Quantum resonant optical bistability with a narrow atomic transition: bistability phase diagram in the bad cavity regime

The experimental setup has been presented in detail in [40]. In short, a cloud of about $N = 200\,000$ ⁸⁸Sr atoms is cooled to temperatures around $1\operatorname{\mu K}$ and transferred into the mode volume of a ring cavity operated near the narrow intercombination line with the transition wavelength $\lambda_\mathrm{a} = 689\operatorname{\,nm}$ connecting the energy levels $(5s)^2~^1S_0$ and $(5s5p)~^3P_1$. The atomic cloud is confined to the mode volume by the optical dipole potential formed by laser beams coupled into a TEM₀₀ mode of the ring cavity (laser 2 in figure 2) under p-polarization, where the finesse of the cavity is low (F ≈ 100). The length of the ring cavity is stabilized to this laser 2 via a Pound–Drever–Hall servo electronics. Laser 2, which is tuned $\Delta_\text{dip}/2\pi\approx -530\Gamma$ to the red of the strontium resonance is, in turn, locked via a digital phase-locked loop (DPLL) to a second laser (laser 1 in figure 2), which itself is stabilized to a stable supercavity and operated near the strontium resonance.

Zoom In Zoom Out Reset image size

Laser 1 serves for the last cooling stage of the atomic cloud, and also as a probe for the atom-cavity interaction. In this latter role, the light of laser 1 is passed through AOMs, driven by a tunable radio-frequency $\nu_\text{rf}$ and permitting to ramp the laser frequency ω with respect to the atomic resonance $\omega_\mathrm{a}$, thus generating a controlled detuning $\Delta_\mathrm{a} = \omega-\omega_\mathrm{a}$, and with respect to the ring cavity resonance $\omega_\mathrm{c}$, generating a detuning $\Delta_\mathrm{c} = \omega-\omega_\mathrm{c}$. On the other hand, we can control the detuning between the cavity and the atomic resonance, $\Delta_\mathrm{ca} = \omega_\mathrm{c}-\omega_\mathrm{a} = \Delta_\mathrm{a}-\Delta_\mathrm{c}$, via the microwave-frequency $\nu_\mathrm{mw}$ of the DPLL [40].

Injected into one of the two counter-propagating modes of the ring cavity, laser 2 generates an optical dipole trap, in which the atomic cloud adopts a smooth density distribution elongated along the cavity's optical axis. Part of the light power of laser 1 is coupled as probe light into a TEM₀₀ mode of the ring cavity using s-polarization, where the finesse of the cavity is high (F ≈ 1200), counter-propagating with laser 2. The probe light transmitted through the pumped mode through a high reflecting mirror (transmission $T_\mathrm{hr}\approx 0.001$) is filtered by a polarizing beamsplitter, coupled into optical fibers and monitored. We infer the intracavity power P from the transmitted probe light power $P_\mathrm{hr} = T_\mathrm{hr}P = T_\mathrm{hr}\hbar\omega\delta_\mathrm{fsr}n$, where $\delta_\mathrm{fsr} = 8.23\operatorname{GHz}$ is the free spectral range of the ring cavity and n the intracavity photon number. Subscripts ± refer to counter-propagating modes of the ring cavity. In the following we assume that only the mode denoted by $\alpha_+$, where $|\alpha_+|^2 = n$, is pumped and that the mode $\alpha_-$ is filled only via photonic backscattering by the atomic cloud.

The cavity pump rate $\eta_+$ is an experimental parameter which can be determined from the photon number in the empty cavity under resonant driving, $\eta_+/\kappa = \alpha_\eta\equiv\alpha_+(N = 0,\Delta_\mathrm{c} = 0)$. It allows to express the ring cavity transmission as, $T = |\alpha_+/\alpha_\eta|^2$. With the saturation parameter given by $s\equiv 2\Omega^2/\Gamma^2$ defined via the Rabi frequency $\Omega\equiv 2g|\alpha_+|$ we find that the transition is saturated by a mean photon number of $|\alpha_\mathrm{sat}|^2\approx 0.087$ corresponding to a power of $P_\mathrm{sat} = \hbar\omega\delta_\mathrm{fsr}|\alpha_\mathrm{sat}|^2 \approx 0.2\operatorname{\,nW}$. In practice, we typically use saturation parameters in the range $s = P/P_\text{sat}\approx 100\ldots 1000$. The single-photon saturation s₁ and the single-atom cooperativity ϒ are important parameters characterizing our coupled atom-cavity system,

$$\begin{equation}s_1 \equiv \frac{8g^2}{\Gamma^2} \approx 11.5 ~~~~~,~~~~~ \Upsilon \equiv \frac{4g^2}{\kappa\Gamma} \approx 0.013~. \end{equation} \tag{ 1 }$$

We record normal mode spectra by scanning $\Delta_\mathrm{a}$ for fixed $\Delta_\mathrm{ca}$. The typical duration of a scan is $\Delta t_\mathrm{scan} = 0.25\operatorname{\,ms}$. Repeating the scan for various $\Delta_\mathrm{ca}$ we obtain the two-dimensional spectra shown in figures 3(a)–(c) with false-color coding. Yellow indicates maximum cavity transmission and blue an impermeable cavity reflecting the pump light completely.

