Photon blockade in weakly-driven cavity QED systems with many emitters

TL;DR

This paper presents a scalable method to analyze photon blockade in multi-emitter cavity QED systems, revealing how emitter number and detuning affect photon blockade quality.

Contribution

The authors develop an efficient scattering matrix approach that reduces computational complexity, enabling analysis of systems with around 50 emitters, and explore effects of detuning and inhomogeneous broadening.

Findings

Photon blockade worsens with more emitters in resonant systems.

Photon blockade improves with more emitters in detuned systems.

Inhomogeneous broadening impacts photon blockade in complex ways.

Abstract

We use the scattering matrix formalism to analyze photon blockade in coherently-driven CQED systems with a weak drive. By approximating the weak coherent drive by an input single- and two-photon Fock state, we reduce the computational complexity of the transmission and the two-photon correlation function from exponential to polynomial in the number of emitters. This enables us to easily analyze cavity-based systems containing $\sim$50 quantum emitters with modest computational resources. Using this approach we study the coherence statistics of polaritonic photon blockade while increasing the number of emitters for resonant and detuned multi-emitter CQED systems --- we find that increasing the number of emitters worsens photon blockade in resonant systems, and improves it in detuned systems. We also analyze the impact of inhomogeneous broadening in the emitter frequencies on both polaritonic and subradiant photon blockade through this system.

Equations

$$\displaystyle H_{\text{sys}}=\omega_{c}a^{\dagger}a+\sum_{i=1}^{N}\bigg{[}\omega_{i}\sigma_{i}^{\dagger}\sigma_{i}+g_{i}\big{(}a\sigma_{i}^{\dagger}+\sigma_{i}a^{\dagger}\big{)}\bigg{]}.$$

$$|{\psi }_{\text{in}}⟩=\exp [{\beta }_0(b_{{\omega }_L}^{†}-b_{{\omega }_L})]|\text{vac}⟩$$

$$\displaystyle T(\omega_{L})=\bigg{|}\int_{t^{\prime}=-\infty}^{\infty}S_{c}(t;t^{\prime})\exp(-\textrm{i}\omega_{L}t^{\prime})\textrm{d}t^{\prime}\bigg{|}^{2}$$

$$g^{(2)}(t_1,t_2;{\omega }_L)=\frac{1}{4T^2({\omega }_L)}\times$$

$$\displaystyle\bigg{|}\int_{t_{1}^{\prime},t_{2}^{\prime}=-\infty}^{\infty}S_{c,c}(t_{1},t_{2};t_{1}^{\prime},t_{2}^{\prime})\exp[-\textrm{i}\omega_{L}(t_{1}^{\prime}+t_{2}^{\prime})]\textrm{d}t_{1}^{\prime}\textrm{d}t_{2}^{\prime}\bigg{|}^{2}$$

$$H_{\text{eff}}=H_{\text{sys}}-\frac{\text{i}\kappa }{2}{a}^{†}a-\sum _{i=1}^N\frac{\text{i}{\gamma }_i}{2}{\sigma }_i^{†}{\sigma }_i$$

$$\displaystyle T(\omega_{L})=\kappa_{b}\kappa_{c}\bigg{|}\sum_{i=1}^{\mathcal{N}_{1}}\frac{(\bra{\text{G}}a\ket{\phi^{(1)}_{i}}_{\text{T}})^{2}}{\lambda_{i}^{(1)}-\omega_{L}}\bigg{|}^{2}$$

$$\displaystyle g^{(2)}(0;\omega_{L})=\bigg{|}\sum_{i=1}^{\mathcal{N}_{2}}\Gamma_{i}(\omega_{L})\bigg{|}^{2}$$

$${\Gamma }_i({\omega }_L)=$$

$$\displaystyle\sum_{j=1}^{\mathcal{N}_{1}}\Bigg{(}\frac{\bra{\phi^{(2)}_{i}}a^{\dagger}\ket{\phi^{(1)}_{j}}_{\text{T}}\bra{\phi^{(1)}_{j}}a^{\dagger}\ket{\text{G}}_{\text{T}}}{\lambda^{(1)}_{j}-\omega_{L}}\Bigg{)}$$

$$\displaystyle\Delta\omega_{1,2}=\min_{i,j}\big{|}\text{Re}[2\lambda^{(1)}_{i}]-\text{Re}[\lambda^{(2)}_{j}]\big{|}.$$

$$\displaystyle\lim_{N\to\infty}g^{(2)}(0;\omega_{L})=\bigg{|}1-\frac{g^{2}}{(\omega_{L}-\lambda_{e})(2\omega_{L}-\lambda_{e}-\lambda_{c})}\bigg{|}^{2}$$

