A new perturbative approach to the adiabatic approximation

Abstract

A new and intuitive perturbative approach to time-dependent quantum mechanics problems is presented, which is useful in situations where the evolution of the Hamiltonian is slow. The state of a system which starts in an instantaneous eigenstate of the initial Hamiltonian is written as a power series which has a straightforward diagrammatic representation. Each term of the series corresponds to a sequence of "adiabatic" evolutions, during which the system remains in an instantaneous eigenstate of the Hamiltonian, punctuated by transitions from one state to another. The first term of this series is the standard adiabatic evolution, the next is the well-known first correction to it, and subsequent terms can be written down essentially by inspection. Although the final result is perhaps not terribly surprising, it seems to be not widely known, and the interpretation is new, as far as we know. Application of the method to the adiabatic approximation is given, and some discussion of the validity of this approximation is presented.

9 pages. Added references, discussion of previous results, expanded upon discussion of main result and application of it

References in corpus (10)

• Quantum Computation by Adiabatic Evolution
• Inconsistency in the application of the adiabatic theorem
• Quantitative conditions do not guarantee the validity of the adiabatic approximation
• An Elementary Proof of the Quantum Adiabatic Theorem
• A note on the geometric phase in adiabatic approximation
• Perturbative Analysis of Possible Failures in the Traditional Adiabatic Conditions
• An Analog Analogue of a Digital Quantum Computation
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• Condition for the adiabatic approximation
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Cited by in corpus (24)

• Bounds for the adiabatic approximation with applications to quantum computation
• Consistency of the Adiabatic Theorem
• Sufficiency Criterion for the Validity of the Adiabatic Approximation
• The quantitative condition is necessary in guaranteeing the validity of the adiabatic approximation
• Improved Error-Scaling for Adiabatic Quantum State Transfer
• Validity of the Adiabatic Approximation
• Improved Error Bounds for the Adiabatic Approximation
• Exploring adiabatic quantum trajectories via optimal control
• Adiabatic Condition and Quantum Geometric Potential
• Self-healing of Trotter error in digital adiabatic state preparation
• Generalized Geometrical Phase in the Case of Continuous Spectra
• Perturbative Analysis of Possible Failures in the Traditional Adiabatic Conditions
• On The Power Of Coherently Controlled Quantum Adiabatic Evolutions
• Why the Quantitative Condition Fails to Reveal Quantum Adiabaticity
• On spin evolution in a time-dependent magnetic field: post-adiabatic corrections and geometric phases
• Optimizing adiabaticity in quantum mechanics
• Adiabatic approximation in the second quantized formulation
• Practicality of quantum adiabatic algorithm for chemistry applications
• Intrinsic Dynamical Fluctuation Assisted Symmetry Breaking in Adiabatic Following
• Quantum annealing showing an exponentially small success probability despite a constant energy gap with polynomial energy
• Numerical Test of Born-Oppenheimer Approximation in Chaotic Systems
• Error-run-time trade-off in the adiabatic approximation beyond scaling relations
• Femtosecond pulses and dynamics of molecular photoexcitation: RbCs example
• General conditions for a quantum adiabatic evolution