FIELD II
Field Uncertainty and Squeezing
A single monochromatic mode:
ˆq = ( ˆb + ˆb†
) /21/2
ˆp = (ˆb − ˆb†
) /i21/2
ˆE
(+)
= i
ω j
2ε0
ˆb u0(z)exp(−iω0 t)
Hermitean operators:
q, p = quadrature operators
They obey:
Uncertainty relation:
[ˆq, ˆp] = i
std(q) std(p) ≥ 1/2
q
p
t
ˆE(+)
(z,t) ∝ ˆqcos(ω0t − k0z) + ˆpsin(ω0t − k0z)
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QUANTIZATION OF THE OPTICAL FIELD III
Coherent State - ideal laser output
) /21/2
) /i21/2quadrature operators:
Equal Uncertainties:
std(q) = std(p) = 1/ 2
q
p
std(p)
std(q)
t
ˆE(+)
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QUANTIZATION OF THE OPTICAL FIELD IV
Field Uncertainty and Squeezing
q
p
std(p)
std(q)
quadrature-
squeezed light:
ˆE(+)
) /21/2
) /i21/2
q noise reduced
p noise increased
t
15 of 42

Coherent and Squeezed States
ˆE(+)
ψ(q) = exp − q − q0( )
2
/2 − i p0 q[ ]
ψ(q) =
exp − q − q0( )2
/(2β2
) − i p0 q[ ]
q0 q
p
std(p)
std(q)
p0
p
std(p)
std(q)
p0
q0 q
β2
= (1/2)e−2s
photon number probability:
Poisson
p(n) = nψ
2
=
nn
e−n
n!
n = α
2
, α =
q0 + ip0
2
photon number
probability?
16 of 42

Quadrature-Squeezed Vacuum States
ˆE(+)
ψ(q) = exp −q2
/(2β2
)[ ]
β2
= (1/2)e−2s
peven (n) = n ψ
2
=
n
n / 2
⎛
⎝⎜
⎞
⎠⎟
1
cosh(s)
1
2
tanh(s)
⎛
⎝⎜
⎞
⎠⎟
n
q
p
podd (n) = 0
vacuum
ψ(q) = exp −q2
/2[ ]
p(n)
q
p
s=2
(pair creation)
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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)
pump (ωp)
χ(2)
z=O z=L
∂
∂ z
ψ = − i ˆH ψ , ˆH = i
g
2
ˆb1
ˆb2 − ˆb1
† ˆb2
†
( )
ˆND = ˆn1 − ˆn2( ) = ˆb1
† ˆb1 − ˆb2
† ˆb2( )
photon difference number ND is a constant of the motion:
ˆND, ˆH[ ]= 0
signal(ω1)
idler (ω2)
signal(ω1)
idler (ω2)
Equal numbers of photons added
to signal and idler beams
18 of 42

Difference-Number Squeezing:
Classical Particle Model
Case 1. Independent
coherent-state fields
(equal means)
photon
difference
number
(ω1)
(ω2) n2
n1
Case 2. Coherent seeds (OPA)
pump (ωp)
χ(2)
signal(ω1)
idler (ω2) j2
j1
ND= n1 - n2
variances add: var(ND)= var(n1)+ var(n2)
=<n1>+<n2>=2<n1>=<nTOT>
SHOT-NOISE LEVEL (SNL)
n1
n2
j1 =n1 +m
j2 =n2 +m
ND= j1 - j2= n1 - n2 (output noise = same as input noise)
19 of 42

Difference-Number Squeezing:
Classical Particle Model II
Coherent seeds (OPA)
χ(2)
signal(ω1)
idler (ω2) j2
j1
n1
n2
j1 =n1 +m
j2 =n2 +m
ND= j1 - j2= n1 - n2 (output noise = same as input noise)
m>> n1 , n2
input: var(ND)=<n1>+<n2> = <nTOT> (input SNL)
output: var(ND)=<n1>+<n2> << <jTOT> (output SNL)
<jTOT>=
<n1>+<n2> +2m
p(ND)
ND
input
output
Gain= <jTOT>/<nTOT>
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(η=0.8)
300ps, 1064nm
KTP
21 of 42

p(ND), (seeds) p(ND), amplified
ND ND
var(ND)
theory
<n1>=
<n2>=106
69%
below SNL
22 of 42

WIGNER DISTRIBUTION
visualize state of a single mode in (q, p) phase space.
ˆE(+)
q
p
q
p
projected distributions: Pr(q), Pr(p)
Underlying Joint Distribution?
23 of 42

WIGNER DISTRIBUTION
in (q, p) phase space.
q
p
q
p
Pr(q) = W (q, p)
−∞
∞
∫ dp , Pr(p) = W (q, p)
−∞
∞
∫ dq
W(q,p) acts like a joint probability distribution.
But it can be negative.
W(q,p)W(q,p)
W (q, p) =
1
2π
ψ (q + q'/ 2) ψ *(q − q'/ 2)
− ∞
∞
∫ exp(−iq' p) dq'
24 of 42

