Abstract: In this paper we investigate the effect of time separation and delay between two f-STIRAP on single-qubit rotation gate based on Lacour et al (2006 Opt. Commun. 264 362). The f-STIRAP is a basic method used to adiabatically transfer population between lower states, where the two pulses terminate simultaneously while maintaining a constant ratio of amplitudes. Furthermore, we obtain numerically the optimal values for the time separation and delay for a perfect single-qubit rotation gate

Azeddine Messikh

Youssouf Hamidou Issoufa

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Australian Journal of Basic and Applied Sciences, 5(6): 360-364, 2011
ISSN 1991-8178
Single-qubit Rotation Gate Using Three-level Lambda Systems
Youssouf Hamidou Issoufa and Azeddine Messikh
Department of Computer Science, Faculty of Information and Communication Technology, IIUM,
53100 Kuala Lumpur, Malaysia.
Abstract: In this paper we investigate the effect of time separation and delay between two f-STIRAP
on single-qubit rotation gate based on Lacour et al (2006 Opt. Commun. 264 362). The f-STIRAP is
a basic method used to adiabatically transfer population between lower states, where the two pulses
terminate simultaneously while maintaining a constant ratio of amplitudes. Furthermore, we obtain
numerically the optimal values for the time separation and delay for a perfect single-qubit rotation
gate.
Key words: Quantum Rotation Gates, Adiabatic Process, STIRAP, f-STIRAP
INTRODUCTION
The physical implementation of quantum computer requires series of accurately controllable quantum
operations. These quantum operations can be implemented using quantum geometric phases (Wilczek, 1984),
where some parameters are controlled around a defined curved. This method is known as holonomic or
geometric quantum computation, and has become one of the key approaches to achieve quantum computation
that is resilient against errors (Erik Sj, 2008). In 1999 Zanardi and Rasetti (1999) laid the theoretical
foundations of holonomic quantum computation by showing that any quantum circuit can be generated by using
suitable Hamiltonians that depend on experimentally controllable parameters, such as those related to the
bosonic mode in a quantum optical systems (Pachos, 2000). At the same time, Jones et al (2000) demonstrated
experimentally a quantum gate based on geometric phase that was able to entangle a pair of nuclear spins in
a nuclear magnetic resonance (NMR) setup. Deutsch presented in 1989 a three qubits quantum gate and
showed that these gates together with arbitrary one-qubit rotations are sufficient to create any quantum network
(Lacour, 2002). Such a set of gates are called universal gates for quantum computation. Since then many sets
of gates were proved to be universal (Barenco, 1995). Recently Lacour et al (2002) have proposed
experimental technique to implement single-qubit quantum gates based on Stimulation Raman Adiabatic Process
(STIRAP) and static laser phases. This technique requires only the control of relative phase of the driving
fields, which can be implemented precisely, and do not involve dynamic or geometric phases. It is well known
that STIRAP technique is very efficient and robust theoretically as well as experimentally. An other important
technique called fractional STIRAP (f-STIRAP). It is a generalization of STIRAP and it is based upon (partial)
adiabatic population transfer between two states through an intermediate state. Geometric phases accumulated
during a STIRAP process were previously investigated for tripod systems and used to implement single-qubit
rotations gates. In this paper we extend the work in Ref. (Lacour et al, 2002) and investigate the effect of time
separation and delay of pulses on the single-qubit rotation gates. We utilize a recent idea by Moller et al
(2007) who applied STIRAP to implement quantum logic gates. We numerically obtain optimal values for the
time separation and delay. The work is organized as follows. In sec. II we present the atomic system under
consideration. In sec.III we review the generalized rotation gate proposed by Lacour et al (2002). In sec. IV
we study the effect of time separation and time delay between two pulses used by the two f-STIRAP. A
conclusion is given in sec. V.
Model and Equations of Motion:
We consider an atom in three-level lambda configuration depicted in Fig.1. The two lower levels |0> and
|1> are long-lived atomic states. They are coupled to the upper state |e> by two non resonant coherent laser
fields with Rabi frequencies Ω+(-) In practice the lower states can be ground Zeeman or hyperfine sub-levels,
and the upper state is an electronically excited state or excited state manifold. The Hamiltonian of the model
can be expressed in the basis states {|0>, |1>, |e>} by
Corresponding Author: Youssouf Hamidou Issoufa, Department of Computer Science, Faculty of Information and
Communication Technology, IIUM, 53100 Kuala Lumpur, Malaysia
E-mail: Benyous005@yahoo.fr,amessikh@yahoo.com
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Aust. J. Basic & Appl. Sci., 5(6): 360-364, 2011
Fig. 1: Three-level lambda system with lower levels driven by two non resonant coherent fields. The detuning
Δ, the common one-photon detuning of the two laser fields, is very large compared to the Rabi
frequencies Ω+, so that the excited state |e> can be adiabatically eliminated.
 
where Δ is the common one-photon detuning of the two laser fields which have time-dependent Gaussian
profile given by
 
where σ is the width, Ωj is the amplitude and tj is the time delay. The detuning Δ is taken to be large enough
so that the excited state |e> is therefore never populated during the coherent pumping process. Thus, our qubit
consists of the two lower states {|0>, |1>}. 
The evolution of the system is governed by the Schrödinger equation
 
