Download quadrature-based teleportation using a fiber-optic entanglement source (PDF).

Multidisciplinary University Research Initiative
Kick-off Meeting: June 13–14, 2000
MIT/NU Collaboration on
Quantum Information Technology:
Entanglement, Teleportation, and Quantum Memory
PI at NU: Prem Kumar, Professor
Department of Electrical and Computer Engineering
Northwestern University, Evanston, IL 60208-3118
Tel: (847) 491-4128; Fax: (847) 4491-4455; E-mail: [email protected]
Co-PI: Horace P. Yuen, Professor
Department of Electrical and Computer Engineering
and Department of Physics and Astronomy
Northwestern University, Evanston, IL 60208-3118
Northwestern University
McCormick School of Engineering and Applied Science
Key Elements of MIT/NU MURI
• Entanglement, teleportation, and quantum storage
using singlet states (mostly based at MIT)
• Entanglement and teleportation using field
quadratures (mostly based at NU)
• New paradigms for quantum communication and
memory (based at both MIT and NU)
Northwestern University
McCormick School of Engineering and Applied Science
Singlet-based Teleportation and
Quantum Storage
One hop
M
L
L
M
P
Two hops
L
L
P
M
M
M
L
P
L
M
B
M = Quantum memory
P = High brightness polarization-entangled photon-pair source
B = Bell state measurement
L = 25km; can reach 100km with two hops
Northwestern University
McCormick School of Engineering and Applied Science
Field-Quadrature based Entanglement
and Teleportation
One hop
in
H
L
F
L
T
out
H
L
F
L
T
L
T
Quantum Teleportation Repeater Line
F = Source of entangled field quadratures
H = Dual quadrature homodyne detection
T = Teleportation completion via modulated mean-field injection
L = 20km
in = Input state
out = Teleported output state
Northwestern University
McCormick School of Engineering and Applied Science
Quadrature Entanglement and
Teleportation using Solitons in Fibers
Objective: Demonstrate fiber-based teleportation of quantum states.
Approach: Use two time and polarization multiplexed Sagnac fiber loops to
produce quadrature-entangled beams.
PM Fiber
Sagnac Loop
Teleported
Ouput State
EPR
Beam 1
to Bob
PM 50/50
Coupler
HWP
Cross
Splice
PBS
Bob's
Measurement
Station
Input
State
1.5 µm
Fiber Laser
Homodyne 1
|in
Homodyne 2
Classical
Information
~
~
PM Fiber
Delay
~
~
EPR
Beam 2
Polarization
Controller
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|out
Differential
Phase
Control
X
Delayed
Pumps
100 km
Fiber
Alice's
Measurement
Station
Phase
Modulator
Amplitude
Modulator
McCormick School of Engineering and Applied Science
Propagation of Light in a
Nonlinear Optical Fiber
Wigner QPD at the Output
(Number--Phase State)
Wigner QPD at the Input
(Coherent State)
Im {u}
Im {u}
u(0)
Re {u}
z)
(
u
Nonlinear Optical Fiber of Length "z"
Re {u}
Solutions
 +
β2 ∂ 2 
Nonlinear Schrödinger Eq. ∂ ˆ
u (T , z ) = i  γ uˆ (T , z )uˆ (T , z ) −
uˆ (T , z )
2 
(moving frame of reference): ∂ z
2 ∂T 

Quasi-CW
iγ uˆ + ( T , 0 ) uˆ ( T , 0 ) z
2
2
[u (T ,0 ) + ∆ uˆ (T ,0 )]
uˆ (T , z ) = e
( β 2 (∂ uˆ ∂ T ) ≈ 0 ):
Linearized around
Fundamental Soliton
(β2 - negative):
 n0
uˆ (T , z ) = 
 2
u(T,0) - pulse envelope
- 2nd order linear dispersion
β2
γ
- nonlinear interaction constant
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

