Driven Open Quantum Systems in the Presence of Thermal Fluctuations

Chakrabarti, Arnab (2019) Driven Open Quantum Systems in the Presence of Thermal Fluctuations. PhD thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

The origin of irreversible phenomena is connected to the observation of the macroscopic properties of thermodynamically large number (ensemble) of entities (atoms, molecules, spins etc.). Although the underlying microscopic dynamics of each entity is governed by time-symmetric laws of motion, their emergent behaviour must possess an irreversible character, in order to account for our day-to-day experiences. The physics of the emergent irreversible behaviour is complicated to say the least and requires the use of physically motivated approximations along with mathematically rigorous techniques. While the usual treatments tend to describe the irreversible journey of an out-of-equilibrium ensemble, towards its equilibrium configuration, it is also essential to understand the dynamical processes through which the ensemble attained its non-equilibrium state. It may be thought of as the effect of an external perturbation which disrupts the equilibrium distribution. In experimental studies on physical ensembles, such perturbations are usually realized by the application of time-dependent external forces (drive) on the ensemble. In response to the applied drive, the ensemble looses its equilibrium state while the irreversible processes tend to restore it back, resulting in a damped-driven dynamics. Driven-dissipative systems serve as a model for many physical processes, a subclass of which forms the main theme of the work presented in this dissertation. Our discussions and results mainly pertain to the theoretical and experimental studies on various features of dissipative quantum mechanical ensembles subjected to a weak external drive. In the first part of the thesis we provide a new theoretical model which suggests that the external drive also serves as an additional source of damping in such systems. We then present a direct experimental proof of the validity of our claim using liquid state nuclear magnetic resonance (NMR) methods. We also describe the ability of our theoretical model, in accurately describing the dynamical steady-states observed in such driven quantum ensembles. In the remaining part of the thesis, we analyze the different driving protocols for suppression of decoherence originating from non-stationary random-walk processes in the ensemble. The thesis consists of 6 chapters, short descriptions of which are presented below. In the introductory chapter (Chapter 1) we provide a broad overview of different methods adopted to describe irreversible phenomena, highlighting the common features of these approaches. Starting from the celebrated Boltzmann equation, we summarize other classical approaches describing dissipative dynamics, v.i.z. the Langevin, Fokker-Planck and Classical Master Equation formalisms. After the brief prelude on classical irreversible dynamics, we consider the quantum version of the problem where irreversible behaviour is generally described in terms of quantum master equations (QME). The first QME was formulated by Pauli using a Repeated Random Phase Approximation (RRPA) which is akin to Boltzmann’s assumption of molecular chaos. Following the discussion on Pauli’s approach we briefly describe the alternate theory proposed by Van Hove. Next we consider Open Quantum Systems and introduce the concept of a dynamical semigroup for the reduced system dynamics. The generator of the dynamical semigroup describes a Markovian QME, the most general form of which has been developed by Gorini, Kossakowski, Lindblad and Sudarshan (GKLS). We present the detailed derivation of the Born-Markov QME to illustrate the regimes in which equations of the GKLS form can emerge from the microscopic Schrödinger equations. As an alternative to the Born-Markov scheme, we introduce Redfield’s semiclassical approach developed in the context of describing relaxation phenomena in NMR experiments. We also present a brief discussion on Non-Markovian quantum master equations. Theoretical and experimental description of driven Open Quantum Systems, form the main theme of this dissertation, with our experimental investigations being restricted of NMR techniques. As such we present a brief introduction to NMR experiments, highlighting its importance as one of the earliest techniques of systematically studying driven-dissipative quantum ensembles. We describe the phenomenological Bloch equations which govern the dynamics of the macroscopic magnetic moment vector of an ensemble of spins, in a typical NMR experiment. The microscopic derivation of the phenomenological Bloch equations, requires the formulation of a QME for the driven spin-ensemble. We present Bloch and Wangsness’ derivation of the QME where only first order effect of the external drive is considered. We then highlight how the Bloch equations can be used to develop pulsed NMR experiments in order to estimate the relaxation rates of spins as well as other parameters related to the spin-bearing molecules. Subsequently we provide a discussion on Bloch-Siegert shifts and drive dependent relaxation in two-level systems (TLS), which clearly suggest the existence of higher order effects of an external drive, contrary to the assumption of Wangsness and Bloch. The construction and experimental verification of a QME having second-order drive terms forms the main goal of the next three chapters. We conclude the current chapter with a brief outline of the organization of the thesis. In Chapter 2, we present the construction of a QME having time non-local second order contributions of a weak external drive. To arrive at this result, we consider the fact that the external drive induces identical effects on every member of the ensemble, while the local environment of each member experiences random fluctuations due to the thermal nature of the heat bath. As such, we introduce these fluctuations as explicit Hamiltonians, in the form of random hermitian matrices, acting on the local environments coupled to the individual ensemble members. For example, in the context of an ensemble of spins, the local environment of a particular spin consists of the molecule in which the spin is embedded. Then the fluctuations originate from random molecular collisions at a finite temperature. Using a physically motivated model for the fluctuation Hamiltonian, we construct a time-coarse-grained QME having time-nonlocal leading second order terms of the drive as well as the system-environment interaction Hamiltonian. The most crucial step in this derivation is the assumption of a time-scale separation between the system and bath dynamics. As such, in the coarse-grained time interval, many instances of the fluctuation take place while only leading order contributions of the system evolution remains significant. The average effect of many fluctuation instances manifest as an explicit exponential regulator in