Download Lecture I

Quantum information in
bright colors
Paulo A. Nussenzveig
Instituto de Física - USP
Paraty – 2009
The Team
Antônio Sales
Felippe Barbosa
Katiúscia Cassemiro
Jonatas César
Alessandro Villar
Luciano Cruz
Marcelo Martinelli
Paulo Valente
Paulo Nussenzveig
Lectures
• 1st Lecture – Continuous variables (CV): entanglement,
squeezing, OPO basics.
• 2nd Lecture – More on OPOs, bipartite entanglement
below and above threshold.
• 3rd Lecture – Direct generation of tripartite three-color
entanglement; “Entanglement Sudden Death” in a CV
system.
Einstein, Podolsky & Rosen’s paper
Einstein, Podolsky & Rosen’s paper
EPR’s example
|y  d(x1 – x2 – L)d(p1 + p2)
(localized in x1 – x2 and p1 + p2)
A measurement of x1 yields x2, just as a measurement of p1 gives
p2. But x2 and p2 don’t commute! ↔ [x, p] = i ħ
EPR’s conclusion
If (1) is false, then (2) is also false! Hence, (1) should be true:
quantum theory, although it allows for correct predictions, must be
incomplete. Measurements should just reveal pre-existing states,
which are not described by this incomplete theory.
Bohr’s reply
Bohr discusses complementarity, but his paper does not give sufficient
arguments to rule out the EPR program. (This story goes on with the
theorems by John Bell and experiments to violate Bell’s inequalities, and
GHZ-states etc.)
Quantum Optics
It is our purpose here to investigate these “spooky”
correlations by using electromagnetic fields. Our bright
beams of light have properties that are described just as
position and momentum observables. We begin by
describing these properties and then we will study them
in a specific system, the optical parametric oscillator
(OPO). Borrowing a line from an anonymous reviewer,
these systems are of “great interest to the quantum
information community, but also to a broad audience
interested in the latest progress on sophisticated optical
systems designed for quantum information applications”.
Field Quantization
Each mode is a harmonic oscillator, with the Hamiltonian
given the commutation relations
The electric field is:
Field Quadratures
The electric field can be decomposed as
And also as
X and Y are the field quadrature operators, satisfying
Thus,
Quantum Optics
Field quadratures behave just as position and momentum
operators. Thus, we can expect to observe phenomena
such as EPR-type correlations among optical fields.
Before we proceed, we notice that the uncertainty relation
sets a minimum bound on the product of the variances of
orthogonal quadratures. For coherent states the variances
are both equal to 1 (the so-called Standard Quantum
Limit – SQL). For squeezed states one variance is
smaller than 1, while the orthogonal quadrature
necessarily has excess noise.
Coherent x Squeezed
Y
Y
X
X
Noise Measurements
±
S.A.
D2
(Balanced) Homodyne Detection
ĉ
BS
b̂
d̂
D1
Â
If field b is strong, we can replace the operator by its mean value
If field b is the vacuum, we can obtain A’s intensity noise by measuring n+
The Optical Parametric
Oscillator (OPO)
New quantum light
from a classic system
Optical Parametric Oscillator (OPO)
OPO
Crystal
Idler
Pump
532 nm
~1064 nm
Signal
Optical Parametric Oscillator (OPO)
Optical Parametric Oscillator (OPO)
Let us describe classical properties of the system before we
analyze quantum properties. We’ll consider a Triply
Resonant OPO (TR-OPO) in a ring cavity (for simplicity).
a0in
a1out
ℓ
R=1
a0out
a2out
The hamiltonian
has three terms:
r0
r1
r2
R=1
Optical Parametric Oscillator (OPO)
The amplitudes will be given in photon flux (photons per
second). If we consider that the single pass gain is small, we
can approximate the equations for the amplitudes, for
propagation inside the crystal as:
And, for a round trip:
Optical Parametric Oscillator (OPO)
If djj is small, we can write:
where the total loss for
each mode is defined
Normalizing the detuning,
we have
Optical Parametric Oscillator (OPO)
A first solution of these equations is a1 = a2 = 0,
corresponding to operation below threshold. We are more
interested in above-threshold operation. Multiplying the
complex conjugate of the third equation by the second, we
have:

