Terahertz Lasing in Ensemble of Asymmetric Quantum Dots - ACS

Article Cite This: ACS Photonics 2017, 4, 2726-2737

pubs.acs.org/journal/apchd5

Terahertz Lasing in Ensemble of Asymmetric Quantum Dots Igor Yu. Chestnov,*,† Vanik A. Shahnazaryan,‡,¶,§ Alexander P. Alodjants,¶ and Ivan A. Shelykh‡,¶ †

Vladimir State University named after A. G. and N. G. Stoletovs, 87 Gorkii St., 600000 Vladimir, Russia Science Institute, University of Iceland, IS-107 Reykjavik, Iceland ¶ ITMO University, St. Petersburg 197101, Russia § Russian-Armenian (Slavonic) University, Yerevan 0051, Armenia ‡

ABSTRACT: We propose a scheme of terahertz laser based on an ensemble of asymmetric quantum dots dressed by an intense electromagnetic field. THz emission originates from the transitions at Rabi energy between the neighboring dressed states. For the amplification of the lasing mode, high-Q photonic crystal cavity tuned to terahertz range can be employed. Within the mean field approximation, the system is described by Maxwell−Bloch type equations, which account for inhomogeneous broadening and decoherence processes. The conditions for the onset of the lasing are determined, and emission intensity and quantum efficiency are obtained by numerical solution of the Langevin type stochastic equations, describing the generation of THz pulses. The energy gap between dressed levels is determined by the driving field intensity that implies the ability of flexible control over lasing parameters. KEYWORDS: asymmetric quantum dot, THz lasing, photonic crystal cavity, dressed states, strong coupling

D

tunability.11 Such devices can be realized in both bulk12 and low-dimensional13 geometries, the latter including planar14 and pillar15 systems together with photonic crystal cavities16 and for a broad variety of materials including wide gap semiconductors (for instance, GaN and ZnO) for which room temperature operation becomes feasible.17−19 Upper and lower polariton states are separated in energy by the gap, which is usually on the order of magnitude of several millielectronvolts for conventional semiconductor systems and of tens of millielectronvolts for wide bandgap semiconductors or organic materials. Radiative transitions between upper and lower polariton states, if possible, will thus generate emission in the THz frequency range, which can be enhanced by the effect of bosonic stimulation in the system. Unfortunately, direct implementation of this scenario faces a fundamental problem: optical transition between two polariton branches is forbidden due to the same parity of their wave functions. There are several attempts to overcome this obstacle, which include the use of intersubband polaritons,20 bosonic cascade lasers in

esigning efficient terahertz (THz) lasing devices is one of major challenges of modern applied physics.1 In recent decades, several proposals for practical realization of THz emitters appeared. They are based on a variety of materials and methods, covering the areas of gas lasers,2,3 single-layer4 and double-layer5 graphene, photonic crystals,6 quantum cascade lasers,7,8 and frequency conversion in nonlinear optical systems.9 However, despite the significant progress achieved over the years, there are variety of fundamental technological problems yet to overcome. As a result, the goal of creation of tunable and stable THz laser sources hitherto remains not achieved. One of the possible routes to realize this goal is operating in the regime of strong light−matter coupling, for which the interaction between material part with light drastically modifies the properties of the matter itself. Striking examples of such systems are low dimensional semiconductor microcavities for which strong resonant coupling between confined photonic cavity mode and excitonic resonance in the active media gives rise to the appearance of the hybrid modes, referred as exciton polaritons.10 Among the notable applications of polaritons are polariton lasers, which have certain advantages with respect to conventional lasers, including extremely low threshold and easy © 2017 American Chemical Society

