Open Access
Add to BibSonomy
Abstract
The creation of nanoscale lasers that operate above a coherent threshold is a challenging problem. We propose a way to circumvent this issue using systems in which a strong coupling regime is achieved between the light and the active medium. In the strong coupling regime, energy oscillations take place between the EM field in the cavity and the atoms. We show that by applying appropriate time modulation to the pumping, it is possible to control these energy oscillations in such a way that coherence in the laser system appears below the lasing threshold. In this approach, the radiation linewidth is two orders of magnitude smaller than the linewidth of a conventional laser for the same photon number. In addition, the second order coherence function of the output radiation is reduced from two to one before the system reaches a positive population inversion. Our results pave the way for the creation of nanoscale sources of coherent radiation that can operate below the lasing threshold.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The creation of nano-sized sources of coherent radiation is a challenging problem. The minimum possible size of dielectric lasers is restricted by the half-wavelength of the generated light. Using plasmonic nanostructures as laser resonators enables us to overcome the size limitations arising due to diffraction limits [1–3]. However, the sub-wavelength localization of the EM field in the plasmonic lasers results in large ohmic losses. To achieve lasing, it is necessary to overcome these losses, which requires high gain in the active medium and high pumping power. However, high pumping power leads to heating and degradation of the active medium, which negatively impacts the performance of nanolasers [4,5]. Moreover, the high losses in plasmonic lasers prevent a reduction in the laser linewidth.
On the other hand, a reduction in the size of a plasmonic laser results in strong localization of the EM field in the plasmonic structures, and consequently an increase in the coupling constant between the EM field and the active atoms. If the coupling constant exceeds the dissipation rate, a system moves to a strong coupling regime [6–12]. In this strong coupling regime, the interaction between the EM field and the atoms or molecules results in the formation of hybrid polaritonic states and the appearance of Rabi splitting in the system spectrum [7–11]. This formation of hybrid states leads to changes in the optical [13,14], electrical [15,16] and chemical [17–19] properties of the medium, which can be used, for example, to control the chemical reactions [18–20] or to tune the electrical conductivity [15] and work function [16]. Strongly coupled light–matter systems show promise for the development of polariton circuits [21], single-photon switches [22,23] and all-optical logic elements [13]. In addition, these systems can serve as building blocks for quantum computers [24–26] and detectors [26].
In a laser system with a single-mode cavity, a strong coupling regime can be achieved with a negative population inversion in the active medium [27,28]. In the strong coupling regime, energy oscillations between the cavity EM field and atoms occur. During oscillations, there are time intervals in which energy flows from the atoms to EM field, resulting in light amplification in the cavity. Notably, this occurs for a negative population inversion of the atoms. Recently, it was demonstrated [27] that time-modulated pumping enables these energy oscillations to be controlled and, as a result, light amplification to be achieved even for a negative population inversion of the active medium [27]. However, light amplification by itself does not guarantee coherence of the emitted light; to achieve coherence in the sense of the first- and second-order coherence functions, it is necessary to overcome incoherent contributions from noise [29,30]. In this context, obtaining a coherent signal below the lasing threshold by utilizing the properties of the system in the strong-coupling regime is important problem.
In this paper, we consider the coherent properties of a laser system in the strong coupling regime with time-modulated pumping. We show that two peaks appear in the system spectrum at the frequencies determined by Rabi splitting. The linewidths of these peaks narrow with an increase in the pump rate. Surprisingly, the radiation linewidth in this regime is two orders of magnitude smaller than the Schawlow-Townes linewidth of a conventional laser at the same photon number, allowing us to achieve a narrow linewidth for a small number of photons. Notably, this is achieved at negative population inversions, where conventional lasers do not provide coherent radiation. We demonstrate that the second-order coherence function of radiation from the system changes from two to one with an increase in pumping, meaning that the radiation becomes coherent below the lasing threshold. A reduction in the pumping power required to produce coherent light can solve the problems of overheating and degradation of the active medium in nanolasers.
2. Model of a single-mode parametric laser without inversion
In this section, we examine the behavior of a system consisting of a single-mode cavity filled with two-level atoms that are subject to periodic variation in pumping. The cavity mode and the atoms form a strongly-coupled system. This can be achieved, for example, in cavities based on photonic [7] or plasmonic [12,31] structures.
