Quantum optical measurement

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 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


[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.


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


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).


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


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)


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



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)


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)


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


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


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


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)


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 (+ )



QUANTIZATION OF THE OPTICAL FIELD IV

Field Uncertainty and Squeezing

q

p

std(p)

std(q)

quadrature-

squeezed light:

�

ˆ E (+ )


�

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


Coherent and Squeezed States

�

ˆ E (+ )


�

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?


Quadrature-Squeezed Vacuum States

�

ˆ E (+ )


�

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)


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

20 of 42

(η=0.8)

300ps, 1064nm

KTP


p(ND), (seeds) p(ND), amplified

ND ND

var(ND)

theory

<n1>=

<n2>=106

69%

below SNL


WIGNER DISTRIBUTIONvisualize state of a single mode in (q, p) phase space.

�

ˆ E (+ )


q

p

q

p

projected distributions: Pr(q), Pr(p)

Underlying Joint Distribution?


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 '


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


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


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


QUANTIZATION OF THE OPTICAL FIELD V

Multimode Fields

�

u j(r) =V−1/2


�

ˆ Φ (+ )

(r,t) = i cj


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


QUANTIZATION OF THE OPTICAL FIELD VI

Wave-packet Modes

�

u j(r) =V−1/2


�

ˆ Φ (+ )

(r,t) = i cj


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:


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:=

+

+

+


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


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


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?

=

+

+

+


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


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

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 biasM.G.Raymer_TTRL2a_V2_2005

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

VCf

Rf

CdFET


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


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)


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 (+ )



PART 2 :

BALANCED HOMODYNE

DETECTION

Michael G. Raymer


[email protected]

M.G.Raymer_TTRL2b_V2_20051 of 31

OUTLINE

PART 1




PART 2



PART 3

6. Quantum State Tomography


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


DC-BALANCED HOMODYNE DETECTION IIintegrator circuit

n1

n2

θ

dt

dt

ND

PD

PD


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


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


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


�

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)


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


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


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


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 -->


LINEAR OPTICAL SAMPLING VI

Phase Sweeping for Indirect Sampling of Mean

Photon Number and Photon Number Fluctuations

�

ˆ q θ≡

ˆ N D(θ)


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)


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


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


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)


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)


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


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.


Measured quadratures

(continuous and dashed

line) on a 10-Gb/s

pulse train.

Waveform obtained by

postdetection squaring

and summing of the two

quadratures.


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


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


at low temp. (10K)

E. Blansett

�

gi, j

(2)(t

1,t

2) =

: 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


Two-Mode DC-HOMODYNE DETECTION VI

Single-time, two-

polarization correlation

measurements on


at room temp.

�

gi, j

(2)(t

1,t

2) =

: 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.


Quantum State Measurement of

Optical Fields and Ultrafast

Statistical Sampling

M. G. Raymer


[email protected]

M.G.Raymer_TTRL2c_V2_20051 of 34

OUTLINE

PART 1




PART 2



PART 3

6. Quantum State Tomagraphy


For introduction, see:

M.R., Contemp. Physics 38, 343 (1997)

PART 3

QUANTUM STATE TOMOGRAPHY


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).


“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.


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

∑


Quantum States Provide Information SecurityQuantum States Provide Information Security

P and Q are

non-Commuting Variables.

Can measure only one.

ψ


Quantum-State Cloning? Quantum-State Cloning?

P and Q are

Both determined?

P

Q

ψ

ψ

ψ


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

∑⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟


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


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.



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.


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



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).


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


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 )


�

ξ = x + px (t /m)

x = 0 x

ξ = 0

x

px

EXPERIMENT WITH ATOMIC BEAM

Joint Probability

Distribution

x

Pr(x;t)


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:


�

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.


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∫


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]


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




Harmonic Oscillator Example - Phase Space

�

Pr(q,t) = P(ψ)ψ

∑ ψ( t)2



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


Harmonic Oscillator Example - Phase Space

�

Pr(q,t) = P(ψ)ψ

∑ ψ( t)2



q

p

q

Phase-Space

Probability Density

W(q,p,t )

�

Pr(q,t) = W (q, p,t)dp∫

Projection =

Probability:


θ = ω t = phase angle

Radon

Transform

Wigner Distr.


Radon

Transform

Wigner Distr.

For squeezed state

Inverse Radon

Transform


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)


Measured Wigner distribution for Single-Photon State

Breitenbach, Schiller, Lvovsky (Uni Konstanz)

Down conversion

Photon pair

trigger

BHD Tomography

vacuum 1-photon

filters


Measured Density Matrix for Quadrature-Squeezed Light

D. Smithey 1993


Determining a Quantum Wave Function from a

Wigner Distribution

BHD Tomography

Density

Matrix ρ

Wigner Distr.

Wave

Function


Determined Quantum Wave Function of Coherent

State from a Laser (Smithey 1993)

Pulsed

Laser BHD

Tomography

attenuator


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)


Science

Quantum optical measurement