Zoom In Zoom Out Reset image size

The figures 3(a)–(c) show transmission spectra measured for different pump rates $\eta_+$. The spectra show a number of interesting features: (i) The normal modes are clearly visible as two resonances describing an avoided crossing. These resonances are emphasized by dashed black lines in the figures. (ii) Near the atomic resonance, around $\Delta_\mathrm{a}\simeq 0$, the spectra develop a narrow ridge of high transmission bridging the normal modes. Normal mode spectra recorded by scanning $\Delta_\mathrm{a}$ near the atom-cavity resonance, i.e. around $\Delta_\mathrm{ca}\simeq 0$, therefore exhibit three peaks instead of two. (iii) The ridge is flanked by discontinuous steep precipices into dark regions appearing as sharp horizontal features in the false color plots. The additional central peak has sharp edges, pointing to bistable behavior of the cavity transmission.

Comparing different spectra taken for different pump rates $\eta_+$ we notice additional features: (iv) The (vertical) width of the bridge between the normal modes in figures 3(a)–(c), that is, the width of the central peak in $\Delta_\mathrm{a}$-scans, increases with stronger pump rates $\eta_+$. (v) As the pump rate is increased, the discontinuous ledges are 'pushed' away from resonance towards higher $|\Delta_\mathrm{ca}|$. At sufficiently strong pumping the avoided crossing of the normal modes turns into a dispersive lineshape. $\Delta_\mathrm{a}$-scans near atom-cavity resonance still show three peaks, but do not any longer exhibit discontinuities arising from bistable behavior.

Figure 4 shows a more detailed study of the dependence of the normal mode spectra on the pump rate $\eta_+$. In figure 4(a) the pump laser was scanned from $\Delta_a = -2.2\kappa$ to $+2.2\kappa$, immediately followed by a scan in the inverse direction exhibited in (b). The features to be extracted from these measurements are: (vi) There is a central peak around $\Delta_\mathrm{a} = 0$ bordered by sharp edges. The central peak corresponds to the 'bridge' of figures 3(a)–(c). (vii) We note a hysteretic behavior of the sharp edges, as the position of the central peak in figure 4(b) is slightly shifted as compared to (a). The latter feature is seen more clearly in a single scan taken at a pump rate of $\eta_+ = 120$ [white dashed lines in figures 4(a) and (b)] and plotted in figure 4(d). (viii) The central peak increases its width at high pump rates. (ix) The distance between the normal mode peaks diminishes when increasing the pump rate very far beyond saturation. Eventually all peaks merge into a single one.

Zoom In Zoom Out Reset image size

We identify in all those features a bistability mechanism of a different kind than previously reported in the literature for atom-cavity systems. In the next sections we develop a semiclassical theory that fits well our data and grasps this new mechanism [41].

The condition of zero phase shift for the intracavity light field $\alpha_+$ with respect to the pump field $\eta_+$ can be found by setting the imaginary part of the right-hand side of equation (3) to zero. It yields,

$$\begin{equation}\Delta_\mathrm{ca} = \Delta_\mathrm{a}-\frac{Ng^2\Delta_\mathrm{a}}{\Delta_\mathrm{a}^2+\Gamma^2/4+\Omega_\eta^2/2}~. \end{equation} \tag{ 4 }$$

In this expression $\Omega_\eta = 2g\eta_+/\kappa$ is the maximum Rabi frequency, i.e. the Rabi frequency if all impinging photons would enter the cavity. The corresponding curve is shown in figure 3 as dash-dotted lines. We also show in dashed lines the corresponding zero-phase shift curve for vanishing pump field, $\Omega_\eta\rightarrow 0$. We note that it follows closely the maxima of the experimental transmission curves for all range of experimental parameters. Although this condition is generic for most experiments dealing with cavities filled with atoms and operated in the strong collective coupling regime, the central peak is commonly suppressed due to strong resonant absorption of the light by the atoms. In our case, the width of the atomic transition is very small, and we can expect that very little light intensity will already saturate the transition, thus changing this scenario. The rate of photon injection into the cavity, $\eta_+ = \kappa n_\eta$, that saturates all atoms roughly equalizes the rate at which they emit photons once they are saturated, $N\Gamma/2$. This means that for $n_\eta\gtrsim N\Gamma/2\kappa$, the atoms become transparent, and a transmission window opens around the empty cavity resonance. This quantum nonlinear feature of the cavity transmission entails the appearance of a bistable spectral region near resonance. The calculations in the appendix [see (46)] indeed show that for $\Delta_\mathrm{a} = \Delta_\mathrm{c} = 0$ a lower bound on the pump rate for the appearance of two stable solutions (the signature of a bistable behavior) is,

$$\begin{equation}n_\eta \gtrsim \frac{N\Gamma\left(1+\sqrt{3}/2\right)}{2\kappa}~, \end{equation} \tag{ 5 }$$

and the width of the bistable region for $\Delta_\mathrm{c} = 0$ is given by equation (52) as,

$$\begin{equation}|\Delta_\mathrm{a}| \simeq \frac{g}{\sqrt{2}}\sqrt{n_\eta-\frac{N\Gamma}{2\kappa}} = \frac{1}{2} \sqrt{\frac{\Omega_\eta^2}{2}-\frac{\Gamma^2\Upsilon_N}{4}}~. \end{equation} \tag{ 6 }$$

We will call this bistability mechanism on-resonance bistability. It is present for example in the NMBC phase shown in figure 1(g).

