quantum optical measurement
TRANSCRIPT
Quantum Optical Measurement:
A Tutorial
Michael G. Raymer
Oregon Center for Optics, University of Oregon
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_20051 of 42
PART 1 :
Quantum Optical Measurement
Michael G. Raymer
Oregon Center for Optics, University of Oregon
-----------------------------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.
M.G.Raymer_TTRL2a_V2_20052 of 42
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
M.G.Raymer_TTRL2a_V2_20053 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).
M.G.Raymer_TTRL2a_V2_20054 of 42
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
M.G.Raymer_TTRL2a_V2_20055 of 42
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
O <n> n
std(n)P(n)
M.G.Raymer_TTRL2a_V2_20056 of 42
Photoelectron Number Probability=Poisson
�
p(n) =n
ne−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
mean rate = r (constant)
M.G.Raymer_TTRL2a_V2_20057 of 42
Photoelectron Number Probability (Semiclassical Theory)
If the mean rate r(t) is not constant.
Define Integrated rate:
�
p(n) =(ηW )ne−ηW
n!0
∞
∫ P(W ) dW
Variance:
�
var(n) = η2var(W ) + n
T
t
T
�
W =1
ηr(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)
M.G.Raymer_TTRL2a_V2_20058 of 42
Photoelectron Number Probability -- Example -
Classical Blackbody Light in Single Mode (frequency, direction)
Short-time integration, T<<1/Δω
�
p(n) =(ηW )
ne−ηW
n!0
∞
∫ P(W ) dW =1
n +1
n
n +1
⎛
⎝ ⎜
⎞
⎠ ⎟
n
T
t
T
�
W =1
ηr(t) dt
0
T
∫
Probability to count n photoelectrons in time T:
(η=QE)
�
I dt0
T
∫ nfilter, Δω
�
P(W ) = <W >−1e−W /<W >
�
n = ηW = η r Tmean number of counts:
Bose-Einstein Distribution
(derived here semiclassically)
M.G.Raymer_TTRL2a_V2_20059 of 42
Photoelectron Number Probability=Bose-Einstein
Mean:
Variance:
�
var(n) = n 2
+ 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 = η r T
M.G.Raymer_TTRL2a_V2_200510 of 42
Breakdown of the Semiclassical Photodetection Theory
�
p(n) =(ηW )ne−ηW
n!0
∞
∫ P(W ) dW
Variance:
�
var(n) = η2var(W ) + n
�
W =1
ηr(t) dt
0
T
∫
There exist light sources for which
(η=QE)
wave noise particle noise
�
var(n) < n
T
t
T
M.G.Raymer_TTRL2a_V2_200511 of 42
QUANTIZATION OF THE OPTICAL FIELD I
�
ˆ E (+ )
(r,t) = iω j
2ε0j
∑ ˆ b j u j (r) exp(−iω j t)
�
u j(r) =V−1/2
ε j exp(i k j ⋅ r)
�
[ ˆ b j ,ˆ b k
†]=δ j k
monochromatic plane-
wave modes:
commutator:
�
ˆ b j,ˆ b k
†photon annilation and
creation operators:
polarization
(ωj > 0)
one-photon state:
�
1ω
= ˆ b ω
†vac
n-photon state:
�
nω
= ( ˆ b ω
†)
nvac
M.G.Raymer_TTRL2a_V2_200512 of 42
QUANTIZATION OF THE OPTICAL FIELD II
Field Uncertainty and Squeezing
A single monochromatic mode:
�
ˆ q = ( ˆ b + ˆ b †) / 2
1/2
�
ˆ p = ( ˆ b − ˆ b †) / i2
1/2
�
ˆ E (+ )
= iω j
2ε0
ˆ b u0(z)exp(−iω0 t)
