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Driving Atoms Into Decoherence-Free States

We describe the decoherence-free subspace of N atoms in a cavity, in which decoherence due to the leakage of photons through the cavity mirrors is suppressed. We show how the states of the subspace can be entangled with the help of weak laser pulses, using the high decay rate of the cavity field and strong coupling between the atoms and the resonator mode. The atoms remain decoherence-free with a probability which can, in principle, be arbitrarily close to unity.


I. INTRODUCTION
Following the theoretical formulation of quantum computing [1] and the first algorithms for problems which can be solved more easily on a quantum computer than on a classical computer [2,3] the practical implementation of such a device has become a challenging task. Initial steps have already been taken. Quantum bits (qubits) can be realised for instance by storing the information in a superposition of the internal states of two-level atoms. To provide the interaction between the atoms necessary to perform operations between the qubits the coupling via vibrational modes [4][5][6][7] or via the single mode inside a cavity [8][9][10] can be used. In other proposals, level shifts due to dipole-dipole interaction [11][12][13] and due to light shifts [14,15] have been considered.
The main limiting factor for quantum computing is decoherence. This normally limits factoring [2], for example, to small numbers [16,17] and demonstrates the necessity for error correcting codes [18,19]. But even with the help of quantum error correction, it remains uncertain as to whether decoherence will still destroy the quantum coherence too rapidly for any practical use if the number of qubits required is of the order of several hundreds or thousands. Indeed, a superposition of two quantum mechanical wave functions loses its coherence very rapidly with the "distance" between the components involved [20].
However, it has recently become clear that decoherence-free subspaces (DFSs) of the total Hilbert space may exist, in which the states are in principle exempt from decoherence [21][22][23][24]. They arise if the coupling to the environment has a certain symmetry. The decoherence-free (DF) states then all acquire the same phase factor, so that arbitrary superpositions of them remain intact in spite of the interaction with the environment [20]. DFSs are promising candidates for quantum computing. The dependence of quantum information processing on error correction schemes is substantially reduced [25]. While the underlying theoretical nature of DFS has received much attention, far less is known about potential realisations (for examples see Refs. [26,27]) and the manipulation of the states inside the DFS in general (see however Refs. [28,29]   In this paper we give an example for a DFS which can be implemented using present technology, at least for small numbers of qubits and we describe how to prepare and to manipulate the states inside a subspace. The system we discuss consists of N macroscopically separated metastable two-level atoms and is shown in Fig. 1. We generate an interaction between the atoms by placing them at fixed positions in a cavity which acts as a resonator for an electromagnetic field. The atoms can be stored between the cavity mirrors in a linear trap or in the nodes of a standing light field. The atomic transition is assumed to be in resonance with a single field mode in the cavity. The atoms should be strongly coupled to the field mode and the interaction between each atom with the field is given by the coupling constant g i . As a simplification we assume g i ≡ g for all i, but the ideas discussed here can also be carried over to the more general case.
The main source of decoherence in this system is that a photon can leak out through the cavity mirrors with a rate κ which is due to the coupling of the resonator mode to the free radiation field. Even if the cavity mode is empty, the atoms will in general transfer excitation into the resonator mode which then can be lost. As we will show later this process does not take place if the cavity mode is empty and the atoms are prepared in a trapped state. As a result an example of a DFS is found. The trapped states of two two-level atoms in a cavity have been discussed in Refs. [30][31][32][33]. They belong to a two-dimensional Hilbert space which includes the ground state and the maximally entangled state We will show below that the trapped states of N atoms create a DFS of dimension for odd and even numbers of atoms, respectively. For large N the dimension roughly equals 2/(πN ) · 2 N and therefore increases with N almost as fast as the dimension of the whole state space, 2 N .
The distance between the atoms should be much larger than an optical wavelength. This allows us to address each atom individually by a single laser pulse. If their Rabi frequencies are much smaller than the constants g and κ, laser pulses can be used to prepare and to manipulate the states inside the DFS. The reason for this is a mechanism which strongly inhibits the transition from trapped to non-trapped states in this parameter regime and which can be understood with the help of the quantum Zeno effect [34][35][36]. We in fact profit from a high decay rate of the resonator field and the results do not depend on precise values of g and κ. Arbitrary unitary operations can be constructed in a DF qubit formed out of two states of two atoms. In particular we show how a maximally entangled Bell state of the two atoms can be generated out of the atomic ground state.
