Noncommutative Dirac oscillator in an external magnetic field

Physics Letters A 376 (2012) 2467–2470

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Noncommutative Dirac oscillator in an external magnetic field Bhabani Prasad Mandal ∗ , Sumit Kumar Rai Department of Physics, Banaras Hindu University, Varanasi-221005, India

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 28 February 2012 Received in revised form 29 June 2012 Accepted 2 July 2012 Available online 4 July 2012 Communicated by P.R. Holland

We show that (2 + 1)-dimensional noncommutative Dirac oscillator in an external magnetic field is mapped onto the same but with reduced angular frequency in absence of magnetic field. We construct the relativistic Landau levels by solving corresponding Dirac equation in (2 + 1)-dimensional noncommutative phase space. All the Landau levels become independent of noncommutative parameter for a critical value of the magnetic field. Several other interesting features along with the relevance of such models in the study of atomic transitions in a radiation field have been discussed. © 2012 Elsevier B.V. All rights reserved.

1. Introduction

spin–orbit coupling. The Dirac Hamiltonian for a free particle with a linear harmonic potential term is given by

The noncommutativity has become a vital field of research owing its development in string theories, quantum field theories and in quantum mechanics [1]. The open strings end points are noncommutative (NC) in the presence of the background NS–NS B-field which indicates that the co-ordinates of D-branes are noncommutative [2]. There has been a lot of research papers based on perturbative and non-perturbative field theories in noncommutative space [3]. An extensive research has also been done on NC quantum mechanical systems [4–6]. In this Letter, we study the Dirac oscillator with a perpendicular magnetic field in an NC plane. NC plane is characterized by the following commutation relations [7,8]

[ xi , x j ] = i θ i j ,

[xi , p j ] = ih¯ δi j ,

[ p i , p j ] = 0,

(1.1)

whereas in the presence of magnetic field the above commutation relations are modified as

[ xi , x j ] = i θ i j ,

[xi , p j ] = ih¯ δi j ,  2 e eB [πi , π j ] = ih¯ B i j + i θ i j , c

2c

(1.2)

where θ is an arbitrary constant parameter. Dirac equation with a linear harmonic potential −imc ωβ α · r was first studied by Ito et al. [10]. Later Moshinsky and Szczepaniak [9] referred to this system as Dirac oscillator as in the nonrelativistic limit it reduces to a harmonic oscillator with a strong

*

Corresponding author. E-mail addresses: [email protected], [email protected] (B.P. Mandal), [email protected] (S.K. Rai). 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.07.001

H = c α · (p − imωβ r) + β mc 2 .

(1.3)

α , β are the usual Dirac matrices and m, c and ω are the rest mass of the particle, speed of light and oscillator frequency respectively. Because of the wide applicability of Dirac oscillator, it has become an extensive field of research in wide range of physics [11,12]. The properties of Dirac oscillator finds interesting connection with quantum optics [12]. Especially, its properties in 2 + 1 dimensions ((2 + 1)-D) is exactly mapped to that of Anti-Jaynes– Cummings (AJC) model [13] which describes the atomic transitions in a two level system. In this present Letter, we solve the (2 + 1)-D Dirac equation in NC phase space1 in which the noncommutativity is considered only in coordinate space [15,16] and construct the relativistic Landau levels (LL). The lowest Landau levels (LLL) are constructed explicitly in noncommutative space which is exactly same as the LLL obtained in commutative space in θ → 0 limit [17]. For a critical value of magnetic field, all the LL become independent of the NC parameter θ . The (2 + 1)-D NC Dirac oscillator in the presence of external magnetic field is mapped onto the same but with the reduced angular frequency in absence of magnetic field. Further, we also show the exact mapping of this relativistic model to the AJC model. In Section 2, we discuss NC Dirac oscillator and construct relativistic Landau levels. In Section 3, we show the connection of NC Dirac oscillator to AJC model. Section 4, is devoted to the conclusions. 1 Relativistic oscillators in the background of magnetic field has also been considered in Ref. [14]. However, their goal differs from our work.

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B.P. Mandal, S.K. Rai / Physics Letters A 376 (2012) 2467–2470



where two component |ψ =

2. NC Dirac oscillator and relativistic Landau levels



|ψ1  . Now we introduce the fol|ψ2 

lowing two sets of operators The Dirac Hamiltonian corresponding to the Dirac oscillator in the presence of external uniform magnetic field B in NC phase space [Eq. (1.2)] is given by

 H = cα · p −

eA c

 2

− imωβ r + β mc .

(2.1)

e is the electronic charge in this case and the magnetic field is produced by the vector potential A. This theory of (2 + 1)-D Dirac oscillator in usual commutative space with an external magnetic field is mapped onto the theory of (2 + 1)-D Dirac oscillator without magnetic field in the same space given by the Hamiltonian [17]

˜ β r) + β mc 2 , H  = c α · (p − imω

˜ =ω− where ω

ωc 2

(2.2)

.

