Photon Molecules in Atomic Gases Trapped Near Photonic Crystal Waveguides

Realizing systems that support robust, controlled interactions between individual photons is an exciting frontier of nonlinear optics. To this end, one approach that has emerged recently is to leverage atomic interactions to create strong and spatially nonlocal interactions between photons. In particular, effective photonic interactions have been successfully created via interactions between atoms excited to Rydberg levels. Here, we investigate an alternative approach, in which atomic interactions arise via their common coupling to photonic crystal waveguides. This technique takes advantage of the ability to separately tailor the strength and range of interactions via the dispersion engineering of the structure itself, which can lead to qualitatively new types of phenomena. For example, much of the work on photon-photon interactions relies on the linear optical effect of electromagnetically induced transparency, in combination with the use of interactions to shift optical pulses into or out of the associated transparency window. Here, we identify a large new class of “correlated transparency windows,” in which photonic states of a certain number and shape selectively propagate through the system. Through this technique, we show that molecular bound states of photon pairs can be created.

Popular Summary

Particles of light—photons—are the information carrier of the world’s communication networks. Photons are particularly suited to this task because they do not strongly interact with materials or with one another, which enables them to travel over long distances. However, this lack of interaction also limits how photons can be used and the type of physical phenomena that occur with light. Producing and controlling interactions between photons could enable exotic behaviors, such as those associated with charged particles. Here, we address this challenge by identifying a method to generate highly tunable, long-range interactions between photons and show how the photonic equivalent of a molecule can be created.

We propose using newly demonstrated systems of cold atoms trapped near nanoscale devices called photonic crystals. The presence of a photonic crystal changes the nearby environment of the atoms in a way that modifies the interactions between the atoms. In particular, the range and spatial shape of the interactions can be adjusted. Photons traveling through the atomic gas then interact with one another as a result of these atom-atom interactions, and the way each photon propagates depends strongly on the presence of other photons in the system and their positions. Using numerical analyses, we show that this mechanism allows photon “molecules” to be created in which two spatially separate photons feel a restoring force that always stabilizes the distance between them, in a manner analogous to the forces that hold a molecule of atoms together.

A first step toward achieving full control of interactions in many-photon states would be to observe the molecular states of photons that we predict can arise from these photon-photon interactions.

Figure 1

(a) Rydberg interactions between atoms mediate photon-photon interactions. Two-photon excitation of atom 1 to Rydberg level $R$ shifts the $R$ level in other surrounding atoms by an amount much larger than the two-photon transition width, making the transition effectively two level for a second photon entering the system. (b) Four-level atom for EIT with interactions. The input quantum probe field $\mathcal{E}$ couples to the transition $|g⟩-|e⟩$, while transition $|s⟩-|e⟩$ couples to a classical drive field. Adding level $|d⟩$ that couples to the band edge modes of a photonic crystal (red curves with blue region indicating the band gap) and driving the transition $|s⟩-|d⟩$ off-resonantly creates an effective interaction between atoms in state $|s⟩$. For simplicity, we assume states $|e⟩$ and $|d⟩$ both have a free-space spontaneous decay rate ${\mathrm{\Gamma }}^{\prime }$. (c) The periodic dielectric structure of a photonic crystal (right) leads to band structure (left) in the dispersion relation for photons propagating in the dielectric, where interference leads to band gaps over which propagation is forbidden. For photonic crystals that support multiple modes, e.g., TE and TM (represented here by the solid and dotted lines), the band gaps of different modes may occur at different frequencies. The $|s⟩-|d⟩$ transition of the atoms (green spheres) trapped near the photonic crystal couples to the photonic crystal bands. When the transition frequency ${\omega }_{sd}$ is in the band gap (with detuning ${\mathrm{\Delta }}_b$ from the band edge) atoms can couple with one another via evanescent fields in the photonic crystal (illustrated in red), yielding a dipole-dipole interaction between the atoms.

