Quantum correlations of light and matter through environmental transitions

One aspect of solid-state photonic devices that distinguishes them from their atomic counterparts is the unavoidable interaction between system excitations and lattice vibrations of the host material. This coupling may lead to surprising departures in emission properties between solid-state and atomic systems. Here we predict a striking and important example of such an effect. We show that in solid-state cavity quantum electrodynamics, interactions with the host vibrational environment can generate quantum cavity-emitter correlations in regimes that are semiclassical for atomic systems. This behaviour, which can be probed experimentally through the cavity emission properties, heralds a failure of the semiclassical approach in the solid-state, and challenges the notion that coupling to a thermal bath supports a more classical description of the system. Furthermore, it does not rely on the spectral details of the host environment under consideration and is robust to changes in temperature. It should thus be of relevance to a wide variety of photonic devices.

The diversity of systems studied in cavity quantum electrodynamics (CQED) places the subject at the heart of many prospective quantum and classical technologies. Examples include single photon sources [1,2], ultrafast optical switches [3,4], and quantum gates [5][6][7], which require the development of robust, scalable, and potentially strongly coupled emitter-cavity systems. Though the quantum strong coupling (QSC) limit and beyond-in which the system eigenstates become lightmatter entangled-have now been attained in the microwave regime [8,9], it remains technically demanding to manufacture optical cavities of sufficiently high quality (Q) factor to unambiguously demonstrate QSC phenomena. Example systems in which great strides have recently been made towards this goal include self-assembled quantum dots (QDs) within optical nano-and microcavites [10][11][12][13][14]. Here, small mode volumes can readily be obtained, resulting in potentially large Q-factors and cavity coupling strengths that are substantial in comparison to the emitter decay rate. In conjunction with their solid state nature, this makes QD-cavity systems excellent candidates for future technological applications.
Nevertheless, it still remains a challenging endeavour to reach the QSC regime due to significant cavity losses [15,16]. The broad cavity lineshape that results masks contributions from higher order dressed states, thus placing the system in an intermediate coupling regime that can be described using semiclassical techniques [17]. In addition to interactions with external electromagnetic fields, many CQED systems are also in contact their host (e.g. thermal) environment; for example, in QDs this influence is often dominated by acoustic phonons [18,19]. In order to explore the effect that such couplings have on the system optical emission, it is necessary to modify the standard quantum optics treatments [20], which may lead to significant departures from atomic-like behaviour [21][22][23][24][25][26][27][28][29][30][31].
Here, we demonstrate that even for a lossy cavity within the semiclassical regime, it is possible to observe signatures of the joint emitter-cavity eigenstates via transitions induced by the host environment. Specifically, we explore the effect of a thermal bath on the emission properties of a CQED system in several important parameter regimes, corresponding to the semiclassical [15,16], Fano [32,33], and QSC limits [34,35]. We show that the quantum mechanical nature of the host environment results in optical emission that is observably sensitive to the joint eigenstructure of the cavity and emitter, even if the optical transitions in its absence are not. This behaviour, which may be probed experimentally through asymmetries in both the cavity reflectivity and emission spectra, also challenges the notion that the addition of a thermal environment should simply decohere our system to a more classical effective description.
We consider a driven cavity coupled to a two level emitter (TLE), as shown schematically in Fig. 1(a). This is a model of wide importance, though later we shall also consider specific parameters relevant to QD-microcavity systems to provide experimental context. Within a frame rotating at the laser driving frequency ω L , and after a rotating-wave approximation, the system Hamiltonian is Here, ν = ω c − ω L and δ = ω X − ω L are, respectively, the cavity-laser and emitter-laser detunings, g is the emittercavity interaction strength, and η is the cavity-laser coupling. σ † = |X 0| and σ = |0 X| are raising and lowering operators for the TLE, while a † and a are creation and annihilation operators for the cavity mode.
To highlight the qualitative changes in behaviour brought about by an explicit inclusion of the thermal environment, we model its impact on the system in two ways. The first simply assumes a phenomenological pure dephasing of the TLE coherences, which may be approximated by a semiclassical description in the appropriate arXiv:1412.6044v2 [cond-mat.mes-hall] 5 Jan 2015 The cavity is one sided, driven by a continuous-wave classical laser of frequency ωL with strength η, and loses excitation through the top with rate κ, and the sides with rate κs. The TLE decay rate is γ, and the TLE-cavity coupling strength is g. (b) The first rung of the dressed state ladder, i.e. the lowest eigenstates of the coupled TLE-cavity system, in the absence of dissipation and driving.