 n0 γ
γ


sech
T + ∆uˆ (T , z ) e


β2

 2 β2 
Ave. number
of photons:
Commutator:
n0 ≡
∞
∫
2
2
n  γ z
i 0 
 2  β2 2
uˆ + (T , z ) uˆ (T , z ) dT
−∞
[uˆ(T, z),uˆ (T′, z)]= δ (T −T′)
+
McCormick School of Engineering and Applied Science
Quasi-CW Propagation
(Linearization Approximation)
Exact mean of the field-operator
(assuming coherent state input):
θmin
u ( z ) = u (0)e
i u (0)
ξ
nl
Im {u}
θmin
ξ
ξ
nl
Re {u}
2
( 1 − cos( γ z ))
γz → 0
[
]
uˆ(T ,ξ ) = u(T ,0) + bˆ(T ,ξ ) eiξnl (T )
Nonlinear length
(phase-shift):
ξ nl ( T ) ≡
Bogoliubov transformation
2
2
(unitary, µ − ν
= 1 ):
z
2
= u (0, T ) γz
L nl
bˆ(ξ nl ) = µ∆uˆ(0) +ν∆uˆ + (0)
µ = 1 + i ξ nl
out
in
∆ Xˆ max
= ( µ + ν )∆ Xˆ max
θmin
e
− u (0)
Linearized Solution
(around u (T ,0 ) e iξ ( T ) ):
Quadrature-noise gains:
nl
sin( γ z )
For any reasonable propagation distance
amplitude remains unsqueezed:
nl
z
2
out
in
∆ Xˆ min
= ( µ − ν )∆ Xˆ min
Minimum uncertainty
product for all ‘’T’’:
g
max/
=
min
≡
ν = i ξ nl
µ
1 + ξ
± ν
2
nl
± ξ
nl
For ξ nl >> 1, g max/min = [2ξ nl ]±1
1
out
out
∆2 Xˆ min
(T ) ∆2 Xˆ max
(T ) =
16
g max (T ) g min (T ) = 1
Northwestern University
McCormick School of Engineering and Applied Science
Nonlinear-Fiber Sagnac Interferometer
(Quasi-CW Classical Description)
Pump
"P"
Fiber
Coupler
50/50
U+
50/50
Beam
Splitter
Fiber
of
Length=z
Pump
"P"
Mirror
Signal
"S"
U-
U+
U-
Mirror
Signal
"S"
Mirror
Ordinary Sagnac Interferometer
Fiber Sagnac Interferometer
Signal power gain (i.e., S = G s (ξ nls , ξ nlp , θ ) S 0 ):
Pump power gain (i.e.,
P = G p (ξ nls , ξ nlp , θ ) P0 ):
ξ nlp
ξ nlp
ξ nls
ξ nls
2
x
x
2
2
x
x
Gs = cos (ξ ) + s sin (ξ nl ) −
sin(2ξ nl ) sin(θ ) G p = cos (ξ nl ) + p sin (ξ nl ) +
sin(2ξ nlx ) sin(θ )
s
p
ξ nl
ξ nl
ξ nl
ξ nl
2
x
nl
Nonlinear phase-shifts:
p
Pump: ξ nl ≡ γ zP0 (T )
Signal: ξ nls ≡ γ zS 0 (T )
Cross: ξ nlx ≡
ξ ξ
Northwestern University
p
nl
s
nl
cos( θ )
Initial phase difference
iθ ( T )
(i.e., u p (T ,0 ) = P0 (T ) e p ; u s (T ,0 ) = S 0 (T ) e iθ s (T ) );
θ (T ) = θ p (T ) − θ s (T )
Note: Signal/pump power gains are nonlinear in
P0(T) & So(T) and phase-sensitive.
McCormick School of Engineering and Applied Science
Im {u}
Quantum Noise in Nonlinear-Fiber
Sagnac Interferometer
Pump
"P"
Coherent
State Input
Nonlinear Optical
Loop Mirror (NOLM)
Fiber
Coupler
50/50
U+
noise in
u(0)
Re {u}
Fiber
of
Length=z
UOptical Fiber
Im {u}
Length=z
Number-Phase
State Output
Signal
"S"
noise out
M. Rosenbluh and R. M. Shelby, Phys. Rev. Lett. 66, 153 (1991);
K. Bergman and H. A. Haus, Opt. Lett. 16, 663 (1991).
z)
(
u
Re {u}
Northwestern University
McCormick School of Engineering and Applied Science
Guided Acoustic Wave Brillouin
Scattering (GAWBS) Noise
In a jacketed fiber every acoustic
mode has a line-width ∆(ωs 2π ) ≈ 2−10MHz
and the entire GAWBS spectrum
spans the range of 20 - 1000 MHz.
Measured GAWBS spectrum
(100 MHz pump, LPF at 70 MHz)
Input:
Light
Wave:
⇒
The three compressional waves create index variation:
(m -radial, n-azimuthal, p-longitudinal)
i (ω ms ,n , p t − k ms ,n , p z )
0
m ,n, p
∆nm ,n , p = ∆n
e
f m , n , p ( r ,θ )
60 MHz
dip
Normalized index variance:
∆2 n
n2
=
ηkT
Mcs2
GAWBS may be cancelled by using a two-pulse phasesensitive detection scheme with short (∆t < 1ns) time delay
between the pulses [K. Bergman, C. R. Doerr, H. A. Haus, and M.
kT - energy in an acoustic mode
cs - speed of sound
M - total mass of fiber
η - photo-elastic constant
⇒
θ+π
∆t
θ
Shirasaki, Opt. Lett. 18, 643 (1993)].
Northwestern University
McCormick School of Engineering and Applied Science
GAWBS-Compensated
Nonlinear Optical Loop Mirror
+/-
balanced
photodetector
N variable
coupler
PBS
M
L
B
heater
lens
HWP
K
O
Signal D
Arm
F-center
laser
A
C
heater
polarization
controllers E
Pump
Arm
differential
F phase
control
J
G
circulator
~50/50
I
phase
H
control
variable
coupler
PM fiber
100 m
Sagnac
interferometer
Northwestern University
McCormick School of Engineering and Applied Science
Experimental Results
Rel. Photocurrent noise (dB)
Rel. photocurrent noise power,
Power gain, Rel. light noise power
1.5
Levandovsky
(a)
1
The two traces are not
“synchronized” left-to-right
0.5
Shot noise
(De)amplified
noise
sin(θ )
-0.5
3
Gain and Fano factor (dB)
and Kumar
Opt. Lett. 24,
0
984 (1999).
Rel. signal
phase
(b)
2
1
≈ −0.6 dB (η = 0.44)
0
λmax ≈ 2 dB
-1
Gsmax = 1.7 dB
-2
Vasilyev
0
0.2
0.4
0.6
Pump strength
(dimensionless units)
Northwestern University
λmin ≈ − 1 .4 dB
min
s
G
≈ −1.87 dB
Almost noiseless amplification
NF = 0.3dB, compared to 1.2dB for
an ideal linear laser amplifier.
Bright sub-Poissonian light
McCormick School of Engineering and Applied Science
Soliton Propagation
(Linearization Approximation)
GAWBS - scales with fiber length and average power (linear phenomenon).
Squeezing - scales with square of the fiber length and peak power.
⇓
For ultra-short (femtosecond) pulses, squeezing dominates GAWBS.
Ultra-short solitons avoid fast temporal spreading, maintain shape & high peak power.
In a fundamental soliton P0T0 = const
2
Normalization of NLSE:
τ = T T0
(P0 - peak power, T0 - pulse width).
ξ = z β 2 T02 aˆ (τ , ξ ) = T0 γ β 2 uˆ (τ , ξ )
⇓
Linearized NLSE
(dimensionless form):
 +
∂
1 ∂2 
aˆ (τ , ξ ) = i  aˆ (τ , ξ ) aˆ (τ , ξ ) +
aˆ (τ , ξ )
2 
∂z
2 ∂τ 