In the system we discuss here one source of decoherence remains. Even if the spontaneous decay rate of the atoms is decreased by the presence of the resonator, photons can still be emitted spontaneously into non-cavity field modes. We therefore propose to use metastable atoms, which have a very small decay rate Γ. Spontaneous emission can be neglected if the duration of the operations performed on the atoms is short compared to 1/Γ. Therefore the applied laser pulses cannot be arbitrarily weak, as is necessary for the scheme to work. Care is thus needed to ensure an overall advantage [17]. Problems arising from this will be discussed in detail.
In principle, one could argue that an even larger Hilbert space of atomic states than the DFS considered here can be obtained by storing atoms (or ions) in free space without a surrounding cavity. For this, atomic decoherence is also due only to spontaneous emission. We should emphasise that the major advantage of the system discussed here is that two qubit entanglement operations can be performed with the help of laser pulses, while laser pulses cannot entangle atoms in the free space case.
One method of entangling atoms via their interaction with a resonator mode is discussed in Ref. [8,10] in which the atoms fly through a high finesse cavity. The time over which the atoms interact with the field is fixed and determined by the atomic velocity. If the atoms leave the cavity their time evolution stops and the prepared state is stable. Using this idea to perform many operations in a sequence and to scale up the system by using many atoms becomes costly in both time and material. In our approach, the system once prepared in a state of the DFS, does not change, because the interaction between the atoms, the cavity mode and the environment of the system is effectively switched off. The atoms can be stored in the cavity over long periods and arbitrarily many operations can be performed.
The paper is organised as follows. In the next Section we give a detailed description of the physical system we deal with. In Section III we review the quantum jump approach [37][38][39] employed to describe the dissipative dynamics. This approach is equivalent to the Monte-Carlo wavefunction approach [40] and to quantum trajectories [41]. It also gives a simple criterion for a state to be DF. We construct the DFS for N atoms in Section IV. How the states in the DFS can be manipulated is explained in the following two Sections. We summarise our results in Section VII.

II. DESCRIPTION OF THE PHYSICAL SYSTEM
The system considered here consists of N metastable two-level atoms (or ions) confined to fixed positions inside an optical cavity. In the following |0 i and |1 i denote the ground and the excited state of atom i, respectively. The Pauli operator σ i = |0 i i 1| is the atomic lowering operator. The atoms with level separationhω 0 are considered to be in resonance with a single mode of the electromagnetic field inside the cavity. The coupling strength for each atom to the cavity mode g is taken as real. The field annihilation operator for the cavity mode is denoted by b. In addition the atoms are weakly coupled to the free radiation field outside the cavity with a coupling constant g (i) kλ for the ith atom and a field mode with wave vector k and polarisation λ. The annihilation operator for this mode is a kλ . This free radiation field provides an environment for the atoms and is responsible for spontaneous emission. We also take into account non-ideal cavity mirrors by coupling the field inside the resonator to the outside with a strengthg kλ , so that single photons can leak out. The annihilation operator of the free radiation field to which the cavity field couples is given byã kλ . Then, in the Schrödinger picture the Hamiltonian of the system and its environment is given by The first four terms give the interaction free Hamiltonian and correspond to the free energy of the atoms, the resonant cavity mode and the electromagnetic fields outside the system. Going over to the interaction picture with respect to the interaction free Hamiltonian gives rise to the interaction Hamiltonian The first term contains the coupling of the atoms to the cavity mode. The second term describes the coupling of the atoms to the free radiation field and is responsible for spontaneous emission with a decay rate Γ (see Fig. 1) as will be shown in the next Section. From the last term the damping of the cavity mode by leaking of photons through the cavity mirrors will arise. The decay rate of a single photon inside the resonator is κ and we assume here i.e. g and κ are of the same order of magnitude.
To prepare and manipulate the states of the atoms inside the DFS, resonant laser pulses are applied, which address each atom individually. The Rabi frequency of the laser which interacts with atom i will be denoted by Ω i . The Hamiltonian describing the effect of the laser in rotating wave approximation and in the interaction picture chosen above is equal to We will assume here for all Ω i = 0, Note, that the frequencies Ω i are in general complex numbers. Their phase factors cannot be compensated by changing the basis of the atomic states, because we have already chosen the coupling constants g i to be the same for all atoms.
To increase the precision of the state preparation, detectors could be used which continuously monitor the free radiation field outside the system. If a photon is emitted spontaneously or leaks out through the cavity mirrors one should stop the experiment and re-initiate the whole process. But even without detectors the experiment can work, in principle, with an arbitrary high success rate. We will show that the probability for the loss of a photon is negligible and only small errors are introduced if it is not recorded.

