Quantum Information Processing as told by: David G. Cory Department of Nuclear Science & Engineering...
-
Upload
laurence-little -
Category
Documents
-
view
216 -
download
0
Transcript of Quantum Information Processing as told by: David G. Cory Department of Nuclear Science & Engineering...
Quantum Information Processing
as told by: David G. CoryDepartment of Nuclear Science & Engineering
Massachusetts Institute of Technology
Neutron Interferometr
yDmitry PushinDmitry PushinPhysics, MITPhysics, MIT
Dr. Sekhar Ramanathan Dr. Sekhar Ramanathan
Dr. Timothy Havel
Professor Seth Lloyd
Dr. Sergio Valenzuela
Dr. Will Oliver
Dr. John Bernard
Dr. M. Arif, Dr. M. Arif, NISTNIST
University of WaterlooUniversity of WaterlooProfessor Joseph EmersonProfessor Joseph EmersonProfessor Raymond LaflammeProfessor Raymond LaflammeDr. Jonanthan BaughDr. Jonanthan Baugh
Dr. Timothy Havel
Professor Seth Lloyd
Dr. Sekhar Ramanathan Dr. Sekhar Ramanathan Dr. Joseph EmersonDr. Joseph EmersonPaola Cappellaro Michael Henry Jonathan HodgesSuddhasattwa Sinha Jamie Yang
1900 1910 1920 1930
Planck - photons
1940 1950 1960
Bell - locality testsBell - locality tests
Bohr - old QT, interpretation Bohr - old QT, interpretation Dirac - relativistic wave-equationDirac - relativistic wave-equation
EinsteinEinstein
Heisenberg - new QTHeisenberg - new QT
Schrödinger - wave equationSchrödinger - wave equation
Landauer - Landauer - information is physicalinformation is physical
1970 1980
1900 1920 1940 1960 1980 2000 2020
Old QTOld QT New QTNew QT Tests Tests QIPQIP
HarocheHarocheAspectAspect ZeilingerZeilinger
Quantum mechanics permits information processing beyond the classical limit
These new possibilities are
Macroscopic Quantum CoherenceMacroscopic Quantum CoherenceInterferometer
Neutron
Beam
Sample
Phase Shifter
Detectors
H-beam
O-beam
Neutron interferometryNeutron interferometry
an example of macroscopic quantum coherencean example of macroscopic quantum coherence
3-blade, interferometer3-blade, interferometer
Size ~ 10 cmSize ~ 10 cm
NeutronsNeutrons
2.1 Å2.1 Å
~ 1800 m/s~ 1800 m/s
~ 50 µs / 10 cm~ 50 µs / 10 cm
1 neutron every 0.35 s1 neutron every 0.35 s
Neutron interferometryNeutron interferometry
an example of macroscopic quantum coherencean example of macroscopic quantum coherence
Bragg scattering Bragg scattering
Each neutron Each neutron
is coherently spread is coherently spread
over two pathsover two paths
€
ψ = 12
upper + lower( )
Neutron interferometryNeutron interferometry
an example of macroscopic quantum coherencean example of macroscopic quantum coherence
Ignore the beam Ignore the beam
that is scattered that is scattered
out of the out of the
interferometerinterferometer
No information lost.No information lost.
The transmitted and reflected beams The transmitted and reflected beams
carry the same information,carry the same information,
Neutron interferometryNeutron interferometry
an example of macroscopic quantum coherencean example of macroscopic quantum coherence
€
ψ = 12
eiφ upper + lower( )
€
ψ = 12
upper + lower( )
Third blade recombines Third blade recombines
the beams and allows the beams and allows
them to interfere.them to interfere.
Neutron interferometryNeutron interferometry
an example of macroscopic quantum coherencean example of macroscopic quantum coherence
€
ψO =12
eiφtrr upper + rrt lower( )t - transmittedt - transmitted
r - reflectedr - reflected
€
ψH =12
eiφtrt upper + rrr lower( )
Neutron interferometryNeutron interferometry
an example of macroscopic quantum coherencean example of macroscopic quantum coherence
€
IO = ψO2 = t 2 r 4 1+ cos φ( )[ ]Measure the neutronMeasure the neutron
Intensity.Intensity.
In this case that is the In this case that is the
number of neutronsnumber of neutrons
per unit time.per unit time.