Gripp et al [26] find a qualitatively different modification of the normal mode splitting when the intensity is increased, reproduced here in figures 1(a)–(e). For their parameters, the bistability emerges off-resonance, at the inner edge of each of the normal mode splitting peaks. It appears for example in the NMBW phase shown in figure 1(c). The underlying mechanism follows a prototypical intensity-driven bistability also found in anharmonic classical mechanical oscillators [45] subject to an intensity-dependent frequency shift of the normal mode splitting resonances. Upon further increase of the pump intensity, this bistability mechanism eventually leads to a merger of the two normal mode peaks into a single transmission peak. For $\kappa,\Gamma\ll g\sqrt{N}$ (i.e. for a normal mode splitting much larger than the width of the normal mode resonances), this happens when the two local extremes of the zero-phase condition curve $\Delta_\mathrm{ca}$ as a function of $\Delta_\mathrm{a}$ as given by equation (4) merge into a single point ($\left.d\Delta_\mathrm{ca}/d\Delta_\mathrm{a}\right|_{\Delta_\mathrm{a} = 0} = 0$). This is the case when

$$\begin{equation}Ng^2 = \Gamma^2/4+\Omega_\eta^2/2~. \end{equation} \tag{ 7 }$$

We may reasonably expect that, in order to have on-resonance bistability, the critical intensity for its appearance, as described by equation (5), must be smaller than the critical intensity for the merging of the off-resonance bistable spectral intervals. For $Ng^2\gg\Gamma^2/4$, this amounts to $\Gamma(1+\sqrt{3}/2)\lesssim\kappa$, which is qualitatively the condition for the bad-cavity regime. Thus, it is the narrowness of the strontium transition that allows us to observe the new bistable mechanism in our experiment. This new mechanism for bistability relies not only on the nonlinearity of the atom-cavity interaction, but also on the complete saturation of the atomic population to cause full transparency of the cavity to light. Thus, it cannot be generated by a simple nonlinear susceptibility.

The calculated spectra in figures 3 and 4 are in good qualitative agreement with the measured ones. For instance, comparing the spectra of figures 4(a) and (b) we notice that the bistability around the central peak is symmetrically inverted when the scan direction is also inverted, and that the maximum of the transmission is roughly independent of the pump rate, showing no anharmonicity in the resonance frequency. Those features are also evident in the theoretical spectra of figures 1(f)–(j).

On the other hand, while the observed spectral width of the normal mode resonances agrees well with the simulations and does not depend on the pump rate, the spectral width of the bistable region is systematically larger than predicted by simulations. As illustrated in figure 4(c), the width of the bistable region should be dominated by power broadening. But from earlier work [26, 28, 29, 32] we know that the onset of bistability is accompanied by oscillations and hysteresis, that can lead to deformations and broadening of the spectra not grasped by our stationary model.

As mentioned in section 1.2, we indeed observe hysteresis behavior. The oscillations are manifestations of a ringing induced by a time lag between the evolution imposed by the scan and the evolution of the degrees of freedom characterizing the system, which becomes relevant when the slowest time constant of our system, which is the atomic decay rate $2\pi/\Gamma = 0.13\operatorname{\,ms}$, is comparable to or slower than the scan duration $\Delta t_\mathrm{scan} = 0.25\operatorname{\,ms}$. We indeed observe deformation of the spectra when shorter scan times $\Delta t_\mathrm{scan}$ are chosen. However, our photodetector, being optimized for high sensitivity to weak light intensities, is too slow to resolve the fast oscillations.

An example for a spectral deformation can be observed in figure 4(d), where the normal mode splitting is found to be larger on the first scan from negative to positive detunings (blue curve) than on the subsequent scan in backward direction (red curve). The deformations of the spectra become stronger when longer scan times are used. A possible reason can be heating of the atomic cloud during the scan causing a reduction of the effective number of atoms interacting with the cavity.

We also note that, in sufficiently strong magnetic fields, a phenomenon called magnetically induced transparency (MIT) [46] induced by the Zeeman substructure of the excited $^3P_1$ state can add a central feature to the normal mode spectra even at very low saturation. Numerical simulations performed with linearized equations of motion, however, show a qualitatively different behavior as the one observed in figure 3. Furthermore, we tested applying different magnetic fields without noticing a dependency.

In order to derive the complete phase diagram we have solved the steady-state equations for the transmission spectrum for various values of the atomic transverse decay rate $\Gamma/2$ and the pump rate $\eta_+$, both scaled to the cavity linewidth κ. Some examples have already been plotted in figure 1. Details on how we solve the equations are given in the appendix A.2.

We classify the bistable solutions by the overall shape of the spectrum, i.e. resembling either a normal mode or an empty cavity spectrum, and by the frequencies at which the bistability occurs, i.e. either in the wings or in the center of the spectrum. In figures 6(a) and (b) we compiled phase diagrams for two different situations: figure 6(a) is done for $g_N = 12.4\kappa$, reproducing the situation of Gripp et al [26], while figure 6(b) is obtained for the collective coupling constant g_N of our experimental system.

Zoom In Zoom Out Reset image size

The parameter region for the occurrence of on-resonance bistability, identified by the existence of three positive real solutions for $\Delta_a = \Delta_c = 0$, is limited to large values of n_η by the condition (5). In contrast, when the collective cooperativity is larger than the saturation parameter, only a single real solution of the intracavity light power exists in resonance. The condition (5) for the onset of on-resonance bistability is represented as dotted black lines in figure 7.

Zoom In Zoom Out Reset image size

For too strong pumping, as seen in figures 5, the bistability disappears for all values of Γ, and we get spectra resembling that of an empty cavity (ECU for empty cavity-like bistable center phase). The maximum pump rate for which bistability occurs is derived from (50) in the appendix A.5) as,

$$\begin{equation}n_\eta \lesssim \frac{N^2g^2}{8\kappa^2}~. \end{equation} \tag{ 8 }$$

which is plotted as continuous black horizontal lines. This upper boundary of the bistable region is met when two out of three solutions become degenerate. The analytically derived phase boundaries agree very well with the simulated phase diagrams.

