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Active Control of Probability Amplitudes in a Mesoscale System via Feedback-Induced Suppression of Dissipation and Noise

We introduce a classical potentiostatic feedback mechanism that attenuates the dissipation in a quantum system arising from coupling to the surrounding thermodynamic bath, preserving the inter-state interference in an electronic excitation transfer (EET) process. A three-terminal potentiostat device applies a low-noise voltage bias to the terminals of the EET system and reduces the physical coupling between the quantum system and its environment. We introduce a classical equivalent circuit to model the environment-coupled excitation transfer in an elementary two-state system. This model provides qualitative insight into how classical feedback action affects the transition probabilities between the states and selectively reduces the dissipative coupling for one of the vibronic energy levels of the transfer system. Furthermore, we show that negative feedback results in persistent spectral coherence between the energy level of the decoupled state and the vibronic levels of the complementary state, making the decoupled vibronic channel a probe for characterizing the vibronic structure of the complementary channel of the EET system.


I. INTRODUCTION
Environment-induced decoherence of quantum states has been extensively investigated in experimental [1][2][3][4] and theoretical literature, [5][6][7] with the consensus that coupling to the degrees of freedom of a thermodynamic bath is sufficient to average out coherent interference in a quantum system. Environmental coupling is largely responsible for the onset of classical behavior in quantum systems. It has been identified as the cause for the extinction of 'glory' oscillations [8,9] in the scattering of magnetic monopoles by charged molecules due to coupling with the rotational degrees of freedom in the molecular bath, as well as for the loss of information in superposed states within a quantum computer, again due to coupling with a surrounding thermodynamic bath with several degrees of freedom. [10][11][12][13] Therefore, preservation of the superposition of quantum probability amplitudes requires reduced interactions between the system and bath, or a reduction in the number of bath modes that can interact with the system, for the case when the quantum system is coupled to a large number of modes. [14] A scheme for preserving interferences between states would enable new room temperature systems exhibiting quantum behavior that could be applied to sensing, computing and energy conversion. As an example, persistent quantum coherent interferences of exciton waves are thought to boost the efficiency of EET processes, [15][16][17] and by extension, the efficiency of an EET transport-mediated photosynthetic process. [18,19] In this paper, we demonstrate how controlling the excitation of an EET charge transfer system with a classical electronic negative feedback loop can prolong the coherence lifetime for the participating electronic states. An environment-coupled molecular system that is comprised of a single level donor and acceptor species, 'dressed' by a collective of bath vibrational modes, is used to model the charge transfer process, which leads to an equivalent circuit model in which the dynamical variables describe wavefunction probability amplitudes. The impact of feedback on wavefunction probability amplitudes can then be described in terms of the dynamical variables of the circuit model.

A. Classical Oscillator
The equations of motion for a one dimensional particle (system) coupled to a bath of damping vibrational modes are given by where γ is the damping kernel, and m , B k , T and ( ) V x are the system mass, Boltzmann constant, bath temperature and conservative potential respectively.
( ) t ξ is a Gaussian function with zero mean and correlation given by in the classical limit. Thus, the amplitude of thermal disturbance acting on the system is related to the dissipative force exerted by the environment, subject to the assumption that the thermal reservoir is large enough such that the bath vibrational modes continue to stay in equilibrium throughout their interaction with the system. [20] These equations of motion are derived from the Hamiltonian description of the system and the environment, in which the environment is modeled as a collection of noninteracting harmonic oscillators (h.o.) and the interaction between the system and the environment is bilinear in the environment h.o coordinates and the system coordinate [21][22][23][24].
Under the assumption that the environment coordinates and momenta values are sampled from an equilibrium Boltzmann distribution, [25] the fluctuation dissipation relationship can be shown to hold. The equivalent bath temperature as seen by the system is given by In the Markovian limit, when the environment-system interaction is The real part of the parameter η would represent an effective viscosity in a mechanical system, or could be interpreted as a linear resistance in an oscillatory electrical circuit. ( Figure 1A) The system in question, whether quantum or classical, is stimulated by the randomized environment-induced thermal disturbances, which are balanced by the dissipative forces as it moves in the field described by the potential. We propose an electronic feedback-based mechanism for the bandwidth-limited control of these thermal disturbances and the related damping forces. The case of electrical oscillators is considered here for demonstration purposes, but the proposed mechanism could be extended to mechanical systems as well. Specifically, a scheme is presented whereby the system is decoupled from the physical reservoir with which it is in contact and coupled to another bath of pre-specified spectral density, ensuring control over the bath's effective 'temperature' and the damping experienced by the system. A sequence of cascaded amplifiers (A 1 , A 2 ) is configured to deliver a high gain, corrective signal proportional to the difference between set V and ref V upon measurement of the reference voltage, ref V ( Figure   1B). The measurement is performed with a buffer amplifier, B 1 that has a high impedance input to minimize leakage currents in the measurement. The physical reference electrode (R.E.) for probing the reservoir voltage, ref V , is deemed ideally to have zero source impedance, as is the physical counter electrode (C.E.) that applies the corrective signal ' x V to the system ( Figure 1B).
In addition, the circuit schematics in Figures 1A, 1B measure the current flowing through the system across dissipative elements r X and m X , respectively, in response to the classical voltage excitation bias applied at the reference electrode. The system impedance is characterized as X+jY, where X represents the dissipative part and Y is the purely imaginary contribution.
The respective transimpedance responses for systems in Figures 1A and 1B are: where s Y is the resonant component of the system, given by where the first term, which includes the principle part (symbol PP ) of the integral over the complex plane, is representative of a resonant frequency shift and the real term is the dissipation experienced by the oscillating system. Thus, the response of the oscillatory system to thermal excitations is dictated by the ensemble-averaged lumped circuit representation of the environment-induced dissipation as well as by the ensemble-averaged 'dressing' down of the resonant frequency of the system, also by the environment vibrational modes. The application of high gain negative feedback cancels the dissipation and the dressing down of the resonance as observed in the LTSpice IV A. C. small-signal simulation in Figure 2. Commercial operational amplifier schematics were used to generate the simulation results in Figure 2.
The thermal disturbances induced by the reservoir on the system are measured at the R. E.
node. These disturbances can be estimated and referred to the input source V excitation for the schematics in 1A and 1B, as is standard practice in noise analysis in electronic circuits.
The input-referred noise at the reference node is obtained by superposing the input referred voltage noise from each source in the feedback loop and referring them to the input: and i f ∆ is the bandwidth of the i-th voltage noise source. For large m X and 1 A , small reference node bandwidth ref f ∆ and a sufficiently quiet feedback network, the system would experience significantly smaller thermal disturbances, or a lower equivalent bath temperature, than in the case without feedback. The equivalent mode temperature for the oscillatory system, when in equilibrium with the reservoir modes, is estimated from the equipartition theorem as assuming 1 2 1 A A >> and is independent of frequency, and for which 2 Integrating over the frequency domain yields In effect, the physical environment around the system is exchanged with the bath of modes with increasing m X , as observed from the reduced area under the curve, and the corresponding equivalent system mode temperature, is also illustrated in Figure 3C.

