Frequency Locking of an Optical Cavity using LQG Integral Control

arXiv:0809.0545v1 [quant-ph] 3 Sep 2008 Frequency Locking of an Optical Cavity using LQG Integral Control S. Z. Sayed Hassen, M. Heurs, E. H. Huntington, I. R. Petersen University of New South Wales at the Australian Defence Force Academy, School of Information Technology and Electrical Engineering, Canberra, ACT 2600, Australia∗ M. R. James Department of Engineering, Australian National University, Canberra, ACT 2600, Australia (Dated: October 23, 2018) Abstract This paper considers the application of integral Linear Quadratic Gaussian (LQG) optimal con- trol theory to a problem of cavity locking in quantum optics. The cavity locking problem involves controlling the error between the laser frequency and the resonant frequency of the cavity. A model for the cavity system, which comprises a piezo-electric actuator and an optical cavity is experi- mentally determined using a subspace identification method. An LQG controller which includes integral action is synthesized to stabilize the frequency of the cavity to the laser frequency and to reject low frequency noise. The controller is successfully implemented in the laboratory using a dSpace DSP board. PACS numbers: 42.60.Da, 42.60.Fc, 42.60.Mi, 42.62.Eh ∗Electronic address: s.sayedhassen@adfa.edu.au 1 I. INTRODUCTION Many future technologies will be based on quantum systems manipulated to achieve engineering outcomes [1, 2]. Quantum feedback control forms one of the key design methodologies that will be required to achieve these quantum engineering objectives [3, 4, 5, 6, 7, 8, 9, 10]. Examples of quantum systems in which quantum control may play a key role include the quantum error correction problem (see [11]) which is central to the development of a quantum computer and also important in the problem of developing a repeater for quantum cryptography systems, spin control in coherent magnetometry (see [12]), control of an atom trapped in a cavity (see [8]), the control of a laser optical quantum system (see [13]), control of atom lasers and Bose Einstein Condensates (see [14]), and the feedback cooling of a nanomechanical resonator (see [15]). Attention is now turning to more general aspects of quantum control, particularly in the development of systematic quantum control theories for quantum systems. For example in [8] and [16] it was shown that the linear quadratic Gaussian (LQG) optimal control approach to controller design can be extended to linear quantum systems. Also, in [17], it was shown that the H∞optimal control approach to controller design can be extended to linear quantum systems. These theoretical results indicate that systematic optimal control methods of modern control theory have the potential of being applied to quantum systems. Such modern control theory methods have the advantage that they are strongly model based and provide systematic methods of designing multivariable control systems which can achieve excellent closed loop performance and robustness. Experimental demonstrations of some of these theoretical results now appear viable. For example, Ref. [18] presents the first experimental demonstration of the design and implementation of a “coherent controller” from within this formalism. One particular systematic approach to control is the LQG optimal control approach to design. LQG optimal control is based on a linear dynamical model of the plant being con- trolled which is subject to Gaussian white noise disturbances; e.g, see [19]. In LQG optimal control, a dynamic linear output feedback controller is sought to minimize a quadratic cost functional which encapsulates the performance requirements of the control system. A fea- ture of the LQG optimal control problem is that its solution involves the use of a Kalman Filter which provides estimates of the internal system variables. Furthermore, in many ap- 2 plications integral action is required in order to overcome low frequency disturbances acting on the system being controlled. This issue is addressed here by using a version of LQG optimal control referred to as integral LQG control which forces the controller to include integral action; see [20]. In this paper, we consider the application of systematic methods of LQG optimal control to the archetypal quantum optical problem of locking the resonant frequency of an opti- cal cavity to that of a laser. Homodyne detection of the reflected port of a Fabry-Perot cavity is used as the measurement signal for an integral LQG controller. In our case, the linear dynamic model used is obtained using both physical considerations and experimen- tally measured frequency response data which is fitted to a linear dynamic model using subspace system identification methods; e.g., see [21]. The integral LQG controller design is discretized and implemented on a dSpace digital signal processing (DSP) system in the laboratory and experimental results were obtained showing that the controller has been ef- fective in locking the optical cavity to the laser fre

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