Time-resolved physical spectrum in cavity quantum electrodynamics

The luminescence spectrum is one of the key subjects in quantum optics to study the fundamental characteristics of quantum emitters [1, 2, 3, 4]. The accessibility to the spectrum is experimentally straightforward, the techniques of its calculation are theoretically well-established, and the consistency between experiment and theory is sufficient, especially when the system of interest is in a steady state. As a result, in the regime of cavity quantum electrodynamics (cQED), a number of quantum optical effects, such as the off-resonant cavity feeding [5, 6, 7, 8, 9, 10, 11, 12, 13], the incoherent pumping effects [14, 15, 16], the resonance fluorescence [17, 18, 19] and the Fano resonance [20, 21], have been revealed in the last decades by the steady-state power spectrum, formally described as

Here, $\breve{O}_{\mu}(t)$ is the system operator linked to the radiation field in the Heisenberg picture, $\langle\breve{O}^{\dagger}_{\mu}(t)\breve{O}_{\mu^{\prime}}(t+\tau)\rangle_{0}=\text{Tr}_{\text{S}}[\breve{O}^{\dagger}_{\mu}(t)\breve{O}_{\mu^{\prime}}(t+\tau)\hat{\rho}_{0}]$ is the correlation function with $\hat{\rho}_{0}$ being the initial density operator, and $\chi_{\mu^{\prime},\mu}$ denotes a coefficient relevant to each emission channel. In such a treatment, one advantage to consider the steady state is the equivalence between the correlation functions, $\langle\breve{O}^{\dagger}_{\mu}(t)\breve{O}_{\mu^{\prime}}(t+\tau)\rangle_{0}=\langle\breve{O}^{\dagger}_{\mu}(t-\tau)\breve{O}_{\mu^{\prime}}(t)\rangle_{0}$, which means that the correlation from the future toward the present is statistically equivalent to that from the present toward the past by assuming $\tau>0$ with the present time $t$. This nature allows us to conveniently forget the causality that the correlations with future quantities cannot be measured until the future comes, and the power spectrum can readily be evaluated through the quantum regression theorem (QRT) [1, 2, 3].

In principle, however, the most fundamental is the time-resolved spectrum because it can naturally resolve dynamical effects [22, 23], while extra care is needed for its description, called the physical spectrum [24, 25, 26, 27], to avoid artifactual results. Earlier studies indeed pointed out that the time-resolved spectrum can exhibit distinctive features, e.g., in the fluorescence of a coherently driven atom [28, 29, 30, 31, 32]. Pump-probe ultrafast spectroscopy has also shown the importance of time-resolved measurements [33, 34, 35, 36, 37, 38, 39, 40]. Nevertheless, the inherent nature of the time-resolved spectrum has not been studied in the context of the causality. In fact, to our knowledge, no attention has been paid thus far to the impact of the causality. Moreover, theoretical predictions have increasing importance especially in recent experiments of cQED [10, 35, 36, 37, 38, 39, 40, 41, 42] with the Fano interference effect [43, 20, 21, 44, 45] although there remain few theoretical studies to directly investigate the time-resolved physical spectrum (TRPS) for such systems.

In this paper, we theoretically study the TRPS of luminescence for an initially excited two-level system (TLS) interacting with a single mode cavity. In its construction, the correlation functions up to the present time are crucial, while the correlations with future quantities are unavailable because of the causality. We find that this causality gives a key insight to understand the TRPS in the strong coupling regime, where the Rabi doublet can never be seen during the time of the first peak of the Rabi oscillation. Furthermore, we demonstrate that the magnitude of the Rabi doublet can vary with time in some situations. These features are in contrast to the common view in the steady-state power spectrum and significant, e.g., for the shaping of the temporal waveform of single photons and the switching of the Rabi oscillations [46, 47, 48, 49], because the spectral nature of single photons is essential for quantum information applications. We also study the dynamics of the Fano anti-resonance, where the difference from the Rabi doublet can be highlighted.