The constant magnetic field decreases the angular frequency of the B |e | Dirac oscillator by half of the cyclotron frequency (ωc = mc ). For the sake of convenience, we consider the magnetic field, B along the z-direction and the vector potential is chosen in the symmetric gauge as A = (− B2 y , B2 x, 0). The Dirac oscillator in NC space can be studied in two alternative ways. Either by considering Hamiltonian in Eq. (2.1) and commutation relations in Eq. (1.2) or by considering the reduced Hamiltonian H  in Eq. (2.2) and the commutation relations in Eq. (1.1). We follow the later approach to write the Dirac Hamiltonian in (2 + 1)-D NC phase space in presence of magnetic field as

 H=

mc 2

˜ + h¯θκ ) z 2c κ Πz¯ − ic (mω

 ˜ + h¯θκ )¯z 2c κ Πz + ic (mω −mc 2

, (2.3)

where κ = (1 + m2ω˜h¯ θ ). Dirac matrices are chosen in terms of 2 × 2 Pauli spin matrices as αx = σx , α y = σ y and β = σz . We have expressed the Hamiltonian in terms of complex coordinate z = x + iy and conjugate momentum in noncommutative phase space Πz is defined as [7,15]

Πz = p z − Πz¯ = p z¯ +

ih¯ 2θ ih¯ 2θ

p z¯ = −ih¯

d dz d d z¯



(2.5)

= ( p x + ip y ),

(2.6)

2 1 2

[ z, Πz¯ ] = 0 = [¯z, Πz ],



2θ h¯ 2

(2.9)

Πz¯ .





a , a † = 1,

b , b † = 1,

(2.10)

where as other commutators are zero. Using these creation and annihilation operators, the coupled equation given by Eq. (2.8) can be expressed as

 







E − mc 2 |ψ1  = ig a† cosh Φ − ib sinh Φ |ψ2 ,







E + mc 2 |ψ2  = −ig a cosh Φ + ib† sinh Φ |ψ1 ,

(2.11)

where

g=c

2

θ

tanh Φ =

(mω˜ θ + h¯ κ )2 − (h¯ κ )2 h¯ κ

˜ θ + h¯ κ mω



and (2.12)

.

It is interesting to note that for a noncommutative parameter θ → 0, the coupling constant g = gc , where gc is the coupling constant in the commutative case [17]. In order to find energy spectra, we further define annihilation and creation operators as follows

C = a cosh Φ + ib† sinh Φ, C † = a† cosh Φ − ib sinh Φ,

(2.13)

satisfying the commutation relation as





C , C † = 1.

(2.14)

Eq. (2.11) can be decoupled as



E 2 − m2 c 4 |ψ1  = g 2 C † C |ψ1 ,







E 2 − m2 c 4 |ψ2  = g 2 C † C + 1 |ψ2 ,

(2.15)

[Πz , Πz¯ ] =

h¯ 2 2θ









˜+ E + mc |ψ2  = c 2κ Πz¯ − i mω

h¯ κ

θ h¯ κ

θ

,

 



 

E 2 − m2 c 4 n = g 2 n + 1 n ,

(2.16)

The relativistic Landau levels or the energy eigenvalues so obtained from Eq. (2.16) are as follows



g2 m2 c 4

n,

n = 0, 1 , 2 , . . .

(2.17)

with normalized negative and positive energy states2 given by

  z¯ |ψ2 ,

     ±   Ψ = c ± n; 1 + d± n − 1; − 1 , n n  n 2 2 2

  z |ψ1 ,





E 2 − m2 c 4 |n = g 2n|n,

E = ± E n = ±mc 2 1 +

(2.7)

˜+ E − mc 2 |ψ1  = c 2κ Πz + i mω  2





1

= ( p x − ip y ),

The time independent planar Dirac equation (H |ψ = E |ψ) corresponding to the Hamiltonian given in Eq. (2.3), can then be written in component form as



b =

Πz ,

Writing the spinors wave function in basis of number operators n = C † C as

[z, Πz ] = [¯z, Πz¯ ] = 0.



h¯

2



†

It can be checked easily that a, a† , b and b† satisfy the following algebra

(2.4)

which satisfy [ z, p z ] = i h¯ = [¯z, p z¯ ] and [ z, p z¯ ] = 0 = [¯z, p z ]. The noncommutative algebra in Eq. (1.1) [15] can then be written as

[z, z¯ ] = 2θ,

2θ

b=



z,

1 a† = √ z¯ , 2θ





z¯ ,

where

p z = −ih¯

1 a = √ z, 2θ

(2.8)

(2.18)

We adopt the notation for the state as |n, 12 ms  ≡ ψn ( z, z¯ )ξms where n is the

eigenvalue for the number operator, C † C and ms = ±1 are the eigenvalues of the operator σz i.e. σz | 12 ms  = ms | 12 ms , and ψn ( z, z¯ ) is the space part of the wave function in the coordinate representation whereas ξms is the spin part of the wave function. These states indicate the entanglement between orbital and spin degrees of freedom in the Dirac oscillator problem.