To describe this cavity-atom system, we employ the Maxwell-Bloch equations with noise terms [29,32]:
Here, $a$ is the electric field amplitude in the cavity mode, and $\sigma$ and $D$ are the average polarization and population inversion of the atoms. ${\omega _0}$ is the transition frequency of the atoms, which coincides with the resonator frequency. ${N_{at}} = {10^6}$ is the number of atoms in the cavity; ${\Omega _R}$ is the coupling constant between an individual active atom and the cavity; ${\gamma _a}$ is the loss rate of the cavity; ${\gamma _D}$ is the longitude relaxation rate of the active atoms; ${\gamma _P}\left ( t \right )$ is the pump rate, which varies in time; and ${\gamma _\sigma }$ is the transverse relaxation rate of the atomic dipole moment, which is determined by other relaxation rates as ${\gamma _\sigma }(t) = \left ( {{\gamma _P}(t) + {\gamma _D}} \right )/2 + {\gamma _{deph}}$, where ${\gamma _{deph}}$ is the dephasing rate of the atom’s polarization. ${F_a}(t)$, ${F_\sigma }(t)$, and ${F_D}(t)$ are the noise terms for the respective variables. These noise terms enable us to take into account the spontaneous emission processes in the laser [29,33]. For the noise terms, we use the following correlations properties: $\left \langle {F_a^ * (t){F_a}(t)} \right \rangle = 0$, $\left \langle {F_D^ * (t){F_D}(t)} \right \rangle = 0$, $\left \langle {F_\sigma ^ * (t){F_\sigma }(t)} \right \rangle = \left ( {{\gamma _D} + {\gamma _P}(t) + {\gamma _{deph}}\left ( {{D_0} + 1} \right )} \right )/2$. Other noise correlators vanish at room temperature [33].
3. Results
3.1 Generation of coherent light below the lasing threshold
It is known that a strong coupling regime can be realized in lasers [27,28]. To demonstrate this, we consider the linearization of the Eqs. (1)–(3) (without noise terms) near stationary solution. Below the lasing threshold, the stationary amplitudes of the electric field ${a_{st}}$ and polarization of atoms ${\sigma _{st}}$ are equal to 0 and the stationary value of population inversion is ${D_0} = \left ( {{\gamma _P} - {\gamma _D}} \right )/\left ( {{\gamma _P} + {\gamma _D}} \right )$. Small deviations of the electric field amplitude $\delta a$ and polarization of atoms $\delta \sigma$ from the stationary values obey the linearized equations [27,34]
The linearized Eqs. (4) and (5) have two hybrid eigenstates with complex eigenfrequencies ${\omega _{1,2}} = {\omega _0} \pm i\sqrt {{{\left ( {{\gamma _\sigma } - {\gamma _a}} \right )}^2}/4 + \Omega _R^2{N_{at}}{D_0}}$ [27,28], where the real parts determine the oscillation frequencies and the imaginary parts determine the relaxation rates (Fig. 1).
Fig. 1. Dependence of oscillation frequencies (blue lines) and relaxation rates (red lines) on the stationary value of population inversion ${D_0}$ calculated by the eigenfrequencies of the Eqs. (4)–(5). The following values of the system parameters are used: ${\gamma _a} = 5 \times {10^{ - 3}}{\omega _0}$, ${\gamma _{deph}} = 5 \times {10^{ - 3}}{\omega _0}$, ${\Omega _R} = 2 \times {10^{ - 4}}{\omega _0}$, ${N_{at}} = {10^6}$. Inset: Sketch of considered system - cavity mode interacting with $N_{at}$ two-level atoms.
In weak coupling regime, two eigenstates have the same oscillation frequencies and different relaxation rates; in the strong coupling regime, the oscillation frequencies of the laser eigenstates are split, while the relaxation rates coincide (Fig. 1). In the considered system, the transition from the strong to the weak coupling regime occurs at a negative population inversion of atoms, when the population inversion ${D_0}$ equals [27]:
This transition from the weak to strong coupling regimes (${D_0} = {D_{EP}}$) corresponds to an exceptional point (EP) at which two eigenstates of the laser system are linearly dependent and their eigenfrequencies are equal to each other [27,34]. The population inversion of atom can only take the values from $- 1$ to $1$, and hence if ${D_{EP}} < - 1$, the strong coupling regime cannot arise in the system. Note that the exceptional point can appear at a positive population inversion, when ${\gamma _a} = {\gamma _\sigma }$ and there is a detuning between the transition frequency of the atoms and the frequency of the EM field mode [28]. Here, we do not consider such a case.
We consider a laser in the strong coupling regime, in which the pump rate is varied over time as ${\gamma _P}\left ( t \right ) = {\bar \gamma _P} + \delta {\gamma _P}\,\sin \left ( {{\omega _M}t} \right )$, where ${\bar \gamma _P}$ is the mean value of the pump rate, and ${\omega _M}$ and $\delta {\gamma _P}$ are the modulation frequency and amplitude, respectively. In Ref. [27], it was demonstrated that a periodic variation in the pump rate enables light amplification even for a negative population inversion in the active medium. This amplification is realized when the system is in the strong coupling regime, ${\bar D_0} = \left ( {{{\bar \gamma }_P} - {\gamma _D}} \right )/\left ( {{{\bar \gamma }_P} + {\gamma _D}} \right ) < {D_{EP}}$, and the modulation frequency ${\omega _M}\left ( {{{\bar D}_0}} \right )$ is equal to the frequency of Rabi splitting, ${\omega _M}\left ( {{{\bar D}_0}} \right ) = \sqrt {\left | {{{\left ( {{\gamma _\sigma } - {\gamma _a}} \right )}^2} + 4\Omega _R^2{N_{at}}{{\bar D}_0}} \right |}$ [27].
