Nonlinear quantum optics mediated by Rydberg interactions

If an optical nonlinearity arises due to a field-dependent shift in the atomic resonance, then to first order the nonlinear response is proportional to the product of the shift and the gradient of frequency dependence of the refractive index, i.e., the dispersion [28]. It is convenient to parameterize the gradient in the refractive index in terms of the group refractive index which is defined as

$$\begin{eqnarray}&&{n}_{{\rm{g}}}=1+\omega \displaystyle \frac{\partial n}{\partial \omega },\end{eqnarray} \tag{ 1 }$$

where ω is the angular frequency of the light. In a dilute medium where the refractive index is close to unity, we can write $n\;=1+\tfrac{1}{2}{\chi }_{{\rm{r}}}$, where ${\chi }_{{\rm{r}}}$ is the real part of the electric susceptibility. Hence for a large group index, ${n}_{{\rm{g}}}=\tfrac{1}{2}\omega \partial {\chi }_{{\rm{r}}}/\partial \omega$.

Next we consider the shift of the Rydberg level, which for an external electric field has the form of a dc or ac electric Stark shift

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{{\rm{Ryd}}}=-\displaystyle \frac{1}{2}\displaystyle \frac{\alpha {{ \mathcal E }}^{2}}{{\hslash }},\end{eqnarray} \tag{ 2 }$$

where α is the atomic polarizabilty at the frequency of the external field, which can be different to the frequency of the optical field. This latter point is the key to origin of large nonlinearities in Rydberg ensembles. The polarizability of Rydberg atoms at optical frequencies is small but the polarizability from dc to microwave frequencies can be enormous [29]. This fact has been exploited in microwave cavity QED experiments for decades, where individual Rydberg atoms are used as ultra-sensitive probes able to monitor few-photon intra-cavity fields [30]. These low frequency susceptibilities scale as the principal quantum number n⁷. To qualitatively understand the Rydberg–Rydberg interaction, consider that another Rydberg atom produces a low frequency field ${ \mathcal E }$ proportional to the induced Rydberg dipole which scales as n². Consequently, when considering only the dipole–dipole term of the interaction Hamiltonian, we obtain a van-der-Waals type interaction scaling as $\alpha {{ \mathcal E }}^{2}\sim {n}^{11}$.

Writing the nonlinear optical response as slope $\partial {\chi }_{{\rm{r}}}/\partial \omega$ times shift ${{\rm{\Delta }}}_{{\rm{Ryd}}}$, we get a term that scales quadratically with the field, i.e., a Kerr-like effect

$$\begin{eqnarray}&&{\chi }^{(3)}{{ \mathcal E }}^{2}=\displaystyle \frac{\partial {\chi }_{{\rm{r}}}}{\partial \omega }{{\rm{\Delta }}}_{{\rm{Ryd}}}=-\displaystyle \frac{{n}_{{\rm{g}}}\alpha }{\omega }{{ \mathcal E }}^{2}.\end{eqnarray} \tag{ 3 }$$

So the Kerr nonlinearity ${\chi }^{(3)}$ is proportional to the product of the group index and the polarizability. To get a large group index we would like to work close to resonance or even on resonance, but this has the disadvantage that the imaginary part of the susceptibility is also large, giving rise to off-axis scattering and hence loss. The solution is EIT, where an additional control field renders the medium transparent on resonance due to destructive interference between excitation pathways [23]. Group indices as large as 10⁶ are possible using EIT in atomic ensembles as first demonstrated in 1999 [31]. A large group index gives rise to the phenomenon of slow light, enabling compression of the light pulse inside the medium. In addition, the ability to control the group index enables 'storing' and retrieving light pulses, which is the basis of quantum memory (see e.g., Hammerer et al for a recent review [32]).

By combining EIT and Rydberg states, we get the best of both worlds, i.e., both the largest possible group index and the exaggerated sensitivity of Rydberg state to low frequency fields either applied externally or induced by other nearby Rydberg atoms.

To see how the large group index arises in an EIT medium, we present a brief derivation of the EIT susceptibility following Gea-Banacloche et al [33] and Fleischhauer et al [23]. Consider the level structure in figure 1(a) of an atom with states $| {\rm{g}}\rangle$ and $| {\rm{e}}\rangle$ that is excited by a probe laser with Rabi frequency ${{\rm{\Omega }}}_{{\rm{p}}}$ and detuning ${{\rm{\Delta }}}_{{\rm{p}}}$. The excited state $| {\rm{e}}\rangle$ is coupled to a highly excited Rydberg state $| {\rm{r}}\rangle$ by a coupling laser with detuning ${{\rm{\Delta }}}_{{\rm{c}}}$ and Rabi frequency ${{\rm{\Omega }}}_{{\rm{c}}}$. The equations for the coherences of this 3-level ladder system are

$$\begin{eqnarray*}{\dot{\tilde{\rho }}}_{{\rm{eg}}} & = & -{\rm{i}}\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{p}}}}{2}({\tilde{\rho }}_{{\rm{gg}}}-{\tilde{\rho }}_{{\rm{ee}}})+{\rm{i}}{{\rm{\Delta }}}_{{\rm{p}}}{\tilde{\rho }}_{{\rm{eg}}}-{\rm{i}}\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{c}}}}{2}{\tilde{\rho }}_{{\rm{rg}}}-\gamma {\tilde{\rho }}_{{\rm{eg}}}\;\\ {\dot{\tilde{\rho }}}_{{\rm{rg}}} & = & {\rm{i}}\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{p}}}}{2}{\tilde{\rho }}_{{\rm{re}}}+{\rm{i}}({{\rm{\Delta }}}_{{\rm{p}}}+{{\rm{\Delta }}}_{{\rm{c}}}){\tilde{\rho }}_{{\rm{rg}}}-{\rm{i}}\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{c}}}}{2}{\tilde{\rho }}_{{\rm{eg}}}-{\gamma }_{{\rm{r}}}{\tilde{\rho }}_{{\rm{rg}}}.\\ {\dot{\tilde{\rho }}}_{{\rm{re}}} & = & -{\rm{i}}\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{c}}}}{2}({\tilde{\rho }}_{{\rm{ee}}}-{\tilde{\rho }}_{{\rm{rr}}})+{\rm{i}}{{\rm{\Delta }}}_{{\rm{c}}}{\tilde{\rho }}_{{\rm{re}}}+{\rm{i}}\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{p}}}}{2}{\tilde{\rho }}_{{\rm{rg}}}-\gamma ^{\prime} {\tilde{\rho }}_{{\rm{re}}},\end{eqnarray*}$$