Received: June 5, 2017 Published: October 27, 2017 2726

DOI: 10.1021/acsphotonics.7b00575 ACS Photonics 2017, 4, 2726−2737

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parabolic traps,21 and systems with asymmetric quantum wells22−25 for which dark exciton states of different parity are admixed to upper and lower polariton modes and thus open a radiative relaxation channel between them. However, the majority of the mentioned proposals suffer from poor tunability. Particularly, in refs 21−25, the THz frequency is rigidly determined by the structure parameters. The later circumstance sufficiently hinders practical applications. The concept of a polariton is closely related to the concept of electromagnetically dressed states (DSs). The latter appears in systems placed in a strong external laser field.26,27 Formation of the DSs was predicted and then experimentally observed in a broad variety of physical systems such as atomic systems28 and solid-state setups, including bulk semiconductors29,30 and various mesoscopic structures.31−43 Characteristic energy separating DSs is known as Rabi splitting. Differently from the case of cavity polaritons (where a similar parameter is also known as vacuum Rabi splitting), it can be easily tuned by the intensity of the dressing field. The possibility of a radiative transition between the states of electromagnetically dressed systems was previously predicted theoretically and later on observed in the experiments for atomic media,44−49 condensed-matter structures,50−52 and superconducting circuits.53−55 In this context, the systems with broken inversion symmetry are of special interest. Formally, in such systems not only transversal (proportional to Pauli pseudospin operators σx and σy) but also longitudinal (proportional to σz operator characterizing population inversion) coupling with electromagnetic field becomes possible.56 The latter opens a radiative channel between adjacent dressed levels at Rabi frequency.57−60 In semiconductor structures where light−matter interaction is sufficiently strong, Rabi splitting can easily reach values of several millielectronvolts for moderate dressing field intensities, which can make them attractive candidates for realization of THz laser sources provided that one could find structures possessing broken inversion symmetry. Among the promising candidates are asymmetric quantum dots (QDs) formed by semiconductor materials of wurtzite type (e.g., GaN QDs), for which giant piezoelectric effect was reported and static dipole momentum routinely appears.61−67 The interaction of this built-in moment with external fields can lead to intriguing optical effects, such as dynamically controllable fluorescence spectra under bichromatic dressing field, found both experimentally68 and theoretically,69 and spontaneous emission at Rabi frequency.60,70 The fluorescence spectra of an ensemble of two level systems in a microcavity was previously widely studied in both weak71 and strong coupling regimes.72 Various aspects of emission were addressed, including the violation of the Cauchy−Schwarz inequality,73 superradiance,71 influence of tunneling between two-level systems,74 etc. In the current investigation, we target an ensemble of two level emitters with simultaneous presence of asymmetry and strong near resonant electromagnetic field. In particular, on the basis of our previous results on optical properties of individual QDs with broken inversion symmetry, in the current paper, we set a goal of construction of the dynamical theory of THz lasing in an ensemble of asymmetric QDs dressed by an intense optical field inside the THz laser cavity. We adopt the method of creation of population inversion between dressed levels implying exploitation of spontaneous decay of an excited eigenstate of the two-level system, see refs 47, 49, and 57. We also take into account the

fact that systems of multiple QDs have certain problems with scalability. The main reason for that is significant inhomogeneous broadening of QD ensembles arising from imperfections of the fabrication process.75,76 The values of inhomogeneous broadening can reach tens of millielectronvolts, which is larger than the frequency of the emitted THz radiation. This situation leads to the qualitative difference of our system and conventional lasers operating at optical frequencies. Thus, the presence of inhomogeneous broadening will possibly challenge the ability of realization of THz laser source based on asymmetric QDs. However, the performed calculations let us conclude that the proposed laser can still operate under realistic system parameters. The paper is organized as follows. In “The Model” section, we describe the geometry of the system. In “Maxwell-Bloch-like Laser Equations in the DS Basis” section, we introduce DSs of QDs and provide the main equations that govern the lasing. “Terahertz Lasing in Dressed Quantum Dots Ensemble” section contains the main results of the modeling, including lasing conditions and efficiency of the proposed scheme. Conclusions summarize the obtained results.

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THE MODEL We consider the system of N asymmetrical QDs placed in the planar THz cavity (Figure 1a). The system is irradiated by an intensive classical CW laser field E⃗ cos(ωdt) with frequency ωd close to the interband transition in the considered QDs. Up-todate technologies allow deposition of QDs with interdot separation of about 30 nm (corresponds to surface density of ∼1011 cm−2), which is at least several times larger than the typical value of individual QD lateral size. Thus, we treat each QD as an individual two level system with ground state |g⟩ν, having all electrons in the valence band, and upper state |e⟩ν, corresponding to the excitation of single electron−hole pair. We denote the interlevel separation energy as ℏω0ν (Figure 1b), where subscript ν numerates QDs in the ensemble. Higher excited states of QD are supposed to lie far from the resonance with the driving field. For the sake of simplicity, we suppose the driving laser field E⃗ to be linearly polarized and neglect polarization and spin degrees of freedom. The generic Hamiltonian written in the basis of QD eigenstates is N

H=

∑ ν

ℏω0ν z σν + 2 N

+

∑ dν⃗ ec⃗ ν

N

∑ Ed⃗ ν⃗ cos(ωdt ) + ℏΩca†a ν

ℏΩc (a + a†) 2ε0Vc

(1)

The first term is the Hamiltonian of the two level QDs, the second term describes coupling with external driving mode, the third term corresponds to the quantized field of the THz cavity, and the fourth term describes the coupling between THz photons and QDs. Here σzν = |e⟩ν⟨e|ν − |g⟩ν⟨g|ν is an operator of the population inversion written in the basis of the bare states of QDs. The quantity d⃗ν corresponds to the dipole momentum operator, with nondiagonal elements describing the interband transition rates. They are determined by the material properties ⃗ = dge ⃗ . The diagonal and thus are equal for all QDs: d⃗egν ≡ deg terms are absent in symmetric systems and emerge as a result of inversion symmetry breaking. In the considered structure the asymmetry originates from giant piezoelectric effect, characteristic for III-nitride semiconductors. The latter gives rise to built2727