The intracavity pump power is easily obtained and we see it
is “clipped”: above-threshold it is always the same
Besides, for
, we also have
The classical equations are already signaling that the intensities
of signal and idler beams should be strongly correlated and that
the pump must be depleted.
Optical Parametric Oscillator (OPO)
From the first equation we can derive the threshold power,
given the intracavity pump field (a1 = a2 = 0)
An important parameter will be the ratio of incident power to
threshold power on resonance:
Substituting a2 in the first equation, we have
Optical Parametric Oscillator (OPO)
Since
and
We get
Solving for aj

Optical Parametric Oscillator (OPO)
This gives the photon flux. Considering, for the sake of the
argument, the frequency-degenerate case (w1=w2=w0/2), we
can obtain the total output power and the efficiency
Where hmax is the maximum efficiency leading to
We will see that the parameter x determines the maximum
squeezing in the above-threshold OPO.
Quantum properties of the OPO
We resume from the Hamiltonian and the master equation:
We will sketch the general method, which consists of choosing
a representation (quasi-probability distribution) and converting
the master equation into a set of stochastic differential
equations. Our choice is to use the Wigner function, in spite of
the fact that higher-order derivatives appear, which we simply
neglect…
Quantum properties of the OPO
The operators
are replaced by amplitudes
and the density operator is replaced by
Using the rules
Quantum properties of the OPO
We obtain
Neglecting higher-order derivatives, we have a Fokker-Planck
equation
Quantum properties of the OPO
Which is equivalent to a set of Langevin equations
The mean values in steady state are the same as in the classical
treatment. Since we will (typically) deal with intense fields, we
proceed by linearizing the fluctuations, neglecting products of
fluctuating terms:
Quantum properties of the OPO
Defining
with
We get
Quantum properties of the OPO
The subspace related to the subtraction of the fields decouples
from the sum and the pump fluctuations. However, q- does not
have any decay term, thus the solutions are not strictly stable.
As a matter of fact, there is phase diffusion and the subtraction
of the phases is unbounded. Nevertheless, this is a slow
process and we will be interested in measuring phases with
respect to the phase of the mean field (in other words, we will
follow “adiabatically” the diffusion).
Instead of solving these equations in the time domain, we look
instead in the frequency domain.
Wiener-Khintchine theorem
• We wish to study a stationary random process z(t)
• We can try
where S(w) would represent the strength of fluctuations associated to a
Fourier component of z(t).
• However, z(t) is nonzero for t  , so the above definition is not
mathematically sound.
• This problem was removed by Wiener and Khintchine who noticed
that the auto-correlation function
is well defined for a large number of functions z(t), approaching zero
when    if z(t) = 0.
From Optical Coherence and Quantum Optics, L. Mandel e E. Wolf
Wiener-Khintchine Theorem
The auto-correlation function and the spectral density (or
power spectrum) are related by Fourier transforms.
Experimentally
Spectrum Analyzer
Time series of
photocurrent
measurements
Auto-correlation
function
WienerKhintchine
theorem
Power
spectrum
Singularities in the power spectrum
• If z  0, then ()  |z|2 for   . This causes singularities to
appear in S(w).
• A well-behaved function can be obtained again by writing z(t) = z +
z0(t), where z0(t) has a zero mean. For z0(t), we have
0(0) gives the variance of
z(t)
Singularity removal in S(w)
Experimentally
light
Our Detector
z(t)
z(t)
DC output
DC Filter
z0(t)
HF output
Oscilloscope
Spectrum
Analyzer
Wait: not so fast!
What do we really measure? How about the commutation
relations? How do we define the SQL?
i(t)
Field operators depend on time, leading to slight changes in the
commutation relations:
Wait: not so fast!
In the frequency domain,
In every detection system, we have a finite bandwidth, so we
measure
OK, back to the OPO
We concentrate on the subtraction subspace:
The Fourier transform is
giving
OK, back to the OPO
The output field fluctuations are
Finally,
OK, back to the OPO
The subtraction subspace gives a minimum uncertainty product,
for D = 0.
Twin beams!