Hermitean operators:
q, p = quadrature operators
They obey:
Uncertainty relation:
�
[ ˆ q , ˆ p ] = i
�
std(q) std(p) ≥ 1 / 2
q
p
t
�
ˆ E (+ )
(z,t) ∝ ˆ q cos(ω0t − k0z) + ˆ p sin(ω0t − k0z)
M.G.Raymer_TTRL2a_V2_200513 of 42
QUANTIZATION OF THE OPTICAL FIELD III
Coherent State - ideal laser output
�
ˆ q = ( ˆ b + ˆ b †) / 2
1/2
�
ˆ p = ( ˆ b − ˆ b †) / i2
1/2quadrature operators:
Equal Uncertainties:
�
std(q) = std(p) = 1 / 2
q
p
std(p)
std(q)
t
�
ˆ E (+ )
(z,t) ∝ ˆ q cos(ω0t − k0z) + ˆ p sin(ω0t − k0z)
M.G.Raymer_TTRL2a_V2_200514 of 42
QUANTIZATION OF THE OPTICAL FIELD IV
Field Uncertainty and Squeezing
q
p
std(p)
std(q)
quadrature-
squeezed light:
�
ˆ E (+ )
(z,t) ∝ ˆ q cos(ω0t − k0z) + ˆ p sin(ω0t − k0z)
�
ˆ q = ( ˆ b + ˆ b †) / 2
1/2
�
ˆ p = ( ˆ b − ˆ b †) / i2
1/2
q noise reduced
p noise increased
t
M.G.Raymer_TTRL2a_V2_200515 of 42
Coherent and Squeezed States
�
ˆ E (+ )
(z,t) ∝ ˆ q cos(ω0t − k0z) + ˆ p sin(ω0t − k0z)
�
ψ(q) = exp − q − q0( )2/ 2 − i p0 q[ ]
�
ψ(q) =
exp − q − q0( )2/ (2β 2
) − i p0 q[ ]
q0 q
p
std(p)
std(q)
p0
p
std(p)
std(q)
p0
q0 q
�
β 2= (1 / 2)e
−2s
photon number probability:
Poisson
�
p(n) = nψ2
=n
ne−n
n!
�
n = α2, α =
q0
+ ip0
2
photon number
probability?
M.G.Raymer_TTRL2a_V2_200516 of 42
Quadrature-Squeezed Vacuum States
�
ˆ E (+ )
(z,t) ∝ ˆ q cos(ω0t − k0z) + ˆ p sin(ω0t − k0z)
�
ψ(q) = exp −q2 / (2β 2)[ ]
�
β 2= (1 / 2)e
−2s
peven(n) = n ψ
2
=
n
n / 2
⎛⎝⎜
⎞⎠⎟
1
cosh(s)
1
2tanh(s)
⎛⎝⎜
⎞⎠⎟n
q
p
�
podd (n) = 0
vacuum
�
ψ(q) = exp −q2/ 2[ ]
p(n)
q
p
s=2(pair creation)
M.G.Raymer_TTRL2a_V2_200517 of 42
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
�
∂
∂ zψ = − i ˆ H ψ , ˆ H = i
g
2
ˆ b 1
ˆ b 2− ˆ b
1
† ˆ b 2
†
( )
�
ˆ N D
= ˆ n 1− ˆ n
2( ) = ˆ b 1
† ˆ b 1− ˆ b
2
† ˆ b 2( )
photon difference number ND is a constant of the motion:
�
ˆ N D, ˆ H [ ] = 0
signal(ω1)
idler (ω2)signal(ω1)
idler (ω2)
Equal numbers of photons added
to signal and idler beamsM.G.Raymer_TTRL2a_V2_2005
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 - n2variances add: var(ND)= var(n1)+ var(n2)
=<n1>+<n2>=2<n1>=<nTOT>
SHOT-NOISE LEVEL (SNL)
n1n2
j1 =n1 +m
j2 =n2 +m
ND= j1 - j2= n1 - n2 (output noise = same as input noise)M.G.Raymer_TTRL2a_V2_2005
19 of 42
Difference-Number Squeezing:
Classical Particle Model II
Coherent seeds (OPA)
χ(2)signal(ω1)
idler (ω2) j2
j1n1n2
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> +2mp(ND)
ND
inputoutput
Gain= <jTOT>/<nTOT>M.G.Raymer_TTRL2a_V2_2005
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(η=0.8)
300ps, 1064nm
KTP
M.G.Raymer_TTRL2a_V2_200521 of 42
p(ND), (seeds) p(ND), amplified
ND ND
var(ND)
theory
<n1>=
<n2>=106
69%
below SNL
M.G.Raymer_TTRL2a_V2_200522 of 42
WIGNER DISTRIBUTIONvisualize state of a single mode in (q, p) phase space.