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depicted in Figure 1a. Such cavities both possess high Q-factors (up to 104 in the best samples77) and allow tight confinement of THz field. The mode volume in this case can be approximated by Vc ≈ (λc/2)3 = π3c3/Ωc3. Another possible alternative is plasmonic THz cavities, which demonstrate subwavelength confinement in one direction dramatically reducing mode volume. However, this usually costs the loss of Q-factor, which is at best on the order of tens in such structures.78,79 The Hamiltonian (eq 1) contains time-dependence in an explicit form. In order to get rid of it, we perform the unitary transformation UR = exp(iωd/2∑νσzν·t), corresponding to the transition to the frame rotating with driving frequency ωd, according to H̃ = URHU†R + iℏU̇ RU†R. Applying the rotating wave approximation and thus omitting quickly oscillating terms containing factors exp(±iωdt) and exp(±2iωdt), we arrive at the simple stationary Hamiltonian: N

HRWA = −∑ ν N

+

∑ ν

ℏχν 2

N

∑ ℏΩ (σν+ + σν−) + ℏΩca†a ν

2

(ν + σνz)(a + a†), (3)

⃗ /ℏ corresponds to the Rabi where Δν = ωd − ω0ν and Ω = E⃗ dge splitting. Note that within rotating wave approximation, the latter is determined by transversal coupling between QD and ⃗ driving field only. Additionally, the dipole matrix element dge has an induced character and in fact is always parallel to the excited field. Thus, the value of Rabi splitting does not depend on the orientation of driving field polarization. On the contrary, the strength of the interaction with THz Ωc cavity mode, χ = dee⃗ ν · ec⃗ , does depend on the mutual

Figure 1. Sketch of the system under consideration. (a) Terahertz cavity (photonic crystal slab) containing an ensemble of N asymmetric QDs. The system is interacting with the optical driving field with frequency close to resonance with the QD interband transition. (b) Schematic illustration of (i) energy levels of an individual QD and (ii) energy states of an electromagnetically dressed QD in the presence of a blue detuned driving field, Δ > 0. The population inversion in DS basis emerges due to spontaneous relaxation processes, which occur in the QD eigenstate basis (see main text for details).

ν

2ε0ℏVc

orientation of the cavity mode polarization ec⃗ and stationary dipole moment d⃗eeν whose direction is determined by the symmetry of the system. Note that if stationary dipoles of QDs in the ensemble are oriented chaotically, the overall interaction with the cavity mode will be suppressed. Fortunately, modern technologies allow growth of QDs along the polar [0001] axis of the wurtzite GaN structure. It guaranties that all permanent dipoles are directed perpendicular to the plane of the substrate. Since the value of the stationary dipole moment is determined by the size of the QD, which is different for all QDs in the ensemble, the absolute values of |χν| are distributed normally around some average value. Besides the values of χν should differ by phase factors. We suppose that when the macroscopic polarization of the medium is set by the lasing process, all the emitters are synchronized. This assumption looks reasonable in the case when the QD ensemble is localized in the region where variation of the cavity mode intensity is negligibly small, compare with ref 47.

in electric field along the asymmetry axis, leading to spatial separation of an electron and a hole and appearance of internal dipole momentum. For the ground state, when there are no ⃗ = 0, while excited electrons in the system, one can assume dggν for the excited state d⃗eeν ≠ 0, and its value depends on individual QD parameters. According to the aforesaid, the dipole operator d⃗ν of the νth QD can be represented as ⃗ ν(σν+ + σν−) dν⃗ = dee⃗ ν(ν + σνz)/2 + dge

ℏΔν z σν + 2

(2)

The Pauli raising and lowering operators acting in the space of ground and excited states of the individual dots read: σ+ν = |g⟩ν⟨e|ν and σ−ν = |e⟩ν⟨g|ν, and ν = |e⟩ν ⟨e|ν + |g ⟩ν ⟨g |ν is a unitary operator. The parameter Ωc in the third term in eq 1 is resonant frequency of the THz cavity, Vc in the last term is cavity mode volume, ε0 is the dielectric constant, and ec⃗ is polarization vector of the THz cavity mode. In order to achieve efficient THz lasing, one should maximize interaction strength between QDs and THz photons and minimize the losses in the cavity. Appropriate candidates that comply with both of these requirements are planar photonic crystal slabs, schematically

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MAXWELL-BLOCH-LIKE LASER EQUATIONS IN THE DS BASIS In the presence of an intense driving field, bare states of QDs become optically dressed. It leads to the formation of the Rabi doublet consisting of the energy eigenstates of the subsystem “νth QD + driving field”, which can be denoted as upper |1⟩ν and lower |2⟩ν DSs. The Hamiltonian of the system written in the DS basis can be obtained by means of rotation in pseudospin space with unitary operator Ud = exp[i∑νθνσyν], 2728