where γ and $\gamma ^{\prime}$ are decoherence rates of the driven transitions (${\rm{g}}\leftrightarrow {\rm{e}}$ and ${\rm{e}}\leftrightarrow {\rm{r}}$), which are often much larger than the decoherence rate ${\gamma }_{{\rm{r}}}$ of the two-photon transition ${\rm{g}}\leftrightarrow {\rm{r}}$. If spontaneous emission is the only decay mechanism then $\gamma ={\rm{\Gamma }}/2$, where Γ is the spontaneous decay rate of state $| {\rm{e}}\rangle$. For the steady-state solution in the weak-probe limit (where the populations ${\rho }_{{\rm{gg}}}=1,{\rho }_{{\rm{ee}}}=0,{\rho }_{{\rm{rr}}}=0$), we find that ${\tilde{\rho }}_{{\rm{re}}}=0$ and then from the second equation

$$\begin{eqnarray*}&&{\tilde{\rho }}_{{\rm{rg}}}=-\displaystyle \frac{{\rm{i}}{{\rm{\Omega }}}_{{\rm{c}}}/2}{{\gamma }_{{\rm{r}}}-{\rm{i}}({{\rm{\Delta }}}_{{\rm{p}}}+{{\rm{\Delta }}}_{{\rm{c}}})}{\tilde{\rho }}_{{\rm{eg}}}\end{eqnarray*}$$

and if we substitute this into the steady-state solution of the first equation, we find an expression for the coherence on the probe transition

$$\begin{eqnarray*}&&{\tilde{\rho }}_{{\rm{eg}}}=-\displaystyle \frac{{\rm{i}}{{\rm{\Omega }}}_{{\rm{p}}}/2}{\gamma -{\rm{i}}{{\rm{\Delta }}}_{{\rm{p}}}+{{\rm{\Omega }}}_{{\rm{c}}}^{2}/[{\gamma }_{{\rm{r}}}-{\rm{i}}({{\rm{\Delta }}}_{{\rm{p}}}+{{\rm{\Delta }}}_{{\rm{c}}})]/4}\quad ,\end{eqnarray*}$$

which determines the induced dipole on the probe transition. The resulting electrical susceptibility is

$$\begin{eqnarray}{\chi }_{3 \mbox{-} \mathrm{level}} & = & \displaystyle \frac{2\gamma }{{{kl}}_{{\rm{a}}}}\displaystyle \frac{{\tilde{\rho }}_{{\rm{eg}}}}{{{\rm{\Omega }}}_{{\rm{p}}}}={\chi }_{2 \mbox{-} \mathrm{level}}\\ & & \times \left[1-\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{c}}}^{2}}{4({\gamma }_{{\rm{r}}}-{\rm{i}}{{\rm{\Delta }}}_{{\rm{p}}}-{\rm{i}}{{\rm{\Delta }}}_{{\rm{c}}})(\gamma -{\rm{i}}{{\rm{\Delta }}}_{{\rm{p}}})+{{\rm{\Omega }}}_{{\rm{c}}}^{2}}\right],\end{eqnarray} \tag{ 4 }$$

with the susceptibility of the bare two-level system (${{\rm{\Omega }}}_{{\rm{c}}}=0$) given by

$$\begin{eqnarray}&&{\chi }_{2 \mbox{-} \mathrm{level}}=\displaystyle \frac{1}{{{kl}}_{{\rm{a}}}}\displaystyle \frac{{\rm{i}}\gamma }{\gamma -{\rm{i}}{{\rm{\Delta }}}_{{\rm{p}}}}.\end{eqnarray} \tag{ 5 }$$

Here ${l}_{{\rm{a}}}={(N\sigma )}^{-1}$ is the resonant attenuation length, with N the number density of atoms and σ the optical cross-section ($\sigma =3{\lambda }^{2}/2\pi$ for a closed two-level transition). These susceptibilities are plotted in figures 1(b) and (c).

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To simplify the presentation, we now focus on the resonance case ${{\rm{\Delta }}}_{{\rm{c}}}=0$. Whereas in the two-level system we have maximum scattering on resonance, in EIT the scattering is suppressed. Minimum scattering is obtained at two-photon resonance ${{\rm{\Delta }}}_{{\rm{p}}}=-{{\rm{\Delta }}}_{{\rm{c}}}=0$, where

$$\begin{eqnarray}&&{\chi }_{3 \mbox{-} \mathrm{level}}({{\rm{\Delta }}}_{{\rm{p}}}=-{{\rm{\Delta }}}_{{\rm{c}}}=0)={\chi }_{2 \mbox{-} \mathrm{level}}\left[1-\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{c}}}^{2}}{4{\gamma }_{{\rm{r}}}\gamma +{{\rm{\Omega }}}_{{\rm{c}}}^{2}}\right],\end{eqnarray} \tag{ 6 }$$