WIGNER DISTRIBUTION
for one-photon state | 1 >
W (q, p) =
2q2
+ 2p2
−1
π
exp(−q2
− p2
)
q
p
ψ(q) = q exp −q2
/2[ ]
q
W(q,p)
negative
25 of 42

WIGNER DISTRIBUTION
for vacuum state | O >
W (q, p) =
1
π
exp(−q2
− p2
)
q
p
ψvac (q) = exp −q2
/2[ ]
q
Wvac(q,p)
positive
26 of 42

Impact of Losses or Detector Inefficiency on the
WIGNER DISTRIBUTION
convolve with a smoothing function:
Wafter(q, p) =
exp −(q − x)2
/ε2
− (p − y)2
/ε2
[ ]
π ε2 W (x,y)
−∞
∞
∫ dx dy
ε2
=1/η − 1 η = Quantum Efficiency
Example: η = 0.5 --> convolve with
vacuum-state Wigner distribution:
Wafter(q, p) =
exp −(q − x)2
− (p − y)2
[ ]
π
W (x,y)
−∞
∞
∫ dx dy
27 of 42

QUANTIZATION OF THE OPTICAL FIELD V
Multimode Fields
u j (r) = V−1/2
ˆΦ
(+)
(r,t) = i c
j
monochromatic plane-
wave modes:
Photon-flux amplitude operator:
Photon flux through a plane at z=O:
ˆI(t) = d2
xDet
∫ ˆΦ
(−)
(x,0,t)⋅ ˆΦ
(+)
(x,0,t)
ˆN =
0
T
∫ ˆI(t) dt
Integrated photon number in time T:
z
x
28 of 42

QUANTIZATION OF THE OPTICAL FIELD VI
Wave-packet Modes
u j (r) = V−1/2
ˆΦ
(+)
(r,t) = i c
j
Non-mon0chromatic Wave-Packet Modes:
ˆΦ
(+)
(r,t) = i c
k
∑ âk vk (r,t)
change of
mode basis:
vk(r,t) =
j
∑ Ck j u j (r) exp(−iω j t)
âk =
m
∑ Ck m
* ˆbmnew annihilation operators: [âj , âk
†
] = δj k
Unitary
Transformation
(temporal
modes)
vk *(r,t)⋅v j (r,t)∫ d3
r = δk jorthogonality:
29 of 42

QUANTIZATION OF THE OPTICAL FIELD
Wave-packet Modes
Non-mon0chromatic (temporal) Wave-Packet Modes:
ˆΦ
(+)
(r,t) = i c
k
∑ ˆak vk (r,t)
v0(r,t) =
j
∑ C0 j u j (r) exp(−iω j t)
k=O: =
+
+
+
30 of 42

QUANTIZATION OF THE OPTICAL FIELD VII
Non-monchromatic Wave-
Packet Modes
ˆΦ
(+)
(r,t) = i c
k
∑ âk vk (r,t)
operator creates one photon in the wave packetâk
†
vk(r,t)
âk
†
vac = 0, 0,...1k, 0, 0...
z
vk(r,t)
one
click
detector
array
31 of 42

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 one-
photon 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
32 of 42

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>
|n>
TBS=
transmission
m counts
j pr(m) =
n!
m!(n − m)!
TBS
m
(1− TBS )n−m
probability(m) = binomial
distribution
quantum state is changed by losses, or by low Q.E.
ideal detector
loss
33 of 42

2. Importance of temporal selectivity:
Multimode Fields
ˆΦ
(+)
(r,t) = i c
k
∑ ˆak vk (r,t)
Non-mon0chromatic
Wave-Packet Modes:
Want to measure the
light in just one of
these packets.
How to select?
=
+
+
+
34 of 42

PROPERTIES OF DETECTORS
1. Photo-Cathodes (photomultiplier, PMT)
sensitivity - single photon Good
quantum efficiency 10-20% Bad
gain noise - 10-20% Bad
dark current - 10-14 amp Good
time response - 10ps-10ns Good
e-light current
i(t)
V
t
gain ~ 106
35 of 42

2. Avalanche Semiconductor (Silicon) Photodiode
sensitivity - single photon Good
quantum efficiency 80% Good
gain noise - 100% Bad
dark current - large Bad
time response - 10ps Fair
V
p-type n-type
electron-hole pair created
+ -
current meter
--> avalanche
gain~106
large reverse bias
36 of 42