�
ˆ E (+ )
(z,t) ∝ ˆ q cos(ω0t − k0z) + ˆ p sin(ω0t − k0z)
q
p
q
p
projected distributions: Pr(q), Pr(p)
Underlying Joint Distribution?
M.G.Raymer_TTRL2a_V2_200523 of 42
WIGNER DISTRIBUTIONin (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
2πψ (q + q '/ 2)ψ * (q − q '/ 2)
− ∞
∞
∫ exp(−i q ' p) dq '
M.G.Raymer_TTRL2a_V2_200524 of 42
WIGNER DISTRIBUTIONfor one-photon state | 1 >
�
W (q, p) =2q
2+ 2p
2−1
πexp(−q
2− p
2)
q
p
�
ψ(q) = q exp −q2/ 2[ ]
q
W(q,p)
negative
M.G.Raymer_TTRL2a_V2_200525 of 42
WIGNER DISTRIBUTIONfor vacuum state | O >
�
W (q, p) =1
πexp(−q
2− p
2)
q
p
�
ψvac (q) = exp −q2/ 2[ ]
q
Wvac(q,p)
positive
M.G.Raymer_TTRL2a_V2_200526 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 ε2W (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[ ]
πW (x, y)
−∞
∞
∫ dx dy
M.G.Raymer_TTRL2a_V2_200527 of 42
QUANTIZATION OF THE OPTICAL FIELD V
Multimode Fields
�
u j(r) =V−1/2
ε j exp(i k j ⋅ r)
�
ˆ Φ (+ )
(r,t) = i cj
∑ ˆ b j u j (r) exp(−iω j t)
monochromatic plane-
wave modes:
Photon-flux amplitude operator:
Photon flux through a plane at z=O:
�
ˆ I (t) = d2x
Det∫ ˆ Φ
(− )
(x,0,t) ⋅ ˆ Φ ( + )
(x,0,t)
�
ˆ N =0
T
∫ ˆ I (t) dt
Integrated photon number in time T:
z
x
M.G.Raymer_TTRL2a_V2_200528 of 42
QUANTIZATION OF THE OPTICAL FIELD VI
Wave-packet Modes
�
u j(r) =V−1/2
ε j exp(i k j ⋅ r)
�
ˆ Φ (+ )
(r,t) = i cj
∑ ˆ b j u j (r) exp(−iω j t)
Non-mon0chromatic Wave-Packet Modes:
�
ˆ Φ (+ )
(r,t) = i c
k
∑ ˆ a k
v k (r,t)change of
mode basis:
�
vk(r,t) =
j
∑ Ck j u j (r) exp(−iω j t)
�
ˆ a k
=
m
∑ Ck m
* ˆ b m
new annihilation operators:
�
[ ˆ a j , ˆ a k†] = δ j k
Unitary
Transformation
(temporal
modes)
�
vk * (r,t) ⋅v j (r,t)∫ d3r = δk jorthogonality:
M.G.Raymer_TTRL2a_V2_200529 of 42
QUANTIZATION OF THE OPTICAL FIELD
Wave-packet Modes
Non-mon0chromatic (temporal) Wave-Packet Modes:
�
ˆ Φ (+ )
(r,t) = i c
k
∑ ˆ a k
v k (r,t)
�
v 0(r,t) =
j
∑ C0 j u j (r) exp(−iω j t)
k=O:=
+
+
+
M.G.Raymer_TTRL2a_V2_200530 of 42
QUANTIZATION OF THE OPTICAL FIELD VII
Non-monchromatic Wave-
Packet Modes
�
ˆ Φ (+ )
(r,t) = i c
k
∑ ˆ a k
v k (r,t)
operator creates one photon in the wave packet
�
ˆ a k
†
�
vk(r,t)
�
ˆ a k
†vac = 0, 0,...1
k, 0, 0...