Figure 2

(a) The excitation of the central atom to $|s⟩$ by a probe pulse (red line) traveling along the waveguide (grey), shifts the level $|s⟩$ of the surrounding atoms and a new pulse (dashed red line) encounters a shifted EIT transparency window. (b–e) EIT with uniform interactions $H_{ss}=\hslash U\sum _{j\ne l}{\sigma }_{ss}^j{\sigma }_{ss}^l$ leads to number-correlated transparency windows. (b) For a single photon, EIT occurs at the normal two-photon resonance $\delta =0$, independently of $U$, as shown in (c) by the simulated output intensity $I_1=⟨{\mathcal{E}}_{\mathrm{o}}^{†}{\mathcal{E}}_{\mathrm{o}}⟩/⟨{\mathcal{E}}_{\mathrm{i}}^{†}{\mathcal{E}}_{\mathrm{i}}⟩$, which for weak input intensity corresponds to the transmittance of single photons. (d) For two photons, EIT occurs for $\delta =U$, in which case transfer to the $|ss⟩$ two-excitation state is resonant, as shown in (e) by the equal-time coincidence rate at output $I_2=⟨{\mathcal{E}}_{\mathrm{o}}^{†}{\mathcal{E}}_{\mathrm{o}}^{†}{\mathcal{E}}_{\mathrm{o}}{\mathcal{E}}_{\mathrm{o}}⟩/⟨{\mathcal{E}}_{\mathrm{i}}^{†}{\mathcal{E}}_{\mathrm{i}}{⟩}^2$, which has a maximum for $\delta \sim U$. In (c) and (e), the system has optical depth $D=400$ and is continuously pumped with $|{\mathcal{E}}_{\mathrm{i}}{|}^2={10}^{-4}{\mathrm{\Gamma }}^{\prime }$ and $\mathrm{\Delta }=0$. (Note that $\delta =U$ can be achieved by adjusting the probe or control field frequencies; here, we adjust the control, which in this case results in higher two-photon coincidence counts for $|U|>0.5$.) (f) Two-photon transmission is also achieved by inputting two pulses with different frequencies ${\omega }_1$ and ${\omega }_2$, where the first pulse is inside the medium when the second arrives [as in (a)]. For ${\delta }_1\sim 0$, a single photon from the first pulse may propagate in the medium, while, taking ${\delta }_2\sim 2U$, a photon from the second pulse may only enter when a photon from the first is inside the medium. (g) The presence of the first photon then switches the probability of transmission $T$ for the second photon from being maximized at ${\delta }_2=0$ (blue curve) to ${\delta }_2=2U$ (green curve), with $U=2{\mathrm{\Gamma }}^{\prime }$. All simulations were done with $\mathrm{\Omega }=2{\mathrm{\Gamma }}^{\prime }$, ${\mathrm{\Gamma }}_{1\mathrm{D}}=2{\mathrm{\Gamma }}^{\prime }$, and $N_a=100$.

Figure 3

EIT in atomic medium with $N_a=200$ atoms on a lattice with lattice constant $z_a$ and interaction $H_{ss}=-\hslash \sum _{j\ne l}{\sigma }_{ss}^j{\sigma }_{ss}^{l}V(z_j-z_l)$. (a) The potential is chosen to be a square well, with an interaction strength of $V(z_j-z_l)=-0.5{\mathrm{\Gamma }}^{\prime }$ for atoms with separation $|z_j-z_l|<50z_a$ and zero otherwise. The colored curves show the three lowest energy states of the step potential. (b) When the system is driven continuously so that $\delta =0$ ($|{\mathcal{E}}_{\mathrm{i}}{|}^2={10}^{-4}{\mathrm{\Gamma }}^{\prime }$), the majority of the steady-state spin population $|{\psi }_{ss}^{j,l}{|}^2$ (arbitrary units) is in pairs of excitations with $|z_j-z_l|>50z_a$. (c) Propagation of a two-photon wave function (arbitrary units) in which the relative coordinate is initially prepared as the third harmonic eigenstate [red curve in (a)] of the square-well potential. Here, the final spin population is multiplied by a factor of 100, so both initial $|{\psi }^i{|}^2$ and final $|{\psi }^f{|}^2$ are visible on the same scale. Other simulation parameters used are $\mathrm{\Omega }=2{\mathrm{\Gamma }}^{\prime }$, ${\mathrm{\Gamma }}_{1\mathrm{D}}={\mathrm{\Gamma }}^{\prime }$ (optical depth $D=400$), and $\mathrm{\Delta }=0$.