Based on the quantum master equation (QME) approach, in the Schrödinger picture with $\hbar=1$, the system density operator, $\hat{\rho}_{t}$, is evolved by

where $t$ is the present time, and $\mathcal{D}$ is given by

with $\mathcal{D}_{\hat{X},\hat{Y}}\hat{\rho}_{t}=2\hat{Y}\hat{\rho}_{t}\hat{X}-\hat{X}\hat{Y}\hat{\rho}_{t}-\hat{\rho}_{t}\hat{X}\hat{Y}$. The system Hamiltonian, $\hat{H}_{\mathrm{S}}=\frac{1}{2}\omega_{21}\hat{\sigma}_{z}+\omega_{\mathrm{c}}\hat{a}_{\mathrm{c}}^{\dagger}\hat{a}_{\mathrm{c}}+(g\hat{\sigma}_{+}\hat{a}_{\mathrm{c}}+g^{*}\hat{a}_{\mathrm{c}}^{\dagger}\hat{\sigma}_{-})$, models the coupling between the TLS and the cavity by a coupling constant $g=|g|e^{\mathrm{i}\pi/2}$, where $\hat{a}_{\mathrm{c}}^{\dagger}$ and $\hat{a}_{\mathrm{c}}$ are the creation and annihilation operators of the cavity photons, $\hat{\sigma}_{\pm}$ are the raising and lowering operators of the TLS, and $\hat{\sigma}_{z}=\hat{\sigma}_{+}\hat{\sigma}_{-}-\hat{\sigma}_{-}\hat{\sigma}_{+}$. In Eq. (3), $\gamma$ and $\kappa$ denote the emission rates of the TLS and the cavity, respectively, while $\gamma_{\mathrm{ph}}$ is the pure dephasing rate of the TLS. The last two terms in Eq. (3) describe the Fano interference between the two emission processes, where $\gamma_{\mathrm{F}}=e^{\mathrm{i}\theta}\sqrt{\eta\gamma\kappa}$ [45]. $\eta$ ($0\leq\eta\leq 1$) is the degree of overlap between the two radiation patterns, while $\theta$ is the phase difference between the two processes.

For the construction of the TRPS, it is essential to remember that the QME is originally derived from the total Hamiltonian including the environment although the QME is a convenient starting point in our analysis. By considering the Heisenberg equations for the radiation modes [2, 45], then, one can arrive at the following expression of the TRPS [24],

where $\delta_{\mathrm{s}}$ is the spectral resolution of the spectrometer and $\mu,\mu^{\prime}\in\{\sigma,a\}$ with $\breve{O}_{\sigma}(t)=\breve{\sigma}_{-}(t)$ and $\breve{O}_{a}(t)=\breve{a}_{\mathrm{c}}(t)$. In the derivation, we have integrated the spectrum over all directions, and as a result, the coefficients $\chi_{\mu^{\prime},\mu}$ can be determined as $\{\chi_{\sigma,\sigma},\chi_{a,a},\chi_{\sigma,a},\chi_{a,\sigma}\}=\{\gamma/\pi,\kappa/\pi,\gamma_{\mathrm{F}}/\pi,\gamma^{*}_{\mathrm{F}}/\pi\}$. Here, $S(\nu,t,\delta_{\mathrm{s}})$ can recover $S_{\SS}(\nu)$ in the limit of $t\to\infty$, apart from the spectral resolution. However, in comparison with Eq. (1), we can explicitly find the causality in Eq. (4), where the correlations are inherently with the past quantities, instead of the future quantities. For transient dynamics, $\langle\breve{O}^{\dagger}_{\mu}(s-\tau)\breve{O}_{\mu^{\prime}}(s)\rangle_{0}$ cannot be replaced by $\langle\breve{O}^{\dagger}_{\mu}(s)\breve{O}_{\mu^{\prime}}(s+\tau)\rangle_{0}$ because $\langle\breve{O}^{\dagger}_{\mu}(s-\tau)\breve{O}_{\mu^{\prime}}(s)\rangle_{0}$ depends not only on the time difference $\tau$ but also on the absolute time $s$ ($=t-t^{\prime}$ in Eq. (4)). This indicates that the causality can have significant impact on the spectrum in general; this is one of the major differences, e.g., from Ref. [22], where the time-resolved spectrum was also discussed. It is further worth noting that the spectral resolution, $\delta_{\mathrm{s}}$, is inevitably included in Eq. (4) and its influence cannot be separated from the TRPS in general. This is another important point although the effect of the resolution is often underestimated.

