1. Implementation of All-Optical Toffoli gate in Λ- systems
V. Chaltykyan, E. Gazazyan, G. Grigoryan
Institute for Physical Research, 0203 Ashtarak-2, Armenia
An optical Toffoli gate is the essential logical element, which permits the implementation of a reversible optical processor. We propose a simple realization of such a gate in films of crystals doped with rare-earth ions. The proposed scheme is based on adiabatic population transfer in a Λ-system by means of counterintuitive and intuitive sequences of short laser pulses. We also discuss possibilities for experimental realization of the proposed gate.
INPUT
OUTPUT
 0 
 1  1   1
Phys. Input
Phys. Output
IPP
ISP
ISA
EPP
ESP
FSA
The adiabaticity for a single atom when the above states are realized can be provided under the following conditions:
When passing to a macroscopic volume, conditions fulfilled for a single atom are no longer sufficient. Because of energy exchange between the pulses and between the pulses and medium the shapes of pulses can be modified essentially and the dynamics of variation of angles considered above may be violated. At long distances of propagation also the interaction adiabaticity can break. Detailed investigation of interaction adiabaticity in the medium and of efficient population transfer was performed in a number of works, where it was shown that at propagation in a medium the changes in shapes of pulses may be neglected at the distances satisfying the conditions
Truth Table of the Toffoli gate. Notations are: IPP and EPP – incident and emerging pump pulse (on – 1, off – 0); ISP and ESP – incident and emerging Stokes pulse (on – 1, off – 0); ISA and FSA – initial and final state of atom (|3 – 1, |1 – 0).
The efficient adiabatic transfer of atomic populations in Λ-system with large detunings of one-photon resonances by the methods of STIRAP and b-STIRAP make the Λ-system fully reversible in interactions with short laser pulses whose durations are much shorter than relaxation times. In this case the Toffoli gate can be organized according to the Truth Table above. If the transverse relaxation width of the system is of the order of 1000 MHz or less and the durations of laser pulses are of the order of 1010 s, the same sequence of pulses can transfer without losses the entire atomic population from one ground state to the other and backwards. In this case, if the optical length of medium satisfies certain conditions, the change in laser pulses during propagation may also be neglected. So, for number density of resonant atoms 1014 cm-3, one-photon detuning Δ=10/T, and the generalized Rabi frequency Ω=20/T, the medium length may be of the order of a millimeter. Under such conditions we have demonstrated the possibility of realization of all-optical reversible universal logic gate in a solid medium with use of coherent cyclic population transfer. By means of circuits of such gates all-optical reversible processor can be designed.
Fig.2. Transfer of population by means of counterintuitive sequence of pulses (STIRAP). Rjj are the populations of corresponding levels. Shapes of pulses are chosen to be Gaussian and the value of the one-photon detuning is Δ=10/T.
Here L is the medium length and q the coupling parameter defined as
with dp,s being the dipole moments of respective transitions, and N is the number density of atoms in the medium
Relaxation processes may lead to mixing of wave functions, therefore for complete analysis of interaction dynamics it is necessary to solve the equations for the density matrix . Numerical solutions of these equations are depicted in Fig.2 (STIRAP) and Fig.3 (b-STIRAP). We see that at sufficiently large one-photon detuning (much larger than the width of the excited level) relaxation processes do not essentially affect the population transfer in case of short pulses (with duration much shorter than the relaxation times), but the transfer efficiency drops considerably if the durations of pulses become of the order of relaxation times.
Fig.3. Transfer of population by means of intuitive sequence of pulses (b-STIRAP). Parameters are the same as in Fig.2.
The work was supported by Volkswagen foundation grant “Optical information processing, driven by adiabatic interactions between light and matter” and is performed in the scope of the International Associated Laboratory IRMAS, ANSEF PS-opt-2911 and Grant of the State Committee of Science of the Ministry of Education and Science of the RA