Figure 4

(a) The photonic-crystal-mediated atomic interactions result in an interaction potential $V_{jl}=V(z_j-z_l)=G[\exp (-|z_j-z_l|/L_u)-\exp (-|z_j-z_l|/L_{\ell })]$ (black solid line) for polaritons separated by distance $|z_j-z_l|$. The interaction potential is the sum of the contribution from the lower and upper photonic crystal band edges (dashed blue lines) that have length scales $L_u=15z_a$ and $L_{\ell }=30z_a$ and strength $G=1.28{\mathrm{\Gamma }}^{\prime }$, which in this case leads to a minimum at a finite atomic separation $r_0\sim 21z_a$. Loss resulting from the photonic-crystal-mediated interactions is calculated assuming a cooperativity of $C_{\lambda }=24000$. The amplitudes $|\psi (r){|}^2$ (arbitrary units) of the two lowest bound-state wave functions are shown for ${\mathrm{\Delta }}_M=2.5{\mathrm{\Gamma }}^{\prime }$, $\mathrm{\Omega }={\mathrm{\Gamma }}^{\prime }$, and ${\mathrm{\Gamma }}_{1\mathrm{D}}=2{\mathrm{\Gamma }}^{\prime }$. (b) For two polaritons propagating with this interaction, a state initiated with a wave function matching that of the ground state, but offset from the ground-state separation at time $t=0$, oscillates in time in the relative coordinate. The oscillation of the expectation value of polariton separation about the ground-state separation (about $24z_a$, indicated by the dotted line) found by simulation the full EIT system (solid line) is closely matched to the oscillation produced by the evolution of the effective Hamiltonian $H_{\mathrm{rel}}=-({\hslash }^2/m)({\partial }^2/\partial r^2)-2\hslash V(r)$ (dashed line). In this case, a full oscillation occurs after the polaritons travel an optical depth of $D=250$ using the parameters in (a). (c) Spin population $|{\psi }_{j,l}^{ss}{|}^2$ (arbitrary units) as the initially offset pair of polaritons propagate through the atomic ensemble, with the dotted line indicating the ground-state separation of the two polaritons. The frames I, II, and III correspond to the turning points of the oscillation marked in (b).

Figure 5

Input and output of an EIT system with molecularlike interactions resulting from the potential shown in Fig. 4. (a) Single-photon component of input (blue line) and delayed output (green line) pulses, normalized by the peak intensity of the input pulse, where the output has been multiplied by a factor of 10 for visibility. (b) Two-photon input and output components, where the output has been scaled by a factor of $10^2$. (c) Second-order correlation function $g^{(2)}(\tau )$ for output two-photon separation time $\tau =|t_1-t_2|$ taken along the white line shown in (b). (d) Oscillation in average two-photon separation time $⟨\tau ⟩$ at output as the interaction strength and renormalized detuning are scaled to become $G^{\prime }=\xi G$ and ${\mathrm{\Delta }}_M^{\prime }=\xi {\mathrm{\Delta }}_M$. The optical depth is 400 for plots (a)–(c) and 800 for plot (d), with $G=1.28{\mathrm{\Gamma }}^{\prime }$, ${\mathrm{\Delta }}_M=1.5{\mathrm{\Gamma }}^{\prime }$, $\mathrm{\Omega }=1.5{\mathrm{\Gamma }}^{\prime }$, $|{\mathcal{E}}_{\mathrm{i}}{|}^2={10}^{-4}{\mathrm{\Gamma }}^{\prime }$, and ${\mathrm{\Gamma }}_{1\mathrm{D}}=2{\mathrm{\Gamma }}^{\prime }$.