BBilayer graphene coherent states
David J. Fern´andez C. ∗ and Dennis I. Mart´ınez-Moreno † Physics Department, Cinvestav, P.O. Box 14-740, 07000 Mexico City, Mexico
July 5, 2020
Abstract
In this paper we consider the interaction of electrons in bilayer graphene with a constanthomogeneous magnetic field which is orthogonal to the bilayer surface. Departing from theenergy eigenstates of the effective Hamiltonian, the corresponding coherent states will beconstructed. For doing this, first we will determine appropriate creation and annihilationoperators in order to subsequently derive the coherent states as eigenstates of the annihilationoperator with complex eigenvalue. Then, we will calculate some physical quantities, as theHeisenberg uncertainty relation, the probabilities and current density as well as the meanenergy value. Finally, we will explore the time evolution for these states and we will compareit with the corresponding evolution for monolayer graphene coherent states.
Carbon is the basis of all organic chemistry; due to the flexibility of their bonds, carbon-basedsystems show different structures with a wide variety of physical properties [1]. Graphene con-sists of a monolayer of carbon atoms arranged in a hexagonal crystal lattice with a distanceof 1.42˚A between nearest neighbour atoms. The discovery of this material and the interestingelectronic properties induced by the low energy excitations have attracted attention of the scien-tific community. In particular, the possibility arises that certain aspects of relativistic quantummechanics can be tested, like the Klein paradox or the anomalous Landau-Hall effect [1–5].The electronic structure of graphene is typically studied using the tight-binding model. As-suming that only next nearest neighbour hopping processes in the low-energy limit take place(close to the Dirac points), the equation ruling the electrons in graphene with applied staticmagnetic fields is the Dirac-Weyl equation [1, 3, 6]. This equation can be solved for a constanthomogeneous magnetic field, leading to the Landau levels of grapehene and the correspondingeigenfunctions, which constitutes the standard quantum mechanical framework [3] . However,an alternative approach exists, in which the system is addressed through coherent states. This ∗ david@fis.cinvestav.mx † dmartinez@fis.cinvestav.mx Let us note that the energy eigenfunctions and eigenvalues for graphene and some of its allotropes have beenas well determined for time-independent magnetic fields which are not necessarily homogeneous [3, 6–14] (seealso [15]). a r X i v : . [ qu a n t - ph ] J u l pproach supplies information which is supplementary to that obtained by the standard method,and it has been recently implemented for monolayer graphene [16].Bilayer graphene in a static magnetic field can be also studied through the tight-binding model. Assuming that the electrons can hop between the next-nearest neighbours in the samelayer or between layers via a perpendicular hopping parameter t ⊥ , an equation with an effectiveHamiltonian different from the monolayer one is obtained [17]. Once again, in the standardapproach this problem can be solved exactly for a constant homogeneous magnetic field, leadingto the Landau levels and the associated eigenfunctions for bilayer graphene, which are differentfrom the monolayer ones.With this in mind, it seems natural to explore the possibility of generating the bilayergraphene coherent states (BGCS), thus addressing the alternative approach already implementedfor monolayer graphene. This is the main goal of this paper, which is organized as follows. Insection 2 some general aspects of bilayer graphene are discussed. Through the tight binding model and some simplifying assumptions, the effective Hamiltonian in the low-energy approx-imation is obtained. The Landau levels for bilayer graphene in a magnetic field orthogonal toits surface are also determined. In section 3 the coherent states will be generated as eigenstatesof the annihilation operator with complex eigenvalue, and several physical quantities for thesestates will be calculated. In section 4 the time evolution for the coherent states of bilayer andmonolayer graphene are studied. Finally, our conclusions will be presented in section 5. Graphene is formed by carbon atoms arranged in a honeycomb hexagonal crystal lattice, withtwo atoms per unit cell belonging to sub-lattices A and B. Each atom of sub-lattice A is sur-rounded by three atoms of sub-lattice B and vice versa, as it is shown in Figure 1.We will consider here bilayer graphene, which is composed by two monolayers of carbonatoms and can be obtained by exfoliation of graphite. Its electronic structure can be studied inthe framework of the tight-binding model [17, 18], while its crystal structure is shown in Figure2. In bilayer graphene, the second layer of carbon atoms is rotated 60 o with respect to the firstlayer. Moreover, the sub-lattices A of the two layers lie exactly on top of one another. Thehopping parameters are: t ≈ t ⊥ ≈ A and the atom A ; t (cid:48) ≈ B and the atom B ; t (cid:48)(cid:48) ≈ − A ( A ) and the atom B ( B ) [1]. The distance between thetwo graphene layers (inter-planar spacing) is d ≈ t (cid:48) = 0) and between crossed