Quantum holonomies for displaced Landau Aharonov Casher states

Transcription

1 Quantum Inf Process 204) 3: DOI 0.007/s Quantum holonomies for displaced Landau Aharonov Casher states J. Lemos de Melo K. Bakke C. Furtado Received: 5 March 204 / Accepted: 2 May 204 / Published online: 27 May 204 Springer Science+Business Media New York 204 Abstract We construct displaced Fock states for a Landau Aharonov Casher system for neutral particles. Abelian and non-abelian geometric phases can be obtained in an adiabatic cyclic evolution using this displaced states. Moreover, we show that a possible logical base related to the angular momenta of the neutral particle with permanent magnetic dipole moment can be defined, and then quantum holonomies for specific paths can be built and used to implement one-qubit quantum gates. Keywords Displaced Fock states, Geometric phase Holonomic quantum computation Aharonov-Casher system Landau levels Introduction Coherent states was introduced by Glauber [] in 963 and have been a topic widely studied in quantum optics. Coherent states arise from the eigenstates of the annihilation operator or the shift of the ground state of the quantum harmonic oscillator in phase space[]. On the other hand, in 985, Venkata Satyanarayana [2] made a theoretical description of highly non-classical states of the harmonic oscillator, which are obtained by acting displacement operators on the Fock states. These states created by the action of the displacement operators on the Fock states are known as displaced Fock states [2] J. L. de Melo K. Bakke C. Furtado B) Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, João Pessoa, PB, Brazil furtado@fisica.ufpb.br J. L. de Melo jilvanlm@fisica.ufpb.br K. Bakke kbakke@fisica.ufpb.br 23

2 564 J. L. de Melo et al. and have received a great deal of attention in recent years. These states have interesting applications and unusual physical properties [3 6]. In Ref. [7], the authors present a direct observation of a classical analogue of the emergence of the displaced Fock states from the eigenstates of the harmonic oscillator in a Glauber-Fock photonic lattice. Recently, displaced single-photon Fock states of the light field have been produced [8] and investigated by means of the quantum optical homodyne tomography [9]. Furthermore, displaced Fock states representations of the Wigner function have been used to study theoretical aspects [0] and experimental investigations [,2] ofstates of instein-podolski-rosen Correlations. The structure of this paper is as follows: In Sect. 2, we start by introducing the displaced Landau Aharonov Casher states. In what follows, we show that Abelian and non-abelian geometric phases can be obtained in an adiabatic cyclic evolution. Finally, we discuss a possible way of building quantum holonomies which can be in the interest of studies of the holonomic quantum computation [3]; in Sect. 3, we present our conclusions. 2 Displaced Landau Aharonov Casher states Let us start our discussion by considering a neutral particle with a permanent magnetic dipole moment interacting with electric and magnetic fields. The Schrödinger-Pauli equation [4 6] that describes this quantum dynamics is i h ψ t = [ ˆp μ A AC ] 2 2m ψ + μ h 2mc ) ψ μ2 2 2mc 2 ψ μ σ B ψ, ) where the magnetic dipole moment is given by μ = μ σ ; thus, μ is the magnitude of the magnetic dipole moment and A ) AC = c 2 σ is an effective vector potential which is called the Aharonov-Casher vector potential σ = σ,σ 2,σ 3) are the Pauli matrices) [4,7]. The Landau quantization for neutral particles with a permanent magnetic dipole moment was proposed in Ref. [7] based on three conditions: The first condition corresponds to the absence of torque on the dipole moment; the second condition is that the external field must satisfy the electrostatic equations; finally, the third condition is the presence of an effective uniform magnetic field given by B AC = A AC, perpendicular to the plane of the motion of the particle. In order to satisfy these conditions, we can consider the magnetic dipole moment of the neutral particle is parallel or antiparallel to the z-axis and an electric field given by = λ 2 x, y, 0), where λ is a uniform volume charge density [7,8]. This field configuration is known as the symmetric gauge and the corresponding energy levels for a neutral particle with the magnetic dipole moment parallel to the z-axis [ moving in the xy-plane by using the analytic method) are given by η,l, + = ω AC η + l ], l + where η = 0,, 2,...is the quantum number associated with the radial modes, l = 0, ±, ±2,... is the momentum angular quantum number, and the label + means the magnetic dipole moment projection parallel to the z-axis [8]. Based on the quantum numbers 23

