Hanbury Brown – Twiss Intensity Interferometry:

Hanbury Brown – Twiss
Intensity Interferometry:
from
stars to nuclei to atoms and electrons
Gordon Baym
University of Illinois, Urbana
RIKEN
April 8, 2014
In 1949 Robert Hanbury Brown
in England, soon after the development of radio astronomy, wanted
to measure the angular sizes of
two radio sources, Cas-A and Cyg-A
using Michelsen interferometry.
If the sources were small he would
need two radio telescopes on opposite
sides of the Atlantic Ocean, connected
to measure relative phase!
Not possible at the time.
1916-2002
He then conceived the idea of intensity interferometry as
a way of measuring the angular sizes of astronomical objects.
Supposing, I thought, there was an another
man many miles away looking at another
identical cathode-ray tube, would he see the
same `noise-like‘ signal? … The next
morning I worked out the answer. … if the
radiation received at two places is mutually
coherent, then the fluctuations in the intensity
of the signals received at those two places is
also correlated. Since the noise on a cathoderay tube corresponds to the low-frequency
fluctuations in the intensity of the signal, the
pictures seen by the two observers must also
be correlated. … To my joy the mathematics
showed that the correlation between their two
pictures is a direct measure of mutual
coherence and can therefore be used to find
the angular size of the source.
R. Hanbury Brown, Boffin (Adam Hilger, 1991)
Angular diameter of Sirius
= 0.0068” ± 0.0005” = 3.1 x
d = 2.7 pc
<I1I2>/<I1><I2>
10-8
rad
λL/R
d
Hanbury Brown – Twiss intensity interferometry
One or many sources (a,b,...)
illuminate 2 detectors (1,2)
Study normalized intensity correlation
between detectors 1 and 2
For photons (and other bosons)
C rises up to 2 at detector separations
d < wavelength (L/R) = λ/θ
λL/R
d
In terms of momentum difference of
particles see (equivalently) rise at
q < /R.
Direct measure of size of source.
Elementary derivation of Hanbury Brown – Twiss interferometry
Two sources at a,b with
φa, φb = random phases
= amplitude at detector 1
intensity at 1
averaged over φ’s
Average intensity at detectors
Correlated normalized
average intensities
at detectors
C(d) varies on scale d = λ /θ
λ = wavelength of light
θ = R/L = angular size of source
λL/R
d
Extended source:
Intensity interferometry measures Fourier transform (squared)
of source distribution
Particle collisions:
Density of sources
is space and time dependent
approximately
4D Fourier transform of dynamical source distribution
BRIEF HISTORY OF HBT INTERFEROMETRY
1949: Robert Hanbury Brown (1916-2002) conceives intensity interferometry. Enlists
Richard Q. Twiss to do mathematical analysis.
1950: 2 radio telescopes (2.4m) used to measure angular diameter of the Sun.
[Arguments with Cambridge “dedicated interferometrophiles”]
1952: Cyg-A and Cas-A radio sources resolved with few km. separation. [“Steam
roller to crack a nut.”]
1955: Sees “photon bunching.”
Test of ideas for((quantum) optics:
1955-1956: Measurement of angular size of Sirius.
¯ p collisions.
1960: G. Goldhaber, S. Goldhaber, W. Lee, A Pais apply to pions from p
Followed by extensive application in heavy ion collisions.
1958: Anti-correlations for fermions (Feynman). Evidence for pp, nn in heavy ion
collisions.
1996: First detection in boson atomic beam experiments.
2007: HBT for 4 He (boson) vs. 3 He (fermion) atoms.
2011: Detection of anti-bunching in electron beam (Kodama, Osakabe &Tonomura)
Measurement of angular diameter of Sirius
Angular diameter of Sirius
= 0.0068” ± 0.0005”
= 3.1 x 10-8 rad
d = 2.7 pc
C(0)-1 = τcoherence/τbin
= 10-14/10-8 = 10-66
Why are signals enhanced at detectors nearby in space and time?
