Cavity QED quantum phase gates for a single longitudinal mode of the intracavity field

A single three-level atom driven by a longitudinal mode of a high-Q cavity is used to implement two-qubit quantum phase gates for the intracavity field. The two qubits are associated to the zero-and one-photon Fock states of each of the two opposite circular polarization states of the field. The three-level atom yields the conditional phase gate provided the two polarization states and the atom interact in a $V$-type configuration and the two photon resonance condition is fulfilled. Microwave and optical implementations are discussed with gate fidelities being evaluated against several decoherence mechanisms such as atomic velocity fluctuations or the presence of a weak magnetic field. The use of coherent states for both polarization states is investigated to assess the entanglement capability of the proposed quantum gates.


I. INTRODUCTION
Along the last decade, cavity Quantum Electrodynamics (cQED) [1,2,3,4,5,6,7,8,9,10] both in the microwave and optical regimes has been used to test the most striking quantum features of single atoms interacting with zero-, one-, or few-photon states. Some relevant examples are the observation of photon trapping states and sub-Poissonian statistics in the micromaser [1], the generation of photon number states on demand [2], the single atom microlaser [3], nonlinear optics with single atoms and photons [4], and non-classical statistics in wave-particle quantum correlations [5]. In particular, a series of seminal papers by S. Haroche et al. [6,7,8,9,10] have shown that cQED with long-lived atomic Rydberg states provides us with one of the most simple systems to unambiguosly test the non-local nature of quantum mechanics. In all these experiments, the Rabi oscillations between a (vacuum) quantum field and an effective twolevel atom were used to entangle the cavity field with the atom. Moreover, the passage of subsequent atoms through the microwave cavity with well controlled velocities was used to entangle them via either real [6] or virtual photons [10] allowing, therefore, the creation of massive EPR pairs. cQED devices hold great promise as basic tools for quantum networks [11] since they provide an interface between computation and communication, i.e., between atoms and photons. In this context, it is a very important task to look for techniques to quantum engineer the state of the intracavity field. Very recently it has been * Electronic address: jordi.mompart@uab.es suggested the use of a three-level atom in a cascade configuration [12] to entangle two different longitudinal modes of the radiation field in one single step. In particular, single and two bit quantum gates were discussed with the number of photons (n = 0 or 1) of each mode being the quantum bit of information. Although this proposal is very interesting its eventual implementation presents two main drawbacks: (i) it requires a high-Q cavity that sustains two different longitudinal radiation modes; and (ii) both modes must be adjusted to very particular frequencies: one mode must be on-resonance with one of the bare atomic transition frequencies while the second one must be tuned to one of the dressed states built up by the first longitudinal mode. We note here that three and multi-level atoms have been extensively investigated in the past as a successful tool for many quantum optics applications [13] and, in particular, for quantum information [4,14,15,16,17].
In this paper, we propose a novel scheme that, while also using a three-level atom, overcomes the disadvantages of the previously discussed proposal [12]. We will make use of a single longitudinal mode of the cavity to implement a quantum phase gate (QPG) between the two qubits associated to the zero and single photon states of the two opposite circular polarization states of this mode. A precise control of the interaction time between the three-level atom and the mode will yield the conditional evolution needed to implement the QPG, provided that the atom and the two polarization states interact in the so-called V -type configuration and that the twophoton resonance condition is fulfilled.
The paper is organized as follows. In Sec. II we will briefly review the interaction of a single three-level atom with a few photons field, discuss the basic ideas of the QPG implementation, and determine the explicit con-ditions for its realization. In Section III we will address some practical considerations for the physical implementation of the QPGs in both microwave and optical regimes. The application of the QPG to the intracavity field with both circular polarizations in a coherent state will be discussed in Section IV as a method to entangle the intracavity field. Finally, we will sumarize the proposal and present the main conclusions in Section V.