Full text

††preprint: APS/123-QED

*Introduction. *Cavity quantum electrodynamics (CQED) is a fundamental model of light and matter which has been experimentally implemented in a variety of physical platforms. Atomic and solid state CQED systems with a few two-level emitters have exhibited a rich set of quantum phenomena in transmission statistics, including, but not limited to, the vacuum Rabi oscillations Brune et al. (1996); Yoshie et al. (2004), the conventional and the unconventional photon blockade Birnbaum et al. (2005); Imamoǧlu et al. (1998); Snijders et al. (2018), and the photon-induced tunneling Faraon et al. (2008). While suitable approximations can provide understanding of the eigenstructure of multi-element CQED systems Diniz et al. (2011); Radulaski et al. (2017b) obtained in experiments Thompson et al. (1992); Zhong et al. (2017), the numerical studies of light-emission and scattering from this system have been limited due to the exponential scaling of the Hilbert space with the number of emitters.

The scattering matrix formalism for quantum-optical systems provides the solution to this problem. Recently, a general formalism for computing this scattering matrix for an arbitrary time-independent and time-dependent Markovian quantum-optical system was developed Xu and Fan (2015); Trivedi et al. (2018), reducing its computation to that of an effective propagator for the quantum-optical system. Use of the scattering matrices allows relating the transmission and two-photon correlation through a system to the single- and two-photon scattering matrix whose computation time scales as $\sim\mathcal{O}(N^{3})$ and $\sim\mathcal{O}(N^{6})$ respectively in the number of emitters $N$.

In this letter, we use the scattering matrix formalism to study multi-emitter CQED systems with a large number of emitters ($N\sim$ 50) driven by weak continuous-wave classical light (e.g. a laser). We show that increasing the number of emitters does not increase the depth of the photon blockade in a resonant multi-emitter CQED systems with identical emitters. However, we find that increasing the number of emitters improves photon blockade if the emitters are detuned from the cavity resonance. Finally, we study the impact of inhomogenous broadening Evans et al. (2016); Zhang et al. (2018); Dory et al. (2018); Radulaski et al. (2017c); Bracher et al. (2017); Banks et al. (2018); Lukin et al. (2019) in the emitter frequencies on photon blockade in the multi-emitter systems.

Simulation method. A schematic of the considered system is shown in Fig. 1 — a cavity, with annihilation operator $a$, is coupled to $N$ two-level emitters, with lowering operators $\sigma_{i}$, $1\leq i\leq N$. The cavity is excited through a waveguide, with a frequency dependent annihilation operator $b_{\omega}$, and the emission from the cavity is collected through another waveguide, with annihilation operator $c_{\omega}$. The emitters, in addition to coupling to the cavity mode, also radiate into loss channels with annihilation operators $l^{(i)}_{\omega}$ — these loss channels model the linewidths of the emitters. The Hamiltonian for the multi-emitter CQED system is given by:

[TABLE]

where $\omega_{c}$ is the cavity resonance frequency, $\omega_{i}$ is the transition frequency of the $i^{\text{th}}$ emitter and $g_{i}$ is the coupling constant between the $i^{\text{th}}$ emitter and the cavity mode. We study the excitation of this system with a continuous-wave coherent state at frequency $\omega_{L}$, described by an input state:

[TABLE]

where $\beta_{0}^{2}$ is the photon flux (number of photons per unit time) in the coherent state. As is detailed in the supplementary material sup , we establish the following relationship of the transmission $T(\omega_{L})$ and two-photon correlation $g^{(2)}(t_{1},t_{2};\omega_{L})$ to the single-photon $[S_{c}(\cdot)]$ and two-photon $[S_{c,c}(\cdot)]$ scattering matrices for continuous-wave input in the limit of small input photon flux $\beta_{0}^{2}$:

[TABLE]

where the $S$ matrices capture scattering of photons propagating in the input-waveguide (with annihilation operator $b_{\omega}$) to the output-waveguide (with annihilation operator $c_{\omega}$). The scattering matrices are functions only of the system operators and external coupling constants $\kappa_{b,c}$ and $\gamma_{n}$.