sublattices ( t (cid:48)(cid:48) = 0). The model which takes into accountthese considerations [17] is characterized by the Hamiltonian2igure 1: The lattice structure of monolayer graphene is made out of the two triangular sub-lattices A (red points) and B (blue points); (cid:126)a i are the lattice vectors and (cid:126)δ i the nearest neighbourvectors. H ( (cid:126)k ) = tS ( (cid:126)k ) t ⊥ tS ∗ ( (cid:126)k ) 0 0 0 t ⊥ tS ∗ ( (cid:126)k )0 0 tS ( (cid:126)k ) 0 , (1)where S ( (cid:126)k ) = (cid:88) (cid:126)δ e i(cid:126)k · (cid:126)δ = 2 exp (cid:18) ik x a (cid:19) cos (cid:32) √ k y a (cid:33) + exp ( − ik x a ) , with (cid:126)k being the wave vector.In the low-energy approximation ( | E | (cid:28) t ⊥ ) the two important eigenvalues of the Hamilto-nian of Eq. (1) are E , ( (cid:126)k ) ≈ ± t | S ( (cid:126)k ) | t ⊥ ≈ ± (cid:126) q m ∗ , (2)where the effective mass is m ∗ = | t ⊥ | / (2 v F ) ≈ m e , with m e being the electron mass and v F the Fermi velocity in graphene. The other two energies E , ( (cid:126)k ) are separated by a gap ofsize 2 | t ⊥ | , thus they are irrelevant for low-energies.The replacement of (cid:126) q x and (cid:126) q y by the operators ˆ p x = − i (cid:126) ∂∂x and ˆ p y = − i (cid:126) ∂∂y respectively,leads to the following effective Hamiltonian for bilayer grapheneˆ H = 12 m ∗ (cid:18) p x − i ˆ p y ) (ˆ p x + i ˆ p y ) (cid:19) . (3)3igure 2: Lattice structure of bilayer graphene. The blue points correspond to the sub-latticesA and the red points to the sub-lattices B. Let us consider now the bilayer graphene placed in a constant homogeneous magnetic field whichis orthogonal to the material surface (the x-y plane). The interaction of the electrons with suchfield is described by the Hamiltonian of Eq. (3) where the momentum operator ˆ (cid:126)p is replaced byˆ (cid:126)p + ec ˆ (cid:126)A , according to the minimal coupling rule [3, 17], namely,ˆ H = 12 m ∗ (cid:104) (ˆ p x + ec ˆ A x ) − i (ˆ p y + ec ˆ A y ) (cid:105) (cid:104) (ˆ p x + ec ˆ A x ) + i (ˆ p y + ec ˆ A y ) (cid:105) . (4)In the Landau gauge the vector potential can be chosen as ˆ (cid:126)A = ˆ A ( x )ˆ e y so that ˆ (cid:126)B = ∇ × ˆ (cid:126)A =ˆ B ( x )ˆ e z . Additionally, in order to get a constant magnetic field orthogonal to the surface, along z direction ˆ (cid:126)B = B ˆ e z , the vector potential is selected as ˆ (cid:126)A = Bx ˆ e y .Let us define the operators (see [17])ˆ (cid:126)π := ˆ (cid:126)p + ec ˆ (cid:126)A, ˆ (cid:126)p = − i (cid:126) ∇ , (5)such that [ˆ π x , ˆ π y ] = − i (cid:126) ec B ˆ1 . The bosonic operators ˆ b − and ˆ b + are introduced now as followsˆ π − := ˆ π x − i ˆ π y = (cid:114) e (cid:126) Bc ˆ b − , (6)4 π + := ˆ π x + i ˆ π y = (cid:114) e (cid:126) Bc ˆ b + , (7)such that [ˆ b − , ˆ b + ] = ˆ1. Thus, the effective Hamiltonian becomesˆ H = 12 m ∗ (cid:18) π − ˆ π (cid:19) = (cid:126) ω ∗ c (cid:18) b − ˆ b +2 (cid:19) , ω ∗ c = eBm ∗ c , (8)where ω ∗ c is the cyclotron frequency for non-relativistic electrons with effective mass m ∗ .In order to find the stationary states, the time independent Schr¨odinger equationˆ H Ψ( x, y ) = E Ψ( x, y ) , (9)must be solved. Taking into account the translational invariance along y direction, the two-component spinor is proposed asΨ( x, y ) = exp( iky ) (cid:18) ψ + ( x ) ψ − ( x ) (cid:19) , (10)where k is the wave number in y -direction and ψ ± ( x ) describe the electron amplitude. Then,Eq. (9) yields the next two equations:( (cid:126) ω ∗ c )(ˆ b − ) ψ − ( x ) = Eψ + ( x ) , ( (cid:126) ω ∗ c )(ˆ b + ) ψ + ( x ) = Eψ − ( x ) . (11)In order to decouple this system, let us calculate ˆ H Ψ( x, y ) = E Ψ( x, y ) so that( (cid:126) ω ∗ c ) (ˆ b − ) (ˆ b + ) ψ + ( x ) = E ψ + ( x ) , ( (cid:126) ω ∗ c ) (ˆ b + ) (ˆ b − ) ψ − ( x ) = E ψ − ( x ) . (12)It is well known that the harmonic oscillator Hamiltonian satisfies:ˆ H HO = ˆ b + ˆ b − + 12 , ˆ H HO ψ n ( x ) = (cid:18) n + 12 (cid:19) ψ n ( x ) , (cid:104) ˆ b − , ˆ b + (cid:105) = ˆ1 . (13)If in Eq. (12) we take ψ + ( x ) := ψ n − ( x ) , ψ − ( x ) := ψ n ( x ) , (14)with ψ n ( x ) being the eigenfunctions of the harmonic oscillator Hamiltonian, it is obtained that E = ( (cid:126) ω ∗ c ) n ( n − . Thus, the eigenvalues of the Hamiltonian in Eq. (8) become E ± n = ± (cid:126) ω ∗ c (cid:112) n ( n − , (15)where the plus (minus) sign characterizes the energy electrons (holes), while the normalizedeigenstates are 5 ( x, y ) = exp( iky ) (cid:18) ψ ( x ) (cid:19) , Ψ ( x, y ) = exp( iky ) (cid:18) ψ ( x ) (cid:19) , Ψ n ( x, y ) = exp( iky ) √ (cid:18) ψ n − ( x ) ψ n ( x ) (cid:19) , n = 2 , , . . . (16)with (see [14]) ψ n ( x ) = (cid:114) n n ! (cid:16) ω π (cid:17) / H n (cid:20)(cid:114) ω (cid:18) x + 2 kω (cid:19)(cid:21) exp (cid:32) − ω (cid:18) x + 2 kω (cid:19) (cid:33) , ω = 2 m ∗ (cid:126) ω ∗ c . (17)We should note that E ± = E ± = 0, i.e., the ground state energy has a fourfold degeneracy(a double degeneracy due to electrons and the same due the holes). On the other hand, thelevels E ± n for n ≥ We will follow a procedure similar to the one used for monolayer graphene [16] in order togenerate the bilayer graphene coherent states. Thus, we need to identify first the appropriateannihilation and creation operators, which will be done next.