3. Linear -Response Semiconductor (Silicon) Photodiode
quantum efficiency 99% Excellent
read-out noise - 100-300 photons Bad
gain noise - small Good
dark current - 10-9 amp Bad
time response - 10ns Bad
V
p-type n-type
electron-hole pair created
+ -
current meter
(no gain)
small reverse bias
37 of 42

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
2. Read-out Noise:
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)
Rc
V
Cf
Rf
Cd
FET
38 of 42

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)
Rc
V
Cf
Rf
Cd
FET
39 of 42

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 linear-
response 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)
40 of 42

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
ˆq = (â + â†
) /21/2
ˆp = (â − â†
) /i21/2
Ê(+)
41 of 42

PART 2 :
BALANCED HOMODYNE
DETECTION
Michael G. Raymer
raymer@uoregon.edu
M.G.Raymer_TTRL2b_V2_2005
1 of 31

OUTLINE
PART 1
PART 2
PART 3
6. Quantum State Tomography
2 of 31

DC-BALANCED HOMODYNE DETECTION I
Goal -- measure quadrature amplitudes with high
Q.E. and temporal-mode selectivity
ES = signal field (ωO), 1 - 1000 photons
EL = laser reference field (local oscillator) (ωO), 106 photons
ND ∝ E1
(−)
(t − τd )∫ E1
(+)
(t) dt
− E2
(−)
(t − τd )∫ E2
(+)
(t) dt
ES (t)
EL (t)
n1
n2
θ
BS
dt
dt
ND
E1 =
ES + EL
PD
PD
E2 =
ES - EL
delay
τd
3 of 31

DC-BALANCED HOMODYNE DETECTION II
integrator circuit
n1
n2
θ
dt
dt
ND
PD
PD
4 of 31

DC-BALANCED HOMODYNE DETECTION III
ΦS = signal amplitude; ΦL = laser reference amplitude
ES (t)
EL (t)
n1
n2
θ
BS
dt
dt
ND
delay
τd
ˆND = dt
0
T
∫ d2
x
Det
∫ ˆΦL
(−)
(x,0,t − τd )⋅ ˆΦS
(+)
(x,0,t) + h.c.
ˆΦS
(+)
(r,t) = i c
k
∑ ˆak vk (r,t)
vk(r,t) =
j
∑ Ck j u j (r) exp(−iω j t)
ΦS
ΦL
overlap
integral
c dt0
T
∫ d2
x v *k (x,0,t)⋅ vm (x,0,t)Det
∫ = δk m
wave-packet
modesM.G.Raymer_TTRL2b_V2_2005
5 of 31

DC-BALANCED HOMODYNE DETECTION IV
ˆND ∝ dt
0
T
∫ d2
x
Det
∫ ˆΦL
(−)
(x,0,t − τd )⋅
k
∑ âk vk (x,0,t) + h.c.
wave-packet modes
The signal field is spatially and temporally gated by the LO field,
which has a controlled shape. Where the LO is zero, that portion
of the signal is rejected. Only a single temporal-spatial wave-
packet mode of the signal is detected.
Assume that the LO pulse is a strong coherent state of a particular
localized wave packet mode:
ˆND(θ) = |αL |( â e−iθ
+ â†
eiθ
)
ˆΦL
(+)
(r,t) ∝ |αL |exp(iθ) vL (r,t) + vacuum
LO phase
â =
k
∑ âk c dt0
T
∫ d2
xDet
∫ v *L (x,0,t − τd )⋅ vk (x,0,t) = âk= L
6 of 31

DC-BALANCED HOMODYNE DETECTION V
wave-packet
modes
signal :
quadrature operators:
ˆqθ ≡
ˆND(θ)
|αL | 2
=
âe−iθ
+ â†
eiθ
2
ˆqθ
ˆpθ
⎛
⎝
⎜
⎞
⎠
⎟ =
cosθ sinθ
−sinθ cosθ
⎛
⎝
⎜
⎞
⎠
⎟
ˆq
ˆp
⎛
⎝
⎜
⎞
⎠
⎟
ˆq = (â + â†
) /21/2
) /i21/2
ˆΦS
(+)
(r,t) ∝ â vL (r,t) +
k
∑ âk vk (r,t)
ˆqθ ≡
ˆND(θ)
|αL | 2
= ˆq cosθ + ˆp sinθ
detected
quantity:
LO phase
7 of 31

ULTRAFAST OPTICAL SAMPLING
Conventional Approach:
Ultrafast Time Gating of Light Intensity by
NON-LINEAR OPTICAL SAMPLING
strong short
pump (ωp )
weak signal(ωs )
sum-frequency (ωp + ωs )
second-order NL crystal
delay
8 of 31

ˆq = (â + â†
) /21/2
) /i21/2
LINEAR OPTICAL SAMPLING I
BHD for Ultrafast Time Gating of Quadrature Amplitudes
ˆqθ ≡
ˆND(θ)
|αL | 2
detected
quantity: LO phase
â =
k
∑ âk c dt0
T
∫ d2
xDet
∫ v *L (x,0,t − τd )⋅ vk (x,0,t) = âk= L
t
signalLO
θ
9 of 31