To calculate the TRPS, however, one difficulty lies in its numerical cost due to the double convolution integral, which must be calculated repeatedly for different $\nu$ and $t$. In our view, this is one reason for the scarcity of theoretical research of the TRPS in cQED systems. For the reduction of the numerical cost, based on the QRT, we write the correlation as

for $s\geq\tau\geq 0$, where ${\mathcal{L}}^{\dagger}$ is the adjoint superoperator of $\mathcal{L}$, $\langle\hat{X}\rangle_{t}\equiv\text{Tr}_{\text{S}}[\hat{X}\hat{\rho}_{t}]$, $C_{\mu^{\prime},i}(\tau)\equiv\text{Tr}_{\text{S}}[(e^{{\mathcal{L}}^{\dagger}\tau}\hat{O}^{\dagger}_{\mu^{\prime}})^{\dagger}\hat{A}^{\dagger}_{i}]$, and $\{\hat{A}_{i}\}$ is a complete set of system operators in the Liouville space. The first equality is helpful in general because $e^{{\mathcal{L}}^{\dagger}\tau}\hat{O}^{\dagger}_{\mu^{\prime}}$ can be computed separately from the evolution of the density operator, $\hat{\rho}_{s-\tau}$. The second equality is further advantageous when $C_{\mu^{\prime},i}(\tau)$ can be obtained analytically. By employing the latter approach, we have performed one of the integrals in Eq. (4) analytically. We also note that $\langle\breve{O}^{\dagger}_{\mu}(s-\tau)\breve{O}_{\mu^{\prime}}(s)\rangle_{0}=0$ either for $s-\tau<0$ or for $s<0$ by definition [50]. As a result, we can successfully analyze the TRPS; see also Appendix A for further details.

Figure 1 shows typical numerical results for different spectral resolutions with $\gamma_{\mathrm{ph}}=0$ and $\eta=0$. The TLS is initially excited and other parameters are assumed $\kappa=50\leavevmode\nobreak\ \mu$eV, $\gamma=0.05\leavevmode\nobreak\ \mu$eV, and $|g|=100\leavevmode\nobreak\ \mu$eV under the resonant condition. Since our setting, $|g|\gtrsim\kappa+\gamma$, indicates the strong-coupling regime, the Rabi doublets should be well resolved when $\delta_{\mathrm{s}}\ll|g|=100\leavevmode\nobreak\ \mu$eV. As a result, for $\delta_{\mathrm{s}}=5\leavevmode\nobreak\ \mu$eV, we can see the typical Rabi doublets in the time-integrated spectra for the cavity and the TLS [Figs. 1(a) and (b)]. However, a single-peak structure can be found in the early stage of the TRPS for the cavity [Fig. 1(c)]. A similar structure can also be seen for the TLS [Fig. 1(d) and the inset] although its appearance is earlier than for the cavity. These features are in contrast to the ordinary understanding of the Rabi doublets. We also note that the dynamics in the TRPS are largely different from the Rabi oscillations of $\langle\hat{a}_{\mathrm{c}}^{\dagger}\hat{a}_{\mathrm{c}}\rangle_{t}$ and $\langle\hat{\sigma}_{+}\hat{\sigma}_{-}\rangle_{t}$, as seen in Fig. 1(e).

By increasing $\delta_{\mathrm{s}}$ to $150\leavevmode\nobreak\ \mu$eV (comparable to $|g|$), however, oscillatory behaviors come out in Fig. 1(j), while the Rabi doublets become blurred in Figs. 1(f) and (g). This is due to the uncertainty between energy and time. Hence, by further increasing $\delta_{\mathrm{s}}$ up to $500\leavevmode\nobreak\ \mu$eV, the energy integrated intensities recover the Rabi oscillations almost completely in Fig. 1(o), while the Rabi doublets cannot be resolved any more in Figs. 1(k) and (l). In fact, in Eq. (4), $\delta_{\mathrm{s}}e^{-\delta_{\mathrm{s}}t^{\prime}}$ gives the time uncertainty as a response function of the spectrometer, while $e^{-\delta_{\mathrm{s}}\tau/2}$ results in the energy uncertainty as a resolution of the spectrometer. It is thus obvious that $\delta_{\mathrm{s}}$ plays an essential role for the consistent description of the physical spectrum. As a result, we note that the initial rising time of the TRPS intensities cannot be faster than $\delta_{\mathrm{s}}^{-1}$ in Fig. 1.