B. Mesoscopic Charge Transfer System
The quantum dynamics of Hermitian Hamiltonians are known to correspond to the coupled motion of classical mechanical or electrical oscillators. [31,32] Specifically, the classical probability amplitudes describing the time-dependent state of an oscillatory system are equivalent to the quantum amplitudes that characterize the evolution of the wavefunction of an The 'momentum' and 'position' coordinates for the system and environment can be suitably non-dimensionalized [31,32] to re-derive the dynamics of the system and the environment from the modified non-dimensionalized Hamiltonian: Based on the dynamical equations of motion (4.1-4.3), we postulate an equivalent circuit description of the inter-coupled, single electronic energy level, donor and acceptor charge transfer system in Figure 4A. The individual energy levels are modeled as resonant elements that are coupled to the physical environment or, in the case of feedback, to the bath reservoir of instrumentation modes dissipatively via resistors at the reference node. The reference node in Figure 4A defines the location where an external bias is applied, or where, in the feedback case, a reference probe is inserted to measure the 'energy' of one of the levels that the feedback loop constrains to a desired setpoint. In this context, the reference is deemed a proxy measure for the second energy level in experimental systems where direct access to the state energy is unavailable. In addition, a capacitor between a resonant unit and the reference probe models the non-dissipative coupling between the energy levels. The 'ground' for the proposed circuit model in Figure   coupling, which is also referred to as a system of p&q coupled oscillators. [ with the parameters which is indicative of a diabatic crossing [25] of the weakly coupled h.o. wells, also for the limit of zero dissipation. The corresponding probability that the system makes a quantum jump from energy surface 1 ω to surface 2 ω at the crossing of the diabatic curves is given by: The two-state charge transfer system, coupled to the electronic feedback loop, is depicted in Figure 4B. Here, the reference probe measures the energy of the quantum state 2, and the feedback sets the energy to a desired setpoint via a corrective signal applied to the energy of state 1. All state energies are measured relative to the system ground as mentioned previously. We shall assume an ideal, dissipation-free reference probe in contact with the participant energy state 2 for the subsequent analysis with 22 22 , 0 ω γ ∆ → . The analysis may be extended to the more general case with dissipation in the reference channel. With the application of feedback, the probability amplitudes for the states of the two participatory species in the charge transfer process are given by: In these descriptors for the probability amplitudes, the excitation signal applied to the EET system via the feedback loop input, seen in Figure 4B, comprises two separable frequency components: a high frequency part that characterizes the energy difference between the two participant states (V ) of the quantum mechanical charge transfer system and a low frequency signal that describes the time response of the electrical feedback mechanism (ω ). In the asymptotic limit of large gain, the dynamic equations, as derived from Equations (5.1a) and (5.1b), governing the evolution of the probability amplitudes are given by: where the significantly slower dynamics of timescales ~ 1/ ω are considered static as the probability amplitudes rapidly evolve towards steady state. The eigenfrequencies for the charge transfer system, in the limit of large gain, and for the specific case of the dissipation-less reference probe are given by: Here V ∆ is the thermal r.m.s. voltage fluctuation of the excitation signal, which is proportional to T .