Qutrit quantum computer with trapped ions A.B. Klimov1, J. C. Retamal2 , y C. Saavedra3 1 2 3

Departamento de F´ısica, Universidad de Guadalajara, Revoluci´on 1500, 44410, Guadalajara, Jal., M´exico Departamento de F´ısica, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile Departamento de F´ısica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile

Resumen We study a physical implementation of a qutrit quantum computer. Qutrits are embodied in electronic levels of trapped ions. We concentrate our attention on a universal two-qutrit gate. Using this gate and a general gate of an individual qutrit, any gate can be decomposed into a sequence of these gates.

The building blocks of a quantum computer are qubits [1]. They can be manipulated collectively or individually using quantum gates, in analogy with their classical counterpart. In the simplest case collective manipulation reduces to bipartite quantum gates, for instance the controlled-NOT gate [2], also called XOR gate. In this gate one qubit acts as a control and the other as the target, so that if the state of the control qubit is in state 1 , the state of the target qubit is flipped, and in other cases the target qubit remains unchanged. The experimental implementation of a two-qubit XOR has been studied in different physical contexts, as for instance in: Trapped ions [3,4], Nuclear Magnetic Resonance [5,6,7], cavity QED systems and Quantum Heterostructures [8,9,10]. In recent works the use of quantum entanglement in higher dimensional quantum systems has been studied in a number of works . Among the subjects of interest within this context we mention for example: The notion of entanglement generation and characterization in the case of three-level quantum systems[11]; The use of qutrits in a quantum key distribution protocol [12]; Generalized version of an XOR, [13]; Bounds on entanglement between qudits [14]; discrimination among Bell states of qudits [15]; entanglement among qudits [16], GHZ paradox for many qudits [17]; quantum computing with qudits[18]; quantum tomography for qudits states[19]; entanglement swaping between multiqudits systems [20]; Quantum communication complexity protocol with two entangled qutrits [21]. Quantum computation requires the possibility to implement gates operating on single and bipartite systems. In this work we present a protocol for a physical implementation of one and two-qutrit gates through coherent manipulation of ions in a linear trap [3]. A two qutrits gate change the state of a target qutrit conditioned to the state of a control qutrit. It has been shown [13] that a unitary GXOR gate between two qudits (d-dimensional elementary quantum systems)is conveniently defined by GXOR12 i where i j denotes the difference i XOR 3 gate given by

1

j

2

i

1

i

j

(1)

2

j, modulo d . In this work we study the realization of a two qutrit, GXOR12 i

1

j

2

i

1

j i

(2)

2

Let us consider is a linear array of trapped ions, where qutrit states are defined using the electronic levels of trapped ions. The schematic diagram for the relevant electronic transitions of ions is given in Fig. 1. In this case the relevant electronic levels are 0 , 1 , 2 3 4 . A key ingredient is the existence of degenerate electric dipole forbidden transitions 0 1 and 0 2 . These transitions can be addressed via Raman transitions through the independent channels associated with orthogonal polarizations, driven by classical fields Ω03 , Ω13 , Ω04 and Ω24 The ion level populations are manipulated by selecting the desired coherent operation. For example we can independently operate with transitions 0 1 , 0 2 , 1 2 by adjusting the parameters. The Hamiltonian describing this configuration, under the standard dipole and rotating wave approximation, is given by H

∑ωj

j j

e iν2t Ω04 0 4 Ω03 0 3

j

e

iν1 t

Ω13 1 3

Ω24 2 4

h c

(3)

236

Concepci´on, 13-15 de Noviembre de 2002

Figura 1. Electronic level structure of trapped ion. Quantum information of qutrits is stored in levels 0 , 1 , and 2 . The transitions involving effective interactions between levels 0 1 and 0 2 are driven by classical fields with different polarizations.