E. Purcell
For thermal sources, statistics of E are given by two dimensional
2+y2)
iφ
iφ
-(x
[Re (Ee ), Im (Ee )] Gaussian distribution. If P(x,y) ~ e
then
<(x2+y2)2> = 2 <x2+y2> 2
vs
for coherent source (laser)
Enhanced fluctuations => larger signal ~ e-(ta-tb)/τcoh
Quantum Mechanical Description of HBT
Classical waves:
In terms of photons sum four processes:
:two photons from each source
:one photon from each source with
qm (bosonic) exchange
Underlying HBT is symmetry of boson wave functions,
e.g.,
Expect similar effect for pairs of
identical particles produced in
high energy collisions
etc.
Generalize to nuclear and particle physics
G. Goldhaber, S. Goldhaber, W,Y. Lee and A. Pais (1960)
=> heavy ion collisions, e+e- annihilation, …
= distribution of single particles (of given species)
averaged over events
= pair correlation in single collision averaged over events
p1 = p/2 +q, p2 = p/2 - q
Correlation function:
Falloff in q measures size
of collision volume
Parameterize correlation function most simply as
-Rinv2qinv2
C(qinv) = 1 + λe
qinv2 = (p1-p 2) 2 - (ε1-ε 2 ) 2
NA44 (SPS) Pi-Pi and K-K interferometry
W+W- correlation function in e+e- annihilation at LEP
√s = 189 Gev
=> rsource ~ 0.5-1 fm
from Alexander and Cohen, 1998
PHENIX (RHIC) Pion interferometry in Au+Au
at √sNN = 200 GeV
A.M. Glenn, Nucl.Phys.A830:833c (2009)
3 dimensional parameterization
x1
p1

q
qside
Rside
x2
p2
qlong
qout
  
q = p2 − p1
 1  
k = (p 2 + p1 )
2
Rout
(Dan Magestro)
Measure correlations as functions of qlong, qside, and qout separately
and parameterize correlation function as:
STAR (RHIC) Pion interferometry from
p+p to d+Au to Au+Au
Rout (fm)
Rlong (fm)
Rside (fm)
Rout / Rout(pp)
Rlong / Rlong(pp)
Rside / Rside(pp)
Why doesn’t the correlation function rise up to 2? (λ < 1)
If the source is coherent, as in a laser, then C(q) ≡1
Long lived resonances give an unresolved peak at small q
ex., ω, η, η’
Uncertainties in the Coulomb correction
Fluctuations from scattering in air and target
Sample contamination: inclusion of non-identical particles through
misidentification
Pion wave functions
HBT is sensitive to wave functions of emerging particles. One of
few quantum phenomenon in high energy measurements where wave
properties of emitted particles play a role
(cf. K regeneration, ν oscillations – q.m. internal degrees of freedom))
For HBT cannot describe particles as little bullets!
Source particles produced in q.m. mixed state:
In general single pion density matrix
(pure state)
(mixed state, with probabilities fi)
Entropy:
in mixed state
Particles are effectively emitted in wave packets:
source at r0,t0 of size Rc, τc emits π of “momenta” p
Pion density matrix:
= distribution of sources
Initial wave packet
spreads
Uncertainty principle and spreading of wave packet:
Transverse:
Longitudinal:
ex.:
L = ct = 10 m from collision to detector,
cp = 1 GeV pions (γ=7), R~ 10 fm = 10-12 cm
Photons:
Particle detection
Detect high energy particles by exciting electrons in atom in detectors:
gas, emulsion, Si, ….
Characteristic excitation time:
= time the detector does quantum mechanics with the pion wave
Incident wave packet:
Find momenta in range p ± Δp
Single particle detection
Probability of detection at first point, a:
Two particle detection
= two particle correlation function
~ |φ(rt, r’t’)|2
For
+cc
usual terms
HBT enhancement (j≠i)
Get enhanced probability of detection when both wave packets
overlap in each of the detectors
HBT does not depend on the history of the particles.
HBT enhancement is a property of the wave functions as
detected by the detectors.
Are correlations property of particles at source?
Simulated emission?
Are correlations property of particles at source?
Simulated emission?
Direction of k1 not correlated with direction of k2
HBT? Enhanced
correlation at detectors
Two point sources
d1 => separation of sources
d2 => Rstar
half-silvered pion mirror
Detection of HBT between pions from RHIC and LHC?
