content_image/427122.jpg
# Photonic Phase Gate via an Exchange of Fermionic Spin Waves in a Spin Chain 

Alexey V. Gorshkov, 1 Johannes Otterbach, 2 Eugene Demler, 1 Michael Fleischhauer, 2 and Mikhail D. Lukin 1 

$^{1}$Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA 

$^{2}$Department of Physics and Research Center OPTIMAS, Technische Universit¨ at Kaiserslautern, 67663, Kaiserslautern, Germany 

(Dated: November 19, 2018) 

We propose a new protocol for implementing the two-qubit photonic phase gate. In our approach, the π phase is acquired by mapping two single photons into atomic excitations with fermionic character and exchanging their positions. The fermionic excitations are realized as spin waves in a spin chain, while photon storage techniques provide the interface between the photons and the spin waves. Possible imperfections and experimental systems suitable for implementing the gate are discussed. 

Strong nonlinear interactions between sing le photons are essential for many potential applications in quantum information processing, such as efficient optica l quantum computing and fast long-distance quantum co mmunica- tion [1, 2]. However, since single-photon no nlinearities are generally very weak [3], strong photon-ph oton inter- actions require elaborate and experimentally c hallenging schemes [4–7] with a reliable, practical appro ach yet to emerge. In principle, a robust and conceptu ally simple gate between two photons can be achieved by temporar- ily storing their quantum information into t wo excita- tions with fermionic character and exchangin g these ex- citations to obtain the π -phase shift [2]. In this Letter, we propose how to implement this scheme w ith the role of fermionic excitations played by spin waves in a one- dimensional (1D) spin chain. 

The main idea of our protocol is illustrate d schemat- ically in Fig. 1(a). Two photonic wave pa ckets ( ˆ E in the Figure), labeled as R (right-moving) a nd L (left- moving), propagate in an optical waveguide [8–10] cou- pled to a 1D optical lattice filled with one bosonic Λ- type atom [Fig. 1(b)] per site and with tunneling be- tween sites turned off [11]. In step (1), photon storage techniques [12–15] are used to map the two photons via an auxiliary control field (labeled Ω) onto two spin waves formed by the collective coherence between atomic states g and s (the amplitude of the spin wave on a given atom is indicated by the darkness of the circle). In step (2), the lattice depth is reduced yielding a nearest-neighbor superexchange spin Hamiltonian [11, 16], which couples states s and g and which can be adjusted in such a way that the spin waves behave as free fermions [17]. In step (3), the spin waves propagate through each other, ex- change places, and, being fermionic, pick up the desired π phase. In step (4), the superexchange is turned off. Fi- nally, in step (5), the spin waves are retrieved back onto photonic modes. 

The gate is thus achieved via an exchange of two free fermionic excitations that temporarily carry the photonic qubits. A closely related idea of endowing photons with 

PACS numbers: 03.67.Lx, 42.65.Hw, 03.67.Ac, 37.10.Jk 

<img src='content_image/427122.jpg'>

fermionic character by means of nonlinear atom-photon interactions is presented in Ref. [2]. Contrary to this, in our proposal, fermionic behavior of photons is achieved by mapping photonic qubits onto atomic states and then relying on atom-atom interactions. Such an approach is similar to that of Refs. [5, 6, 14], where photonic quantum information is also processed by mapping it onto atomic states and using atom-atom interactions. 

Since our gate is based on the relative positions of the fermions, it is robust in that the nonlinear phase is ex- 

