Bloch space structure of cascade, lambda and vee type of three-level systems and qutrit wave function

0.11 MB·English

by  Surajit Sen

Bloch space structure of cascade, lambda and vee type
of three-level systems and qutrit wave function
0
1 Surajit Sen,1 Mihir Ranjan Nath2 and Tushar Kanti Dey 3
0
Department of Physics
2
Guru Charan College
n
Silchar 788004, India
a
J

8 Gautam Gangopadhyay 4
2
S N Bose National Centre for Basic Sciences

] JD Block, Sector III
h
Salt Lake City, Kolkata 700098, India
p
-
t
n
a
u
q
[
Abstract

3
Thecascade,lambdaandveetypeofthree-levelsystemsareshowntobedescribed
v
8 by three different Hamiltonians in the SU(3) basis. We investigate the Bloch space
4 structureofeachconfigurationbysolvingthe correspondingBlochequationandshow
2 that at resonance, the seven-dimensional Bloch sphere 7 is broken into two distinct
3 subspaces 2 4 due to the existence of a pair of quadSratic constants. We also give
. S ×S
1 a possible representation of the qutrit wave function and discuss its equivalence with
0 the three-level system.
9
0
:
v
i
X
r
a
PACS No.: 42.50.Ct, 42.50.Pq, 42.50.Ex
Key words: SU(3) group; Bloch equation; Three-level system; Qutrit
[email protected]
2mrnath 95@rediffmail.com
3tkdey54@rediffmail.com
[email protected]
1
The atomic system interacting with two or multiple number of lasers provides an
unique opportunity to manipulate them because of its well-defined level structure. To
develop a systematic theory of the coherent control mechanism, it is necessary to study
the dynamics of the two-, three- or multi-level systems which may pave the way to the
experimental realization of the quantum computer [1-3] and associated phenomena such
as quantum teleportation [4], quantum cryptography [5], dense coding [6,7], quantum
cloning [8] etc. In the quantum information theory parlance where the qubit is treated
as the primary building block, the Bloch space representation of the two-level system is
significant for many reasons. Most importantly, a qubit state q > is codified as the
B
|
superposition of the computational basis 0 >, 1 > and it corresponds to various points
{| | }
on the two-dimensional unit Bloch sphere 2 [3,9]. Thus a quantum mechanical gate,
S
which indeed represents an unitary operator A, corresponds to the transformation of the
qubit state as q >=Aq >, defined on the surface of that sphere. A natural extension
| B′ | B
of the qubit is the qutrit system, where the computational basis are defined as 0 >, 1 >
{| |
and 2 > , respectively. Physically a qutrit possesses much complex but richer structure
| }
than the ordinary qubit and there exists several suggestions that it may be implemented
either as a three-level system [10,11], or transverse spatial modes of single photons [12], or
polarization states of the biphoton field [13,14], respectively. In recent years considerable
progress has been made to understand various aspects of the qutrit system treating it a
three-level system driven by bichromatic laser fields. These studies include, the quantum
information transfer between qutrits [15], separability among joint states of qutrits [16],
entanglement sudden death for qubit-qutrit composite system [17], cooperative mode in
the qutrit system with complex dipole-dipole interaction [18], teleportation between two
unknown entangled qutrits [19], cloning the qutrit [20], possible algebraic and geometric
structure[21-24],newquantumkeydistributionprotocolsdealtwithqutrits[25,26]etc[27].
In spite of these progress, understanding the Bloch space of all three-level configurations
and the wave function of the qutrit system is the necessary prerequisite to define various
logic gates and the associated circuits, which perhapsform the basis of developing a richer
form of quantum information theory beyond qubit.
AsystematicstudyoftheN-level systemanditsconnectionwiththeSU(N)groupwas
first initiated by Eberly and Hioe [28-30] and later it was investigated in detail in context
with the three-level system [31-38]. From these studies it is revealed that the quadratic
casimir for the SU(3) group is manifested through the existence of a set of nonlinear
constants which gives rise to the non-trivial structure of the Bloch space, i.e., 0 2 4.