z
�
vk(r,t)
one
click
detector
array
M.G.Raymer_TTRL2a_V2_200531 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 detectionM.G.Raymer_TTRL2a_V2_2005
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
M.G.Raymer_TTRL2a_V2_200533 of 42
2. Importance of temporal selectivity:
Multimode Fields
�
ˆ Φ (+ )
(r,t) = i c
k
∑ ˆ a k
v k (r,t)Non-mon0chromatic
Wave-Packet Modes:
Want to measure the
light in just one of
these packets.
How to select?
=
+
+
+
M.G.Raymer_TTRL2a_V2_200534 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
M.G.Raymer_TTRL2a_V2_200535 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 biasM.G.Raymer_TTRL2a_V2_2005
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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 biasM.G.Raymer_TTRL2a_V2_2005
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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
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
VCf
Rf
CdFET
M.G.Raymer_TTRL2a_V2_200538 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
VCf
Rf
CdFET
M.G.Raymer_TTRL2a_V2_200539 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)
M.G.Raymer_TTRL2a_V2_200540 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
�
ˆ q = ( ˆ a + ˆ a †) / 2
1/2
�
ˆ p = ( ˆ a − ˆ a †) / i2
1/2
�
ˆ E (+ )
(z,t) ∝ ˆ q cos(ω0t − k0z) + ˆ p sin(ω0t − k0z)
M.G.Raymer_TTRL2a_V2_200541 of 42
PART 2 :
BALANCED HOMODYNE
DETECTION
Michael G. Raymer
Oregon Center for Optics, University of Oregon
M.G.Raymer_TTRL2b_V2_20051 of 31
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
M.G.Raymer_TTRL2b_V2_20052 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∝ E
1
(− )(t − τ
d)∫ E
1
(+)(t) dt
− E2
(− )(t − τ
d)∫ E
2
(+)(t) dt
ES(t)
EL(t)
n1
n2
θ
BS
dt
dt
ND
E1 =ES + EL
PD
PD
E2 =ES - EL
delay
τd
M.G.Raymer_TTRL2b_V2_20053 of 31
DC-BALANCED HOMODYNE DETECTION IIintegrator circuit
n1
n2
θ
dt
dt
ND
PD
PD
M.G.Raymer_TTRL2b_V2_20054 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
�
ˆ N D
= dt0
T
∫ d2x
Det∫ ˆ Φ L
(− )
(x,0,t − τd) ⋅ ˆ Φ S
( + )
(x,0,t) + h.c.
�
ˆ Φ S(+ )
(r,t) = i c
k
∑ ˆ a k
v k (r,t)
�
vk(r,t) =
j
∑ Ck j u j (r) exp(−iω j t)
ΦS
ΦL
overlap
integral
�
c dt0
T
∫ d2x v *
k(x,0,t) ⋅ vm (x,0,t)
Det∫ = δ
k m
wave-packet
modesM.G.Raymer_TTRL2b_V2_20055 of 31
DC-BALANCED HOMODYNE DETECTION IV
�
ˆ N D∝ dt
0
T
∫ d2x
Det∫ ˆ Φ L
(− )
(x,0,t − τd) ⋅
k
∑ ˆ a k
v k (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:
�
ˆ N D(θ) = |α
L| ( ˆ a e
− iθ+ ˆ a
†e
iθ)
�
ˆ Φ L(+ )
(r,t) ∝ |αL | exp(iθ) vL (r,t) + vacuum
LO phase
�
ˆ a =
k
∑ ˆ a k
c dt0
T
∫ d2x
Det∫ v *
L(x,0,t − τ
d) ⋅ vk (x,0,t) = ˆ a
k= L
M.G.Raymer_TTRL2b_V2_20056 of 31
