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arXiv:1704.02008

From Hamiltonians to complex symplectic transformations

Gianfranco Cariolaro, Gianfranco L. Pierobon

Published Apr 6, 2017• 0 citations
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Abstract

Gaussian unitaries are specified by a second order polynomial in the bosonic operators, that is, by a quadratic polynomial and a linear term. From the Hamiltonian other equivalent representations of the Gaussian unitaries are obtained, such as Bogoliubov and real symplectic transformations. The paper develops an alternative representation, called complex symplectic transformation, which is more compact and is comprehensive of both Bogoliubov and real symplectic transformations. Moreover, it has other advantages. One of the main results of the theory, not available in the literature, is that the final displacement is not simply given by the linear part of the Hamiltonian, but depends also on the quadratic part. In particular, it is shown that by combining squeezing and rotation, it is possible to achieve a final displacement with an arbitrary amount.

DOI
10.48550/arxiv.1704.02008

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Gaussian unitaries are specified by a second order polynomial in the bosonic operators, that is, by a quadratic polynomial and a linear term. From the Hamiltonian other equivalent representations of the Gaussian unitaries are obtained, such as Bogoliubov and real symplectic transformations. The paper develops an alternative representation, called complex symplectic transformation, which is more compact and is comprehensive of both Bogoliubov and real symplectic transformations. Moreover, it has other advantages. One of the main results of the theory, not available in the literature, is that the final displacement is not simply given by the linear part of the Hamiltonian, but depends also on the quadratic part. In particular, it is shown that by combining squeezing and rotation, it is possible to achieve a final displacement with an arbitrary amount.
Gaussian unitaries play a fundamental role in the field of continuous variables. In the general n mode, they may formulated by a second-order polynomial in the bosonic operators. Another important role related to Gaussian unitaries is played by the symplectic transformations in the phase space. The paper investigates the links between the two representations: the link from Hamiltonian to symplectic, governed by an exponential, and the link from symplectic to Hamiltonian, governed by a logarithm. Thus, an answer is given to the non-trivial question: which Hamiltonian produces a given symplectic representation? The complex instead of the traditional real symplectic representation is considered, with the advantage of getting compact and elegant relations. The application to the single, two, and three modes illustrates the theory.
Gaussian unitaries, generated by quadratic Hamiltonians, are fundamental in quantum optics and continuous-variable computing. Their structures correspond to symplectic (bosons) and orthogonal (fermions) groups, but physical realizations give rise to their respective double covers, introducing phase and sign ambiguities. The homogeneous (quadratic-only) case has been resolved through a parameterization constructed in a recent work [arXiv:2409.11628]. We extend the previous framework to inhomogeneous Gaussian unitaries parameterized by (M,z,Ψ)(M,z,Ψ)(M,z,Ψ). The Baker-Campbel-Hausdorff formula allows us then to factor any Gaussian unitary into a squeezing and a displacement transformation, from which we derive the group multiplication law.
Gaussian unitary transformations are generated by quadratic Hamiltonians, i.e., Hamiltonians containing quadratic terms in creations and annihilation operators, and are heavily used in many areas of quantum physics, ranging from quantum optics and condensed matter theory to quantum information and quantum field theory in curved spacetime. They are known to form a representation of the metaplectic and spin group for bosons and fermions, respectively. These groups are the double covers of the symplectic and special orthogonal group, respectively, and our goal is to analyze the behavior of the sign ambiguity that one needs to deal with when moving between these groups and their double cover. We relate this sign ambiguity to expectation values of the form ⟨0∣exp⁡(−iH^)∣0⟩\langle 0|\exp{(-i\hat{H})}|0\rangle⟨0∣exp(−iH^)∣0⟩, where ∣0⟩|0\rangle∣0⟩ is a Gaussian state and H^\hat{H}H^ an arbitrary quadratic Hamiltonian. We provide closed formulas for ⟨0∣exp⁡(−iH^)∣0⟩\langle 0|\exp{(-i\hat{H})}|0\rangle⟨0∣exp(−iH^)∣0⟩ and show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space. Our construction relies on an explicit parametrization of these two groups (metaplectic, spin) in terms of symplectic and orthogonal group elements together with a twisted U(1) group.
The aim of this chapter is to construct the quantum unitary propagator for Hamiltonians which are quadratic in position and momentum with time-dependent coefficients. We show that the quantum evolution is exactly solvable in terms of the classical flow which is linear. This allows to construct the metaplectic transformations which are unitary operators in L2(Rn)L^2(\mathbb R^n)L2(Rn) corresponding to symplectic transformations. Simple examples of such metaplectic transformations are the Fourier transform, which corresponds to the symplectic matrix J defined in (3.4) and the propagator of the harmonic oscillator, corresponding to rotations in the phase space. The main results of this chapter are computations of the quantum evolution operators for quadratic Hamiltonians acting on coherent states. We show that the time evolved coherent states are still Gaussian states which are recognized to be squeezed states centered at the classical phase space point (see Chap. 8). From these computations we can deduce most of properties concerning quantum quadratic Hamiltonians. In particular we get the explicit form of the Weyl symbols of the metaplectic transformations. These formulae are generalizations of the Mehler formula for the harmonic oscillator. Quadratic Hamiltonians are very important in quantum mechanics because more general Hamiltonians can be considered as non trivial perturbations of time dependent quadratic ones as we shall see in Chap. 5.
Gaussian quantum states of bosonic systems are an important class of states. In particular, they play a key role in quantum optics as all processes generated by Hamiltonians up to second order in the field operators (i.e. linear optics and quadrature squeezing) preserve Gaussianity. A powerful approach to calculations and analysis of Gaussian states is using phase-space variables and symplectic transformations. The purpose of this note is to serve as a concise reference for performing phase-space calculations on Gaussian states. In particular, we list symplectic transformations for commonly used optical operations (displacements, beam splitters, squeezing), and formulae for tracing out modes, treating homodyne measurements, and computing fidelities.
Abstract Quadratic bosonic Hamiltonians and their associated unitary transformations form a fundamental class of operations in quantum optics, modelling key processes such as squeezing, displacement and beam-splitting. Their Heisenberg-picture dynamics simplifies to linear (or possibly affine) transformations on quadrature operators, enabling efficient analysis and decomposition into optical gate sets using matrix operations. However, this formalism discards a phase, which, while often neglected, is essential for a complete unitary characterization. We present efficient methods to recover this phase directly from the vacuum-to-vacuum amplitude of the unitary, using calculations that scale polynomially with the number of modes and avoid Fock space manipulations. We reduce the general problem for time-dependent Hamiltonians to integration, and provide analytical results for key cases including time-independent Hamiltonians which are positive definite, passive, active or single-mode. Finally, we show that our results can be easily used to obtain the phase associated with any Gaussian state, be it mixed or pure.