The dominant cost for computing these scattering matrices is that of diagonalizing the effective Hamiltonian $H_{\text{eff}}$ sup :

[TABLE]

where $\kappa=\kappa_{b}+\kappa_{c}$ is the total decay rate for the optical cavity. Since $H_{\text{eff}}$ conserves the total excitation number ($a^{\dagger}a+\sum_{n=1}^{N}\sigma_{n}^{\dagger}\sigma_{n})$, this diagonalization can be performed separately within the excitation conserving subspaces of the full Hilbert space. When computing the single- and two-photon scattering matrices, it is only necessary to diagonalize the effective Hamiltonian within the single- and two-excitation subspaces the cost of which approximately as $\sim\mathcal{O}(N^{3})$ and $\sim\mathcal{O}(N^{6})$ respectively. We note that when the emitters are identical (i.e. $\omega_{i}=\omega,\gamma_{i}=\gamma$ and $g_{i}=g$  for all $i\in\{1,2\dots N\}$), by utilizing the Clebsh-Gordan series this diagonalization can be mapped to the diagonalization of $3\times 3$ and $2\times 2$ complex matrices sup .

Having diagonalized $H_{\text{eff}}$, the transmission $T(\omega_{L})$ and equal-time two-photon correlation $g^{(2)}(0;\omega_{L})=g^{(2)}(t,t;\omega_{L})$ can be expressed as sup :

[TABLE]

where $\ket{\text{G}}$ is the ground state of the multi-emitter CQED system, $\langle\cdot\rangle_{\text{T}}$ denotes a ‘transpose’ inner product between two states, $\mathcal{N}_{i}$ is the dimensionality of the $i^{\text{th}}$ excitation subspace of the multi-emitter CQED system, $(\lambda^{(i)}_{j},\ket{\phi^{(i)}_{j}})$ are the eigenvalues and eigenstates of $H_{\text{eff}}$ within the $i^{\text{th}}$ excitation subspace and $\Gamma_{i}(\omega_{L})$, given below, can be interpreted as the contribution of the $i^{\text{th}}$ two-excitation eigenstate to the equal time two-photon emission:

[TABLE]

These expressions for $T(\omega_{L})$ and $g^{(2)}(0;\omega_{L})$ explicitly show their dependence on the energy eigenvalues [$\sim\text{Re}(\lambda_{i}^{(j)})$], linewidths [$\sim\text{Im}(\lambda_{i}^{(j)})$] as well as the eigenstates ($\ket{\phi_{j}^{(i)}}$) of the multi-emitter CQED systems.

Results. Using a large number of identical emitters coupling coherently to the same cavity mode is a potential strategy to achieve strong coupling between the emitters and the cavity in a situation where an individual emitter only weakly couples to the cavity mode. Figure 2(a)-(b) shows the transmissivity $T(\omega_{L})$ and equal-time correlation $g^{(2)}(0;\omega_{L})$ for multi-emitter CQED systems with 1–100 emitters. Consistent with the result obtained on a direct diagonalization of $H_{\text{sys}}$, we observe that the splitting between the polaritonic peaks in the transmissivity scales as $\sqrt{N}$. We also observe that the minimum two-photon correlation $g^{(2)}(0;\omega_{L})$, which is achieved at the polaritonic frequencies, tends towards unity with an increase in the number of emitters for both strongly-coupled emitters [Fig. 2(a)] and weakly-coupled emitters [Fig. 2(b)]. This trend can be easily explained by looking closely at the ‘anharmonicity’ ($\delta\omega_{1,2}$) between the single- and two-excitation eigenenergies of the multi-emitter CQED system:

[TABLE]

