Bosonic density matrix

For femiions (especially) and bosons diere are additional problems. Let /Jbe one of the pemuitations of particle labels. Then the femiion density matrix has the symmetry... 

G. Gidofalvi and D. A. Mazziotti, Boson correlation energies via variational minimization with the two-particle reduced density matrix exact iV-representabihty conditions for harmonic interactions. Phys. Rev. A 69, 042511 (2004). 

Remarkably, when our general ME is applied to either AN or PN in Section 4.4, the resulting dynamically controlled relaxation or decoherence rates obey analogous formulae provided the corresponding density matrix (generalized Bloch) equations are written in the appropriate basis. This underscores the universality of our treatment. It allows us to present a PN treatment that does not describe noise phenomenologically, but rather dynamically, starting from the ubiquitous spin-boson Hamiltonian. 

So far, one can be much more successful in calculating a rate constant when one knows in advance that it exists, than in answering the question of whether it exists. A considerable breakthrough in this area was the solution of the spin-boson problem, which, however, has only limited relevance to any problem in chemistry because it neglects the effects of intrawell dynamics (vibrational relaxation) and does not describe thermally activated transitions. A number of attempts have been made to go beyond the two-level system approximation, but the basic question of how vibrational relaxation affects the transition from coherent oscillations to exponential decay awaits a quantitative solution. Such a solution might be obtained by numerical computation of real-time path integrals for the density matrix using the influence functional technique. 

Among the technical methods proper to the one dimensional geometry, one may cite the Bethe ansatz [19], the bosonization techniques [18], and, more recently, the Density-Matrix Renormalization Group (DMRG) method (20, 21] and a closely related scheme which is directy considered in this note, the Recurrent Variational Approach (RVA) [22, 21], The two first methods are analytical and the third one is numerical the RVA method is in between. 

As for the case of a bosonic bath, the starting point here is the TL approach and a TL QME based on a second-order perturbation theory in the molecule-lead coupling was developed for the reduced density matrix p(t) of the molecule [38,65]... 

The main goal in the development of mixed quantum classical methods has as its focus the treatment of large, complex, many-body quantum systems. While applications to models with many realistic elements have been carried out [10,11], here we test the methods and algorithms on the spin-boson model, which is the standard test case in this field. In particular, we focus on the asymmetric spin-boson model and the calculation of off-diagonal density matrix elements, which present difficulties for some simulation schemes. We show that both of the methods discussed here are able to accurately and efficiently simulate this model. 

The chapter is organized as follows The quantum-classical Liouville dynamics scheme is first outlined and a rigorous surface hopping trajectory algorithm for its implementation is presented. The iterative linearized density matrix propagation approach is then described and an approach for its implementation is presented. In the Model Simulations section the comparable performance of the two methods is documented for the generalized spin-boson model and numerical convergence issues are mentioned. In the Conclusions we review the perspectives of this study. 

In this section we present results using the two approaches described in the previous sections the Trotter factorized QCL (TQCL), and iterative linearized density matrix (ILDM) propagation schemes, to study the spin-boson model consisting of a two level system that is bi-linearly coupled to a bath with Mh harmonic modes. This popular model of a quantum system embedded in an environment is described by the following general hamiltonian ... 

For bosons, n should run from - d - l)/2 to [d - l)/2, taking integers values between them, and from —d/2 to d — )I2, taking half odds and integers values for fermions. However, in the fermionic systems, a convention establishes that the density matrix elements of half odd should be taken as zero. 

Finally, one notes the important role played by the momentum distribution, which is related to the one-body density matrix in the coordinate representation through a Fourier transform. The one-body density matrix in this case contains the correlations between the ends of the PI particle path which, as a result of bosonic exchange, becomes an open string. For details on PI calculations of properties such as the momentum distribution, condensate fraction, and superfluid density the reader is referred to Refs. 28,70,215,231,232. 

As a last application of ensemble theory to the quantum mechanical ideal gas, we obtain the equation of state for fermions and bosons. To this end, the most convenient approach is to use the grand canonical partition function and the momentum representation, in which the matrix elements of the density are diagonal. This gives for the canonical partition function... 

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