III. THE CONDITIONAL TIME EVOLUTION
One necessary requirement for quantum computing is the ability to manipulate the qubits in a controlled way. In any quantum algorithm, a system in an arbitrary pure state has to be transformed into another pure state by appropriate coherent unitary operations. In general the system considered here interacts with its environment, stochastically loses a photon and after a short time has to be described by a density matrix. To avoid this we consider in the following only the specific time evolution under the condition that no decay takes place, which can easily be determined from a quantum jump approach description [37,38] of the system. In this Section we summarise the main results of this approach.
With the help of the quantum jump approach a conditional Hamiltonian H cond can be obtained, which describes the time evolution of the system provided no photon is emitted, either by spontaneous emission or by leakage of photons through the cavity mirrors. This Hamiltonian can be evaluated by second order perturbation theory from the expression using Eq. (3) and (5). Here |0 ph is defined as the vacuum state of the free radiation fields outside the system. In a similar way to that used in Ref. [32], where the case of two atoms in a cavity was discussed, one finds The corresponding conditional time development operator, U cond (t, 0) = exp(−iH cond t/h), is non-unitary because H cond is non-Hermitian. This leads to a decrease of the norm of the vector developing with U cond and is connected to the waiting time distribution for emission of a (next) photon. If at t = 0 the state of the system is |ψ 0 , the state at time t is given by the normalized state [37,38] |ψ The probability P 0 to observe no photon in (0, t) by a broadband detector (over all space) is In a real experiment, the emitted photons are actually registered with an efficiency η smaller than 1, or even η = 0. Then the system is in case of no photon detection prepared in a statistical mixture of the form Here ρ ⊥ describes the state of the system for the case of photon emissions, which is in general different from the state |ψ 0 we want to prepare.

IV. CONSTRUCTION OF THE DECOHERENCE-FREE SUBSPACE
With the help of the quantum jump approach we easily find a necessary and sufficient criterion to establish a decoherence free subspace (DFS). For all states |ψ of a DFS, the probability for no photon emission for all times t has to remain unity, i.e.
This condition is fulfilled if the system does effectively not interact with the environment [22]. In addition, our criterion demands that the system's own time evolution does not move the state out of the DFS. In this Section we neglect spontaneous emission (Γ = 0) and determine all states which fulfill condition (12). In the following |n denotes a states with n photons in the cavity field mode, |ϕ corresponds to a state of the atoms only and we define |n ⊗ |ϕ ≡ |nϕ . Let us first investigate under what condition the probability density for the loss of a photon by a system in a state |ψ is equal to zero. This is the case if dP 0 (t, ψ)/dt| t=0 = 0 and leads, using Eq. (9) and (10), to the condition Therefore each state of the DFS must be of the form As expected, only if the cavity mode is empty no photon leaks out through the resonator mirrors. But condition (14) is not yet a sufficient criterion for the states of a DFS. To assure that P 0 (t, ψ) ≡ 1 for all times t, the cavity mode must never become populated. All matrix elements of the conditional Hamiltonian of the form nϕ ′ | H cond |0ϕ have to vanish for n = 0. Using Eq. (8) we find that this is the case, iff Under this condition the system's own time evolution does not drive the state out of the DFS. The states defined by Eq. (14) and (15) are also known in the literature as trapped states [30][31][32][33]. An explicit expression for the trapped states of N = 2, 3 and 4 atoms is given in Ref. [26]. Atomic states which fulfill condition (15) are well known in quantum optics as the Dicke states, of the form |l, −l in the usual |j, m notation [42]. They are eigenstates of the total Pauli spin operator. The quantum number l can take on the values 1/2, 3/2, . . . , N/2 for N odd and 0, 1, . . . , N/2 for N even. The states |l, −l are highly degenerate, namely The Dicke states with a fixed quantum number l are also eigenstates of the operator i σ † i σ i which measures the excitation n in the system [42]. The relation between n and l is given by n = N/2 − l. We describe now how an orthonormal basis for such a subset of states can be found which are orthogonal to all other Dicke states. Using the notation and Eq. (15), it can be proven that each state of the form in which for instance the first and third atom are in an antisymmetric state, the second one is in the ground state and so on, is a Dicke state. Writing down all possible states in which n pairs of atoms are in the antisymmetric state and all others in the ground state gives a subset of Dicke states. They all have the same excitation number n and cover uniformly the whole subspace of Dicke states |l, −l with n = N/2 − l. Now these states can be orthogonalised. An orthonormal basis for the DFS of N atoms can be obtained by joining together all atomic subbases for fixed n combined with the vacuum state of the cavity field. Let us define analog to Eq. (16) Then, for instance, an orthonormal basis of the trapped states of four atoms can be obtained by orthogonalising the states |g 12 g 34 , |g 12 a 34 , |g 13 a 24 , |g 14 a 23 , |g 23 a 14 , |g 24 a 13 , |g 34 a 12 and |a 12 a 34 and one finds |gg , |ga , |ag , |aa , |x 1 ≡ (|sg − |gs )/ √ 2 and |x 2 ≡ (|eg + |ge − |ss )/ √ 3 .