all second order terms (drive and interaction), making them time-nonlocal in a suitable limit. We also highlight the fact that our time-coarse-grained QME having second-order drive terms can be expressed in the GKLS form and as such describes a Markovian dynamics of the quantum ensemble. Chapter 3 concerns with the application of our QME to an ensemble of driven TLS. As an example, we consider the dynamics of a spin-1/2 ensemble in a Zeeman field, subjected to a resonant transverse excitation (drive). Using our QME we arrive at a modified version of the Bloch equations for such an ensemble whereby, the predicted decay rates and frequency shifts are dependent on the drive strength. Specifically, the decay rates and dynamic frequency shifts manifest as absorptive and dispersive Lorentzians proportional to the square of the drive amplitude. We show that in appropriate limits, the dispersive Lorentzians converge to the well-known forms of the Bloch-Siegert shift and the frequency shift obtained at large detuning. Unlike the prediction of Floquet theoretic approaches, these frequency shifts do not incur any divergence in the on-resonance condition, due to their Lorentzian nature. The absorptive Lorentzians which provide for the drivedependent decay rates in our modified Bloch equations, are Kramers-Kronig pairs of these frequency shifts. Using these absorptive Lorentzians, we demonstrate how one can arrive at the Redfield limit of drive-dependent relaxation rates, in a suitable limit, confirming the observations in solid-state NMR and optical spectroscopic measurements. Further, the very nature of these drive-induced damping rates suggest that such effects should be observable in all phases of matter. The Lorentzian profile of the drive-induced frequency shifts and damping coefficients, explicitly depend on the correlation-times of the bath operators, indicating the temperature dependence of these quantities. In Chapter 4 we report the detection of drive-dependent decay rates of Rabi oscillations in liquid NMR experiments, in line with the predictions of Chapter 3. Previous experimental observations of drive-dependent of decay rates have been restricted to solid state samples. As such, several theoretical models pertaining to solid-state media, have been put forward to explain such phenomena. But our model suggests that the drive-dependence of decay rates is an unavoidable feature of driven-dissipative dynamics, irrespective of the nature of the environment. Thus the detection of such an effect in an ensemble of spin-1/2 nuclei having negligible coupling to other spins, in isotropic liquid environment, provides a direct evidence for the validity of our theory. The presence of drive inhomogeneities in the sample volume offer the main challenge in an accurate estimation of the decay rates of Rabi oscillations. We present a novel, broad-band, super-cycled rotary-echo protocol to eliminate signal dephasing due to drive-inhomogeneities in the sample. Further, we use spatial encoding techniques during signal detection to remove the effects of susceptibility jumps near the sample edges. We show that the decay rates of Rabi oscillations in our experiment, are proportional to the square of the drive amplitude, the proportionality factor being of the order of rotational correlation times in liquids – confirming our theoretical construction. We also present the experimental data showing the temperature sensitivity of the drive-dependent Non-Bloch decay rates obtained from our measurements. Chapter 5 deals with the application of our QME to an ensemble of two spin-1/2 nuclei coupled through their mutual dipole-dipole interactions. Our main aim is to provide a dynamical theory describing the origin of spin-locking in dipolar solids. Spin-locking refers to the persistence of transverse magnetization in dipolar spinnetworks, for a period much longer than the corresponding transverse relaxation time, under the influence of an in-phase drive. The traditional theories developed to account for this phenomenon rely on the assumption of a Spin-Temperature, where the dipolar and Zeeman spin Hamiltonians define different temperatures for the spin ensemble. Spin-locking is then explained as a steady state condition when a common equilibrium temperature is attained by the ensemble. The presence of dipolar couplings between the spins, mediate the energy exchanges between them, facilitating the emergence of this steady state. In this chapter we trace the origin of spin-locking to the presence of drive-dipole cross-correlations in our QME. Defining the two-spin observables in a suitable frame, we provide a simplified picture which captures the effect of spin-locking through a set of coupled differential equations for these observables. Numerical solutions of these equations illustrate the spin-locking effect, thereby providing a deeper understanding into its possible origin. Also the steady-state behaviour of the in-phase magnetization obtained from these coupled differential equations conform with the predictions of the Spin-Temperature theory. In the final chapter, (Chapter 6) we focus on a different source of thermal fluctuations in spin ensembles, which lead to a loss of coherence. Our main aim is to analyze the efficiency of different dynamic decoupling protocols, in eliminating these decohering effects. Specifically, we consider the decoherence due to non-stationary phase fluctuations originating from translational diffusion of spin-bearing molecules under a constant Zeeman field gradient. We choose the traditional Carr-Purcell- Meiboom-Gill (CPMG) scheme and Uhrig’s Dynamic Decoupling (UDD) protocol for a comparative study. It is known that UDD is an optimal choice for minimizing loss of coherence originating from stationary Gaussian phase fluctuations with a sharp high-frequency cut-off. For soft cut-offs and also for stationary telegraph noise processes, CPMG turns out to provide optimal suppression of decoherence. The random phases picked up by the decohering spins due to diffusive motion of their respective molecules, have a Gaussian character and as such, the resulting decoherence rate is proportional to the phase auto-correlation. Using mathematical induction we calculate the general expression for the autocorrelation of the diffusion induced random phases, acquired after the application of an arbitrary sequence of n - π pulses. Finding the optimal protocol then becomes a problem of choosing the pulse instances which minimize the phase autocorrelation. We show that CPMG pulse instances minimize this autocorrelation, for up to n = 55. Also, we find that the UDD scheme results in higher values of the auto-correlation in comparison to CPMG for a wide range of n. As such we conclude that CPMG is more efficient than UDD, in suppressing loss of coherence due to non-stationary phase fluctuations originating from random walk processes. We verify this result by performing liquid NMR experiments on the protons of water.

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