PHYSICAL REVIEW A VOLUME 47, NUMBER 1 JANUARY 1993 Quantum-noise reduction in a driven cavity with feedback A. Liebman and G. J. Milburn Department of Physics, University of Queensland, St. Lucia $0/8, Australia (Received 5 June 1992) We show that amplitude-squeezed states may be produced by driving a feedback-controlled cav- ity with a coherent input signal. The feedback controls the transmissivity of one output from the cavity and is essentially equivalent to nonlinear absorption. The cavity electively acts as a nonlin- ear reflector. Hence, amplitude-squeezed states with arbitrarily strong coherent intensities can be obtained. PACS number(s): 42.50. Dv, 42. 50.Ar, 42. 50. Lc I. INTRODUCTION Feedback has long been used to stabilize the operation of optical cavities and lasers. Recently Yamamoto and co-workers [1] demonstrated that feedback could be used to reduce the intensity fluctuations in the light emitted by a semiconductor laser. In their experiment, the light from the laser illuminated a photodetector and the result- ing photocurrent was fed back to the injection current of the laser. The in-loop field, that is, the light between the laser and the photodetector, does exhibit sub-shot-noise statistics, however, it is difBcult to extract this field to exploit the reduced noise properties in applications. To overcome this problem Wiseman and Milburn [3] pro- posed a model in which the transmissivity of one output coupler from a laser cavity is controlled by a feedback circuit similar to that of Yamamoto and co-workers. The light leaving the cavity through the feedback-controlled output coupler falls on a photodetector and the result- ing current used to control the transmissivity at the out- put. However, unlike the scheme of Yamamoto and co- workers, this scheme includes another cavity output port from which a sub-shot-noise light field may be extracted. In both the laser feedback schemes described above the light produced has reduced intensity noise but, as is typical of lasing devices, the noise is phase independent. Squeezed states are a more general class of states of the field for which the noise is very phase dependent. Such states can exhibit either reduced intensity fluctuations or reduced phase fluctuations. It is the purpose of this paper to show that the feedback model of Wiseman and Mil- burn can be used to produce squeezed states. To achieve this we consider an empty, externally driven feedback- controlled cavity. The cavity thus becomes a nonlinear reflector as far as the input light is concerned. The non- linearity, in fact, appears as nonlinear absorption. The cavity, of course, may be designed to operate at whatever the laser input frequency is. This flexibility is one of the great attractions of producing squeezed states in this way. One does not need to rely on the frequency restrictions of a particular nonlinear optical material. Feedback enables one to engineer the nonlin- earity at any frequency. The other advantage comes from the fact that the squeezing is induced on a field which has a large coherent amplitude. The theoretical description of the model is based on a model of feedback control of a travelling wave proposed by Shapiro and co-workers [2], and the general theory of photodetection from an optical cavity, given by Srinivas and Davies [4]. II. QUANTUM THEORY OF FEEDBACK- CONTROLLED CAVITY TRANSMISSIVITY In Fig. 1 we indicate schematically the feedback sys- tem. The cavity has two outputs. The transmissivity of one output is controlled by a current derived from a photodetector illuminated by the light leaving through that port. The current controlled beam splitter may be realized in a number of ways using acousto-optic mod- ulators and perhaps polarizing filters. A more detailed discussion of the experimental realization will be given in a future paper. The other output of the cavity is illu- minated by a strong coherent field. As we now show, the input field sees a cavity containing an effective nonlinear absorber due to the feedback, where the ouput transmis- sivity is increased with increasing detection rate at the photodetector. The zero-feedback damping rates at the input coupler and feedback controlled coupler are p2 and pq, respectively. The count rate at the photodetector is determined by a count superoperator A(t), which depends on the counting history through (2. 1) sn FIG. 1. A schematic representation of the cavity with feedback model where p. d. denotes the photodetector. The modes b;„and b „t denote the input and output fields, respec- tively. 47 634 1993 The American Physical Society