Figure 3(a) shows $\Delta\omega_{1,2}$ as a function of $N$ along with the linewidth $\delta\omega_{2}=\text{Im}[2\lambda^{(2)}_{i}]$ of the most harmonic eigenstate in the two-excitation subspace — increasing the number of emitters makes the system’s energy levels more equally spaced while saturating their linewidths, thereby worsening photon blockade. It is worth noting that with the weakly coupled emitters the value of the min$[g^{(2)}(0;\omega_{L})]$ has an initial decrease, before monotonically increasing with the number of emitters consistent with previously reported results Radulaski et al. (2017b). We also observe a pronounced ‘bunching’ peak in $g^{(2)}(0;\omega_{L})$ for strongly coupled emitters [Fig. 2(a)], near the anti-bunching dip — this corresponds to $2\omega_{L}$ being resonant with the two-excitation eigenstates. $g^{(2)}(0;\omega_{L})$ at the bunching peak also tends to $1$ as $N\to\infty$ due to the system eigenstates becoming increasingly harmonic. Moreover, $g^{(2)}(0;\omega_{L})$ can be analytically evaluated in the limit of $N\to\infty$ to obtain sup :

[TABLE]

where $\lambda_{e}=\omega_{e}-\textrm{i}\gamma/2$ and $\lambda_{c}=\omega_{c}-\textrm{i}\kappa/2$. As can be seen from Figs. 2(a) and (b), in the limit of large number of emitters, the multi-emitter system does not show any blockade — photon bunching can be seen at $\omega_{L}\sim\omega_{c}$ as a consequence of near zero single-photon transmission.

We next study the impact of detuning between the emitters and the optical mode on the polaritonic photon blockade [Fig. 2(c)]. Consider the transmission lineshape: there is a distinct Fano dip to nearly zero transmission at $\omega_{L}\approx\omega_{e}$ where the single-photon transmission through the two single-excitation eigenstates (one being more cavity-like and the other being more emitter-like) exactly cancels. Towards the right of this Fano dip, the light antibunches ($g^{(2)}(0;\omega_{L})<1$) owing to the standard polaritonic photon blockade of a detuned system Laussy et al. (2012); Müller et al. (2015). Due to the multi-emitter system becoming more harmonic with an increase in the number of emitters [Fig. 3(a)], this blockade effect degrades similar to that in the resonant CQED system. Photon bunching ($g^{(2)}(0;\omega_{L})>1$) is observed exactly at the Fano dip due to the single-photon transmission becoming nearly zero — within the framework of scattering theory, this is equivalent to the contribution of the unconnected (linear or frequency-preserving) part of the scattering matrix being small in the output state, and the scattering happening almost entirely from the connected (nonlinear or frequency-mixing) part of the two-photon $S$ matrix Shen and Fan (2007).

Slightly left of the Fano dip, we again observe photon antibunching — moreover, unlike polaritonic blockade, the blockade depth increases with increasing $N$. This blockade occurs due to a destructive interference between the two-photon emissions from different two-excitation eigenstates. More insight into this phenomena can be obtained by closely studying the two-excitation eigenstates as well as their contribution to two-photon emission. Note from Eqs. 5b and Photon blockade in weakly-driven cavity QED systems with many emitters that only the two-excitation eigenstates which have non-zero overlap with two-photons in the cavity (i.e. $\bra{\text{G}}a^{2}\ket{\phi_{i}^{(2)}}\neq 0$) have a non-zero contribution to $g^{(2)}(0)$. When all the emitters are identical, there are three such two-excitation eigenstates — $\ket{\phi^{(2)}_{\pm}}$ which have a probability of $1/4$ of having 2 photons in the cavity in the limit of $N\to\infty$, and $\ket{\phi^{(2)}_{0}}$ which when $N\to\infty$ has a probability of $1/2$ of having 2 photons in the cavity sup . Figure 3(b)-(c) shows the amplitude and phase of the contribution of these three eigenstates to equal-time two-photon emission (i.e. $\Gamma_{i}(\omega_{L})$ defined by Eq. Photon blockade in weakly-driven cavity QED systems with many emitters) at the blockade frequency — it can easily be seen that individually the eigenstates have significant two-photon emission, with the amplitude of emission being proportional to the probability of the eigenstate having 2 photons in the cavity mode. However, the two-photon emission from $\ket{\phi_{0}^{(2)}}$ is out of phase from the emission of $\ket{\phi_{\pm}^{(2)}}$, with the phase difference between them approaching $\pi$ as the number of emitters increases. This explains the interference based character of this blockade, as well as its dependence on $N$. The limit of $g^{(2)}(0;\omega_{L})$ as $N\to\infty$, given by Eq. 8 and plotted in Fig. 2(c), also shows a pronounced interference based antibunching, along with a disappearance of the polaritonic blockade. Moreover, the interference based blockade can be made deeper by increasing the detuning between the emitters and the cavity mode sup . We also note that the transmission $T(\omega_{L})$ at the interference based photon blockade is small — it scales in inverse proportion to $N^{2}$ sup — there thus exists a tradeoff between the purity of the single-photon state emitted by these systems and the brightness of the single-photon state.