Linearized solution
aˆ(τ,ξ) =a(τ)+∆aˆ(τ,ξ)
Fundamental Soliton
(canonical form):
a = sech(τ )eiξ / 2
Northwestern University
[∆aˆ(τ,ξ)∆aˆ (τ′,ξ)] =δ(τ −τ′)
+
Commutation relation
McCormick School of Engineering and Applied Science
Solution via Perturbation Expansion
Approach H.Haus and Y.Lai, J. Opt. Soc. Am. B 7, 386 (1990)
Based on: H.Haus, W. Wong, F. Khatri, J. Opt. Soc. Am. B 14, 304 (1997)
∂
i ∂ 2
∆ aˆ =
∆ aˆ + 2 i a (τ , ξ
Linearized NLSE:
2
∂ξ
2 ∂τ
) 2 ∆ aˆ + ia 2 (τ , ξ )∆ aˆ +
Expansion based on a Complete set of Orthogonal Eigenmodes of Linearized NLS:
 iξ 2

dΩ ˆ
iξ 2
ˆ
ˆ
ˆ
{
∆aˆ(τ ,ξ ) ≡ b(τ ,ξ )e =  ∑ Vi (ξ ) fi (τ ) + ∫
Vc (Ω,ξ ) fc (Ω,τ ) + Vs (Ω,ξ ) f s (Ω,τ )}e
2π