so we require ${{\rm{\Omega }}}_{{\rm{c}}}^{2}\gg 4{\gamma }_{{\rm{r}}}\gamma$ to induce significant transparency. In this regime, the EIT linewidth ${{\rm{\Gamma }}}_{{\rm{EIT}}}={{\rm{\Omega }}}_{{\rm{c}}}^{2}/(4\gamma )$ is dominated by power broadening (${{\rm{\Gamma }}}_{{\rm{EIT}}}\gg {\gamma }_{{\rm{r}}}$). Assuming an EIT transparency window much narrower than the one-photon absorption line (${{\rm{\Gamma }}}_{{\rm{EIT}}}\ll \gamma$), we find from equation (4)

$$\begin{eqnarray}&&{\chi }_{3 \mbox{-} \mathrm{level}}\approx {\chi }_{2 \mbox{-} \mathrm{level}}\left(1-\displaystyle \frac{{{\rm{\Gamma }}}_{{\rm{EIT}}}}{{{\rm{\Gamma }}}_{{\rm{EIT}}}-{\rm{i}}{{\rm{\Delta }}}_{{\rm{p}}}}\right)\;\end{eqnarray} \tag{ 7 }$$

for probe tuned within the EIT linewidth ($| {{\rm{\Delta }}}_{{\rm{p}}}| \ll {{\rm{\Gamma }}}_{{\rm{EIT}}}$ and ${{\rm{\Delta }}}_{{\rm{c}}}=0$). If we now linearize the susceptibility around ${{\rm{\Delta }}}_{{\rm{p}}}=0$, we find a real part ${\chi }_{{\rm{r}}}\approx {({{kl}}_{{\rm{a}}})}^{-1}{{\rm{\Delta }}}_{{\rm{p}}}/{{\rm{\Gamma }}}_{{\rm{EIT}}}$ and a corresponding group index

$$\begin{eqnarray}&&{n}_{{\rm{g}}}\approx \displaystyle \frac{\omega }{2}\displaystyle \frac{\partial {\chi }_{{\rm{r}}}}{\partial {{\rm{\Delta }}}_{{\rm{p}}}}\approx \displaystyle \frac{1}{{{kl}}_{{\rm{a}}}}\displaystyle \frac{\omega }{2{{\rm{\Gamma }}}_{{\rm{EIT}}}}.\end{eqnarray} \tag{ 8 }$$

Thus narrow EIT resonances enable a large group index and hence large optical nonlinearity. To obtain a narrow resonance (small ${{\rm{\Gamma }}}_{{\rm{EIT}}}$) while satisfying the above requirement ${{\rm{\Gamma }}}_{{\rm{EIT}}}\gg {\gamma }_{{\rm{r}}}$ to guarantee significant transparency, we require a long-lived state $| {\rm{r}}\rangle$ (small ${\gamma }_{{\rm{r}}}$). This is indeed the case if $| {\rm{r}}\rangle$ is another ground state (Λ-EIT) or a Rydberg state.

The nonlinear response arises from a level shift which changes ${{\rm{\Delta }}}_{{\rm{c}}}$, as illustrated in figure 1. The success of Rydberg nonlinear optics relies on the ability to map the large shifts arising from low frequency fields onto an optical field using EIT. The first experiment demonstrating a large Kerr effect due to an external electric field using Rydberg EIT was reported in 2008 [34].

The probe field propagates in the Rydberg-EIT medium as a so-called Rydberg polariton with a group velocity

$$\begin{eqnarray}&&{v}_{{\rm{g}}}=\displaystyle \frac{c}{{n}_{{\rm{g}}}}.\end{eqnarray} \tag{ 9 }$$

A large group index thus implies a small photonic component, on order v_g/c, and correspondingly a large Rydberg component. The interaction between the Rydberg atoms shifts the Rydberg level, effectively altering the control-field detuning ${{\rm{\Delta }}}_{{\rm{c}}}$. The term Rydberg blockade refers to the case where the interaction-induced shift is much larger than the EIT linewidth. In this case, the nonlinearity can be considered as a switch from the 3-level EIT susceptibility ${\chi }_{3 \mbox{-} \mathrm{level}}$ to the 2-level susceptibility ${\chi }_{2 \mbox{-} \mathrm{level}}$ [35–37], as illustrated in figure 1(c).

The volume around a Rydberg polariton in which EIT is suppressed is known as the blockade sphere. Its radius is found from the requirement $V({r}_{{\rm{b}}})=2{\hslash }{{\rm{\Gamma }}}_{{\rm{EIT}}}$, where V(r) is the Rydberg–Rydberg interaction potential. For a van der Waals interaction $V(r)={C}_{6}/{r}^{6}$, we find

$$\begin{eqnarray}&&{r}_{{\rm{b}}}=\sqrt[6]{{C}_{6}/(2{\hslash }{{\rm{\Gamma }}}_{{\rm{EIT}}})}.\end{eqnarray} \tag{ 10 }$$

If there are enough atoms contributing to the 2-level susceptibility within the blockade sphere, the effect of a single Rydberg polariton on the transmission of nearby photons can be dramatic. This is the quantum nonlinear regime that will be the focus of the rest of this article and requires that the medium has a high optical depth per blockade sphere.