While the previous analysis was primarily done under the assumption of identical emitters, emitters in practical systems are inhomogenously broadened i.e. they have slightly different transition frequencies. For solid-state color centers, the distribution of the emitter frequencies can be modelled as a normal distribution with standard deviation $\Delta\lesssim$ 20 GHz Radulaski et al. (2017c); Banks et al. (2018); Lukin et al. (2019). Results of a Monte–Carlo analysis on the transmission and equal-time two-photon correlation through the multi-emitter system are shown in Fig. 4(a) for resonant emitters and Fig. 4(b) for detuned emitters. We observe an emergence of a large number of very narrow linewidth dips in $g^{(2)}(0;\omega_{L})$ which correspond to the subradiant photon blockade that has been studied in CQED systems with two non-identical emitters Radulaski et al. (2017a). The occurrence of these dips is due to subradiant states. These highly entangled states that did not overlap with the cavity mode when the emitters were identical, now do overlap with the cavity mode and hence contribute to light emission from the system. These blockades reach very low $g^{(2)}(0;\omega_{L})$ values even for emitters that individually couple to the cavity only weakly. Moreover, for the resonant system, the distribution of the frequencies of the blockade dips ($\omega_{B}$) reveal that the spread in the frequencies of the subradiant photon blockade is of the order of the inhomogeneous broadening in the emitter frequencies, whereas the frequencies of polaritonic photon blockade are significantly more robust to inhomogeneous broadening in the emitter frequencies albeit with a much larger value of $g^{(2)}(0;\omega_{B})$. A similar trend is observed in the detuned system [Fig. 4(b)], with the polaritonic dip being robust to inhomogeneous broadening, and the interference-based dips (identified as the first dip which smoothly plateaus to 1 as $|\omega_{L}|\to\infty$) are much more sensitive to the inhomogeneous broadening while reaching very low $g^{(2)}(0;\omega_{B})$ values ($\sim 0-0.1$) similar to the identical-emitter system.

Finally, our study has uncovered two fundamental tradeoffs in multi-emitter CQED systems which can help inform future experiments and their suitability for quantum information processing applications. Firstly, for a given emitter-cavity coupling strength and cavity decay rate, there exists a tradeoff between the achievable transmission and the depth of photon blockade [measured as $g^{(2)}(0;\omega_{B})$]. Increasing either the cavity-emitter detuning or the number of emitters increases the depth of photon blockade, but also reduces transmission at the blockade frequency. Secondly, a tradeoff exists between the depth of achievable blockade and robustness of the blockade frequency to inhomogeneous broadening — polaritonic photon blockade, which typically has $g^{(2)}(0;\omega_{B})\sim 1$, is robust to inhomogenous broadening, while detuning the emitters from the cavity resonance can allow the multi-emitter CQED system to exhibit the interference-based photon blockade with significantly lower $g^{(2)}(0;\omega_{L})$. However, relying on destructive interference of two-photon emissions from various two-excitation eigenstates makes the blockade sensitive to the emitter frequencies. Moreover, the subradiant dips in the photon blockade also provide very low $g^{(2)}(0;\omega_{B})$, but the blockade frequencies $\omega_{B}$ are difficult to engineer without precise control over the emitter frequencies.