Dissipative dynamics of a solid-state qubit coupled to surface plasmons: from non-Markov to Markov regimes
We theoretically study the dissipative dynamics of a quantum emitter placed near the planar surface of a metal supporting surface plasmon excitations. The emitter-metal coupling regime can be tuned by varying some control parameters such as the qubit-surface separation and/or the detuning between characteristic frequencies. By using a Green’s function approach jointly with a time-convolutionless master equation, we analyze the non-Markovian dissipative features on the qubit time evolution in two cases of interest: i) an undriven qubit initially prepared in its excited state and ii) the evolution towards a steady-state for a system driven by a laser field. For weak to moderate qubit-metal coupling strength, and on timescales large compared to the surface plasmon oscillation time, a Markovian approximation for the master equation results to be adequate to describe the qubit main optical properties: surface enhancements of rate emission, optical spectra and time-dependent photon-photon correlation functions. The qubit decay shows a crossover passing from being purely dissipative for small qubit-surface distances to plasmon emission for larger separations.
Surface plasmons (SP) on metals, a topic extensively studied from many years agoRaether (1988), has recently received a strongly renewed attention due to significant advances in new experimental capabilities and numerical developmentsBarnes et al. (2003); García-Vidal et al. (2010). Great attention has been focused on the emerging field of quantum plasmonic with the goal of making devices for quantum information processingChang et al. (2006, 2007a) as single-photon transistor Chang et al. (2007b) or lasersOulton et al. (2009). As a requisite for this goal, a lot of effort has been devoted to get coherent coupling between plasmons and a quantum emitter made of a solid state qubit (SSQ) as, for instance, a quantum dot, a single NV center or a single molecule among others.
Strong coupling signatures of SSQ and SP have been experimentally observed both in spectroscopic as well as in time-resolved studies. In spectroscopy, the anticrossing between exciton and plasmon features in optical spectra have already been reported in systems such as organic semiconductorsBellessa et al. (2004), organic molecules placed in subwavelength hole arraysDintinger et al. (2005), metallic nanowiresAkimov et al. (2007), hybrid metal-semiconductor nanostructuresVasa et al. (2008) and even in carbon nanotubesBondarev et al. (2009). These anticrossings have been claimed to be a manifestation of strong coupling between SSQ and SP. On the other hand, ultrafast time-resolved signatures of strong coupling in SSQ-SP systems have also been reported. An enhancement of several orders of magnitude for the spontaneous emission rate in a time-resolved photoluminescence measurement on a InGaN heterostructure close to a silver thin layer has been reported Neogi et al. (2002); Okamoto et al. (2005). Additionally, recent experiments which operate simultaneously with both Raman and fluorescence signals coming from a single molecule in very close proximity to a metal surface have allowed the indirect measurement of ultrafast ( fs) dynamical features in such SSQ-SP systemGalloway et al. (2009).
On the theoretical side, some progress has been made to understand SSQ-SP coupling in different geometries using different approaches. The first attempts were devoted to computing the spontaneous emission rate enhancement of an atom near an absorbing surface as given by the atom self-energy in a near field limitYeung and Gustafson (1996). More recently, a hydrodynamic modelTrügler and Hohenester (2008) has been used to study a single molecule coupled to metallic nanoparticles. A transfer matrix method has also been used for simulating attenuated-reflection experimentsTame et al. (2008). However, the experimental setup which has risen the highest interest has been the quantum emitter coupled to a metallic nanowireAkimov et al. (2007); Kolesov et al. (2009), where the generation of a single optical plasmon can be achieved. Several theoretical studies on this system have considered the full quantum behavior of plasmon modes Klimov and Ducloy (2004); Chen et al. (2009); Dzsotjan et al. (2010). In particular, some attention has been devotedChen et al. (2009) to non-Markovian effects that can be important in the SSQ-nanowire system because the spectral density (carefully discussed in the present work) is highly structured due to a divergence at the edge of the SP density of states.
