Quantum controlled-phase-flip gate between a flying optical photon and a Rydberg atomic ensemble

Quantum controlled-phase-flip (CPF) gate between a flying photon qubit and a stationary atomic qubit could allow the linking of distant computational nodes in a quantum network. Here we present a scheme to realize quantum CPF gate between a flying optical photon and an atomic ensemble based on cavity input-output process and Rydberg blockade. When a flying single-photon pulse is reflected off the cavity containing a Rydberg atomic ensemble, the dark resonance and Rydberg blockade induce a conditional phase shift for the photon pulse, thus we can achieve the CPF gate between the photon and the atomic ensemble. Assisted by Rydberg blockade interaction, our scheme works in the N-atoms strong-coupling regime and significantly relaxes the requirement of strong coupling of single atom to photon in the optical cavity.

we can achieve the CPF gate between the photon and the atomic ensemble. Assisted by Rydberg blockade interaction, our scheme works in the N-atoms strong-coupling regime, i.e., the collective cooperativity C NC 1 ′ ≈ > . With a large number of atoms (N 1  ), our scheme can work in the single-atom weak-coupling regime, i.e., C 1  , which significantly relaxes the requirement of the optical cavity for realization of the quantum CPF gate.

Results
As illustrated in Fig. 1(a), the basic building block of our scheme is an ensemble of N Rydberg atoms trapped inside a single-sided optical cavity, which reflects off a flying single-photon pulse. The relevant atomic level structure and transitions are shown in Fig. 1(b). Each atom has a stable ground state g , an excited state e , and two Rydberg states r 1 and r 2 . The atomic transition g e ↔ is resonantly coupled to the cavity mode a h with horizontal (h) polarization, while a classical control field with Rabi frequency c Ω drives the transition e r 1 ↔ . Thus they form the standard three-level electromagnetically induced transparency (EIT) configuration [21][22][23] in terms of the collective states, here r r 1 2 ∆ is additional energy shift when two atoms are excited to Rydberg states r 1 and r 2 , respectively 20 . Then the total Hamiltonian for the combined system (atoms + cavity mode + free space) has the following form in the rotating frame 25 where b ω ( ) denotes the annihilation operator of free-space modes with the commutation relation , κ is the decay rate of the cavity mode, and γ is the spontaneous emission rate of the atomic excited state, and the spontaneous emissions of Rydberg states are neglected due to their long coherence time.
In this paper, two initial states for atomic qubit are considered: i) state  photon is reflected off the cavity containing the atoms in state 0 | or 1 | , the whole state of the system at arbitrary time can be described by t ) are one-atom (two-atom) excitation states of the atomic ensemble. According to the Schrödinger equation , ,  where 0 1 ξ = , denotes that the initial state of atoms is ξ . Equations (6-9) determine the evolution of the combined system, and can be solved without further approximation through numerical simulation. However, we can attack this problem analytically with some rough approximations to reveal the underlying physics. Then we find that the cavity output c out is connected with the input c in by (see Methods) To achieve the condition in Eq. (12), we could set, for example, Therefore, assisted by Rydberg blockade interaction, our scheme can work in the single-atom weak-coupling regime, i.e., C 1  , when the number of atoms N 1  . Based on above analysis, when the photon pulse is reflected off the cavity, it achieves a conditional phase shift π, i.e., when the atoms are in state 0 | , the photon experiences a phase shift π, while there is no phase shift if the atoms are in state 1 | . The physical understanding of these results can be seen from the so-called dark resonance 26 . As shown in Fig. 1(c), there are three resonant peaks for three-level cavity-EIT system. The central peak results from dark resonance 27 . When the atoms are in state 0 | , the Rydberg blockade interaction does not work (H 0 II = ). Thus the system of atoms and cavity mode is a typical three-level Ξ-type system and its Hamiltonian H I ′ has a dark state θ Ω Ω = / + . This dark state is decoupled from state e due to quantum interference in this three-level system. When the single photon is reflected off the cavity, the effect of dark resonance is equivalent to that of no atom coupled to the cavity 6 . Then the photon pulse will enter the cavity and leave it with a phase shift π. When the atoms are in state 1 | , the Rydberg blockade interaction shifts the level R 1 and moves the atomic system out of the dark state Dark . Therefore, the photon pulse, under certain conditions, will bounce back with no phase shift. Now we describe in detail how to realize the photon-atom CPF gate. Initially, the atoms are prepared in an arbitrary superposition state of two logical states, i.e., , on atoms in cavity and the photon pulse, so that there is a phase shift π only when the atoms are in the state 0 | and the photon is in the polarization h . We quantify the quality of the CPF gate between the flying optical photon and the Rydberg atomic ensemble through the numerical simulation. Following the method of Ref. [28], we perform numerical simulations with the assumption that the single-photon pulse is a Gaussian pulse, i.e., the pulse shape ∆ κ ≥ , as depicted in Fig. 2. Note that there are some symmetrical phase jumps for the π phase on both sides of center frequency, which was also observed in the single atom case 10 , however, the influence of these small phase jumps on the CPF gate is small, because most of the population of the photon pulse are around the center frequency when T 1 κ /  . Second, this conditional phase factor is very insensitive to the variation of C Ng 2 κγ ′ = / . For instance, its variation is smaller than 10 3 − for Ng 2 κγ / varying from 50 to 5, so that we do not need to know the exact number N of the atoms in the optical cavity. Third, the phase shift has a high fidelity F 0 99 > . in the typical parameter region, i.e.,  Due to atomic spontaneous emission, the noise arises from photon loss which leads to a vacuum-state output. This noise yields a leakage error which means that the final state is outside of the qubit Hilbert . Figure 3 shows the probability P of spontaneous emission loss as a function of C′ for the atomic states 0 | and 1 | . When the atoms are in state 0 | , the numerical results show P is smaller than 10 3 − . The physical reason for the results is that the dark state Dark has no contribution from the excited state e and the dark resonance process does not involve the state e . Since the population in state e is zero, there is no spontaneous emission and hence no absorption. If the atoms are in state 1 | , the curve is well simulated by the empirical formula P C 1 1 2 ≈ /( + ′). Other sources of photon loss come from the mirror scattering and absorption [16][17][18] . Note that these leakage errors only affect the probability to register a photon from each pulse and has no influence on the fidelity of its polarization state if a photon is registered for each qubit. So, the leakage errors induce small inefficiency of the CPF gate used for scalable quantum computation 8,9 .

