Cavity QED with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures

We study the nonlinear dynamics of an ensemble of cold trapped atoms with a hyperfine transition magnetically coupled to a resonant microwave cavity mode. Despite the minute single-atom coupling, one obtains strong coupling between collective hyperfine qubits and microwave photons, enabling coherent transfer of an excitation between the long-lived atomic qubit state and the mode. Evidence of strong coupling can be obtained from the cavity transmission spectrum even at finite thermal photon number. The system makes it possible to study further prominent collective phenomena such as superradiant decay of an inverted ensemble or the building of a narrowband stripline micromaser locked to an atomic hyperfine transition.

Figure 1

$(\text{a})$ Steady-state number of photons $\langle a^{†}a\rangle {}_{\mathrm{ss}}$ in the pumped cavity for different detunings ${\Delta }_m={\omega }_m-{\omega }_l$ and different temperatures, obtained from Hamiltonian $H_{\mathrm{TC}}$. The parameters chosen were $\kappa =1$, $N=2$, $g=3$, ${\gamma }_a=0.05$, $\eta =0.1$. With increasing temperature the two peaks indicating strong coupling are superimposed by thermal photons. To compare the dynamics of $H_{\mathrm{TC}}$ and $H$, we plot both results for $T=0.04\hspace{0.5em}\mathrm{K}$ ($\mathrm{n̄}=3\times {10}^{-4}$) in $(\text{b})$.

Figure 2

Effects of higher temperatures: For $T=0.1\hspace{0.5em}\mathrm{K}$, $(\text{a})$ shows the steady-state number of photons, whereas $(\text{b})$ and $(\text{c})$ show the real and imaginary part of the steady state field in the cavity. The peaks in the photon number start to broaden and finally vanish. The amplitude of the field shows similar behavior. Panels $(\text{d})$, $(\text{e})$, and $(\text{f})$ show the same quantities for $T=0.7\hspace{0.5em}\mathrm{K}$. The remaining parameters were chosen as in Fig. 1.

Figure 3

The steady-state field and photon number in the cavity for different frequencies of the pump. The size of the ensemble was chosen to be $N={10}^5$ and we set $\kappa =7\times {10}^3$, $\eta =5\times {10}^5$, ${\gamma }_a=0.3$, and $g=40$. Panel $(\text{a})$ shows the steady-state number of photons, whereas $(\text{b})$ and $(\text{c})$ show the real and imaginary parts of the steady-state field in the cavity for $T=0.1\hspace{0.5em}\mathrm{K}$, which corresponds to $\mathrm{n̄}=0.04$. The remaining figures show the same quantities for $T=0.7\hspace{0.5em}\mathrm{K}$, corresponding to $\mathrm{n̄}=1.67$. The results show that for a sufficiently large number of atoms strong coupling remains observable despite of the presence of thermal photons.

Figure 4

To simplify matters, we depict the cavity as a Fabry-Perot cavity which can be pumped through a mirror with high reflectivity. The observation of the dynamics is carried out using the second mirror, which has a lower reflectivity. Additionally, we can pump the ensemble incoherently from the side.

Figure 5

Overview of the transmitted spectrum $S$ for different sizes of the ensemble (a). The temperature of the cavity is set to $T=0.1\hspace{0.5em}\mathrm{K}$ ($\mathrm{n̄}=0.04$). The remaining parameters were chosen to be $\kappa =7\times {10}^3$, ${\gamma }_a=0.3$, $g=40$, ${\omega }_a={\omega }_m=2\pi \times 6.83\hspace{0.5em}\mathrm{GHz}$. The dips in the spectrum indicate that thermal photons are absorbed by the ensemble and re-emitted into modes other than the cavity mode. The increasing distance between the absorption dips reflects the increasing number of atoms. In panels (b)–(d) we depict the spectra at $N_1=3.2\times {10}^6$, $N_2=1\times {10}^6$, and $N_3=3.4\times {10}^5$, indicated in (a) by the dashed horizontal lines.

Figure 6

Loss of photons from the cavity mode. (a) Dynamics of the occupation of the mode for different temperatures and ${\gamma }_a=5\times {10}^4$. A constant fraction of the photons is removed from the cavity mode. $(\text{b})$ For fixed $T=4\hspace{0.5em}\mathrm{K}$ ($\mathrm{n̄}=11.7$), the loss rate of the atoms is varied between ${\gamma }_a=1\times {10}^3$ and ${\gamma }_a=2\times {10}^5$. The steady-state number of photons shows that there is an optimal loss rate. The red curve (with diamond markers) corresponds to the expected number of photons according to Eq. (21). The remaining parameters were chosen to be $\kappa =7\times {10}^3$, $g=40$, ${\omega }_a={\omega }_m=2\pi \times 6.83\hspace{0.5em}\mathrm{GHz}$, $N={10}^5$.

Figure 7

Steady state of the photon number (a) and the inversion (b) for a loss rate ${\gamma }_a=\kappa =7\times {10}^3$. Increasing the number of atoms $N$ leads to a more effective removal of thermal photons. The remaining parameters are chosen to be $g=40$, ${\omega }_a={\omega }_m=2\pi \times 6.83\hspace{0.5em}\mathrm{GHz}$, $T=4\hspace{0.5em}\mathrm{K}$ ($\mathrm{n̄}=11.7$).