LINEAR OPTICAL SAMPLING II
Ultrafast Time Gating of Quadrature Amplitudes
vL (x,0,t) ∝ αL vL (x) fL (t − τd )
ˆND(τd ) = −i cαL
*
dt
0
T
∫ fL
*
(t − τd ) φS (t) + h.c.
φS (t) = d2
x
Det
∫ vL *(x)⋅ ˆΦS
(+)
(x,0,t)
ˆND(τd ) ∝ αL
* ˜fL
*
(ν)
dω
2πν −B /2
ν +B /2
∫ exp(−iωτd ) ˜φS (ω) + h.c.
∝ αL
* ˜fL
*
(ν) φS (τd ) + h.c.
LO mode:
fL (t) ∝(1/ t)sin(Bt /2)
if signal is band-limited and
LO covers the band, e.g.
ν−Β/2 ν+Β/2 ω
signal
LO
exact samplingM.G.Raymer_TTRL2b_V2_2005
10 of 31

LINEAR OPTICAL SAMPLING III
M. E. Anderson, M. Munroe, U. Leonhardt, D. Boggavarapu, D. F. McAlister and M. G. Raymer, Proceedings of
Generation, Amplification, and Measurment of Ultrafast Laser Pulses III, pg 142-151 (OE/LASE, San Jose, Jan.
1996) (SPIE, Vol. 2701, 1996).
ˆqθ (t) ψ
Ultrafast
Laser
Spectral
Filter
Time
Delay
Signal
Source
Balanced
Homodyne
Detector
Computer
LO
Signal
Phase
Adjustment
(optical or
elect. synch.)
n1
n
2
mean quadrature
amplitude in sampling
window at time t
θτd
Reference (LO)
Signal
11 of 31

LINEAR OPTICAL SAMPLING IV
Sample: Microcavity
exciton polariton
scan LO
delay τd
840 nm, 170 fs
ˆqθ (t) ψ
θ
LO
Balanced
Homodyne
detector
coherent
signal
12 of 31

LINEAR OPTICAL SAMPLING V
Mean Quadrature Measurement - sub ps Time Resolution
0.01
0.1
1
10
100
1000
10000
<n(t)>
121086420
Time (ps)
5
4
3
2
1
0
- 1
g
(2)
(t,t)
mean
quadrature
amplitude
<q> at
time t
LO delay τd (ps)
Sample: Microcavity
exciton polaritonˆqθ (t) ψ
ˆqθ +π /2(t) ψ
= ˆpθ (t) ψ
≅ 0coherent field --> M.G.Raymer_TTRL2b_V2_2005
13 of 31

LINEAR OPTICAL SAMPLING VI
Phase Sweeping for Indirect Sampling of Mean
Photon Number and Photon Number Fluctuations
ˆqθ ≡
ˆND(θ)
|αL | 2
detected
quantity: (θ = LO phase)
Relation with photon-number operator:
ˆn = ˆa†
ˆa =
1
2
ˆq − i ˆp( ) ˆq + i ˆp( ) = ˆq2
+ ˆp2
+
1
2
Phase-averaged quadrature-squared:
ˆqθ
2
θ
=
1
π
ˆqθ
2
dθ0
π
∫ =
1
π
ˆq cosθ + ˆp sinθ( )
2
dθ0
π
∫ =
1
2
ˆq2
+ ˆp2
( )
ˆn = ˆqθ
2
θ
−
1
2
ˆn(t) ψ
= ˆqθ
2
(t)
θ ψ
−
1
2
ensemble
average
works also for incoherent field (no fixed phase)M.G.Raymer_TTRL2b_V2_2005
14 of 31

LINEAR OPTICAL SAMPLING VII
Phase Sweeping --> Photon Number Fluctuations
ˆqθ ≡
ˆND(θ)
|αL | 2
detected
quantity:
n(r)
ψ
= [n(n −1)...(n − r +1)]
n= 0
∞
∑ p(n) = (â†
)r
( â)r
ψ
=
(r!)2
2r
(2r)!
dθ
2π0
2π
∫ H2r (ˆqθ ) ψ
Richter’s formula for Factorial Moments:
ˆn(t) ψ
= ˆqθ
2
(t) θ ψ
−
1
2
H0(x) =1, H1(x) = 2x, H3(x) = 4x2
− 2Hermite Polynomials:
n(1)
= â†
â =
1
4
dθ
2π0
2π
∫ 4 ˆqθ
2
− 2
ψ
n(2)
= â†2
â2
=
dθ
2π0
2π
∫
2
3
ˆqθ
4
− 2 ˆqθ
2
+
1
2 ψ
15 of 31

