INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S28–S31 doi:10.1088/1464-4266/7/3/004 A quantum optical shutter Samuel L Braunstein Computer Science, University of York, York YO10 5DD, UK Received 19 May 2004, accepted for publication 16 December 2004 Published 1 March 2005 Online at stacks.iop.org/JOptB/7/S28 Abstract A time-dependent dielectric cmedium is used to model a time-varying beam-splitter inside a cavity. The time-varying boundary conditions smoothly evolve from a highly transmitting to a highly reflecting beam-splitter. These approximately correspond to a transformation from a single cavity to a pair of cavities. Quantum field-theoretic calculations show that such a smooth change yields non-singular evolution of the field. However, it predicts a production of photons up to frequencies comparable to the rate of change of the transition. We find that a time-varying beam-splitter operating at optical frequencies would produce an observable number of photons. Keywords: time-varying dielectric, beam-splitter, optical shutter In this paper we deal with a model problem in a second- quantized field theory: an ideal laser cavity with a variable reflectivity mirror at its centre. Such a partial mirror is called a beam-splitter. Absorption is neglected. We are interested in changing the reflectivity of the central ‘partition’ in order to study the evolution from a single cavity to a pair of disconnected cavities. The outer mirrors remain perfectly reflecting. We start by reviewing the quantum theory of static beam- splitters and give a heuristic explanation for why time-varying beam-splitters might be expected to yield photon production. Then we consider a field-theoretic formulation of a time- varying beam-splitter. Static beam-splitters Earlier treatments of the quantized time-stationary beam- splitter [1–5] all rely on the behaviour of the classical, time- stationary beam-splitter [6]. In this case, the beam-splitter is a two-input, two-output device that obeys relations like ˆ a out (t ) = cos θ ˆ a in (t ) +e iϕ sin θ ˆ b in (t ) ˆ b out (t ) = cos θ ˆ b in (t ) − e −iϕ sin θ ˆ a in (t ), (1) where ˆ a and ˆ b are the annihilation operators of the incoming and outgoing modes, t denotes time and where the transmission and reflection amplitude coefficients are cos θ and e iϕ sin θ , respectively. One of these papers [4] introduces an inner product to confirm that the modes are orthogonal in order to derive the above input–output relations (or so-called Bogoliubov transformations), though their choice of inner product appears to be somewhat ad hoc. In this paper we seek to incorporate dynamics into the beam-splitter element by allowing its transmission and reflection coefficients to change in time in some prescribed manner [7]. We show that the above equations are no longer sufficient as they stand. One effect of these time-varying boundary conditions is particle creation. A heuristic way to see that particle creation might result from a time-varying beam-splitter is to consider the effect of a time-varying reflection coefficient on equation (1). The simplest time-dependence we might add is harmonic θ(t ) = t . Decomposing the output modes into their frequency components, we find two predominant effects: each incoming frequency ω produces a pair of sidebands at ω ± ; negative frequency contributions appear when ω<. The former effect is the expected sideband modulation from the time- varying transmission. The latter is a purely quantum effect. By making the replacement ˆ a(−ω) →ˆ a † (ω), the creation of quanta is expected for frequencies up to the beam-splitter modulation frequency. We confirm this qualitative behaviour for our model of a time-varying beam- splitter. 1464-4266/05/030028+04$30.00 © 2005 IOP Publishing Ltd Printed in the UK S28