3 Quantum holonomies for displaced Landau Aharonov Casher states 565 {η, l, +}, each energy level n of the Landau Aharonov Casher quantization is given by the relation: n = η + l l. Observe an interesting characteristic of the Landau Aharonov Casher quantization: The angular momentum quantum number l defines the energy sublevels; thus, for each quantum number η, we have that the energy sublevels given by l 0 collapse. This corresponds to the well-known infinity) degeneracy characteristic of the Landau levels. On the other hand, by following Ref. [7], then, we neglect terms of O 2) and consider the magnetic dipole moment being initially parallel to the z-axis, that is, μ = μ ˆn = μ 0, 0, ). Therefore, we can define the creation and annihilation operator as a = γ π x isπ y ) ; a = γ π x + isπ y ), 2) where π = p μ A AC and γ = 2m h ω AC ) /2. The parameter ω AC = s μλ /mc 2 corresponds to the cyclotron frequency, where s = ± indicates the direction of revolution. Observe that the operators defined in q. 2) satisfy the commutation relation [ a, a ] =, which allows us to consider a neutral particle as a boson. Thereby, the time-independent Schrödinger equation can be written as h ω AC [a a + 2 ] + s) n + ˆp2 z 2m n = n, 3) where a n = n + n+, a n = n n and n = h ω AC [n + /2 + s)] + 2m k2. Since there is no torque on the magnetic dipole moment, we can take k = 0 and reduce the system to a planar system. It is worth mentioning that we can write the z-component of the angular momentum ˆL z by introducing the following raising and lowering operators: b =γ [ π x 2p x )+i s π y 2p y )] ; b =γ [ π x 2p x ) is π y 2p y )]. 4) Note that [ b, b ] = and the operator b b commutes with the Hamiltonian operator Ĥ = h ω AC [ a a s)]. Therefore, these operators share a set of eigenstates, that is, Ĥ n, l = n n, l and b b n, l = n + l) n, l, where the states n, l are orthonormal. In this way, the operator ˆL z is defined in terms of the operators given in qs. 2) and 4) as ˆL z = hb b a a). Moreover, we have that the Hamiltonian operator Ĥ commutes with ˆL z ;thus,wehave ˆL z n, l = l h n, l, where l = 0, ±, ±2,... Besides, by using the algebraic method, one can note that the ground state 0, 0 is defined by a 0, 0 = b 0, 0 = 0; thus, we can write a ) n b ) n+ l n, l = n! l +n)! 0, 0. From now on, we follow Refs. [9 2] in order to define the displaced Landau Aharonov Casher states. The displaced Landau Aharonov Casher states are defined by introducing a unitary transformation ˆD ν) = exp ν a ν a ), where ν = ν x + 23

4 566 J. L. de Melo et al. iν y. Thereby, the displaced Landau Aharonov Casher states are defined as ) n ν), l = ˆD ν) n, l = exp ν a ν a ) n b ) n+ l a 0, 0. 5) n! l + n)! The time evolution of the displaced Landau Aharonov Casher states is governed by the Hamiltonian operator which is given by Ĥ ν) = ˆD ν) Ĥ ˆD ν), that is [ Ĥ ν) = h ω AC a ν ) a ν) + 2 = 2m [ π x ν ) 2 x + π y + ν y γ γ ] + s) ) ] 2. 6) quation 6) shows us that each component of the Aharonov-Casher vector potential acquires a new term which is constant, whose meaning is related to the experimental setup. For instance, from the point of view of an experiment, new contributions to the Aharonov-Casher vector potential can be achieved by adding a constant electric field parallel to the plane of motion of the neutral particle. This is reasonable since the conditions for achieving the Landau Aharonov Casher quantization ) remains satisfied. Hence, by including a constant electric field = x, y, 0, the Aharonov- Casher vector potential obtain a new contribution given by A = c 2 ˆn ) = c 2 y, x, 0), then, the parameters ν x and ν y given in q. 6) becomes ν x = μγ c 2 y ; ν y = μγ c 2 x. 7) The change in the Hamiltonian operator 6) given by the presence of the parameters 7) corresponds to the transformation x, y) x + δx, y + δy) in the wave function, wherewehaveδx = x λ and δy = y λ. Thereby, the wave function of the ground state ψ 00 x, y) = x, y 0, 0 ψ 00 x + δx, y + δy). Hence, in a way analogous to the Landau states discussed in Ref. [2], the displaced Landau Aharonov Casher states defined in q. 5) correspond to shifted states of the Landau Aharonov Casher quantization )[7] in the phase space, whose shift is caused by the electric field = x, y, 0 we also assume that this shift is yielded in a real space). In recent years, the interest in building quantum holonomies has attracted a great deal of works [3,22 27] as a possible way to implement the quantum computation is due to geometric quantum phases to be robust to the environment noise [23,24]. For instance, quantum holonomies have been built for neutral particles with permanent magnetic and electric dipole moments in Refs. [23,24] based on the interaction between the permanent magnetic electric) dipole moment with external electric magnetic) field in the presence of topological defects. The logical basis was defined by the projections of the magnetic electric) dipole moment on the z-axis of the topological defect. In the present case, based on the correspondence n = η + l l discussed 23