An orthonormal basis states for the Dicke states of two atoms is {|g 12 , |a 12 }.
In general, to obtain a simple form of the states which form the DF qubits, one can combine the atoms into pairs. The ground states and the antisymmetric states of each pair can then form one qubit. Thus for instance the first four states in Eq. (19) could be used to obtain two qubits. In this way we find N/2 qubits for an even number of atoms. They belong to a 2 N/2 dimensional subspace of the total DFS. The additional states can serve as auxiliary levels to realise certain logical operations.

V. MANIPULATION OF THE DF STATES OF TWO ATOMS
We now know how DF qubits can be constructed resulting from the states of N atoms in a cavity. But to do quantum computing one also has to be able to perform operations inside the DFS. In this Section we discuss using the example of two atoms how DF states can be manipulated. To do so a weak laser pulse is applied to create Rabi frequencies Ω 1 and Ω 2 which obey condition (6). We discuss the effect of the pulse on the system with the help of a quantum jump approach description (see Section III) which also gives the probability for no photon emission, e.g. the success rate of the proposed experiment. It will be shown that the atoms remain DF with a success rate which can, in principle, be arbitrarily close to 1. This is due to a mechanism which decouples trapped states from non-trapped ones, which we will explain in detail. A generalisation of the scheme to higher atom numbers is given in the next Section. In the following we use the same notation as given in Eq. (16) and (18), but suppress the index 12 for simplicity. As shown above the two trapped states of two atoms are |g and |a . The states |s and |e complete a basis for the atomic states. From Eq. (8) and with the abbreviations the conditional Hamiltonian, which describes the time evolution of the system under the condition of no photon losses, becomes during the laser interaction The first term describes the exchange of excitation between the field mode and the atoms, while the laser pulses change only the atomic states, as shown in Fig. 2. Terms proportional to Γ and κ are responsible for a decrease in the norm of the state vector, if higher modes of the cavity are populated or spontaneous emission of the atoms can take place. Let us assume that the system is in the ground state |0g at time t = 0 when a laser pulse of length T is applied. The unnormalised state of the system under the condition of no photon losses |ψ 0 (t) at time t is denoted in the following by To describe the time evolution of the coefficients c nx we obtain from the time dependent Schrödinger equation ih d/dt|ψ 0 (t) = H cond |ψ 0 (t) a system of differential equations, 2n g c n−1 e + 2(n + 1) g c n+1 g − (Γ + nκ) c nṡ which will be solved to a good approximation in the following.

A. Simplified discussion
First we discuss the case where the spontaneous emission by the atoms can be neglected and we set Γ = 0. The simplified calculation given in this Subsection describes already the main behaviour of the system due to the laser interaction -the one-qubit rotation.