Variance of Photon Number in Sampling Time
Window: var(n)=< n 2 > - < n >2
LINEAR OPTICAL SAMPLING VIII
Phase Sweeping --> Photon Number Fluctuations
var(n) =
dθ
2π0
2π
∫
2
3
ˆqθ
4
− ˆqθ
2
− ˆqθ
2 2
+
1
4
⎡
⎣⎢
⎤
⎦⎥
Second-Order Coherence of Photon Number in
Sampling Time Window:
g(2)(t,t )=[< n 2 > - < n >]/< n >2
corresponds to thermal light, i.e. light produced
primarily by spontaneous emission.
corresponds to light with Poisson statistics, i.e., light
produced by stimulated emission in the presence of gain saturation.
g(2)
(t,t) = 2
g(2)
(t,t) =1
16 of 31

LINEAR OPTICAL SAMPLING IX
Photon Number Fluctuations
PBS1
LO
Signal
PBS2
PhotodiodesComputer
n1
n2 Shaper
Charge-Sensitive
Pre-Amps
Stretcher
Balanced Homodyne Detector
λ/2
λ/2
80MHz 1-50kHz
Ti:Sapphire
Shaper
AD/DA
λ/2
Alt. Source
Electronic
Delay
Voltage
Pulser
Trigger Pulse
GPIB controller
Regen.
Amplifier
Sample
M.
Munroe
if the signal is incoherent, no phase sweeping is required
17 of 31

LINEAR OPTICAL SAMPLING X
Superluminescent Diode (SLD) Optical Amplifier
M. Munroe
~
~
~
~
6
o
SiO 2
p-contact layer
metal cap
n-GaAs substrate
p-clad layer
undoped, graded
confining layers
quantum
wells
n-clad layer
3 µm
600 µm
(AR)
Superluminescent
Emission(Sarnoff Labs) M.G.Raymer_TTRL2b_V2_2005
18 of 31

LINEAR OPTICAL SAMPLING XI
M. Munroe
25
20
15
10
5
0
OutputPower(mW)
2001000
Drive Current (mA)
1.0
0.5
0.0
Intensity(a.u.)
880840800760
Wavelength (nm)
(b)
1.0
0.8
0.6
0.4
0.2
0.0
Intensity(a.u.)
850840830820810
Wavelength (nm)
(a)
(a)
(b)
(no cavity)
19 of 31

LINEAR OPTICAL SAMPLING XII
SLD in the single-pass configuration
Photon Fluctuation
is Thermal-like,
within a single time
window (150 fs)
M. Munroe
3.0
2.5
2.0
1.5
1.0
0.5
<n(t)>
20151050
time (ns)
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
g(2)
(t,t)
<n(t,t)>
g
(2)
(t,t)
20 of 31

LINEAR OPTICAL SAMPLING XIII
SLD in the double-pass with grating configuration
14
12
10
8
6
4
2
0
<n(t)>
20151050
time (ns)
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
g(2)
(t,t)
<n(t)>
g
(2)
(t,t)
Photon Fluctuation
is Laser-like, within
a single time
window (150 fs)
M. Munroe
21 of 31

Single-Shot Linear Optical Sampling I
-- Does not require phase sweeping.
Measure both quadratures simultaneously.
Dual- DC-Balanced Homodyne Detection
π/2 phase
shifter
BHD
BHD
signal
LO1
LO2
q
p
50/50
q2 + p2 = n
22 of 31

Fiber Implementation of Single-shot Linear Optical
Sampling Of Photon Number
MFL: mode-locked Erbium-doped fiber laser. OF: spectral filter.
PC: polarization controller. BD: balanced detector.
23 of 31

Measured quadratures
(continuous and dashed
line) on a 10-Gb/s
pulse train.
Waveform obtained by
postdetection squaring
and summing of the two
quadratures.
24 of 31

Two-Mode DC-HOMODYNE DETECTION I
BHD
signal
Q
LO is in a Superposition of two wave-packet modes, 1 and 2
1 2
ˆΦL
(+)
(r,t) = i c |αL |exp(iθ) v1(r,t)cosα + v2(r,t)exp(−iζ)sinα[ ]
ˆQ = cos(α) ˆq1 cosθ + ˆp1 sinθ[ ]+ sin(α) ˆq2 cosβ + ˆp2 sinβ[ ]
ˆq1θ
β = θ −ζ
Dual temporal modes:
ˆq2β
quadrature of mode 1 quadrature of mode 2
(temporal,
spatial, or
polarization)
Dual LO
25 of 31

