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Journal of Optics B Quantum and Semiclassical Optics

Symplectic groups in quantum optics

Alfred Wünsche

Published Apr 1, 2000• 40 citations
Publisher

Abstract

Starting from the symplectic Lie groups Sp (2 n , ) of an n -mode system in classical and quantum mechanics we discuss, in particular, Sp (2, )~ SO (2,1) for a single mode and Sp (4, )~ SO (3,2) with some of its important subgroups such as SU (1,1)~ Sp (2, ) and SU (2). We then look at their applications in quantum optics (squeezing, phase states, beamsplitter and polarization). We explicitly construct the root scheme of the Lie algebra sp (4, ) in two different bases as it is an important means for the investigation of the group properties and the construction of the irreducible representations. Apart from the well known realizations of Sp (2, ) by single-mode and two-mode systems, we discuss some other realizations of this group by fractal combinations of boson operators.

DOI
10.1088/1464-4266/2/2/302

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Starting from the symplectic Lie groups Sp (2 n , ) of an n -mode system in classical and quantum mechanics we discuss, in particular, Sp (2, )~ SO (2,1) for a single mode and Sp (4, )~ SO (3,2) with some of its important subgroups such as SU (1,1)~ Sp (2, ) and SU (2). We then look at their applications in quantum optics (squeezing, phase states, beamsplitter and polarization). We explicitly construct the root scheme of the Lie algebra sp (4, ) in two different bases as it is an important means for the investigation of the group properties and the construction of the irreducible representations. Apart from the well known realizations of Sp (2, ) by single-mode and two-mode systems, we discuss some other realizations of this group by fractal combinations of boson operators.
Four-mode squeezed states: two-field quantum systems and the symplectic group Sp(4,R)\mathrm {Sp}(4,{\mathbb {R}})Sp(4,R)
Abstract We construct the four-mode squeezed states and study their physical properties. These states describe two linearly-coupled quantum scalar fields, which makes them physically relevant in various contexts such as cosmology. They are shown to generalise the usual two-mode squeezed states of single-field systems, with additional transfers of quanta between the fields. To build them in the Fock space, we use the symplectic structure of the phase space. For this reason, we first present a pedagogical analysis of the symplectic group Sp(4,R)\mathrm {Sp}(4,{\mathbb {R}})Sp(4,R) Sp ( 4 , R ) and its Lie algebra, from which we construct the four-mode squeezed states and discuss their structure. We also study the reduced single-field system obtained by tracing out one of the two fields. This procedure being easier in the phase space, it motivates the use of the Wigner function which we introduce as an alternative description of the state. It allows us to discuss environmental effects in the case of linear interactions. In particular, we find that there is always a range of interaction coupling for which decoherence occurs without substantially affecting the power spectra (hence the observables) of the system.
In this paper, we show the relation between sp(4,R)sp(4,\mathbb{R})sp(4,R), the Lie algebra of the symplectic group, and the elements of Sp(4,R)Sp(4,\mathbb{R})Sp(4,R). We use this result to obtain some special cases of symplectic matrices relevant to the study of squeezed states. In this regard, we provide some applications in quantum mechanics and analyze the squeezed polymer states obtained from the polymer representation of the symplectic group. Remarkably, the polymer's dispersions are the same as those obtained for the squeezed states in the usual representation.
A quick review of some Lie algebras related to well-known groups is given. We start with the Heisenberg-Weyl algebra and after the definitions of the Fock states we give the definition of the coherent state of this group. This is followed by the exposition of the SU(2) and SU(1,1) algebras and their coherent states. From there we go on to describe the binomial states and their extensions as the finite dimensional pair coherent states and their nonlinear versions as realizations of the SU(2) group. This is followed by considering the negative binomial states, the single mode and two-mode squeezed states and their variants as realizations of the SU(1,1) group. Generation schemes based on physical systems are considered for some of these states.
It is shown that the SU(1,1)-like and SU(2)-like two-photon coherent states can be combined to form a O(3,2)-like two-photon states. Since the O(3,2) group has many subgroups, there are also many new interesting new coherent and squeezed two-photon states. Among them is the two-photon sheared state whose symmetry property is like that for the two-dimensional Euclidean group. There are now two-phonon coherent states which may exhibit symmetries not yet observed for photons, including sheared states. Let us note that both SU(1,1) and S(3,2) are isomorphic to the symplectic groups Sp(2) and Sp(4) respectively, and that symplectic transformations consist of rotations and squeeze transformations.
We present an explicit construction of coherent states for an arbitrary irreducible representation of the unitary symplectic group USp (4). Three different families of coherent states are obtained, corresponding to the subgroups U (1) × U (1), U (2) and SU (2) × SU (2). The symplectic structure on the manifold of coherent states is obtained, and canonical coordinates are used to express the classical limit of quantum observables. One of the families is seen to provide a trivial classical limit.
The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space S q,p ≊ 2 . In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space S ϕ,I corresponds to the multiply connected punctured plane 2 – {0}, on which the 3-dimensional symplectic group Sp(2, ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U (1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator K o of U (l) represents the classical action variable I . It has the possible spectra n + k, n = 0,1,...; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK 0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K 0 etc. of Sp (2, ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.