Beyond such complementarity, we can notice that the single peaks observed in Figs. 1(c) and (d) are more highlighted in Figs. 1(h) and (i) and finally form the first peaks of the Rabi oscillations in Figs. 1(m) and (n). This indicates that the mechanism of the single peaks is particularly related to the first peaks of the Rabi oscillations. This is consistent with the earlier appearance of the single peak for the TLS than for the cavity. However, it is still unclear why such a peculiar role is given only for the first peak. For a detailed discussion, the cavity spectra for $\delta_{\mathrm{s}}=5\leavevmode\nobreak\ \mu$eV are separately plotted at different times in Figs. 2(a)–(d), corresponding to the white dashed lines in Fig. 1(c). As a result, we can further find that there are weak satellite series around the main peaks, which are absent in the time-integrated spectrum.

To clarify the underlying physics, it is essential to directly analyze the correlation functions. For this purpose, in Figs. 2(e)–(h), we plot the cavity correlations $\langle\breve{a}_{\mathrm{c}}^{\dagger}(s+\tau^{\prime})\breve{a}_{\mathrm{c}}(s)\rangle_{0}e^{-\mathrm{i}\omega_{\mathrm{c}}\tau^{\prime}}$ with $\tau^{\prime}\equiv-\tau$, at absolute times $s$ coincident with the individual times $t$ of Figs. 2(a)–(d). These correlations at $s=t$ ($t^{\prime}=0$) give the major contributions in the $t^{\prime}$ integral of Eq. (4) when Figs. 2(a)–(d) are calculated, respectively. In Figs. 2(e)–(h), we can see that the correlations show oscillatory dynamics at half the Rabi frequency both for $\tau^{\prime}<0$ and $\tau^{\prime}>0$. However, we note that the causality is satisfied only for $\tau^{\prime}<0$ when calculating the individual spectra [Fig. 2(a)–(d)]. Hence, the correlations become relevant only within the time windows of $-s<\tau^{\prime}<0$ (the gray shaded areas), as they vanish for $\tau^{\prime}<-s$ [50]. In fact, this is the reason for the appearances of the satellite series in Figs. 2(a)–(d). Based on the Fourier analysis, such a time window yields the satellite peaks separated by $2\pi/t$, as indicated by the dashed lines in Figs. 2(c) and (d). This understanding is consistent with our results that such a phenomenon can be still observed even without the cavity (Appendix B). Furthermore, we can also notice that the Rabi doublets can be observed in Figs. 2(a)–(d) only when the correlations can show their oscillations at least for one period during the time windows in Figs. 2(e)–(h). In Figs. 2(g) and (h), the correlations oscillate more than one period during the time windows, and therefore, we can see the Rabi doublets in Figs. 2(c) and (d). In contrast, in Figs. 2(e) and (f), the correlations cannot show their oscillations within the time windows of the causality. This means that the spectrometer cannot know the oscillating behaviors of the correlations in advance, before it really detects the oscillations at least over one period. As a result, the single peaks are obtained in Figs. 2(a) and (b). This is the very reason why the single peaks appear in Figs. 1(c), (d), (h), and (i). Since the correlations oscillate at half the Rabi frequency, we can conclude that the Rabi doublets can never be seen during the times of the first peaks of the Rabi oscillations, either for the cavity or for the TLS. Such unique impact of the causality is not present in the steady-state spectrum. These results are important by considering that spectral features are, in general, essential for single photons to elicit their fundamental performance in quantum information applications.

