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\title{Coupled Parametric Oscilators Proof of Concept in Radio Frequency}
\author{Leon Bello}
\affil{Affiliation not available}
\date{\today}
\maketitle
\section{Overview}
Optical Paramteric Oscillators (OPO) oscillate at a specific frequency by means of a non-linear interaction.
A pump input laser with frequency $\omega_p$ is converted through the non-linear interaction into a signal and idler frequencies, such that $\omega_p = \omega_i + \omega_s$. Both frequencies can resonate in the cavity (doubly-resonant) or just one (singly-resonant).
The light generated by the OPO is "squeezed" - one quadrature is attenuated and the other amplified, which is useful for various scientific purposes.
By coupling two oscillators together, we create more modes. Unlike regular oscillators, where usually only one mode can oscillate due to mode competition, paramteric oscillators can (and must) oscillate in multiple modes. Active mode-locking can be used to create a source for broadband squeezed-light.
Demonstration of these effects using optical components is difficult and time consuming, we use radio components to simulate the same behavior in Radio Frequency domain (RF). RF components are easy and quick to set up, and can capture the same behavior.
\section{Theoretical Background}
\subsection{Parametric Oscillator}
An optical parametric oscillator consists of an optical resonator and a non-linear medium, a pump wave is inserted into the resonator, the resonator is noisy and thus many frequencies are present. Through the non-linear interaction the pump wave amplifies resonant frequency-pairs that satisfy the conditions
\[\omega_p = \omega_s + \omega_i\]
\[\varphi_p = \varphi_s + \varphi_i\]
Where $\omega_j$ is the frequency of the wave and $\varphi_j$ is the phase.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/nloo2/nloo2}
\end{center}
\end{figure}
The optical resonator resonates in at least one of the generated frequencies. The non-linear interaction in the medium leads to amplitude gain for the signal and idler waves and attenuation of the pump wave (parametric amplification). If the amplification is enough to overcome the losses, the resonator will resonate in one of the generated frequencies (singly-resonant) or both (doubly-resonant).
A special case is when the idler and signal frequencies are degenerate.
\[\omega_s = \omega_i = \omega_p/2\]
Our setup is doubly resonant, meaning that the pump, signal and idler frequencies are all a multiple of the base frequency of the resonator.
\[\omega_0 = \frac{c}{nL}\]
The generated frequency can be tuned by changing the phase-matching conditions of the OPO, thus making the OPO a source of tunable radiation, also, as explained in the next section, the light produced is "squeezed".
\subsection{Quadrature Noise}
The motion of a Harmonic Oscillator can be described in phase space, i.e. the system's state is defined by its position and momentum.
A point in phase space, together with the time-evolution equations, completely determines the motion of the Harmonic Oscillator, however, due to noise, it is impossible to determine this point exactly - there is an amount of uncertainty as to where exactly the oscillator is in phase space.
In normalized phase space, the position and momentum only differ in phase and the curve follows a perfect circle.
The same analysis can be applied to signals - suppose we have some signal $f(t)$, the signal can be written in the same form, with a time-dependent "position" and "momentum", or "in-phase" and "quadrature" components, respectively.
\[f(t) = X(t) cos(\omega_c t) + Y(t) sin(\omega_c t)\]
Where $X(t)$ is the in-phase component, $Y(t)$ is the quadrature component and $\omega_c$ is the "carrier" frequency.
It is common convention to use a complex representation of the signal,
\[f(t) = A(t) e^{i \omega_c t}\]
Where the quadrature components are simply the real and imaginary parts of the complex amplitude $A(t)$.
As in the case of the harmonic oscillator, it is impossible to exactly determine $X(t)$ and $Y(t)$ for a given $t$ - there is an amount of uncertainty that can be due to classical or quantum noise.
The quadrature components usually satisfy the uncertainty relation,
\[\Delta X \Delta Y \geq 1 \]
States that minimize this relation are called coherent states. It is possible to decrease the uncertainty in one quadrature component at the expense of the other - these are called squeezed-coherent states.
We may write the generated light signal in the following form,
\[f(t) = A_s e^{i(\omega_s t + \varphi_s)} + A_i e^{i(\omega_i t + \varphi_i)} \]
In steady state near threshold,
\[ A \equiv A_s \approx A_i\]
Also, we may write the signal and idler frequencies in the form,
\[ \omega_{s, i} = \omega_p/2 \pm \delta \]
\[ \varphi \equiv \varphi_s = -\varphi_i \]
From the above equations we get,
\[f(t) = A e^{i\omega_p/2 t}(e^{i(\delta t + \varphi) } + e^{- i (\delta t + \varphi)}) = 2A cos(\delta t + \varphi) e^{i\omega_p/2 t} \]
The signal is squeezed - the quadrature component is attenuated and the in-phase component is amplified.