However, even in the 'classical'—or partially blockaded—regime, the nonlinearities can be enormous! For a weak probe, ${{\rm{\Omega }}}_{{\rm{p}}}\ll {{\rm{\Omega }}}_{{\rm{c}}}$, as the probe intensity increases there is a gradual transition from the 3-level to the 2-level response as each successive Rydberg excitation converts a fraction ${({{\rm{\Omega }}}_{{\rm{p}}}/{{\rm{\Omega }}}_{{\rm{c}}})}^{2}$ of nearby atoms to 2-level scatterers. It follows that the classical Rydberg nonlinearity scales as the 2-level response times the fraction of blockading excitations [35]:

$$\begin{eqnarray}&&{\chi }_{3 \mbox{-} \mathrm{level}}=N\displaystyle \frac{4\pi }{3}{r}_{{\rm{b}}}^{3}\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{p}}}^{2}}{{{\rm{\Omega }}}_{{\rm{c}}}^{2}}{\chi }_{2 \mbox{-} \mathrm{level}}.\end{eqnarray} \tag{ 11 }$$

Substituting ${{\rm{\Omega }}}_{{\rm{p}}}=d{{ \mathcal E }}_{{\rm{p}}}/{\hslash }$, where d is the dipole matrix element for the 2-level transition, this equation gives a Kerr-like optical nonlinearity. For ${r}_{{\rm{b}}}=5\;\mu {\rm{m}}$ (typical for principal quantum numbers 50–100 and ${{\rm{\Omega }}}_{{\rm{c}}}$ on order of a few MHz, satisfying ${{\rm{\Omega }}}_{{\rm{c}}}^{2}\gg 4{\gamma }_{{\rm{r}}}\gamma$) and $N=3\times {10}^{12}\;{{\rm{cm}}}^{-3}$ in Rb, one obtains an estimate of the Rydberg nonlinearity of ${\chi }^{(3)}\sim 5\times {10}^{-2}\;{{\rm{V}}}^{-2}{{\rm{m}}}^{2}$, which is 5 orders of magnitude larger than conventional EIT media [23]. This classical nonlinearity was first demonstrated experimentally in 2010 using rubidium atoms prepared at densities of $N=2\;\times {10}^{10}\;{{\rm{cm}}}^{-3}$ using a magneto-optical trap [27].

Having reviewed the classical nonlinearity arising from Rydberg EIT we now move on to the quantum limit.

It follows from the above expressions that the strength of the nonlinearity is governed by ${\mathrm{OD}}_{{\rm{b}}}$, and we require ${\mathrm{OD}}_{{\rm{b}}}\geqslant 1$ for the quantum nonlinear limit, in both the dissipative and the dispersive regimes. A medium with ${\mathrm{OD}}_{{\rm{b}}}\approx 5$ was realized in 2012 by Peyronel et al [2]. In this experiment, an elongated optical trap compressed a cloud of ultracold rubidium to an atomic density of $N=2\times {10}^{12}\;{{\rm{cm}}}^{-3}$. The Rydberg-EIT had a (half) linewidth of ${{\rm{\Gamma }}}_{{\rm{EIT}}}/(2\pi )\approx 10\;{\rm{MHz}}$ when tuned to the Rydberg level 100S${}_{1/2}$, yielding a blockade radius on order ${r}_{{\rm{b}}}\approx 10\;\mu {\rm{m}}$ and approximately 10 000 atoms within a single blockade sphere. The probe field was focused throughout the $100\;\mu {\rm{m}}$-long cloud to a diameter 2w₀ on the order of ${r}_{{\rm{b}}}$. By that, the condition ${w}_{0}\ll {r}_{{\rm{b}}}$, for keeping the dynamics one-dimensional and blocking two polaritons from propagating side-by-side, was nearly fulfilled.

In these conditions, nonlinear transmission was measured at probe powers as low as 0.25 pW, corresponding to less than two photons in the medium—see figure 2(a) and caption. For incoming photon rates above $2\;\mu {{\rm{s}}}^{-1}$, the outgoing photon rate became constant (figure 2(b)), realizing a photonic version of an hourglass which allows the transmission of only about one photon per μs. The quantum nature of the nonlinearity is evidenced by a strong photon anti-bunching of the outgoing light (figure 2(c)), measured with single-photon detectors. In the regime of the experiment, the (linear) transmission bandwidth of the medium, which sets a lower limit on the duration of probe pulses, sets a similar lower limit on the temporal extent of the anti-bunching feature (figure 2(d)).

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Changing the nonlinearity in a Rydberg-EIT experiment from dissipative to dispersive is straight forward: one simply detunes the probe and control fields from the intermediate state. Firstenberg et al realized this in 2013 with detunings ${{\rm{\Delta }}}_{{\rm{p}}}\approx -{{\rm{\Delta }}}_{{\rm{c}}}\approx 5\gamma$ [7]. The measured transmission and phase-shift spectra, corresponding respectively to the imaginary and real parts of the susceptibility, are shown in figure 3. At the chosen detuning, the 3-level and 2-level responses differ only in their phase, yielding purely dispersive nonlinearity.

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Similarly to the case of dissipative nonlinearity, the measured spectra alone do not suffice to characterize the quantum nonlinearity, and 2-photon correlation measurements are needed. Since now it is a phase, rather than transmission, that is of interest, an interferometric setup is required. To this end, one introduces a reference photonic mode, e.g., with a different polarization or frequency, for which the transmission is linear. By measuring the correlations between the probe and the reference photons when interfered in different bases, one can utilize quantum tomography techniques to reconstruct the full two-photon density matrix $\rho ({t}_{1},{t}_{2})$ (with t₁ and t₂ being the detection times of the two photons). Normalizing this density matrix by the one-photon density matrices $\rho ({t}_{1})\otimes \rho ({t}_{2})$ renders an interaction matrix $\tilde{\rho }({t}_{1},{t}_{2})$ that generalizes the standard ${g}^{(2)}({t}_{1},{t}_{2})$. The interaction matrix, $\tilde{\rho }({t}_{1},{t}_{2})$, yields not only the conditional phase ϕ, but also information on the decoherence and entanglement generation during the process. Experimental results extracted from $\tilde{\rho }({t}_{1},{t}_{2})$ in the 2013 experiment [7] are shown in figures 4(a) and (b). A conditional phase shift as high as $| \phi | =\pi /4$ was observed in this experiment at a linear transmission of 50%.

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The optical nonlinearity we observe originates from the strong dipolar interaction between Rydberg atoms. A reciprocal and complimentary notion is the effective interaction between photons that can be used to describe the nonlinearity at the quantum level. In fact, this effective photon–photon interaction inherits from the dipolar atomic interaction, but is regulated by the optical response of the medium. Dissipative and dispersive nonlinearities thus result from effective dissipative or dispersive photon–photon interactions.