where j 0 1 2 3 4. The spatial profiles of Raman fields have been implicitly considered as phase factors. In a first step, only the carrier transition in the ion is considered, so that no explicit effects on the center of mass of the ion are included. Assuming the following conditions: ∆ ω 4 ω0 ν2 ω3 ω0 ν2 ω4 ω2 ν1 ω3 ω1 ν1 and ∆ Ω04 , Ω03 Ω31 Ω42 ; then levels 3 and 4 can be adiabatically eliminated. Thus the evolution operator in the restricted three dimensional space 1 , 2 0 is given by:

U ϕ

1

g 2 C ϕ gg C ϕ g g C ϕ 1 g 2 C ϕ ig sin ϕ ig sin ϕ

ig sinϕ

ig sin ϕ

(4)

cos ϕ

where ϕ Ωt is an adimensional interaction time, C ϕ cos ϕ 1 , and Ω2 κ 2 κ 2 . The coupling constants are defined as g κ Ω and g κ Ω, where κ Ω04 Ω24 ∆ and κ Ω03 Ω13 ∆. This evolution operator can be specialized to particular cases to implement all the required coherent operations between particular transitions. For example: to activate the transition 1 2 , we assume ϕ π in Eq. (4), so that

U1 where we have defined cos α

eiβ1 sin α 0 cos α 0 e iβ1 sin α cos α 1 0 0

κ κ

κ2

2 2

κ2

(5)

eiβ1 κκ κκ

(6)

2 and 0 1 can be similarly addressed by assuming κ 0 and κ 0 resThe transition 0 pectively, leading to operators U2 and U3 . In addition the combination of two dispersive processes in the 0 1 and the 0 2 channels leads to the dispersive evolution

UD

eiρ 0 0 0 eiε 0 0 0 e i ρ ε

(7)

This decomposition gives eight independent parameters, which allows generating any SU(3) operator [22] which represent a single qutrit gate. In particular we can calculate the decomposition F iUDU2U3U1 of the quantum Fourier transform for qutrits, defined by F j

1 2 2iπl j ∑e 3 l 0

3

l

(8)

In order to get entangled states of two systems, a quantum channel has to be established between them. In the ion quantum computer [3] the quantum channel between ions is the center of mass motion, which is addressed by adjusting a Raman transition to a given red sideband. We proceed along the same line of

XIII Simposio Chileno de F´ısica

237

reasoning, considering the ion model depicted above. Let us assume that we adjust the field amplitudes such that Ω04 Ω24 0 and Ω31 Ω03 0; or Ω04 Ω24 0 and Ω31 Ω03 0 . In both cases, after eliminating the upper excited level and adjusting to the first red sideband transition, we obtain the Hamiltonian describing the ion center of mass coupled to the transition 0 q :

Hn q

Ωq0 η q n 0 ae 2 N

iδt iφ

a† 0 n q eiδt

iφ

(9)

where a and a† are the annihilation and creation operators of the CM phonons, respectively; Ω q0 is the Rabi frequency; φ is the laser phase, δ is the detuning respect to the vibrational redsideband frequency, and

η kθ2 2Mνx is the Lamb Dicke parameter (kθ l cos θ with k the laser wave vector and θ the angle between the X axis and the direction of laser propagation). The subscript q 1 2 refers to the transition excited by the laser in a n-th ion. The following operations are necessary to implement a conditional phase shift: The resonant evolutions

Uml q φ 0

m

0

Uml q φ 0

m

1

Uml q φ q

m

0

0

0 lπ cos 0 2 lπ cos q 2 m

m m

lπ q m 0 2 lπ 0 ieiφ sin 0 m 1 2 1 ie

iφ

sin

where we have defined Ωq0 ηt 2 N lπ 2. These operations allows the interaction between distant ions through an ancillary system (phonons). Besides, the Hamiltonian (9) in the dispersive regime of the first red sideband gives rise to † † Dqm φ e iφaa q m q eiφa a g m g

In addition we can define a dispersive interaction associated to the carrier transition D m ϕq . These operations allows for an intensity dependent phase shift of the electronic levels. In the original proposal of an XOR gate, a conditional phase shift is implemented in the state 1 1 1 1 , which originates the transition 1 0 1 1 0 1 . Thus, the XOR 3 is implemented in the same way. The conditional phase shift on the n-th qutrit depending of the state of the m-th qutrit that we consider is given by: q