Discussion
Next we briefly give some discussion of our scheme. First, as shown in Fig. 2, there are some symmetrical phase jumps, which remain an open question. We will further study it in the future. Second, when the atoms are in state 0 | , the photon can resonate to the cavity as it is under the Ξ-type cavity-EIT condition. Note that the cavity linewidth with this cavity-EIT dark resonance is reduced by a factor Ng Ng cos θ Ω = / + Ref. [30]. Therefore, the pulse duration T of the photons needs to satisfy the condition T 1 cos 2 κ θ /( )  . In our scheme, we assume N g c Ω ≈ , thus the pulse duration T Ng Ng 1 cos 2 . Then we address the experiment feasibility of the proposed scheme. For a potential experimental system, we consider an optical cavity traps a ensemble of ultracold atoms within the volume V m 2 3 µ ≈ ( ) 31,32 . For the high n-s (n 60 ≥ ) Rydberg states, one could achieve the strong blockade interaction with 2 25 r r MHz and the small decay rate 2 1kHz r γ π / ≈ Ref. [20]. Typically, the relevant cavity parameters are 2 8 5 2 κ γ π ( , )/ ≈ ( , . ) MHz Ref. [29] and thus 3 r r 1 2 ∆ κ > . In the optical cavity, the cavity-atom coupling strength depends on the atomic position through the relation 28 where g 0 is the peak coupling strength in the antinodes, R and k 0 are, respectively, the waist and the wave vector of the Gaussian cavity mode, and z is assumed to be along the axis of the cavity. For the experimental realistic parameters of the cavity 29 , R m 23 9µ = .
being the wavelength of cavity mode. Assume that the atomic number density of the atomic ensemble is 10 14 ρ ≈ /( cm 3 ) and thus about N 2000 ≈ atoms within the volume V m 2  is the peak cooperativity for a single atom coupled to the cavity. In summary, we have proposed a scheme that realizes the CPF gate between a flying optical photon and an atomic ensemble. When a flying single-photon pulse is reflected off the cavity containing a Rydberg atomic ensemble, the dark resonance and Rydberg blockade induce a conditional phase shift π, thus we can achieve the CPF gate between the photon and the atomic ensemble. Assisted by Rydberg blockade interaction, our scheme works in the N -atoms strong-coupling regime, i.e., the collective cooperativity C NC 1 ′ ≈ > . With a large number of atoms (N 1  ), our scheme can work in the single-atom weak-coupling regime, i.e., C 1  , which significantly relaxes the requirement of the optical cavity for realization of the quantum CPF gate.