To illustrate this point, we define the 2-photon wavefunction $\psi ({z}_{1},{z}_{2})$ [7], with z₁ and z₂ the photon coordinates inside the medium. $| \psi {| }^{2}$ and ${\rm{arg}}(\psi )$ are respectively the probability to find the two photons and their phase-shift relatively to the non-interacting (Poissonian) case. In the absence of nonlinearity, $\psi =1$. Moving to the center-of-mass $R=({z}_{1}+{z}_{2})/2$ and relative $r={z}_{1}-{z}_{2}$ coordinates, we can approximately relate $\psi (R,r)$ to the outgoing light by

$$\begin{eqnarray}&&\psi (R=L,r={v}_{{\rm{g}}}\tau )=\sqrt{{g}^{(2)}(\tau )}{{\rm{e}}}^{{\rm{i}}\phi (\tau )}.\end{eqnarray} \tag{ 13 }$$

It can then be shown that the evolution of a stationary Poissonian input $\psi (R=0)=1$ in a dispersive nonlinear medium, assuming ${{\rm{\Omega }}}_{{\rm{c}}}\ll {{\rm{\Delta }}}_{{\rm{p}}}$, is given approximately by a Schrödinger-like equation [7]

$$\begin{eqnarray}&&{\rm{i}}\displaystyle \frac{\partial \psi }{\partial R}=\displaystyle \frac{4{l}_{{\rm{a}}}{{\rm{\Delta }}}_{{\rm{p}}}}{\gamma }\displaystyle \frac{{\partial }^{2}\psi }{\partial {r}^{2}}+\displaystyle \frac{\gamma }{{l}_{{\rm{a}}}| {{\rm{\Delta }}}_{{\rm{p}}}| }U(r)\psi .\end{eqnarray} \tag{ 14 }$$

The center-of-mass coordinate, varying from R = 0 to R = L, plays the role of time in this Schrödinger evolution. The first term on the right-hand side accounts for an effective photon mass. It stems from the finite bandwidth of the linear EIT transmission, rendering a quadratic dispersion of the individual polaritons. The second term describes an effective potential

$$\begin{eqnarray}U(r)=\displaystyle \frac{1}{{(r/{r}_{{\rm{b}}})}^{6}+{\rm{sign}}({{\rm{\Delta }}}_{{\rm{p}}})}=\left\{\begin{array}{cc}{\rm{sign}}({{\rm{\Delta }}}_{{\rm{p}}}) & r\ll {r}_{{\rm{b}}},\\ 0 & r\gg {r}_{{\rm{b}}},\end{array}\right.\end{eqnarray} \tag{ 15 }$$

where ${\rm{sign}}({{\rm{\Delta }}}_{{\rm{p}}})={{\rm{\Delta }}}_{{\rm{p}}}/| {{\rm{\Delta }}}_{{\rm{p}}}|$ and ${r}_{{\rm{b}}}=\sqrt[6]{2{C}_{6}| {{\rm{\Delta }}}_{{\rm{p}}}| /({\hslash }{{\rm{\Omega }}}_{{\rm{c}}}^{2})}$ (in an off-resonance EIT, where ${{\rm{\Gamma }}}_{{\rm{EIT}}}={{\rm{\Omega }}}_{{\rm{c}}}^{2}/| 4{{\rm{\Delta }}}_{{\rm{p}}}|$). Equation (15) assumes a repulsive van-der-Waals interaction between the Rydberg atoms, ${C}_{6}/{r}^{6}$ with ${C}_{6}\gt 0$.

We thus find that the behavior of a photon pair in a Rydberg-EIT medium resembles that of a pair of interacting massive particles. Since ${{\rm{\Delta }}}_{{\rm{p}}}$ determines the sign of both the mass and the potential, the overall behavior is that of an attractive potential well, or an attractive force, irrespective of the sign of ${{\rm{\Delta }}}_{{\rm{p}}}$. However, the well U(r) is well-behaved only for ${{\rm{\Delta }}}_{{\rm{p}}}\gt 0;$ for ${{\rm{\Delta }}}_{{\rm{p}}}\lt 0$, the denominator in equation (15) vanishes at the boundaries ($| r| \approx {r}_{{\rm{b}}}$) due to a resonant Raman absorption, creating local resonance-like features in U(r).

For ${{\rm{\Delta }}}_{{\rm{p}}}\gt 0$, bound-state solutions of equation (14) may be termed 'molecules' of two photons. In experiments so far [7], the well was shallow and supported only a single bound state ${\psi }_{{\rm{bound}}}$. The 'finite-time' evolution (from R = 0 to R = L) following equation (14) is then governed by ${\psi }_{{\rm{bound}}}$. The initial state describing the lack of photon–photon correlations at the entrance to the medium is $\psi (R=0,r)=1$ (i.e., a uniform distribution in the relative coordinate r), which is a superposition of a bound-state component ${\psi }_{{\rm{bound}}}$ and a scattering component $1-{\psi }_{{\rm{bound}}}$. The difference in the accumulated phase between these two components leads to constructive interference between them at r = 0, and hence to photon bunching [7]. This bunching and a reminiscence of the bound state are shown in figures 4(a) and (c).