 i=n, p,τ ,θ
Projection based on orthogonality relation:
Vˆi (ξ ) = Re
ˆ (τ , ξ ) f * (τ )d τ
b
i
∫
Hermitian Operators and Respective Modes Express Perturbations of:
Symmetric:
f (τ ) = f * (− τ )
Vˆn (ξ ) f n (τ )
Vˆ (ξ ) f (τ )
p
p
- Photon Number
- Momentum (frequency)
Vˆc (Ω, ξ ) f c (Ω,τ ) - Continuum at frequency Ω
Northwestern University
Antisymmetric:
Vˆθ (ξ ) fθ (τ )
Vˆ (ξ ) f (τ )
τ
τ
f (τ ) = − f * (− τ )
- Phase
- Position (time)
Vˆs (Ω,ξ ) f s (Ω,τ ) - Continuum at frequency Ω
McCormick School of Engineering and Applied Science
Asymmetric Sagnac Interferometer
(Asymmetric NOLM)
Fundamental soliton
Im {ain}
f n (ω ) ≡ πsech (πω 2 )
∆asol
Fiber Sagnac Interferometer
âin
Fiber Coupler
( transmittance = T )
âsol
a
gv
d
âout
Fiber
of
Length = z
l
a so
a out
ξ/2
Re {ain}
Experiments:
S. Schmitt et al., Phys. Rev. Lett. 81, 2446 (1998) -- 3.9dB;
D. Krylov, K. Bergman, Opt. Lett. 23,1390 (1998) -- 5.7dB.
Output mean field
aout = ALO e iξ
â gvd
To direct
detection
setup
∆aout
2
1 − T − i (ω 2 +1)ξ 2  iξ

e
e
= f n T 1 −

T


2
[
Northwestern University
2
T − ∆a gvd (ξ ) 1 − T
]
aˆ sol = f n + bˆsol (τ , ξ ) e iξ / 2
CCW wave
Output quantum noise
∆aˆ out = bˆsol (ξ )e iξ
CW wave
aˆ gvd = i
2
T
f n e − iω ξ
1−T
2
+ i ∆ aˆ gvd (ω , ξ )
McCormick School of Engineering and Applied Science
Noise Reduction in an
Asymmetric NOLM
Levandovsky, Vasilyev, and Kumar, Opt. Lett. 24, 89 (1999).
Periodicity: ∆ ξ/2=2π
Noise reduction is
limited by (1-T)
losses.
(1-T) limit
Without dispersion
With dispersion
Northwestern University
McCormick School of Engineering and Applied Science
MURI Fellow: Polarization Entangled Photon Pairs
using Microstructure (Holey) Fibers
DSF Fiber
Sagnac Loop
Entangled
Photons at λ1
to Bob
Standard
Fiber
50/50
Coupler
PM Fiber
X
X
X
Splice
Initial experiments with
standard dispersionshifted fiber
Splice with
90o rotation
Splice
WDM
Entangled
Photons at λ2
to Alice
Polarization
Controler
Delayed
Pumps
PM Fiber
1.5 µm Mode-Locked
Fiber Laser
Main experiments near 800nm
wavelength will be with microstructure
(holey) fiber obtained from Lucent
Northwestern University
McCormick School of Engineering and Applied Science
Wavelength-Tunable Picosecond Pulse Source for
Ultrahigh Speed WDM Communications
1541 nm
ML-EDFL
Pr
Serkland and Kumar,
OL 24, 92 (1999).
Signal Spectrum over Tunable Range
-30
20
FS
FPC1
50
-35
FPC2
S
dB
Pump
NFSI
Pi
-40
-45
Grating
10
PBS AL
DSF
-50
1490
1510
1530
1550
1570
1590
Wavelength(nm)
S+I
Pulse Shape (Auto-correlation)
• Signal wavelength tunable over 72nm. 3nm per
channel, total 24 WDM channels
• 2ps FWHM pulse width, which corresponds to
100 Gb/s NRZ transmission data rate
• 24 channels at 100Gb/s, provide total system
capacity up to 2.4 Tb/s !
• Insensitive to pump polarization
• All fiber device, a compact and rugged source
Northwestern University
1.2
Normalized Amplitud
Main Results with EDFL Pumping:
1
0.8
0.6
0.4
0.2
0
-6
-4
-2
0
2
Time (ps)
4
6
McCormick School of Engineering and Applied Science
Classical FOPA Theory for CW Gain
Theoretical Gain
CW FOPA Gain Equations
2
Gain curves of the FOPA
 δk 
 sinh 2 ( gL)
Gs = cosh 2 ( gL) + 
 2g 
Zero dispersion wavelength 1537 nm, pump power 5 W
( )
2
2