S ×S ×S
Latertheseconstantswerestudiedbysolvingthepseudo-spinequation[35]andalsobythe
Floquet theory technique [37,38]. However, all three-level configurations, namely, cascade,
lambda and vee type of system, are found to exhibit same set of nonlinear constants. The
primaryhindranceofgettingdifferentconstantsfordifferentconfigurationsisthefollowing:
It was pointed out by Hioe and Eberly that if the position of the intermediary energy
level E is changed with respect to other two levels shown in Fig.1, then all three-level
2
configurations canbeexpressedby similarHamiltonian [29]. Sincetheproposalwas made,
ithasbecomeawidelyacceptedconventionthatthecascade,lambdaandveetypeofthree-
level systems are described by similar Hamiltonian with the energy hierarchy conditions,
namely, E > E > E , E > E > E and E > E > E , respectively. Similar structure
3 2 1 2 3 1 1 2 3
of the Hamiltonian leads to same Bloch space for three systems and therefore forbids the
existence of different non-linear constants for different configurations. However, it is now
2
well-known that each configuration of the three-level system is associated with a diverse
rangeofcoherentphenomenaandthereforemustbedescribedbythedistinctHamiltonian.
For example, the lambdaconfiguration exhibits thephenomenasuch as STIRAP[39], EIT
[40] etc, while the vee configuration is associated with quantum jump [41], quantum zeno
effect [42], quantum beat [43] and others, respectively. Taking the interacting fields to be
either classical or quantized bichromatic fields, the necessity of the inequivalence between
the lambda and vee systems is also studied [44]. In a recent work, we have proposed a
novel scheme to write the Hamiltonians of the three-level systems in the SU(3) operator
basis where the energy levels are arranged as Ea > Ea > Ea (a = Ξ,Λ and V) shown
3 2 1
in Fig.2 [45]. In particular, we have discussed the exact solution of the semiclassical and
quantized three-level systems and have compared the corresponding Rabi oscillations.
The primary objective of this paper is to discuss the structure of the Bloch space of
the three-level systems, while taking distinct Hamiltonian for different configuration. It is
shown that at vanishing detuning frequency, the Bloch space of each three-level system is
broken into two distinct sectors 2 4 and they are different for different configurations.
S ×S
Finally we have constructed the wave function of the qutrit system and discuss its various
properties.
It is customary to view the three-level system as two two-level systems with one of
their levels overlaps with each other. Thus in the dipole approximation, the Hamiltonian
of the semiclassical cascade system can be expressed as [45]
HΞ =h¯(ω1U3+ω2T3)+dˆ12 E1(Ω1)+dˆ23 E2(Ω2), (1)
· ·
where, dˆ12 = (U++U ) and dˆ23 = (T++T ) be the dipole operators representing the
− −
transitions 1 2 and 2 3 and E1(Ω1) = E1+ +E1 and E2(Ω2) = E2+ +E2 be the
↔ ↔ − −
bichromatic classical electric fields with frequencies Ω and Ω , respectively. Similarly we