Loss of interferometry signal if particles arrive at detectors with greater
separation than the detector time scale
ex. Directly produced pion vs. pion from Λ → p + π
Signal
HBT two atom correlations
M. Yasuda & F. Shimizu, PRL 77 (1996)
20Ne
beams
⌧0 ⇠ ~/kB Tsource
Interference of matter (atom) waves
Upper condensate
Lower condensate
Condensate has
long range
coherence
Andrews et al. (MIT) 1997 Bose-condensed 23Na
Single run of experiment with detection of many particles
Each condensate separately prepared
No initial correlations between the two.
Do not expect
6 <ρ(r)> to show interference
~ 10 atoms in each condensate
But multiprobe (many photons in detector laser)
measurement of single event (single run of experiment)
does show interference. (Castin & Dalibard, Javanainen)
The phase φ is random from event to event =>
HBT
Multiphoton measurement on single event reveals full structure
Experiment demonstrates multiparticle (~104) HBT correlations
HBT interferometry with fermions:
same spin => anti-correlations:
However –
e e, p p: strong Coulomb repulsions mimics effect of statistical
correlations for same spin identical fermions
n n: strong low energy final state interactions also preclude
direct observation of HBT anti-bunching
atomic experiments show anti-bunching
HBT with 4He (bosons) vs 3He (fermions) beams
T. Jeltes et al. (Orsay + Amsterdam), Nature 445, 402 (2007)
Neutral spin polarized
excited He atoms
4He
3He
Definitive demonstration of effect of statistics in HBT
Bunching and anti-bunching of
neutral cold atoms in optical lattice
87Rb (boson) released
from optical lattice shows
bunching.
40K (fermion) released
from optical lattice shows
anti-bunching
Fölling et al., Nature 434, 481 (2005)
Rom et al., Nature 444, 733 (2006)
HBT with electronic anyons
(fractional statistics in 2D mesoscopic semiconductors -- GaAs)
S. Vishveshwara, PRL91, 196803 (2003)
G. Campagnano et al., PRL 109, 106802 (2012)
Phase factor
e
i⇡⌫/2
e3i⇡⌫/2
ν = filling factor (ex. 1/3, 2/5, ...)
Not yet realized!
BRIEF HISTORY OF HBT INTERFEROMETRY
1949: Robert Hanbury Brown (1916-2002) conceives intensity interferometry. Enlists
Richard Q. Twiss to do mathematical analysis.
1950: 2 radio telescopes (2.4m) used to measure angular diameter of the Sun.
[Arguments with Cambridge “dedicated interferometrophiles”]
1952: Cyg-A and Cas-A radio sources resolved with few km. separation. [“Steam
roller to crack a nut.”]
1955: Sees “photon bunching.”
Test of ideas for((quantum) optics:
1955-1956: Measurement of angular size of Sirius.
¯ p collisions.
1960: G. Goldhaber, S. Goldhaber, W. Lee, A Pais apply to pions from p
Followed by extensive application in heavy ion collisions.
1958: Anti-correlations for fermions (Feynman). Evidence for pp, nn in heavy ion
collisions.
1996: First detection in boson atomic beam experiments.
2007: HBT for 4 He (boson) vs. 3 He (fermion) atoms.
2011: Detection of anti-bunching in electron beam (Kodama, Osakabe &Tonomura)
HBT experiment with electrons -Akira Tonomura et al. 1992-2011
1992
April 25, 1942 – May 2, 2012
1995
2011
incoherent
partially
coherent
coherent
Transmission electron
microscope
Two particle correlation functions
vs. time delay of pairs for different
illumination of detectors.
Role of Coulomb?
HBT searches with electrons in free space
Energetic ions + atom (He, Ne) è double
ionization (3.6 MeV/A Au53+ , 100 MeV/A C6+).
Study correlations of pair of electrons.
Schulz et al., PRL84, 863 (2000)
\
Free electrons from tungsten tip
Kiesel et al., Nature 418, 382 (2002)
Data not corrected for
Coulomb repulsion
between electrons –
results not definitive
F. Hasselbach, Repts. Prog.
Phys. 73, 016101 (2010)
Coulomb interactions vs. quantum statistics
Coulomb interaction between a pair of charged can
mimic the HBT effect from quantum statistics.
Coulomb repulsion between like charged particles reduces
chance of their hitting near-by detectors, reducing the
correlation function, independent of quantum statistics.
Anti-bunching even for distinguishable particles.