k
δ
g = (γPp ) −
2 

Gain (dB)
18
δk = 2γPp + (k s + ki − 2k p )
δk = 2γPp + 2∑
l = 1535 nm
l = 1539 nm
l = 1541 nm
24
12
6
β 2m
(ω s − ω p ) 2 m
2m!
0
1500
1510
1520
1530
1540
1550
1560
1570
1580
Wavelength (nm)
Experimental Gain Results
2E-5
1
2E-5
• The CW theory provides insight into the interplay
Modulation (A.U.)
2E-5
2E-5
between FOPA gain and various experimental
parameters (pump power, pump and signal
wavelength, fiber length, fiber dispersion and
nonlinearity, and phase mismatch).
2E-5
1E-5
1E-5
1E-5
8E-6
6E-6
• Although accurate data is difficult to obtain over the
4E-6
0
2E-6
0E0
-20
-10
0
10
20
entire gain bandwidth, we see that our system is
behaving as expected.
Detuning (nm)
Northwestern University
McCormick School of Engineering and Applied Science
Fiber-optic Parametric Amplifier
(FOPA) Apparatus
FWM Spectra Results
Principles of Operation
• FWM occurs between synchronous pump, signal,
and idler pulses within the Sagnac loop
Intensity [au]
• Phase matching is achieved by operating near the
zero dispersion wavelength of the fiber in the loop
• The pump is filtered from the signal and idler by
adjusting FPC1 so that the loop mirror reflects
• Polarization control for efficient mixing is achieved
Idler
Amp. Sig.
Pump
Unamp.
Sig
by adjusting FPC2
• Signal and idler are detected separately by
1530
1540
Wavelength [nm]
dispersing them with a diffraction grating
+/-
Simplified
Schematic
Grating
300 m
DSF lo=1537 nm
λo
Filter
60
dB
FPC1
50
90
Pump
Input
50
10
FPC2
Elect. Spec.
Analyzer
Signal
Input
Northwestern University
McCormick School of Engineering and Applied Science
Quantum Properties of the FOPA Radiation
Experimental Noise Reduction Results
pump approximation the
FOPA is equivalent to a
nondegenerate OPA.
• The expected noise reduction
for such a system is then as
in [Aytür and Kumar, PRL 65,
1551 (1990)].
Photon correlation in a FOPA
0.8
Experiment
Theory (E=0.25)
Relative Noise (dB)
0.6
• Within the strong, undepleted
0.4
0.2
0.0
-0.2
Theoretical Noise Reduction
-0.4
-0.6
-0.8
-1.0
1.0
1.2
1.4
1.6
1.8
2.0
2.2
E
R = 1− E +
2g − 1
Gain
E = overall detection efficiency
J. E. Sharping, M. Fiorentino, and P. Kumar,
“Observation of twin-beams type quantum correlation in optical fiber,”
to be submitted to Optics Letters.
Northwestern University
McCormick School of Engineering and Applied Science
Fiber Characteristics
Gain vs. pump power
with fits to theory
Observations
• While gain as a function of λ is difficult to measure,
gain as a function of pump power is easily obtained
8
• Fitting theory with data indicates that the CW
theory is adequate to describe our pulsed system
parameters that are consistent with expectations
6
Gain
• We obtain experimental values for the relevant fiber
Signal Wavelength
1544 nm
1546 nm
1548 nm
Lines = Theory
7
5
4
3
2
Dispersion vs. λ
1
0
Data
0.3
0.5
1
1.5
2
2.5
3
Peak Pump Power [W]
Line = Trend
D {ps/(nm km)}
0.2
λo
0.1
Fiber Nonlinear Coefficient
γ = 1.8 [kW m]-1
0.0
Zero Dispersion Wavelength
λo = 1535.6 nm
-0.1
-0.2
1533
1534
1535
1536
1537
1538
1539
1540
1541
Wavelength (nm)
Northwestern University
McCormick School of Engineering and Applied Science
Multidisciplinary University Research Initiative
Kick-off Meeting: June 12–13, 2000
MIT/NU Collaboration on
Quantum Information Technology:
Entanglement, Teleportation, and Quantum Memory
PI at NU: Prem Kumar, Professor
Department of Electrical and Computer Engineering
Northwestern University, Evanston, IL 60208-3118
Tel: (847) 491-4128; Fax: (847) 4491-4455; E-mail: [email protected]
Co-PI: Horace P. Yuen, Professor
Department of Electrical and Computer Engineering
and Department of Physics and Astronomy
Northwestern University, Evanston, IL 60208-3118
Northwestern University
McCormick School of Engineering and Applied Science