Two-Mode DC-HOMODYNE DETECTION II
ultrafast two-time number correlation measurements using dual-
LO BHD; super luminescent laser diode (SLD)
two-time second-
order coherence
g(2)
(t1,t2) =
: ˆn(t1) ˆn(t2):
ˆn(t1) ˆn(t2)
BHD
signal
Dual LO
Q
1 2
SLD
t1 t2
D. McAlister M.G.Raymer_TTRL2b_V2_2005
26 of 31

Two-Mode DC-HOMODYNE DETECTION III
two-pol., two-time
second-order
coherence
BHD
signal
LO
Q
gi, j
(2)
(t1,t2) =
: ˆni (t1) ˆn j (t2):
ˆni (t1) ˆn j (t2)
source
polarization rotator
Alternative Method using a Single LO.
Signal is split and delayed by different times.
Polarization rotations can be introduced.
A. Funk M.G.Raymer_TTRL2b_V2_2005
27 of 31

Two-Mode DC-HOMODYNE DETECTION IV
E. Blansett
Single-time, two-polarization correlation measurements on
emission from a VCSEL
0-2π phase
sweeping
and time
delay
0-2π relative phase sweepingM.G.Raymer_TTRL2b_V2_2005
28 of 31

Two-Mode DC-HOMODYNE DETECTION V
Single-time, two-
polarization correlation
measurements on
at low temp. (10K)
E. Blansett
gi, j
(2)
(t1,t2) =
: ˆni (t1) ˆn j (t2):
gi,i
(2)
(t1,t2) =
: ˆni (t1) ˆni(t2):
ˆni (t1) ˆni (t2)
uncorrelated
29 of 31

Two-Mode DC-HOMODYNE DETECTION VI
Single-time, two-
polarization correlation
measurements on
at room temp.
gi, j
(2)
(t1,t2) =
: ˆni (t1) ˆn j (t2):
gi,i
(2)
(t1,t2) =
: ˆni (t1) ˆni(t2):
ˆni (t1) ˆni (t2)
anticorrelated
Spin-flip --> gain competition M.G.Raymer_TTRL2b_V2_2005
30 of 31

SUMMARY: DC-Balanced Homodyne Detection
1. BHD can take advantage of: high QE and ultrafast time
gating.
2. BHD can provide measurements of photon mean
numbers, as well as fluctuation information (variance,
second-order coherence).
3. BHD can selectively detect unique spatial-temporal
modes, including polarization states.
31 of 31

Quantum State Measurement of
Optical Fields and Ultrafast
Statistical Sampling
M. G. Raymer
raymer@uoregon.edu
M.G.Raymer_TTRL2c_V2_2005
1 of 34

OUTLINE
PART 1
PART 2
PART 3
6. Quantum State Tomagraphy
2 of 34

For introduction, see:
M.R., Contemp. Physics 38, 343 (1997)
PART 3
QUANTUM STATE TOMOGRAPHY
3 of 34

Niels Bohr:
"In quantum physics ... evidence about atomic objects
obtained by different experimental arrangements
exhibits a novel kind of complementarity relationship.
... Such evidence, which appears contradictory when
combination into a single picture is attempted, exhausts
all conceivable knowledge about the object. ...
Moreover, a completeness of description like that aimed
at in classical physics is provided by the possibility of
taking every conceivable experimental arrangement into
account."
from "Quantum Physics and Philosophy" (1958), reprinted in Niels Bohr
Collected Works , Foundations of Quantum Physics II, Vol.7, ed. by J.
Kalckar (Elsevier, Amsterdam, 1996).
4 of 34

“To know the quantum mechanical state of a system
implies, in general, only statistical restrictions on the
results of measurements.”
-- John S. Bell, 1966
Converse--
To know the statistical restrictions on the results of
measurements is to know the quantum mechanical
state of a system.
5 of 34

1
2 3 4
5
Ω
V
ψ
R
ψ
I
Volt - Ohm - Psi - Meter
ψ Single
System
Pat. Pend.
The state of an individual system cannot, even in
principle, be measured. The state of an ensemble of
identically prepared systems can be measured.
6 of 34

A
Realistic
Psi -
Meter
A “Quorum”
of Variables
M.R., Contemp. Physics 38, 343 (1997)
1
2 3 4
5
ψ
R
ψ
I
1
2 3 4
5
a b c d e f g h i j k l m
ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ
A PHYSICAL EMSEMBLE OF SYSTEMS
ψ = aj φj
j
∑
7 of 34

Quantum States Provide Information SecurityQuantum States Provide Information Security
P and Q are
non-Commuting Variables.
Can measure only one.
ψ
8 of 34

Quantum-State Cloning?Quantum-State Cloning?
P and Q are
Both determined?
P
Q
ψ
ψ
ψ
9 of 34

No - Cloning Theorem (Wootters and Zurek)No - Cloning Theorem (Wootters and Zurek)
ψ SYS
ψ AUXψ SYS
χref AUX
φJ SYS
χref AUX
φJ SYS
φJ AUX
(basis state)
(arbitrary state)
aj φj SYS
j
∑
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟ χref AUX
aj φj SYS
j
∑
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟ aj φj AUX
j
∑
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟≠
?
aj φj SYS
φj AUX
j
∑
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
10 of 34