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where σyν = i(σ−ν − σ+ν ), and the rotation angles are determined from the condition tan 2θν = −Ω/Δν. This gives N

H

DS

ℏΩ Rν z ςν + ℏΩca†a + (a + a†) 2

∑

=

ν N

∑

Here κν = χνΩ/2ΩRν is an effective strength of the coupling between the THz cavity mode and the νth dressed QD. This is a key parameter determining the efficiency of THz lasing in our system. The parameter

ℏχν

ν

2

⎛ Ω2 1⎞ ⎟ Γdν = Γ⎜ + 2 2⎠ ⎝ 4Ω Rν

[ν + cos(2θν)ςνz − sin(2θν)(ςν+ + ςν−)]

in eq 9b is the effective dephasing rate responsible for the relaxation of DS polarization Sν. The impact of the spontaneous processes on the population inversion Szν of Rabi levels in eq 9c is twofold. First, it contributes to the relaxation from the excited state to the ground state with the rate

(4)

where ς are spin operators acting in the space of the Rabi doublet states (see Figure 1b) z,±

|1⟩ν = sin θν|g ⟩ν + cos θν|e⟩ν

(5a)

|2⟩ν = cos θν|g ⟩ν − sin θν|e⟩ν

(5b)

split by frequency Ω Rν = Δν 2 + Ω2 . It is clearly seen from the Hamiltonian eq 4 that the presence of the stationary dipole (χν ∝ deeν ≠ 0) allows energy exchange between the dressed QD states and the cavity mode. Further consideration of the THz emission needs accounting for the spontaneous processes, which play a crucial role in lasing and, as it will be shown later, also play the role of the effective pump in the considered system. For this purpose, we use the master equation approach for the density matrix of the system ϱ i ϱ̇ = − [HDS , ϱ] + 3R ϱ + 3Cϱ (6) ℏ

Γzν =

N

Szstν =

(7)

describes the radiative decay of QD excitations with rate Γ (supposed to be identical for all QDs). Note that for the sake of simplicity we present the decay terms in the basis of bare states. In order to investigate the dynamics of DSs, these terms should be rewritten in the DSs basis. The term 3Cσ = Γc(2aϱa† − a†ϱa − ϱa†a)

(8)

N ν

(9a)

Sν̇ = −iκνSzνα − i Ω RνSν − ΓdνSν

(9b)

Sż ν = 2iκν(Sν*α − Sνα*) − ΓzνSzν + Γsν

(9c)

(11)

−1 2Δν ⎛ Δ2 ⎞ ⎜1 + ν 2 ⎟ Ω Rν ⎝ Ω Rν ⎠

(12)

From this equation it is clearly seen that when the driving field is blue-detuned from the QD, that is, Δν > 0, the term Γsν acts as an effective pump and the population inversion in the DS basis establishes Sstzν > 0 (see Figure 2b). That is, incoherent processes of spontaneous relaxation occurring in the QD bare state basis not only deteriorate DS lasing due to the relaxation of DS polarization Sν but also favor DS population inversion Szν > 0, which is a necessary prerequisite for the lasing. The sketch of this process is shown in Figure 1b. Note that a similar pumping mechanism was intensively investigated both theoretically and experimentally in atomic gases44−49 and in superconductor quantum circuits.53−55 It is important to note that the subsequent analysis does not take into account the interaction with a phonon bath,80 which drives the system to thermodynamic equilibrium where the populations of the Rabi levels are Szν = −tanh[ℏΩRν/2kBT] < 0, see refs 80 and 81. We qualitatively estimate the influence of the phonon bath on the properties of the THz lasing in the last section. The discussed effect of spontaneous relaxation-induced population inversion will thus occur only if the rate of DS thermalization due to the interaction with the phonon bath is slower than Γsν, which is possible if the lattice temperature T in the system is sufficiently low and excited state lifetime Γ−1 is short enough to overcome quick phonon assisted relaxation. In our calculations, we take Γ = 0.02 ps−1. Such condition can be reached in the small QDs for which the overlap integral between electron and hole wave functions is high enough. This leads to the high decay rate of the excited state,82 which allows quick gain of the DS inversion.

is responsible for the THz cavity mode net damping with rate Γc related to the cavity Q-factor, Q = Ωc/2Γc. To analyze the dynamics of the system, we use mean-field approximation, which means that all the mean values of the products of the operators are factorized into products of their mean values, ⟨AB⟩ = ⟨A⟩⟨B⟩. To account for the inhomogeneous broadening provided by variations in the Rabi splittings ΩRν and couplings to THz mode χν for each QD we write down individual equations for all QDs in the ensemble introducing averaged values of DS spinor operators Sν = ⟨ς−ν ⟩, Sν* = ⟨ς+ν ⟩, and Szν = ⟨ςzν⟩. The corresponding system of Maxwell−Blochtype equations for the ensemble of N individual QDs interacting with a single THz cavity mode characterized by complex amplitude α = ⟨a⟩ reads α̇ = i∑ κνSν − i Ωcα − Γcα

