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Fidelity between two Hamiltonians

J. M. Zhang

Published Mar 1, 2016• 0 citations
OpenAlex

Abstract

Here N is the size of the Hilbert space. If g1 is very close to g2, one might expect that ψ (1) i ∼ ψ (2) i for 1 ≤ i ≤ N . That is, the eigenstates of H(g1) are not so much mixed to form the eigenstates of H(g2). However, if g1 is far away from g2, say, they are on the opposite side of each other to some critical point, one would expect each state in the second spectrum is a superposition of “a lot” (to be quantified somehow below) of states in the first spectrum. Intuitively, we say, the spectrum is greatly twisted as g is varied from g1 to g2. We want to characterize the extent of twisting. The strategy is like this. We want to find a one-to-one mapping between the two spectra, such that the sum of the overlaps inside each pair is maximized. That is, we want

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Here N is the size of the Hilbert space. If g1 is very close to g2, one might expect that ψ (1) i ∼ ψ (2) i for 1 ≤ i ≤ N . That is, the eigenstates of H(g1) are not so much mixed to form the eigenstates of H(g2). However, if g1 is far away from g2, say, they are on the opposite side of each other to some critical point, one would expect each state in the second spectrum is a superposition of “a lot” (to be quantified somehow below) of states in the first spectrum. Intuitively, we say, the spectrum is greatly twisted as g is varied from g1 to g2. We want to characterize the extent of twisting. The strategy is like this. We want to find a one-to-one mapping between the two spectra, such that the sum of the overlaps inside each pair is maximized. That is, we want
The Dicke model describes an ensemble of N identical two-level atoms (qubits) coupled to a single quantized mode of a bosonic field. The fermion Dicke model should be obtained by changing the atomic pseudo-spin operators by a linear combination of Fermi operators. The generalized fermion Dicke model is defined introducing different coupling constants between the single mode of the bosonic field and the reservoir, g1 and g2 for rotating and counter-rotating terms respectively. In the limit N → ∞, the thermodynamic of the fermion Dicke model can be analyzed using the path integral approach with functional method. The system exhibits a second order phase transition from normal to superradiance at some critical temperature with the presence of a condensate. We evaluate the critical transition temperature and present the spectrum of the collective bosonic excitations for the general case (g1 ̸ = 0 and g2 ̸ = 0). There is quantum critical behavior when the coupling constants g1 and g2 satisfy g1 + g2 = (ω0 Ω) 1 2, where ω0 is the frequency of the mode of the field and Ω is the energy gap between energy eigenstates of the qubits. Two particular situations are analyzed. First, we present the spectrum of the collective bosonic excitations, in the case g1 ̸ = 0 and g2 = 0, recovering the well known results. Second, the case g1 = 0 and g2 ̸ = 0 is studied. In this last case, it is possible to have a superradiant phase when only virtual processes are introduced in the interaction Hamiltonian. Here also appears a quantum phase transition at the critical coupling g2 = (ω0 Ω) 1 2, and for larger values for the critical coupling, the system enter in this superradiant phase with a Goldstone mode.
The cavity field spectrum of two atoms interacting with two-mode fields with different coupling constants(g_1≠g_2) was studied.By solving the eigen equation,the cavity field spectrum expression of the system was given out.Some numerical results for the fields in pure number state and the atoms in the upper level were presented initially.The effects of the relative coupling constant R=g_2/g_1 or photon number on the spectrum structure were analyzed.It is found that R plays a different role in the spectrum with field in vacuum state,weak state and strong state.The frequency of one mode field spectrum is tuned by the intensity of the other mode.The frequencies of two modes spectra can be changed by the relative coupling constant.
Abstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrix