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Time-dependent Quantum Systems

Jaakko Lehto

Published Jun 2, 2017• 8 citations
OpenAlex

Abstract

This thesis discusses different aspects of time-dependent two-level quantum systems and their coherent dynamics. In the introductory chapters, we present some basic models used to study them and some basic results related to the level-crossing problems generally. As these systems are only rarely analytically solvable, some emphasis is put on the existing approximative theories. In particular, the generalization of the method of Dykhne, Davis and Pechukas (DDP), originally developed for problems in near-adiabatic region, is studied further. In the DDP theory, the final populations of different states are determined by the complex zero points of the eigenenergies. Superparabolic level-crossing models are introduced in this thesis and they prove to be simple but versatile test models for these studies. It is shown that, by considering all of the zero points, one can obtain accurate approximations also for highly non-adiabatic regions. In Chapter 4, a general differential geometric framework for time-dependent level-crossing models is developed and its basic character is discussed. A natural way of associating a plane curve to a time-dependent two-level model is introduced. This association allows one to use all of the mathematical results concerning plane curves to study these models. As an example, we use the Four-vertex theorem to discuss the adiabatic limit of certain type of systems. We also consider the prospect of enhancing the population transfer by transforming the time-dependent coupling pulse into a zero-area coupling. This is done either smoothly or by an abrupt jump in the phase of the coupling. It has been shown earlier, that one can obtain a complete population inversion (CPI) in a robust way with strong, non-resonant zero-area pulses. We study this in the more general time-dependent setting in Chapter 5, where also the energy levels are driven in a time-dependent fashion. This allows to transport the CPI phenomenon from the strong coupling region towards more moderate couplings. As an example of the smooth case, we consider the SechTanh-model, and show that even the nearest-zero DDP approximation can be used, despite the highly non-adiabatic origin of the phenomenon. We develop an approximative scheme also for the phase-jump case, using the parabolic model as an example. As a limiting case, we derive an approximative formula for the transition probability, which has some universal character and which proves to be useful in the most interesting case where the robust CPI is obtained. It is shown that even the highly suppressed transitions in tunnelling cases can be strongly enhanced in the phase-jump scenario.

Related papers

Heavy-particle collisions and quantum optics: The parabolic noncrossing model

B. S. Nesbitt, Derrick S. F. Crothers, S F C O'Rourke +1 more

The problem of deriving analytic formulas for transition probabilities in two-level systems is studied. The two-level systems are described by a pair of first-order differential equations coupled by a time-dependent potential. One such model is given by ${\mathrm{da}}_{m}/dt=\ensuremath{-}i\ensuremath{\beta}{f(t)a}_{n}{e}^{(\ensuremath{-}{1)}^{n}i\ensuremath{\alpha}t}$ $(m,n=1,2$; $m\ensuremath{\ne}/n)$, which describes certain types of ion-atom collisions and some quantum-optics two-level problems. It will be shown that the correct approach in solving the coupled equations is to adopt a Zwaan-Stueckelberg phase-integral analysis of the four-transition-point problem based on the parabolic noncrossing model of Crothers [J. Phys. B 9, 635 (1976)]. Alternatively, one may obtain an approximation by employing adiabatic perturbation theory, but such an approach can at best provide only weak-coupling solutions and can never guarantee unitarity in the probability amplitudes. The advantage of the phase-integral method is that it produces a strong-coupling approximation by embracing the appropriate asymptotic expansions for cylinder functions of large order and argument [D. S. F. Crothers, J. Phys. A 5, 1680 (1972)] and it also ensures analyticity, unitarity, and symmetry.

Physical Review A•Aug 1, 1997•5 citations

Analytical Treatment of Some Quantum Transitions in Molecular Dynamic Processes

Lukáš Pichl

They have analytically tested several quantum transitions in molecular dynamics. In addition, they have introduced applications of theoretical formalism in physics to economics. First, they have completed the semiclassical theory of nonadiabatic transitions in the exponential potential model. They treated a two-state quantum system, which commonly appears as a subsystem in many-state problems. They were concerned with one-dimensional Schroedinger equation, although the type or semiclassical approach which they used can be applied to higher-dimension potential surfaces by means of Fourier transform. In particular, they solved the case of repulsive exponentially growing potentials, that is the forward-back scattering. They re-expressed the nonadiabatic transition matrix, which describes the transition between the two states for one-direction passage, in terms of contour integrals in the complex coordinate plane. The integrands are adiabatic momenta derived from the eigenvalues of potential matrix. Such type of concept has the advantage that it can be applied to other models than just the exponential potential model. Based on this, they have obtained an analytical tool for describing one-transition point class of potential curve systems. Examples are Rosen-Zener-Demkov model and Landau-Zener-Stueckelberg model. The formalism, which they contributed to, is useful in describing physical processes such as the charge transfer, and it is suitable for shaping external fields or modulating their frequencies in order to achieve a desired state or a product with the theoretical 100% probability. Second, they have derived the first exact analytical solution for a diabatically avoided crossing model. The physical system is described with one-dimensional coupled potential well and barrier which do not intersect in the whole coordinate range. On the basis of special mathematical functions they derived the complete quantum mechanical solution for the wave functions and the transmission coefficient. They also analyzed this model semiclassically and compared it with the exact results. This comparison yields interesting implications on the validity and accuracy of semiclassical approximation. In particular, they solved the case of dramatically avoided crossing with exponentially steep potential curves, and obtained the probability of one-way passage through the interaction region. The overall behavior corresponds to the probability of tunneling through the single potential barrier. In addition, there are complete reflection resonances at discrete energies, resulting from the trapping of a wave in the potential well. The formalism of diabatically avoided crossing model is useful in laser control of molecular reactions. This is because the potential energy curves in realistic molecular systems can be shifted by the photon energy; the curves are coupled by the field intensity via the dipole moment. That is why the new topology of potential energy curves, which was so far broadly omitted, becomes practically interesting due to the development in lasers techniques. Third, they treated several two-state quantum systems common in molecular physics. The semiclassical conditions for the complete transmission and the complete reflection in one-dimensional scattering were given analytically. They discovered the possibility of these phenomena in energy ranges in which the complete reflection or the complete transmission are not commonly expected to occur. In particular, they have made use of the broadly shared assumption that the transitions between the states are localized phenomena. This enables them to separate the whole coordinate region into distinct subsets and to analyze the physics in each subset separately. Applying this approach, they obtained a fully algebraic problem, which can be treated e.g. by diagrammatic techniques. Let them also mention that the complete transmission or complete reflection conditions simplify the wave function, and hence the mathematical solution. The formalism which they used while deriving the semiclassical conditions for the resonant phenomena mentioned above can be further generalized. This will probably yield a whole spectrum of tools in the laser control of molecular processes. Fourth, they developed a powerful technique for solving integral equations with singular kernel. One of the physical formulations of scattering is the time-independent approach based on the Lippmann-Schwinger equation for the Green operator. They have developed a detailed technique how to treat the singularity in this equation. In particular, any scattering problem can be divided into two parts. The first one is the geometry of the system and the static interaction of its components. The second one is more dynamical: the singular equation for the Green operator including the variable energy of the scattering. They fix the most costly calculations in the static part. Then they solve the singularity analytically and thus obtain the cross section as a function of energy. The so far common approach is quite opposite: to fix the singularity first and then to recalculate heavily all the static part for any new value of energy. Obviously, their approach saves a considerable amount of computational time; therefore it is a good standpoint for treating large systems. The formalism above was proved to be extremely efficient taking the example of dissociative recombination H2+ + e- → H + H* because this is a diatomic system, quantum chemical data are easily available in the literature and they could demonstrate the superiority of their approach to the other approaches. Fifth, they suggested applications of basic physical concepts to economics. They introduced aggregate quantities in financial markets and interpreted these quantities as potential energy surfaces. They also formalized interactions between cyclic processes in terms of state to state transitions. Summarizing, they treated the exponential potential model of nonadiabatic transitions, introduced a new exactly solvable model of diabatically avoided crossing, discussed the semiclassical conditions for complete transmission and complete reflection from the viewpoint of laser control, contributed to scattering calculations, and introduced some interdisciplinary applications.

•Sep 29, 2000•0 citations

Population transfer in a level-crossing model with two time scales

Kalle‐Antti Suominen, B. M. Garraway

We set up a level-crossing model with two time scales and compare the numerically calculated probability transfer with several analytic results. We also examine the probability transfer in the adiabatic limit and make a comparison with a calculation based on the approach of Dykhne [Sov. Phys.---JETP 11, 411 (1960); 14, 941 (1962)] and of Davis and Pechukas [J. Chem. Phys. 64, 3129 (1976)]. We find that we can get an improved agreement if we modify their approach.

Physical Review A•Jan 1, 1992•86 citations

Time-dependent properties of solids : non-perturbative Floquet Hamiltonians and beyond

Michael Rudolf Vogl

In the different chapters of this dissertation we investigate multiple non-perturbative approaches that allow the study of solids subjected to time dependent fields. In the second chapter, we develop a very general non-perturbative and flexible approach that allows to study periodically driven quantum systems via an effective time-independent theory. It makes use of renormalization-group-like flow equations to find highly accurate effective Hamiltonians[Phys. Rev. X 9, 021037 (2019)]. The range of validity of the approach is checked numerically for various 1D spin systems and confirmed to be beyond the reaches of ordinary perturbative methods. It also compares favourably to another non-perturbative approach. In the third chapter we examine the wealth of different approximations to the time evolution operator that are known from the literature. Inspired by Hamilton-Jacobi theory we find a useful reformulation of the equation for the time evolution operator [Phys. Rev. A 100, 012132 (2019)]. This allows us to find the various known approximations of the time evolution operator in a unified fashion and new ones. Interestingly the RG-like flow equations from the second chapter are found to be one specific limit of the more general approach developed here. A periodically driven Ising model is used to verify the range of validity of the different approximations. This allows us to put them into a hierarchy. Up until here the methods that were explored are most valid in the high to mid frequency regime. In the fourth chapter we show how for the case of a weakly driven system a simple approach can be used to treat the low frequency regime [Phys. Rev. B 101, 024303 (2020)]. To achieve this we derive a quasi-energy dependent Hamiltonians and very crucially require self-consistency for the quasi-energies. For the simple two band example of single layer graphene it is found to be highly accurate approach that also allows for accurate descriptions of even sensitive quantities like Chern numbers For the last fifth chapter of the thesis we switch gears and consider a specific system of much recent interest - twisted bilayer graphene. We investigate how light that is altered by the boundary conditions of a metallic waveguide can change the strength of interlayer couplings [arXiv:2001.04416 (2020)]. We investigate how this change affects the twist angles at which flat bands appear and find very simple analytic results that are in good agreement with a numerical treatment. In the sixth chapter of the thesis, we investigate the effects circularly polarized light has on twisted bilayer graphene[ arXiv:2002.05124 (2020)]. We derive two effective Hamiltonians valid in different driving regimes. One is valid for weak drives but low frequencies and the other for relatively strong drives. The Hamiltonian for the strongly driven regime is compared to exact numeric results and found to perform better than other common methods. We analyse the newly generated terms in both Hamiltonians for their symmetry.

•May 1, 2020•0 citations

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Synopsis: The time development of quantum systems with time-dependent Hamiltonians or potentials will be discussed in this chapter. First, we set up the formalism of time development in a very general way and discover that there are two basic Hilbert space descriptions of these processes. As discussed earlier, exact solutions for physical systems are rare, and we therefore find approximate ways of describing fast and slow time developments. Concentrating on two-state systems, we investigate the important role complex phases play in the slow evolution of quantum systems. In this context, we learn how to apply perturbation theory to our formalism, resulting in wonderfully practical formulas that can readily be applied in the lab, including elastic scattering and spontaneous emission rates. Finally, the important topic of time decay of systems and associated uncertainties in the energies of such states will be discussed.

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•Aug 1, 2006•0 citations
This thesis discusses different aspects of time-dependent two-level quantum systems and their coherent dynamics. In the introductory chapters, we present some basic models used to study them and some basic results related to the level-crossing problems generally. As these systems are only rarely analytically solvable, some emphasis is put on the existing approximative theories. In particular, the generalization of the method of Dykhne, Davis and Pechukas (DDP), originally developed for problems in near-adiabatic region, is studied further. In the DDP theory, the final populations of different states are determined by the complex zero points of the eigenenergies. Superparabolic level-crossing models are introduced in this thesis and they prove to be simple but versatile test models for these studies. It is shown that, by considering all of the zero points, one can obtain accurate approximations also for highly non-adiabatic regions. In Chapter 4, a general differential geometric framework for time-dependent level-crossing models is developed and its basic character is discussed. A natural way of associating a plane curve to a time-dependent two-level model is introduced. This association allows one to use all of the mathematical results concerning plane curves to study these models. As an example, we use the Four-vertex theorem to discuss the adiabatic limit of certain type of systems. We also consider the prospect of enhancing the population transfer by transforming the time-dependent coupling pulse into a zero-area coupling. This is done either smoothly or by an abrupt jump in the phase of the coupling. It has been shown earlier, that one can obtain a complete population inversion (CPI) in a robust way with strong, non-resonant zero-area pulses. We study this in the more general time-dependent setting in Chapter 5, where also the energy levels are driven in a time-dependent fashion. This allows to transport the CPI phenomenon from the strong coupling region towards more moderate couplings. As an example of the smooth case, we consider the SechTanh-model, and show that even the nearest-zero DDP approximation can be used, despite the highly non-adiabatic origin of the phenomenon. We develop an approximative scheme also for the phase-jump case, using the parabolic model as an example. As a limiting case, we derive an approximative formula for the transition probability, which has some universal character and which proves to be useful in the most interesting case where the robust CPI is obtained. It is shown that even the highly suppressed transitions in tunnelling cases can be strongly enhanced in the phase-jump scenario.
The problem of deriving analytic formulas for transition probabilities in two-level systems is studied. The two-level systems are described by a pair of first-order differential equations coupled by a time-dependent potential. One such model is given by ${\mathrm{da}}_{m}/dt=\ensuremath{-}i\ensuremath{\beta}{f(t)a}_{n}{e}^{(\ensuremath{-}{1)}^{n}i\ensuremath{\alpha}t}$ (m,n=1,2(m,n=1,2(m,n=1,2; $m\ensuremath{\ne}/n)$, which describes certain types of ion-atom collisions and some quantum-optics two-level problems. It will be shown that the correct approach in solving the coupled equations is to adopt a Zwaan-Stueckelberg phase-integral analysis of the four-transition-point problem based on the parabolic noncrossing model of Crothers [J. Phys. B 9, 635 (1976)]. Alternatively, one may obtain an approximation by employing adiabatic perturbation theory, but such an approach can at best provide only weak-coupling solutions and can never guarantee unitarity in the probability amplitudes. The advantage of the phase-integral method is that it produces a strong-coupling approximation by embracing the appropriate asymptotic expansions for cylinder functions of large order and argument [D. S. F. Crothers, J. Phys. A 5, 1680 (1972)] and it also ensures analyticity, unitarity, and symmetry.
They have analytically tested several quantum transitions in molecular dynamics. In addition, they have introduced applications of theoretical formalism in physics to economics. First, they have completed the semiclassical theory of nonadiabatic transitions in the exponential potential model. They treated a two-state quantum system, which commonly appears as a subsystem in many-state problems. They were concerned with one-dimensional Schroedinger equation, although the type or semiclassical approach which they used can be applied to higher-dimension potential surfaces by means of Fourier transform. In particular, they solved the case of repulsive exponentially growing potentials, that is the forward-back scattering. They re-expressed the nonadiabatic transition matrix, which describes the transition between the two states for one-direction passage, in terms of contour integrals in the complex coordinate plane. The integrands are adiabatic momenta derived from the eigenvalues of potential matrix. Such type of concept has the advantage that it can be applied to other models than just the exponential potential model. Based on this, they have obtained an analytical tool for describing one-transition point class of potential curve systems. Examples are Rosen-Zener-Demkov model and Landau-Zener-Stueckelberg model. The formalism, which they contributed to, is useful in describing physical processes such as the charge transfer, and it is suitable for shaping external fields or modulating their frequencies in order to achieve a desired state or a