The case ${{\rm{\Delta }}}_{{\rm{p}}}\lt 0$ was theoretically analyzed by Maghrebi, Gullans, et al [39]. The resonance-like features in U(r) resemble a Coulomb potential at $| r| \lesssim {r}_{{\rm{b}}}$. They support a continuum of metastable bound states with an hydrogen-like energy spectrum, with the two polaritons separated by a finite distance $\sim {r}_{{\rm{b}}}$. In the latter sense, this type of photonic 'molecule' perhaps resembles a real molecule more than the one at ${{\rm{\Delta }}}_{{\rm{p}}}\gt 0$ (described above), which peaks at zero separation. However, the metastable states propagate in the medium with negative group velocity while decaying to a pair of Rydberg atoms at a rate $| {{\rm{\Omega }}}_{{\rm{c}}}^{2}/{{\rm{\Delta }}}_{{\rm{p}}}| /{\phi }^{2}$ (where $\phi \;=-({\mathrm{OD}}_{{\rm{b}}}/2)(\gamma /{{\rm{\Delta }}}_{{\rm{p}}})$ as defined above).

The successful realization of quantum nonlinearity with Rydberg-EIT prompted great theoretical efforts to better describe and understand the system. A full description of a uniform EIT system involves one dark and two bright polariton branches, obtained by diagonalizing the non-interacting Hamiltonian in momentum space. In this basis, the Rydberg interaction appears as a non-trivial local scatterer, coupling between the different branches. The scattering properties for the case of two photons in one dimension were derived by Bienias et al [9] using quantum scattering theory. This approach requires very little assumptions and is thus applicable for a wide range of parameters, including for large blockade radii and strong control fields ${{\rm{\Omega }}}_{{\rm{c}}}$, on the order of or even much larger than ${{\rm{\Delta }}}_{{\rm{p}}}$. In particular, Bienias et al show that increasing ${{\rm{\Omega }}}_{{\rm{c}}}$ can modify the 1D scattering length from attractive, as in equation (14), to repulsive. Such transition occurs at a scattering resonance similar to a Feshbach resonance. Furthermore, this approach provides a generalization of equation (14) to the non-stationary regime: for fields slowly varying in time, one only needs to replace $\partial /\partial R\to \partial /\partial R+\partial /({v}_{{\rm{g}}}\partial t)$, rendering a spatio-temporal Schrödinger-like dynamics. Finally, an effective many-body Hamiltonian can be formulated in terms of the 1D scattering length for low energies.

An alternative approach to scattering theory is the input–output formalism. In this formalism, the system is described in the Heisenberg picture by defining a set of operators for the optical modes, including special operators for the input and output modes. For many quantum optics systems, this approach is more natural and proves extremely useful [40, 41]. Recently, Caneva et al introduced a generalized input-output formalism for describing the Rydberg-EIT system [42]. They effectively model the system as a one-dimensional chain of interacting three-level atoms (a spin model) that is tailored to reproduce the mean-field parameters of the real continuous medium, such as OD and ${v}_{{\rm{g}}}$. Calculating high-order correlations of the outgoing photonic state is then done in a relatively straight-forward procedure by solving for higher moments of the Heisenberg operators.

An exact many-body formulation of the continuous system was recently presented by Moos et al [10]. The model includes photon loss from the bright polariton branches, arising from dark–bright coupling due to the Rydberg–Rydberg interaction. The effect of finite beam size (paraxial propagation) is also considered. Even so, an effective one-dimensional, many-body, model for only the dark polaritons is still valid under certain conditions, as verified by exact numerical simulations [10].

An exciting prospect for quantum nonlinear optical systems is their potential to realize strongly correlated states and dynamical many-body phases with photons. While experiments are being setup to pursue this regime, there have been a few theoretical predictions for a many-body behavior.

Honer et al proposed a single-photon absorber based on an effective two level system (large ${{\rm{\Delta }}}_{{\rm{p}}}$), with the Rydberg state sufficiently interacting such that a single excitation blockades the complete optical medium [43]. In this case, after the first absorption event, the blockaded medium becomes completely transparent for all subsequent signal photons. If the optical depth is sufficient to absorb the first photon with large probability, this removes with high fidelity exactly one photon from an arbitrary input state. In the opposite limit of dissipative nonlinearity (${{\rm{\Delta }}}_{{\rm{p}}}=0$), the system transmits single photons while scattering the multi-photon components. The back-action of this scattering on the properties of the transmitted photon was investigated by Gorshkov et al [44]. This work implies that when the medium is smaller than one blockade sphere, it will transform an intense coherent input into a stream of single photons with a well-defined separation.

Such a photonic state, a so-called one-dimensional crystal of photons, promises to be a valuable resource for metrology and quantum computation. In the absence of dissipation, with only dispersive (conservative) interaction between the photons, crystallization of an incoming coherent field is akin to Wigner crystallization of electrons. This many-body process, which requires dynamic control of the group-velocity, was studied by Otterbach et al using Luttinger liquid theory in the dilute, low-energy regime [8] and afterwards validated by Moos et al using numerical simulations of their many-body model [10].

The concept of a Rydberg-mediated non-classical light source is to convert a weak coherent input pulse into non-classical output via the effective interaction inside the medium. In other words, the Rydberg medium acts as a filter for the input light, changing its photon statistics based on how the Rydberg interaction in the medium is coupled to the signal photons. The Rydberg blockade mechanism enables such operation with the aid of EIT and slow-light, but also outside the EIT regime.

The first quantum light source originating from a Rydberg excitation was demonstrated by Dudin and Kuzmich [1] in 2012. A coherent signal pulse with large detuning ${{\rm{\Delta }}}_{{\rm{p}}}$ creates a collective Rydberg excitation inside a medium that is larger than a single blockade. The long-range interaction between different excitations results in strong dephasing of the initial many-atom state [1, 53]. As a consequence, when a resonant (${{\rm{\Delta }}}_{{\rm{p}}}=0$) readout field is turned on after some time, the retrieved signal photons are scattered in random directions. On the other hand, if only a single excitation is created by the input beam, no dephasing occurs and a single photon is retrieved in the phase-matched direction. The single-photon character of the signal output was characterized in the experiment by measuring the second order correlation function. The reported value of ${g}^{(2)}(0)=0.040(14)$ shows the extreme suppression of readout containing more than one photon. Furthermore, for an optical medium smaller than a single blockade volume, Dudin et al showed that this optical readout can be used to probe the dynamics of the collective Rydberg excitation [54].