Δ2 ⎞ Γ⎛ ⎜1 + ν 2 ⎟ 2⎝ Ω Rν ⎠

Further, spontaneous transitions between bare states, |e⟩ν and |g⟩ν also feed populations of both Rabi levels (see definitions in eq 5a). This is reflected in the last term of eq 9c, Γsν = ΓΔν/ ΩRν, which plays a role in pumping or dissipation for DS population inversion Szν depending on which of the Rabi states, upper or lower, is pumped more efficiently. If the amplitude of cavity mode is negligibly small, α ≈ 0 (e.g., at the start of the lasing), the populations of the Rabi levels are governed by the relaxation processes only. In this case, the population inversion for the νth QD approaches stationary level, which reads

The term ⎛ ⎞ 1 3R ϱ = Γ∑ ⎜σν−ϱσν+ − (σν+σν−ϱ + ϱσν+σν−)⎟ ⎝ ⎠ 2 ν

(10)

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function of QD size: ωqν ≃ ω̅ q(1 − 2(lν − l0)/l0), where ω̅ q = ℏ/(2μl02). This gives for the detunings Δν = Δ0 + δν, where δν = 2ω̅ q(lν − l0)/l0 is the deviation of the interband transition frequency of the νth QD from the corresponding ensemble averaged value Δ0. The magnitude of the static dipole moment deeν, which is proportional to electron−hole separation, can also be supposed to depend linearly on the QD size lν. It allows us to write the following estimate for the coupling strength with THz cavity mode (see eq 3 and comments below it): χν = χ0(1 + (lν − l0)/ l0) = χ0(1 + δν/2ω̅ q), where χ0 is the ensemble averaged value. Our results demonstrate that generally the effects of the broadening in the stationary dipole moments have a minor effect in comparison with inhomogeneous broadening of the interband transition. However, both of them were accounted for in the calculation. The Net Gain from QD Ensemble. The effect of lasing occurs when the regime of positive feedback established in the active media, and the amplification of the cavity field provided by the ensemble of emitters exceeds the losses. In the ideal situation, all emitters should be equivalent. However, even in the case when inhomogeneous broadening is present, all emitters will contribute to the filling of the cavity mode although their contribution is not equivalent. In order to derive the net gain for the THz field in the system, we adiabatically eliminate variables corresponding to QD subsystem from eqs 9. For that purpose we use the substitution Sν(t) = Sν e−iΩct in eq 9b thus supposing that the lasing occurs at the eigenfrequency of the THz cavity. Note that in the general case, the frequency of lasing is not exactly equal to the cavity frequency Ωc due to the frequency pulling effect. However, in the case of large inhomogeneous broadening, this effect is negligible (see ref 85). For the occupancy of the cavity mode, we get

Figure 2. (a) Population inversion of dressed QD in the absence of THz field (α = 0). Blue line is determined by eq 12 for ℏΩ = 4 meV. The position of the cavity frequency corresponds to Ωc/2π = 2 THz. In the inset, the profile of spectral density of population inversion, S̅z, in the presence of lasing is shown. The curve corresponds to the profile along the vertical dashed line in Figure 5b. (b) Spectral distribution of QDs governed by spectral density ρ(Δ) (gray curve). Green curve corresponds to homogeneous line L(Ωc, Δ) the width of which was multiplied by 50 for illustrative purposes. The upper axis shows the corresponding values of Rabi splitting.

N

∂|α|2 = (∑ gν (|α|2 , Δν ) − 2Γc)|α|2 ∂t ν

(13)

where the effective gain from an individual νth QD is given by

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TERAHERTZ LASING IN DRESSED QUANTUM DOTS ENSEMBLE Inhomogeneous Ensemble of QDs. Due to technical reasons, it is extremely difficult to fabricate a large amount of identical QDs. In real samples, QD sizes always vary over a wide range. As a result, the main parameters characterizing an individual QD such as interband transition frequency ω0ν and the value of permanent dipole deeν are distributed normally in the ensemble. This leads to the inhomogeneous broadening of the QD frequency ω0ν (and the detuning Δν from driving field as a consequence). The typical values of the full width at halfmaximum (fwhm) of this distribution are on the order of tens of millielectronvolts.83,84 We attribute inhomogeneous broadening of the interband transition to the size quantization energy ℏωqν = ℏω0ν − Eg, where Eg denotes the semiconductor bandgap. Without loss of generality we can use the following estimate: ℏωqν ≈ ℏ2/ (2μlν2), where μ ≈ 0.1m0, with m0 denoting free electron mass. Here lν is the size of the νth QD. Let us suppose that the values of lν are distributed around mean value l0. Assuming that the inhomogeneous width is much smaller than ωqν, which is reasonable for small QDs (lν on the order of nanometers) we can approximate the interband transition frequency by a linear

gν (|α|2 , Δν ) =

2κν 2 Γsν Lν(Ωc) 2 Γzν 1 + 4κν |α|2 Lν(Ωc)/Γzν

(14)

and the homogeneous line shape of the DS transition is Lν(ω) =

Γdν 2

Γdν + (Ω Rν − ω)2

(15)

