Implementation of quantum state manipulation in a dissipative cavity

We discuss a method to perform dissipation-assisted quantum state manipulation in a cavity. We show that atomic spontaneous emission and cavity decay might be exploited to drive many atoms into many-body steady-state entanglement. Our protocol offers a dramatic improvement in fidelity when noise strength increases. Moreover, the dephasing noise is suppressed effectively by showing that high-fidelity target state can be obtained in a dissipative environment.

where subscript j corresponds to the jth identical atom. Ω is the Rabi frequency of classical pulse. a and b are the annihilation operators for cavity modes. g and j Ω are the atom-cavity coupling constants. In the system, the source of decoherence originates from cavity photon decay and atomic spontaneous emission. With considering the dissipation within Markovian approximation, the time evolution operator is given by a master equation with a Lindblad form . j 1 τ , and j 2 τ , are the spontaneous emission rates which are related to the decay channels E L → and E R → , respectively. κ and κ′ are the photon decay rates (for the sake of simplicity, we set that κ κ = ′). The central idea of our work can be understood by considering three atoms in an open cavity. When the ground state L L L 1 2 3 is initially populated and the strength of classical field Ω is sufficiently weak, there is only one single excitation in the whole system. The transition from 0 0 The subscripts a and b denote the cavity modes a and b, respectively. Then the excited manifold may show energy splitting. Under the condition that g ∆ = , the state L L L 1 2 3 is resonantly coupled to the state 0 0 1 0 (χ is a normalization parameter). The atoms decay through the cavity mode a from the ground state to the state 2 ψ which is our wanted state. With considering the homogeneous collective spontaneous emission of atoms, the net transfer is only possible from the ground state to state 2 ψ . A similar behavior has been reported for the system of an optical cavity containing many atoms in the presence of collective spontaneous emission 17 . When the other atomic decay channels are included 18  1 } = , , ..., − . We can use this method to drive many atoms into the state with single excitation.
In Fig. 1(a), we consider the case that three atoms in a cavity are initially in ground state. By a direct numerical simulation of master equation with the Hamiltonian in Eq. (1), the temporal evolution of density matrix is calculated numerically. As expected, the numerical results show that the fidelity of state 2 ψ can reach 0.92. In addition, the fidelity of four-qubit state is about 0.9 if j k τ , is g 0 02 . and the other parameters are chosen the same as those in three-qubit case. Without loss of generality, in the following, we will use three atoms as an example. With the increasing of j k τ , and κ, the Lindblad operators corresponding to noise terms will drive the transition from 2 ψ to the other states, which makes the stationary state be away from 2 ψ , so the fidelity is reduced greatly. Similar problem arises in Refs. [15][16][17][18]. Is there any way to improve the fidelity under strong dissipation?In Fig. 1(b), one can observe that the density matrix can be expressed as L L L L L L approximately because the other matrix elements have negligible amplitudes. The stronger the noise strengths are, the larger the parameter α is. To improve the fidelity, the single qubit operations x j σ will be performed on the atom j Here F is an auxiliary ground state which is not coupled to the cavity mode.
Then we set the parameters j Ω to be equal to Ω′. Through the driving H c , the atoms in state R will evolve to state L . The photon in mode b need be detected by a detector. The process is described by the following master equation where d denotes the annihilation operator of a detector mode. The master equation describes an irreversible detection process. By detecting the photon at time t (t t d > ), the system is projected to the subspace where the detector clicks. Correspondingly, the density matrix is expressed by  Fig. 2, the fidelity of three-qubit is improved from 0.8 to 0.98 when the cooperativity g 2 κτ / is about 20. It is very important to discuss the variation of interaction time to reach the stationary state when the number of atoms increases. For many-atom system, we cannot simulate the dynamics evolution directly because of the increased complexity. In order to get additional insight, we use Monte Carlo wave function method 23,24 to calculate the time evolution of four-and five-atom states. The simulations are performed  under 200 quantum trajectories for each time point. Because the time needed to complete the preparation process is longer than its corresponding detection time, we will consider the time evolution in the preparation process. In Fig. 3(a), one observes that the fidelity of four-or five-atom state is about 0.9. The equilibration time corresponding to four-and five-atom systems is slightly different. However, with increasing of atomic number, the interaction time should increase. In Figs. 3(b,c), the density matrix elements are shown. The steady-state density matrix of the atoms can still be written as . As the interaction time decreases, the ratio α β / becomes large. Then high-fidelity state can still be obtained in the detection process. For example, when the interaction time is chosen as 3000/g and the other parameters are the same as those in Fig. 3, the final fidelities of four-and five-atom states are 0.99 and 0.97, respectively.