Figure 8

The temperature of the cavity is now set to $T=0.001\hspace{0.5em}\mathrm{K}$ ($\mathrm{n̄}=0$) and an incoherent pump of the atoms with $w=0.05$ is switched on. The spectrum now shows increased emission at the frequencies of the coupled system. Again figures (b)–(d) depict the spectra at $N_1=3.2\times {10}^6$, $N_2=1\times {10}^6$, and $N_3=3.4\times {10}^5$, indicated in (a) by the dashed horizontal lines.

Figure 9

Steady-state field in the driven cavity, real part (a) and imaginary part (b). The size of the ensemble is chosen to be $N={10}^5$. The lines for $T=0.01\hspace{0.5em}\mathrm{K}$ ($\mathrm{n̄}=0$) and $T=10\hspace{0.5em}\mathrm{K}$ ($\mathrm{n̄}=30$) coincide.

Figure 10

Steady-state field in the driven cavity, real part (a) and imaginary part (b), for $T=0.01\hspace{0.5em}\mathrm{K}$ ($\mathrm{n̄}=0$) (solid lines) and $T=10\hspace{0.5em}\mathrm{K}$ ($\mathrm{n̄}=30$) (dashed lines). For the small ensemble with $N={10}^2$ atoms we recover the effects of the thermal photons. To compensate for the lower number of atoms, the coupling is chosen to $g=1200$. Otherwise, the splitting would be covered by the cavity linewidth.

Figure 11

Incoherent spectrum of the mode $(\text{a})$ and of the fluorescence of the ensemble $(\text{b})$ for $T=0.025\hspace{0.5em}\mathrm{K}$, $\kappa =7\times {10}^3$, $\gamma =0.3$, $N={10}^5$, $\eta =9\times {10}^5$ in red (gray), and $\eta ={10}^6$ in black. The pump driving the system is on resonance with the cavity and the ensemble. Insets $(\text{c})$ and $(\text{d})$ show a magnification of the central peak of each spectrum. The lower red (gray) line represents the result for $\eta =9\times {10}^5$.

Figure 12

Dynamics of superradiant emission: numerical solutions of the dynamical equations for an ensemble of $N={10}^5$ atoms. The rapid drop of the inversion $\langle {\sigma }_1^z\rangle$ during the emission can be seen in $(\text{a})$. The exchange of excitations between the ensemble and the cavity is characterized by $\langle a{\sigma }_1^{+}\rangle$ in $(\text{b})$. Negative imaginary part of $\langle a{\sigma }_1^{+}\rangle$ indicates emission from the ensemble into the cavity, where a positive imaginary part indicates absorption of cavity photons by the ensemble. In $(\text{c})$ the number of photons in the cavity is depicted. Panel $(\text{d})$ shows the spin-spin correlation $\langle {\sigma }_1^{+}{\sigma }_2^{-}\rangle$. In this example the temperature of the mode was chosen to be $T=4\hspace{0.5em}\mathrm{K}$.

Figure 13

Dynamics of the photon number in the cavity with increasing temperature. The onset of the superradiant emission is shifted to earlier times since the initially present thermal photons contribute to the fluctuations that trigger the emission process.

Figure 14

Steady-state inversion of the ensemble $(\text{a})$ and occupation of the cavity mode $(\text{b})$ for varying ensemble size $N$ and pump strength $w$. The temperature was chosen to be $T_1=0.001\hspace{0.5em}\mathrm{K}$, which corresponds to an empty cavity ($\mathrm{n̄}=0$). Vertical red dashed lines mark the masing threshold $w={\gamma }_a=0.3$. The horizontal red dashed lines at $N={10}^5$ indicate the position of the curves shown in $(\text{c})$ and $\text{(d})$. From $(\text{c})$ we recover the passage of the inversion through zero for $w={\gamma }_a=0.3$. At this point we see in $(\text{d})$ a rapid increase of the photon number in the mode.

Figure 15

$(\text{a})$ Linewidth of the spectrum $S(\omega )$. For each set of parameters we numerically determine the linewidth of the spectrum. The parameters were chosen to be $\kappa =7\times {10}^5$, ${\gamma }_a=0.3$, $g=40$, ${\omega }_m={\omega }_a=2\pi \times 6.83\times {10}^9$, $T=0.001$. $(\text{b})$ Exemplary spectrum for $N={10}^5$ and $w=0.55$, marked in $(\text{a})$ by the white cross.

Figure 16

Finite temperature effects in the spectrum. Panel $(\text{a})$ again shows the linewidth of the spectrum. Below the critical pump strength we recover the linewidth of the cavity. The parameters were chosen to be $\kappa =7\times {10}^5$, ${\gamma }_a=0.3$, $g=40$, ${\omega }_g={\omega }_a=2\pi \times 6.83\times {10}^9$, $T=0.1$. $(\text{b})$ Exemplary spectrum for $N={10}^5$ and $w=0.55$, marked in $(\text{a})$ by the white cross.

Figure 17

Same as Fig. 14 with $T_2=0.1$, which corresponds to $\mathrm{n̄}=0.04$. In $(\text{d})$ the thermal excitations in the mode appear as a constant background from which the increase due to the pump stands out.

Figure 18

$(\text{a})$ Dynamics of the photon number in the cavity mode for an initially fully inverted ensemble. The solid line shows the dynamics produced by the set of equations where $\langle a^{†}a{\sigma }_1^z\rangle {}_c$ was neglected. The dashed curve is the result of the full set of 13 equations in which $\langle a^{†}a{\sigma }_1^z\rangle {}_c$ is kept. $(\text{b})$ Numerically obtained cumulant $\langle a^{†}a{\sigma }_1^z\rangle {}_c$ in the steady state. With increasing loss rate of the cavity $\kappa$ the correlation between photon number and inversion decreases.