An open quantum system strongly coupled to a reservoir displays a complex dynamics which, in general, requires a description beyond simple Markovian theoriesBreuer et al. (1999); Rodríguez et al. (2008). In order to clarify the relevance of non-Markovian effects in SSQ-SP systems, we concentrate in a quantum emitter close to a planar surface of a dissipative metal, a system conceptually simpler than wires because it only has a single band of plasmons with a density of states having a singularity at a frequency . We study the properties of the light emitted by the system depicted in figure 1: a SSQ close to the planar metallic surface which supports a plasmon field as well as some dissipation mechanism. Strong SSQ-SP coupling could be expected when the qubit-surface distance is small compared with a typical length scale as, for instance, the wavelength of the emitted light. In order to understand the fundamental mechanisms of SSQ-SP strong coupling, we restrict ourselves to consider just a single quantum emitter. However, collective effects of many emitters coupled to the same plasmon field have been recently proposedSavasta et al. (2010) as responsible for the detection of the Rabi vacuum splitting in these systems.
We start by analyzing non-Markovian features in the SSQ-SP optical features by using a time-convolutionless approachBreuer and Petruccione (2002). We show that SSQ-SP dissipative evolution is determined by the rapidly varying structure of the reservoir spectral function at a frequency close to of the order of a few in a normal metal. Therefore, the timescale for non-Markovian effects reduces, at most, to a few hundreds times , i.e. typical times under a picosecond. Consequently, we conclude that the observation of non-Markovian signatures in SSQ-SP systems made with normal metals will indeed demand experimental set ups at the edge front of present state-of-the-art ultrafast technology. Beyond this short timescale, rate emissions stay constant and a Markovian approach becomes adequate for computing population dynamics, optical spectrum and second order coherence function. Here, both a Markovian and a non-Markovian analysis are applied to a SSQ-SP system under two different excitation schemes: firstly, we analyze the spontaneous emission of a SSQ initially prepared in the excited state. Secondly, the time evolution of a SSQ initially in its ground state and driven by means of a coherent laser field up to a stationary state is studied.
We take along this paper which is organized as follows: in Section II we introduce the Green’s tensor of the layered system and study its main properties, in particular the spectral density function. In Section III the time-convolutionless method is briefly reviewed and the non-Markovian effects on the SSQ-SP system dynamics are considered. In Section IV we use the Markovian limit to calculate the optical properties of the system. Finally, in Section V we summarize our results and draw some conclusions.
Ii Green’s tensor and spectral density
Electrodynamics of a dissipative medium is described by the Green’s tensor which satisfies the Maxwell equation:
We study the system depicted in figure 1: a SSQ in the upper-half space is embedded within a dielectric matrix with a dielectric function that can be taken as real and constant, , in the range of frequencies of interest. In the lower half-space, , a dissipative metal is characterized by a complex dielectric function that we take in a renormalized Drude approximation:
is the high-frequency limit of the metal dielectric function, is the bulk plasmon frequency and is the Landau damping constant.
The Green’s tensor for this layered geometry has two contributions:
where the first term is the free-space solution given byScheel and Buhmann (2008)
with , , , is the identity matrix and represents the dyadic product between the vectors . The second term in Eq.(II), , is the reflection Green’s tensor with on-site nonzero componentsScheel and Buhmann (2008)
where for .
All the parameters of the absorbing medium relevant to the SSQ dissipative dynamics appear in the Green’s tensor , where denotes the SSQ location. The action of the absorbing medium on a SSQ with dipole moment , is completely described by the spectral density
which is related with the qubit-environment coupling and the density of states of the environment by means of . In order to compute the spectral function of a representative system, we use parameters for silver in the range of frequencies of interest where , and Johnson and Christy (1972), while for the dielectric constant at the upper-half-space we take . Since the density of states has a singularity at non-Markovian effects associated with the structured reservoir can be expected to occur around that inverse frequency. Thus, we consider a SSQ with a dipole oriented along the -direction and an energy splitting slightly detuned with respect to the singularity at the SP edge . In particular, we calculate for where the frequencies have been normalized to the bulk plasmon frequency and .
