Balanced homodyne detection

PART 2 :
BALANCED HOMODYNE
DETECTION
Michael G. Raymer
Oregon Center for Optics, University of Oregon
raymer@uoregon.edu

M.G.Raymer_TTRL2b_V2_2005
1 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

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
n1
E1 = dt
ES (t) ES + E L PD
ND
BS n2
PD dt
EL (t) θ E2 =
ES - EL
ND ∝ ∫ E1(− )(t − τ d ) E1(+) (t) dt
τd
delay − ∫ E 2(− )(t − τ d ) E 2(+) (t) dt
3 of 31

DC-BALANCED HOMODYNE DETECTION II
integrator circuit

n1
dt
PD
ND
n2
PD dt
θ

4 of 31

DC-BALANCED HOMODYNE DETECTION III
ΦS = signal amplitude; ΦL = laser reference amplitude
n1
dt
ΦS
ES (t)
ND
BS n2
dt
Φ
EL (t)
L
θ
τd
delay

overlap
∫ dt ∫ Det d x ΦL
T
ˆ
ND = ˆ (− ) (x,0,t − τ d ) ⋅ Φ(+) (x,0,t) + h.c.
2 ˆS
0 integral
ˆ (+ ) (r,t) = i c
ΦS ∑ ˆ
ak v k (r,t)
k
v k (r,t) = ∑ Ck j u j (r) exp(−iω j t)
j
wave-packet
c ∫ 0 dt ∫ Det d x v *k (x,0,t) ⋅ v m (x,0,t) = δ k m
T 2
modes
5 of 31

DC-BALANCED HOMODYNE DETECTION IV

∫ ˆ (− ) (x,0,t − τ d ) ⋅ ∑ ak v k (x,0,t) + h.c.
dt ∫ Det d x ΦL
T
ˆ
ND ∝ 2
ˆ
0
k
wave-packet modes
Assume that the LO pulse is a strong coherent state of a particular
localized wave packet mode: LO phase
ˆ (+ ) (r,t) ∝ | α | exp(i θ ) v L (r,t) + vacuum
ΦL L

N D (θ ) = | α L | ( a e−iθ + a† e iθ )
ˆ ˆ ˆ

a = ∑ ak c ∫ 0 dt ∫ Det d 2 x v *L (x,0,t − τ d ) ⋅ v k (x,0,t) = ak= L
T
ˆ ˆ ˆ
k

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.
6 of 31

DC-BALANCED HOMODYNE DETECTION V

wave-packet
signal : ΦS (r,t) ∝ a v L (r,t) + ∑ ak v k (r,t)
ˆ (+ ) ˆ ˆ
k
modes

quadrature operators: q = ( a + a† ) / 21/2
ˆ ˆ ˆ
p = (a − a† ) / i21/2
ˆ ˆ ˆ

detected N D (θ ) a e−iθ + a† e iθ
ˆ ˆ ˆ LO phase
qθ ≡
ˆ =
quantity: |αL | 2 2

ˆ
N D (θ )
qθ ≡
ˆ = q cosθ + p sin θ
ˆ ˆ
|αL | 2

⎛qθ ⎞ ⎛ cos θ sin θ ⎞⎛ q ⎞
ˆ ˆ
⎜ ⎟=⎜ ⎟⎜ ⎟
⎝ pθ ⎠ ⎝ −sin θ cos θ⎠⎝ p⎠
ˆ ˆ
7 of 31

ULTRAFAST OPTICAL SAMPLING

Conventional Approach:
Ultrafast Time Gating of Light Intensity by
NON-LINEAR OPTICAL SAMPLING

strong short
pump (ωp )
delay sum-frequency (ωp + ωs )

weak signal(ωs ) second-order NL crystal

8 of 31

LINEAR OPTICAL SAMPLING I
BHD for Ultrafast Time Gating of Quadrature Amplitudes

detected ˆ
N D (θ )
qθ ≡
ˆ = q cosθ + p sin θ
ˆ ˆ
quantity: |αL | 2 LO phase
q = ( a + a† ) / 21/2
ˆ ˆ ˆ p = (a − a† ) / i21/2
ˆ ˆ ˆ

a = ∑ ak c ∫ 0 dt ∫ Det d 2 x v *L (x,0,t − τ d ) ⋅ v k (x,0,t) = ak= L
T
ˆ ˆ ˆ
k
LO signal

t

θ

9 of 31

LINEAR OPTICAL SAMPLING II
Ultrafast Time Gating of Quadrature Amplitudes

LO mode: v L (x,0,t) ∝ α L v L (x) f L (t − τ d )

∫
T
ˆ
N D (τ d ) = −i c α * dt f L* (t − τ d ) φS (t) + h.c.
L 0

φS (t) = ∫ Det d x v L * (x) ⋅ ΦS
2 ˆ (+) (x,0,t)

if signal is band-limited and signal
LO covers the band, e.g. LO
f L (t) ∝ (1 / t)sin(B t / 2)
ν−Β/2 ν+Β/2 ω

ˆ D (τ d ) ∝ α * f˜L* (ν ) ∫ ν +B /2 dω exp(−i ω τ d ) φ S (ω ) + h.c.
N ˜
L ν −B /2 2π

∝ α L f˜L* (ν ) φ S (τ d ) + h.c.
*

exact sampling
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).