The annihilation operator ˆ A − for the Hamiltonian in Eq. (8) is proposed as followsˆ A − := (cid:18) f ( ˆ N )ˆ a − f ( ˆ N + ˆ1)ˆ a − (cid:19) , (18)where the operators ˆ a ± and ˆ N are given byˆ a ± = 1 √ (cid:18) ξ ∓ ddξ (cid:19) , ˆ N = ˆ a + ˆ a − ,ξ = (cid:114) ω (cid:18) x + 2 kω (cid:19) , and f , f are two auxiliary real functions. The operators ˆ a ± and ˆ b ± arerelated as follows i ˆ a + = ˆ b + , i ˆ a − = − ˆ b − . Note that ˆ A − Ψ n ( x, y ) = exp( iky ) √ (cid:18) √ n − f ( n − ψ n − ( x ) √ n f ( n ) ψ n − ( x ) (cid:19) . (19)In order to ensure that ˆ A − Ψ n ( x, y ) = c n Ψ n − ( x, y ), the functions f , f must fulfill √ n − f ( n −
3) = √ nf ( n ) ≡ c n . (20)6hus, the explicit expression for the annihilation operator ˆ A − becomesˆ A − = √ ˆ N +ˆ3 √ ˆ N +ˆ1 f ( ˆ N + ˆ3)ˆ a − f ( ˆ N + ˆ1)ˆ a − . (21)Finally, the creation operator ˆ A + is just the Hermitian conjugate of ˆ A − , i.e.,ˆ A + = ˆ a + √ ˆ N +ˆ3 √ ˆ N +ˆ1 f ( ˆ N + ˆ3) 00 ˆ a + f ( ˆ N + ˆ1) , (22)such that ˆ A + Ψ ( x, y ) = f (1)Ψ ( x, y ) , ˆ A + Ψ ( x, y ) = √ f (2) exp( iky ) (cid:18) ψ ( x ) (cid:19) , ˆ A + Ψ n ( x, y ) = √ n + 1 f ( n + 1)Ψ n +1 ( x, y ) , n = 2 , , . . . (23) ˆ A − The coherent states Ψ α ( x, y ), discovered for the first time by Schr¨odinger in 1926 [19] andrediscovered later by Klauder, Glauber and Sudarshan in the early 1960s [20–22], are quantumstates very close to classical states [23]. They can be defined as eigenstates of the previousannihilation operator ˆ A − with complex eigenvalue α , namely,ˆ A − Ψ α ( x, y ) = α Ψ α ( x, y ) , α ∈ C . (24)By expressing now the states Ψ α ( x, y ) as a linear combination of the eigenstates Ψ n ( x, y ),Ψ α ( x, y ) = ∞ (cid:88) n =0 a n Ψ n ( x, y ) = e iky (cid:34) a (cid:18) ψ ( x ) (cid:19) + a (cid:18) ψ ( x ) (cid:19) + ∞ (cid:88) n =2 a n √ (cid:18) ψ n − ( x ) ψ n ( x ) (cid:19)(cid:35) , (25)and substituting this expression into equation (24) we obtain a recurrence relationship for thecoefficients a n a = αa f (1) ,a = αa f (2) ,a n +1 = α √ n + 1 f ( n + 1) a n , n = 2 , , . . . (26)We can identify two different cases. 7 .2.1 Coherent states when f (1) (cid:54) = 0Suppose that f ( n ) (cid:54) = 0 ∀ n ∈ N , so that the coefficients a n become a n = √ α n √ n ! [ f ( n )]! a , n = 2 , , . . . (27)where, for any function q ( s ) such that s ∈ N , the generalized factorial function is defined by[ q ( s )]! := (cid:26) s = 0 ,q (1) · · · q ( s ) for s > . Thus, the coherent states in this case areΨ α ( x, y ) = (cid:34) | α | f (1) + 2 ∞ (cid:88) n =2 | α | n n ! ([ f ( n )]!) (cid:35) − × (cid:34) Ψ ( x, y ) + αf (1) Ψ ( x, y ) + √ ∞ (cid:88) n =2 α n √ n ! [ f ( n )]! Ψ n ( x, y ) (cid:35) . (28)Note that the states of Eq. (28) were normalized using the free coefficient a . f (1) = 0If f (1) = 0 it should happen that a = 0. Once again, two different subcases arise. A. Case with f (2) (cid:54) = 0 . If f (2) (cid:54) = 0 the free parameter becomes now a . The recurrence relationship for the coefficients a n leads to a n +1 = √ α n (cid:112) ( n + 1)! [ g ( n )]! a , n = 1 , , . . . (29)were g ( n ) := f ( n + 1). Substituting this result in Eq. (25), the normalized coherent states areΨ α ( x, y ) = (cid:34) ∞ (cid:88) n =1 | α | n ( n + 1)! ([ g ( n ]]!) (cid:35) − (cid:34) Ψ ( x, y ) + √ ∞ (cid:88) n =1 α n (cid:112) ( n + 1)! [ g ( n )]! Ψ n +1 ( x, y ) (cid:35) . (30) B. Case with f (2) = 0 . On the other hand, if f (2) = 0 it should happen that a = 0, therefore a is the free parameternow. By defining h ( n ) := f ( n + 2) ∀ n ∈ N , the normalized coherent states become nowΨ α ( x, y ) = (cid:34) ∞ (cid:88) n =0 | α | n ( n + 2)! ([ h ( n )]!) (cid:35) − ∞ (cid:88) n =0 α n (cid:112) ( n + 2)! [ h ( n )]! Ψ n +2 ( x, y ) . (31)8quations (28, 30, 31) contain three different sets of bilayer graphene coherent states (BGCS),all of them depending on the particular choice of the function f ( n ). Note that these BGCS looksimilar to the monolayer graphene coherent states (MGCS) derived in [16]. However, as we willsee later on the BGCS evolve in time in a completely different way as the MGCS do (see Section4). The Heisenberg uncertainty relation (HUR) has been useful for studying the standard coherentstates, since it is one of the most important quantities available to analyse the possible classicalbehaviour of a given quantum state [23].In order to introduce the HUR, the dimensionless position and momentum operators in termsof ˆ a ± are required, i.e., ˆ q = 1 √ a + + ˆ a − ) , ˆ p = i √ a + − ˆ a − ) . Then, the Heisenberg uncertainty relation for ˆ q and ˆ p in a coherent state Ψ α is given by( σ q ) α ( σ p ) α ≥ , (32)where the standard deviation for an arbitrary observable ˆ S is defined as follows σ S := (cid:113) (cid:104) ˆ S (cid:105) − (cid:104) ˆ S (cid:105) . We are going to calculate next the HUR for the BGCS built in the previous section. f (1) (cid:54) = 0First of all, let us make the particular choice f ( ˆ N ) := ˆ1. Thus, Eq. (28) reduces toΨ α ( x, y ) = 1 (cid:112) e r − r − (cid:34) Ψ ( x, y ) + α Ψ ( x, y ) + √ ∞ (cid:88) n =2 α n √ n ! Ψ n ( x, y ) (cid:35) , (33)where α = r exp( iθ ). 