Frequency and time... dispersion-cancellation, optical phase, etc.

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Frequency and time... dispersion-cancellation,
optical phase, etc.
(AKA A bunch of loosely connected stuff I
decided I need to cover before I get to some
other stuff...)

• Dispersion cancellation in an HOM interferometer
• (more "collapse versus correlations")
• (useful for time measurements)
• What are time measurements?
• (no time operator)
• (indirect measurements)
• (energy-time "uncertainty relation")
• States of an electromagnetic mode
• (number-phase "uncertainty relation")
• (homodyne measurements, et cetera)
• Phase of a single photon...

28 Oct 2003
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Entangled photon pairs(spontaneous parametric
down-conversion)
The time-reverse of second-harmonic generation. A
purely quantum process (cf. parametric
amplification) Each energy is uncertain, yet
their sum is precisely defined. (For a
continuous-wave pump!) Each emission time is
uncertain, yet they are simultaneous.
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Hong-Ou-Mandel interferometer
Remember if you detect only one photon, the
other photon "knows" where yours came from.
Hence there is no interference (each detector
sees 1/2 of the photons, irrespective of any
phases or path-length differences).
But if you detect both photons, there is no way
to tell whether both were reflected or both were
transmitted. r2t2 (i2 12)/2 0. (any
lossless symmetric beam splitter has a p/2 phase
shift between r and t.)
CAVEAT there must be no way to tell which
occurred. If the paths aren't aligned right, no
interference occurs. If one photon reaches the
beam splitter before the other, no interference
occurs.
How long is a photon?
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the famous dip
In every experiment to date, the width of this
feature is limited only by the bandwidth of the
photons in other words, the photons are
as tightly correlated as they could possibly be
given their own uncertainty in time (Dt gt 1/2Dw).
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What's the speed of a photon?
Silly questions about group velocity, phase
velocity, "collective" nature ofthe index of
refraction, precursors, et cetera. More serious
questions about how to quantize electromagnetic
fields in a dispersive medium. Longstanding
debate about "superluminal" tunneling.
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Problem with propagation measurements
Quadratic term leads to group-velocity
dispersion, broadening, chirp.
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But wait!
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So, what can we predict?
Detectors are "infinitely slow" (ns 106 fs)...
The physical meaning calculate the probability
for each pair of frequencies which might reach
the two detectors, and then integrate. Why? No
interference between paths leading to different
frequencies at the detectors, because in
principle one could go back and measure how much
energy had been absorbed. Note it took a long
time-integral to enforce this. If the detector
had been open only for 1 fs, it would be
impossible to tell what frequency it had seen.
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No, we're not done yet...
The probability of detecting a given frequency
pair
But we only started with 2 photons, so if we
annihilate 2, there will be 0 left
the phase difference is all that concerns us
Note 1 the GVD terms (quadratic in freq.)
cancelled out. Note 2 the pattern moves at the
group velocity. Note 3 the shape is the Fourier
transform of f(w), like the pulse itself.
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Nonlocal cancellation of dispersion
(Oh, and by the way... yeah, single-photon
pulses travel at the group velocity.)
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The Feynman path picture
REMEMBER interference only occurs between
two paths which yield the same w1 and the same
w2. If in the TT path, a "blue" photon is
detected at D1 (and a "red" at D2), then I must
compare with the phase of an RR path with the
same outcome...
The two paths are indistinguishable even though
in each case one of the pulses was
broadened. Perfect interference occurs.
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The collapse picture
After D1 fires, it projects the light in arm 2
into a superposition of two identical chirped
wave packets-- these two packets exhibit
perfect interference.
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But how do you really measure time?
There is no such thing as a direct time
measurement, even classically.
"The position of an electron" has some
meaning... but what is "the time of an electron"?
In QM (as in CM), time is a free parameter, not
an operator/observable.
Time is always measured by observing a pointer
which is believed to evolve at a constant rate,
and "triggering" its evolution (or at least
readout).
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What's the shortest time you can measure?
For this "clock" system to evolve appreciably in
a time t, it must have an energy uncertainty
greater than h/t. "Energy-time uncertainty
principle."
Warning oddly, the reverse is not true. You can
measure energy arbitrarily well in an
arbitrarily short time Aharonov Bohm, Phys Rev
122, 1649 (1961).
Why no rigorous uncertainty principle? Because
time is not an operator. Energy (the Hamiltonian)
is the generator of time-translations. A
canonically conjugate time operator would
generate energy-translations. But energy is
bounded (at least from below) no Hermitian
T-operator exists.
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Can you use photons as a clock?
Of course Electric field changes very rapidly,
on the order of 1015 Hz (even microwave -- Cs
atomic clock -- yields 9109 Hz)
But wait... the energy of a photon is completely
certain.
Energy eigenstate stationary state
In a single-photon state, there "are"
oscillations with a known amplitude... but since
their time origin is completely uncertain, so is
E(t).
Compare an eigenstate of the harmonic
oscillator- the maximum X is related to the
energy, but the particle's equally likely to be
at X or -X.
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What does this mean?
Optical phase actually refers (roughly) to the
quantum phase difference between the amplitude to
have n photons and the amplitude to have n1.
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Uncertainty relations?
What is phase, except time? What is
photon-number, except energy?
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Phase-space picture
complex E E eif e-iwt E1 cos wt E2 sin
wt
The vacuum isn't empty
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Electric field of a coherent state
E
t
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How does one measure these things?
With a radio-frequency field, of course, one
measures E with an antenna, in real time.
Not possible at optical frequencies - we can only
detect power/energy/photons.
Homodyning/heterodyning interfere a signal with
a strong oscillator.
Es eif ELO 2

ELO2 2 ELOEs cos f ...
By varying phase of local oscillator, can measure
different quadratures just by measuring intensity.
Note not so different from RF your oscilloscope
has a local oscillator too.
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E
t
E1
E2
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n
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Back to Bohr Einstein(and Young and Taylor,...)
So, does a single photon exhibit interference, or
not?!
1gt
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Some references (incomplete for now!)
Dispersion cancellation
http//home.t-online.de/home/gerd.breitenbach/gall
ery/ for info pictures about squeezed-states
and their measurement. Uncountable textbooks
review articles... see for instance Loudon Walls
Milburn Scully Zubairy.
Glauber for quantum optics formalism Clauser for
early work on classical versus quantum
effects Grangier et al. for experiment on true
single-photon interference. Thought for next
week atom number is conserved (unlike photon
number). Can it be uncertain? Should two
atom clouds interfere with one another or not?