DC-BALANCED HOMODYNE DETECTION V
wave-packet
modessignal :
quadrature operators:
�
ˆ q θ≡
ˆ N D(θ)
|αL | 2=
ˆ a e− iθ
+ ˆ a †e
iθ
2
�
ˆ q θ
ˆ p θ
⎛
⎝ ⎜
⎞
⎠ ⎟ =
cosθ sinθ−sinθ cosθ⎛
⎝ ⎜
⎞
⎠ ⎟
ˆ q
ˆ p
⎛
⎝ ⎜
⎞
⎠ ⎟
�
ˆ q = ( ˆ a + ˆ a †) / 2
1/2
�
ˆ p = ( ˆ a − ˆ a †) / i2
1/2
�
ˆ Φ S(+ )
(r,t)∝ ˆ a vL (r,t) +
k
∑ ˆ a k
v k (r,t)
�
ˆ q θ≡
ˆ N D(θ)
|αL | 2= ˆ q cosθ + ˆ p sinθ
detected
quantity:LO phase
M.G.Raymer_TTRL2b_V2_20057 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
M.G.Raymer_TTRL2b_V2_20058 of 31
�
ˆ q = ( ˆ a + ˆ a †) / 2
1/2
�
ˆ p = ( ˆ a − ˆ a †) / i2
1/2
LINEAR OPTICAL SAMPLING I
BHD for Ultrafast Time Gating of Quadrature Amplitudes
�
ˆ q θ≡
ˆ N D(θ)
|αL | 2= ˆ q cosθ + ˆ p sinθ
detected
quantity: LO phase
�
ˆ a =
k
∑ ˆ a k
c dt0
T
∫ d2x
Det∫ v *
L(x,0,t − τ
d) ⋅ vk (x,0,t) = ˆ a
k= L
t
signalLO
θ
M.G.Raymer_TTRL2b_V2_20059 of 31
LINEAR OPTICAL SAMPLING II
Ultrafast Time Gating of Quadrature Amplitudes
�
vL(x,0,t) ∝ αL vL (x) fL (t − τ d )
�
ˆ N D(τ d ) = −i cαL
*dt
0
T
∫ fL
*(t − τ d ) φS (t) + h.c.
�
φS(t) = d
2x
Det∫ vL * (x) ⋅ ˆ Φ S
( + )
(x,0,t)
�
ˆ N D(τ d ) ∝ αL
* ˜ f L*(ν )
dω2πν −B /2
ν + B /2
∫ exp(−iωτ d ) ˜ φ S (ω) + h.c.
∝ αL
* ˜ f L*(ν ) φS (τd ) + h.c.
LO mode:
�
fL(t)∝ (1 / t)sin(B t / 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_200510 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).
�
ˆ q θ (t)ψ
Ultrafast
Laser
Spectral
Fi lter
Time
Delay
Signal
Source
Balanced
Homodyne
Detector
Computer
LO
Signal
Phase
Adjustment
(optical or
elect. synch.)
n1
n2
mean quadrature
amplitude in sampling
window at time t
θτd
Reference (LO)
Signal
M.G.Raymer_TTRL2b_V2_200511 of 31
LINEAR OPTICAL SAMPLING IV
Sample: Microcavity
exciton polariton
scan LO
delay τd840 nm, 170 fs
�
ˆ q θ (t)ψ
θ
LO
Balanced
Homodyne
detector
coherent
signal
M.G.Raymer_TTRL2b_V2_200512 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 polariton
�
ˆ q θ (t)ψ
�
ˆ q θ +π /2(t)
ψ= ˆ p θ (t)
ψ≅ 0coherent field -->
M.G.Raymer_TTRL2b_V2_200513 of 31
LINEAR OPTICAL SAMPLING VI
Phase Sweeping for Indirect Sampling of Mean
Photon Number and Photon Number Fluctuations
�
ˆ q θ≡
ˆ N D(θ)
|αL | 2= ˆ q cosθ + ˆ p sinθ
detected
quantity: (θ = LO phase)
Relation with photon-number operator:
n = a†a =
1
2q − i p( ) q + i p( ) = q
2+ p
2+1
2
Phase-averaged quadrature-squared:
�
ˆ q θ2
θ=
1
πˆ q θ
2
dθ0
π
∫ =1
πˆ q cosθ + ˆ p sinθ( )
2
dθ0
π
∫ =1
2ˆ q
2+ ˆ p
2( )
�
ˆ n = ˆ q θ
2
θ−
1
2
�
ˆ n (t)ψ
= ˆ q θ2(t)
θ ψ−
1
2ensemble
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
�
ˆ q θ≡
ˆ N D(θ)
|αL | 2= ˆ q cosθ + ˆ p sinθ
detected
quantity:
�
n(r )
ψ= [n(n −1)...(n − r + 1)]
n= 0
∞
∑ p(n) = ( ˆ a †)
r( ˆ a )
r
ψ
=(r!)