B.P. Mandal, S.K. Rai / Physics Letters A 376 (2012) 2467–2470

where



cn± = ±



E n+ ± mc 2

and dn± = ∓

2E n+

E n+ ∓ mc 2 2E n+

.

(2.19)

From Eq. (2.17), it is clear that energy levels are non-equidistant  ˜| in energy. For a large value of the magnetic field, g varies as |ω √ ˜ | ∝ B. Therefore, the level separation varies as B. and |ω The lowest Landau levels [LLL] are obtained by using the condition,

C ψ0 ( z, z¯ ) = 0.

(2.20)

Equivalently in coordinate representation, the space part wave function (ψ0 ( z, z¯ )) of the LLL satisfies



 ˜ mω ∂ + z ψ0 ( z, z¯ ) = 0. ∂ z¯ 2h¯ κ

(2.21)

phase space which depends on the strength of the magnetic field as well as on the noncommutative parameter θ . Hence, it is quite interesting to note that how two different theories are interlinked. This interconnection has been shown for an arbitrary strength of magnetic field and for arbitrary parameter θ . It provides an alternative approach to study the atomic transitions in two level systems using Dirac oscillator in noncommutative phase space and in the presence of magnetic field. The connection between quantum optics and relativistics quantum mechanics have been realized experimentally in [19] but is yet to be realized in noncommutative phase space. The connection of NC Dirac oscillator to AJC model can also be established by using Hamiltonian in Eq. (2.1) and the commutation relations in Eq. (1.2). The Hamiltonian in Eq. (2.1) can be written in the matrix form in NC space as



H=

mc 2 A ( p x + ip y ) − i B (x + iy )

By substituting

ψ0 ( z, z¯ ) = e

2469

A ( p x − ip y ) + i B (x − iy ) −mc 2



, (3.5)

− 2mh¯ωκ˜ z z¯

u 0 ( z, z¯ ),

(2.22)

we further obtain

where A and B are constants given as

A=c+

∂ u 0 ( z, z¯ ) = 0, ∂ z¯

(2.23)

as the defining rule for the LLL. We obtain the LLL in the coordinate representation, which is infinitely degenerate, as ˜ mω

ψ0 ( z, z¯ ) = zl e − 2h¯ κ z z¯ ,

l = 0, 1 , 2 , . . . ,

(2.24)

as u 0 ( z, z¯ ) is analytic function and the monomials zl with l = 0, 1, 2, . . . can serve as a linearly independent basis. The first excited state and other higher states in the coordinate space can be obtained by applying C † on the LLL repeatedly. The usual LLL in commutative space is obtained by considering θ → 0 limit, which is exactly same as in Ref. [17]. A very interesting situation occurs ˜ = 0. The Dirac for a critical value of magnetic field B = 2ω|emc | , i.e. ω oscillator stops oscillating and all the Landau levels become independent of θ .

e Bθ 4h¯

+

mωc θ 2h¯

B=

,

eB 2

+ mωc .

(3.6)

Now, it can be easily seen that this Hamiltonian is equivalent to



H=

mc 2 g˜ ∗ D

g˜ D † −mc 2



(3.7)

,

with [ D , D † ] = 1. This implies that Hamiltonian for NC Dirac oscillator in Eq. (2.1) is mapped to AJC model with same coupling constant g˜ . The commutation relations in Eq. (1.2) have been used to show the above result. This justifies the equivalence of two approaches mentioned in Section 2. It is further shown that the Zitterbewegung frequency for (2 + 1)-dimensional Dirac oscillator in an external magnetic field depends on the strength of the magnetic field as well as noncommutative parameter θ when calculated in noncommutative phase space. To show this, we eliminate the state |n − 1, − 12  from Eq. (2.18) and then express the initial pure state at t = 0 as

3. NC Dirac oscillator and AJC model

  +     1   n, (0) = E n d+ Ψ − − d− Ψ + . n n n n  2 2 mc

Now we would like to show the connection of this model to the AJC model [13] so widely used in the study of quantum optics. The Hamiltonian corresponding to the simple version of AJC model is given as,

Eq. (3.8) shows that the starting initial state is a superposition of both the positive and negative energy solutions. The time evolution of this state can be written as

−

˜∗

H A J C = g˜ σ a + g

+ †

2

σ a + σz mc .