0 0 0 1 1 1 2 2 2 where

1 Pmn

and

2 Pmn

m m m m m m m m m

0 1 2 0 1 2 0 1 2

n n n n n n n n n

2 Pmn 1 e4iπ Pmn e2iπ

e2iπ e4iπ

3 3 3 3

0 0 0 1 1 1 2 2 2

m m m m m m m m m

0 n 1 n 2 n 0 n 1 n

2 n 0 n 1 n 2 n

are defined as follows:

(10)

P1 Um1 1 3π 2 Dn2 ξ2 D2n φ2 Dn1 ξ1 D1n φ1 Um1 1 π 2

P2 Um1 2 3π 2 Dn1 ξ1 D2n φ1 Dn1 ξ2 D1n φ2 Um1 2 π 2

with ξi 2π φi 2. Finally, the effective conditional change of the state of the target qutrit gives rise to the XOR 3 gate. In brief, 3 Fn 1 Pmn 2 Pmn 1 Fn XORmn (11) Universal quantum computation requires, in addition, a measurement scheme in the computational basis. This can be accomplished by connecting resonant interactions from 1 , 2 to states 3 , 4 respectively. Fast decay of excited optical levels through separated polarization channels allows us to discriminate between occupation of levels 1 , 2 , when fluorescence is observed; or level 0 when nothing is observed. This work was partially supported by CONACYT (Mexico) and CONICYT (Chile) Informaci´on Cu´antica e Interacciones Colectivas, Milenio ICM P99-135F and Grant No. FONDECYT 1010010.

238

Concepci´on, 13-15 de Noviembre de 2002

Referencias 1. M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information” (Cambridge Univ. Press., Cambridge, 2000). 2. C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett 75, 4714 (1995). 3. J. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). 4. Q. A. Turchette et al., C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, Phys. Rev. Lett. 81, 3631 (1998). 5. I. L. Chuang, N. Gershenfeld, and M. Kubinec, Phys. Rev. Lett. 80, 3408 (1998). 6. R. Laflamme, E. Knill, W. H. Zurek, P. Catasti, and S. V. S. Mariappan, Philos. Trans. R. Soc. London, Ser. A 356, 1941 (1998). 7. D. G. Cory, M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, Phys. Rev. Lett. 81, 2152 (1998). 8. D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). 9. B. E. Kane, Nature (London) 393, 133 (1998). 10. G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59, 2070 (1999). 11. Carlton M. Caves and Gerard J. Milburn, Opt. Comm. 179, 439 (2000). 12. D. Bruss and C. Macchiavello, Phys. Rev. Lett 88, 127901 (2002); N. J. Cerf, m. Bourennane, A. Karrlson, and N. Gisin, Phys. Rev. Lett. 88, 127902 (2002). 13. Alber G., Delgado A., Gisin N., Jex I, Journal of Physics A 34, 42 (2001). 14. Vivien M Kendon, Karol Zyczkowski, William J Munro, quant-ph/0203037. 15. Miloslav Dusek, quant-ph/0107119. 16. Kenneth A. Dennison, William K. Wootters, quant-ph/0106058. P. Rungta , W. J. Munro , K. Nemoto , P. Deuar , G. J. Milburn , C. M. Caves, quant-ph/0001075. 17. Nicolas J. Cerf , Serge Massar , Stefano Pironio, quant-ph/0107031. 18. Stephen D. Bartlett, Hubert de Guise, Barry C. Sanders, quant-ph/0109066. 19. R.T.Thew, K.Nemoto, A.G.White, W.J. Munro, quant-ph/0201052. 20. Jan Bouda, Vladimir Buzek, Journal of Phys. A Math. Gen. 34 4301-4311 May 25 2001. 21. Caslav Brukner, Marek Zukowski, and Anton Zeilinger quant-ph/ 0205080. 22. N. Vilenkin and A. Klimyk, Representation of Lie Group and Special Functions (Kluwer Academic, Dordrecht, 1991), Vols. 1-3.