In figure 2 we plot, for different values of the qubit-surface renormalized separation , the spectral density in units of the spontaneous decay rate of the SSQ in free space . Two main results can be identified:
i) A strong reduction of when the qubit gets farther from the surface. This is a consequence of the exponential reduction of the coupling, as a function of , as indicated in Eq.(4).
ii) For small separations is highly structured presenting a strong peak close to the frequency (vertical line in Fig.(2)). This is a consequence of the singularity of the density of SP states at small detunings . For increasing separation, becomes much smoother and a reduced maximum separates from .
Iii Time convolutionless method and non-Markovian effects
The spectral density for the SSQ-SP system computed within a purely classical scheme, can be used within a quantum framework describing the dissipative dynamics of an open quantum system. As it is well known, when the time correlation between the system and the environment decay much faster than the characteristic inverse dissipation rate, memory effects can be neglected in the so called Markovian approximation, and the observables of the system are given by analytical expressions. However, this timescale does not represent the only one relevant to determine the system’s evolution. When the environment correlation time is longer than the inverse rate of the system-bath coupling, new physics can arise at very short times. A SSQ in close proximity to a dissipative metal surface supporting SP modes sees a highly structured reservoir to which it might be strongly coupled. Thus, one can expect non-Markovian effects to be significant in the qubit time evolution. Many worksChen et al. (2009); Breuer and Petruccione (2002); Lewenstein and Mossberg (1988); Imamoglu (1994); Stenius and Imamoglu (1996); Wilson-Rae and Imamoglu (2002); Breuer and Vacchini (2008); Ma and John (2009); Burkard (2009); Chruscinski and Kossakowski (2010); Kaer et al. (2010); Roy and John (2010); Haikka and Maniscalco (); Vacchini and Breuer (2010). have been devoted to treat this problem at different levels of precision and sophistication. Here, we chose to work within a time-convolutionless (TCL) frameworkBreuer and Petruccione (2002) to capture non-Markovian effects to the lowest order in the SSQ-SP coupling strength. This method has already been applied to consider the spontaneous decay of a two-level system coupled to a general structured reservoirBreuer and Petruccione (2002). For SSQ-SP systems, the strong variation of occurring for frequencies close to , implies that dynamical features in time scales from femtoseconds to picoseconds are expected.
iii.1 Time-convolutionless method
What is of interest for us of the TCL method can be sketched as followsBreuer and Petruccione (2002). It consists in transforming the typical non-Markovian integro-differential equation for the reduced density matrix into a local in time evolution equation by making use of a power expansion technique of the Nakajima-Zwanzig type. As a result, a master equation for a qubit is obtained with time dependent decay rate and Lamb shift :
Time-dependent rates can be calculated within a perturbative expansion. In order to calculate them, a first step is to Fourier transform the spectral density:
The lowest order non-Markovian effects, i.e. the so-called post-Markovian behavior, are contained in the second order contributions to and given by:
where and denote real and imaginary parts, respectively.
iii.2 SSQ spontaneous decay
We start by considering the situation where an undriven SSQ is prepared in its excited state from which it decays emitting a photon to the vacuum or to the SP field. There are two possible situations depending on the sign of the detuning between SP and the SSQ renormalized frequencies. The time evolution of the excited state population, is given byBreuer and Petruccione (2002)
with and the decay rate obtained from Eq.(9).
At this stage, we want to analyze the importance of memory effects. Therefore, in the calculations reported in this subsection we do not include the free space part of the Green’s tensor, Eq.(II), which involves a much slower dynamics than the one associated to the reflection contribution , as given in Eq.(4).
