Oscillatory multiband dynamics of free particles: The ubiquity of<i>zitterbewegung</i>effects

We derive a general and simple expression for the time-dependence of the position operator of a multi-band Hamiltonian with arbitrary matrix elements depending only on the momentum of the quasi-particle. Our result shows that in such systems the Zitterbewegung like term related to a trembling motion of the quasi-particle, always appears in the position operator. Moreover, the Zitterbewegung is, in general, a multi-frequency oscillatory motion of the quasi-particle. We derive a few different expressions for the amplitude of the oscillatory motion including that related to the Berry connection matrix. We present several examples to demonstrate how general and versatile our result is.PACS numbers: 71.70.Ej,73.22.Pr,03.65.Vf Schrödinger in his original paper has predicted a 'trembling' or in other words a rapid oscillatory motion of the center of the free wave packet for relativistic electron [1]. However, the Zitterbewegung is not strictly a relativistic effect [2-6] but can be observed in spintronic systems as well [7]. This work has initiated many other works with an aim to demonstrate the appearance of the Zitterbewegung not only for relativistic Dirac electrons. In these works a common feature is that the oscillatory motion of the free particles can be described only by one frequency.In our previous work we showed that for a wide class of Hamiltonians related to for example spintronic systems and graphene, the Zitterbewegung can be treated in a unified way [33]. Here the basic idea was that the Hamiltonian of several systems can be mapped to that modelling the precession of a virtual spin in an effective magnetic field. The coupled equations for this virtual spin precession and the orbital motion of the quasi-particle can easily be solved. Thus, the Zitterbewegung is arising because the virtual spin and the orbital motion for the quasi-particle are coupled. Our work suggests as natural question whether the phenomenon of the Zitterbewegung also arises for an even more general Hamiltonian.In the present work we extend the Zitterbewegung phenomena to a broader class of quantum Hamiltonians for free (quasi-) particles. In particularly, we derive a general and simple expression for the time-dependence of the position operator x(t) for a multi-band Hamiltonian given bywhere each matrix element is a differentiable function of the momentum p of the particle itself and n ≥ 2 is the number of degrees of freedom of the system. From our general expression for the position operator x(t) we shall show that i) the Zitterbewegung always appears for systems given by the Hamiltonian (1), i) for n > 2 the Zitterbewegung is in fact a multi-component oscillatory motion of the free quasi-particle, ii) for n = 2 we recover the results obtained earlier in the above mentioned references. To find the time dependence of the position operator x(t) of the quasi-particle in Heisenberg picture one needs to calculatewhere x(0) is the position operator at t = 0, ie, it equals to the position operator in Schrödinger picture. Because the mom...

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“…[19], and by J. Schliemann in a private communication using a direct calculation of the right hand side of Eq. (2).…”

Section: Luttinger Hamiltonianmentioning

confidence: 99%

“…[33] using a different approach. For many systems [5,[7][8][9][10][11][12][13][14][15][16][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] the Hamiltonian can be written in a quite general form:…”

Section: Spin In An Effective Magnetic Fieldmentioning

confidence: 99%

General theory of Zitterbewegung

Dávid

Cserti

2010

Phys. Rev. B

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“…26 In fact, there is a wide class of Hamiltonians ͑where the corresponding wave function is a spinor͒ which all exhibit ZB. 27 Neverthelessalthough almost omnipresent-the phenomenon has never been observed in nature. Therefore, it is desirable to propose FIG.…”

Section: Introductionmentioning

confidence: 99%

Photon-assisted electron transport in graphene: Scattering theory analysis

2007

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Photon-assisted electron transport in ballistic graphene is analyzed using scattering theory. We show that the presence of an ac signal ͑applied to a gate electrode in a region of the system͒ has interesting consequences on electron transport in graphene, where the low energy dynamics is described by the Dirac equation. In particular, such a setup describes a feasible way to probe energy dependent transmission in graphene. This is of substantial interest because the energy dependence of transmission in mesoscopic graphene is the basis of many peculiar transport phenomena proposed in the recent literature. Furthermore, we discuss the relevance of our analysis of ac transport in graphene to the observability of zitterbewegung of electrons that behave as relativistic particles ͑but with a lower effective speed of light͒.

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“…However, it is shown that there exist Zitterbewegung-like effects in graphene [2,3], superconductors [4], photonic crystal [5], single trapped ion [6,7] and semiconductor quantum wells [8,9]. Hence, the Zitterbewegung-like effect has attracted great attentions recently, both theoretically and experimentally in various physics fields [10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Zitterbewegungeffect in spin-orbit-coupled spin-1 ultracold atoms

et al. 2013

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The Zitterbewegung effect in spin-orbit coupled spin-1 cold atoms is investigated in the presence of the Zeeman field and a harmonic trap. It is shown that the Zeeman field and the harmonic trap have significant effect on the Zitterbewegung oscillatory behaviors. The external Zeeman field could suppress or enhance the Zitterbewegung amplitude and change the frequencies of oscillation. A much slowly damping Zitterbewegung oscillation can be achieved by adjusting both the linear and quadratic Zeeman field. Multi-frequency Zitterbewegung oscillation can be induced by the applied Zeeman field. In the presence of the harmonic trap, the subpackets corresponding to different eigenenergies would always keep coherent, resulting in the persistent Zitterbewegung oscillations. The Zitterbewegung oscillation would display very complicated and irregular oscillation characteristics due to the coexistence of different frequencies of the Zitterbewegung oscillation. Numerical results show that, the Zitterbewegung effect is robust even in the presence of interaction between atoms.

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Oscillatory multiband dynamics of free particles: The ubiquity ofzitterbewegungeffects

Cited by 71 publications

References 48 publications

General theory of Zitterbewegung

General theory of Zitterbewegung

Photon-assisted electron transport in graphene: Scattering theory analysis

Zitterbewegungeffect in spin-orbit-coupled spin-1 ultracold atoms

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