Note that adiabatical elimination of the QD variables performed above is justified in the limit of Γdν, Γzν ≫ Γc, when the dissipation in the QD subsystem dominates over the cavity loss rate.53 In the system under consideration, those QDs that effectively interact with the THz field are characterized by large positive detunings, Δν. In this case, both Γdν and Γzν are on order of Γ/2. Thus, the adiabaticity condition is satisfied in the case of a good enough THz cavity, for which Γc ≪ Γ, that is, with Q-factor about 103 and more for used parameters. To derive the total gain from whole QD ensemble, we assume that the number of QDs is so large that continuous approximation can be done. Thus, we replace summation over distinct QDs in eq 13 with integration over QD transition frequency in the rotating frame using the substitution ∑Nν → ∫ ρ(Δ) dΔ, where for the spectral density ρ(Δ) of QDs we take the Gaussian distribution 2730


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⎡ (Δ − Δ )2 ⎤ N 0 ⎥ exp⎢ − 2π ξ 2ξ 2 ⎣ ⎦

(16)

with fwhm equal to 2 2 ln(2) ξ . For the net gain of the THz field, G = ∑Nν gν(|α|2) we get this way: G=

N 2π ξ

+∞

∫−∞

⎡ (Δ − Δ )2 ⎤ 0 ⎥ dΔ g (|α|2 , Δ) exp⎢ − 2ξ 2 ⎦ ⎣ (17)

where all parameters that were labeled with subscript ν heretofore are now functions of the detuning Δ, that is, κ ≡ κ(Δ), ΩR ≡ ΩR(Δ), Lν(ω) ≡ L(ω, Δ) etc. DS Laser Conditions. The term containing the THz mode occupancy |α|2 in the denominator of eq 14 is responsible for the saturation of DS inversion and leads to the depletion of the gain. If we aim at obtaining the amplification condition, we can neglect this term as lasing starts when |α|2 ≈ 0. So, if the condition G(|α|2 = 0) > 2Γc

Figure 3. The domain of existence of lasing in the parameter space of cavity Q-factor, inhomogeneous broadening width ξ, and resonant Rabi splitting Ω. The remaining parameters are Ωc/2π = 2 THz, nqd = 1013 cm−2, deg = 10 D, Γ = 0.02 ps−1, χ0 = 3.16 × 10−5 ps−1 (corresponds to the ensemble averaged value of permanent dipole moment, dee = 75 D66). At the each point, value of average detuning, Δ0, is chosen in such a way that the condition of resonance,

(18)

is satisfied (see eq 13), any fluctuation of the cavity field grows and lasing occurs. Fulfillment of this condition is determined by several independent parameters of the system. First of all, a cavity with a high quality-factor Q is necessary. Second, the number of QDs effectively interacting with the THz mode should be maximized. According to eq 14, the gain is substantial only for those QDs for which DS splitting ΩRν is close to the THz cavity eigenfrequency, Ωc, that is, lies within the homogeneous profile L(ω) centered around Ωc, see Figure 2b. However, the number of QDs belonging to this domain is rather low because the value of inhomogeneous broadening is big. The shape of inhomogeneous broadening is shown by the gray curve in Figure 2b (cf. with green curve denoting homogeneous line). As a result, a minor part of the whole amount of QDs in the ensemble effectively contributes to amplification of the cavity field. Analogously to eq 17 we can estimate this value as Neff = ∫ ∞ −∞ΓdL(Ωc, Δ)ρ(Δ) dΔ. For inhomogeneous broadening parameter ℏξ = 5 meV, which corresponds to fwhm about 14 meV, we obtain Neff/N ≈ 10−3. Hence, in order to maximize gain one needs, first, to engineer properly the QD ensemble (or take the cavity with the specific frequency Ω c ) in such a way that the maximum of inhomogeneous distribution of Rabi splitting ΩRν coincides with Ωc, see Figure 2b. Besides, it is necessary to use dense QD ensembles. The value of the surface density, nqd, of QDs fabricated by the epitaxy method is limited to 1011 cm−2.86 We expect that consistent depositions of QDs in several vertical stacked layers (cf. with refs 87 and 88) can allow us to increase this value by at least an order of magnitude. Next, the key parameter, which can be easily tuned experimentally, is the intensity of driving field proportional to the resonant Rabi splitting Ω. The latter explicitly determines the strength of dressed QD−cavity field interaction κ and also main parameters of the DS dynamics, such as Γd, Γs, and Γz, see eqs 10 and 11. Consequently, the driving field affects the gain properties of dressed QDs in a rather nontrivial way. In order to reveal the role of the most important parameters of the system, we plot the phase diagram in the parameter space of (Q, ξ, Ω) (Figure 3). The region inside the blue domain corresponds to self-amplification of the THz cavity field, that is, condition 18 is satisfied. One can easily recognize that lasing in