\subsection{Coupled Parametric Oscillators}
A system of two coupled harmonic oscillators has two modes of oscillation and the general oscillation of the system is a linear combination of the two modes. This phenomenon is called beats - it seems as if the energy is flowing from one oscillator to the other at a rate that depends linearly on the strength of the coupling.
Beats can form in any resonator that has more than one mode. Suppose we have two, one-dimensional resonators that are coupled together so energy from one flows into the other, we can write the evolution of the system in each cycle.
\[A_{n+1} = T A_n + R B_n\]
\[B_{n+1} = - R A_n + T B_n \]
If the system is lossless,
\[ T^2 + R^2 = 1\]
In Matrix form,
\[M =
\begin{pmatrix}
T & R \\
-R & T
\end{pmatrix}\]
With A (B) being the signal in loop A (B).
The eigen-states being,
\[\lambda_\pm = T \pm i R \]
\[v_\pm = \frac{1}{\sqrt{2}}
\begin{pmatrix}
1 \\
\pm i
\end{pmatrix} \]
In phasor notation,
\[ \lambda_\pm = e^{\pm i \theta} \]
\[ \tan(\theta) = \frac{R}{T} \]
As the system evolves, the eigen-states only accumulate phase.
\[ \varphi = \theta N\]
Where N is the number of cycles inside the resonator.
By substituting $N = t/\tau$, where $\tau$ is the round trip time in the cavity, and taking the time derivative we get the angular frequency of each mode,
\[ \Omega_\pm = \pm \theta/\tau \equiv \pm \Omega\]
The system will beat with this frequency, that depends on the coupling strength.
Demostration of the effect in optical resonators is difficult due to losses - some gain mechanism must be added to the system for the effect to be noticeable, but since different modes experience different gain, one mode will always dominate and the beats won't be noticeable.
Parametric gain offers a way to avoid this problem altogether, since it can only amplify frequency-pairs that satisfy the conditions above, the signal in each loop must have the same phase as the pump wave.
This can only be achieved by a equal combination of the two modes.
We may write the signal in the oscillator as a linear sum of the eigen-modes,
\[f(t) =
\begin{pmatrix}
A(t) \\
B(t)
\end{pmatrix}
= \alpha \pmb{v_+} + \beta \pmb{v_-} =
\begin{pmatrix}
\alpha e^{i \Omega t} + \beta e^{-i \Omega t} \\
i (\alpha e^{i \Omega t} - \beta e^{-i \Omega t})
\end{pmatrix} \]
A and B are in-phase with the pump only for $\alpha = \beta$, for which we get,
\[A(t) = A(0)cos(\Omega t)\]
\[B(t) = -B(0)sin(\Omega t)\]
\section{Experimental Setup}
Our oscillator comprised of a frequency mixer that we pumped using a signal generator, an attenuator and an amplifier to work near threshold, and a coupler to inspect the signal on a Spectrum Analyzer. The whole setup was connected onto itself to form a resonator, as shown in the figure. We pumped the mixer with AC voltage and some DC offset that allowed us to use the mixer as variable attenuator to work as close as possible to the threshold.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/rpo/rpo}
\end{center}
\end{figure}
The key component in the setup is the frequency mixer, acting as the non-linear amplifier. Two signals are applied to the mixer - the pump wave and a signal frequency provided by the noise inside the loop, producing a difference frequency idler wave (as well as other frequency components).
Coupling is currently done using another set of mixers that are used as variable attenuators, as shown in the figure. Changing the voltage applied to the mixers affects the strength of the coupling (linearly for small values).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/coupledrpo1/coupledrpo1}
\end{center}
\end{figure}
For weak coupling, the evolution matrix is the same as that of the matrix of coupled resonators above.
\[A_{n+1} = \alpha A_n + \beta B_n\]
\[B_{n+1} = - \beta A_n + \alpha B_n\]
Where $\alpha$ is the coupling coefficient and $\beta$ is the attenuation coefficient of the mixers. $\beta$ can be controlled by changing the applied voltage on the mixers.
\section{Results and Conclusions}
Unfortunately, no meaningful results were obtained until due date.
Both systems were built, but the uncoupled one produced very noisy results due to use of unsuitable amplifiers, and I was not yet able to get results from the coupled oscillators.
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