Ultrafast Signal
Laser (optical or Source
elect. synch.)

Spectral Signal
Filter
Signal

Reference (LO)
Time Phase LO Balanced
Delay Adjustment Homodyne
Detector
τd θ
n1 n2
Computer

mean quadrature
amplitude in sampling ˆ
qθ (t) ψ
window at time t
11 of 31

LINEAR OPTICAL SAMPLING IV

LO
scan LO
840 nm, 170 fs θ delay τd
Sample: Microcavity
exciton polariton
coherent
signal

Balanced
Homodyne
detector

ˆ
qθ (t) ψ
12 of 31

LINEAR OPTICAL SAMPLING V
Mean Quadrature Measurement - sub ps Time Resolution

Sample: Microcavity
ˆ
q (t)
10000θ ψ
exciton polariton 5

1000 4

mean 100 3

quadrature

g
< n(t) >

(2)
10 2

(t,t)
amplitude
<q> at 1 1
time t
0.1 0

0.01 -1
0 2 4 6 8 10 12
Time (ps)
LO delay τd (ps)

ˆ
coherent field --> qθ + π /2 (t) ψ
= pθ (t) ψ ≅ 0
ˆ
13 of 31

LINEAR OPTICAL SAMPLING VI
Phase Sweeping for Indirect Sampling of Mean
Photon Number and Photon Number Fluctuations
detected ˆ
N D (θ )
qθ ≡
ˆ = q cosθ + p sin θ (θ = LO phase)
ˆ ˆ
quantity: |αL | 2
Relation with photon-number operator:
1 1
n = a a = ( q − i p )( q + i p ) = q + p +
ˆ †
ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2
ˆ 2

2 2
Phase-averaged quadrature-squared:
1 π 2 1 π 1 2
qθ θ = ∫ 0 qθ dθ = ∫0 ˆ(q cosθ + p sin θ ) dθ = (q + p 2 )
2 2
ˆ ˆ ˆ ˆ ˆ
π π 2

1 ensemble 1
n = qθ
ˆ ˆ 2 − n (t) ψ = qθ (t)
ˆ ˆ 2 −
θ 2 θ ψ 2
average

works also for incoherent field (no fixed phase)
14 of 31

LINEAR OPTICAL SAMPLING VII
Phase Sweeping --> Photon Number Fluctuations
detected ˆ
N D (θ )
quantity: qθ ≡ | α | 2 = q cosθ + p sin θ
ˆ ˆ ˆ
L

Richter’s formula for Factorial Moments:
∞
n (r ) ψ
= ∑ [n(n −1)...(n − r + 1)] p(n) = ( a† ) r ( a) r
ˆ ˆ ψ
n= 0

(r!) 2 2 π dθ
= r
2 (2r)!
∫ 0 2π H 2r (qθ ) ψ
ˆ

Hermite Polynomials: H 0 (x) = 1, H1 (x) = 2x, H 3 (x) = 4 x 2 − 2
1 2π dθ
∫
1
n (1)
= a
ˆ a =
ˆ †
ˆθ 2 − 2
4q ˆ (t) ψ = qθ 2 (t)
n ˆ −
4 0 2π ψ θ ψ 2
2π dθ 2 4 1
n (2)
= a a
ˆ ˆ †2 2
= ∫ 0 2π 3
qθ − 2 qθ +
ˆ ˆ2
2 ψ
15 of 31

LINEAR OPTICAL SAMPLING VIII
Phase Sweeping --> Photon Number Fluctuations

Variance of Photon Number in Sampling Time
Window: var(n)=< n 2 > - < n >2
2π dθ ⎡ 2 4 1⎤
∫
2
var(n) = qθ − qθ − qθ
ˆ ˆ2 ˆ2 + ⎥
0 2π ⎢ 3
⎣ 4⎦

Second-Order Coherence of Photon Number in
Sampling Time Window:
g(2)(t,t )=[< n 2 > - < n >]/< n >2

g(2) (t,t) = 2 corresponds to thermal light, i.e. light produced
primarily by spontaneous emission.
g(2) (t,t) = 1 corresponds to light with Poisson statistics, i.e., light
produced by stimulated emission in the presence of gain saturation.
16 of 31

LINEAR OPTICAL SAMPLING IX
Photon Number Fluctuations
if the signal is incoherent, no phase sweeping is required
80MHz 1-50kHz
Ti:Sapphire Regen.
Amplifier
λ/2
Electronic Trigger Pulse Sample LO
Delay
λ/2 Signal

Alt. Source PBS1
λ/2
Voltage Charge-Sensitive PBS2
Pulser Pre-Amps
Computer Photodiodes
n1 Shaper
AD/DA Stretcher
n2 Shaper M.
GPIB controller Balanced Homodyne Detector Munroe
17 of 31