Ωc = Ω R0 =

Δ0 2 + Ω2 , holds.

a strongly inhomogeneously broadened ensemble demands both high Q-factor of the THz cavity (up to 104) and high intensity of the driving field. The first condition, although it represents a challenge for practical implementation, is nevertheless reachable in photonic crystal slabs.77 The requirement of having strong driving fields on the other hand involves certain practical obstacles. The analysis of the complicated dependence of the unsaturated net gain G(|α|2 = 0) on the resonant Rabi splitting Ω, given by eqs 14 and 17, reveals that the most appropriate conditions for lasing are achieved for a high values of Ω, which are close to Ωc, see Figure 4a. This is valid for a wide range of values of inhomogeneous broadening, as it is illustrated in the inset to

Figure 4. Properties of the THz lasing. (a) Dependence of the unsaturated net gain G(|α|2 = 0) on the resonant Rabi splitting for different values of inhomogeneous broadening. For all curves, the maximum of spectral distribution of QDs coincides with cavity mode frequency Ωc/2π = 2 THz. Gray dashed line corresponds to the cavity loss rate. Below this line lasing cannot occur. In the inset, the value of resonant Rabi splitting corresponding to maximal gain vs inhomogeneous broadening parameter ξ is shown. (b) Dependence of the occupancy of the THz cavity mode on the value of inhomogeneous broadening for different resonant Rabi splittings ℏΩ. For both panels Γc = 1/2000 ps−1. Other values are the same as for the Figure 3. Black points label parameters corresponding to Figure 5. 2731


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Figure 5. Pulse dynamics of the THz laser. (a) Time dependence of THz photon number |α|2 (red curve, left axis). The magnitude of resonant Rabi splitting ℏΩ(t) is shown with blue curve (right axis). Initial conditions: Szν = Sstzν, S = 0, population of cavity mode α corresponds to the thermal occupation under T = 30 K. THz cavity frequency is Ωc/2π = 2 THz. (b) Dynamics of the spectral density of DS population inversion S̅z(Δ). Spectral position of QDs participating in lasing being currently in resonance with THz cavity mode, ΩRν(t) = Ωc, is shown with white dashed curve. (c) Dependence of the quantum efficiency η on duration of the Gaussian driving pulse τ for three different fixed values of total energy (per unit area) accumulated in the pulse. In the inset, the domain of short pulses is magnified. (d) The same as in panel c but for fixed τ and varying energy density Jd.

drops since the number of QDs, Neff, effectively participating in lasing decreases. Evidently, the maximum possible value of THz laser intensity corresponds to idealized homogeneous system of N identical QDs with Rabi splitting ΩR, being in resonance with the THz cavity mode Ωc. This value can be obtained from the steady state solution of the full set of eqs 9, assuming that all parameters are independent of ν (cf. with ref 49):

Figure 4a, where the position of the maximum of the net gain is shown. Thus, one needs to use high intensities of dressing laser to provide efficient gain of the THz mode. For realistic values of ξ and Q, the necessary resonant Rabi splitting ℏΩ should be on the order of millielectronvolt. For the typical value of offdiagonal dipole moment for GaN QDs, which is deg = 10 D,89 such splitting corresponds to the driving laser intensities in the diapason of MW/cm2 (cf. with ref 59). Such intensities in the case of CW laser excitation would inevitably lead to melting of the sample. Thus, pulsed excitation is the only possible way to reach THz lasing in the system under consideration. Properties of DS Lasing. One of the most important output characteristics of every light source is intensity, that is, the number of emitted photons per unit time. In our case, it is proportional to the occupancy of the THz cavity mode |α|2. Let us consider first the case of the quasi-CW-regime corresponding to excitation by a rectangular pulse. In this simplest case, it is possible to estimate the maximum value of |α|2, which is achieved in the steady-state regime when saturated gain is equalized by losses, that is when the condition

G(|α|2 ) = 2Γc

|α|max 2 = N

Γs ΓΓ − z 2d 4Γc 4κ

(20)

Since the values shown in Figure 4b correspond to the steady state regime, they are reachable if the driving pulse is long enough. The necessary duration of the driving pulse can be estimated from eq 13 supposing that the gain G remains unsaturated until |α|2 reaches its stationary value. For instance, if ℏΩ = 4 meV and ℏξ = 5 meV (black points in Figure 4a.b) this rough estimate gives for pulse duration the value about 2 ns. Consequently, operation in the nanosecond regime is most appropriate for the considered THz laser source. According to Figure 4b, for every given driving field intensity, some critical value of ξ exists above which lasing is impossible to achieve. The bigger is the value of Ω, the wider is the region where steady-state lasing is possible. For parameters under consideration, the Rabi splitting of about ℏΩ = 2 meV is enough to obtain lasing for ℏξ = 5 meV. However, larger values of driving intensity yield much better output laser characteristics.

