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. PACS numbers: 03.67.Lx, 42.65.Hw, 03.67.Ac, 37.10.Jk 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 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- FIG. 1: (color online) (a) Schematic diagram for the imple- mentation of the photonic two-qubit phase gate. (b) The Λ-type atomic level diagram used for interfacing photons and spin waves. (c) The fermionic dispersion relation ϵ ( k ) in units of J (solid line), and, schematically, the amplitudes of the two spin waves in k -space (dashed lines). Π GLYPH<Slash1> 2 Π GLYPH<Minus>Π GLYPH<Slash1> 2 GLYPH<Minus>Π 1 2 GLYPH<Minus> 1 GLYPH<Minus> 2 arXiv:1001.0968v3 [quant-ph] 18 Sep 2010