€
IH = ψH2 = r 2 t 4 + r 4
( ) − r 2 t 2 cos φ( )[ ]
Interference
€
re iφ2 ′ k
242
0 trA ψ=
][6242
0 rrtB +=ψ
A:
B:
|path I |path II
€
t e iφ1 k
€
k
C:
€
rre i ˜ φ 2 k
€
rt e i ˜ φ 1 ′ k
D:
3He detectors
H-beam
O-beampath I
path II
ABCD
O-beam:
€
rrte i ˜ φ 1 k + trre i ˜ φ 2 k
H-beam:
€
trte i ˜ φ 1 ′ k + rrre i ˜ φ 2 ′ k
€
IO = rrte i ˜ φ 1 k + trre i ˜ φ 2 k2
= A[1+ cos(Δφ)]
€
IH = trte i ˜ φ 1 ′ k + rrre i ˜ φ 2 ′ k 2
= B − Acos(Δφ)
Neutron interferometryNeutron interferometry
an example of macroscopic quantum coherencean example of macroscopic quantum coherence
Clothier et al, (1991) PRA 44, 5357Clothier et al, (1991) PRA 44, 5357
Neu
ton
s/ 3
min
Neu
ton
s/ 3
min
phasephase
Neutron interferometryNeutron interferometry
an example of macroscopic quantum coherencean example of macroscopic quantum coherence
A simple exampleA simple example
of probability amplitudes.of probability amplitudes.
Set Set so that I so that IHH=0.=0.
€
IH ∝ 1− cos φ( )[ ]
€
IO ∝ 1+ cos φ( )[ ]
€
IO ∝ 2
€
IO ∝1
€
IO ∝1
€
IH ∝1
Coherent Neutron ImagingCoherent Neutron Imaging
€
ψ phase ⏐ → ⏐ ⏐ ψ0eiλ N (z)bc (z)dz∫
z − path
N(z) - particle density
bc z( ) - coherent scattering length
€
IO ∝ 1+ cos kz( )[ ]
€
wedge - linear phase ramp
φ(z) = λNbcz tan ϑ( )
ϑ - wedge angle
k = λNbc tan ϑ( )
Spatially encoding of the neutron beam
3He detector
position sensitive detector
wedge
Neutron beam
By spatially encoding beam we are introducinga new degree of freedom. By tracing this degree we can:• measure spatial properties of materials (softmatter)• use it as controlled decoherence in QIP
Coherent Neutron ImagingCoherent Neutron Imaging
€
IO k( )∝ 1+ w(z)cos kz( )∫ dz[ ]
€
sample, assume bc is spatially invariant.
φs z( ) = λbcNw(z)
w(z) - sample width
€
wedge - linear phase ramp
φ(z) = λNbcz tan ϑ( )
ϑ - wedge angle
k = λNbc tan ϑ( )
Vary k to collect a complete set of Vary k to collect a complete set of
Fourier components. The resolutionFourier components. The resolution
depends on S/N not the detector.depends on S/N not the detector.
Spatial encoding
- No sample
- Step-like sample
The fit is to the known sample geometry, parameters are step location and size.Notice that each point is 50 minutes of averaging.
Spin Polarized Neutrons
Polarizer
Analyzer
Detector
π
π/2
+ = not
Interference and spin
€
re iφ2 ′ k ↓
242
0 trA ψ=
][6242
0 rrtB +=ψ
A:
B:
|path I |path II
€
t e iφ1 k ↑
€
k ↑
C:
€
rre i ˜ φ 2 k ↓
€
rt e i ˜ φ 1 ′ k ↑
D:
H-beam
O-beam path I
path II
ABCD
O-beam:
€
1
2r( rte i ˜ φ 1 + trre i ˜ φ 2 ) k ↑
H-beam:
€
trte i ˜ φ 1 ′ k ↓ + rrre i ˜ φ 2 ′ k ↑
€
IO = 12 rrte i ˜ φ 1 + trre i ˜ φ 2
2= 1
2 A[1+ cos(Δφ)]
€
IH = trte i ˜ φ 12
+ rrre i ˜ φ 22
= B
/2
Polarizer
Analyzer
+ = not
Spin based phase gratingSpin based phase grating
Coherence Measurements
Fussed Silica Wedgesused to move vertically one beam with respect to another Phase Flag
Neutron beam
To the detectors
Contrast measurements directly yields the coherence function
A neutron interferometer is a macroscopicA neutron interferometer is a macroscopic
quantum coherence device, we will measure quantum coherence device, we will measure
the coherence length of the neutrons wave-function.the coherence length of the neutrons wave-function.