Light amplification is not sufficient to obtain coherent radiation with a narrow linewidth [35]. To achieve coherence in the sense of the first- and second-order coherence functions, it is necessary to overcome the incoherent contributions from noise [29,30]. In a conventional laser, this is realized only above the lasing threshold [29]; however, in a strongly coupled laser system, radiation can become coherent when the average pump rate ${\bar \gamma _P} < {\gamma _D}$ and the population inversion of atoms is negative. Numerical simulation of Eqs. (1)–(3) shows that the EM field intensity has a threshold behavior that depends on the modulation amplitude $\delta {\gamma _P}$ (Fig. 2(a)). Two peaks in the system spectrum can be observed at frequencies ${\omega _0} \pm {\omega _M}/2$ (Fig. 2(b)). The linewidth of each peak narrows with an increase in $\delta {\gamma _P}$ (Fig. 2(a)), and the radiation becomes coherent. For this reason, we will refer to this system as a new type of laser: a strong coupling laser (SCL or SC laser).
Fig. 2. (a) Dependence of the EM field intensity (black solid line) and the radiation linewidth (red dashed line) on the modulation amplitude $\delta {\gamma _P}$, when the pump rate ${\gamma _P}\left ( t \right ) = {\bar \gamma _P} + \delta {\gamma _P}\,\sin \left ( {{\omega _M}t} \right )$, the average pump rate ${\bar \gamma _P} = 0.91{\gamma _D}$ and the modulation frequency ${\omega _M} = 8.6 \times {10^{ - 2}}{\omega _{TLS}}$ are fixed; (b) spectrum of the radiation. The following values of the system parameters are used: ${\gamma _a} = 5 \times {10^{ - 5}}{\omega _0}$, ${\gamma _{deph}} = 5 \times {10^{ - 4}}{\omega _0}$, ${\gamma _D} = 5 \times {10^{ - 4}}{\omega _0}$, ${\Omega _R} = 2 \times {10^{ - 5}}{\omega _0}$, ${N_{at}} = {10^6}$.
3.2 Comparison of the coherence characteristics of radiation from a strong coupling laser and a conventional laser
For a more detailed study of the radiation properties of an SCL, we compare the coherence characteristics of an SCL and a conventional laser with continuous wave (CW) pumping for the same parameters of the cavity and active medium. In a CW laser, the pump rate does not depend on time, i.e. ${\gamma _P}\left ( t \right ) = {\bar \gamma _P}$, whereas in an SCL, the pump rate varies over time as ${\gamma _P}\left ( t \right ) = {\bar \gamma _P} + \delta {\gamma _P}\sin ({\omega _M}t)$. To compare an SCL with a CW laser, we fix the modulation depth, $\delta {\gamma _P} = 0.6{\gamma _D}$ and change the average pump rate ${\bar \gamma _P}$ and modulation frequency ${\omega _M}\left ( {{{\bar D}_0}} \right ) = \sqrt {\left | {{{\left ( {{\gamma _\sigma } - {\gamma _a}} \right )}^2}/4 + \Omega _R^2{N_{at}}{{\bar D}_0}} \right |}$. We examine the dependences of the output intensity, linewidth and second-order coherence function of radiation on the average pump rate ${\bar \gamma _P}$. In this case, the average energy supplied by the pump to both lasers is the same.
Both the CW laser and the SCL exhibit a sharp increase in the average number of photons (and thus in the output power) at a certain threshold pump rate (Fig. 3). However, the threshold for the SCL is considerably smaller than that for the CW laser. The radiation linewidth and coherence function ${g^{\left ( 2 \right )}}\left ( 0 \right )$ abruptly decrease for both lasers above their respective thresholds (Fig. 4). Since the threshold for the SCL is lower than that for the CW laser, the SC laser reaches the same values of coherence at a lower value of the average pump rate (Fig. 4).
Fig. 3. Number of photons in the SC laser (black curve) and a conventional laser (red curve) versus the average pump rate ${\bar \gamma _P}$. The vertical dashed line shows ${\bar \gamma _P}$, for which ${\bar D_0} = {D_{EP}}$. When ${\bar D_0} > {D_{EP}}$, there is no splitting between the eigenfrequencies (${\omega _M}\left ( {{{\bar D}_0}} \right ) = 0$), and SCL operates as a conventional laser. The system parameters are the same as in Fig. 2.