At pump rates below the limit imposed by (8), the spectra present either on-resonance bistability (NMBC for normal mode bistable center phase), off-resonance bistability (NMBW for normal mode bistable wing phase), both separately (NMBWC for normal mode bistable wing and center phase) or a merging of both bistable spectral regions (ECBC for empty cavity-like bistable center phase), as we will show below. For the onset of off-resonance bistability close to the normal mode splitting resonances Gripp and Orozco [27] derived the condition,

$$\begin{equation}n_\eta > \frac{4\sqrt{N}}{3\sqrt{3}} \frac{\left(\kappa+\Gamma/2\right)^3}{g\kappa^2}~, \end{equation} \tag{ 9 }$$

shown as dash-dotted black lines. We note that for a collective coupling strength comparable to the cavity decay rate, as in our experimental situation and illustrated in figure 6(b), this limit is higher than the critical pump rate according to the condition (8), and thus no off-resonance bistability can occur, which leaves the NMBC bistable phase as the only option. For larger collective coupling strength as shown in figure 6(a), the orange area delimits a NMBW phase where only off-resonance bistability exists (as shown in figure 1(c)), and the green-shaded area delimits a region where both bistabilites coexist in a NMBWC phase; an example of such spectra is also shown in figure 1(h). In the region below both conditions (5) and (9), only a unistable solution exists, which characterizes a normal mode unistable (NMU) phase.

When further increasing the pump rate, the bistable regions of the spectrum merge into a ECBC phase. The lower bound of this region is described by the empirically guessed expression,

$$\begin{equation}n_\eta \gtrsim \frac{N\left(\kappa+2\Gamma\right)}{2\kappa}~, \end{equation} \tag{ 10 }$$

identified as a dashed black line.

The atomic saturation is an important observable to characterize the different phases of the atom-cavity system. Although more difficult to measure experimentally than the cavity transmission, it gives further insight on the difference between the on-resonance and off-resonance bistability. Figures 7(a) and (b) show the maximum atomic excited state population achieved when scanning the laser frequency, for the same parameters as for figures 6(a) and (b). We learn from the graphs that, while the presence of off-resonance bistability does not require full saturation of the atomic population [see for example the region in figure 7(a) that corresponds to the orange region in figure 6(a)], it is on the contrary required for the emergence of on-resonance bistability [yellow areas in figures 7(a) and (b) and corresponding areas in figures 6(a) and (b)]. We also note that, while the maximum atomic population is achieved off-resonance for the points below the condition represented by equation (5), it is achieved on-resonance for the points above it, where the full saturation is found. This further confirms that the underlying mechanism for this bistability is the appearance of a transparency window caused by full saturation of the atoms. Moreover, the atomic population presents a discontinuity when going from lower to higher pump rates, as the threshold for the on-resonance bistability is crossed. This is also evident from the two insets of figures 7(a) and (b), calculated for the parameters of [26] and for our experiment, respectively. Since it relies on the full saturation of a quantum two-level transition, the on-resonance bistability cannot stem only from a nonlinear susceptibility and is fully quantum in its origin, at odds with the off-resonance bistability.

The reason why bistability could be observed in the resonantly coupled regime ($|\Delta_\mathrm{c}|\ll\kappa$) in our experiment is that we are deep in the bad-cavity limit. As seen from figure 5, bistability around $\Delta_\mathrm{c} = 0$ appears over wide ranges of pump rates $\eta_+$ only if $\Gamma\ll\kappa$.

Previous experiments [25, 30–32] carried out with alkali atoms were all performed in the good-cavity limit, due to the large spontaneous decay width Γ of the atomic transition. Another experiment carried out by Gothe et al [28] employed ytterbium atoms coupled to a cavity via a narrow transition. However, the finesse of the cavity was so high that the atomic decay rate was still twice the cavity decay rate. Consequently, on resonance, spontaneous emission remained the main decay channel. In order to avoid spontaneous emission leading to heating and trap loss they coupled the atoms dispersively to their cavity, so that the bistable features appeared far from the atomic resonance.

Using the Hamiltonian (11) and for the decay processes $\hat L$ the Lindblad superoperator expression $\mathcal{L}_{\gamma,\hat L}^\dagger\hat A = \gamma(2\hat L^\dagger\hat A\hat L-\hat L^\dagger\hat L\hat A-\hat A\hat L^\dagger\hat L)$, we derive the equations of motion for the individual atomic operators,

$$\begin{align}\dot{\hat\sigma}_i^- & = -\imath\left[\hat\sigma_i^-,\hat H\right]+\mathcal{L}_{\Gamma/2,\hat\sigma_i^-}^\dagger\hat\sigma_i^-\nonumber\\ & = \left(\imath\Delta_\mathrm{a}-\tfrac{\Gamma}{2}\right)\hat\sigma_i^-+\imath g\left(e^{\imath kz_i}\hat a_+ +e^{-\imath kz_i}\hat a_-\right)\hat\sigma_i^z~, \end{align} \tag{ 14 }$$

and

$$\begin{align}\dot{\hat\sigma}_i^z & = -\imath\left[\hat\sigma_i^z,\hat H\right] +\mathcal{L}_{\Gamma/2,\hat\sigma_i^-}^\dagger\hat\sigma_i^z\nonumber\\ & = -2\imath g\left(e^{\imath kz_i}\hat a_++e^{-\imath kz_i}\hat a_-\right)\hat\sigma_i^+\nonumber\\ & \quad +2\imath g\left(e^{-\imath kz_i}\hat a_+^\dagger+e^{\imath kz_i}\hat a_-^\dagger\right)\hat\sigma_i^- -\Gamma\left(\mathbb{I}_2+\hat\sigma_i^z\right)\nonumber \end{align} \tag{ 15 }$$

and for the field operators,

$$\begin{align}\dot{\hat a}_\pm & = -\imath\left[\hat a_\pm,\hat H\right]+\mathcal{L}_{\kappa,\hat a}^\dagger\hat a_\pm\nonumber\\ & = \left(\imath\Delta_\mathrm{c}-\kappa\right)\hat a_\pm -\imath g\sum_j\hat\sigma_j^-e^{\mp\imath kz_j}+\eta_\pm~.\nonumber \end{align} \tag{ 16 }$$

