P7: Amplitude-mode duality of non-Markovian, fermionic, open systems
As with closed quantum systems, the time-dependent state density operator |ρ(t)) = Σk λk |mk)(ak|ρ(0)) of a Markovian open quantum system is governed by the eigenvalues λk, of its evolution generator, the eigenmodes |mk), and the amplitudes (ak| of an initial state. However, even in the Markovian limit, there is no simple relation between amplitudes and modes, unlike closed systems in which they are simply Hermitian adjoints. Moreover, for non-Markovian open systems, all of these quantities become frequency dependent due to memory effects. This causes the above simple formula to break down. Nevertheless, we have recently discovered a very general duality which relates these amplitudes and modes of fermionic open systems and allows for arbitrary strong coupling in the wide-band limit, resulting in strongly non-Markovian dynamics . The most striking immediate implication is that many properties of an open system “know” about their dual partner system in which all energies are inverted, in particular, the sign of the interaction.
In the “simple” Markovian master equation limit, already, this reveals a deep origin of many ill-understood features, such as the transient charge, spin- and energy currents after a quench [1,2], as well as the full stationary thermoelectric response  of a quantum dot. Importantly, this duality gives rise to a completely new method of constructing time-dependent solutions to Markovian master equations  which simplifies calculations and clarifies the results. Although it is more powerful than dualities based on time-reversal symmetry  or electron-hole mappings , our duality is limited to fermion systems by relying on the fermion-parity superselection postulate of many-body quantum physics. Notably, the general validity of the duality can only be shown  by combing the “superfermion” method  originating from the real-time renormalization group approach  with “ordinary” perturbative calculations .
In this project, we will first systematically investigate the implications of this general duality for quantum dots with moderate coupling to electrodes, such that memory, level-shift and broadening effects play a role. For a quantum dot subject to a sudden quench we will compute the time-evolution kernel to 2nd order in the coupling and combine our duality approach to construct the explicit frequency-dependent eigenvalues, modes and amplitudes. This will allow us to compute time-dependent transport currents for various quantities of repulsive quantum dots, in particular, the heat current which is sensitive to surprising resonances of the dual attractive system . Within the Markovian limit it is also of interest to study the exact time-dependent solutions of the driven master equation by combining duality with the Floquet method and the related exact adiabatic-iteration approach of Berry.
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