Departamento de F´ısica, Universidad de Guadalajara, Revoluci´on 1500, 44410, Guadalajara, Jal., M´exico Departamento de F´ısica, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile Departamento de F´ısica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile

Resumen We study a physical implementation of a qutrit quantum computer. Qutrits are embodied in electronic levels of trapped ions. We concentrate our attention on a universal two-qutrit gate. Using this gate and a general gate of an individual qutrit, any gate can be decomposed into a sequence of these gates.

The building blocks of a quantum computer are qubits [1]. They can be manipulated collectively or individually using quantum gates, in analogy with their classical counterpart. In the simplest case collective manipulation reduces to bipartite quantum gates, for instance the controlled-NOT gate [2], also called XOR gate. In this gate one qubit acts as a control and the other as the target, so that if the state of the control qubit is in state 1 , the state of the target qubit is flipped, and in other cases the target qubit remains unchanged. The experimental implementation of a two-qubit XOR has been studied in different physical contexts, as for instance in: Trapped ions [3,4], Nuclear Magnetic Resonance [5,6,7], cavity QED systems and Quantum Heterostructures [8,9,10]. In recent works the use of quantum entanglement in higher dimensional quantum systems has been studied in a number of works . Among the subjects of interest within this context we mention for example: The notion of entanglement generation and characterization in the case of three-level quantum systems[11]; The use of qutrits in a quantum key distribution protocol [12]; Generalized version of an XOR, [13]; Bounds on entanglement between qudits [14]; discrimination among Bell states of qudits [15]; entanglement among qudits [16], GHZ paradox for many qudits [17]; quantum computing with qudits[18]; quantum tomography for qudits states[19]; entanglement swaping between multiqudits systems [20]; Quantum communication complexity protocol with two entangled qutrits [21]. Quantum computation requires the possibility to implement gates operating on single and bipartite systems. In this work we present a protocol for a physical implementation of one and two-qutrit gates through coherent manipulation of ions in a linear trap [3]. A two qutrits gate change the state of a target qutrit conditioned to the state of a control qutrit. It has been shown [13] that a unitary GXOR gate between two qudits (d-dimensional elementary quantum systems)is conveniently defined by GXOR12 i where i j denotes the difference i XOR 3 gate given by

1

j

2

i

1

i

j

(1)

2

j, modulo d . In this work we study the realization of a two qutrit, GXOR12 i

1

j

2

i

1

j i

(2)

2

Let us consider is a linear array of trapped ions, where qutrit states are defined using the electronic levels of trapped ions. The schematic diagram for the relevant electronic transitions of ions is given in Fig. 1. In this case the relevant electronic levels are 0 , 1 , 2 3 4 . A key ingredient is the existence of degenerate electric dipole forbidden transitions 0 1 and 0 2 . These transitions can be addressed via Raman transitions through the independent channels associated with orthogonal polarizations, driven by classical fields Ω03 , Ω13 , Ω04 and Ω24 The ion level populations are manipulated by selecting the desired coherent operation. For example we can independently operate with transitions 0 1 , 0 2 , 1 2 by adjusting the parameters. The Hamiltonian describing this configuration, under the standard dipole and rotating wave approximation, is given by H

∑ωj

j j

e iν2t Ω04 0 4 Ω03 0 3

j

e

iν1 t

Ω13 1 3

Ω24 2 4

h c

(3)

236

Concepci´on, 13-15 de Noviembre de 2002

Figura 1. Electronic level structure of trapped ion. Quantum information of qutrits is stored in levels 0 , 1 , and 2 . The transitions involving effective interactions between levels 0 1 and 0 2 are driven by classical fields with different polarizations.

where j 0 1 2 3 4. The spatial profiles of Raman fields have been implicitly considered as phase factors. In a first step, only the carrier transition in the ion is considered, so that no explicit effects on the center of mass of the ion are included. Assuming the following conditions: ∆ ω 4 ω0 ν2 ω3 ω0 ν2 ω4 ω2 ν1 ω3 ω1 ν1 and ∆ Ω04 , Ω03 Ω31 Ω42 ; then levels 3 and 4 can be adiabatically eliminated. Thus the evolution operator in the restricted three dimensional space 1 , 2 0 is given by:

U ϕ

1

g 2 C ϕ gg C ϕ g g C ϕ 1 g 2 C ϕ ig sin ϕ ig sin ϕ

ig sinϕ

ig sin ϕ

(4)

cos ϕ

where ϕ Ωt is an adimensional interaction time, C ϕ cos ϕ 1 , and Ω2 κ 2 κ 2 . The coupling constants are defined as g κ Ω and g κ Ω, where κ Ω04 Ω24 ∆ and κ Ω03 Ω13 ∆. This evolution operator can be specialized to particular cases to implement all the required coherent operations between particular transitions. For example: to activate the transition 1 2 , we assume ϕ π in Eq. (4), so that

U1 where we have defined cos α

eiβ1 sin α 0 cos α 0 e iβ1 sin α cos α 1 0 0

κ κ

κ2

2 2

κ2

(5)

eiβ1 κκ κκ

(6)

2 and 0 1 can be similarly addressed by assuming κ 0 and κ 0 resThe transition 0 pectively, leading to operators U2 and U3 . In addition the combination of two dispersive processes in the 0 1 and the 0 2 channels leads to the dispersive evolution

UD

eiρ 0 0 0 eiε 0 0 0 e i ρ ε

(7)

This decomposition gives eight independent parameters, which allows generating any SU(3) operator [22] which represent a single qutrit gate. In particular we can calculate the decomposition F iUDU2U3U1 of the quantum Fourier transform for qutrits, defined by F j

1 2 2iπl j ∑e 3 l 0

3

l

(8)

In order to get entangled states of two systems, a quantum channel has to be established between them. In the ion quantum computer [3] the quantum channel between ions is the center of mass motion, which is addressed by adjusting a Raman transition to a given red sideband. We proceed along the same line of

XIII Simposio Chileno de F´ısica

237

reasoning, considering the ion model depicted above. Let us assume that we adjust the field amplitudes such that Ω04 Ω24 0 and Ω31 Ω03 0; or Ω04 Ω24 0 and Ω31 Ω03 0 . In both cases, after eliminating the upper excited level and adjusting to the first red sideband transition, we obtain the Hamiltonian describing the ion center of mass coupled to the transition 0 q :

Hn q

Ωq0 η q n 0 ae 2 N

iδt iφ

a† 0 n q eiδt

iφ

(9)

where a and a† are the annihilation and creation operators of the CM phonons, respectively; Ω q0 is the Rabi frequency; φ is the laser phase, δ is the detuning respect to the vibrational redsideband frequency, and

η kθ2 2Mνx is the Lamb Dicke parameter (kθ l cos θ with k the laser wave vector and θ the angle between the X axis and the direction of laser propagation). The subscript q 1 2 refers to the transition excited by the laser in a n-th ion. The following operations are necessary to implement a conditional phase shift: The resonant evolutions

Uml q φ 0

m

0

Uml q φ 0

m

1

Uml q φ q

m

0

0

0 lπ cos 0 2 lπ cos q 2 m

m m

lπ q m 0 2 lπ 0 ieiφ sin 0 m 1 2 1 ie

iφ

sin

where we have defined Ωq0 ηt 2 N lπ 2. These operations allows the interaction between distant ions through an ancillary system (phonons). Besides, the Hamiltonian (9) in the dispersive regime of the first red sideband gives rise to † † Dqm φ e iφaa q m q eiφa a g m g

In addition we can define a dispersive interaction associated to the carrier transition D m ϕq . These operations allows for an intensity dependent phase shift of the electronic levels. In the original proposal of an XOR gate, a conditional phase shift is implemented in the state 1 1 1 1 , which originates the transition 1 0 1 1 0 1 . Thus, the XOR 3 is implemented in the same way. The conditional phase shift on the n-th qutrit depending of the state of the m-th qutrit that we consider is given by: q