5 Quantum holonomies for displaced Landau Aharonov Casher states 567 previously, we have that the state defined by η = 0 and l = and the state defined by η = and l 0 are degenerated, whose corresponding energy level is n =. In this way, let us make use of the fact that the energy sublevels given by l 0 collapse, and defined a possible logical basis based on the states n, l as 0 =, ; = 0 l=, l. 8) Hence, we can build quantum holonomies involving the displaced Landau Aharonov Casher states by using the logical states 8). It is well known in the current literature that the geometric quantum phase that arises from a cyclic evolution in a degenerate space has a non-abelian nature [28,29]. Then, according to the adiabatic theorem, the non-abelian connection one-form is given by [28 32] n τ) = i n, k n, l. 9) τ For the displaced Landau Aharonov Casher states 5), the corresponding control parameters τ are the components x and y of the electric field and the uniform volume charge density λ. From these definitions, we have the following non-abelian connections one-form for n = ): ) x = ζ 2 y λ δ k, l ; ) y = ζ 2 x λ δ k, l ; λ) = i ζ 2 2 x ) 2 + y λ ) l δ k, l i ζ 2 2 λ 3/2 ) λ /2 [ ) l y i x δ k, l ) ] l + y + i x δ k, l+, 0) where we have defined ζ = μ components of the connections one-form.fromq.0), we can see that the non-null 2 hc 2 ) ) x, A k, l y and λ) for the energy level n = and k = l are proportional to the identity matrix; therefore, these connections one-form correspond to Berry connections or the Mead-Berry vector potential [29,33]. As example, by keeping the control parameter λ constant and by taking the closed path of the adiabatic evolution of x and y in the x y plane given in Fig. ), then, the geometric phase [29,33 35] associated with these connections one-form is 23

6 568 J. L. de Melo et al. Fig. Closed path of the adiabatic evolution of the control parameters x and y in the x y plane 2x φn= = A l, l x ) d x + 2y A l, l x ) y d y + A l, l x ) d x x y 2x + y 2y A l, l ) y d y = 2 ζ 2 S, ) λ where S is the area enclosed by x and y as shown in Fig. ). The result ) corresponds to an analogue effect of the Aharonov-Bohm effect [36]. Furthermore, we can see in q. 0) that the non-abelian connection one-form λ) is a matrix which possesses non-diagonal terms for each energy level n = for k = l) that stem from the displacement parameters 7). Hence, we have that the geometric phase associated with the vector potential λ) has a non-abelian nature, which is of the interests of studies of the holonomic quantum computation [3,22 25]. The non-abelian geometric phase associated with the displaced Landau Aharonov Casher states is given by φ 2 n= = C λ) dλ, 2) for k = l ork = l + aswecanseeinq.0). From qs. 0), ) and 2), we can see that each holonomy can be written as U n C) = ˆP exp iφn) 2, where ˆP is the path ordering operator [3,29] and n =. However, we cannot obtain a general expression for the non-abelian geometric phase 2) without specifying the path. Let us first consider n = in order to make a cyclic adiabatic evolution and thus choose the closed path C 2 A B C D A as shown in Fig. 2). We can see, in the path A B, that the control parameters x = x and y = y are kept constants, while the control parameter λ varies from λ to λ 2.In this path, thus, we have a non-null contribution to the geometric phase 2); in the path B C, we have that the control parameters λ = λ 2 and x = x are kept constants while the control parameter y varies from y to 2y. Note that no contribution to the 23