II. MODEL
The model we will use in this paper is sketched in Fig. 1(a) and consists of a high-Q cavity with a single longitudinal mode at angular frequency ω c and a three-level atom with its two allowed transition frequencies denoted by ω ac and ω bc . These two atomic transitions couple to the longitudinal cavity mode via the two opposite circular polarizations σ ± with coupling rates g ± and detunings ∆ + = ω ac − ω c and ∆ − = ω bc − ω c . For simplicity, in what follows we will consider the completely symmetric case given by g + = g − ≡ g and ∆ + = ∆ − ≡ ∆. Later on, in Section III, we will address the case ∆ + = ∆ − due to the presence of a stray magnetic field.

A. Setting-up the idea
It is very well known [18] that under the two-photon resonance condition, the V -type system under investigation can be appropriately described in the bright-dark states basis where the ground state |c couples only to a particular combination of the atomic bare states |a and |b , namely the bright state |B , while remaining uncoupled to the orthogonal superposition, i.e., the dark state |D . In this case, the three-level atom in interaction with the two polarization modes becomes effectively a two-level system, i.e., the atom exhibits Rabi oscillations between the atomic ground state |c and the bright state |B . Thus, a single complete Rabi oscillation takes the system back to the initial state with a global phase that depends on the detuning. On resonance a π phase is attained. These features of the V -type three-level system will be used later on to quantum engineer the intracavity field.
In what follows we will define a qubit by the vacuum and single photon Fock states. Thus, a single longitudinal cavity mode allows to hold two qubits, one for the right-hand (σ + ) and one for the left-hand (σ − ) circular polarization. To drive the conditional evolution between these two qubits, a single three-level atom is initially prepared into the internal ground state |c and then the interaction is switched on for a controlled period of time.
The particular mechanism to swith on/off the interaction will depend on the physical implementation and, accordingly, will be discussed in the next Section. Using the notation |i ⊗ |j ⊗ |k ≡ |i, j, k where, respectively, i denotes the atomic state while j and k the number of σ + and σ − polarized photons, the final state of the system after the interaction can be written, in general, as (see Fig. 1 where the c's, a 00 , b 00 , and B 01 are probability amplitudes whose explicit value depends on the detuning and the interaction time. For the symmetric case considered here c 01 = c 10 and a 00 = b 00 . |B 01 ≡ 1 √ 2 (|a, 0, 1 + |b, 1, 0 ), (|D 01 ≡ 1 √ 2 (|a, 0, 1 − |b, 1, 0 )) is the bright (dark) state combination of |a, 0, 1 and |b, 1, 0 . For interaction times such that complete Rabi oscillations occur, i.e., times for which the atom is brougth back to the internal state |c , only phases are left, whose explicit value depends on the number of oscillations and the cavity detuning. Therefore, looking for interaction times that yield complete Rabi oscillations both when the initial state is |c, 1, 0 (and, therefore, also for |c, 0, 1 ) and when it is |c, 1, 1 , two different QPGs can be implemented: Each of these two QPGs together with arbitrary single qubit gates [12] for both circular polarization states yields a universal set of quantum gates and, therefore, these two QPGs can be used to entangle the two polarization states. We want to note here that both V -and Λ-type three-level atomic configurations are suitable for the proposal here discussed. The additional advantatge of using a V -type scheme is that the common state |c has lower energy than the other two atomic states and can be radiatively stable if it is the ground state.