The stationary solution follows from the expectation values of the equations (14)–(16),

$$\begin{align}\text{(i)} \qquad 0 & = \left(\imath\Delta_\mathrm{a}-\tfrac{\Gamma}{2}\right)\left\langle\hat\sigma_i^-\right\rangle+\imath g\left(e^{\imath kz_i} \left\langle\hat a_+\hat\sigma_i^z\right\rangle+e^{-\imath kz_i}\left\langle\hat a_-\hat\sigma_i^z\right\rangle\right)\nonumber\\ \text{(ii)} \qquad 0 & = -2\imath g\left(e^{\imath kz_i}\left\langle\hat a_+\hat\sigma_i^+\right\rangle +e^{-\imath kz_i}\left\langle\hat a_-\hat\sigma_i^+\right\rangle\right)+2\imath g\left(e^{-\imath kz_i}\left\langle\hat a_+^\dagger\hat\sigma_i^-\right\rangle +e^{\imath kz_i}\left\langle\hat a_-^\dagger\hat\sigma_i^-\right\rangle\right)-\Gamma\left(\mathbb{I}_2+\left\langle\hat\sigma_i^z\right\rangle\right)\nonumber\\ \text{(iii)} \qquad 0 & = \left(\imath\Delta_\mathrm{c}-\kappa\right)\left\langle\hat a_\pm\right\rangle-\imath g\sum_je^{\mp\imath kz_j}\left\langle\hat\sigma_j^-\right\rangle +\eta_\pm \end{align} \tag{ 17 }$$

Neglecting all correlations by setting, e.g. $\langle\hat a_\pm\hat\sigma_i^z\rangle = \langle\hat a_\pm\rangle\langle\hat \sigma_i^z\rangle$, and analogously for all operator products, we loose phenomena rooted in entanglement, such as spin-squeezing. Using the abbreviations,

$$\begin{equation}U_\gamma \equiv U_0-\imath\gamma_0 \equiv \frac{g^2\left(\Delta_\mathrm{a}-\imath\tfrac{\Gamma}{2}\right)}{\Delta_\mathrm{a}^2+\tfrac{\Gamma^2}{4}} ~~~~~\text{and}~~~~~ \Delta_\kappa \equiv \Delta_\mathrm{c}+\imath\kappa~, \end{equation} \tag{ 18 }$$

where U₀ has the meaning of a single-photon light shift and γ₀ of single-photon scattering rate, equation (17)(i) resolved by $\langle\hat\sigma_i^-\rangle$ becomes,

$$\begin{equation}\left\langle\hat\sigma_i^-\right\rangle = -\tfrac{U_\gamma}{g}\left(e^{\imath kz_i}\alpha _++e^{-\imath kz_i}\alpha_-\right)\left\langle\hat\sigma_i^z\right\rangle~. \end{equation} \tag{ 19 }$$

Substituting $\langle\hat\sigma_i^\pm\rangle$ in (17)(ii),

$$\begin{equation}\left(1+2\left|\tfrac{U_\gamma}{g}\right|^2\left|e^{\imath kz_i}\alpha_++e^{-\imath kz_i}\alpha_-\right|^2\right)\left\langle\hat\sigma_i^z\right\rangle = -1~, \end{equation} \tag{ 20 }$$

and in (iii),

$$\begin{equation}\Delta_\kappa\alpha_\pm+\sum_jU_\gamma\left(e^{\imath kz_j\mp\imath kz_j}\alpha_++e^{-\imath kz_j\mp\imath kz_j}\alpha_-\right)\left\langle\hat\sigma_j^z\right\rangle = \imath\eta_\pm~. \end{equation} \tag{ 21 }$$

Eliminating $\langle\hat\sigma_j^z\rangle$ from the expressions (20) and (21),

$$\begin{equation}\boxed{\sum_j\frac{-U_\gamma\left(\alpha_\pm+e^{\mp 2\imath kz_j}\alpha_\mp\right)} {1+2|U_\gamma/g|^2|e^{\imath kz_j}\alpha_++e^{-\imath kz_j} \alpha _-|^2} = \imath\eta_\pm-\Delta_\kappa\alpha_\pm}~, \end{equation} \tag{ 22 }$$

which is the expression (2) given in the main text.

It is obvious that the steady-state expression (22) will generate bistable behavior. With the aim of deriving an analytic solution accounting for our experimental situation we assume that the atomic cloud is homogeneously distributed over the cavity's mode volume. The other extreme case that the atoms are perfectly bunched, $b\equiv\tfrac{1}{N}\sum_je^{2\imath kz_j} = 1$, which is the case when the atoms are arranged in an optical lattice, can be treated analogously and produces similar results except from the fact that the mode $\alpha_-$ counter-propagating to the pumped mode $\alpha_+$ may receive a considerable number of photons due to Bragg-enhanced backscattering.