Fig. 4. (a) Spectral width of SCL emission (black curve) and conventional laser emission (red curve) versus the average pump rate ${\bar \gamma _P}$; (b) second-order autocorrelation function of SCL emission (black curve) and conventional laser emission (red curve) versus the average pump rate ${\bar \gamma _P}$. The vertical dashed lines show ${\bar \gamma _P}$, for which ${\bar D_0} = {D_{EP}}$. When ${\bar D_0} > {D_{EP}}$, there is no splitting between the eigenfrequencies (${\omega _M}\left ( {{{\bar D}_0}} \right ) = 0$), and SCL operates as a conventional laser. The system parameters are the same as in Fig. 2.
Moreover, the SC laser displays unusual coherence properties. The radiation linewidth of a conventional laser has a lower limit defined by the Schawlow-Townes relation, in which the minimal linewidth $\Delta \omega$ is inversely proportional to the average number of photons $\left \langle n \right \rangle$ [29]:
where ${A_{CW}}$ is a proportionality factor for the CW laser [32]. Hence, to achieve a narrow radiation linewidth, a conventional laser must operate well above the lasing threshold when $\left \langle n \right \rangle > > 1$.In the SCL, however, the radiation linewidths are two orders smaller than the Schawlow-Townes linewidth for the same photon number. Indeed, when the average pump rate ${\bar \gamma _P}$ exceeds the threshold for the CW laser, the number of photons in the CW laser is about two orders of magnitude greater than that for the SCL (Fig. 3). At the same time, the radiation linewidth of the CW laser is about the same as that for the SCL (Fig. 4(a)). Thus, the SCL becomes a generator of coherent radiation at a smaller average value of the pump rate and at a much smaller number of photons than conventional CW lasers.
Up to now, the parameters used in calculations correspond to a high-Q system ($Q = 2 \times {10^4}$). Note that the conclusion, that the SC laser is a generator of coherent radiation at a smaller average value of the pump rate than one in conventional CW lasers, remains true also for low-Q systems, e.g., plasmonic nanolaser and spaser. To illustrate this fact, we present the results of calculations for low-Q system ($Q=200$), see Figs. 5 and 6. In this case, similar to the high-Q system, the radiation of SC laser becomes coherent at a smaller average value of the pump rate than CW lasers.
Fig. 5. Number of photons in an SCL (black curve) and a conventional laser (red curve) versus the average pump rate ${\bar \gamma _P}$. The vertical dashed line shows ${\bar \gamma _P}$, for which ${\bar D_0} = {D_{EP}}$. When ${\bar D_0} > {D_{EP}}$, there is no splitting between the eigenfrequencies (${\omega _M}\left ( {{{\bar D}_0}} \right ) = 0$), and SCL operates as a conventional laser. The following values of the system parameters are used: ${\gamma _a} = 5 \times {10^{ - 3}}{\omega _0}$,${\gamma _{deph}} = 5 \times {10^{ - 3}}{\omega _0}$, ${\gamma _D} = 5 \times {10^{ - 3}}{\omega _0}$, ${\Omega _R} = 2 \times {10^{ - 4}}{\omega _0}$, ${N_{at}} = {10^6}$.
Fig. 6. (a) Spectral width of SCL emission (black curve) and conventional laser emission (red curve) versus the average pump rate ${\bar \gamma _P}$; (b) second-order autocorrelation function of SCL emission (black curve) and conventional laser emission (red curve) versus the average pump rate ${\bar \gamma _P}$. The vertical dashed lines show ${\bar \gamma _P}$, for which ${\bar D_0} = {D_{EP}}$. When ${\bar D_0} > {D_{EP}}$, there is no splitting between the eigenfrequencies (${\omega _M}\left ( {{{\bar D}_0}} \right ) = 0$), and SCL operates as a conventional laser. The system parameters are the same as in Fig. 5.
4. Discussion
We have shown that the SCL becomes a generator of coherent radiation for much smaller numbers of photons than for a CW laser. This is because spontaneous emission affects the generation of coherent light differently in the cases of a CW laser and an SCL.
In a CW laser, above the lasing threshold, the spontaneous emission of atoms leads to excitation of the EM field oscillations, whose phase is not consistent with the current phase of the generated EM field. Consequently, the phase of the EM field changes chaotically over time, resulting in a finite linewidth of laser radiation. The linewidth is determined by the ratio of the intensity of the EM field excited by the spontaneous emission to the total intensity of the EM field in the cavity [29].