1 2
have
HΛ = h¯(ω1V3+ω2T3)+dˆ13 E1(Ω1)+dˆ23 E2(Ω2), (2)
· ·
forthelambdasystem, wheredˆ13 = (V++V ), dˆ23 =(T++T )bethedipoleoperators
− −
representing transitions 1 3 and 2 3, respectively. Finally for the vee system we have
↔ ↔
HV = h¯(ω1V3+ω2U3)+dˆ13 E1(Ω1)+dˆ12 E2(Ω2), (3)
· ·
where dˆ13 = (V+ +V ) and dˆ12 = (U+ +U ) characterize the transitions 1 3 and
1 2, respectively. I−n Eq.(1-3), ¯hω (= EΞ−),¯h(ω ω )(= EΞ) and h¯ω (=↔EΞ) be
↔ 1 − 1 1 − 2 2 2 3
the energies of the three levels of the cascade system, h¯ω (= EΛ),¯hω (= EΛ) and
1 − 1 2 − 2
¯h(ω +ω )(= EΛ) be the energies of the lambda system and h¯ω (= EV),¯hω (= EV) and
2 1 3 1 1 2 2
¯h(ω +ω )(= EV) be those of the vee system shown in Fig.2, respectively. We note that
− 2 1 3
in all cases, unlike the conventional scheme shown Fig.1 [29], the energy levels maintain
the hierarchy, Ea > Ea > Ea (a = Ξ,Λ and V). In Eqs.(1-3), we have defined the SU(3)
3 2 1
shift vectors,
T = 1(λ iλ ), U = 1(λ iλ ), V = 1(λ iλ ),
± 2 1± 2 ± 2 6± 7 ± 2 4± 5
T = λ , U = (√3λ λ )/2, V = (√3λ +λ )/2, (4)
3 3 3 8 3 3 8 3
−
respectively, where the Gellmann matrices are given by,
3
0 1 0 0 i 0 1 0 0
   −   
λ = 1 0 0 , λ = i 0 0 , λ = 0 1 0 ,
1 2 3
     − 
 0 0 0   0 0 0   0 0 0 
0 0 1 0 0 i 0 0 0
   −   
λ = 0 0 0 , λ = 0 0 0 , λ = 0 0 1 ,
4 5 6
     
 1 0 0   i 0 0   0 1 0 
0 0 0 1 0 0
   
λ = 0 0 i , λ = 1 0 1 0 , (5)
7  −  8 √3 
 0 i 0   0 0 2 
−
respectively. These matrices follow the commutation and anti-commutation relations
[λ ,λ ]= 2if λ , λ ,λ = 4δ +2d λ , (6)
i j ijk k { i j} 3 ij ijk k
where d and f (i,j = 1,2,..,8) represent completely symmetric and completely anti-
ijk ijk
symmetric structure constants, respectively, which characterize SU(3) group [46].
In the rotating wave approximation (RWA), the Hamiltonians can be written as
HΞ = h¯(Ω ω ω )U +h¯(Ω ω ω )T +h¯(∆ΛU +∆ΛT )+
1− 1− 2 3 2− 1− 2 3 1 3 2 3
¯hκ U exp( iΩ t)+h¯κ T exp( iΩ t)+h.c. , (7)
1 + 1 2 + 2
− −
for the cascade system,
HΛ = h¯(Ω ω ω )V +h¯(Ω ω ω )T +h¯(∆ΛV +∆ΛT )+
1− 1− 2 3 2− 1− 2 3 1 3 2 3
¯hκ V exp( iΩ t)+h¯κ T exp( iΩ t)+h.c. , (8)
1 + 1 2 + 2
− −
for the lambda system,
HV = h¯(Ω ω ω )V +h¯(Ω ω ω )U +h¯(∆ΛV +∆ΛU )+
1− 1− 2 3 2− 1− 2 3 1 3 2 3
¯hκ V exp( iΩ t)+h¯κ U exp( iΩ t)+h.c. , (9)
1 + 1 2 + 2
− −
for the vee system, respectively. In Eq.(7-9), κ (i = 1,2) be the coupling parameters and
i
∆a = (2ω +ω Ω )and∆a = (ω +2ω Ω )betherespectivedetuningfrequencies. Thus
1 1 2− 1 2 1 2− 2
we note that the Hamiltonian of any specific three-level configuration can be expressed by
a subset of SU(3) matrices, namely,
Cascade system : U ,U ,T ,T , i.e., λ ,λ ,λ ,λ ,λ ,λ ,
3 3 1 2 3 6 7 8
± ± { }
Lambda system : T ,T ,V ,V , i.e., λ ,λ ,λ ,λ ,λ ,λ , (10)
3 3 1 2 3 4 5 8
± ± { }
Vee system : U ,U ,V ,V , i.e., λ ,λ ,λ ,λ ,λ ,λ ,
3 3 3 4 5 6 7 8
± ± { }
respectively [45], rather than all eight matrices [28-30]. Given with three well-defined
models, we proceed to develop the Bloch equation of all three-level systems and discuss
the constraints involving their solutions.
