Charge and Heat Transport in Low-dimensional Quantum Systems
Transport of electrons through low-dimensional quantum conductors like nano-junctions has been actively pursued research area for the last few decades. Experimental studies on nano-junctions are nowadays routinely carried out around the globe thanks to the invention of nanoscale analysis and manipulation techniques like the scanning tunneling microscope and the atomic force microscope. The research on charge and heat transport in nano-junctions is not only motivated by the miniaturization of electronic devices relevant for applications, but also by the intriguing quantum mechanical transport phenomena which differ from that in macroscopic conductors. Motivated by the feasibility of experimentally testing the theoretical predictions of various aspects of transport through quantum junctions, the theoretical work on charge and heat transport through nano-junctions presented in this thesis is carried out using quantum master equation and non-equilibrium Green functions approaches. In chapter 1, experimental works that have motivated the theoretical work presented in this thesis are briefly discussed, followed by a summary of key theoretical techniques which are used to rationalize these experiments. A survey of important works on fluctuations of charge and heat transport in quantum systems and fluctuation theorems satisfied by these fluctuations is also presented in this chapter. In chapter 2, the effect of system-reservoir coupling on charge currents flowing through nano- junctions is studied using non-equilibrium Green functions method applied to two double-quantum dot circuits. It is found that the charge currents do not always increase with the increasing system- reservoir coupling strength. Further, this behavior depends on the way nanosystem is coupled to the reservoirs. For the case when two quantum dots are serially coupled, that is, when the two dots are coupled to two different reservoirs, the current exhibits a non-monotonic behavior and diminishes for large coupling strengths. While for the case where two quantum dots are side-coupled to the two reservoirs, that is, only one of the dots is coupled to the two reservoirs, the current increases monotonically and saturates for asymptotically large system-reservoir coupling strength. To rationalize this behavior, the charge currents flowing between system and reservoirs at steady state are partitioned into the currents carried by populations in the single-particle energy eigenstates of the nanosystem and the currents carried by the coherent superposition of these eigenstates. It is shown that the currents carried by populations are always positive (in the direction of applied bias) and increases monotonically with system-reservoir coupling strength. But the currents carried by the coherent superposition of single-particle eigenstates becomes negative (opposite to the direction of the applied bias) for large system-reservoir coupling. For the serially coupled case, this coherent contribution becomes equal in magnitude to the sum of currents carried by the populations for asymptotically large system-reservoir coupling strengths making the net current zero. While for the side coupled case, this cancellation is not complete, making net current saturate to finite value asymptotically. Further, this behavior is found to be robust to weak Coulomb interactions. These theoretical predictions can be tested using either quantum dot junctions where system-reservoir coupling strengths can be tuned using gate potentials, or using molecular junctions where system- reservoir coupling strengths can be tuned by various means like chemical gating, by tuning Fermi level of the metals or by tuning overlap of molecular orbitals with electronic states of the metals. An important aspect of charge transport through nano-junctions is local currents flowing inside the quantum systems. This is discussed in chapter 3. This work is motivated by the possibility of inferring the pathways electrons can take through the molecule while flowing between the two reservoirs, either using inelastic electron tunneling spectroscopy or by measuring local magnetic fields generated by these currents. To understand the local currents flowing inside quantum systems, two simple models, a four quantum dot cyclic molecular junction symmetrically coupled to two reservoirs in the presence of applied magnetic field and a similar junction (in the absence of magnetic field) but with an extra quantum dot substituted on one of the arms coupled to two reservoirs, are studied using non-equilibrium Green functions method. For the symmetric junction, the currents flowing through the two paths or branches inside the ring are found to have two contributions, one due to the applied bias and the other due to the applied magnetic field. It is shown that these two contributions can be tuned by adjusting the applied bias and the applied magnetic field such that one of the branches is conducting selectively. For the asymmetric junction case, it is shown that by tuning the applied bias and the system-reservoir coupling strength, a circulating current can be induced inside the molecular junction. Chapter 4 is a slight deviation from the general theme of the thesis in the sense that it is not about the transport of electrons through quantum junctions but deals with a temporally driven isolated quantum system. In this chapter, the statistics of work that is performed while preparing displaced squeezed thermal state of a quantum optical oscillator from a thermal state by driving with two classical electric fields (one-photon resonant and two-photon resonant drives) is explored.3 This work is motivated by recent theoretical and experimental works which showed the possibility of using squeezed thermal reservoirs for designing heat engines whose efficiencies can surpass the classical Carnot bound. Using standard quantum optics technique of Weyl generating function, the moment generating function for the work done by two classical drives on the quantum optical oscillator is obtained analytically. Using this, Jarzynski-Crooks fluctuation theorems for the work distribution function are verified. It is found that the work statistics is qualitatively different for the cases when only one of the drives is present. Further, it is found that the phase difference between the two drives affects the work statistics in a non-trivial fashion where the most probable work can shift to negative values (although on an average work is performed on the oscillator conforming with the second law of thermodynamics) due to interference effects. Chapter 5 investigates the possibility of pumping charge across a quantum junction which is coupled to two unbiased reservoirs by temporally modulating system-reservoir coupling in an adiabatic fashion. This work is motivated by the discrepancy between a recent theoretical work which argued that the adiabatic modulation of system-reservoir coupling cannot break the detailed balance and hence no net flux can flow between two unbiased reservoirs and a two-decade-old experiment which found that net flux can flow between the reservoirs. By considering a simple (system-reservoir coupling) driven resonant level model junction, the moment generating function for the charge flowing between two unbiased reservoirs is computed analytically in the adiabatic limit using non-equilibrium Green functions approach. It is found that the level broadening (lifetime broadening) which is absent in the simple second-order quantum master equation approaches is crucial for the finite charge flux. Further, the direction of charge flux is found to be dependent on the phase difference between the two system-reservoir drivings and the alignment of the resonant level relative to the chemical potentials of the two reservoirs. The probability distribution function for the charge flow is shown to satisfy a fluctuation theorem. In chapter 6, the possibility of heat transport mediated by Coulomb interactions is explored. This work is motivated by recent breakthroughs in experimental nano calorimetry. To understand the role of Coulomb interactions on the heat transport in nano-junctions, a simple capacitively coupled double quantum dot junction is studied using quantum master equation and non-equilibrium Green functions approaches. It is found that when the Coulomb interactions are treated within the mean-field level, at steady-state no heat flows between the two reservoirs. This is because, at this level of approximation, the system decouples into two independent resonant levels which are at equilibrium. For going beyond the simple mean-field approximation, two approximate approaches are employed; the Lindblad quantum master equation approach which treats system-reservoir coupling approximately and the non-equilibrium Green functions approach within the renormalized random phase approximation which treats Coulomb interaction strength approximately. Using these approaches, it is found that heat flux and its fluctuations are non-monotonic functions of the Coulomb interaction strength and the system-reservoir coupling strength. Further, the thermodynamic consistency of these predictions is asserted by verifying the steady-state fluctuation theorem for stochastic heat flux flowing between the two reservoirs using both the approximations employed. A summary of the work presented in this thesis is given in chapter 7.