B. Conditions for the gate operation
To look for the conditions needed to implement (2) and (3) we start by writing down the Hamiltonian of the system. In the rotating wave approximation and the interaction picture, the truncated Hamiltonian of the system restricted to the computational basis reads (h = 1): where g is the vacuum Rabi frequency. It will be shown in Section III that the cavity decay and the spontaneous emission in modes other than the cavity mode can be neglected, based on a time-scale arguments, for a suitably chosen set of parameters. Accordingly, we now solve the Schrödinger equation for this Hamiltonian which, in our case, will provide the same information as the density matrix of the corresponding master equation. It is clear from Eq. (4) that we deal with three uncoupled two-level systems and a simple analytical solution can be obtained by integrating the corresponding Schrödinger equation. In each of these three cases, the probability amplitude of state |c, j, k evolves in time according to with Ω 01 = Ω 10 = 4g 2 + ∆ 2 and Ω 11 = 8g 2 + ∆ 2 , and where we have assumed c jk (t = 0) = 1 for j, k = 0, 1. Note from (4) that c 00 ≡ c, 0, 0| ψ(t) does not evolve in time. Therefore, to implement the first gate transformation (2) one needs: ; ; where the integers m, n, and p should fulfill the inequality 2p + 1 > 2n > 2m ≥ 0. From the definition of Ω 01 and Ω 11 it follows from Eq. (6) that which is a Diophantine-type equation whereby for a fixed n the value of m is determined through the detuning ∆ according to the relation ∆/g = 2m/ √ n 2 − m 2 . The problem then reduces to find a p closest to an integer value that satisfies Eq. (7). Similarly, the implementation of the second quantum phase gate (3) requires ∆t 2 = 2πm; Ω 01 t 2 = (2n + 1) π; Ω 11 t 2 = (2p + 1) π, (8) leading to the equation The corresponding inequality reads 2p + 1 > 2n + 1 > 2m ≥ 0, while the relation between m, n and the detuning is ∆/g = 2m/ (2n + 1) 2 − m 2 . We report in Table   1 the best numerical solutions of (7) and (9) that, in addition, minimize the interaction time gt. Note that the values for the detuning and the interaction time given in Table I are made dimensionless through the vacuum Rabi frequency g which means that these results are general in the sense thay they do not rely on any specific physical implementation.   (7) and (9), sorted by the required interaction time in dimensionless units.
In order to check the validity of our proposal, we have numerically integrated the Schrödinger equation for the four different input states in (1) and checked for both QPGs how much the output states deviate in amplitude and phase from the exact phase gate transformation. We have characterized the deviation by the following fidelity where c out jk ≡ c, j, k| ψ out jk , δφ jk is the phase difference between the phase acquired during the gate time and the exact phase of the gate defined in (2)

III. PHYSICAL IMPLEMENTATIONS
Up to this point we have discussed the realization of the QPGs for the intracavity field in general terms. Below we will give some practical considerations regarding the physical implementation of the previous ideas in both microwave and optical regimes.

A. Microwave regime
In the microwave regime, single three-level Rydberg atoms crossing the high-Q cavity could be used to implement the QPGs for the intracavity field. The interaction time can be controlled by an accurate selection of the velocity of the incident atoms. Typically, Rydberg atoms used in cQED yield vacuum-Rabi frequencies on the order of g/2π ≃ 50 kHz [6] which would imply gate times of a few tenths of a millisecond for the QPGs discussed in this paper. This time must be compared with the lifetime of a photon in high-Q microwave cavity that can be as large as few ms, and with the atomic lifetime in Rydberg states that can be tens of ms [6].
We have investigated the robustness of the previously discussed QPGs against some of the experimental imperfections existing in the microwave regime. Figure 3 shows the fidelity for the implementation for the first gate in Table I as a function of both the atomic velocity through the cavity and the intensity of a uniform stray magnetic field along the cavity axis. The magnetic field plays in general a negative role as it breaks the degeneracy between atomic states |b and |c , i.e., yields ∆ + = −∆ − = 0, so that the closed two-level picture for the |c, 1, 1 ↔ |B 01 transition is no longer valid. The relationship between the cavity detuning and the strength of the magnetic field shown in Fig. 3 is given byh∆ + = µ B g J m J B where g J is the gyromagnetic factor and µ B is the Bohr magneton. To be specific, we have choosen the following values m J = 1 and g J = 3/2 corresponding to J = 1, L = 1, and S = 1.
Thus, gate realizations with fidelities F > 0.99 demand an accuracy of the atomic velocities on the order of a few tenths of meters per second and magnetic fields smaller than a few tenths of mG. Both requirements can been achieved in present cQED experiments with Rydberg atoms [10]. Similar results are obtained for the rest of the QPGs shown in Table I.