Let the solution of the Schro¨dinger equation of a generic three-level system described
by the Hamiltonian Eq.(7-9) is given by,
Ψa(t) = Ca(t) 0 +Ca(t) 1 +Ca(t) 2 , (11)
0 | i 1 | i 2 | i
4
where Ca(t), Ca(t) and Ca(t) be the normalized amplitudes with basis states 0 , 1 and
0 1 2 | i | i
2 ,respectively. Theexactevaluationoftheprobabilityamplitudesenablesustocalculate
| i
the density matrix of any system given by
ρa(t) = Ψa(t)> < Ψa(t). (12)
| ⊗ |
To start with, we consider the dressed wave function of the cascade system obtained by a
unitary transformation
Ψ˜Ξ = U†(t)ΨΞ, (13)
Ξ
where the unitary operator is given by
i
U (t) = exp[ ((Ω +2Ω )T +(2Ω +Ω )U )t]. (14)
Ξ 1 2 3 1 2 3
−3
Thecorrespondingtime-independent Hamiltonian in Eq.(7) of the cascade system is given
by
H˜Ξ(0) = (−¯hUΞ†U˙Ξ+UΞ†HΞ(t)UΞ)
1¯h(∆Ξ+2∆Ξ) ¯hκ 0
 3 1 2 2 
= ¯hκ 1¯h(∆Ξ ∆Ξ) h¯κ . (15)
 02 3 h¯1κ−1 2 −13¯h(2∆Ξ11+∆Ξ2) 
Similarly, the unitary operator of the lambda system described by the Hamiltonian
Eq.(8) is,
i
U (t)= exp[ ((2Ω Ω )T +(2Ω Ω )V )t], (13)
Λ 2 1 3 1 2 3
−3 − −
and the corresponding time-independent Hamiltonian is given by,
1¯h(∆Λ+∆Λ) ¯hκ ¯hκ
 3 1 2 2 1 
H˜Λ(0) = ¯hκ 1¯h(∆Λ 2∆Λ) 0 . (17)
 ¯hκ21 3 10− 2 31¯h(∆Λ2 −2∆Λ1) 
Also the Hamiltonian of the vee system in Eq.(6) can be made time-independent by
using the transformation operator
U (t) = exp[ i((2Ω Ω )U +(2Ω Ω )V )t], (18)
V −3 2− 1 3 1− 2 3
and corresponding Hamiltonian is,
1¯h(2∆V ∆V) 0 h¯κ
H˜V(0) =  3 10− 2 1¯h(2∆V ∆V) ¯hκ1 . (19)
 ¯hκ1 3 ¯h2κ2− 1 −13¯h(∆V12+∆V2 ) 
Thus we have three distinct Hamiltonians with different non-vanishing entries for three
different configurations.
To obtain the Bloch equation, we define the generic SU(3) Bloch vectors
Sa(t) = Tr[ρa(t)λ ], (20)
i i
5
where ρa be the density matrix of the three-level systems given by Eq.(12) which satisfies
the Lioville equation, namely,
dρa i
= [ρa,H˜a(0)]. (21).
