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quantum computation employs the principle of coherent superposition and quantum entanglement to solve certain problems , such as factoring large integers and searching data in an array , much faster than a classical computer @xcite .
the basic building blocks of a quantum computer are quantum logic gates .
it was shown that any quantum computation can be reduced to a sequence of two classes of quantum gates , namely , universal two - qubit logic gates and one - qubit local operations universal .
the standard paradigm of quantum computation is the dynamical one where the local interactions between the qubits are controlled in such a way so that one can enact a sequence of quantum gates . on the other hand
, it has been recognized that the quantum gate operations can also be implemented through the geometric effects on the wave function of the systems , this is the so - called geometric quantum computation @xcite .
compared with the dynamical gates , the geometric quantum computation possesses practical advantages .
it is well known that geometric phases depend only on some global geometric features , and do not depend on the details of the path , the time spent , the driving hamiltonian , and the initial and final states of the evolution @xcite .
therefore the geometric quantum computation is largely insensitive to local inaccuracies and fluctuations , and thus provides us a possible way to achieve fault - tolerant quantum gates .    in the implementation of geometric quantum computation ,
one practical question we usually meet is how to remove or avoid the dynamical phases since geometric phases are generally accompanied by dynamical ones which are not robust against local inaccuracies and fluctuations . to this end
one simple method is to choose the dark states as qubit space , thus the dynamical phase is always zero @xcite .
another general method is to let the evolution be dragged by the hamiltonian along several special closed loops , then the dynamical phases accumulated in different loops may be canceled , with the geometric phases being added @xcite .
this is the so - called multi - loop scheme .
the geometric quantum computation which is based on the cancelation of dynamical phases is referred to as conventional geometric quantum computation .
correspondingly several schemes have been presented recently to realize the so - called unconventional geometric quantum computation leibfried03,unzhu , zheng , feng .
the central idea of the unconventional geometric quantum computation is that for certain quantum evolution of a quantum system of interest one can implement fault - tolerant quantum computation by using the total phase accumulated in the evolution if it depends only on global geometric features of the evolution . in comparison with conventional geometric gates
, unconventional geometric gates do not require additional operations to cancel the dynamical phases and thus simplify the realization operations .
schemes for implementing the unconventional geometric gate have been proposed in trapped ion systems leibfried03,unzhu and in cavity qed systems@xcite . in the schemes of cavity qed systems@xcite , the excited states are utilized as the computational bases , thus the spontaneous emission can not be avoided in such schemes .    in this paper
we make use of the two - channel raman interaction in cavity qed @xcite to realize the unconventional geometric gate . in our scheme the atomic excited states
are adiabatically eliminated and never excited during the quantum gate construction , therefore atomic spontaneous emission can be avoided in our scheme .
we consider two identical three - level atoms in @xmath0-configuration placed in a high-_q _ cavity .
the level structure of the atoms is shown  in fig .
[ fig1 ] , where @xmath1 , @xmath2 ( @xmath3 ) are metastable states and @xmath4 is an excited state .
the transitions @xmath5 and @xmath6 and @xmath7  with the frequencies @xmath8 and @xmath9  respectively .
the second channel contains a classical external field @xmath10 with a frequency @xmath11 and a quantized cavity field of frequency @xmath12 .
both channels are assumed to satisfy the usual raman resonance , that is , @xmath13
@xmath14 @xmath15 where @xmath16 is the energy difference between levels @xmath17 and @xmath18 .
the electric field operator of the cavity mode can be expressed as @xmath19 , where @xmath20 and @xmath21 are respectively the annihilation and creation operators , @xmath22 is the polarization vector and @xmath23 is a constant determined by the quantization volume . in the case that the detunings @xmath24 and @xmath25 are sufficiently large , the atomic excited state @xmath26 can be adiabatically eliminated and we can obtain an effective hamiltonian [ fig1 ] .
the detunings @xmath27 and @xmath28  are assumed to be sufficiently large , so that the excited state @xmath26 can be eliminated .
@xmath29 + \sum_{j=1}^{2}\left [ g(t)a\sigma _ { j}^{+}+g^{\ast } ( t)a^{\dagger } \sigma _ { j}^{-}\right ] , \label{h1}\ ] ]    where @xmath30 and @xmath31 are atomic operators , @xmath32 and @xmath33 are respectively the effective classical and quantum couplings and they take the following form@xmath34 \left [ \langle e|\overrightarrow{d}%    g(t ) & = & -\frac{\left [ \langle c|\overrightarrow{d}|g\rangle \cdot \overrightarrow{e}_{g}(t)\right ] \left [ \langle e|\overrightarrow{d}%    _ { 2}}.\end{aligned}\]]here @xmath35 ( @xmath36 ) denote the atomic dipole matrix elements . from above equations
it is easy to note that the effective coupling parameters @xmath32 and @xmath33 can be controlled by adjusting the driving light fields . based on such a feature of the hamiltonian ( [ h1 ] ) a scheme to generate arbitrary quantum states of the cavity fields was proposed @xcite .
now we further make the following transformation on the hamiltonian ( [ h1])@xmath37 \exp \left ( -ih_{0}t\right ) , \]]where @xmath38 $ ] . for simplicity , we assume @xmath32 is real , then @xmath39 .$ ] after a simple calculation we obtain@xmath40 g(t)a+h.c.,\label{e5}\end{aligned}\]]where @xmath41 are eigenstates of @xmath42 with eigenvalues @xmath43 , respectively . in the strong effective classical driving regime
@xmath44 the terms in eq .
