Inserting a resolution of the identity between two operators turns a matrix element of their product into a phase-space integral. For unbounded operators the product need not exist on the vector at hand, while the integral may still converge. We follow this gap through the chain $e^{μa^2}e^{λa^{\dagger2}}$ of two-photon exponentials, where it can be traced completely. Five regions of parameter space appear: where the Bochner conditions of the strong formulation hold, where the product exists as a vector, where it exists as a sesquilinear form, where the coherent-state integral converges, and where the Fock-space series converges. The first three are strictly nested inside the intersection of the last two, and these two overlap without either containing the other. A single Hermite-type function agrees with the matrix element where it exists and with the integral and the series wherever they converge, so the integral carries the analytic continuation of the matrix element beyond the domain of the product and beyond the convergence of its Fock series. On the real slice of parameter space, an iterated form of the integral also reaches every point where the series converges. The analysis rests on a weak composition lemma whose hypotheses concern only the bra, the ket and the resolving family. For bounded chains the same identity is unconditional, and worked examples of that kind are given in a companion paper.
Inserting a resolution of the identity between two operators turns a matrix element of their product into a phase-space integral. For unbounded operators the product need not exist on the vector at hand, while the integral may still converge. We follow this gap through the chain eμa2eλa†2 of two-photon exponentials, where it can be traced completely. Five regions of parameter space appear: where the Bochner conditions of the strong formulation hold, where the product exists as a vector, where it exists as a sesquilinear form, where the coherent-state integral converges, and where the Fock-space series converges. The first three are strictly nested inside the intersection of the last two, and these two overlap without either containing the other. A single Hermite-type function agrees with the matrix element where it exists and with the integral and the series wherever they converge, so the integral carries the analytic continuation of the matrix element beyond the domain of the product and beyond the convergence of its Fock series. On the real slice of parameter space, an iterated form of the integral also reaches every point where the series converges. The analysis rests on a weak composition lemma whose hypotheses concern only the bra, the ket and the resolving family. For bounded chains the same identity is unconditional, and worked examples of that kind are given in a companion paper.
A resolution of the identity due to canonical coherent states is often proven in the weak operator topology. However, such a resolution with an integral symbol is typically supposed to hold in the strong operator topology associated with the framework of the spectral theorem. We provide an elementary proof of the strong convergence for the resolution of the identity due to canonical coherent states starting with a mostly familiar setup. Further, we enjoy a different proof and show that the relevant uniform limit does not exist.
When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation T of a real algebraic Lie group G. This requires defining an inner product on the Hilbert space mathbbH that carries the representation T. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of T, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet (mathfrakFmathbbHD, mathbbHD, mathfrakFmathfrakHD) of Bargmann spaces. Coherent state wave functions -- the elements of mathfrakFmathbbHD and of mathfrakFmathbbHD -- are used to define the inner product on mathbbHD in a way that simplifies the calculation of the overlaps. This inner product and the group action Gamma of G on mathfrakFmathbbHD are used to formulate expressions for the matrix elements of T with coefficients from the given subrepresentation.\n\nThe process of finding an explicit expression for Gamma relies on matrix factorizations in the complexification of G even though the representation T does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and Gamma action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in mathbbH. The scalar and vector coherent state methods will both be applied to Sp(n) and Sp(n,mathbbR).
This paper develops a unified kernel-based method for computing exact matrix elements of composite quantum operations by inserting resolutions of the identity after individual operators or between successive factors of an operator chain. A single insertion gives an exact scalar identity, and nested insertions between the factors give multiple integral identities for chains of closed operators under explicit Bochner-integrability conditions. The procedure is carried through on polynomial and analytic functions of the ladder operators, on one- and two-mode Gaussian unitaries, and on three composite operations: a Kerr phase followed by squeezing or by a cubic phase gate, a two-mode squeezer followed by a beam splitter, and a heralded cat-state circuit with two beam splitters. At rational Kerr phases the Kerr-squeezed amplitudes reduce to finite sums of squeezed coherent-state amplitudes weighted by quadratic Gauss sums, with an explicit error bound at the remaining phases; the two-mode network gives a bivariate Hermite polynomial; and the heralded circuit gives closed forms for the amplitudes, the heralding probabilities and their extension to an arbitrary passive network fed by one squeezed source. The examples are set against the reach of four established techniques: operator ordering, the Berezin symbol calculus, Lie disentangling and the symplectic formalism. Each technique reaches a subset of the examples directly, while the insertion procedure reaches all of them under the same hypotheses. A comparison with direct Fock summation identifies the regime in which the finite Kerr sums are efficient, and the choice of resolution at each boundary of a chain follows a selection rule set by the action of the adjacent factor. Each closed form agrees with an independent numerical computation.
Let Γ(H) be the boson Fock space over a finite dimensional Hilbert space H. It is shown that every gaussian symmetry admits a Klauder-Bargmann integral representation in terms of coherent states. Furthermore, gaussian symmetries, gaussian states and second quantization contractions, all of these operators belong to a weakly closed, selfadjoint semigroup E2(H) of bounded operators in Γ(H). This yields, a new parametrization of gaussian states, which is a very fruitful alternative to the customary parametrization by position-momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries:
(i) every gaussian state ρ admits a factorization ρ=Z1†Z1, where Z1 is an element of E2(H) and has the form Z1=cΓ(Λ)exp∑r=1nλrar+∑r,s=1nαrsaras on the dense linear manifold generated by all exponential vectors, Λ being a positive operator in H, ar,1≤r≤n are the annihilation operators corresponding to the n different modes in Γ(H), λr∈C and [αrs] is a symmetric matrix in Mn(C);
(ii) an explicit particle basis expansion of an arbitrary mean zero pure gaussian state vector along with a density matrix formula for a general gaussian state in terms of its E2(H)-parameters;
(iii) an easy test for the entanglement of pure gaussian states and a class of examples of pure n-mode gaussian states which are completely entangled;
(iv) tomography of an unknown gaussian state in Γ(Cn) by the estimation of its E2(Cn)-parameters using O(n2) measurements with a finite number of outcomes.
Operating in Laplace language and making use of a representation based on photon-number states, we find the exact solution for the density operator that belongs to the Jaynes-Cummings model with cavity damping. The detuning parameter is set equal to zero and the optical resonator does not contain any thermal photons. It is shown that the master equation for the density operator can be replaced by two algebraic recursion relations for vectors of dimension 2 and 4. These vectors are built up from suitably chosen matrix elements of the density operator. By performing an iterative procedure, the exact solution for each matrix element is found in the form of an infinite series. We demonstrate that all series are convergent and discuss how they can be truncated when carrying out numerical work. With the help of techniques from function theory, it is proved that our solutions respect the following conditions on the density operator: conservation of trace, Hermiticity, convergence to the initial state for small times, and convergence to the ground state for large times. We compute some matrix elements of the density operator for the case of weak damping and find that their analytic structure becomes much simpler. Finally, it is shown that the exact atomic density matrix converges to the state of maximum von Neumann entropy if the time, the square of the initial electromagnetic energy density, and the inverse of the cavity-damping parameter tend to infinity equally fast. The initial condition for the atom can be chosen freely, whereas the field may start from either a coherent or a photon-number state.