Figure 3 shows for different positive detunings, i.e. when the SSQ is resonant with the continuum stripe of SP modes(). In order to have a highly structured reservoir, we have taken a small qubit-surface separation, (very close to the blue line spectral density in figure 2). For large detuning , oscillates around a constant (Markovian) value. At some time intervals, takes on negative values, a fact that tends to slow down the decay of the excited state population. Physically, this behavior can be understood as due to the back-action of the reservoir on the SSQ re-exciting it. When the SSQ splitting energy gets closer in resonance with the top SP energy, e.g. , the oscillations slow down, the envelope of the oscillatory decay rate becomes smaller and the negative parts of the decay rate tend to vanish producing only few oscillations before the spontaneous decay becomes almost exponential. For further smaller detunings, e.g. , the SSQ sees an even more structured reservoir with a decay rate modifying completely its behavior: its value increases considerably and it just oscillates slightly around a large positive value, producing a monotonous decay of the SSQ excited state population. This last result indicates that the second-order TCL method is approaching its limit of validity. Physically, this behavior is a consequence of the fast transfer of the SSQ energy to the SP field, an energy which is irreversibly lost and the quantum emitter ends up in its ground state.
On the other hand, a physically different situation occurs for negative detunings, i.e. when the SSQ energy falls within the gap where no SP states exist. Figure 4 is similar to figure 3 with the same and detunings with just a change of sign with respect to the ones in figure 3. When is far above the edge of the SP dispersion, the SSQ basically remains in its excited state as the spectral density for these energies is practically zero, so there are not accessible SP modes to which decay into. Nevertheless, as shown in the inset of figure 3, one may observe some non-Markovian oscillations for very short times. When the SSQ energy is tuned closer in resonance with , e.g. , a very interesting phenomenon occurs: the emitter undergoes the so-called fractional decay in which the population tends to a finite, non-zero, value at long times. Including the effect of free space emission () produces a decay of in a time scale () much larger than that of the figure. As the emission frequency is further scanned closer to the band edge, the behavior changes again dramatically: the decay rate, instead of oscillating around zero, oscillates slightly around a positive value, which results into an irreversible exponential decay.
A very important result must be drawn from all these results: the timescale of these non-Markovian effects is a few hundred times . For normal metals, this means times below . Beyond that short timescale, becomes constant just at the value it takes in a Markovian description as discussed in the following Section IV.
iii.3 Coherently driven SSQ
After having studied the effect of the structured reservoir on the SSQ spontaneous emission, now we turn our attention to the case where the system is coherently driven by a laser field. The SSQ emits and absorbs photons simultaneously. The system can achieve a stationary state in which light absorbed from the laser ends being transferred to plasmons. The laser can be treated as a classical field included in the, local in time, coherent part of the master equation (III.1) through the Hamiltonian . The Rabi frequency measures the strength of the coupling of the SSQ dipole with the laser field. The time evolution of the SSQ excited state population is obtained from the solution of Eq.(7) represented in a rotating frame at the laser frequency . is plotted in figure 5, for the case of perfect resonance between the laser and the SSQ. We assume that remains unaffected by the laser field, so that the main effect of the laser is to bring the system to a stationary state in a timescale which is similar to that of the spontaneous decay discussed in the previous subsection.
Iv Optical properties in the Markov approximation
The results of the previous section show that a SSQ presents significant non-Markovian effects in a timescale a couple of orders of magnitude larger than . Hereafter, we concentrate in the usual case of having a resolution in time larger than a picosecond. Then, the system can be described by a Markovian dynamics given by a Master equation like Eq. (III.1) but now with a Lamb-shift and a constant decay rate including both the free space and the reflection contributions to the dissipative dynamics. Since the only effect of the Lamb-shift is a constant energy shift, from now on we do not pay attention to it.
iv.1 Decay rate
The decay rate in the Markovian regime coincides with the long time limit of , , allowing to identify as simply the spectral function at the SSQ frequency:
where the two terms corresponding to the free space () and the reflection part () of the dissipative dynamics are included in .
The SSQ decay rate to the SP reservoir of the metallic surface, , is shown in figure 6, in a parameter space , where lighter blues correspond to high decay rates with a variation of four orders of magnitude between the highest and the lowest values. In order to discuss these results, it is better to plot vs. the SSQ-interface distance (in logarithmic scales) for different SSQ energies as depicted in figure 7. It is worth noticing two important features: First, at a large value the assisted decay rate is smaller than the vacuum one for a certain range of distances, due to the fact that the reflected part of the Green’s tensor is interfering destructively with the direct one. This effect is evident when the SSQ frequency approaches while it moves to larger separations , and it weakens, when is far from the SP band edge.