2
2r(2r)!
dθ2π0
2π
∫ H2r ( ˆ q θ ) ψ
Richter’s formula for Factorial Moments:
�
ˆ n (t)ψ
= ˆ q θ2(t)
θ ψ−
1
2
�
H0(x) =1, H
1(x) = 2x, H
3(x) = 4x
2− 2Hermite Polynomials:
�
n(1)
= ˆ a † ˆ a =
1
4
dθ2π0
2π
∫ 4 ˆ q θ2 − 2
ψ
�
n(2)
= ˆ a †2
ˆ a 2
=dθ2π0
2π
∫ 2
3ˆ q θ
4 − 2 ˆ q θ2
+1
2 ψM.G.Raymer_TTRL2b_V2_2005
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
3ˆ q θ
4− ˆ q θ
2− ˆ q θ
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
M.G.Raymer_TTRL2b_V2_200516 of 31
LINEAR OPTICAL SAMPLING IX
Photon Number Fluctuations
PBS1
LO
Signal
PBS2
PhotodiodesComputer
n1
n2 Shaper
Charge-SensitivePre-Amps
Stretcher
Balanced Homodyne Detector
λ/2
λ/2
80MHz 1-50kHz
Ti:Sapphire
Shaper
AD/DA
λ/2
Alt. Source
ElectronicDelay
VoltagePulser
Trigger Pulse
GPIB controller
Regen.Amplifier
Sample
M.
Munroe
if the signal is incoherent, no phase sweeping is required
M.G.Raymer_TTRL2b_V2_200517 of 31
LINEAR OPTICAL SAMPLING XSuperluminescent Diode (SLD) Optical Amplifier
M. Munroe
~~
~~
6o
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
0Ou
tpu
t P
ow
er (
mW
)
2001000
Drive Current (mA)
1.0
0.5
0.0
Inte
nsi
ty (
a.u
.)
880840800760
Wavelength (nm)
(b)
1.0
0.8
0.6
0.4
0.2
0.0
Inte
nsi
ty (
a.u
.)
850840830820810
Wavelength (nm)
(a)(a)
(b)
(no cavity)
M.G.Raymer_TTRL2b_V2_200519 of 31
LINEAR OPTICAL SAMPLING XIISLD 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)
M.G.Raymer_TTRL2b_V2_200520 of 31
LINEAR OPTICAL SAMPLING XIIISLD 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. MunroeM.G.Raymer_TTRL2b_V2_2005
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
M.G.Raymer_TTRL2b_V2_200522 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.
M.G.Raymer_TTRL2b_V2_200523 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.