In an SCL, two hybrid states of the cavity electric field and matter polarization participate in the generation of coherent radiation. The electric field has the form:
Here, ${\omega _{1,2}}$ are the oscillation frequencies of the two hybrid states, and ${a_{1,2}}\left ( t \right )$ and ${\phi _{1,2}}\left ( t \right )$ are the slowly varying amplitudes and phases of these two hybrid states, respectively. Spontaneous emission (noise) results in fluctuation in the phases ${\phi _{1,2}}\left ( t \right )$, which is the reason for the nonzero radiation linewidths of both peaks. Eq. (8) can be rewritten as:
Modulation of pumping causes a periodic change in the relative phase between the states, forming an effective potential for $\Delta \phi$. The equations for a strongly coupled laser system with modulated pumping are reduced to the equation for a parametric oscillator [27] (Mathieu’s equation), which is equivalent to the Helmholtz equation with a periodic potential. This periodic potential creates a correlation between the phases of the two hybrid states, and can result in the suppression of diffusion of $\Delta \phi$. This type of suppression mechanism is realized in a holographic laser [29,36–38] where suppression of diffusion of the relative phase between the states leads to a decrease in the linewidths. The diffusion of the common phase $\phi \left ( t \right )$ in holographic laser prevents the linewidths from being reduced to zero.
In the case of an SC laser, in addition to suppression of fluctuations in the relative phase, $\Delta \phi$, suppression of the common phase, $\phi$, takes place. The reason for this is the strong non-orthogonality of the eigenstates near the EP, due to which the spontaneous emission of atoms predominantly leads to fluctuations in the relative phase, $\Delta \phi$, rather than the common phase, $\phi$ (see Appendix).
Thus, in the SC laser, modulation of pumping leads to suppression of the fluctuations in the relative phase, $\Delta \phi$, while operating near the EP results in suppression of the fluctuations in the common phase, $\phi$. As a result, significant narrowing of both peak linewidths takes place in the SCL. This narrowing is observed from numerical simulation of Eqs. (1)–(3).
Note that the mechanisms resulting in suppression of phase fluctuations in the SCL have no equivalent in a CW laser. For this reason, a larger number of photons are required to achieve light coherence in a CW laser.
5. Conclusion
Localization of light at the nanoscale can result in a regime of strong coupling between the light and the active atoms. In this regime, hybrid states of light and active atoms appear, and energy oscillations occur between the EM field in the cavity and the atoms. Time-modulated pumping can be used to control these energy oscillations and to achieve light amplification at a negative population inversion of the active medium. In this paper, we demonstrate that the light amplification caused by time-modulated pumping can result in the generation of coherent light in the laser system below the lasing threshold. In this regime, the spectrum for the laser system has two peaks at frequencies determined by Rabi splitting. The linewidths of these peaks are two orders of magnitude smaller than the Schawlow-Townes linewidth for a conventional laser with the same photon number. In this way, it becomes possible to achieve a narrow linewidth for a small number of photons and sub-threshold pumping. The generation of coherent light in the sub-threshold regime paves the way for the creation of nanoscale lasers, which are less prone to overheating and degradation.
Note that the possibility of generating coherent radiation without inversion has been previously discussed in the context of polariton lasers [39–44]. These devices are also based on systems with a strong coupling between light and active medium. However, the physical mechanisms, leading to generation of the coherent light, are completely different in the SC laser and the polariton laser. In the polariton laser, strong light-active medium interaction in a multimode microcavity results in formation of polariton branches with dispersion relations for energy and momentum, $E\left ( k \right )$ [40,42]. The pumping initially excites high-energy states and then the process of Bose-Einstein condensation (BEC) results in macroscopic occupation of the ground state of polariton modes [40,43]. As a result, the radiation from the system becomes coherent [40]. Though modulation of pumping can be used to control temporal response of the polariton laser [45], for the generation of coherent light pump modulation is not necessary (see also [46,47]).
Contrary, the SC laser is based on a single-mode cavity, which can have subwavelength dimensions in all three directions, such as a plasmon cavity. There are only two polaritonic modes, which is typical situation for single-mode cavity. That is, there is no continuous branches of the polariton modes and no BEC in our system. For the SC laser, which we propose, time modulation of pump rate is crucial. Light coherence in the SC laser is achieved due to the suppression of phase fluctuations near the exceptional point by appropriate time modulation of pumping (see section "Discussion" in our manuscript). Without pump modulation at the same net pumping, the SC laser does not generate coherent light.
Appendix
The spontaneous emission of atoms excites oscillations in both the eigenstates of the laser system. The contribution of a single act of spontaneous emission can be represented as ${\delta\textbf{f}} = \delta {f_1}\,{\textbf{e}_1} + \delta {f_2}\,{\textbf{e}_2}$, where ${\textbf{e}_{1,2}}$ are the eigenstates of the laser system and $\delta {f_{1,2}}$ are expansion coefficients determined by the fluctuation.