(3.1)

On the other hand the JC model is described by the Hamiltonian,

H J C = g˜ σ − a† + g˜ ∗ σ + a + σz mc 2 ,



H=

mc 2 g˜ ∗ C

g˜ C † −mc 2



(3.3)

,

where g˜ = ig. This Hamiltonian in the Eq. (3.3) can be expressed as + †

˜∗

H = g˜ σ C + g

−

2

σ C + σz mc ,

  +     1   n, (t ) = E n d+ Ψ − e i ωn t − d− Ψ + e −i ωn t , n n n n  2 2 mc

ωn =

E n+ h¯

=

mc 2 h¯



1+

g2 m2 c 4

which is exactly same as H A J C in Eq. (3.1). g˜ is the coupling between spin and orbital degrees of freedom in noncommutative

n.

(3.10)

ωn depends on both magnetic field and the noncommutative parameter θ . Substituting |Ψ +  and |Ψ −  and the constants dn+ and dn− in Eq. (3.9) we obtain,      2  1  n, (t ) = cos (ωn t ) − imc sin (ωn t ) n, 1 (0)  2  2 E n+

  2  E n+ − m2 c 4 n − 1, − 1 (0). sin ( ω t ) +i n  2 4 m c

(3.4)

(3.9)

where ωn is the frequency of oscillation between positive and negative energy solutions and given as

(3.2)

which is quite different from Hamiltonian of AJC model as the positions of creation and annihilation operators are interchanged and defined as σ ± = 12 (σx ± i σ y ). σ + and σ − are usual spin raising and lowering operators respectively. The Hamiltonian given in Eq. (2.3) can be expressed as follows

(3.8)

2

(3.11)

This equation shows the oscillatory behavior between the states |n, 12  and |n − 1, − 12  which is exactly similar to atomic Rabi oscillations [13,20] occurring in the JC/AJC models. Rabi frequency is

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B.P. Mandal, S.K. Rai / Physics Letters A 376 (2012) 2467–2470

given by ωn given in Eq. (3.10). We further observe that the zitterbewegung frequency for the (2 + 1)-dimensional Dirac oscillator in the external magnetic field depends on the strength of the magnetic field. 4. Conclusion We have studied Dirac oscillator in 2 + 1 dimension in presence of transverse external magnetic field in noncommutative space by constructing suitable creation and annihilation operators in terms of properly chosen canonical pairs of coordinates and its corresponding momenta in a complex noncommutative phase space. We have shown that this theory maps onto Dirac oscillator without magnetic field but with different frequency in the same noncommutative space. The frequency is reduced by half of cyclotron frequency (ωc ). We have further solved Dirac equation in 2 + 1 dimension in noncommutative space to find explicit solutions. Lowest Landau levels are constructed explicitly in the co-ordinate representations which show the infinite degeneracy of such state in a simple manner. Landau levels √ are non-equidistant in energy and the level separation varies as B for large magnetic field. This feature is quite similar to those shown by graphene [18]. For a critical value of magnetic field B = 2ω|emc | , the Dirac oscillator stops oscillations and all the Landau levels become independent of θ . The coupling between orbital and spin degrees of freedom depends on the strength of the magnetic field and the noncommutative parameter. Certain specific value of magnetic field, the result reduces to that of Dirac oscillator in commutative plane. We have further shown that this model can be mapped onto AJC model, so widely used in the study of quantum optics. It is a good indication to experimentally observe some properties of noncommutativity. Zitterbewegung frequency has been calculated for noncommutative Dirac oscillator in transverse magnetic field. It depends on the strength of magnetic field and noncommutative parameter θ . For large magnetic √ field it varies as ωn ∝ nB. This oscillation between negative and positive energy is similar to the Rabi oscillations in the two level systems. Acknowledgements We thankfully acknowledge the financial support from the Department of Science and Technology (DST), Government of India, under the SERC project sanction grant No. SR/S2/HEP-29/2007. One of us (S.K.R.) would also like to thank CSIR, New Delhi for its financial support. References [1] N. Seiberg, E. Witten, JHEP 9909 (1999) 032; A. Connes, M. Douglas, A.S. Schwarz, JHEP 9802 (1998) 003; M.R. Douglas, N.A. Nekrasov, Rev. Mod. Phys. 73 (2001) 977. [2] C.S. Chu, P.M. Ho, Nucl. Phys. B 550 (1999) 151; C.S. Chu, P.M. Ho, Nucl. Phys. B 568 (2000) 447; F. Ardalan, H. Arfaei, M.M. Sheikh-Jabbari, Nucl. Phys. B 576 (2000) 578; J. Jing, Z.-W. Long, Phys. Rev. D 72 (2005) 126002.

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