0 0 0 1 1 1 2 2 2 where

1 Pmn

and

2 Pmn

m m m m m m m m m

0 1 2 0 1 2 0 1 2

n n n n n n n n n

2 Pmn 1 e4iπ Pmn e2iπ

e2iπ e4iπ

3 3 3 3

0 0 0 1 1 1 2 2 2

m m m m m m m m m

0 n 1 n 2 n 0 n 1 n

2 n 0 n 1 n 2 n

are defined as follows:

(10)

P1 Um1 1 3π 2 Dn2 ξ2 D2n φ2 Dn1 ξ1 D1n φ1 Um1 1 π 2

P2 Um1 2 3π 2 Dn1 ξ1 D2n φ1 Dn1 ξ2 D1n φ2 Um1 2 π 2

with ξi 2π φi 2. Finally, the effective conditional change of the state of the target qutrit gives rise to the XOR 3 gate. In brief, 3 Fn 1 Pmn 2 Pmn 1 Fn XORmn (11) Universal quantum computation requires, in addition, a measurement scheme in the computational basis. This can be accomplished by connecting resonant interactions from 1 , 2 to states 3 , 4 respectively. Fast decay of excited optical levels through separated polarization channels allows us to discriminate between occupation of levels 1 , 2 , when fluorescence is observed; or level 0 when nothing is observed. This work was partially supported by CONACYT (Mexico) and CONICYT (Chile) Informaci´on Cu´antica e Interacciones Colectivas, Milenio ICM P99-135F and Grant No. FONDECYT 1010010.

238

Concepci´on, 13-15 de Noviembre de 2002

Referencias 1. M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information” (Cambridge Univ. Press., Cambridge, 2000). 2. C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett 75, 4714 (1995). 3. J. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). 4. Q. A. Turchette et al., C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, Phys. Rev. Lett. 81, 3631 (1998). 5. I. L. Chuang, N. Gershenfeld, and M. Kubinec, Phys. Rev. Lett. 80, 3408 (1998). 6. R. Laflamme, E. Knill, W. H. Zurek, P. Catasti, and S. V. S. Mariappan, Philos. Trans. R. Soc. London, Ser. A 356, 1941 (1998). 7. D. G. Cory, M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, Phys. Rev. Lett. 81, 2152 (1998). 8. D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). 9. B. E. Kane, Nature (London) 393, 133 (1998). 10. G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59, 2070 (1999). 11. Carlton M. Caves and Gerard J. Milburn, Opt. Comm. 179, 439 (2000). 12. D. Bruss and C. Macchiavello, Phys. Rev. Lett 88, 127901 (2002); N. J. Cerf, m. Bourennane, A. Karrlson, and N. Gisin, Phys. Rev. Lett. 88, 127902 (2002). 13. Alber G., Delgado A., Gisin N., Jex I, Journal of Physics A 34, 42 (2001). 14. Vivien M Kendon, Karol Zyczkowski, William J Munro, quant-ph/0203037. 15. Miloslav Dusek, quant-ph/0107119. 16. Kenneth A. Dennison, William K. Wootters, quant-ph/0106058. P. Rungta , W. J. Munro , K. Nemoto , P. Deuar , G. J. Milburn , C. M. Caves, quant-ph/0001075. 17. Nicolas J. Cerf , Serge Massar , Stefano Pironio, quant-ph/0107031. 18. Stephen D. Bartlett, Hubert de Guise, Barry C. Sanders, quant-ph/0109066. 19. R.T.Thew, K.Nemoto, A.G.White, W.J. Munro, quant-ph/0201052. 20. Jan Bouda, Vladimir Buzek, Journal of Phys. A Math. Gen. 34 4301-4311 May 25 2001. 21. Caslav Brukner, Marek Zukowski, and Anton Zeilinger quant-ph/ 0205080. 22. N. Vilenkin and A. Klimyk, Representation of Lie Group and Special Functions (Kluwer Academic, Dordrecht, 1991), Vols. 1-3.