The same contribution can be written as ${\delta\textbf{f}} = {b_1}{\textbf{e}_1} + {b_ \bot }{\textbf{e}_ \bot }$, where ${\textbf{e}_ \bot }$ is a state orthogonal to ${\textbf{e}_\textbf {{1}}}$, i.e., the scalar product $\left \langle {{{\textbf{e}_1}}} \mathrel {\left | {\vphantom {{{\textbf{e}_1}} {{\textbf{e}_ \bot }}}} \right .} {{{\textbf{e}_ \bot }}} \right \rangle = 0$. Due to stochastic nature of ${\delta\textbf{f}}$ and orthogonality of ${\textbf{e}_\textbf {{1}}}$ and ${\textbf{e}_ \bot }$, the values of ${b_1}$ and ${b_ \bot }$ are independent stochastic variables. The eigenstate ${\textbf{e}_2}$ is written as ${\textbf{e}_2} = \left \langle {{{\textbf{e}_2}}} \mathrel {\left | {\vphantom {{{\textbf{e}_2}} {{\textbf{e}_1}}}} \right.} {{{\textbf{e}_1}}} \right \rangle \,{\textbf{e}_1} + \left \langle {{{\textbf{e}_2}}} \mathrel {\left | {\vphantom {{{\textbf{e}_2}} {{\textbf{e}_ \bot }}}} \right.} {{{\textbf{e}_ \bot }}} \right \rangle \,{\textbf{e}_ \bot }$ and ${\delta\textbf{f}}$ takes form
If $\left | {{b_ \bot }\left \langle {{{\textbf{e}_2}}} \mathrel {\left | {\vphantom {{{\textbf{e}_2}} {{\textbf{e}_1}}}} \right.} {{{\textbf{e}_1}}} \right \rangle /\left \langle {{{\textbf{e}_2}}} \mathrel {\left | {\vphantom {{{\textbf{e}_2}} {{\textbf{e}_ \bot }}}} \right.} {{{\textbf{e}_ \bot }}} \right \rangle } \right | > > \left | {{b_1}} \right |$ and $\arg \left \langle {{{\textbf{e}_2}}} \mathrel {\left | {\vphantom {{{\textbf{e}_2}} {{\textbf{e}_1}}}} \right.} {{{\textbf{e}_1}}} \right \rangle = 2\pi n,\,n = 0,1,\ldots$, then the expansion coefficients of ${\delta\textbf{f}}$ are approximately equal in modulus and have opposite signs [38], i.e., $\delta {f_1} = - \,\delta {f_2} = \delta f$. Near the exceptional point, ${\textbf{e}_1}$ almost coincides with ${\textbf{e}_2}$ (i.e., $\left | {\left \langle {{{\textbf{e}_1}}} \mathrel {\left | {\vphantom {{{\textbf{e}_1}} {{\textbf{e}_2}}}} \right.} {{{\textbf{e}_2}}} \right \rangle } \right | \sim 1$ and $\left | {\arg \left \langle {{{\textbf{e}_2}}} \mathrel {\left | {\vphantom {{{\textbf{e}_2}} {{\textbf{e}_1}}}} \right.} {{{\textbf{e}_1}}} \right \rangle } \right | < < 1$) and $\left | {\left \langle {{{\textbf{e}_2}}} \mathrel {\left | {\vphantom {{{\textbf{e}_2}} {{\textbf{e}_ \bot }}}} \right.} {{{\textbf{e}_ \bot }}} \right \rangle } \right | < < 1$. Thus, due to independencies of ${b_1}$ and ${b_ \bot }$, the conditions $\left | {{b_ \bot }\left \langle {{{\textbf{e}_2}}} \mathrel {\left | {\vphantom {{{\textbf{e}_2}} {{\textbf{e}_1}}}} \right.} {{{\textbf{e}_1}}} \right \rangle /\left \langle {{{\textbf{e}_2}}} \mathrel {\left | {\vphantom {{{\textbf{e}_2}} {{\textbf{e}_ \bot }}}} \right.} {{{\textbf{e}_ \bot }}} \right \rangle } \right | > > \left | {{b_1}} \right |$ and $\arg \left \langle {{{\textbf{e}_2}}} \mathrel {\left | {\vphantom {{{\textbf{e}_2}} {{\textbf{e}_1}}}} \right.} {{{\textbf{e}_1}}} \right \rangle = 2\pi n,\,n = 0,1,\ldots$ are satisfied for most possible values of ${\delta\textbf{f}}$.
Thus, near the EP, the expansion coefficients in $\delta {f_1}$ and $\delta {f_2}$ for almost all ${\delta\textbf{f}}$ are approximately equal in modulus and have opposite signs [48], i.e., $\delta {f_1} = - \,\delta {f_2} = \delta f$. In this case, taking into account the contribution from spontaneous emission to the electric field, we obtain:
If ${a_1} = {a_2}$ and $\Delta \phi = 2\pi n,\,\,\,n = 0,1,\ldots$, then $\Delta {\phi _1} = \Delta {\phi _2}$ and the spontaneous emission leads only to a change in the relative phase ($\Delta \phi \to \Delta {\phi _1}$) while the common phase $\phi$ is unchanged. In an SCL, the first condition (${a_1} \approx {a_2}$) holds true at all moments in time. The second condition is satisfied at the maxima of the electromagnetic field, which make the main contribution to the laser radiation and determines the linewidths.