product with the theoretical 100% probability. Second, they have derived the first exact analytical solution for a diabatically avoided crossing model. The physical system is described with one-dimensional coupled potential well and barrier which do not intersect in the whole coordinate range. On the basis of special mathematical functions they derived the complete quantum mechanical solution for the wave functions and the transmission coefficient. They also analyzed this model semiclassically and compared it with the exact results. This comparison yields interesting implications on the validity and accuracy of semiclassical approximation. In particular, they solved the case of dramatically avoided crossing with exponentially steep potential curves, and obtained the probability of one-way passage through the interaction region. The overall behavior corresponds to the probability of tunneling through the single potential barrier. In addition, there are complete reflection resonances at discrete energies, resulting from the trapping of a wave in the potential well. The formalism of diabatically avoided crossing model is useful in laser control of molecular reactions. This is because the potential energy curves in realistic molecular systems can be shifted by the photon energy; the curves are coupled by the field intensity via the dipole moment. That is why the new topology of potential energy curves, which was so far broadly omitted, becomes practically interesting due to the development in lasers techniques. Third, they treated several two-state quantum systems common in molecular physics. The semiclassical conditions for the complete transmission and the complete reflection in one-dimensional scattering were given analytically. They discovered the possibility of these phenomena in energy ranges in which the complete reflection or the complete transmission are not commonly expected to occur. In particular, they have made use of the broadly shared assumption that the transitions between the states are localized phenomena. This enables them to separate the whole coordinate region into distinct subsets and to analyze the physics in each subset separately. Applying this approach, they obtained a fully algebraic problem, which can be treated e.g. by diagrammatic techniques. Let them also mention that the complete transmission or complete reflection conditions simplify the wave function, and hence the mathematical solution. The formalism which they used while deriving the semiclassical conditions for the resonant phenomena mentioned above can be further generalized. This will probably yield a whole spectrum of tools in the laser control of molecular processes. Fourth, they developed a powerful technique for solving integral equations with singular kernel. One of the physical formulations of scattering is the time-independent approach based on the Lippmann-Schwinger equation for the Green operator. They have developed a detailed technique how to treat the singularity in this equation. In particular, any scattering problem can be divided into two parts. The first one is the geometry of the system and the static interaction of its components. The second one is more dynamical: the singular equation for the Green operator including the variable energy of the scattering. They fix the most costly calculations in the static part. Then they solve the singularity analytically and thus obtain the cross section as a function of energy. The so far common approach is quite opposite: to fix the singularity first and then to recalculate heavily all the static part for any new value of energy. Obviously, their approach saves a considerable amount of computational time; therefore it is a good standpoint for treating large systems. The formalism above was proved to be extremely efficient taking the example of dissociative recombination H2+ + e- → H + H* because this is a diatomic system, quantum chemical data are easily available in the literature and they could demonstrate the superiority of their approach to the other approaches. Fifth, they suggested applications of basic physical concepts to economics. They introduced aggregate quantities in financial markets and interpreted these quantities as potential energy surfaces. They also formalized interactions between cyclic processes in terms of state to state transitions. Summarizing, they treated the exponential potential model of nonadiabatic transitions, introduced a new exactly solvable model of diabatically avoided crossing, discussed the semiclassical conditions for complete transmission and complete reflection from the viewpoint of laser control, contributed to scattering calculations, and introduced some interdisciplinary applications.
In the different chapters of this dissertation we investigate multiple non-perturbative approaches that allow the study of solids subjected to time dependent fields. In the second chapter, we develop a very general non-perturbative and flexible approach that allows to study periodically driven quantum systems via an effective time-independent theory. It makes use of renormalization-group-like flow equations to find highly accurate effective Hamiltonians[Phys. Rev. X 9, 021037 (2019)]. The range of validity of the approach is checked numerically for various 1D spin systems and confirmed to be beyond the reaches of ordinary perturbative methods. It also compares favourably to another non-perturbative approach. In the third chapter we examine the wealth of different approximations to the time evolution operator that are known from the