Coherent states for dispersive pseudo-Landau-levels in strained honeycomb lattices

Abstract

Dirac fermions in graphene may experiment dispersive pseudo-Landau levels due to a homogeneous pseudomagnetic field and a position-dependent Fermi velocity induced by strain. In this paper, we study the (semi-classical) dynamics of these particles under such a physical context from an approach of coherent states. For this purpose we use a Landau-like gauge to built Perelomov coherent states by the action of a non-unitary displacement operator $D(\alpha )$ on the fundamental state of the system. We analyze the time evolution of the probability density and the generalized uncertainty principle as well as the Wigner function for the coherent states. Our results show how x-momentum dependency affects the motion periodicity and the Wigner function shape in phase space.

Appendices

Orthogonality and completeness relation

The PCSs satisfy the following relation

$$\begin{aligned} \vert \langle \varPsi _{z'}\vert \varPsi _{z}\rangle \vert =\left| \frac{\exp (z'^{*}z)+1}{\sqrt{(\exp (\vert z\vert ^{2})+1)(\exp (\vert z'\vert ^{2})+1)}}\right| \ne \delta (z'-z). \end{aligned}$$

(A.1)

This implies that these states are not orthogonal for $z\ne z'$, so we can say that the set of states $\varPsi _{\alpha }$ is overcomplete.

On the other hand, it is worth to remark that the PCS do not satisfy a completeness relation in the usual sense, since the superposition considers positive energy states only [83]. In order to clarify this, let us consider the following expression:

$$\begin{aligned} \int _{\mathbb {C}}\vert \varPsi _{z}\rangle \langle \varPsi _{z}\vert \frac{\mathrm{d}\mu (z)}{\pi }+\frac{1}{2}\sum _{n=1}^{\infty }\vert \varPsi _{n}\rangle \langle \varPsi _{n}\vert , \end{aligned}$$

(A.2)

where $\mathrm {d}\mu (z)$ is the measure defined in the complex plane as

$$\begin{aligned} \mathrm{d}\mu (z)=\frac{\vert z\vert \left( \exp \left( \vert z\vert ^{2}\right) +1\right) }{2\exp \left( \vert z\vert ^{2}\right) }\mathrm{d}\vert z\vert \,\mathrm{d}\theta . \end{aligned}$$

(A.3)

Now, by defining the variable $r=\vert z\vert$, Eq. (A.2) can be rewritten as:

$$\begin{aligned}&\int _{0}^{\infty }\int _{0}^{2\pi }\frac{e^{-r^{2}}r}{4\pi }\left[ \vert \varPsi _{0}\rangle \langle \varPsi _{0}\vert +\sum _{m=0}^{\infty }\frac{r^{m}e^{-im\theta }}{\sqrt{m!}}\vert \varPsi _{0}\rangle \langle \varPsi _{m}\vert +\sum _{n=0}^{\infty }\frac{r^{n}e^{in\theta }}{\sqrt{n!}}\vert \varPsi _{n}\rangle \langle \varPsi _{0}\vert \right. \nonumber \\&\quad \left. +\sum _{n,m=0}^{\infty }\frac{r^{n+m}e^{i(n-m)\theta }}{\sqrt{n!\,m!}}\vert \varPsi _{n}\rangle \langle \varPsi _{m}\vert \right] \mathrm{d}\theta \mathrm{d}r+\frac{1}{2}\sum _{n=1}^{\infty }\vert \varPsi _{n}\rangle \langle \varPsi _{n}\vert , \end{aligned}$$

(A.4)

which, after applying the results

$$\begin{aligned} \int _{0}^{2\pi }\exp \left( i(n-m)\theta \right) \mathrm {d}\theta =2\pi \delta _{mn}, \quad \int _{0}^{\infty }2r^{2n+1}\exp \left( -r^{2}\right) \mathrm {d}r=\varGamma (n+1)=n!,\nonumber \\ \end{aligned}$$

(A.5)

yields to

$$\begin{aligned} \sum _{n=0}^{\infty }\vert \varPsi _{n}\rangle \langle \varPsi _{n}\vert \equiv \mathbb {I}, \end{aligned}$$

(A.6)

where $\mathbb {I}$ denotes the identity operator in the Hilbert space $\mathcal {H}$ of Landau levels in the conduction band ($\lambda =+$).

Occupation number distribution

On the other hand, the probability of a PCS of being in an eigenstate $\varPsi _{n}$ is given by

$$\begin{aligned} P_{\alpha }(n)=\vert \langle \varPsi _{n}\vert \varPsi _{\alpha }\rangle \vert ^{2}=\left( \frac{2}{\exp \left( 2\mu \right) +1}\right) \times {\left\{ \begin{array}{ll} 1, &{}\quad n=0, \\ \frac{(2\mu )^{n}}{2n!}, &{}\quad n>0, \end{array}\right. } \end{aligned}$$

(B.1)

where $\mu =\vert \alpha \vert ^{2}$.

This occupation number distribution is compared with that of the CSs of the harmonic oscillator with eigenvalue $z_\mathrm{CS}\in \mathbb {C}$ in Fig. 12. For the harmonic oscillator case, $P_{z}(n)$ is a Poisson distribution, namely, $P_{z}(n)=\exp (-\tau )\tau ^{n}/n!$ with mean $\tau =\vert z_\mathrm{CS}\vert ^{2}$. In our case, as $\mu$ increases, we have that $P_{\alpha }(n)\sim P_{z}(n)$ with $\tau =2\mu$.

Fig. 12

Occupation number distribution $P_{\alpha }(n)$ in Eq. (B.1) for the coherent states $\varPsi _{\alpha }$ for different values of $\mu =\vert \alpha \vert ^{2}$. $P_{\alpha }(n)$ adjusts to Poisson distribution (solid curves) as $\mu$ grows