7 Quantum holonomies for displaced Landau Aharonov Casher states 569 Fig. 2 Possible closed paths which yield a non-abelian geometric phase geometric phase 2) is yielded in this path since we have considered k = l. Hence, we can change the control parameter y, keeping the parameters λ = λ 2 and x = x constants, without changing the geometric phase; in the path C D, the control parameters x = x and y = 2y are kept constants while the control parameter λ varies from λ 2 to λ. In this path, we have a new contribution to the geometric phase 2) that differs from the first path because y = 2y. Finally, in the path D A we complete the circuit by keeping the control parameters λ = λ and x = x constants, while the control parameter y varies from 2y to y.again,wehaveno contribution to the geometric phase 2) since we have considered k = l. Hence, the quantum holonomy associated with the path C 2 A B C D A is U C 2 ) = cos τ 2 Î iσ 2 sin τ 2, 3) ) where Î is the 2 2 identity matrix, τ 2 = ζ 2 y2 y, = λ2 + λ + λ 2 λ and ζ = μ. A particular case is to consider τ 2 hc 2 2 = π/2. In this case, the unitary transformation 3) acts on the computational basis 8) by changing the logical states 0 and yielding a phase shift, which corresponds to an analogue of the Y -Pauli gate [37,38], up to a global phase factor. Let us discuss a new closed path: C A H G B A. Observe that in the path A H we have that the control parameters λ = λ and y = y are kept constants while the control parameter x varies from x to 2x. Note that no contribution to the geometric phase 2) is yielded in this path since we have considered k = l. Hence, we can change the control parameter x, keeping the parameters λ = λ and y = y constants, without changing the geometric phase; in the path H G, the control parameters x = 2x and y = y are kept constants while the control 23

8 570 J. L. de Melo et al. parameter λ varies from λ to λ 2. In this path, thus, we have a non-null contribution to the geometric phase 2); in the path G B, we have that the control parameters λ = λ 2 and y = y are kept constants while the control parameter x varies from 2x to x. Again, no contribution to the geometric phase 2) is yielded in this path since we have considered k = l; the last path is B A; then, we have the control parameters y = y and x = x are kept constants while the control parameter λ varies from λ 2 to λ. In this path, we have a new contribution to the geometric phase 2) that differs from the first path because x = 2x. Hence, the quantum holonomy associated with the path C A H G B A is U C ) = cos τ Î i σ sin τ, 4) where Î is the 2 2 identity matrix, τ = ζ 2 x2 x ), = λ2 + λ + λ2 λ and ζ = μ. Another particular case occurs by considering τ 2 hc 2 = π/2. Then, by applying the holonomy transformation 4), with τ = π/2, on the quantum states 8) we obtain that the logical states 0 are swapped, which corresponds to an analogue of the X-Pauli gate [37,38], up to a global phase factor. Another interesting example is given when we take the path C by considering τ = 3π/2 and, in the following, by taking the path C 2 by considering τ 2 = π/4. In this case, the corresponding holonomy matrix is given by U C 3 ) = U C 2 ) τ2 =π/4 U C ) ) τ =3π/2 = eiπ/2, 5) 2 which corresponds to a Hadamard quantum gate up to a global phase factor. Hence, we have shown a new way of building quantum holonomies for displaced Landau Aharonov Casher states and a new method of implementing one-qubit quantum gates. The logical states are made by the lowest Landau Aharonov Casher states as shown in q. 8), where we can perform analogues of the X-Pauli, the Y -Pauli and the Hadamard gates by changing the parameters related to the electric field and the uniform volume charge density λ. 3 conclusions The method proposed in the work is promising because it allows us to work with the holonomic quantum computation [3] and a well-known quantum system in the literature: the Landau quantization [39]. In particular, we have worked with the Landau quantization based on the Aharonov Casher effect [4], which is called the Landau Aharonov Casher quantization [7], because it is simpler than the Landau system [39] to build the displaced Fock states due to the need of including only a constant electric field. Thus, we have built the displaced Landau Aharonov Casher states and shown that a contribution to each component of the Aharonov-Casher vector potential arises from the presence of this class of quantum states. Such states allow us to find Abelian and non-abelian geometric phases, while the Landau Aharonov Casher states allow 23

