Download Introduction to Quantum Optics for Cavity QED Quantum correlations

Introduction to Quantum Optics for
Cavity QED
Quantum correlations of the
intensity and the field.
Luis A. Orozco
Joint Quantum Institute
Department of Physics at UMD
and NIST
State Equation of the
dimensionless fields.
⎡
2C ⎤
y = x ⎢1+
2 ⎥,
⎣ 1+ x ⎦
2C
x=y−
2
1+ x
Output=Input-Atomic polarization that can saturate
€
Atomic Polarization
-2Cx
1+x2
y
x
Drive
Transmission
Steady State:
Exchange of Excitation:
Intensity response to step down of the atoms-cavity system.
Steady State:
Exchange of Excitation:
Jaynes-Cummings Model
1
+
− +
+
H = ω aσ z + ω c a a + g(σ a + σ a)
2
No interaction term:
ground state
With interaction term:
ground state
degenerate
excited states
excited states
This interaction splits the degenerate excite states.
This is the so-called vacuum
Rabi splitting.
Same Physics as two coupled oscillators (field mode and
atomic polarization). We only have one excitation in the
system, there is no difference between boson and fermion.
The study of noisy signals is done with correlation
functions. We have learned a lot about characteristic
times, sizes and some dynamics with such correlation
functions in statistical mechanics. They have the
following form: Example of a Noisy photocurrent:
<F(t) F(t+τ) >
<F(t) G(t+τ)>
For Optical signals the
variables we want to
correlate to themselves or to
each other can be:
Field and Intensity
How do we measure these functions?
G(1)(t+τ) = <E(t) E*(t+τ)> field-field
G(2)(t+τ) = <I(t) I(t+τ)> intensity-intensity Η(t+τ) = <I(t) Ε(t+τ)> intensity-field
Wave-Wave Correlation
Michelson Interferomenter
g (1) (τ ) =
E * (t ) E (t + τ )
I (t )
Spectrum of the source
F (ω ) =
1
(1)
e
xp
(
i
ωτ
)
g
(τ )dτ
∫
2π
Basis of Fourier Transform Spectroscopy
Classical Intensity Correlation Functions: Hanbury
Brown and Twiss. (HTB) two persons only!
At equal time:
With the variance:
The equal time correlation function is related to the
variance, and will be greater than 1.
Discretize the time series:
Ii
Multiply by the delayed time series
Ii+n
Average the result and see the “correlation time”
Another way to calculate the correlation function is with the
waiting time distribution.
Measure the separation between two consecutive pulses (start
and stop)
Histogram the distribution of separations.
This is the same as g(2)(τ) if the fluctuations are very rare.
Make sure the intensities are low, so that seldom you get
coincidences.
Intensity (photons)
time
Comparison of g (2) (τ) from photon counting (a), and
from a time series of intensities measured with a single
PMT (b).
Correlation functions in quantum optics are conditional
measurements.
The detection of the the first photon prepares a state that then
evolves in time.
The correlation functions can be
Field-Field: Mach-Zehnder, Interferogram.
Intensity-Intensity: g(2)(t) Hanbury-Brown and Twiss.
Use a digital
Storage oscilloscope
Start = trigger
Stop = conditional
average
Very brief and incomplete history of the intensity correlation functions.
1955 Forrester measures “Photoelectric mixing of Incoherent light”
1956 Hanbury-Brown and Twiss; astronomy to measure the size of a star
looking at the intensity with two different detectors, not interfering the
fields as Michelson had proposed and doneo (bunching).
1963 Glauber and others (Mandel, Sudarshan, Wolf) formalize the
quantum correlation funcions “Quantum theory of Coherence”.
1976 Experiment and Theory of single atom resonance fluorescence
Kimble, Dagenais, Mandel; Carmichael and Walls.
Shows the non-classical effect of antibunching.
Earlier experiments for Bell Inequalities by Clauser with a cascaded
atomic source can be interpreted as well as measurements of nonclassical properties of light.
