Quantum information in bright colors. Paulo A. Nussenzveig. Instituto de Física - USP. Paraty – 2009. The Team. Antônio Sales Felippe Barbosa Jonatas César Luciano Cruz Paulo Valente. Katiúscia Cassemiro Alessandro Villar Marcelo Martinelli Paulo Nussenzveig. Lectures.
Paulo A. Nussenzveig
Instituto de Física - USP
|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 p2don’t commute! ↔ [x, p] = i ħ
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 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.)
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”.
Each mode is a harmonic oscillator, with the Hamiltonian
given the commutation relations
The electric field is:
The electric field can be decomposed as
And also as
X and Y are the field quadrature operators, satisfying
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.
(Balanced) Homodyne Detection
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)
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).
The hamiltonian has three terms:
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:
If djj is small, we can write:
where the total loss for each mode is defined
Normalizing the detuning, we have
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.
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
Solving for aj
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.
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…
The operators are replaced by amplitudes
and the density operator is replaced by
Using the rules
Neglecting higher-order derivatives, we have a Fokker-Planck equation
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:
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.
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.Wiener-Khintchine theorem
where S() would represent the strength of fluctuations associated to a Fourier component of z(t).
From Optical Coherence and Quantum Optics, L. Mandel e E. Wolf
The auto-correlation function and the spectral density (or power spectrum) are related by Fourier transforms.
Time series of photocurrent measurements
0(0) gives the variance of z(t)
Singularity removal in S()
What do we really measure? How about the commutation relations? How do we define the SQL?
Field operators depend on time, leading to slight changes in the commutation relations:
In the frequency domain,
In every detection system, we have a finite bandwidth, so we measure
We concentrate on the subtraction subspace:
The Fourier transform is
The output field fluctuations are
The subtraction subspace gives a minimum uncertainty product, for D = 0.