Quantum Optics
Quantum optics is the study of the quantum theory of light at low energies and interactions with bound electronic systems. We discuss physically achievable states of the electromagnetic field, including squeezed states and single photons states, as well a
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Quantum Opt 14. Quantum Optics
14.1 Quantum Fields .................................... 1053 14.2 States of Light ...................................... 1055 14.3 Measurement ....................................... 1058 14.3.1 Photon Counting .......................... 1059 14.3.2 Homo-/Heterodyne Detection ........ 1060 14.4 Dissipation and Noise ........................... 1061 14.4.1 Quantum Trajectories.................... 1063 14.4.2 Simulating Quantum Trajectories.... 1066 14.5 Ion Traps ............................................. 1066 14.6 Quantum Communication and Computation ................................. 1070 14.6.1 Linear Optical Quantum Computing 1072 References .................................................. 1077
14.1 Quantum Fields Quantum optics is the study of the quantum theory of light at low energies. It is a special case of quantum electrodynamics (QED) for the electromagnetic field with frequencies ranging from microwave to ultraviolet and electrons bound in atomic systems. We consider first of all the free electromagnetic field described classically by the vector potential A(x, t). As this obeys the wave equation it may be expanded in terms of plane wave states with two orthogonal transverse polarisations. If we assume the field is bound in a box of volume V , with Dirichlet boundary conditions, we find that A(x, t) =
2o ωn V n,ν i(k .x−ω ) −i(kn .x−ωn t) ∗ nt α × en,ν e n αn,ν ,(14.1) n,ν + e
where en,ν are two orthonormal polarisation vectors (ν = 1, 2) which satisfy kn · en,ν = 0, as required for a transverse field, and the frequency is given by the dispersion relation ωn = c|kn |. The positive- and
negative-frequency Fourier amplitudes are, respectively, ∗ . The corresponding electric field is given αn,ν and αm,ν by E(x, t) =i
n,ν
ωn 2o V
∗ .(14.2) × en,ν ei(kn .x−ωn t) αn,ν − e−i(kn .x−ωn t) αn,ν Canonical quantisation [14.1] is now carried out by promoting these Fourier amplitudes to the operators ∗ → a† with bosonic commutation αn,ν → an,ν , αn,ν n,ν relations †
[an,ν , an ,ν ] = δνν δnn
(14.3)
with all other commutations relations zero (we are assuming Coulomb gauge quantisation). The canonical quantisation procedure then gives the Hamiltonian for the free field as † H= ωk ak ak , (14.4) k
Part D 14
Quantum optics is the study of the quantum theory of light at low energies and interactions with bound electronic systems. We discuss physically achievable states of the electromagnetic field, including squeezed states and single photons states, as well as schemes by which they may be generated and measured. Measured systems are necessarily open systems and we discuss how dissipation, noise and decoherence is treated in quantum optics in terms of Markov master equations, quantum trajectories and quantum stochastic differential equations. Quantum optics has recently proved a valuable test-bed to implement new communication protocols such as teleportation and quantum information processing and we discus some of these new schemes including
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