dt ¯h
From Eq.(20), the density matrix written in terms of Bloch vector is,
8
1 3
ρa(t) = (1+ Sa(t)λ ). (22)
3 2X i i
i=1
Substituting Eq.(22) and the Hamiltonians given by Eqs.(15), (17) and (19) in Eq.(21),
we obtain the Bloch equation
dSa
i = MaSa, (23)
dt ij j
where Ma be the eight dimensional anti-asymmetric matrix. For the Hamiltonian of the
ij
cascade system given by Eq.(15), the matrix MΞ reads
ij
0 ∆Ξ 0 0 κ 0 0 0
 2 − 1 
∆Ξ 0 2κ κ 0 0 0 0
− 2 2 1
 0 2κ 0 0 0 0 κ 0 
 2 1 
MΞ =  0 −−κ1 0 0 (∆Ξ1 +∆Ξ2) 0 κ2 0 . (24)
ij  κ 0 0 (∆Ξ+∆Ξ) 0 κ 0 0 
 1 − 1 2 − 2 
 0 0 0 0 κ 0 ∆Ξ 0 
 2 1 
 0 0 κ κ 0 ∆Ξ 0 √3κ 
 − 1 − 2 − 1 1 
 0 0 0 0 0 0 √3κ1 0 
−
Similarly for Eq.(17), the matrix of the lambda system is given by
0 ∆Λ 0 0 0 0 κ 0
 2 − 1 
∆Λ 0 2κ 0 0 κ 0 0
− 2 2 1
 0 2κ 0 0 κ 0 0 0 
 2 1 
MΛ =  0 −0 0 0 −∆Λ1 0 κ2 0 , (25)
ij  0 0 κ ∆Λ 0 κ 0 √3κ 
 1 − 1 − 2 1 
 0 κ 0 0 κ 0 (∆Λ ∆Λ) 0 
 κ 01 0 κ 02 (∆Λ ∆Λ) 1 −0 2 0 
 1 − 2 − 1 − 2 
 0 0 0 0 √3κ1 0 0 0 
−
and finally from Eq.(19) for the vee system we have,
0 (∆V ∆V) 0 0 κ 0 κ 0
 1 − 2 − 2 − 1 
(∆V ∆V) 0 0 κ 0 κ 0 0
− 1 − 2 − 2 − 1
 0 0 0 0 κ 0 κ 0 
 1 2 
MV =  0 −κ2 0 0 −∆V1 0 0 0 ,
ij  κ 0 κ ∆V 0 0 0 √3κ 
 2 1 − 1 1 
 0 κ 0 0 0 0 ∆V 0 
 1 − 2 
 κ1 0 −κ2 0 0 −∆V2 0 √3κ2 
 0 0 0 0 √3κ1 0 √3κ2 0 
− −
(26)
6
respectively.
The algebraic structure of the SU(3) group allows the existence of a set of quadratic
casimirs which will appear in form of the quadratic constants. However, the searching of
the exact tuple of the Bloch vectors forming such constants is quite cumbersome because
of the large number of such combinations. More specifically, for eight Bloch vectors, we
have to search 8! (= 8,28,56,70,...,1) number of combinations forming such tuples
(8 n)!n!
