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Geometric Phase and Related Phenomena in Quantum Nanosystems

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Owing to the rapid growing of the nanotechnology, now it ispossible to design and fabricate different types of nanostructureswhich have nanometer length scales in one, two, or threedimensions. Examples of nanostructures are: quantumwells [12],quantum wires [33], and quantumdots [15] inwhich the motion of charge carriers is restricted within nanometerlength scale in one, two, and three dimensions, respectively. Usuallyquantum wells, wires, and dots are made of semiconductor materials andbased on the related designing and fabricating technology. Otherexamples are those of carbon‐based nanomaterials, such asgraphene [26],carbon nanotubes [17], and fullerenesC60 orC70 [10], in which the motion ofelectrons is also restricted within nanometer scale in one, two, andthree dimensions, respectively. The electric, optical, and magneticproperties of these nanostructures are mainly depending on the statesof electrons. So here we define the electron system in...

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Abbreviations

Target system:

Thesystem under investigation. Usually the target system is a partof degrees of freedom in a nanostructure whose state is describedquantum mechanically.

Environment:

Thesystem which interacts with the target system and influences itsstate. The environment may be the parts of degrees of freedom in thenanostructure other than the target, the systems which surround thenanostructure and interact with the target system, or an externalelectric or magnetic field which is exerted on the targetsystem.

Hamiltonian:

Thequantum operator that describes the energy of the system and acts onthe Hilbert space for the quantum states. It is usually denoted by\({ \hat{H} } \) with subscriptindicating the described system.

Eigen energy and eigen wave function:

For time‐independent(stationary) Hamiltonian the system can only take specific values ofenergy, called the eigen energies. Corresponding every eigen energythere is a state of the system described by the eigen wavefunction. All eigen wave functions are normalized and orthogonal toeach other, forming a complete linear space.

Hilbert space :

A complete linear space spanned by all eigenwave functions of the time‐independent Hamiltonian. Any statesof the related system can be expressed as a vector in this linearspace.

Parameter space:

The space spanned by parameters which specify theinteractions of the environment on the target system. Ina dynamical process in which the geometric phase is investigated,the parameters are assumed to be periodically varying in time. Theperiod is denoted as T.

Evolution of wave functions:

During a period of variation of parameters,the Hamiltonian transverses a circle and returns to its initialsituation at the end of the period. In this process the wave functionof the target system also undergoes an evolution with thetime.

Dynamical phase :

Even though the Hamiltonian is time independent,the wave function of the target system still has a time dependentphase factor denoted as \( { {\text{e}}^{\text{i}\gamma_{\text{d}}(t)} }\). Ina period T ofa periodic dynamical process the phase \( {\gamma_{\text{d}} } \) acquiredby the target wave function, which is calculated under the assumptionthat the Hamiltonian is the averaged one and stationary in thisperiod, is called the dynamical phase .

Geometric phase :

If the Hamiltonian is periodicallytime‐dependent, the phase γ acquired by the targetwave function in a period of the evolution is different from thedynamical phase \( { \gamma_{\text{d}} }\). The difference\({ \gamma_{\text{g}}= \gamma-\gamma_{\text{d}} }\) is called the geometricphase .

Adiabatic approximation :

If the evolution of the Hamiltonian is slow enoughso that there is no transition between different eigen wave functionsof the target system during the evolution, this evolution is regardedas adiabatic. The approximation based on the adiabatic assumption ofthe process is called adiabatic approximation.

Berry phase :

Thegeometric phase calculated in the adiabatic approximation is calledthe Berry phase .

Quantum interference :

In the case where the states ofa nanostructure are quantum mechanically described by wavefunctions, the amplitude of a resultant wave function can beconstructively enhanced when its components have the same phase, orcan be destructively weakened when its components have oppositephases. Generically the resultant wave function depends not only onthe amplitudes of its components, but also depends on the phases ofthe components. This phenomenon is called quantuminterference.

Dephasing :

Theprocesses or mechanisms which cause spatial or temporal uncertainty ofphase of the target wavefunctions.

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Acknowledgments

Thiswork was supported by the State Key Programs for BasicResearch of China (2005CB623605 and 2006CB921803), andby National Foundation of Natural Science in ChinaGrant Nos. 10474033 and 60676056.

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Authors and Affiliations

  1. National Laboratory of Solid StateMicrostructures and Department ofPhysics, Nanjing University, Nanjing, China

    Shi-Jie Xiong

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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Xiong, SJ. (2009). Geometric Phase and Related Phenomena in Quantum Nanosystems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_247

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