limits. We shall show that such a procedure is generally inadequate, regardless of whether the semiclassical limit is taken or not. Instead, it is necessary to follow a second, more rigorous approach, and retain a detailed, quantum mechanical description of the thermal environment to properly understand its influence on the system dynamics. This will be achieved through a polaron representation master equation [22,[24][25][26]30], which can then be related to the cavity optical emission through the input-output formalism [36,37]. Here we consider the environment to be described by a collection of harmonic oscillators, with free Hamiltonian is the creation (annihilation) operator for mode k. The TLE-environment interaction is given by and its effect on the system may be described by the spectral density, which we take to have a super-Ohmic form appropriate to bulk acoustic phonon processes, J(ν) = k g k δ(ν − ν k ) = αν 3 e −ν 2 /Λ 2 , with α the coupling strength and Λ a high frequency cut-off [18,19]. Applying the polaron transformation, to the full Hamiltonian H = H S + H B + H I allows us to derive a master equation valid beyond the weak TLE-environment coupling regime [38]. This unitary generates a displaced representation of the thermal bath, thus removing the linear coupling term of Eq. (2) to givẽ Here, the TLE-cavity coupling strength has been renormalised by the average displacement of the oscillator environment, g → gB, with B = tr (B ± ρ th ) denoting the expectation of the displacement operators Moving into the interaction picture with respect to the coupled TLE-cavity Hamiltonian in the polaron frame, H s = δσ † σ +gB σ † a + σa † +η(a † +a)+νa † a, we derive a master equation for their reduced state, ρ(t), by tracing out the environment within a second-order Born-Markov approximation. In essence, this procedure may be thought of as a perturbative expansion about the parameter g/Λ [25], and is thus non-perturbative in the TLEenvironment coupling strength. Crucially, as the cut-off Λ is usually much larger than any other energy scale in the problem (not just g) and the renormalised coupling term gB σ † a + σa † appears explicitly in the system HamiltonianH S , this expansion does not preclude the exploration of TLE-cavity dressed states. It simply enables us to move beyond weak TLE-environment couplings by retaining multi-boson processes [39]. Including photon emission from both the TLE and the cavity, also within a Born-Markov approximation and assuming the radiation field to have a flat spectrum [20,40,41], our master equation takes the Schrödingier picture form [25,42]: Here, L x [ρ] = 2xρx † −{x † x, ρ}, γ is the TLE spontaneous emission rate, and κ (κ s ) is the photon loss rates from the top (sides) of the cavity. The superoperator , accounts for interactions with the thermal bath as just described. When referring to the phenomenological treatment in which the bath is assumed to lead only to pure dephasing of the TLE coherences, the Hamiltonian in Eq. (4) reduces to that given in Eq.
, where Γ defines the pure dephasing rate.
Having outlined our theoretical approach, let us focus our analysis first on what we shall term the intermediate coupling regime, as it has particular experimental relevance to QDs in optical microcavities [15,16]. Here, κ + κ s g > γ, such that cavity leakage is significant. For an atomic system, this regime is characterised by a symmetric double peak structure in both the steady state TLE population and cavity occupation on scan-  ning a weak driving field through resonance. This can be seen by the dashed curves in Figs. 2(a) and (b) which treat the TLE-cavity coupling fully quantum mechanically (through H s ), but include the host environment only through a phenomenological pure dephasing process with rate Γ. The peaks lie at resonances of the first two eigenstates of the TLE-cavity system, that is, the first rung of the dressed state ladder [see Fig. 1(b)]. However, unlike the true QSC regime (in which g > κ + κ s , γ), the broad cavity transition obscures contributions from higher order dressed states, allowing an effective semiclassical description to be derived [16,17]. The double peak structure may then be interpreted simply as a normal mode splitting between two classical oscillators. This can be shown by considering the relevant optical Bloch equations, obtained from the pure dephasing master equation: with Ô = tr(Ôρ). Applying a mean-field approximation between the cavity and TLE, such that σ z a ≈ σ z a , neglects any quantum correlations accumulated between them. In the weak driving limit, we may further assume that on average the TLE remains close to its ground state, such that σ z ≈ −1 [17,33]. For a sufficiently lossy cavity, the resulting semiclassical theory agrees perfectly with the pure dephasing approach, as also demonstrated in Fig. 2. Discrepancies become apparent, however, when comparing to the full (polaron) master equation which treats both the TLE-cavity coupling and the thermal environment quantum mechanically. We now see a shift in the peak positions due to bath renormalisation of the TLE-cavity coupling, and asymmetries in the TLE and cavity populations that were entirely absent in either the pure dephasing or semiclassical calculations. Clearly, such simplified approaches are unable to properly capture the physics of our system in the presence of the host environment. Importantly, this implies that the semiclassical description breaks down here even within the lossy cavity regime.