In the case of a homogeneous cloud (no bunching b = 0) and with one-sided probing, $\eta_- = 0$, we may neglect backscattered light altogether, $\alpha_- = 0$. Re-scaling via $\bar\Delta_\mathrm{c}\equiv\Delta_\mathrm{c}/\kappa$ and $\bar\Delta_\mathrm{a}\equiv 2\Delta_\mathrm{a}/\Gamma$ and using the definitions of the single-atom cooperativity and the single-photon saturation parameter,

$$\begin{equation}\Upsilon \equiv \frac{\Omega_1^2}{\Gamma\kappa} = \frac{4g^2}{\Gamma\kappa} ~~~~~\text{and}~~~~~ s_1 \equiv \frac{2\Omega_1^2}{\Gamma^2} = \frac{8g^2}{\Gamma^2}~, \end{equation} \tag{ 23 }$$

the expression (22) simplifies to,

$$\begin{equation}\alpha_+ = \frac{\eta_+}{\kappa}\frac{1}{1-\imath\bar\Delta_\mathrm{c}+\frac{N\Upsilon \left(1+\imath\bar\Delta_\mathrm{a}\right)}{2\left(1+s_1n+\bar\Delta_\mathrm{a}^2\right)}}~, \end{equation} \tag{ 24 }$$

so that for the intracavity mean photon number we get,

$$\begin{equation}\boxed{n = \frac{\eta_+^2/\kappa^2}{\left(1+\frac{N\Upsilon/2} {1+s_1n+\bar\Delta_\mathrm{a}^2}\right)^2+\left(\bar\Delta_\mathrm{c}-\frac{\bar\Delta_\mathrm{a}N\Upsilon/2} {1+s_1n+\bar\Delta_\mathrm{a}^2}\right)^2}}~. \end{equation} \tag{ 25 }$$

The 'good-cavity' regime, $s_1\lt\Upsilon$, and the 'bad-cavity' regime, $s_1\gt\Upsilon$, are delimited by the ratio between the single-photon saturation and the single-atom cooperativity.

The expression (25) can be written in terms of a third order polynomial,

$$\begin{equation}0 = An^3+Bn^2+Cn+D~, \end{equation} \tag{ 26 }$$

with the coefficients,

$$\begin{equation}\boxed{\begin{array}{rcl} A & = & s_1^2\left(1+\bar\Delta_\mathrm{c}^2\right) \\ B & = & 2s_1\left(\bar\Delta_\mathrm{a}^2+\tfrac{\Upsilon_N}{2}\right)+2s_1\bar\Delta_\mathrm{c} \left[\bar\Delta_\mathrm{c}\left(1+\bar\Delta_\mathrm{a}^2\right)-\tfrac{N\Upsilon}{2}\bar\Delta_\mathrm{a}\right] -s_1s_\eta \\ C & = & \left(\bar\Delta_\mathrm{a}^2+\tfrac{\Upsilon_N}{2}\right)^2 +\left[\bar\Delta_\mathrm{c}\left(1+\bar\Delta_\mathrm{a}^2\right)-\tfrac{N\Upsilon}{2}\bar\Delta_\mathrm{a}\right]^2 -2s_\eta\left(1+\bar\Delta_\mathrm{a}^2\right) \\ D & = & -n_\eta\left(1+\bar\Delta_\mathrm{a}^2\right)^2 \end{array}}~, \end{equation} \tag{ 27 }$$

where we defined the collective cooperativity, the maximum photon number, and the maximum saturation parameter,

$$\begin{equation}\Upsilon_N \equiv N\Upsilon+2 ~~~~~\text{and}~~~~~ n_\eta \equiv \frac{\eta_+^2}{\kappa^2} ~~~~~\text{and}~~~~~ s_\eta \equiv s_1n_\eta~. \end{equation} \tag{ 28 }$$

The roots of the cubic equation are given by,

$$\begin{equation}R \equiv \root{3}\of{36CBA-108DA^2-8B^3 +12\sqrt{3}A\sqrt{4C^3A-C^2B^2-18CBAD+27D^2A^2+4DB^3}} \end{equation} \tag{ 29 }$$

and

$$\begin{equation}X_\pm \equiv \frac{R}{6A}\pm\frac{6AC-2B^2}{3AR} \end{equation} \tag{ 30 }$$

so that,

$$\begin{equation}n_1 = X_-\frac{B}{3A} ~~~,~~~ n_{2,3} = -\frac{1}{2}X_-\frac{B}{3A}\pm\frac{\imath\sqrt{3}}{2}X_+~, \end{equation} \tag{ 31 }$$

from which we obtain the roots $n = |\alpha_+|^2$ and the field amplitude from (24).

The atomic state can be obtained from equations (19) and (20),

$$\begin{align} \left\langle\hat\sigma_i^z\right\rangle & = - \frac{1}{1+2|U_\gamma/g|^2|\alpha_++\alpha_-|^2}\nonumber\\ \left\langle\hat\sigma_i^-\right\rangle & = \frac{U_\gamma}{g}\left(\alpha_++\alpha_-\right)\left\langle\hat\sigma_i^z\right\rangle~.\nonumber \end{align} \tag{ 32 }$$

All numerical curves shown in this work are obtained by evaluating the solutions (31) discarding all solutions with photon numbers $n\notin\mathbb{R}^+$. Figures 5 and 8 exhibit spectral ranges characterized by the presence of several solutions, while the theoretical spectra of figures 3(d)–(f) and 4(c) only show the solution corresponding to the highest photon number.