9igure 3: Heisenberg uncertainty relation ( σ q ) α ( σ p ) α as function of α for the BGCS in the casethat f ( n ) = 1.The mean values for ˆ q , ˆ p and their squares become (cid:104) ˆ q (cid:105) α = √ α )2 e r − r − (cid:34) exp( r ) + ∞ (cid:88) n =2 √ n − r n (cid:112) n ! ( n + 1)! (cid:35) , (34a) (cid:104) ˆ p (cid:105) α = √ α )2 e r − r − (cid:34) exp( r ) + ∞ (cid:88) n =2 √ n − r n (cid:112) n ! ( n + 1)!) (cid:35) , (34b) (cid:104) ˆ q (cid:105) α = 14 e r − r − (cid:34) r + 2 ∞ (cid:88) n =2 (2 n − r n n ! + 2 (cid:0) [Re( α )] − [Im( α )] (cid:1) × (cid:32) exp( r ) + ∞ (cid:88) n =2 r n (cid:112) ( n − n + 2)! (cid:33)(cid:35) , (34c) (cid:104) ˆ p (cid:105) α = 14 e r − r − (cid:34) r + 2 ∞ (cid:88) n =2 (2 n − r n n ! − (cid:0) [Re( α )] − [Im( α )] (cid:1) × (cid:32) exp( r ) + ∞ (cid:88) n =2 r n (cid:112) ( n − n + 2)! (cid:33)(cid:35) . (34d)From these the Heisenberg uncertainty relation ( σ q ) α ( σ p ) α can be calculated (see a plot inFigure 3). Note that ( σ q ) α ( σ p ) α → / α →
0, which is due to the BGCS in Eq. (33) tendto the eigenstate with minimum proper energy involved in the expansion, i.e., Ψ ( x, y ). Thisis the lowest value that the Heisenberg uncertainty relation can have, and it coincides with theone obtained for the standard coherent states [24].10 .3.2 Mean values when f (1) = 0 A. Case with f (2) (cid:54) = 0 . Let us choose now f ( ˆ N + ˆ1) := g ( ˆ N ) := √ ˆ N √ ˆ N +ˆ1 , so that f ( n + 1) (cid:54) = 0 ∀ n = 1 , , . . . . Then, thecoherent states of Eq. (30) becomeΨ α ( x, y ) = 1 (cid:112) e r − (cid:34) Ψ ( x, y ) + √ ∞ (cid:88) n =1 α n √ n ! Ψ n +1 ( x, y ) (cid:35) . (35)The mean values for ˆ q , ˆ p and their squares in this state are given by (cid:104) ˆ q (cid:105) α = √ α )2 e r − (cid:34) ∞ (cid:88) n =0 √ n + 2 r n (cid:112) n ! ( n + 1)! + ∞ (cid:88) n =1 r n (cid:112) ( n − n + 1)! (cid:35) , (36a) (cid:104) ˆ p (cid:105) α = √ α )2 e r − (cid:34) ∞ (cid:88) n =0 √ n + 2 r n (cid:112) n ! ( n + 1)! + ∞ (cid:88) n =1 r n (cid:112) ( n − n + 1)! (cid:35) , (36b) (cid:104) ˆ q (cid:105) α = 14 e r − (cid:34) ∞ (cid:88) n =0 (2 n + 1) r n n ! + 2 (cid:0) [Re( α )] − [Im( α )] (cid:1) × (cid:32) ∞ (cid:88) n =1 √ n + 1 r n (cid:112) ( n − n + 2)! + ∞ (cid:88) n =0 √ n + 3 r n (cid:112) n ! ( n + 1)! (cid:33)(cid:35) , (36c) (cid:104) ˆ p (cid:105) α = 14 e r − (cid:34) ∞ (cid:88) n =0 (2 n + 1) r n n ! − (cid:0) [Re( α )] − [Im( α )] (cid:1) × (cid:32) ∞ (cid:88) n =1 √ n + 1 r n (cid:112) ( n − n + 2)! + ∞ (cid:88) n =0 √ n + 3 r n (cid:112) n ! ( n + 1)! (cid:33)(cid:35) . (36d)Note that ( σ q ) α ( σ p ) α → / α → ( x, y ) in this limit, and the Heisenberg uncertainty relation reaches now amaximum for α = 0. B. Case with f (2) = 0 . Finally, let us consider that f ( ˆ N + ˆ2) = h ( ˆ N ) = ˆ N √ ˆ N +ˆ1 √ ˆ N +ˆ2 , thus the normalized coherent statesof Eq. (31) turn out to beΨ α ( x, y ) = 1 (cid:112) F (1 , r ) (cid:34) ∞ (cid:88) n =0 α n n ! (cid:112) ( n + 1)! Ψ n +2 ( x, y ) (cid:35) , (37)were p F q is the generalized hypergeometric function defined by p F q ( a , . . . , a p , b . . . , b q ; x ) = Γ( b ) · · · Γ( b q )Γ( a ) · · · Γ( a p ) ∞ (cid:88) n =0 Γ( a + n ) · · · Γ( a p + n )Γ( b + n ) · · · Γ( b q + n ) x n n ! . (38)11he mean values for the position and momentum operators and their squares are now (cid:104) ˆ q (cid:105) α = 1 √ α ) F (1 , r ) (cid:34) ∞ (cid:88) n =0 r n ( n + 1)! (cid:112) ( n + 2)! ( n !) + ∞ (cid:88) n =0 √ n + 3 r n n ! (cid:112) ( n + 2)! [( n + 1)!] (cid:35) , (39a) (cid:104) ˆ p (cid:105) α = 1 √ α ) F (1 , r ) (cid:34) ∞ (cid:88) n =0 r n ( n + 1)! (cid:112) ( n + 2)! ( n !) + ∞ (cid:88) n =0 √ n + 3 r n n ! (cid:112) ( n + 2)! [( n + 1)!] (cid:35) , (39b) (cid:104) ˆ q (cid:105) α = 12 F (1 , r ) (cid:34) ∞ (cid:88) n =0 (2 n + 3) r n ( n !) ( n + 1)! + (cid:0) [Re( α )] − [Im( α )] (cid:1) × (cid:32) ∞ (cid:88) n =0 √ n + 2 r n ( n + 2)! (cid:112) ( n + 3)! ( n !) + ∞ (cid:88) n =0 √ n + 4 r n n ! (cid:112) ( n + 1)! [( n + 2)!] (cid:33)(cid:35) , (39c) (cid:104) ˆ p (cid:105) α = 12 F (1 , r ) (cid:34) ∞ (cid:88) n =0 (2 n + 3) r n ( n !) ( n + 1)! − (cid:0) [Re( α )] − [Im( α )] (cid:1) × (cid:32) ∞ (cid:88) n =0 √ n + 2 r n ( n + 2)! (cid:112) ( n + 3)! ( n !) + ∞ (cid:88) n =0 √ n + 4 r n n ! (cid:112) ( n + 1)! [( n + 2)!] (cid:33)(cid:35) . (39d)Note that ( σ q ) α ( σ p ) α → / α → ( x, y ). Once again, the Heisenberg uncertainty relation reaches amaximum in this limit. The probability density is defined by ρ α ( x ) := Ψ † α ( x, y )Ψ α ( x, y ) . (40)For the states given in Eq. (16) we will use the notation ρ n,m ( x ) := ψ n − ( x ) ψ m − ( x ) + ψ n ( x ) ψ m ( x ) = ρ m,n ( x ) , n = 2 , , . . . (41)Note that for n = 0 and n = 1 we will have simply that ρ ( x ) = ψ ( x ) ψ ( x ) = | ψ ( x ) | ,ρ ( x ) = ψ ( x ) ψ ( x ) = | ψ ( x ) | . (42)12igure 4: Heisenberg uncertainty relation ( σ q ) α ( σ p ) α as function of α for the BGCS in the casethat f ( n ) = √ n − / √ n .Figure 5: Heisenberg uncertainty relation ( σ q ) α ( σ p ) α as function of α for the BGCS in the casethat f ( n ) = ( n − √ n − / √ n . 