No Cloning <---> No State Measurement of Single ParticleNo Cloning <---> No State Measurement of Single Particle
(NO)
1
234
5
ψ
R
ψ
I
1
234
5
A Quorum
ψ
ψ
ψ
ψ
ψ
ψ
11 of 34

Systems measured include:
Optical squeezed states: D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev.
Lett. 70, 1244 (1993). G. Breitenbach, S. Schiller, and J. Mlynek, Nature 387, 471 (1997). K.
Banaszek, C. Radzewicz, K. Wodkiewicz, and J. S. Krasinski, Phys. Rev. A 60, 674 (1999).A. I.
Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, Phys. Rev. Lett. 87,
050402 (2001).
Polarization states of photon pairs: A. G. White, D. F. V. James, P. H. Eberhard, and P. G.
Kwiat, Phys. Rev. Lett. 83, 3103 (1999).
Angular momentum states of electrons: J. R. Ashburn, R. A. Cline, P. J. M. van der Burgt,
W. B. Westerveld, and J. S. Risley, Phys. Rev. A 41, 2407 (1990).
Molecular vibrational states: T. J. Dunn, I. A. Walmsley, and S. Mukamel, Phys. Rev. Lett.
74, 884 (1995).
Motional states of trapped ions: D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M.
Itano, and D. J. Wineland, Phys. Rev. Lett. 77, 4281 (1996).
Motional states of atomic beams: C. Kurtsiefer, T. Pfau, and J. Mlynek, Nature 386, 150
(1997).
Nuclear spin states: I. L. Chuang, N. Gershenfeld, M. G. Kubinec, and D. W. Leung, Proc. R.
Soc. London A 454, 447 (1998).
Physics and Astronomy Classification Scheme (PACS) code:
03.65.Wj, State Reconstruction and Quantum Tomography
12 of 34

Why Measure the State?
Once the state is obtained, distributions or moments of quantities can
be calculated, even though they have not been directly measured (or
are not measurable, in principle).
Examples -- The ability to measure photon number distributions at the single-
photon level, even though the detectors used had noise levels too large to directly
measure these distributions.
-- Distributions of optical phase have been determined, even though there is no
known experimental apparatus capable of directly measuring this phase.
-- Expectation values of the number-phase commutator have been measured, even
though this operator is not Hermitian, and thus cannot be directly observed.
Even if the full quantum state is not measured, performing many
measurements corresponding to different observables can yield
information about optical fields. (Linear Optical Sampling)
-- For example, one can obtain information, with high time-resolution (on the time
scale of 10’s of fs), on the photon statistics of light –
propagation in scattering media ,
light emitted by pulsed lasers,
performance of optical communication systems.
13 of 34

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 (or recycled).
• A mathematical transformation is applied to the
data in order to reconstruct, or infer, the state of the
ensemble.
-------------------------------------------------------------
• A single member of the ensemble is described by
the state, but the state cannot be determined by
measuring only that individual.
14 of 34

PURE STATES AND MIXED STATES
How is the state of a quantum system represented?
• Pure State Case -- Every system is prepared using exactly
the same procedure. Described by
• Mixed State Case -- Each system is prepared using one of
many procedures, with probability . The density
operator is an ensemble average
or, in wave-function representation:
ψ , or x ψ = ψ(x)
P(ψ)
ˆρ = P(ψ)
ψ
∑ ψ ψ
ρ(x,x') = P(ψ)
ψ
∑ ψ(x)ψ * (x') = ψ(x)ψ * (x') ens
15 of 34

How many different variables need to be statistically
characterized? (For a one-dimensional system)
• Pure State Case -- Often it is sufficient to
characterize just two variables. (for example: q, p)
• Mixed State Case -- Need to characterize many
variables.
(for example: q, p, q+p, q-p, q cosθ + p sinθ, etc).
16 of 34

State Reconstruction Schemes
1. Limit of infinite amount of perfect data:
Deterministic inversion of data to yield density
matrix. (Here we restrict ourselves to this case.)
2. Finite amount of noisy data: Estimation of the
density matrix by:
• least-squares estimation
• maximum-entropy estimation
• maximum-likelihood estimation
17 of 34

x
z
Px
Pz
ξ
UMRK
muffins
Joint Probability:
How to Determine a Statistical state of an Ensemble of
Classical Particles if you are allowed to measure only one
variable (say position x, but not momentum px)?
ξ = x + px (t / m)
Pr(ξ;t) = ∫ Pr(x, px )∫ δ(ξ −[x + px (t / m)])dxdpx
Pr(x, px )
18 of 34