Second feature is even more important. When the SSQ-surface distance varies, the decay rate suffers a transition from a behavior to a much slower decrease. In order to understand the physics behind this behavior, we analyze the much simpler case and , i.e. a rate for losses one order of magnitude smaller than the one we have used in previous cases. Large dots in figure 8 depict the decay rate as calculated with the full Green’s tensor for two different SSQ frequencies. At very short distances the decay rate is highly enhanced with respect to and it shows a dependence, which can be obtained (dotted lines in the figure) by means of a model which only contains non-radiative processes as the creation of electron-hole pairs in the metallic medium. A crossover to a different behavior occurs at a critical distance, which depends on the SSQ-SP detuning . Beyond this critical distance, a single plasmon pole approximation (dashed lines in the figure) in the Green tensor in Eq.(12) is able to reproduce the numerical result with the complete Green’s tensor. In other words, for separations beyond the crossover, the SSQ decay just produces the emission of surface plasmons while other metallic losses become negligible. In the case of the SSQ embedded in a dielectric or a metal with very large losses this crossover can be hindered by other physical effects such as those coming from local dissipative circulating currentsDzsotjan et al. (2010). As the SP channel contribution increases when the SSQ energy gets closer to the plasmon band edge, this crossover effect can be exploited in designing coherent plasmonic devicesVasa et al. (2008).
iv.2 Spectrum and correlation functions
Hereafter, we consider the case in which the system is coherently driven by a laser so that the system reaches a stationary state with partial occupation of the two levels of the SSQ. The master equation in the rotating frame at the laser frequency, , is:
with . The problem reduces to the study of the SSQ resonance fluorescence Walls and Milburn (1994) near the planar surface of a dissipative metal. The vacuum resonance fluorescence has been widely studied in the literature in the case of resonant excitation for which an analytical solution for the population, spectrum and second order coherence function exists. Here, we extend such analysis to the non-resonant case and pay special attention to the effect of the SP reservoir. We present our analysis in the way the experiments can be performed either by tuning the laser resonantly with the SSQ energy and varying the laser intensity or by fixing the laser intensity and scanning the laser frequency. Figure 9 explores the former alternative whereas in figure 10 we consider the later one.
From the master equation (IV.2) one may derive the equations of motion for the expectation values , and arriving to the well-known optical Bloch equations(OBE). The steady-state solution for the excited state population is:
In general, the OBE must be solved numerically in order to get the population dynamics , except for the resonant case () for which an analytical solution exists:
where , labeled as Rabi splitting at resonanceKhitrova et al. (2006), is the parameter characterizing the strength of the effective coupling. There is a threshold for the laser intensity at . For below this threshold, the solutions are monotonically decaying functions of time so that the system is said to be in the weak coupling (WC) regime. Above that threshold, the populations exhibit oscillations, and the system is said to be in the strong coupling (SC) regime. In figure 9(a) we plot the real part of in the parameter space . The bluest region corresponds to , which means that the Rabi splitting at resonance is purely imaginary and consequently the system is in the WC regime. For the regions in which blue becomes lighter, the values correspond to positive and higher values of .
In order to clarify these results, we show the population dynamics in figure 9(b) for three different points highlighted in part (a) of the same figure: the green curve corresponds to a configuration where the laser is weakly coupled to the system, so no oscillations are observed in the population. The red point corresponds to the region of transition from WC to SC where just one clear oscillation occurs before practically arriving to the steady state. Finally the blue point corresponds to a configuration where the laser is strongly coupled to the SSQ and several oscillations are observed before the steady state is achieved.
Another experimental alternative is to keep constant and vary the laser frequency as it is plotted in figure 10. In this case, the laser is out of resonance and the Rabi splitting must be redefined asKhitrova et al. (2006):
Figure 10(a) shows in the parameter space . The population dynamics is shown in panel (b) of the same figure, where one can observe the transition from strong coupling (oscillations, solid blue) to weak coupling (monotonous in time, green dotted) for non-resonant excitation of the SSQ.