This approach to filtering a coherent input field to achieve a non-classical output was further explored by Maxwell et al [3, 52]. In their experiment, the signal photons were stored as a collective excitations of a Rydberg S-state by turing off the control field during the probe pulse. The Rydberg excitation was subsequently coupled to a neighbouring P-state using a resonant microwave field before retrieving the stored light by turing the control field back on (figure 6). Due to the microwave field, the interaction between excitations was tuned from van-der-Waals to dipolar, which changes both the strength and the angular dependence [55–57]. The resulting dynamics of the interacting Rydberg excitations [58] became visible in the photon statistics of the retrieved signal light, as is shown in figure 6. From this first demonstration it becomes apparent that the extreme tunability of the Rydberg interaction opens further possibilities for tailoring the stored light and for realizing more complex non-classical light states.

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To implement a two-photon phase gate, the dispersive interaction (section 4.2) between polaritons is used. The case of two counter-propagating polaritons was first discussed by Friedler et al [25] and further explored and extended to co-propagating and stored pulses by Gorshkov et al [36]. The basic idea is the same in all configurations: the interaction between two individual polaritons results in a π-phase shift imprinted on the transmitted light, while the large detuning from the intermediate state results in (ideally) zero scattering of the photons. To turn such a conditional phase shift into a quantum gate, the mechanism needs to be state-dependent with regard to whatever basis is used to encode the quantum information in individual photons. A straightforward example is the common polarization encoding, in which case selection rules can be used to couple only one specific combination of signal and gate photon polarization to the Rydberg state [7].

A Rydberg-interaction mediated phase shift of an optical pulse was first reported by Parigi et al in 2012 employing an atomic ensemble inside an optical resonator [59]. Tiarks et al have very recently reported a conditional phase shift for single photons exceeding π, by making use of an optical medium with large optical density ${\rm{OD}}=25$ and storage and retrieval of the gate photon [49]. As explained in section 4.2, the characterization of the conditional phase-shift requires an interferometric measurement. Tiarks et al employ quantum state tomography in the polarization basis, similarly to that used by Firstenberg et al in the first demonstration of Rydberg-mediated dispersive interactions [7].

Besides achieving record single-photon phase shifts, the Rydberg-mediated approach may overcome fundamental limitations of single-photon π phase gates in conventional nonlinear media [5, 60]. The key point here is that due to the long-range interaction the nonlinearity is no longer local, and the phase-shift can be uniform over the full size of the stored spin wave [61], which should allow a high-fidelity phase gate without the unavoidable pulse distortion for conventional nonlinearities.

In a single photon switch the transmission of a target photon is controlled by a single gate photon. The first Rydberg-based switch was demonstrated by Baur et al in 2014 [46]. The scheme is similar to the phase gate described in the previous section: the gate photon is converted into a stationary Rydberg excitation in the medium, either via storing of a Rydberg polariton or via direct excitation. The target photon is subsequently sent into the medium on resonant EIT coupled to a different Rydberg state. The gate excitation changes the optical response inside its blockade volume to that of an effective two-level system, see equation (12), resulting in scattering of the target photon from ground state atoms inside this volume. Since the blockaded optical depth ${\mathrm{OD}}_{{\rm{b}}}$ can be (much) larger than one, the Rydberg-based switch can achieve a very high on/off contrast $C\gt 0.9$ for a single gate excitation (figure 7).

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This concept was then used to demonstrate for the first time single photon transistors by Gorniaczyk et al [47] and Tiarks et al [48], where a single gate photon attenuates a stronger source input beam. This transistor performance is quantified by the optical gain G, which shows how many photons are removed from the source input by a single gate photon [21]. Both experiments achieved a gain of $G\sim 15...20$.

Key to this achievement is the immense flexibility of the Rydberg interaction [55, 56, 62]. For the transistor experiments in particular, employing Rydberg states with different principal quantum numbers coupled to gate and source photons turned out to be an essential step. Practically, this avoids cross-talk between gate and source photons and excitations, but more importantly it enables tuning of the different interaction strengths involved in the scheme. For an optimal transistor, the interaction between the gate and source Rydberg state should be maximal, while the interaction among source photons is ideally small. One particular interesting feature for tuning the Rydberg interaction are Förster resonances [63–69], which occur when two dipole-coupled pairs of Rydberg states are resonant with each other. This results in the transition from the van der Waals interaction regime ${V}_{\mathrm{vdW}}\sim {C}_{6}/{r}^{6}$ to dipolar interaction ${V}_{\mathrm{dd}}\sim {C}_{3}/{r}^{3}$. Tiarks et al showed that the transistor performance improves when choosing gate and source Rydberg states which are close to such a resonance even in zero field (figure 8). Tuning the Förster defect to exact zero by applying external fields has recently enabled further improvement of the transistor, demonstrating an optical gain $G\sim 200$ [51].

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In these experiments, the transistor is operated classically, meaning that the gate photon is lost in the process. Similar to the electronic transistor, this classical device enables the amplification of signals, one example application being the high-fidelity all-optical detection of single Rydberg excitations [70, 71]. With the optical gain now demonstrated, spatially resolved single-shot imaging of Rydberg excitations embedded in a background gas is feasible. In contrast, retrieving the gate photon after the transistor operation constitutes the first step towards a quantum transistor. Such a quantum device with $G\gt 2$ would enable quantum circuits with gain and feedback or creation of entangled multi-photon states. The finite coherence time of the stored gate spin-wave reduces the possible transistor operation time, but with the recent improvements, a gain $G\gt 2$ could be demonstrated even if the gate photon is retrieved afterwards [51]. At the moment, the fidelity of this coherent transistor is limited because the scattering of source photons results in projection of the stored gate spin wave [50]. Similarly to the phase gates discussed in the last section, the long-range character of the interaction can in principle avoid this problem. For this, the blockade volume of the single gate excitation must exceed the total system size (and ${\mathrm{OD}}_{{\rm{b}}}\gg 1$), an experimentally challenging task.