We note, however, that the importance of the causality is not limited to the appearances of the satellite series and/or the disappearances of the Rabi doublets. To illustrate this, we here find that the separations of the Rabi doublets are changed with time, especially for $\delta_{\mathrm{s}}=150\leavevmode\nobreak\ \mu$eV [Figs. 1(h) and (i)], where the separations are gradually decreased during the individual Rabi cycles. This is in contrast to the intuition that the separations are determined only by the magnitude of the coupling constant, $|g|$. In fact, however, this characteristics can be understood simply from a one-sided Fourier transform, $F_{\phi}(\nu)\equiv\int^{\infty}_{0}\mathrm{d}\tau\cos(|g|\tau+\phi)e^{\mathrm{i}\nu\tau-\frac{\delta_{\mathrm{s}}}{2}\tau}$. For $\delta_{\mathrm{s}}$ comparable to $|g|$, $\sin(|g|\tau)e^{-\frac{\delta_{\mathrm{s}}}{2}\tau}$ is close to a single pulse, while $\cos(|g|\tau)e^{-\frac{\delta_{\mathrm{s}}}{2}\tau}$ remains more likely to be oscillatory. Hence, the separation of the doublet in $|F_{\phi}(\nu)|^{2}$ can be changed by its phase, $\phi$, as well as its frequency, $|g|$ (see also Appendix C). The transient Rabi doublets can then be understood essentially in the same manner. The important points here are that the corresponding phase is determined by $\langle\breve{O}^{\dagger}_{\mu}(s+\tau^{\prime})\breve{O}_{\mu^{\prime}}(s)\rangle_{0}$ not for $\tau^{\prime}>0$ but for $\tau^{\prime}<0$ due to the causality, and that these two phases are different, as seen in Figs. 2(e)–(h), for example. These facts mean that the transient Rabi doublets are also strongly influenced by the causality, even if the frequencies of the correlations are the same between $\tau^{\prime}>0$ and $\tau^{\prime}<0$. For example, the dynamics of the cavity Rabi doublet could become very different from Fig. 1(h) if $\langle\breve{a}_{\mathrm{c}}^{\dagger}(s-\tau)\breve{a}_{\mathrm{c}}(s)\rangle_{0}$ was replaced by $\langle\breve{a}_{\mathrm{c}}^{\dagger}(s)\breve{a}_{\mathrm{c}}(s+\tau)\rangle_{0}$ in Eq. (4) by neglecting the causality.

We have thus clarified the fundamental aspects of the TRPS beyond the conventional view of the Rabi doublet and oscillation. However, we finally remark that the dynamics of the spectra can largely depend on the inherent mechanism of the doublet in general. To see this, in Fig. 3, the Fano interference is studied by setting $\eta=1$ with $\gamma_{\mathrm{ph}}=30\leavevmode\nobreak\ \mu$eV and $|g|=1\leavevmode\nobreak\ \mu$eV. In this situation, the Fano anti-resonance yields a doublet in the time-integrated (total) spectrum for $\delta_{\mathrm{s}}=5\leavevmode\nobreak\ \mu$eV in Fig. 3(a), while it becomes unclear by increasing $\delta_{\mathrm{s}}$ in Figs. 3(b) and (c), as expected. In contrast to Fig. 1, however, there is no oscillation relevant to the doublet in Figs. 3(d)–(g) because the doublet is caused by the destructive interference. Hence, the presence of the doublet does not necessarily indicate the oscillation. On the other hand, the single peaks can again be found in the early stages of the TRPS. This is essentially due to the broadening by the time windows of the causality, but cannot be related to oscillatory behaviors in the time domain. It follows that the separation of the doublet does not depend on time, even when the doublet is partly blurred by the resolution [Figs. 3(b) and (e)]. In this context, the time-resolved spectrum is indeed helpful to resolve the dynamical effects.

To summarize, we have demonstrated that the TRPS can exhibit distinctive features beyond the steady-state spectrum, where key viewpoints are given by the causality. Furthermore, the TRPS can also highlight the difference between the Fano anti-resonance and the Rabi doublet. The approach presented here would have a wide range of applications, such as the shaping of the temporal waveform of single photons and the real-time switching of Rabi oscillations toward quantum information processing [46, 47, 48, 49]. Especially, it provides a useful framework in analyzing the Fano interference from quantum optical systems [51, 52, 53, 54]. In addition, despite our focus on the TRPS, we stress that the significance of the causality is not limited to such a case because of its conceptual generality. Our results suggest that the pump-probe absorption spectrum [33, 34] and out-of-time-order correlations in photodetection [55], for example, might be non-trivial. Our findings provide fundamental insight into the time-domain spectroscopy in cQED systems.