literature. Inspired by Hamilton-Jacobi theory we find a useful reformulation of the equation for the time evolution operator [Phys. Rev. A 100, 012132 (2019)]. This allows us to find the various known approximations of the time evolution operator in a unified fashion and new ones. Interestingly the RG-like flow equations from the second chapter are found to be one specific limit of the more general approach developed here. A periodically driven Ising model is used to verify the range of validity of the different approximations. This allows us to put them into a hierarchy. Up until here the methods that were explored are most valid in the high to mid frequency regime. In the fourth chapter we show how for the case of a weakly driven system a simple approach can be used to treat the low frequency regime [Phys. Rev. B 101, 024303 (2020)]. To achieve this we derive a quasi-energy dependent Hamiltonians and very crucially require self-consistency for the quasi-energies. For the simple two band example of single layer graphene it is found to be highly accurate approach that also allows for accurate descriptions of even sensitive quantities like Chern numbers For the last fifth chapter of the thesis we switch gears and consider a specific system of much recent interest - twisted bilayer graphene. We investigate how light that is altered by the boundary conditions of a metallic waveguide can change the strength of interlayer couplings [arXiv:2001.04416 (2020)]. We investigate how this change affects the twist angles at which flat bands appear and find very simple analytic results that are in good agreement with a numerical treatment. In the sixth chapter of the thesis, we investigate the effects circularly polarized light has on twisted bilayer graphene[ arXiv:2002.05124 (2020)]. We derive two effective Hamiltonians valid in different driving regimes. One is valid for weak drives but low frequencies and the other for relatively strong drives. The Hamiltonian for the strongly driven regime is compared to exact numeric results and found to perform better than other common methods. We analyse the newly generated terms in both Hamiltonians for their symmetry.
Synopsis: The time development of quantum systems with time-dependent Hamiltonians or potentials will be discussed in this chapter. First, we set up the formalism of time development in a very general way and discover that there are two basic Hilbert space descriptions of these processes. As discussed earlier, exact solutions for physical systems are rare, and we therefore find approximate ways of describing fast and slow time developments. Concentrating on two-state systems, we investigate the important role complex phases play in the slow evolution of quantum systems. In this context, we learn how to apply perturbation theory to our formalism, resulting in wonderfully practical formulas that can readily be applied in the lab, including elastic scattering and spontaneous emission rates. Finally, the important topic of time decay of systems and associated uncertainties in the energies of such states will be discussed.
The Chapter 2 is devoted to the study of the transport properties of a quantum wire, described by the Tomonaga-Luttinger model, in the presence of a backscattering potential provided by several point-like or extended time-dependent impurities (barriers) and find regimes of conductance enhancement. In Chapte 3 we study a set of crossed 1D systems, which are coupled with each other via tunnelling at the crossings. Concentrating on the case of a single crossing, we were able to explain recent experiments on crossed metallic and semiconducting nanotubes, which showed the presence of localized states in the region of crossing. The Chapter 4 treats the effects of disorder and interactions in a quantum Hall ferromagnet. We study the problem by projecting the original fermionic Hamiltonian into magnon states, which behave as bosons in the vicinity of the ferromagnetic ground state. The approach permits the reformulation of a strongly interacting model into a non-interacting one. The latter is a non-perturbative scheme that consists in treating the two-particle neutral excitations of the electron system as a bosonic single-particle. Indeed, the employment of bosonization facilitates the inclusion of disorder in the study of the system. It has been shown previously that disorder may drive a quantum phase transition in the Hall ferromagnet. However, such studies have been either carried out in the framework of the nonlinear sigma model, as an effective low-energy theory, or included the long-range Coulomb interaction in a quantum description only up to the Hartree-Fock level. Here, we establish the occurrence of a disorder-driven quantum phase transition from a ferromagnetic 2DEG to a spin glass phase by taking into account interactions between electrons up to the random phase approximation level in a fully quantum description. In the Chapter 5 we derive and discuss an experimentally realistic model describing ultracold atoms in an optical lattice including a commensurate, but staggered, Zeeman field. The resulting band structure is quite exotic; fermions in the third band have an unusual rounded picture-frame Fermi surface (essentially two concentric squircles), leading to imperfect nesting. We develop a generalized theory describing the spin and charge degrees of freedom simultaneously, and show that the system can develop a coupled spin-charge-density wave order. This ordering is absent in studies of the Hubbard model that treat spin and charge density separately.