As shown in Fig. 2, only the amplitudes c 0g and c 0a change slowly in time, on a time scale proportional to 1/|Ω − |. Here we are interested in exactly this time evolution. All other levels change on a time scale 1/g and 1/κ which is much shorter due to condition (6). If the system is initially in a DF state the laser pulse excites also the states |0s and |0e . Then the excitation of these levels is transfered with the rate g into states in which the cavity mode is populated. These states are immediately emptied by one of the following two mechanisms. One possibility is that a photon leaks out through the cavity mirrors. But, as long as the population of the cavity field is small, the leakage of a photon through the cavity mirrors is very unlikely to take place. With a much higher probability the excitation of the cavity field vanishes during the conditional time evolution due to the last term in the conditional Hamiltonian in Eq. (21). No population can accumulate in non-DF states and we can assume c nx ≡ 0 for all states outside the DFS and to zeroth order the differential equation (23) simplifies tȯ This equation describes the time evolution of the DF states to a very good approximation. If once only the trapped states are populated, the system remains inside the DFS. It behaves like a two-level system with the states |g and |a driven by a laser with Rabi frequency 2Ω − . If the system is initially, when the laser pulse of length T is applied, in the ground state |0g the atomic state at the end of the pulse, is given by By varying the length T of the laser pulses and control over the phase of Ω − any arbitrary rotation between the two states |0g and |0a can be realised. Due to Eq. (10) and (25), the probability to find no photon, P 0 (T, ψ 0 ), is unity. Note that the qualitative behaviour is independent of the Rabi frequencies Ω 1 and Ω 2 , as long as Ω 1 = Ω 2 . To a very good first approximation the atomic states do not move out of the DFS. The quantitative behaviour of the atoms does not depend on the precise values of g and κ, which simplifies possible realisations of the proposed experiment. The mechanism which decouples the DFS of the two atoms from the other states works better, the larger the parameters g and κ are compared to Ω ± which is why condition (6) has been chosen. In addition, we assumed κ and g to be of the same order of magnitude (see Eq. (4)) [43]. Here we use the presence of leaky cavity mirrors, to ensure that no photon is emitted while the laser pulse is applied! The cavity mode does not become populated during the process which entangles the two atoms with each other and prepares them in the entangled state (25). Another example, in which the no-photon time evolution has been used to entangle atoms without a coupling between them via a populated field mode is described in Ref. [32]. In Ref. [44] it is described how the state of an atom in a cavity can be teleported to an atom inside another distant cavity only by observing emitted photons.

B. A more detailed discussion
In this subsection we discuss the effect of the laser pulse in more detail and assume again Γ = 0. To solve the differential equations (23) we make use of an adiabatic elimination suggested by the separation of the frequency scales (4) and (6). Again, Eq. (23) shows that the only coefficients that do not evolve on the fast time scale g or κ are c 0g and c 0a . They change with the small rates Ω ± and Γ. Their time evolution is given bẏ The amplitudes of all other states, which evolve on the fast time scale g or κ, follow the slowly varying coefficients c 0g and c 0a . Therefore we can neglect their derivatives compared to the fast rates g and κ. Setting the derivatives of c 0s , c 0e , c 1g , c 1s and c 2g in Eq. (23) equal to zero we obtain the equations From Fig. 2 and Eq. (23) we can see that all other coefficients corresponding to non-DF states are smaller by at least one factor of |Ω ± |/g, because they can only be excited via driving with the weak laser pulse if the states |1s and |2g are populated. The amplitudes of these higher states can therefore be neglected in Eq. (27) and we obtain a closed set of equations which can be solved easily for the coefficients of the DF states. We find with The eigenvalues of M are Making use of the formula which can be checked by applying it to the eigenvectors of M [45] we find which are the coefficients of the DF states at time T under the condition of no photon emission. After the laser pulse is turned off at time T the excitation of all non-DF states vanishes during a short transition time of the order 1/g and 1/κ due to the conditional time evolution. Therefore the state of the atoms shortly after T and under the condition that no photon was emitted can be obtained by normalising the state c 0g (T ) |0g +c 0a (T ) |0a . It equals The probability of a successful operation is given by the probability for no photon emission in (0, T ). According to Eq. (10) it is given by |c 0g (T )| 2 + |c 0a (T )| 2 and leads to The state |ψ 0 (T ) belongs to the DFS. Using Eq. (8), (14) and (15) one can show H cond |ψ 0 (T ) = 0 and |ψ 0 (T ) is now -without the laser interaction -stable in time. If one neglects again all terms proportional to Γ and |Ω ± |/g Eq. (25) agrees with the result given in Eq. (25). The laser pulse performs a rotation on the DF qubit. As can be seen from Eq. (34), the sum k 1 + k 2 can be interpreted as the decay rate of the system. As long as this rate is much smaller than 1/T the probability for a successful preparation is close to 1.
C. Preparation of a maximally entangled state of the atoms Finally, we discuss as an example the preparation of the maximally entangled atomic state |a while the cavity is empty. Due to Eq. (33) this can be done by choosing the length of the laser pulse equal to Fig. 3 shows the success rate P 0 for this scheme and results from a numerical solution of Eq. (23). The result agrees in the chosen parameter regime very well with P 0 (T, 0g) given in Eq. (34). For zero spontaneous emission, success rates arbitrarily close to unity can be achieved by reducing the Rabi frequency Ω 1 . However, for Γ = 0 this is not possible. If the laser pulse becomes very long the probability of occurance of a spontaneously emitted photon increases and is no longer negligible. For finite values of Γ there is an optimal value of Ω 1 for which the success rate of the preparation scheme has a maximum. If all outcoming photons are registered and the experiment is repeated in case of an emission the fidelity of the prepared state can, for a very wide parameter regime, be very close to 1. For the parameters given in Fig. 3 it is always higher than 99 %. If the photons are registered only with an efficiency η smaller than 1, this fidelity has to be multiplied with P 0 /(1 − η(1 − P 0 )) as can be seen from Eq. (11) to then give the fidelity of the prepared state in the case of no photon detection.