13n the other hand, the probability current for electrons described by the Hamiltonian (8)takes the form [25] J l ,α = (cid:126) m ∗ Im (cid:104) Ψ † α ( x, y ) j l Ψ α ( x, y ) (cid:105) , l = x, y, (43)where j x = σ x ∂ x + σ y ∂ y , (44a) j y = σ y ∂ x − σ x ∂ y , (44b)with σ i being the Pauli matrices.These quantities are time independent for stationary states, and they are also independentof y due to the translational symmetry along this direction. f (1) (cid:54) = 0For the BGCS of Eq. (33), with f ( n ) = 1, we obtain the following probability density andprobability current, respectively: ρ α ( x, r, θ ) = 12 e r − r − (cid:40) ∞ (cid:88) n =2 ∞ (cid:88) m =2 r n + m cos[( n − m ) θ ] √ n ! m ! ρ n,m ( x ) + [ ψ ( x )] + r [ ψ ( x )] +2 r cos θ ψ ( x ) ψ ( x ) + 2 ∞ (cid:88) n =2 r n √ n ! [cos( nθ ) ψ ( x ) + r cos [( n − θ ] ψ ( x )] ψ n ( x ) (cid:41) , (45) J α,x ( x, r, θ ) = (cid:126) m ∗ (2 e r − r − (cid:40) ∞ (cid:88) n =2 ∞ (cid:88) m =2 r n + m sin[( n − m ) θ ] √ n ! m ! (cid:110) ψ m ( x ) (cid:104)(cid:112) ( n − ω ψ n − ( x ) − (cid:16) ωx k (cid:17) ψ n − ( x ) (cid:105) + ψ m − ( x ) (cid:104) √ nω ψ n − ( x ) − (cid:16) ωx (cid:17) ψ n ( x ) (cid:105)(cid:111) + ∞ (cid:88) n =2 r n √ n ! (cid:104)(cid:112) ( n − ω ψ n − ( x ) − (cid:16) ωx k (cid:17) ψ n − ( x ) (cid:105) [ ψ ( x ) sin( nθ )+ ψ ( x ) r sin [( n − θ ]] − ∞ (cid:88) m =2 r m √ m ! (cid:8) √ ω ψ ( x ) ψ m − ( x ) r sin[( m − θ ] − (cid:16) ωx (cid:17) ψ m − ( x ) [ ψ ( x ) sin( mθ ) + ψ ( x ) r sin [( m − θ ]] (cid:111)(cid:111) . (46)14igure 6: (a) Probability density ρ α ( x ) and (b,c) probability currents ( m ∗ / (cid:125) ) J α,x,y ( x ) for f ( n ) =1 with ω = 1, k = 1 and r = 1. The blue, black and red colors correspond to θ = { , π/ , π/ } ,respectively. J α,y ( x, r, θ ) = (cid:126) m ∗ (2 e r − r − (cid:40) ∞ (cid:88) n =2 ∞ (cid:88) m =2 r n + m cos[( n − m ) θ ] √ n ! m ! (cid:110) ψ m ( x ) (cid:104)(cid:112) ( n − ω ψ n − ( x ) − (cid:16) ωx k (cid:17) ψ n − ( x ) (cid:105) − ψ m − ( x ) (cid:104) √ nω ψ n − ( x ) − (cid:16) ωx (cid:17) ψ n ( x ) (cid:105)(cid:111) + ∞ (cid:88) n =2 r n √ n ! (cid:104)(cid:112) ( n − ω ψ n − ( x ) − (cid:16) ωx k (cid:17) ψ n − ( x ) (cid:105) [ ψ ( x ) cos( nθ )+ ψ ( x ) r cos [( n − θ ]] − ∞ (cid:88) m =2 r m √ m ! (cid:8) √ ω ψ ( x ) ψ m − ( x ) r cos[( m − θ ] − (cid:16) ωx (cid:17) ψ m − ( x ) [ ψ ( x ) cos( mθ ) + ψ ( x ) r cos [( m − θ ]] (cid:111)(cid:111) . (47)Some graphs of these quantities, for different values of θ , are shown in Figure 6.15 .4.2 Probability density and probability current for f (1) = 0 A. Case with f (2) (cid:54) = 0 . For the case when f ( n ) = √ n − / √ n the BGCS of Eq. (35) lead to ρ α ( x, r, θ ) = 12 e r − (cid:40) [ ψ ( x )] + ∞ (cid:88) n =1 ∞ (cid:88) m =1 r n + m cos[( n − m ) θ ] √ n ! m ! ρ n +1 ,m +1 ( x )+2 ∞ (cid:88) n =1 r n cos( nθ ) √ n ! ψ ( x ) ψ n +1 ( x ) (cid:41) , (48) J α,x ( x, r, θ ) = (cid:126) m ∗ (2 e r − (cid:40) ∞ (cid:88) n =1 ∞ (cid:88) m =1 r n + m sin[( n − m ) θ ] √ n ! m ! (cid:110) ψ m +1 ( x ) (cid:104)(cid:112) ( n − ω ψ n − ( x ) − (cid:16) ωx k (cid:17) ψ n − ( x ) (cid:105) + ψ m − ( x ) (cid:104)(cid:112) ( n + 1) ω ψ n ( x ) − (cid:16) ωx (cid:17) ψ n +1 ( x ) (cid:105)(cid:111) + ∞ (cid:88) n =1 r n sin( nθ ) √ n ! ψ ( x ) (cid:104)(cid:112) ( n − ω ψ n − ( x ) − (cid:16) ωx k (cid:17) ψ n − ( x ) (cid:105) − ∞ (cid:88) m =1 r m sin( mθ ) √ m ! ψ m − ( x ) (cid:104) √ ω ψ ( x ) − (cid:16) ωx (cid:17) ψ ( x ) (cid:105)(cid:41) , (49) J α,y ( x, r, θ ) = (cid:126) m ∗ (2 e r − (cid:40) ∞ (cid:88) n =1 ∞ (cid:88) m =1 r n + m cos[( n − m ) θ ] √ n ! m ! (cid:110) ψ m +1 ( x ) (cid:104)(cid:112) ( n − ω ψ n − ( x ) − (cid:16) ωx k (cid:17) ψ n − ( x ) (cid:105) − ψ m − ( x ) (cid:104)(cid:112) ( n + 1) ω ψ n ( x ) − (cid:16) ωx (cid:17) ψ n +1 ( x ) (cid:105)(cid:111) + ∞ (cid:88) n =1 r n cos( nθ ) √ n ! ψ ( x ) (cid:104)(cid:112) ( n − ω ψ n − ( x ) − (cid:16) ωx k (cid:17) ψ n − ( x ) (cid:105) − ∞ (cid:88) m =1 r m cos( mθ ) √ m ! ψ m − ( x ) (cid:104) √ ω ψ ( x ) − (cid:16) ωx (cid:17) ψ ( x ) (cid:105)(cid:41) . (50)Some graphs of these quantities, for different values of θ , are shown in Figure 7. B. Case with f (2) = 0 . Finally, when f ( n ) = ( n − √ n − / √ n the BGCS of Eq. (37) produce the following probabilitydensity and probability current ρ α ( x, r, θ ) = 12 F (1 , r ) ∞ (cid:88) n =0 ∞ (cid:88) m =0 r n + m cos[( n − m ) θ ] n ! m ! (cid:112) ( n + 1)! ( m + 1)! ρ n +2 ,m +2 ( x ) , (51)16igure 7: (a) Probability density ρ α ( x ) and (b,c) probability currents ( m ∗ / (cid:125) ) J α,x,y ( x ) for f ( n ) = √ n − / √ n with ω = 1, k = 1 and r = 1. The blue, black and red colors correspond to θ = { , π/ , π/ } , respectively. J α,x ( x, r, θ ) = (cid:126) m ∗ F (1 , r ) ∞ (cid:88) n =0 ∞ (cid:88) m =0 r n + m sin[( n − m ) θ ] n ! m ! (cid:112) ( n + 1)! ( m + 1)! (cid:8) ψ m +2 ( x ) (cid:2) √ nω ψ n − ( x ) − (cid:16) ωx k (cid:17) ψ n ( x ) (cid:105) + ψ m ( x ) (cid:104)(cid:112) ( n + 2) ω ψ n +1 ( x ) − (cid:16) ωx (cid:17) ψ n +2 ( x ) (cid:105)(cid:111) , (52) J α,y ( x, r, θ ) = (cid:126) m ∗ F (1 , r ) ∞ (cid:88) n =0 ∞ (cid:88) m =0 r n + m cos[( n − m ) θ ] n ! m ! (cid:112) ( n + 1)! ( m + 1)! (cid:8) ψ m +2 ( x ) (cid:2) √ nω ψ n − ( x ) − (cid:16) ωx k (cid:17) ψ n ( x ) (cid:105) − ψ m ( x ) (cid:104)(cid:112) ( n + 2) ω ψ n +1 ( x ) − (cid:16) ωx (cid:17) ψ n +2 ( x ) (cid:105)(cid:111) . (53)Some graphs of these quantities, for different values of θ , are shown in Figure 8.17igure 8: (a) Probability density ρ α ( x ) and (b,c) probability currents ( m ∗ / (cid:125) ) J α,x,y ( x ) for f ( n ) =( n − √ n − / √ n with ω = 1, k = 1 and r = 1. The blue, black and red colors correspond to θ = { , π/ , π/ } , respectively.As can be seen in Figures (6 - 8) the maximum of the probability density moves along x direction when θ increases. On the other hand, non-null probability currents along both x and y directions are obtained for our BGCS. However, for θ = 0 only current along y - directionappears, as it happens for the monolayer graphene coherent states. The energy of the system is another quantity useful to characterize the coherent states. Inaddition, in the next section we will use it to explain the time evolution of the monolayergraphene coherent states, as well as of our bilayer graphene coherent states. This quantity iscalculated as follows: E = (cid:104) ˆ H (cid:105) α = (cid:104) Ψ α | ˆ H | Ψ α (cid:105) , (54)i.e., it is the mean value of the effective Hamiltonian of Eq. (8) in the states Ψ α ( x, y ). We willcalculate next this mean value for each set of coherent states previously derived.18igure 9: Left: Mean energy as function of α for the BGCS with f ( n ) = 1 and B = 1 /
8. Right:Mean energy as function of r = | α | for different magnetic field intensities: the blue, black andred colors correspond to B = { / , / , / } , respectively. (cid:104) ˆ H (cid:105) α for f (1) (cid:54) = 0For the coherent states of Eq. (33) we obtain (see Figure 9) (cid:104) Ψ α | ˆ H | Ψ α (cid:105) = (cid:18) (cid:126) ω ∗ c e r − r − (cid:19) ∞ (cid:88) n =2 (cid:112) n ( n − r n n ! . (55) (cid:104) ˆ H (cid:105) α for f (1) = 0 A. Case with f (2) (cid:54) = 0 . The coherent states of Eq. (35) lead to (see Figure 10) (cid:104) Ψ α | ˆ H | Ψ α (cid:105) = (cid:18) (cid:126) ω ∗ c e r − (cid:19) ∞ (cid:88) n =1 (cid:112) n ( n + 1) r n n ! . (56) B. Case with f (2) = 0 . Finally, for the coherent states of Eq. (37) we arrive to (see Figure 11) (cid:104) Ψ α | ˆ H | Ψ α (cid:105) = (cid:18) (cid:126) ω ∗ c F (1 , r ) (cid:19) ∞ (cid:88) n =0 (cid:112) ( n + 1)( n + 2) r n ( n !) ( n + 1)! . (57)19igure 10: Left: Mean energy as function of α for the BGCS with f ( n ) = √ n − / √ n and B = 1 /
8. Right: Mean energy as function of r = | α | for different magnetic field intensities: theblue, black and red colors correspond to B = { / , / , / } , respectively.Figure 11: Left: Mean energy as function of α for the BGCS with f ( n ) = ( n − √ n − / √ n and B = 1 /
8. Right: Mean energy as function of r = | α | for different magnetic field intensities:the blue, black and red colors correspond to B = { / , / , / } , respectively.20n Figures (9 - 11) we are showing the mean energy value for the three sets of BGCS builtpreviously. As can be seen, it is a growing function of α , but in the last case (Eq. (57)) it growsmore slowly than in the other two cases, since the structure is different for each set of coherentstates. Finally, we can also note that (cid:104) ˆ H (cid:105) α grows as the magnetic field intensity does, and thisincrease is proportional to B . The time evolution of a quantum state is obtained by acting the unitary operator ˆ U ( t, t ) asfollows [26] | Ψ( t ) (cid:105) = ˆ U ( t, t ) | Ψ( t ) (cid:105) , t > t , (58)where ˆ U ( t, t ) is known as the evolution operator . Note thatˆ U ( t , t ) = ˆ I, (59)with ˆ I being the identity operator.If we substitute Eq. (58) into the time-dependent Schr¨odinger equation we get i (cid:126) ∂ ˆ U ( t, t ) ∂t = ˆ H ˆ U ( t, t ) . (60)In particular, for time-independent Hamiltonians the last equation can be simply integrated,with the initial condition of Eq. (59), in order to obtain:ˆ U ( t , t ) = exp (cid:104) − i ( t − t ) ˆ H/ (cid:126) (cid:105) ⇒ | Ψ( t ) (cid:105) = exp (cid:104) − i ( t − t ) ˆ H/ (cid:126) (cid:105) | Ψ( t ) (cid:105) . (61) f (1) (cid:54) = 0One of the most important properties of the standard coherent states is their stability undertime-evolution i.e., a standard coherent state evolves into a standard coherent state at anytime [24]. We calculate next the time evolution of the BGCS of Eq. (33) for t = 0 (see Eq.(61)), which leads toΨ α ( x, y ; t ) = 1 (cid:112) e r − r − (cid:34) Ψ ( x, y ) + α Ψ ( x, y ) + ∞ (cid:88) n =2 √ α n √ n ! e − iω ∗ c √ n ( n − t Ψ n ( x, y ) (cid:35) . (62) f (1) = 0 A. Case with f (2) (cid:54) = 0 . For the states of Eq. (35) it is obtainedΨ α ( x, y ; t ) = 1 (cid:112) e r − (cid:34) Ψ ( x, y ) + ∞ (cid:88) n =2 √ α n − (cid:112) ( n − e − iω ∗ c √ n ( n − t Ψ n ( x, y ) (cid:35) . (63)21igure 12: Left: Probability density | Ψ α ( x, y ; t ) | for the BGCS with f ( n ) = 1 (Eq. (62)), r = 1, θ = 0 and ω ∗ c = 1. Right: Probability density | Ψ α ( x, y ; t ) | for some fixed times (the suggestedapproximate period and some of its multiples). The blue, green and orange lines correspond to τ = (cid:8) , √ π, √ π (cid:9) , respectively. B. Case with f (2) = 0 . Finally, for the BGCS of Eq. (37) we arrive atΨ α ( x, y ; t ) = 1 (cid:112) F (1 , r ) (cid:34) ∞ (cid:88) n =2 α n − ( n − (cid:112) ( n − e − iω ∗ c √ n ( n − t Ψ n ( x, y ) (cid:35) . (64)The probability densities for the evolving states of Eqs. (62, 63, 64) are shown in Figures12, 13 and 14, respectively. It is well known that the time stability of the standard coherent states comes from the factthat the energy levels for the harmonic oscillator are equally spaced [24]. For quantum systemswithout equally spaced energy levels this stability, in general, does not exist [27–29].For our BGCS the energy spectrum is not equidistant for all n (see Eq. (15)). However,starting from a certain integer (for n (cid:38)