Funding
Russian Science Foundation (20-72-10057).
Acknowledgments
The study was supported by a grant from Russian Science Foundation (project No. 20-72-10057). I.V.D., Z.A.A. and E.S.A. thank foundation for the advancement of theoretical physics and mathematics "Basis".
Disclosures
The authors declare no conflicts of interest.
References
1. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003). [CrossRef]
2. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009). [CrossRef]
3. M. Noginov, G. Zhu, A. Belgrave, R. Bakker, V. Shalaev, E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009). [CrossRef]
4. J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012). [CrossRef]
5. J. B. Khurgin and G. Sun, “Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence,” Opt. Express 20(14), 15309–15325 (2012). [CrossRef]
6. F. Benz, M. K. Schmidt, A. Dreismann, R. Chikkaraddy, Y. Zhang, A. Demetriadou, C. Carnegie, H. Ohadi, B. De Nijs, R. Esteban, J. Aizpurua, and J. J. Baumberg, “Single-molecule optomechanics in “picocavities”,” Science 354(6313), 726–729 (2016). [CrossRef]
7. R. Chikkaraddy, B. De Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535(7610), 127–130 (2016). [CrossRef]
8. T. Hummer, F. Garcia-Vidal, L. Martin-Moreno, and D. Zueco, “Weak and strong coupling regimes in plasmonic qed,” Phys. Rev. B 87(11), 115419 (2013). [CrossRef]
9. P. Torma and W. L. Barnes, “Strong coupling between surface plasmon polaritons and emitters: a review,” Rep. Prog. Phys. 78(1), 013901 (2015). [CrossRef]
10. G. Zengin, M. Wersall, S. Nilsson, T. J. Antosiewicz, M. Kall, and T. Shegai, “Realizing strong light-matter interactions between single-nanoparticle plasmons and molecular excitons at ambient conditions,” Phys. Rev. Lett. 114(15), 157401 (2015). [CrossRef]
11. D. G. Baranov, M. Wersall, J. Cuadra, T. J. Antosiewicz, and T. Shegai, “Novel nanostructures and materials for strong light–matter interactions,” ACS Photonics 5(1), 24–42 (2018). [CrossRef]
12. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, “Vacuum rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004). [CrossRef]
13. D. Sanvitto and S. Kena-Cohen, “The road towards polaritonic devices,” Nat. Mater. 15(10), 1061–1073 (2016). [CrossRef]
14. D. D. Smith, H. Chang, K. A. Fuller, A. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004). [CrossRef]
15. E. Orgiu, J. George, J. Hutchison, E. Devaux, J. Dayen, B. Doudin, F. Stellacci, C. Genet, J. Schachenmayer, C. Genes, G. Pupillo, P. Samorì, and T. W. Ebbesen, “Conductivity in organic semiconductors hybridized with the vacuum field,” Nat. Mater. 14(11), 1123–1129 (2015). [CrossRef]
16. J. A. Hutchison, A. Liscio, T. Schwartz, A. Canaguier-Durand, C. Genet, V. Palermo, P. Samori, and T. W. Ebbesen, “Tuning the work-function via strong coupling,” Adv. Mater. 25(17), 2481–2485 (2013). [CrossRef]
17. J. A. Hutchison, T. Schwartz, C. Genet, E. Devaux, and T. W. Ebbesen, “Modifying chemical landscapes by coupling to vacuum fields,” Angew. Chem., Int. Ed. 51(7), 1592–1596 (2012). [CrossRef]
18. J. Galego, F. J. Garcia-Vidal, and J. Feist, “Suppressing photochemical reactions with quantized light fields,” Nat. Commun. 7(1), 13841 (2016). [CrossRef]
19. B. Munkhbat, M. Wersall, D. G. Baranov, T. J. Antosiewicz, and T. Shegai, “Suppression of photo-oxidation of organic chromophores by strong coupling to plasmonic nanoantennas,” Sci. Adv. 4(7), eaas9552 (2018). [CrossRef]
20. N. Nefedkin, E. Andrianov, and A. Vinogradov, “The role of strong coupling in the process of photobleaching suppression,” J. Phys. Chem. C 124(33), 18234–18242 (2020). [CrossRef]
21. X. Liu, T. Galfsky, Z. Sun, F. Xia, E.-c. Lin, Y.-H. Lee, S. Kena-Cohen, and V. M. Menon, “Strong light–matter coupling in two-dimensional atomic crystals,” Nat. Photonics 9(1), 30–34 (2015). [CrossRef]
22. I.-C. Hoi, C. Wilson, G. Johansson, T. Palomaki, B. Peropadre, and P. Delsing, “Demonstration of a single-photon router in the microwave regime,” Phys. Rev. Lett. 107(7), 073601 (2011). [CrossRef]
23. T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6(9), 605–609 (2012). [CrossRef]
24. R. Schoelkopf and S. Girvin, “Wiring up quantum systems,” Nature 451(7179), 664–669 (2008). [CrossRef]
25. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464(7285), 45–53 (2010). [CrossRef]
26. Z.-L. Xiang, S. Ashhab, J. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85(2), 623–653 (2013). [CrossRef]
27. I. Doronin, A. Zyablovsky, E. Andrianov, A. Pukhov, and A. Vinogradov, “Lasing without inversion due to parametric instability of the laser near the exceptional point,” Phys. Rev. A 100(2), 021801 (2019). [CrossRef]
28. J. B. Khurgin, “Exceptional points in polaritonic cavities and subthreshold fabry–perot lasers,” Optica 7(8), 1015–1023 (2020). [CrossRef]