M.G.Raymer_TTRL2b_V2_200524 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
�
ˆ Φ L(+ )
(r,t) = i c |αL
|exp(iθ) v1(r,t)cosα + v 2(r,t)exp(−iζ )sinα[ ]
�
ˆ Q = cos(α) ˆ q 1 cosθ + ˆ p 1 sinθ[ ] + sin(α) ˆ q 2 cosβ + ˆ p 2 sinβ[ ]
�
ˆ q 1θ
�
β = θ −ζ
Dual temporal modes:
�
ˆ q 2β
quadrature of mode 1 quadrature of mode 2
(temporal,
spatial, or
polarization)
Dual LO
M.G.Raymer_TTRL2b_V2_200525 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,t
2) =
: ˆ n (t1) ˆ n (t
2):
ˆ n (t1) ˆ n (t
2)
BHD
signal
Dual LO
Q
1 2
SLD
t1 t2
D. McAlisterM.G.Raymer_TTRL2b_V2_2005
26 of 31
Two-Mode DC-HOMODYNE DETECTION III
two-pol., two-time
second-order
coherence
BHD
signalLO
Q
�
gi, j
(2)(t
1,t
2) =
: ˆ n i(t1) ˆ n j (t
2):
ˆ n i(t1) ˆ n 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_200527 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
emission from a VCSEL
at low temp. (10K)
E. Blansett
�
gi, j
(2)(t
1,t
2) =
: ˆ n i(t1) ˆ n j (t2):
ˆ n i(t1) ˆ n j (t2)
�
gi, i
(2)(t
1,t
2) =
: ˆ n i(t1) ˆ n i(t2):
ˆ n i(t1) ˆ n i(t
2)
uncorrelated
M.G.Raymer_TTRL2b_V2_200529 of 31
Two-Mode DC-HOMODYNE DETECTION VI
Single-time, two-
polarization correlation
measurements on
emission from a VCSEL
at room temp.
�
gi, j
(2)(t
1,t
2) =
: ˆ n i(t1) ˆ n j (t2):
ˆ n i(t1) ˆ n j (t2)
�
gi, i
(2)(t
1,t
2) =
: ˆ n i(t1) ˆ n i(t2):
ˆ n i(t1) ˆ n i(t
2)
anticorrelated
Spin-flip --> gain competition M.G.Raymer_TTRL2b_V2_200530 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.
M.G.Raymer_TTRL2b_V2_200531 of 31
Quantum State Measurement of
Optical Fields and Ultrafast
Statistical Sampling
M. G. Raymer
Oregon Center for Optics, University of Oregon
M.G.Raymer_TTRL2c_V2_20051 of 34
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 Tomagraphy
M.G.Raymer_TTRL2c_V2_20052 of 34
For introduction, see:
M.R., Contemp. Physics 38, 343 (1997)
PART 3
QUANTUM STATE TOMOGRAPHY
M.G.Raymer_TTRL2c_V2_20053 of 34
Niels Bohr:
"In quantum physics ... evidence about atomic objectsobtained by different experimental arrangementsexhibits a novel kind of complementarity relationship.... Such evidence, which appears contradictory whencombination into a single picture is attempted, exhaustsall conceivable knowledge about the object. ...
Moreover, a completeness of description like that aimedat in classical physics is provided by the possibility oftaking every conceivable experimental arrangement intoaccount."
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).
M.G.Raymer_TTRL2c_V2_20054 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.
M.G.Raymer_TTRL2c_V2_20055 of 34
12 3 4
5
Ω
V
ψR
ψI
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.M.G.Raymer_TTRL2c_V2_2005
6 of 34
A
Realistic
Psi -
Meter
A “Quorum”
of Variables
M.R., Contemp. Physics 38, 343 (1997)
12 3 4
5
ψR
ψI
12 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 φ j
j
∑
M.G.Raymer_TTRL2c_V2_20057 of 34
Quantum States Provide Information SecurityQuantum States Provide Information Security
P and Q are
non-Commuting Variables.
Can measure only one.
ψ
M.G.Raymer_TTRL2c_V2_20058 of 34
Quantum-State Cloning? Quantum-State Cloning?
P and Q are
Both determined?
P
Q
ψ
ψ
ψ
M.G.Raymer_TTRL2c_V2_20059 of 34
No - Cloning Theorem (Wootters and Zurek) No - Cloning Theorem (Wootters and Zurek)
ψSYS
ψAUX
ψSYS
χref AUX
φJ SYSχref AUX
φJ SYS
φJ AUX
(basis state)
(arbitrary state)
aj φ j SYSj
∑⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ χref AUX
aj φ j SYSj
∑⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ aj φ j AUXj
∑⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ ≠
?
aj φ j SYSφ j AUX
j
∑⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
M.G.Raymer_TTRL2c_V2_200510 of 34
No Cloning <---> No State Measurement of Single ParticleNo Cloning <---> No State Measurement of Single Particle
(NO)
12
34
5
ψR
ψI
12
34
5A Quorum
ψ
ψψ
ψ
ψ
ψM.G.Raymer_TTRL2c_V2_2005
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
M.G.Raymer_TTRL2c_V2_200512 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.