B. Optical regime
By working in the optical regime, all the cryogenic complications encountered in the microwave case needed to reduce the thermal photon noise can be avoided. In the optical regime, a possible atomic candidate to drive the QPG in the intracavity field is Strontium [19]. Depending on the specific fine-structure component and on its four natural isotopes (three of which are bosonic, 88 Sr (82%), 86 Sr (10%) and 84 Sr (0.5%)), a wide choice of transitions with different g's, linewidths and wavelengths are possible. The inter-combination line 5 1 S 0 − 5 3 P 1 of 88 Sr, in particular, spans two transitions that couple the ground state with vanishing nuclear spin with two fairly long-lived (τ = 20 µsec) degenerate J = 1 states. They both fall in the visible range (λ = 689 nm), and hence are easily accessible with common semiconductor lasers; and both can acquire a large vacuum Rabi frequency g/2π ≃ 25 MHz when high finesse micro-cavities (F ≃ 3×10 6 ) are used [20]. The coherent atom-field interaction needs to dominate over decoherence rates, hence the strong coupling regime of cQED is required. For the cavity decay rate κ and spontaneous emission rate γ, this means that g ≫ (κ, γ). As the vacuum Rabi frequency g is not constant throughout the cavity mode volume, optimal results will be obtained for an atom trapped at the antinode of the cavity field. This is experimentally viable, as shown e.g. in Refs. [21], where trapping times  Table I in the parameter plane atomic velocity versus the intensity of a uniform magnetic field along the cavity axis. The parameter setting is ∆ = 0, g = 2π · 50 kHz, t = 0.12 ms, and L = 4 cm being L the effective cavity length.
up to 1 sec have been reported in the strong coupling regime. Results shown in Table I suggest that the gate operation can be implemented over times t gate of the order of one microsecond. Due to the very high finesse of microcavities [20], lifetimes τ cav of the order of few microseconds can possibly be achieved, implying t gate ≤ τ cav , which is in turn appreciably less than the atomic natural lifetime τ . For the realization of gate fidelities F > 0.99, a condition t gate ≪ τ cav needs to be satisfied. This is feasible, given the rapid progress of optical micro-cavities over the recent years [20,21].

IV. ENTANGLING THE INTRACAVITY FIELD
Application of a two qubit QPG to an intracavity field with well defined photon number clearly produces only a global phase. For quantum computation purposes one also needs to produce single qubit (local) operations for both polarizations. Methods for the implementation of single qubit operations for the intracavity field have been considered in [12]. To only demonstrate the entanglement capability, there is however a very simple and accessible approach to entangle the two circular polarizations of the intracavity field. We will not start with a well defined photon number in each mode, but initially inject into the hight-Q cavity a product state consisting of a coherent state for both circular polarizations: Here |α + | 2 (|α − | 2 ) is the mean photon number of the σ + (σ − ) polarization. This state is clearly separable. In Fig. 4 we use the concurrence [22] C ≡ 2 |c 00 c 11 − c 01 c 10 | to quantify the entanglement between the two polarization states after the interaction with the atom as a function of the mean photon number of each polarization. Notice that C only takes into account the Hilbert space of zero and one photons, while other correlations are ignored. The maximum concurrence, C = 0.73, is reached when the mean photon number of both left and right polarization is equal to N = 0.5.  Table I.

V. CONCLUSIONS
We have proposed a novel cQED technique to quantum enginneer the intracavity field. Single three-level atoms are used to implement two different QPGs between the vacuum and single photon states of the two opposite circular polarizations of a single longitudinal mode. QPG realizations with fidelities above 0.99 and gate times of around a tenths of a ms for the microwave regime and a few µs for the optical regime can be attained. Some practical considerations such as the role of atomic velocity fluctuations or the presence of a uniform stray magnetic field along the cavity axis have been addressed, showing that the QPGs here discussed can be implemented with state of the art technology. We have analyzed a QPG realization where each circular polarization of the intracavity field is initially in a coherent state which, as it has been demonstrated, constitutes a simple method to entangle the two polarization states of the intracavity field. We want to note that the ideas here discussed to quantum engineer the intracavity field could be extended to other cQED physical systems of current interest, e.g., solid state devices [23], such as superconducting electrical circuits, and SQUIDs [24,25].
Finally, we would like to note also that the use of threelevel atoms in the optical regime has been previously discussed in the cQED literature [4,15]. For instance, single three-level atoms in a Λ-type configuration have been used recently to deterministically produce a singlephoton source [15].