Remote Hamiltonian interactions mediated by light

We address a fundamental question of quantum optics: Can a beam of light mediate coherent Hamiltonian interactions between two distant quantum systems? This is an intriguing question whose answer is not a priori clear, since the light carries away information about the systems and might be subject to losses, giving rise to intrinsic decoherence channels associated with the coupling. Our answer is affirmative and we derive a particularly simple sufficient condition for the interactions to be Hamiltonian: The light field needs to interact twice with the systems and the second interaction has to be the time reversal of the first. We demonstrate that, even in the presence of significant optical loss, coherent interactions can be realized and generate substantial amounts of entanglement between the systems. Our method is directly applicable for building hybrid quantum systems, with relevant applications in the fields of optomechanics and atomic ensembles.

Figure 1

Experimental schemes considered in this article. (a) Standard cascaded setup where two systems $S_1$ and $S_2$ interact sequentially with a 1D optical mode $a$ and realize a unidirectional interaction $1\to 2$. (b) Looped cascaded setup where system 1 couples to the light field twice, once before system 2 and once after, thus realizing the interaction $1\to 2\to 1$. (c) Setup with double passes through both systems, realizing the interaction $1\to 2\to 1\to 2$.

Figure 2

Overview of suitable experimental systems to build cascaded systems as shown in Fig. 1. (a) Membrane-in-the-middle optomechanical cavity coupled to a free-space laser beam [18, 19], (b) integrated optomechanical crystal device coupled to an optical waveguide [20, 21], (c) collective atomic spin ensemble probed by a free-space laser beam [22], and (d) atoms coupled to a nano-fiber [23, 24].

Figure 3

Sketch of the cascaded light-matter interactions.

Figure 4

Detailed schematic of the single-pass and double-pass coupling schemes. The counter-propagating mode $a_{-}$ is relevant only for the double-pass scheme.

Figure 5

Detailed schematic of the coupling scheme involving a loop on system 1.

Figure 6

Detailed schematic of the double-loop coupling scheme.

Figure 7

Cooperativity as a function of optical power loss between systems. Cooperativity for the single-loop Eq. (42) (solid red line) and the double-loop Eq. (44) (solid blue line). Also shown are the maximum achievable cooperativities with zero losses (dotted lines). These amount to $4c_1/(1/2+1/c_2)$ for the single loop and to $4c_1{c}_2$ for the double loop. For $c_2<1$, the cooperativities of these two geometries would almost coincide; for large $c_2$, they differ by a factor of $\sim c_2/2$. The dashed lines correspond to the limiting cases of infinite single-system cooperativities as given by Eqs. (42) for the single loop and (44) for the double loop.

Figure 8

Steady-state phonon numbers of both oscillators in the sympathetic cooling scenario described in the main text comparing the performance with losses ($\eta =0.9$, dashed lines) to the one without ($\eta =1$, solid lines).

Figure 9

Entanglement as characterized by the EPR variance (red color) and measured by the logarithmic negativity (blue color) in four relevant cases: (i) the looped geometry 1-2-1 (solid lines), (ii) the looped geometry with reverse order 2-1-2 (dashed lines), (iii) the double loop with interaction order 1-2-1-2 (dot-dashed lines), and (iv) the steady state of the simple cascaded scheme 1-2 (dotted lines). (a) Entanglement measures vs interaction time in the lossless case $\eta =1$. (b) Steady-state logarithmic negativity vs optical loss $1-{\eta }^2$.