Zoom In Zoom Out Reset image size

Let us now discuss the resonant case, $\Delta_\mathrm{c} = 0 = \Delta_\mathrm{a}$, for which equation (24) becomes,

$$\begin{equation}\alpha_+ \simeq \frac{\eta_+}{\kappa}\frac{ns_1+1} {\frac{1}{2}N\Upsilon+ns_1+1}~. \end{equation} \tag{ 33 }$$

From this formula we can easily evaluate the relative importance of collective cooperativity and saturation. Since $N\Upsilon\gg 1$, we get below saturation,

$$\begin{equation}\alpha_+ \overset{n\rightarrow 0}{\longrightarrow} \frac{\eta_+}{\kappa}\frac{2}{N\Upsilon}~, \end{equation} \tag{ 34 }$$

that is, the excitation of the cavity field is heavily suppressed by the normal mode splitting. On the other hand, at high saturation,

$$\begin{equation}\alpha_+ \overset{n\rightarrow\infty}{\longrightarrow} \frac{\eta_+}{\kappa}~, \end{equation} \tag{ 35 }$$

the cavity becomes fully transparent.

For intermediate saturation we evaluate the coefficients (27) of the cubic equation (decorated by an apostrophe),

$$\begin{align}A^{^{\prime}} & = s_1^2\nonumber\\ B^{^{\prime}} & = s_1\left(\Upsilon_N-s_\eta\right)\nonumber\\ C^{^{\prime}} & = \tfrac{1}{4}\Upsilon_N^2-2s_\eta\nonumber\\ D^{^{\prime}} & = -n_\eta~.\nonumber \end{align} \tag{ 36 }$$

The cubic equation (26) with the resonant coefficients (36) has three degenerate real solutions if for some coefficient $n_\mathrm{t}$,

$$\begin{equation}n^3+\tfrac{B^{^{\prime}}}{A^{^{\prime}}}n^2+\tfrac{C^{^{\prime}}}{A^{^{\prime}}}n+\tfrac{D^{^{\prime}}}{A^{^{\prime}}} = \left(n-n_\mathrm{t}\right)^3~. \end{equation} \tag{ 37 }$$

In other words,

$$\begin{equation}\tfrac{B^{^{\prime}}}{A^{^{\prime}}} = -3n_\mathrm{t} ~~~,~~~ \tfrac{C^{^{\prime}}}{A^{^{\prime}}} = 3n_\mathrm{t}^2 ~~~,~~~ \tfrac{D^{^{\prime}}}{A^{^{\prime}}} = -n_\mathrm{t}^3~. \end{equation} \tag{ 38 }$$

From (38) we derive,

$$\begin{align}3s_1n_\mathrm{t} & = s_\eta-\Upsilon_N\nonumber\\ \tfrac{3}{4}\Upsilon_N^2-6s_\eta & = \left(s_\eta-\Upsilon_N\right)^2\nonumber\\ 27s_\eta & = \left(s_\eta-\Upsilon_N\right)^3~.\nonumber \end{align} \tag{ 39 }$$

The equations are solved for

$$\begin{equation}\left(s_\eta,\Upsilon_N,s_1n_\mathrm{t}\right) = \left(0,0,0\right), \left(-1,2,-3\right), \left(27,18,3\right)~, \end{equation} \tag{ 40 }$$

the only non-trivial physical solution being the last one. This means, that the onset of bistability occurs when,

$$\begin{equation}N\Upsilon = 16 ~~~~~\text{and}~~~~~ n_\eta = 27/s_1 ~~~~~\text{and}~~~~~ n_\mathrm{t} = 3/s_1~. \end{equation} \tag{ 41 }$$

The onset is entirely governed by an interplay between the saturation parameter s_η , tuned via the rate at which the cavity is pumped, and the collective cooperativity $\Upsilon_N$, which can be controlled via the number of atoms fed into the cavity mode volume. For our experiment this means, that we observe bistability if more that N > 1232 atoms interact with the cavity and the pump rate is high enough to inject $n_\eta\gt 2.29$ photons resonantly into the cavity in the absence of atoms. Under these circumstances, we actually find $n_\mathrm{t}\gt 0.25$ photons in the cavity in the presence of 1232 atoms.

The condition $0 = \frac{\mathrm d}{\mathrm dn}(A^{^{\prime}}n^3+B^{^{\prime}}n^2+C^{^{\prime}}n+D^{^{\prime}})$ yields the positions of the turning points of the cubic curve parameterized by (26),

$$\begin{equation}n_\mathrm{turn} = -\frac{B^{^{\prime}}}{3A^{^{\prime}}}\pm\frac{B^{^{\prime}}}{3A^{^{\prime}}}\sqrt{1-\frac{3A^{^{\prime}}C^{^{\prime}}}{B^{^{\prime} 2}}}~. \end{equation} \tag{ 42 }$$

In order for the cubic curve to have three (not necessarily degenerate) real roots, the cubic curve must have two distinct maxima. That is, (42) must yield two real solutions (called turning points), which requires $B^{^{\prime} 2}\lt 3A^{^{\prime}}C^{^{\prime}}$. In order for the cubic curve to have three positive real roots, both turning points must be positive. Since $A^{^{\prime}}\gt 0$ we necessarily need $B^{^{\prime}}\lt 0$, which leads to,

$$\begin{equation}\boxed{\Upsilon_N < s_1n_\eta}~. \end{equation} \tag{ 43 }$$

Furthermore, we must satisfy,

$$\begin{equation}0 < 1-\frac{3A^{^{\prime}}C^{^{\prime}}}{B^{^{\prime} 2}} < 1~. \end{equation} \tag{ 44 }$$