It is necessary to consider the influence of inhomogeneous dephasing on the creation of quantum state. The super-operator for dephasing noise is given as follows The fidelity is calculated by solving the master equation including the dephasing noise term. Unfortunately, the fidelity is significantly decreased when the dephasing noise is taken into account, i.e., the fidelity is only 0.1 (0.8) before (after) detecting the photon, if we chose g 0 01 γ = .
in Fig. 5(a). How do we reduce the undesired effect of dephasing noise in the whole process?Base on our method, a simple modification can be made to resist the influence of inhomogeneous dephasing noise. In the beginning, the state of atoms is driven by H d within a time interval t δ . Then the x σ operations drive the atomic transition between the states L and R . In the following, we apply H 0 = for the same time interval t δ . Another x σ operation is performed on each atom again. That is to say, after each coherent driving over a time interval t δ , the operations x σ , e iH t δ − , and x σ are done on the system sequentially. The basic unit of the whole process can be described by open system dynamical maps shown in Fig. (4). From Fig. 5(b), we show that the final fidelity is about 0.97 in the presence of dephasing noise. The physical mechanism can be understood as follows: the dephasing noise will induce an unwanted phase fluctuation that destroys the coherence of our target state. The unwanted phase can be cancelled by applying the operations x σ , e iH t δ − , and x σ in the noisy environment. In addition, the dephasing noise results in a larger or smaller energy shift which does not change the dynamics evolution dramatically within a short time interval. We also must point out that, with the increasing of time interval, the influence of phase noise on the dynamics will no longer be neglected. However, the fidelity is more than 0.9 if the time interval is less than g 50/ .
Next we will discuss the influence of atomic spontaneous emission and photon decay on the manipulation of quantum state. Because the noise is not easy to control, the strength of noise parameter may vary within a wide range. In our work, the dissipation term for the channel E L → plays a neglectable role due to the fact that the atoms are driven to ground state along the channel. On the other hand, the varied spontaneous emission rate for the channel E R → incoherently changes the distribution of populations. Thus we will fix the parameter j 1 τ , and consider the effect of the variation of spontaneous emission rate j 2 τ , corresponding to the channel E R → . In Fig. 6(a), one finds that the final    fidelity is about 0.974 when the noise parameters vary in a wide range. In Fig. 6(b), when photon decay rate κ is changed from g 0 1 . to g 0 3 . , we observe that the fidelity is 0.975. The results show that our method can be robust against the variation of noise parameters. Now we would like to give a brief analysis of the experimental implementation. The configuration of atom may be realized with existing atom-cavity system in experiment 25,26 . In our proposal, we chose the parameters as g 100Mhz , then the target quantum state is obtained with a fidelity of 0.98. The total interaction time is about 73 s µ , which is smaller than the lifetime of metastable state. In the detection process, the conventional detector is only required to distinguish the vacuum and non-vacuum Fock number states because the total excitation number is less than or equal to 1 under the condition of weak driving. In addition, does the imperfect efficiency of detector influence the implementation? To evaluate the effect of detection efficiency on the fidelity, the dissipative term is arranged into the Lindblad form as If the efficiency is low, the photon might leak into the environment. Then the detector will not be clicked. As a result, the success probability will be decreased. However, the fidelity is almost not affected, i.e., when the efficiency of detector η is 0.8, g 0 34 κ′ = . , and the other parameters are the same as in Fig. 2, the fidelity and success probability for three-qubit state are about 0.98 and 0.81, respectively. Therefore, the method might be used to obtain a high-fidelity quantum state in an open system.
In conclusion, we have studied the dissipative dynamics of many atoms in a cavity. We showed that the target quantum state can be obtained by engineering the source of noise. Both cavity decay and atomic spontaneous emission have been changed from a detrimental source to a useful resource. The fidelity inevitably drops with the increasing of the noise strengths. However, the fidelity of steady state can be further improved by detecting the photon in cavity. Furthermore, a slight modification of our method allows the creation of target quantum state in the presence of dephasing noise. Thus our protocol might open up a promising perspective for manipulating quantum state in a noisy environment.