M.G.Raymer_TTRL2c_V2_200513 of 34
QUANTUM STATE TOMOGRAPHY
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.
M.G.Raymer_TTRL2c_V2_200514 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)
�
ˆ ρ = P(ψ)ψ
∑ ψ ψ
�
ρ (x,x ') = P(ψ )ψ
∑ ψ(x)ψ * (x ') = ψ(x)ψ * (x ')ens
M.G.Raymer_TTRL2c_V2_200515 of 34
QUANTUM STATE TOMOGRAPHY
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).
M.G.Raymer_TTRL2c_V2_200516 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
M.G.Raymer_TTRL2c_V2_200517 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 )
M.G.Raymer_TTRL2c_V2_200518 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)
M.G.Raymer_TTRL2c_V2_200519 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:
M.G.Raymer_TTRL2c_V2_200520 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.
M.G.Raymer_TTRL2c_V2_200521 of 34
Joint Probability DistributionPr(x;t) Data
T. Pfau, C.
Kurtseiferx
NEGATIVE!
�
W (q, p) =1
πψ(q + x /2)
−∞
∞
∫ ψ * (q + x /2)exp(−i px)dx
“Joint Probability Distribution” -> Wigner Distribution
�
Pr(x, px ) = Pr(ξ;t)K(x, px;ξ;t)dξ dt∫
M.G.Raymer_TTRL2c_V2_200522 of 34
Harmonic Oscillator Example
Deterministic inversion of statistical data to yielddensity 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]
M.G.Raymer_TTRL2c_V2_200523 of 34
Harmonic Oscillator Example
Deterministic inversion of statistical data to yielddensity 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]
M.G.Raymer_TTRL2c_V2_200524 of 34
Harmonic Oscillator Example - Phase Space
�
Pr(q,t) = P(ψ)ψ
∑ ψ( t)2
= ρµν ψµ (q)µ ,ν
∑ ψν (q)exp[i(ν − µ)ω t]
q
p
q
Phase-Space
Probability Density
W(q,p,t )
�
Pr(q,t) = W (q, p,t)dp∫
Projection =
Probability:
θθ = ω t = phase
angle
M.G.Raymer_TTRL2c_V2_200525 of 34
Harmonic Oscillator Example - Phase Space
�
Pr(q,t) = P(ψ)ψ
∑ ψ( t)2
= ρµν ψµ (q)µ ,ν
∑ ψν (q)exp[i(ν − µ)ω t]
q
p
q
Phase-Space
Probability Density
W(q,p,t )
�
Pr(q,t) = W (q, p,t)dp∫
Projection =
Probability:
M.G.Raymer_TTRL2c_V2_200526 of 34
θ = ω t = phase angle
Radon
Transform
Wigner Distr.
M.G.Raymer_TTRL2c_V2_200527 of 34
Radon
Transform
Wigner Distr.
For squeezed state
Inverse Radon
Transform
M.G.Raymer_TTRL2c_V2_200528 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)
M.G.Raymer_TTRL2c_V2_200529 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
M.G.Raymer_TTRL2c_V2_200530 of 34
Measured Density Matrix for Quadrature-Squeezed Light
D. Smithey 1993
M.G.Raymer_TTRL2c_V2_200531 of 34
Determining a Quantum Wave Function from a
Wigner Distribution
BHD Tomography
Density
Matrix ρ
Wigner Distr.
Wave
Function
M.G.Raymer_TTRL2c_V2_200532 of 34
Determined Quantum Wave Function of Coherent
State from a Laser (Smithey 1993)
Pulsed
Laser BHD
Tomography
attenuator
M.G.Raymer_TTRL2c_V2_200533 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)
M.G.Raymer_TTRL2c_V2_200534 of 34