Coulomb attraction between oppositely charged particles
enhances probablility of their landing in nearby detectors.
Mimics bosonic HBT for distinguishable particles.
Raw data of pion correlations in heavy ion collisions
D. Miskowiec, E877 collaboration, Nucl. Phys. A590, 473c (1995)
Identical particles
Distinguishable particles
Coulomb attraction mimics HBT between distinguishable particles:
enhancement for π +π - and π-p, and anti-bunching for π +p.
Note similarity between π +π +, π –π - and π +π - raw data
HBT with electrons in 2D semiconductors (GaAs):
mesoscopic interferometers
M. Henny et al., Science 284, 296 (1999);
W.D. Oliver, J. Kim,. R.C. Liu & Y. Yamamoto, Science 284, 299 (1999)
Time delay of detections
at different points
Screening in matter decreases effects of Coulomb repulsion
Screening length (~ 5nm) << Fermi wavelength (~ 40nm)
Correcting data for Coulomb
Coulomb problem electron pair characterized by length scales:
1) electron Bohr radius
~2
a0 =
me2
2) classical turning point of the pair:
e2
q2
=
rtp
2mred
q = relative momentum of the two particles, and mred = m/2
3) typical separation of particles along beam direction
4) size of emitting region transverse to beam
Gamow correction
When initial separation of particles is well within turning point
x0 << rtp
divide observed rate by square of Coulomb wavefunction:
c (0)
=
✓
2⇡⌘
e2⇡⌘
1
◆1/2
⌘ = e2 /~vrel
vrel = relative velocity
Coulomb-corrected rate = observed rate / |
c (0)|
2
When two electrons in electron beam experiment are
emitted outside their classical turning point,
classical physics is more relevant.
Toy model of classical Coulomb corrections
GB & P. Braun-Munzinger, Nucl. Phys. A 610 (1996）
K. Shen and GB, Tonomura Memorial vol, World Sci, 2013
Consider only a pair of electrons emitted independently.
Calculate their classical trajectories, distorted by Coulomb repulsion,
and compute distribution of arrival times at the detector(s).
2
1
e
2
Energy conservation mred vrel
=
2
vti
Coulomb hole
Effects of Coulomb on arrival times
Close departures => large Coulomb force
Large initial interval => Coulomb negligible
Classical trajectories of pair
See formation of
Coulomb hole
in distribution
mimicking HBT
Coulomb corrections for pion and kaon interferometry in
ultrarelativistic heavy ion collisions. E877 experiment at BNL
q
Cintrinsic (q0 ) =
Cobserved (q)
q0
q2
q20
e2
=
±
2mred
2mred
r0
q0= initial relative momentum; q = final
r0 = initial transverse separation
Cint (q0 ) = 1
X = raw data, solid = Gamow, dotted = toy model
Distribution of final arrival times:
Include
1) Poisson distribution of initial time separations
2) average over distribution of initial transverse
positions (here assumed uniform over beam width)
3) Gaussian finite time resolution
R(t
1
t)= p
e (t
2⇡tr
0
2
t0 ) /2t2
r
blue = classical
Coulomb alone –
Coulomb hole
red = with finite
time resolution
Detecting HBT with electrons
Coulomb hole in pair arrival times:
⌧coul =
✓
2
4e
Lmv2
◆1/3
L
L2/3
⇠ 5/6
v
E
HBT time scale: ⌧HBT ⇠ ~/Te↵
⌧HBT
E
⇠
⌧coh
( E)
⌧coul
L2/3
⇠ 11/6
⌧HBT
E
1
E = mv2
2
Te↵
( p)2
( E)2
⇠
⇠
m
E
⌧coh = ~/ E
Larger beam energy
a) decreases effects of Coulomb
b) increases HBT timescale – cf.
coincidence time (~0.2 ns)
In experiment of Kodama, Osakabe & Tonomura (2011)
⌧HBT ⇠ 1ns
cf. ⌧coul ⇠ 1 ps
Coincidence time window ~ -.2 ns
Experiment sees HBT effects of statistics on arrival times.
Spatial HBTt
Spatial Coulomb “hole” sc ~ 2 X hole , sHBT , from HBT
Here correlations are dominated by Coulomb repulsion.
Must include Coulomb in analyzing experiments.
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