Abstract
The dynamics of an open quantum system is rarely exactly solvable due to the often infinite number of degrees of freedom associated with the thermal environment. It is well-known that performing a reduction over the bath variables results in a reduced system Hamiltonian which is non-Hermitian, capturing the effects of particle and energy transfer between the system and bath, and inelastic scattering. Using the Feynman-Vernon influence functional approach, and building on the work of Grabert, Schramm and Ingold, we improve upon the recently derived Extended Stochastic Liouville von Neumann (ESLN) equations for the density matrix in which an exact two-time Hubbard Stratonovich transformation applies to the influence functional to replace the system-bath coupling with a set of complex stochastic fields whose correlative properties encode all the information about the bath. These noises exist in both real time and imaginary time and appear in the resulting effective Hamiltonian, making its non-Hermicity explicit.In this approach, the combined system of the open system fully coupled to its environment are jointly thermalised at a finite temperature. This is achieved via an imaginary-time evolution procedure which initializes the combined system in the correct canonical equilibrium state rather than being initially decoupled. We apply the theory to a spin-boson system and derive several transformed SDEs with the aim of reducing the effect of numerical divergences associated with both the negative and complex weights present in the path integral. After verifying the equilibrium properties of the system, we consider non-equilibrium Landau-Zener (LZ) driving for which the long-time asymptotic limit is known, and show that initially thermalising the combined system in the finite past is a better approximation of the true LZ initial state than starting in the normal pure state. To extend the accessible simulation time, we derive a systematic method for exploiting an inherent freedom of the noises to optimise the noise generation procedure.
We then move on to fermionic systems and their description within the Non-Equilibrium Green’s Function (NEGF) formalism. Motivated by the non-Hermitian dynamics of open quantum systems, we derive a complete generalization of the NEGF formalism to account for a fully branch-dependent Hamiltonian, that is, for dynamics which is sensitive to the
branch of the Konstantinov-Perel contour in the complex plane. This generalization leads us to define two new kinds of NEGF: the three-time NEGF defined across the whole contour, and the sub-branch NEGF defined within the vertical branch, whose equations of motion use the Generalized Langreth rules in place of the normal Langreth rules. We consider the properties of the NEGF in detail, and derive a series expansion in terms on the normal NEGF with which it can be calculated, before making the connection between branch dependence, non-Hermitian Hamiltonains, and the broader class of PT -symmetry breaking Hamiltonians.
We then consider a molecular junction consisting of a central region in which electrons are coupled to a phonon environment, coupled to multiple electronic leads. Following our bath-reduction technique, we arrive at an effective non-Hermitian Hamiltonian for the central region. Thee generalisation to the three-time NEGF then becomes necessary, including some additional complexity since the three-time NEGF is now also stochastic, as the non-Hermicity of the effective Hamiltonian causes the dynamics to differ on both the upper and lower horizontal branches according to the real-time noises, with the Matsubara branch inheriting dynamics from the imaginary-time noises. We derive expressions for the current response of the system to a time-dependent external potential in terms of the three-time NEGF, and expand upon a procedure for calculation by relating the three-time NEGF which accounts for inelastic phonon scattering to the normal NEGF in the absence of phonons via a perturbative expansion of the noises.
This approach is exact and fully general, describing the non-equilibrium driven quantum dynamics from an initial thermal state while subject to inelastic scattering. Finally, we verify that in limit that the phonon coupling goes to zero, the standard expression for the current is recovered.
| Date of Award | 1 Jan 2024 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Lev Kantorovich (Supervisor) & Nicola Bonini (Supervisor) |