for n= 1,2,3,..,8−, respectively. We have developed a Mathematica program to search the
exact combination of the Bloch vectors obtained by solving the Bloch equation in Eq.(23)
and after a rigorous search obtain the following results for n = 3 and n = 5 only:
For the cascade system at resonance (∆Ξ = 0= ∆Ξ), the Bloch space is split into two
1 2
parts; one being of two sphere 2,
S
SΞ2(t)+SΞ2(t)+SΞ2(t) = SΞ2(0)+SΞ2(0)+SΞ2(0), (27a)
1 5 6 1 5 6
and other is the four sphere 4,
S
SΞ2(t)+SΞ2(t)+SΞ2(t)+SΞ2(t)+SΞ2(t) =
2 3 4 7 8
SΞ2(0)+SΞ2(0)+SΞ2(0)+SΞ2(0)+SΞ2(0), (27b)
2 3 4 7 8
respectively, where SΞ(0) be the constants at t = 0 which are to be evaluated in terms of
i
the probability amplitudes. By noting the fact that the density matrix can be written as
ρΞ(t) = U (t)ρΞ(0)U(t), Eq.(20) becomes,
†
SΞ(0) = Tr[ρΞ(0)λ ]. (28)
i i
Plucking back Eqs.(5) and (12) into Eq.(28), different values of the constants SΞ(0) can
a
be evaluated in terms of probability amplitudes CΞ (0) at t = 0 and Eqs.(27) become
1,2,3
SΞ2+SΞ2+SΞ2 = 4CΞ(0)2 CΞ(0)2 +4CΞ(0)2 CΞ(0)2 (29a)
1 5 6 | 1 | | 2 | | 2 | | 3 |
and
SΞ2+SΞ2+SΞ2+SΞ2+SΞ2 = 4(CΞ(0)2 + CΞ(0)2 + CΞ(0)2)2
2 3 4 7 8 3 | 1 | | 2 | | 3 |
3C Ξ(0)2 C Ξ(0)2 3C Ξ(0)2 C Ξ(0)2, (29b)
1 2 2 3
− | | | | − | | | |
respectively. Similarly at resonance (∆Λ = 0 = ∆Λ), for the lambda system we have two
1 2
Bloch spheres,
SΛ2+SΛ2+SΛ2 = 4CΞ(0)2 CΞ(0)2 +4CΞ(0)2 CΞ(0)2 (30a)
1 4 7 | 1 | | 2 | | 3 | | 1 |
and
SΛ2+SΛ2+SΛ2+SΛ2+SΛ2 = 4(CΛ(0)2 + CΛ(0)2 + CΛ(0)2)2
2 3 5 6 8 3 | 1 | | 2 | | 3 |
3C Λ(0)2 C Λ(0)2 3C Λ(0)2 C Λ(0)2. (30b)
1 2 3 1
− | | | | − | | | |
7
Also for the vee system at resonance (∆V = 0 = ∆V), we have
1 2
SV2+SV2+SV2 = 4CV(0)2 CV(0)2 +4CV(0)2 CV(0)2 (31a)
2 4 6 | 2 | | 3 | | 3 | | 1 |
and
SV2+SV2+SV2+SV2+SV2 = 4(CV(0)2 + CV(0)2 + CV(0)2)2
1 3 5 7 8 3 | 1 | | 2 | | 3 |
3C V(0)2 C V(0)2 3C V(0)2 C V(0)2, (31b)
2 3 3 1
− | | | | − | | | |
respectively. Thuswe notethat the3-and 5-tupleof theBloch vectors form two quadratic
constants which are different for different three-level systems. From Eqs.(29-31) it is
furtherevidentthatat resonance, theBloch space 7 isbrokenintotwo-subspaces 2 4,
S S ×S
which indicates that each configuration has distinct norm. It is instructive to note that at
offresonance(∆a = 0 = ∆a),thesolutionsoftheBlochequationofallthreeconfigurations
1 6 6 2
satisfy,
4
Sa2+Sa2+Sa2+Sa2+Sa2+Sa2+Sa2+Sa2 = (Ca(0)2+ Ca(0)2+ Ca(0)2)2. (32)
1 2 3 4 5 6 7 8 3 | 1 | | 2 | | 3 |
This shows that the normalization condition Ca(0)2+ Ca(0)2+ Ca(0)2 = 1 gives that
| 1 | | 2 | | 3 |
the radius-squared of the seven-dimensional Bloch sphere 7 will be 4 regardless of the
S 3
configurations.