Inserting the coefficients (36) we obtain,

$$\begin{equation}s_\eta^2-2\Upsilon_Ns_\eta+6s_\eta+\tfrac{1}{4}\Upsilon _N^2 > 0 > 6s_\eta-\tfrac{3}{4}\Upsilon_N^2~. \end{equation} \tag{ 45 }$$

For large collective cooperativity, $\Upsilon_N\gg 1$, we may simplify,

$$\begin{equation}\boxed{\Upsilon_N\left(1+\tfrac{\sqrt{3}}{2}\right) < s_1n_\eta < \tfrac{\Upsilon_N^2}{8}}~. \end{equation} \tag{ 46 }$$

The criterion (46) clarifies, that an essential condition for resonant bistability is a sufficiently large collective cooperativity $\Upsilon_N\gg 8+4\sqrt{3}$. If this is satisfied, one can always find a pump rate $\eta_+$ such that bistability can be reached.

The theoretical curve shown in figure 8 shows the photon numbers calculated in resonance, $\Delta_\mathrm{a} = 0 = \Delta_\mathrm{ca}$ as a function of pump strengths. Bistable phases are recognized by the presence of several solutions marked in different colors. The transitions between different phases are delimited by the conditions (43) and (46) and are marked in figure 8 as vertical black dashed or dash-dotted lines.

In order to derive an analytic expression for the boundary near $s_1n_\eta = 432\,000$ (vertical black dashed line in figure 8) we note that the region above the boundary is characterized by the existence of three solutions, two of which are degenerate. These solutions can be found by a procedure similarly to the one previously implemented in equation (37). We plug the coefficients (36) into the ansatz,

$$\begin{equation}n^3+\tfrac{B^{^{\prime}}}{A^{^{\prime}}}n^2+\tfrac{C^{^{\prime}}}{A^{^{\prime}}}n+\tfrac{D^{^{\prime}}}{A^{^{\prime}}} = \left(n-n_\mathrm{d}\right)^2\left(n-n_\mathrm{s}\right) \end{equation} \tag{ 47 }$$

and compare the coefficients. This yields a set of equations in the photon numbers $n_\mathrm{s}$, $n_\mathrm{d}$, and n_η ,

$$\begin{align}0 & = s_1n_\mathrm{s}+2s_1n_\mathrm{d}+\Upsilon_N-s_1n_\eta\nonumber\\ 0 & = 8s_1^2n_\mathrm{d}n_s+4s_1^2n_\mathrm{d}^2-\Upsilon_N^2+8s_1n_\eta\nonumber\\ 0 & = -s_1^3n_\mathrm{d}^2n_\mathrm{s}+s_1n_\eta~.\nonumber \end{align} \tag{ 48 }$$

To find an approximate analytic solution of the set of equations (48), we assume, $\Upsilon_N\gg 1$ and $n_\mathrm{s}\gg n_\mathrm{d}$, which simplifies the set to,

$$\begin{align}0 & = s_1n_\mathrm{s}-s_1n_\eta\nonumber\\ 0 & = 8s_1^2n_\mathrm{d}n_s-\Upsilon_N^2+8s_1n_\eta\nonumber\\ 0 & = -s_1^3n_\mathrm{d}^2n_\mathrm{s}+s_1n_\eta~.\nonumber \end{align} \tag{ 49 }$$

The solution is,

$$\begin{equation}\boxed{s_1n_\eta \simeq \tfrac{\Upsilon_N^2}{16}}~. \end{equation} \tag{ 50 }$$

Here, we want to calculate the spectral width of the bistable regime, that is, the width of the ridge connecting the normal modes of figure 3 and the width of the central feature of figure 4. Unfortunately, the condition $\Delta_\mathrm{ca} = \Delta_\mathrm{c}-\Delta_\mathrm{a} = 0$ entails cumbersome formulas. For simplicity we use instead $\bar\Delta_\mathrm{c} = 0$, which corresponds to a cut along a diagonal emphasized by dotted lines in figure 3. Assuming a small spontaneous decay rate, $|\bar\Delta_\mathrm{a}|\gg 1$ and a large collective cooperativity, $\Upsilon_N\gg 1$, the coefficients (27) become (decorated by a double apostrophe),

$$\begin{align}A^{^{\prime\prime}} & = s_1^2\nonumber\\ B^{^{\prime\prime}} & \simeq s_1\left(2\bar\Delta_\mathrm{a}^2+\Upsilon_N-s_\eta\right)\nonumber\\ C^{^{\prime\prime}} & \simeq \bar\Delta_\mathrm{a}^4+\left(\tfrac{\Upsilon_N^2}{4}-2s_\eta\right)\bar\Delta_\mathrm{a}^2\nonumber\\ D^{^{\prime\prime}} & \simeq -n_\eta\bar\Delta_\mathrm{a}^4~.\nonumber \end{align} \tag{ 51 }$$

The coefficient B^ʹʹ determines the slope of the cubic curve in its turning point. For $A^{^{\prime\prime}}\gt 0$ it must be negative in order to allow for more than one real solution. Thus, we can use the condition $B^{^{\prime\prime}} = 0$ to deduce that, for small enough spontaneous decay rate, the spectral width of the bistable regime is proportional to the Rabi frequency,

$$\begin{equation}|\Delta_\mathrm{a}| = \tfrac{\Gamma}{4}\sqrt{s_\eta-\Upsilon_N} = \tfrac{1}{2}\sqrt{\tfrac{\Omega_\eta^2}{2}-\tfrac{\Gamma^2\Upsilon_N}{4}} \simeq \tfrac{\Omega_\eta}{2\sqrt{2}}~. \end{equation} \tag{ 52 }$$