A clear manifestation of the transition from WC to SC appears in the optical spectrum at the stationary regime. It can be calculated through the Wiener-Khintchine formula:
The calculation of the two-time correlator in Eq. (17) requires the use of the Quantum Regression TheoremWalls and Milburn (1994) by using the steady state populations as initial values for the second time dynamics. In the resonance fluorescence problem there are always two main contributions to the spectra: the Rayleigh scattering coherent part and the one coming from the incoherent scattering. The former contribution is just a delta function at that we ignore in our results. We are mainly interested in the contribution coming from the inelastic scattering which is shown in figures 9(c) and 10(c) for the resonant and non-resonant cases, respectively. As it occurred with the population, under resonant excitation the spectrum admits an analytical expression:
In the WC regime (green point) , the light emitted simply produces a Lorentzian curve peaked about with linewidth . For the intermediate regime (red point), on top of the Lorentzian peaked at the qubit frequency, some satellites start to appear at the laser Rabi frequency . For a strong-driving field situation these two sidebands appear at frequencies . For the non-resonant case, the threshold changes but the behavior remains qualitatively unaffected: even though the dressed state structure is slightly modified by the detuning, at the end, a triplet is obtained in the resonant case. The existence of this Mollow’s triplet is a manifestation of the SC of the laser to the SSQ-SP system.
Another magnitude of experimental interest is the second order coherence function:
with correlation functions
We evaluate these magnitudes at the stationary state. In the resonant case, the second order coherence function can be analytically expressed as:
It clearly exhibits photon anti-bunching: . Figure 9(d) shows for zero detuning for the three different points considered above for the other magnitudes. Apart from the antibunching, the case of SC shows a remarkable oscillatory behavior. Once more, qualitatively similar results are obtained with laser-SSQ detuning as shown in figure 10(d).
In this work we have studied the properties of the coupling of light with a SSQ, embedded in a dielectric, in the presence of a SP field supported in the interface between this dielectric matrix and a dissipative metal. Using a time-convolutionless approach, we provide a theoretical description of the non-Markovian features for this kind of systems and discuss its relevance in possible observations. In a spontaneous decay situation, different behaviors occur depending on both the sign and the absolute value of the SSQ-SP detuning: from a monotonous (almost exponential) decay for very small detunings, to population oscillations due to reabsorptions in the case of positive detuning. Even fractional decays can be observed, when negative detunings are present and the SSQ energy is not too close to the SP edge band.
In experimental situations, non-Markovian features can be hard to detect due to practical difficulties in getting the adequate time-resolution. Therefore, we have also considered a Markov approximation to study the electrodynamics of the SSQ coupled to a reservoir of SP modes. The whole information of the planar metallic surface is embedded in the decay rate constant, which depends on both the SSQ frequency and distance to the surface. The excitation of the system by a laser allows the existence of a steady state as well as the analysis of different measurable properties of the SSQ-SP system as, for instance, surface enhancements of rate emission, optical spectra and time-dependent photon-photon correlation functions. Our main result is that the qubit decay shows a crossover passing from being purely dissipative for small qubit-surface distances to plasmon emission for larger separations. As the SP emission channel increases when the SSQ energy gets closer to the plasmon band edge, this crossover effect can be exploited in designing coherent plasmonic devices. Our next task, beyond the scope of the present work, is to treat the plasmonic part of the system not as a reservoir but as an ingredient coherently coupled to one or more SSQsVasa et al. (2008).
Helpful discussions with F.J. Garcia-Vidal and L. Martin-Moreno are acknowledged. Work supported in part by the Spanish MEC under contracts Consolider-Ingenio2010 QOIT-CSD2006-00019 and MAT2008-01555, and by the CAM under contract S-0505/ESP-0200. A.G-T acknowledges funding from AP2008-00101 grant of Spanish Education Ministry. L.Q. was partially supported by Faculty of Sciences-Research Funds 2009 (UniAndes). F.J.R. was partially supported by Banco de la Republica (Colombia).
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