Finally we consider the wave function of the qutrit system. A convenient parametriza-
tion of the normalized qutrit wave function is
θ θ θ θ
q >= cos 0 0 > +sin 0 sin 1 sin 2eiφ 1 > +
T
| 2 | 2 2 2 |
θ θ θ θ θ
0 1 0 1 2
(sin cos +isin sin cos )2 >, (33)
2 2 2 2 2 |
where the three coordinates θ , θ and θ lie between 0 to π with arbitrary value of the
0 1 2
phase angle φ. From Eq.(33) it is evident that, the pure qutrit states 0 >, 1 > and
| |
2 > can be obtained by choosing the coordinates (θ ,θ ,θ ) to be (0,θ ,θ ), (π,π,π) and
0 1 2 1 2
|
(π,0,θ ), respectively. This indicates that for a qutrit system, the state 0 > may be any
2
|
point on the (θ ,θ ) plane, 1 > corresponds to a well-defined point and 3 > lies on a line
1 2
| |
forany valueof θ , respectively. Inother words,inthethree-level scenario, thedegeneracy
3
of the 0 > and 2 > states shows that the Bloch space structure of the qutrit is highly
| |
non-trivial. This is in contrast with the two-level system representing qubit, where the
basis states 0 > and 1 > correspond to the north and south poles of the Bloch sphere,
| |
respectively [3,9]. Using Eq.(12), various elements of the density matrix corresponding to
the qutrit wave function in Eq.(33) are given by
θ
ρ11 = cos2 1
T 2
θ θ θ
ρ22 = sin2 0 sin2 1 sin2 2
T 2 2 2
8
1 θ
ρ33 = (3+cosθ +cosθ cosθ cosθ )sin2 0
T 4 1 2− 1 2 2
1 θ θ
ρ1T2 = ρ2T1∗ = 2eiφsinθ0sin 21 sin 22
θ θ θ θ θ θ
ρ2T3 = ρ3T2∗ = e−iφsin2 20 sin 21(cos 21 +isin 21 cos 22)sin 22
1 θ θ θ
ρ1T3 = ρ3T1∗ = 2sinθ0(cos 21 +isin 21 cos 22) (34)
It is worth noting that this qutrit wave function is normalized, i.e., Tr[ρ ] = 1 and the
T
density matrix satisfies the pure state condition, namely, ρ2 = ρ . Finally plucking back
T T
ρ in Eq.(20) we obtain the Bloch sphere on S7,
T
4
Sa2+Sa2+Sa2+Sa2+Sa2+Sa2+Sa2+Sa2 = . (35)
1 2 3 4 5 6 7 8 3
This is precisely same as Eq.(32) obtained from different configurations of three-level
system indicating the equivalence between the qutrit system with the three-level system.
In this paper we have discussed different semiclassical three-level configurations and
obtainthesolutionofcorrespondingBlochequations. Itisshownthat,iftheenergiesofthe
threelevels follow adefinitehierarchy prescription, it is possibleto writetheHamiltonians
in the SU(3) basis which are different for different configurations. At zero detuning, the
Bloch sphere 7 of each configuration is broken into two disjoint sectors 2 4 due to
S S ×S
the existence of different constraints which are not alike. A significant outcome of our
approach is that we have given a possiblerepresentation of the qutrit wave function which
can be identically represented by a three-level system. Although our formulation gives a
possible representation of the qutrit wave function, but it is quite difficult to ascertain the
complicated structure of the Bloch space at this stage. Apart from that, the mechanism
of minimizing the information loss while teleporting between two or multiple number of
qutrits, developing appropriate search algorithm, the qutrit based protocol transfer etc
are still open issues which requirefurtherexploration of the qutritwave function proposed
here. Although the quantum information processing has made significant progress with
the conventional qubit as the primitive identity, butthe potential application of the qutrit
can not be properly addressed unless we do not understand the wave function of the
quatrit and its relation with the Bloch space structure of the three-level system in greater
detail. More exploration along this direction may provide some interesting results which
are within reach of the future high-Q cavity as well as nanomaterial based experiments.
Acknowledgement
SS is thankful to Department of Science and Technology, New Delhi for partial support
and to S N Bose National Centre for Basic Sciences, Kolkata, for supporting his visit to
the centre through the ‘Associateship’ program.
9
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