( [ e5 ] ) which oscillate with high frequencies can be eliminated in the rotating - wave approximation , and eq .
( [ e5 ] ) can thus be simplified as @xmath45 ,   \nonumber \\ & = & \frac{1}{2}\left [ g(t)a+g^{\ast } ( t)a^{\dagger } \right ] \left ( \sigma _ { 1}^{x}+\sigma _ { 2}^{x}\right ) .\label{e6}\end{aligned}\]]similar hamiltonians have been derived in the strongly driving jaynes - cummings model @xcite and the two - channel raman interaction in cavity qed @xcite . in comparison with
the trapped - ion model proposed to realize the unconventional geometric quantum computation unzhu , the hamiltonian in our model is of a similar form to that of the trapped - ion model . in the next section
we will show how to construct an unconventional geometric two - qubit phase gate based on the above hamiltonian .
we choose the eigenstates of @xmath46 , that is , @xmath47 ) will not give rise to any population changes in such a computational basis when the system is governed by the hamiltonian ( [ e6 ] ) . in the computational basis \{@xmath48 , @xmath49 , @xmath50 , @xmath51 }
the hamiltonian ( [ e6 ] ) is diagonal and takes the form @xmath52
\times \mathrm{diag}[\lambda _ { + + } , \lambda _ { + -},\lambda _ { -+},\lambda _ { --}],\label{e7}\]]where @xmath53 @xmath54 are the eigenvales of @xmath55 and @xmath56 , @xmath57 the time evolution matrix @xmath58 is thus diagonal @xmath59 , \]]where the diagonal matrix elements @xmath60 can be derived from eq .
( [ e7 ] ) , @xmath61 dt } ,   \notag \\ & = & \lim_{n\rightarrow \infty } \prod_{n=1}^{n}e^{-i\frac{1}{2}\lambda _ { kl}% \left [ g(t_{n})a+g^{\ast } ( t_{n})a^{\dagger } \right ] \delta t } ,   \notag \\ & = & \lim_{n\rightarrow \infty } \prod_{n=1}^{n}d\left [ \delta \alpha _ { kl}(t_{n})\right ] , \label{e9}\end{aligned}\]]where @xmath62 is the time ordering operator , @xmath63 is the time interval , and @xmath64 @xmath65 is the displacement operator which takes the form @xmath66.$ ] the displacement operators satisfy the following relation @xmath67based on the above formula , the eq .
( [ e9 ] ) can be further simplified as @xmath68with @xmath69 and @xmath70for a closed path @xmath71 , @xmath72 here @xmath73 is the total phase acquired by the state @xmath74 @xmath54 in the cyclic evolution from @xmath75 to @xmath76 .
the total phase @xmath73 consists of two parts , one part is geometric phase @xmath77 , and the other part is the dynamical phase @xmath78 @xcite . according to the coherent - state path integral methods @xcite , the geometric phase @xmath79 and the dynamical phase @xmath78 can be calculated in the following way @xmath80with @xmath81from eq . ( [ e11 ] )
we obtain @xmath82substituting eq . ( [ e7 ] ) and eq .
( [ e15 ] ) into eq .
( [ e14 ] ) , we get @xmath83 ,   \notag \\ & = & -\frac{i}{4}\lambda _ { kl}^{2}g(t),\label{e16}\end{aligned}\]]here for the sake of simplicity we have set @xmath84 . with eq .
( [ e15 ] ) and eq .
( [ e16 ] ) , the geometric phase @xmath77 and the dynamical phase @xmath85 can be calculated according to the formulas ( [ e12 ] ) and ( [ e13]),@xmath86and the total phase is given by @xmath87comparing the above three equations , we have,@xmath88the relations between the total phase @xmath73 , the dynamical phase @xmath85 and the geometric phase @xmath77 indicate that in the system examined here the total phase @xmath89 and the dynamical phase @xmath78 possess the global geometric features as the the geometric phase @xmath77 does .
therefore the cyclic evolution @xmath90\]]with @xmath91 is a two - qubit phase gate operation which is robust against some lacal inaccuracies and fluctuations , this gate is nontrivial if @xmath92 as described in the preceding section , the effective coupling parameter @xmath33 can be controlled by adjusting the driving light field , so that the cyclic evolution and certain total phase @xmath91 can be achieved .
it is worth noting that in our scheme the atomic excited state is adiabatically eliminated and never populated .
the quantum phase gate operation only involves atomic metastable states , therefore the effect of the spontaneous emission can be ignored .
in this paper , we present a scheme for implementing the unconventional geometric two - qubit phase gate with nonzero dynamical phase by using the two - channel raman interaction of two atoms in a cavity .
we show that the dynamical phase acquired in a cyclic evolution is proportional to the geometric phase acquired in the same cyclic evolution , hence the the total phase possesses the same geometric features as the geometric phase does . in our scheme the atomic excited state
is adiabatically eliminated and the operation of the proposed logic gate involves only in the metastable states of the atom and hence is not affected by spontaneous emission .
we present a scheme for implementing the unconventional geometric two - qubit phase gate with nonzero dynamical phase by using the two - channel raman interaction of two atoms in a cavity . 
 we show that the dynamical phase acquired in a cyclic evolution is proportional to the geometric phase acquired in the same cyclic evolution , hence the the total phase possesses the same geometric features as the geometric phase . in our scheme the atomic excited state 
 is adiabatically eliminated and the operation of the proposed logic gate involves only in the metastable states of the atom and hence is not affected by spontaneous emission .