Quantum Mechanically (time and normal order):
The variance of the number of photons is related to the probability
of coincident photons given by g(2)(0). Light with Poissonian
statistics has g(2)(0) = 1. Light with a super-Poissonian statistical
distribution has g(2)(0)> 1. A sub-Poissonian distribution has g(2)(0)
< 1, a clear signature of a nonclassical field.
Intensity correlation function measurements:
g ( 2 ) (τ ) =
Iˆ(t ) Iˆ(t + τ )
Iˆ(t )
2
Gives the probability of detecting a photon at
time t + τ given that one was detected at time t.
This is a conditional measurement:
g ( 2 ) (τ ) =
Iˆ(τ )
Iˆ
c
7 663 536 starts
1 838 544 stops
Regression of the field to steady state after the
detection of a photon.
Steady State:
Exchange of Excitation:
The state of the cavity QED system for N atoms is:
The values of the coefficients are:
The probability density of two simultaneous transmission
of photons is then:
2
00 aˆ ψ
2
This can be zero if p is zero
2
= α pq
2
The conditional field prepared by the click is:
A(t)|0> + B(t)|1> with A(t) ≈ 1 and B(t) << 1
Mostly one prepares the vacuum!
The main object of interest is the FIELD
Can we measure it?
One photon over the coherence time of the cavity has a
field of about 5 mV/m.
HOMODYNE and record as a function of the phase of the
local oscillator to obtain the field. We need interference.
However, the random arrivals of the photons do not
permit us to obtain it CONDITIONAL detection.
Homodyne Measurement
There is an
interference between
the Local Oscillator
and the signal that is
proportional to the
amplitude of the signal
|L.O. × cos(φ) + S.|2 = |L.O.|2 + 2 L.O.× S. × cos(φ) + |S.|2
Perfect detector I(t) = |α + Δα|2
I(t) = |α| 2 + 2 α Δα + |Δα|2 ; <α*α>=n
DC~n
Shot noise ~n1/2
neglect.
Intensity-Field correlations
H(t) = <I(t)E(t)> = <E(t)>conditioned
Detection: Homodyne in a Mach Zehnder interferometer
Conditional Measurement: Only measure when we know there
is a photon.
Source: Cavity QED
Condition on a Click
Measure the correlation function of the Intensity and the Field:
<I(t) E(t+τ)>
Normalized form:
hθ(τ) = <E(τ)>c /<E>
From Cauchy Schwartz inequalities:
0 ≤ h0 (0) − 1 ≤ 2
h0 (τ ) − 1 ≤ h0 (0) − 1
Photocurrent average into 50 Ω after one average,
note the size of the fluctuations (about 100 mV)
After 6,000 averages
After 18,000 averages
A signal has appeared with a size of about 3 mV peak
to peak out of an original noise of 100 mV
After 59,000 averages
Photocurrent average with random conditioning
Flip the phase of the Mach-Zehnder by 146o
The fluctuations of the electromagnetic field are
measured by the spectrum of squeezing. Look at the noise
spectrum of the photocurrent out of the Mach Zehnder
interferometer.
∞
S (ν ,0 ) = 4 F ∫ cos(2πντ )[h0 (τ ) − 1]dτ ,
0
F is the photon flux into the correlator.
The spectrum of squeezing is the Fourier transform of the
normalized correlation function h(t) when third order moments
are negligible.
Regression of the field to steady state after the
detection of a photon.
Monte Carlo simulations of the wave-particle correlation and
the spectrum of squeezing in the low intensity limit for an
atomic beam.
Spectrum of Squeezing from the Fourier Transform of h0(t)
This is the conditional evolution of the field of a fraction of
a photon [B(t)] from the correlation function.
hθ(τ) = <E(τ)>c /<E>
The conditional field prepared by the click is:
A(t)|0> + B(t)|1> with A(t) ≈ 1 and B(t) << 1
We measure the field of a fraction of a photon!
Fluctuations are very important.