ξ = x + px (t / m)
x = 0 x
ξ = 0
x
px
EXPERIMENT WITH ATOMIC BEAM
Joint Probability
Distribution
x
Pr(x;t)
19 of 34

x
px
INVERSION OF EXPERIMENTAL DATA
Joint Probability
Distribution
x
Pr(x;t)
Pr(ξ;t) = ∫ Pr(x, px )∫ δ(ξ −[x + px (t / m)])dxdpx
Pr(x, px ) = Pr(ξ;t)K(x, px;ξ;t)dξ dt∫
Mathematical Inversion:
20 of 34

ξ = x + px (t / m)
x
EXPERIMENT WITH ATOMIC BEAM
T. Pfau, C. Kurtseifer (Uni Konstanz)
Detection plane
Double slit
Atomic source,
velocity
distribution
Measure Pr(x;t)
for many
different flight
times t.
21 of 34

Joint Probability DistributionPr(x;t) Data
T. Pfau, C.
Kurtseifer
x
NEGATIVE!
W (q, p) =
1
π
ψ(q + x /2)
−∞
∞
∫ ψ *(q + x /2)exp(−ipx)dx
“Joint Probability Distribution” -> Wigner Distribution
Pr(x, px ) = Pr(ξ;t)K(x, px;ξ;t)dξ dt∫
22 of 34

Harmonic Oscillator Example
Deterministic inversion of statistical data to yield
density matrix. Probability to observe position = q at
time t :
ρµν = P(ψ)
ψ
∑ µ ψ ψ ν
q
1
2
3
4
5
6
Pr(q,t) = P(ψ)
ψ
∑ ψ(t)
2
= ρµν ψµ (q)
µ,ν
∑ ψν (q)exp[i(ν − µ)ω t]
23 of 34

Harmonic Oscillator Example
Deterministic inversion of statistical data to yield
density matrix. Probability to observe position = q at
time t :
ρµν = P(ψ)
ψ
∑ µ ψ ψ ν
q
1
2
3
4
5
6
Pr(q,t) = P(ψ)
ψ
∑ ψ(t)
2
= ρµν ψµ (q)
µ,ν
24 of 34

Harmonic Oscillator Example - Phase Space
Pr(q,t) = P(ψ)
ψ
∑ ψ(t)
2
= ρµν ψµ (q)
µ,ν
q
p
q
Phase-Space
Probability Density
W(q,p,t )
Pr(q,t) = W (q, p,t)dp∫
Projection =
Probability:
θ
θ=ωt = phase
angle
25 of 34

Harmonic Oscillator Example - Phase Space
Pr(q,t) = P(ψ)
ψ
∑ ψ(t)
2
= ρµν ψµ (q)
µ,ν
q
p
q
Phase-Space
Probability Density
W(q,p,t )
Pr(q,t) = W (q, p,t)dp∫
Projection =
Probability:
26 of 34

θ=ωt = phase angle
Radon
Transform
Wigner Distr.
27 of 34

Radon
Transform
Wigner Distr.
For squeezed state
Inverse Radon
Transform
28 of 34

Measuring Wigner Distributions for Optical Fields
DC-Balanced Homodyne Dection
Tomography
In Phase Space
ES (t)
EL (t)
n1
n2
θ
BS
dt
dt
W(q,p)
29 of 34

Measured Wigner distribution for Single-Photon State
Breitenbach, Schiller, Lvovsky (Uni Konstanz)
Down conversion
Photon pair
trigger
BHD Tomography
vacuum 1-photon
filters
30 of 34

Measured Density Matrix for Quadrature-Squeezed Light
D. Smithey 1993
31 of 34

Determining a Quantum Wave Function from a
Wigner Distribution
BHD Tomography
Density
Matrix ρ
Wigner Distr.
Wave
Function
32 of 34

Determined Quantum Wave Function of Coherent
State from a Laser (Smithey 1993)
Pulsed
Laser BHD
Tomography
attenuator
33 of 34

Quantum state tomography has entered into the standard bag of
tricks in quantum optics and quantum information research.
For elementary review see:
"Measuring the quantum mechanical wave function," M. G. Raymer,
Contemp. Physics 38, 343 (1997).
For detailed reviews see:
“Experimental Quantum State Tomography of Optical Fields and Ultrafast
Statistical Sampling,” (60 pages) Michael G. Raymer and Mark Beck, in
Quantum State Estimation, eds. M. Paris and J. Rehacek, Springer Verlag
(2004).
“Quantum State Tomography of Optical Continuous Variables,” with Alex
Lvovsky, to appear in Quantum Information with Continuous Variables of
Atoms and Light, eds. Nicolas Cerf, Gerd Leuchs, and Eugene Polzik
(Imperial College Press, 2005)
34 of 34

Quantum optical measurement

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