2) this spectrum is practically lineal, thus making sta-ble in time (to a good approximation) the BGCS for which the contribution of the eigenstatesΨ ( x, y ) and Ψ ( x, y ) is small compared with the contribution of all other eigenstates. This canbe seen clearly in Figure 14, where the BGCS of Eq. (64) are stable in time, with the sameperiod as the auxiliary harmonic oscillator ( τ (cid:39) π/ω ∗ c ). For the other two examples, Eqs. (62,63), the coherent states could involve the eigenstate Ψ ( x, y ), Ψ ( x, y ) or both in a non-trivialway, thus making that their time evolution in general would not be stable (see Figures 12 and 13).22igure 13: Left: Probability density | Ψ α ( x, y ; t ) | for the BGCS with f ( n ) = √ n − / √ n (Eq.(63)), r = 1, θ = 0 and ω ∗ c = 1. Right: Probability density | Ψ α ( x, y ; t ) | for some fixed times(the suggested approximate period and some of its multiples). The blue, green and orange linescorrespond to τ = { , π, π } , respectively.Figure 14: Left: Probability density | Ψ α ( x, y ; t ) | for the BGCS with f ( n ) = ( n − √ n − / √ n (Eq. (64)), r = 1, θ = 0 and ω ∗ c = 1. Right: Probability density | Ψ α ( x, y ; t ) | for some fixedtimes (the suggested approximate period and some of its multiples). The blue, green and orangelines correspond to τ = { , π, π } , respectively.23espite the BGCS of Eqs. (62, 63) do not have always the period 2 π/ω ∗ c , however we proposea way to find a possible approximate period τ for these states. First of all let us note that, fora given α , τ is closely related to the mean energy value and to the eigenvalues bounding thisaverage. Hence, by setting α we must calculate first the value (cid:104) ˆ H (cid:105) α , then we must determinethe interval in which it lies, bounded by two consecutive proper energies E j +1 and E j , i.e., E j < (cid:104) ˆ H (cid:105) α < E j +1 . Finally, let us propose the next expression for the possible approximateperiod for the time-evolution of our BGCS: τ = 2 π (cid:126) E j +1 − E j . (65)For the states of Eq. (62) with | α | = 1 (such that 0 < (cid:104) ˆ H (cid:105) α = 0 . (cid:126) ω ∗ c < E ) we obtaina possible approximate period τ (cid:39) √ π/ω ∗ c and for the states of Eq. (63) with E < (cid:104) ˆ H (cid:105) α =1 . (cid:126) ω ∗ c < E we obtain τ (cid:39) π/ω ∗ c . In Figures 12 and 13 we also plot (right) the probabilitydensity for such a τ and some of its multiples in each case. The monolayer graphene coherent states (MGCS) for a constant homogeneous magnetic fieldwere recently derived in [16]. Let us calculate next the time evolution of these states, in similarcases that for bilayer graphene. f (1) (cid:54) = 0When taking f ( n ) = 1 the evolving MGCS becomeΨ α ( x, y ; t ) = 1 (cid:112) e r − (cid:34) Ψ ( x, y ) + ∞ (cid:88) n =1 √ α n √ n ! e − iv F √ nω t Ψ n ( x, y ) (cid:35) , (66)where ω have dimensions of (lenght) − . f (1) = 0 A. Case with f (2) (cid:54) = 0 . For f ( n ) = √ n − / √ n it is obtainedΨ α ( x, y ; t ) = e − r / ∞ (cid:88) n =1 α n − (cid:112) ( n − e − iv F √ nω t Ψ n ( x, y ) . (67) B. Case with f (2) = 0 . For f ( n ) = ( n − √ n − / √ n we arrive atΨ α ( x, y ; t ) = 1 (cid:112) F (1 , r ) ∞ (cid:88) n =2 α n − ( n − (cid:112) ( n − e − iv F √ nω t Ψ n ( x, y ) . (68)The probability densities for the states in Eqs. (66, 67, 68) are shown in Figures 15, 16 and17, respectively. 24igure 15: Left: Probability density | Ψ α ( x, y ; t ) | for the MGCS with f ( n ) = 1 (Eq. (66)), r = 1, θ = 0 and ( v F ω ) / = 1. Right: Probability density | Ψ α ( x, y ; t ) | for some fixed times(the suggested approximate period and some of its multiples). The blue, green and orange linescorrespond to τ = { , π, π } , respectively.Figure 16: Left: Probability density | Ψ α ( x, y ; t ) | for the MGCS with f ( n ) = √ n − / √ n (Eq.(67)), r = 1, θ = 0 and ( v F ω ) / = 1. Right: Probability density | Ψ α ( x, y ; t ) | for some fixedtimes (the suggested approximate period and some of its multiples). The blue, green and orangelines correspond to τ = { , π, π } , respectively.25igure 17: Left: Probability density | Ψ α ( x, y ; t ) | for the MGCS with f ( n ) = ( n − √ n − / √ n (Eq. (68)), r = 1, θ = 0 and ( v F ω ) / = 1. Right: Probability density | Ψ α ( x, y ; t ) | for somefixed times (the suggested approximate period and some of its multiples). The blue, green andorange lines correspond to τ = { , π, π } , respectively. For monolayer graphene the energy levels E n = (cid:126) v F √ nω are never equally spaced, thus we can-not approximate them in general by a linear expression. Nevertheless, the graphs of | Ψ α ( x, y ; t ) | for the MGCS show a certain periodicity, then we can try to find an approximate period in thesame way as for the BGCS (see Eq. (65) and the related discussion).For the states of Eq. (66) with | α | = 1, such that, E < (cid:104) ˆ H (cid:105) α = 0 . (cid:126) v F √ ω < E weobtain the possible approximate period τ (cid:39) π/v F √ ω . For the states of Eq. (67) we obtain that E < (cid:104) ˆ H (cid:105) α = 1 . (cid:126) v F √ ω < E and therefore a possible approximate period is τ (cid:39) π/v F √ ω .Finally, for the states of Eq. (68) with E < (cid:104) ˆ H (cid:105) α = 1 . (cid:126) v F √ ω < E we obtain τ (cid:39) π/v F √ ω .In Figures (15-17) we plot (right) the probability density for τ and some of its multiples in eachcase. Dirac electrons in monolayer graphene interacting with magnetics fields have been studied in [3]in terms of eigenstates and eigenvalues of the effective Hamiltonian, and more recently throughcoherent states [16]. Motivated by these works, in this paper we have derived as well the coherentstates for electrons in bilayer graphene interacting with a constant, homogeneous magnetic fieldorthogonal to the graphene layers. One of the main differences (perhaps the most importantone) between bilayer and monolayer graphene has to do with their energy spectrum, or Landaulevels, which defines quite clearly the different time evolution they will show.26e identified first the annihilation and creation operators for bilayer graphene, and then weconstructed the BGCS as eigenstates of the annihilation operator with complex eigenvalue α ,which involve an arbitrary function f of the number operator that can be chosen at convenience.This function leaves us a lot of freedom in the choice of the annihilation operator, and thus dif-ferent sets of BGCS can be built up.Several quantities useful to study our BGCS have been calculated, the most important onebeing the Heisenberg uncertainty relation. For the BGCS with f ( n ) = 1 the HUR has a min-imum, equal to 1 /
2, for α tending to zero while for the other two cases ( f ( n ) = √ n − / √ n and f ( n ) = ( n − √ n − / √ n ) this quantity reaches a maximum at the same limit, equal to3 / f allowed us to exclude selectively the states with minimum energy from such expansion, whichare annihilated by both operators ˆ A ± on the Hilbert space H generated by the eigenstates ofthe Hamiltonian in Eq. (8).The probability density and probability current for the BGCS have been as well calculatedfor different values of α = re iθ . We observe that as θ increases the probability density reachesa maximum which moves along x -direction, while the probability current shows a random be-haviour, depending on the set of BGCS under consideration. We calculated also the mean energyvalue, which grows as the magnetic field amplitude does. For the BGCS of Eqs. (33, 35) thebehaviour of (cid:104) ˆ H (cid:105) α is similar, unlike the BGCS of Eq. (37) for which the mean energy valuegrows more slowly. This quantity turned out to be useful to explain the quasi-periodic behaviourseen in the time evolution of our BGCS.Specifically, the time evolution of the BGCS indicates that for linear combinations such that n (cid:38)
2, where n labels the eigenstates of the superposition, such time evolution is stable (seeFigure 14), as for the standard coherent states [24], with the same period as for the harmonicoscillator involved ( τ = 2 π/ω ∗ c ). This is so since the Landau levels for bilayer graphene areapproximately equidistant for n (cid:38)
2, but for BGCS where the relative contribution of the eigen-states Ψ ( x, y ) and Ψ ( x, y ) is significant, the time evolution turns out to be quasi-stable, ascan be seen in Figures (12, 13).In this work we went further and calculated as well the time evolution of the MGCS derivedin [16]. An important point of these states is that, since the energy spectrum goes as √ n itis not possible in general to approximate this spectrum linearly. Despite, the time evolution isapproximately periodic for the three sets of coherent states built for monolayer graphene (seeFigures (15, 16, 17)). Thus, in this work we have proposed as well a way to calculate a possibleapproximate period for these states which are showing a quasi-stable motion, despite the systemdoes not have an equidistant spectrum.Finally, let us point out that some other approaches have been implemented recently toaddress these kind of two-dimensional systems. For example, in [33] the coherent states were de-rived for uniaxially strained graphene with non-equidistant Landau levels, and the corresponding27igner functions (WF) were evaluated. The time dependent WF for these coherent states showfluctuations between classical and quantum behaviour, showing as well a quasi-periodic motionwhen they evolve in time. On the other hand, some even and odd superpositions of MGCS havebeen recently addressed [34]. In our opinion, all these examples indicate a path to follow inthe future, when the coherent states approach could be applied to the so-called two-dimensionalDirac materials. References [1] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov and A.K. Geim,
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