29. M. O. Scully and M. Zubairy, Quantum optics, Cambridge Press, London (1997).
30. L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge university press, 1995).
31. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef]
32. A. E. Siegman, “Lasers university science books,” Mill Valley, CA 37, 169 (1986).
33. H. Haken, Laser light dynamics, vol. 1 (North-Holland Amsterdam, 1985).
34. A. Zyablovsky, I. Doronin, E. Andrianov, A. Pukhov, Y. E. Lozovik, A. Vinogradov, and A. Lisyansky, “Exceptional points as lasing prethresholds,” Laser Photonics Rev. 15, 2000450 (2021). [CrossRef]
35. I. V. Doronin, E. S. Andrianov, A. A. Zyablovsky, A. A. Pukhov, Y. E. Lozovik, A. P. Vinogradov, and A. A. Lisyansky, “Second-order coherence properties of amplified spontaneous emission,” Opt. Express 27(8), 10991–11005 (2019). [CrossRef]
36. J. Krause and M. O. Scully, “Theory of the holographic laser: Correlated emission in a ring cavity,” Phys. Rev. A 36(4), 1771–1778 (1987). [CrossRef]
37. M. O. Scully, “Correlated-emission laser gyroscope,” Phys. Rev. A 35(1), 452–455 (1987). [CrossRef]
38. M. Raja, S. Brueck, M. Scully, and C. Lee, “Resonant periodic-gain surface-emitting semiconductor lasers and correlated emission in a ring cavity,” Phys. Rev. A 44(7), 4599–4607 (1991). [CrossRef]
39. A. Imamog, R. Ram, S. Pau, and Y. Yamamoto, “Nonequilibrium condensates and lasers without inversion: Exciton-polariton lasers,” Phys. Rev. A 53(6), 4250–4253 (1996). [CrossRef]
40. H. Deng, G. Weihs, D. Snoke, J. Bloch, and Y. Yamamoto, “Polariton lasing vs. photon lasing in a semiconductor microcavity,” Proc. Natl. Acad. Sci. 100(26), 15318–15323 (2003). [CrossRef]
41. J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. Marchetti, M. Szymanska, R. Andre, J. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and L. S. Dang, “Bose–Einstein condensation of exciton polaritons,” Nature 443(7110), 409–414 (2006). [CrossRef]
42. H. Deng, H. Haug, and Y. Yamamoto, “Exciton-polariton bose-einstein condensation,” Rev. Mod. Phys. 82(2), 1489–1537 (2010). [CrossRef]
43. T. Byrnes, N. Y. Kim, and Y. Yamamoto, “Exciton–polariton condensates,” Nat. Phys. 10(11), 803–813 (2014). [CrossRef]
44. D. Sanvitto and S. Kena-Cohen, “The road towards polaritonic devices,” Nat. Mater. 15(10), 1061–1073 (2016). [CrossRef]
45. M. Z. Baten, T. Frost, I. Iorsh, S. Deshpande, A. Kavokin, and P. Bhattacharya, “Small-signal modulation characteristics of a polariton laser,” Sci. Rep. 5(1), 11915 (2015). [CrossRef]
46. T. Doan, H. T. Cao, D. T. Thoai, and H. Haug, “Condensation kinetics of microcavity polaritons with scattering by phonons and polaritons,” Phys. Rev. B 72(8), 085301 (2005). [CrossRef]
47. T. Doan, H. T. Cao, D. T. Thoai, and H. Haug, “Coherence of condensed microcavity polaritons calculated within boltzmann-master equations,” Phys. Rev. B 78(20), 205306 (2008). [CrossRef]
48. A. Zyablovsky, E. Andrianov, and A. Pukhov, “Parametric instability of optical non-hermitian systems near the exceptional point,” Sci. Rep. 6(1), 29709 (2016). [CrossRef]