This Hamiltonian H has three eigenvalues 
 
They are the instantaneous adiabatic eigenvalues. Their correspondent eigenstates are
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Aust. J. Basic & Appl. Sci., 5(6): 360-364, 2011
where φ is the relative phase between the  two  coherent lasers,          is the  dark  state,          are the
bright states, and
 
 
In the next section, we review the implementation of the generalized rotation gate proposed in Ref.
(Lacour, 2002).
Generalized Single-qubit Rotation Gate:
To construct the generalized rotation gate
 
we briefly review the technique given by Lacour et al in Ref. (Lacour, 2002). Two f-STIRAP processes
separated by T in time are used to transfer population among the lower levels. Each f-STIRAP process has
two pulses separated by τ in time. The first f-STIRAP process is a reversed f-STIRAP with elliptic and σ-
pulses starting with a constant ratio Ω+ / Ω- ÷ cot α and ending such that the σ- pulse vanishes first 
 
The second f-STIRAP process is a standard f-STIRAP where the pulses are switched on counter intuitively
and switched off in a given constant ratio Ω+ / Ω- ÷ tan α
As stated before, for large detuning, Δ>>1, the excited state |e> can be adiabatically eliminated. The
evolution of the system from the initial time ti to the end of the first f-STIRAP t`f is given in the basis {|0>,
|1>} by
where A- is the dynamical phase acquired by the eigenstate           during the first f-STIRAP. From the time
t`f to the end of the second f-STIRAP the evolution of the system is given by 
where A`- is the dynamical phase acquired by the eigenstate           during the second f-STIRAP. Combining 
U1 and U2 the resulting operation from ti to tf leads to the propagator
If the two f-STIRAP have the same pulse shapes with the same delay, the dynamical phases acquired by
the dark state         in the two f-STIRAP are the same,            . Thus, a compensation of the dynamic
phase  is  achieved.  When  the  pulse  sequences  is applied to our model, the process leads to the rotation
gate R(2α, φ) up to a global phase. In Fig. 2 we plot the time-depend Rabi f requencies. In the next section 
we discuss the effect of the two parameters T and τ on the quantum rotation gate R(π/4φ).
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Aust. J. Basic & Appl. Sci., 5(6): 360-364, 2011
Fig. 2: Double f-STIRAP process with Gaussian pulses. Solid (dashed) line represents the time dependence
of  Ω+(-). The parameters are as follows. Δ = 350, A0 = 100, τ = 0.487, T = 2 and a = π/8.
The Effect of T and τ on the Rotation Gate:
To illustrate the effect of T on the single-qubit rotation gate we choose a special gate R(π/4,0), we
numerically get the solutions of Schrödinger equation and plot the populations ρ11 and ρ11 as a function of T,
assuming the atom is prepared initially in the state *0>, Δ = 350, A0 = 100, τ = 0.487, a = π/8 and the final
time tf = 6. We can clearly see from Fig. 3 that there are four points, T =0.17, 0.36, 0.46, 2 at which ρ11 =
ρ22 = 0.5. One can see that at the first three points the slopes are high, which means that any small deviation
from these points will result in imperfect rotation gate. At the last point which corresponds to T=2 the slope
is small. This is the optimal value of T that can be used to get a perfect rotation gate for τ = 0.487.
Fig. 3: The population as a function of T at the end of the second f-STIRAP. Solid line is for ρ11 and dashed
line for ρ22. The parameters are the same as Fig.2.
In Fig. 4 we plot the evolution of the populations as a function of time t for the optimal value T = 2. One
can see from these figures that at the end of the second f-STIRAP we obtain a perfect single-qubit rotation
gate R(π/4, 0).
Fig. 4: Numerical solutions for the populations as a function of time. The parameters are: the same as Fig.2,
the upper figure, the atom is initially prepared in the state |0>. The lower figure, the atom is initially
prepared in the state |1>. Solid line for ρ11 and dashed line for ρ22.
Now we turn our attention to the effect of time delay τ. In Fig. 5 we plot the populations of the lower
levels as a function of τ. We can clearly see that there are three points to be selected τ = 0.40, 0.487, 0.56
where ρ11 = ρ22 = 0.5. The point at τ = 0.487 is the optimal one because at this point the slope is small
compared to the slope at other points. 
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Aust. J. Basic & Appl. Sci., 5(6): 360-364, 2011
Fig. 5: The population as a function of τ at the end of the second f-STIRAP. Solid line is for ρ11 and dashed
line for ρ22. The parameters are the same as Fig. 2.
 
Conclusions:
In conclusion, we have investigated the effect of time separation and time delay of two f-STIRAP on the
single-qubit rotation gate. One has to be careful when selecting the values of T and τ to obtain a perfect
single-qubit rotation gate. We give optimal values τ = 0.487, T=2 for the special rotation gate R(π/4, 0).
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Single-qubit Rotation Gate Using Three-level Lambda Systems

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Single-qubit Rotation Gate Using Three-level Lambda Systems

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