LINEAR OPTICAL SAMPLING X
Superluminescent Diode (SLD) Optical Amplifier
metal cap

o
6

600 µm
3 µm

(AR) SiO 2
p-clad layer
p-contact layer
quantum
wells

~ ~ undoped, graded

~ ~
n-clad layer confining layers
n-GaAs substrate

Superluminescent
(Sarnoff Labs) Emission
M. Munroe
18 of 31

LINEAR OPTICAL SAMPLING XI

(no cavity) 1.0
(a)
(a) 0.8

Intensity (a.u.)
0.6

0.4

0.2

25 0.0
Output Power (mW)

810 820 830 840 850
Wavelength (nm)
20

15

10

5
1.0 (b)
0

Intensity (a.u.)
0 100 200 0.5
Drive Current (mA)
(b) 0.0

760 800 840 880
Wavelength (nm)

M. Munroe
19 of 31

LINEAR OPTICAL SAMPLING XII
SLD in the single-pass configuration

3.0
<n(t,t)> 2.4
(2)
g (t,t)

2.5 2.2

2.0

2.0
1.8

g(2)(t,t)
<n(t)>

1.6
1.5

1.4
Photon Fluctuation
is Thermal-like,
1.0 1.2
within a single time
1.0 window (150 fs)
0.5

0 5 10
time (ns)
15 20 M. Munroe
20 of 31

LINEAR OPTICAL SAMPLING XIII
SLD in the double-pass with grating configuration

4.0
<n(t)>
14 (2)
g (t,t)
3.5
12

3.0
10

2.5

g(2)(t,t)
<n(t)>

8

2.0
6
Photon Fluctuation
1.5
4
is Laser-like, within
2 1.0 a single time
0 0.5
window (150 fs)
0 5 10 15 20
time (ns) 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

LO1

BHD q
signal 50/50
q2 + p2 = n
BHD p

π/2 phase LO2
shifter

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

LO is in a Superposition of two wave-packet modes, 1 and 2
ˆ (+ ) (r,t) = i c | α L |exp(iθ ) [v1 (r,t)cosα + v 2 (r,t)exp(−iζ )sin α ]
ΦL
Dual temporal modes: 1 2 (temporal,
Dual LO spatial, or
signal polarization)

BHD Q
β = θ −ζ

Q = cos(α )[q1 cosθ + p1 sin θ ] + sin(α )[q2 cos β + p2 sin β ]
ˆ ˆ ˆ ˆ ˆ

ˆ
q1θ ˆ
q2 β
quadrature of mode 1 quadrature of mode 2
25 of 31

Two-Mode DC-HOMODYNE DETECTION II
ultrafast two-time number correlation measurements using dual-
LO BHD; super luminescent laser diode (SLD)
1 2
Dual LO
signal t1 t2
SLD BHD Q two-time second-
order coherence
: n (t1 ) n (t2 ):
ˆ ˆ
g (t1,t2 ) =
(2)

n (t1 ) n (t2 )
ˆ ˆ

D. McAlister M.G.Raymer_TTRL2b_V2_2005
26 of 31

Two-Mode DC-HOMODYNE DETECTION III
Alternative Method using a Single LO.
Signal is split and delayed by different times.
Polarization rotations can be introduced.

signal
LO
source

BHD Q
polarization rotator

two-pol., two-time : n i (t1 ) n j (t2 ):
ˆ ˆ
second-order g (t1,t2 ) =
(2)
i, j
coherence n i (t1 ) n j (t2 )
ˆ ˆ
A. Funk M.G.Raymer_TTRL2b_V2_2005
27 of 31

Two-Mode DC-HOMODYNE DETECTION IV

Single-time, two-polarization correlation measurements on
emission from a VCSEL

0-2π phase
sweeping
and time
delay
0-2π relative phase sweeping E. Blansett
28 of 31

Two-Mode DC-HOMODYNE DETECTION V

Single-time, two-
polarization correlation
measurements on
at low temp. (10K)

: n i (t1 ) n i (t2 ):
ˆ ˆ
g (t1,t2 ) =
(2)
i, i
n i (t1 ) ni (t2 )
ˆ ˆ

: n i (t1 ) n j (t2 ):
ˆ ˆ uncorrelated
g (t1,t2 ) =
(2)
i, j
n i (t1 ) n j (t2 )
ˆ ˆ

E. Blansett M.G.Raymer_TTRL2b_V2_2005
29 of 31

Two-Mode DC-HOMODYNE DETECTION VI

Single-time, two-
polarization correlation
measurements on
at room temp.

: n i (t1 ) n i (t2 ):
ˆ ˆ
g (t1,t2 ) =
(2)
i, i
n i (t1 ) ni (t2 )
ˆ ˆ

: n i (t1 ) n j (t2 ):
ˆ ˆ anticorrelated
g (t1,t2 ) =
(2)
i, j
n i (t1 ) n j (t2 )
ˆ ˆ

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

Balanced homodyne detection

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