9 Quantum holonomies for displaced Landau Aharonov Casher states 57 us to obtain just Abelian geometric phases. Moreover, we have seen that both Abelian and non-abelian geometric phases can be obtained in an adiabatic cyclic evolution in the parameter space, where the control parameters are the intensity of the field and uniform volume charge density λ. Finally, we have shown that quantum holonomies can be built in such a way that these unitary transformations give rise to analogues of the well-known one-qubits quantum gates, such as the X-Pauli, the Y -Pauli and the Hadamard gates. Acknowledgments We would like to thank CNPq, CAPS/NANOBIOTC, CNPQ/PNPD, CNPQ/ Universal for financial support. References. Glauber, R.J.: Phys. Rev. 3, ) 2. Venkata Satyanarayana, M.: Phys. Rev. D 32, ) 3. Král, P.: J. Mod. Opt. 37, ) 4. de Oliveira, F.A.M., Kim, M.S., Knight, P.L., Buẑek, V.: Phys. Rev. A 4, ) 5. Moya-Cessa, H.: J. Mod. Opt. 42, ) 6. Obada, A.-S.F., Abd Al-Kader, G.M.: J. Mod. Opt 46, ) 7. Keil, R., Perez-Leija, A., Dreisow, F., Heinrich, M., Moya-Cessa, H., Nolte, S., Christodoulides, D.N., Szameit, A.: Phys. Rev. Lett. 07, ) 8. Lvovsky, A.I., Babichev, S.A.: Phys. Rev. A 66, 080R) 2002) 9. Smithey, D.T., et al.: Phys. Rev. Lett. 70, ) 0. Banaszek, K., Wodkiewicz, K.: Phys. Rev. A 58, ). Kuzmich, A., Walmsley, I.A., Mandel, L.: Phys. Rev. Lett. 85, ) 2. Kuzmich, A., Walmsley, I.A., Mandel, L.: Phys. Rev. A 64, ) 3. Zanardi, P., Rasetti, M.: Phys. Lett. A 264, ) 4. Aharonov, Y., Casher, A.: Phys. Rev. Lett. 53, ) 5. Anandan, J.: Phys. Lett. A 38, ) 6. Anandan, J.: Phys. Rev. Lett. 85, ) 7. ricsson, M., Sjöqvist,.: Phys. Rev. A 65, ) 8. Bakke, K., Furtado, C.: Phys. Rev. A 80, ) 9. Wünsche, A.: Quantum Opt. 3, ) 20. Keil, R., et al.: Phys. Rev. Lett. 07, ) 2. Yang, W.-L., Chen, J.-L.: Phys. Rev. A 75, ) 22. kert, A., ricsson, M., Hayden, P., Inamory, H., Jones, J.A., Oi, D.K.L., Vedral, V.: J. Mod. Opt. 47, ) 23. Vedral, V.: Int. J. Quantum Inf., 2003) 24. Kuvshinov, V.I., Kuzmin, A.V.: Phys. Lett. A 36, ) 25. Margolin, A.., Strazhev, V.I., Tregubovich, A.Y.: Phys. Lett. A 303, ) 26. Bakke, K., Furtado, C.: Quantum Inf. Comput., ) 27. Bakke, K., Furtado, C.: Phys. Lett. A 375, ) 28. Wilczek, F., Zee, A.: Phys. Rev. Lett. 52, 2 984) 29. Bohm, A., Mostafazadeh, A., Koizumi, H., Niu, Q., Zwanziger, J.: The Geometric Phase in Quantum Systems: Foundations, Mathematical Concepts and Applications in Molecular and Condensed Matter Physics. Springer, New York 2003) 30. Mead, C.A., Truhlar, D.G.: J. Chem. Phys. 77, ) 3. Mead, C.A.: J. Chem. Phys. 78, ) 32. Mead, C.A.: Rev. Mod. Phys. 64, 5 992) 33. Berry, M.V.: Proc. R. Soc. Lond. Ser. A 392, ) 34. Mead, C.A., Truhlar, D.G.: J. Chem. Phys. 70, ) 35. Mead, C.A.: Chem. Phys. 49, ) 36. Aharonov, Y., Bohm, D.: Phys. Rev. 5, ) 23

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