A Rydberg EIT medium is a complex quantum many-body system where atoms couple collectively to a continuum of photonic modes. Light propagation, interactions and nonlinearity are all coupled. Currently, exact theoretical treatments are only able to describe interactions between two photons. Theoretical developments tend to advance hand in hand with experimental progress which provides a direct validation of approximations. With experiments starting to explore beyond pair-wise interaction, theory also must evolve from effective mean-field descriptions to true many-body treatment of the system [9, 10, 42].

Much of the theoretical challenge in quantum nonlinear optics, as opposed, for example, to cold atoms and condensed-matter systems, stems from the optical nature of the system. It is naturally a nonequilibrium system, which is constantly driven and constantly monitored. It is also naturally dissipative, with the particle number (here, the photon number) not necessarily conserved. Therefore, to describe the evolution of highly correlated states and nonequilibrium phases in a quantum nonlinear medium, modern methods of quantum field theory are required. These include scattering and input-output formalisms for quantized electromagnetic fields with multiple photons and multiple scatterers, dynamical many-body simulations, and other tools adopted from strongly correlated condensed-matter theory. In turn, Rydberg-EIT systems of growing size could provide ideal testing grounds offering flexibility encountered in few other systems.

On the experimental side, probably the most fundamental challenge encountered so far is the limitation of the atomic density in which Rydberg excitations can be embedded without affecting their coherence or even lifetime [46, 72]. Due to the large size of each Rydberg atom, at high atomic densities ($\gtrsim {10}^{13}\;{\mathrm{cc}}^{-1}$) surrounding ground state atoms are not only found inside the blockade volume, but even inside the orbit of the Rydberg electron. While this opens up access to highly interesting new physics such as low-energy electron–atom collisions and Rydberg molecule formation [73, 74], in the context of quantum nonlinear optics this imposes a major obstacle for increasing the strength of the effective photon–photon interaction. The increasing rate of Rydberg electron–atom collisions at higher atomic densities has been found to result in shorter coherence times [46] and even reduced Rydberg lifetimes [75].

The key parameter encountered in virtually any application of Rydberg nonlinearities is ${\mathrm{OD}}_{{\rm{b}}}$. Many figures of merit depend strongly on ${\mathrm{OD}}_{{\rm{b}}}$ and only weakly on the total optical depth OD, such as the blockade probability for co-propagating photons $1-{{\rm{e}}}^{-{\mathrm{OD}}_{{\rm{b}}}}/\sqrt{{\rm{OD}}}$ [2] or their conditional phase shift $\propto \sqrt{{\rm{OD}}}\cdot {\mathrm{OD}}_{{\rm{b}}}$ [7]. Other merits depend solely on ${\mathrm{OD}}_{{\rm{b}}}$, such as the attenuation of 'signal' polaritons by a stored 'gate' polariton in the photon transistor which equals $1-{{\rm{e}}}^{-{\mathrm{OD}}_{{\rm{b}}}}$ [47]. The parameters dominating the optical depth per blockade sphere ${\mathrm{OD}}_{{\rm{b}}}$ are the atomic density and the principle quantum number. Since the atomic density is fundamentally limited by the electron–atom collisions, the remaining knob for experimentalists is the principal quantum number. But this raises the same problem of electron–atom collisions, and there are practical issues such as available laser power and ability to compensate stray electric fields that have limited Rydberg-EIT experiments to principle quantum numbers $n\approx 100$. While this number will be pushed further upwards in future experiments, the practically achievable optical depth per blockade region will not grow much beyond ${\mathrm{OD}}_{{\rm{b}}}\sim 20$.

To circumvent the limitations on ${\mathrm{OD}}_{{\rm{b}}}$, various attempts are currently in progress to implement more intricate schemes: Including, for example, the idea of enhancing the optical cross section using magic monolayers [76]. Another option to further increase the photon–photon interaction is the introduction of an optical cavity around the atomic ensemble. A ground-breaking experiment by Parigi et al provided a proof of concept for such a system in 2012 [59], recently followed up by the demonstration of long-lived cavity-Rydberg polaritons by Ningyuan et al [77] and detailed study of Rydberg-induced nonlinearities inside a resonator by Boddeda et al [78]. In parallel, theoretical analyses of this system suggest promising predictions for controlling the quantum statistics of light [79–81], high-fidelity conditional-phase gates [82], and the realization of quantum crystals and fractional quantum Hall states of light [83, 84]. The latter make use of the rich 2D transverse-mode spectra of multimode cavities. While the introduction of an optical resonator makes experimental setups somewhat more complicated again, the new physics offered by these combined systems easily justifies this addition. It seems more than likely that fundamental new steps will emerge from these systems.

Another exciting research direction is to transfer the concepts discussed in this review and explored in ultra-cold atomic samples to room-temperature vapour cells. This could result in a great simplification of the experimental setup and pave the way for employing Rydberg-based nonlinearities in future devices. Rydberg interaction effects have already been observed in room-temperature experiments [85–87], suggesting that the concepts of quantum nonlinear optics with Rydberg atoms can be transferred from the ultra-cold. So far, experimental demonstration of light manipulation on the quantum level is still outstanding, but it seems safe to expect further progress in the future.

Based on these open challenges and future steps, the field of Rydberg-based quantum nonlinear optics will keep both theorists and experimentalists busy for quite some time. Certainly we can expect both fundamental breakthroughs, as well as paradigm shifting applications in photonic quantum information processing in the upcoming years.