Optimal amplitude quadrature control of an optical squeezer using an integral LQG approach S. Z. Sayed Hassen and I. R. Petersen Abstract— Squeezed states of light possess an asymmetric noise distribution whereby the uncertainty in one quadrature is less than the standard quantum noise limit. These states can be generated through a nonlinear optical process in an optical parametric oscillator (OPO). In this paper, we model such an optical squeezer and design an LQG controller which includes integral action to optimize the level of squeezing achieved in one of the quadratures of the fundamental optical field. I. I NTRODUCTION Nonlinear processes arising in quantum optics have proved useful in checking the counter-intuitive predictions of the theory of quantum mechanics. One of the simplest nonlinear optical processes, second harmonic generation (SHG), has been extensively studied and found to be a potential source of squeezed states of light. Squeezed light has an asymmetric quadrature noise distribution, with one quadrature having less noise than the standard quantum noise limit (SQL). This feature of squeezed states makes them particularly useful for applications such as gravitational wave detection (see [1]), where the use of squeezed light allows for a reduction of the laser power necessary to achieve a given signal to noise ratio. Other promising applications include quantum cryptography and optical communications. One of the main problems associated with the generation of squeezed states is due to what is known in the physics literature as “dephasing”; see [2]. Dephasing adds considerable phase noise to both the fundamental and second-harmonic fields inside an optical resonator cavity, and if unaccounted for, can completely destroy any squeezing achieved. This noise (also referred to as laser phase noise) occurs predominantly at low frequencies and can be modelled as a constant DC offset disturbance acting on the system. In this paper, we address the problem of optimally squeez- ing a specific quadrature of light using an optimal parametric oscillator (OPO). We apply integral LQG control to an OPO driven by two optical fields ˆ A in and ˆ B in ; see Fig. 1. We include integral action as the standard LQG technique is not able to deal with the type of disturbance present in our application. One of the optical input fields ( ˆ B in ) is controlled by a mirror connected to a piezoelectric actuator. The actuator adjusts the phase quadrature of the optical field, thus minimizing the effect of phase noise and other classical sources of noise. This in turn regulates the phase angles of the fundamental and second-harmonic intra-cavity This work was supported by the Australian Research Council S. Z. Sayed Hassen and I. R. Petersen are with the School of Engineer- ing and Information Technology at the University of New South Wales, Canberra, ACT 2600, Australia. sayed.hassen@gmail.com fields, allowing for squeezing of light in a spatial quadrature. The available measurement is the phase quadrature of the output second harmonic field ˆ B out which is measured using the homodyne detection method; see [3]. One important feature of our approach is that we propose a control law that minimizes the noise in one quadrature of the fundamental output field using both measurement and actuation on the second harmonic field. The complete model we use for the controller design and subsequent simulations is derived from the linearized quantum dynamics of the OPO (optical subsystem) and from the model of a piezoelectric actuator (mechanical subsystem) typically found in a quantum optics laboratory. A similar control design approach to the one presented here was investigated in [4] where the laser phase noise was modelled as (approximately) integrated white noise. Whilst this approximation of the phase noise was suitable for simulation purposes and to expose our ideas with regards to the specific problem, it turns out that the controller obtained in [4] is not appropriate in practice as it does not include integral action. Integral action is necessary to counteract any DC offset. The requirement to minimize the variance of the quadrature of the fundamental output field is reflected in an LQG cost functional and the need for integral action is included by modifying the control signals and the cost functional appropriately. Typical parameter values for an experimental squeezer are used for the controller design and we validate our design through simulation. II. THE OPTICAL PARAMETRIC AMPLIFICATION PROBLEM The optical system under consideration consists of a second-order nonlinear optical medium enclosed within an optical resonator [3]. Materials showing second-order non- linearities χ (2) have the ability to couple a fundamental field (f ) to a second harmonic field (2f ). This coupling forms the basis of operation of the OPO through nonlinear optical interaction and feedback resulting in the build-up of the waves in a process similar to that seen in a laser cavity. The output beams thus become quantum correlated producing squeezed light; ; see, e.g., [5], [6] for more details. Fig. 1 shows our setup used for the OPO control problem. The aim is to maximize the amplitude quadrature squeezing of the fundamental optical field observed at a given operat- ing point by optimally suppressing the different sources of noise feeding into the system. This aim is translated into a specific quadratic cost functional to define an LQG optimal control problem, the underlying assumption being that the 2010 IEEE International Conference on Control Applications Part of 2010 IEEE Multi-Conference on Systems and Control Yokohama, Japan, September 8-10, 2010 978-1-4244-5363-4/10/$26.00 ©2010 IEEE 286