VI. MANIPULATION OF THE DFS IN GENERAL
In the last Section we have shown that a weak enough laser pulse does not move the state of the system of two atoms out of the DFS. In this Section we want to point out a physical principal behind this fact which allows a straightforward generalisation of the preparation scheme to higher numbers of atoms in the cavity and other kinds of interaction. To do so we shortly review the quantum Zeno effect [34]. We also derive an effective Hamiltonian to describe the effect of a weak interaction in general.
The quantum Zeno effect [34] is a theoretical prediction for the behaviour of a system under rapidly repeated ideal measurements. It is a consequence of the projection postulate of von Neumann and Lüders [46] which describes the effect of a single measurement and predicts that the probability to measure whether the state of a system belongs to a certain subspace of states is given by its overlap with the subspace. If the outcome of the measurement is "yes" the state of the system changes during the measurement process. It becomes projected onto the subspace. The quantum Zeno effect predicts that if the time between subsequent measurements equals zero the outcome of each following measurement is the same, even if an additional interaction is applied which is intended to move the system into a complementary subspace. The system can only change inside the subspace.
We now reconsider the system of N atoms inside the cavity and assume first that no laser pulse is applied to the atoms. Let us define ∆T as a time, in which a photon is emitted with probability very close to unity, if the system is prepared in a non-DF state. Then the observation of the free radiation field outside the system over a time interval of the length ∆T can be interpreted as a measurement of whether the system is DF or not. If a photon is emitted, the system has not been in a DF state. Otherwise, its state belongs to the DFS. In the presence of a laser pulse the state of the system can be driven out of the DFS during ∆T , but as long as this effect can be negelected and the observation of the free radiation field over a time interval ∆T can still be interpreted as a measurement of whether the atoms are DF or not to a very good approximation. This is the case in the scheme we discuss here. As it has been shown in the previous Section, ∆T has to be at least of the order 1/g and 1/κ and condition (36) leads to condition (6) given in the Introduction.
In the scheme we propose the free radiation field outside the cavity is observed continuously, i.e., the time between two subsequent measurements is zero. Therefore the quantum Zeno effect can be used to predict the effect of the laser pulse on the time evolution of the system. It suggests, that the system always remains DF if it is once prepared in a state of the DFS.
A generalisation of the proposed scheme to other forms of state manipulation is straightforward. As long as the interaction is weak enough the state of the system does not move out of the DFS. The interpretation of the behaviour of the system with the help of the quantum Zeno effect can also be used to derive an effective Hamiltonian H eff which describes the effect of a weak laser pulse on the system. We know that the state of the system can change only inside the DFS due to rapidly repeated measurements whether the system is still DF. Therefore the time development operator for a short time interval ∆T is to a good approximation given by where IP DFS is the projector onto the DFS. This leads to the effective Hamiltonian If we assume that spontaneous emission by the atoms is negligible (Γ = 0) the definition of the DF state by Eq. (14) and (15) where H laser I describes the laser interaction and is given in Eq. (5). The effect of the laser on the system considered here is very different from its effect on atoms in free space. It confines the system inside the DFS and can be used to generate entanglement between the atoms in the cavity. The effective Hamiltonian for a single laser pulse depends on N different Rabi frequencies which can be chosen arbitrarily. This allows to perform a wide range of operations like the CNOT quantum gate between the qubits of a DFS. A concrete proposal for quantum computation using dissipation which is based on the idea discussed here in detail can be found in Ref. [47].
In the case of two atoms which has been discussed in the previous Section the effective Hamiltonian (39) equals and leads directly to Eq. (24) in the previous Section. The DFS of four atoms is six dimensional. Using the notation given in Eq. (19) we find

VII. CONCLUSIONS
In summary, we have given an example of a DFS suitable for quantum computing and have identified a mechanism for the manipulation of states within the DFS which can be understood in terms of the quantum Zeno effect and allows for generalisation to other forms of manipulation. This concept was demonstrated in detail for the example of two two-level atoms, which lead to an efficient method of entangling them and was genereralized to N two-level atoms.