€
η=1− λ2 Nbc
2π
€
coherence function
Γ Δ( ) = ψ 0( ) ψ Δ( )
Radius of neutron = 0.7 fmRadius of neutron = 0.7 fm
€
Δc =1
2δk≡ coherence length
δk ≡ spread in momentum
Coherent neutron scatteringCoherent neutron scattering
3-blade interferometer with 3-blade interferometer with
prisms to vertically shift the beam.prisms to vertically shift the beam.
€
ψout =12
ψ0 (z) + ψ0(z + Δ)( )
Adjust phase for only O-beam.Adjust phase for only O-beam.Add second interferometer.Add second interferometer.
€
IO Δ( )∝ S z( )∫ S z + Δ( )dz
S z( ) = N z( )bc z( )
First example of coherent neutron wave-funtion First example of coherent neutron wave-funtion over two interferometersover two interferometers
Interferometer
Neutron
Beam
Sample
Phase Shifter
Detectors
H-beam
O-beam
When will we have a neutron Interferometer at MIT?
Sample
Top View
Neutron wave functioncoherently split byBragg diffraction.
3Hedetectors
Phase Shifter
Δ
H-beam
O-beampath I
path II
10 cm
Each crystal blade acts as a beam splitter.
Neutron interferometryNeutron interferometry
with vibrationswith vibrations
Vibrations change theVibrations change the
momentum of the n andmomentum of the n and
thus the Bragg angle.thus the Bragg angle.
Note, the twoNote, the two
paths changepaths change
in opposite in opposite
directions.directions.
€
IH ∝ 1− cos φ( )[ ]
€
IO ∝ 1+ cos φ( )[ ]
Even low frequency vibrations are deadly.Even low frequency vibrations are deadly.
Neutron interferometryNeutron interferometry
with vibrationswith vibrations
€
IH ∝ 1− cos φ( )[ ]
€
IO ∝ 1+ cos φ( )[ ]
Low frequency vibrations are OK.Low frequency vibrations are OK.
No No interferenceinterference
€
τc = 50μs
υ c = 20kHz
Interferometer
Neutron
Beam
Sample
Phase Shifter
Detectors
H-beam
O-beam
When will we have a neutron Interferometer at MIT?
• • Multiple interferometers for controlling neutron information.Multiple interferometers for controlling neutron information.
• • Multiple paths to code for errors.Multiple paths to code for errors.
• • Spin dependent measurements to correct for momentum spread.Spin dependent measurements to correct for momentum spread.
Stern-Gerlach (details)
π/2 π/2π
Gradient magnets
AnalyzerPolarizer
Sample
Detector
II
III IV
V VI
x↑
)( xx2
1↓+↑
( ) ( ) )( xxexe2
1 xxixi ΔτΔστσ +↓+↑ +
( ) ( ) )( xxexe2
1 xxixi ΔτΔστσ +↑+↓ +
( ) ( ) )( xexe2
1 xxixi ↑+↓ + τΔστσ
( )[ ( ) ] xe xxxi ↑+− τΔστσ
( ) ( ) ( )[ ( ) ]∫ +−= Ωρ τΔστσ dexqS xxxi
where 2
m
vL
dxdB
x ⎟⎠⎞
⎜⎝⎛=γΔ
I
I
II
III
IV
V
VI
1998 1999 2000 2001 2002
J. C. Gore C. BreenS. Kumaresan
N. Seiberlich J. S. Hodges K. EdmondsJ. Yang
30 60 900.4
0.6
0.8
1
Noise Strength
Ent
angl
emen
t Fid
elit
y
Decoherence Free Subspace
22
12
211121 |2
|22
1|
2|
2|
2|
2 xyyxyx JNOTC −−−− −−−−−−=
Construction and Implementation of Logic Gates on two Spins
0
0.5
1
Quantum and Classical Channel
2003
A. GorshkovM. Henry
0
0
0 H
H( )n
H
Journal of Magnetic Resonance
Concepts in Magnetic Resonance
New Journal of Physics
Physical Reviews A
Center for Materials Science and Engineering Summer Students (NSF)
Physical Reviews A
2004
D. Khanal
EIT
Dr. Timothy Havel
Professor Seth Lloyd
Dr. Sekhar Ramanathan Dr. Sekhar Ramanathan Dr. Joseph EmersonDr. Joseph EmersonDr. Grum TeklemariamDr. Greg BoutisNicolas BoulantPaola Cappellaro Zhiying (Debra) ChenHyung Joon ChoDaniel Greenbaum Michael Henry Jonathan HodgesSuddhasattwa Sinha Jamie Yang