To relate population asymmetries directly to observable experimental signatures we note that in most emitter-cavity setups it is not the system expectation values that are directly probed, but rather the cavity emission. For example, we can obtain the cavity reflectivity spectra from Eq. (4) by way of the inputoutput formalism [36,37], which draws a formal connection between system operators and those of the cavity emitted field. Heisenberg-Langevin equations are first used to define collective field operators for both the input,â in , and output,â out through the top of the cavity, leading to the famous input-output relation [36,37]: It is then straightforward to derive an expression for the complex cavity reflectivity in the steady state: r = â out / â in = 1 − i(κ/η) â . Here, R = |r| gives the reflectivity coefficient and φ = arg(r) is the phase shift of the reflected light. Figs. 2(c) and (d) compare the cavity reflectivity spectra and associated phase shift for the same intermediate coupling parameters as the populations just described. We see that while the semiclassical theory once again agrees with the pure dephasing master equation, capturing the characteristic normal mode splitting as expected, in the polaron treatment the suppression of the cavity and TLE populations for ω L − ω c > 0 is exhibited also in asymmetries within the cavity reflectivity and phase shift. We can attribute these asymmetric features to the quantum mechanical nature of the host environment, which plays a vital role in determining the system dynamics. In Fig. 2, where we have chosen parameters relevant to the acoustic phonon environment common in QD-microcavity systems, bath-induced transitions occur on a faster timescale than other coherence destroying processes, i.e. cavity leakage and spontaneous emission. The phonon bath is thus sensitive to coherence shared between the cavity and QD emitter, with the result that it mediates transitions directly between the emitter-cavity dressed states. Specifically, when we tune the driving field to the upper dressed state resonance (ω L −ω c ≈ 1 ps −1 ), phonon emission allows population to transfer from the upper to the lower dressed state, with an associated loss of energy to the environment. This leads to a suppression of the upper dressed state popu- lation and also of the reflected light. Provided that the temperature is not too high, the inverse process, which raises population from the lower to the upper dressed state by phonon absorption, is comparatively weaker. Thus, in Fig. 2, we see that for negative detunings between the cavity and driving field (ω L − ω c < 0), the lineshapes of the polaron and semiclassical theories are very similar. The resulting asymmetries herald a failure of the semiclassical theory, which by definition cannot be sensitive to the coherence shared between the TLE and cavity. This is a somewhat counterintuitive point. Naively one might think of phonons purely as a source of decoherence, that is, as giving rise to processes that should push the system towards a more classical description. However, here we see that the sensitivity of the host environment to quantum correlations in fact results in the breakdown of the semiclassical description of our system, and we must instead reinstate a quantum mechanical explanation.
To confirm that phonon transitions do indeed drive population between the dressed states, we can look at the spectra of photons emitted from the cavity when driving either the lower or upper dressed state resonantly. In the former case, we should expect emission around the lower state only, while in the latter, bath-mediated transitions should lead to emission from both the lower and upper states, in stark contrast to the conventional expectation. In Fig. 3 we plot the cavity incoherent emission spectrum, inc (τ ) = lim t→∞ a † (t + τ )a(t) − | a † ss | 2 obtained from our master equation using the quantum regression theorem [20]. Here a † ss is the steady state expectation value of the cavity operator, which describes coherent emission . Fig 3(a) shows the expected resonant response from a normal mode, with a single dominant peak around the driven lower dressed state transition. The spectra calculated either with the phenomenological (pure dephasing) or polaron master equations remain largely unchanged, with little noticeable effect of including the thermal environment in a rigorous manner. However, the spectra become markedly different when we drive the upper dressed state resonance, as shown in Fig. 3(b). While the phenomenological theory again shows a single dominant peak, now at the upper dressed state resonance and symmetric to the previous case, in the polaron theory we see the emergence of an additional feature. In fact, phonon-mediated population transfer leads here to a strong suppression of the peak around the upper dressed state, and correspondingly to substantial emission at the lower dressed state frequency. Hence, by measuring the spectra of light emitted from the cavity, we can unambiguously demonstrate the presence of bath-mediated transitions between the joint eigenstates of the emitter-cavity system, evidencing also the quantum mechanical nature of the host environment.
The sensitivity of the environment to the underlying emitter-cavity eigenstructure is not restricted to the intermediate coupling regime, but is in fact present over wider regions of parameters space. Fig 4 shows two such important cases, corresponding to the classical Fano (a,b), and the QSC (c,d) limits. The Fano regime, for which κ + κ s g γ, is characterised by a sharp peak in the cavity emission at the TLE resonance, which is the result of classical interference between two competing decay pathways [32]. Though less pronounced than in the intermediate coupling regime, environment induced asymmetries become apparent when comparing the cavity lineshape for an emitter tuned above the cavity resonance (ω c − ω X = 0.5) and below (ω c − ω X = −0.5).
In the QSC limit, the coupling strength g becomes the dominant energy scale, g κ+κ s , γ, which allows contributions from higher order dressed states to be resolved. Additionally, the long cavity lifetime results in phonon influences becoming particularly significant. This is especially true at strong driving, shown in Fig. 4(d), where in the polaron theory the asymmetric broadening is so pronounced that it prevents us from resolving any higher order contributions to the upper dressed state resonance.
In summary, we have shown that the presence of a thermal environment allows one to probe the dressed states of a CQED system within an otherwise semiclassical regime. Sensitivity of the environment to the coherence shared between the TLE and cavity leads to direct transitions between their joint eigenstates, and consequently a breakdown of the semiclassical approach. The resulting observable effects persist over a broad range of parameters, encompassing the Fano, intermediate, and strong coupling limits. They should thus be applicable to a variety of CQED systems, such as QDs in micropillar and photonic crystal cavities [3,[10][11][12][13][14][15][16]43], diamond colour centres [44][45][46], and superconducting circuits [8,9,35].