(19)

is satisfied. The THz cavity occupancy can be obtained from the solution of this transcendental equation. In Figure 4b, dependence of the value of |α|2 versus the inhomogeneous broadening parameter ξ is shown for different values of resonant Rabi splitting Ω. As expected, with increase of inhomogeneous broadening, the amplitude of cavity field 2732


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Generation of THz Laser Pulses. The mostly realistic case corresponds to the excitation of a QD ensemble by a pulse of driving field having a Gaussian time profile. In this case, complete information about output laser characteristics can be extracted from the numerical solution of eqs 9. In order to simulate dynamics of a huge number of different QDs participating in lasing, we discretize spectral distribution of QDs over a finite number of domains, which are small enough to consider all QDs belonging to the single domain as identical and characterize them by only a couple of eqs 9b and 9c. Then for summation in eq 9a, we multiply the values of Sν by the corresponding weighting coefficients, which are equal to the number of QDs in the νth domain. We also add to the rhs of eq 9a the Langevin term ζ(t) accounting for the fluctuations due to thermal noise and necessary to switch on the lasing. We suppose that ⟨ζ(t)ζ*(t′)⟩ = 2ΓcNthδ(t − t′), where Nth is a thermal occupation of the cavity mode. Formation of the THz laser pulse induced by the Gaussian pulse of the driving field with amplitude E = E0 exp[−(t − t0)2/ 2τ2] and duration τ = 2 ns is shown in Figure 5a. The intensity of THz radiation emitted from the area S = πλc2/16 is determined as ITHz(t) = 8Γc|α|2ℏΩc3/(π3c2). Thus, the total energy density accumulated in the THz pulse JTHz = ∫ ITHz(t) dt is about 9 × 10−6 J/m2 for the case shown in Figure 5a. At the same time, the energy density in the driving pulse is Jd ≃ 1.7 × 103 J/m2. It allows the expectation of the typical value of the quantum efficiency (the ratio of emitted THz photons NTHz = JTHzS/(ℏΩc) to number of optical photons Nd = JdS/(ℏωd) in the driving pulse) of the proposed THz laser source η = NTHz/ Nd on the order of 10−5 to 10−6. The more detailed analysis demonstrates that the exact value of quantum efficiency η is determined by the parameters of the driving pulse. Because of the purely dynamical nature of the onset of the lasing even for fixed driving energy density Jd, an output THz energy JTHz varies with duration τ of the pulse. Multiple solution of eqs 9 for fixed Jd shows that quantum efficiency η and therefore output THz energy reduces with the increase of pulse duration, see Figure 5c. The best values of η are possible to achieve with subnanosecond pulses, corresponding to large resonant Rabi splittings Ω. It confirms our conclusion that higher gain is obtained for Ω that is close (but not exactly equal) to Ωc, see Figure 4a. However, the efficiency dramatically reduces with further decrease of pulse duration (up to few hundreds of picoseconds), see inset to Figure 5c. It is because the amplitude of Ω(t) is higher than the lasing frequency for these pulses. Hence the condition of resonance between the QDs and the cavity is not satisfied continuously during the action of the driving field, and the total amplification of the THz field diminishes. A similar decline of quantum efficiency occurs for low-energy pulses (small Jd) when Ω is small and THz gain is too low, see Figure 5d. At the same time, the appropriate values of driving energy densities Jd are limited by the effect described above, which prevents the use of too short or too high-energy pulses (see curve corresponding to τ = 0.5 ns in Figure 5d, which falls down for Jd > 1500 J/m2). So the optimal values of Jd are about units of kJ/m2. Amplification of the THz mode is accompanied by reduction of DS inversion, which is connected with transfer of the excitation from the subsystem of dressed QDs to the THz field.

Behavior of the QD subsystem can be illustrated by spectral density of inversion of dressed QDs determined as Δ+ dΔ

Sz̅ (Δ) =

∫Δ

Sz(δ)ρ(δ) dδ

Δ+ dΔ

∫Δ

ρ (δ ) d δ

(21)

Time dynamics of Sz̅ (Δ) is shown in Figure 5b. When the THz field increases substantially, a gap forms in the distribution of the population inversion. The corresponding spectral profile of Sz̅ (Δ) at the moment when driving field reaches its maximum (vertical dash-dotted line in Figure 5b) is shown in the inset to Figure 2. This phenomenon is also known as a spectral hole burning effect. The specific shape of this hole in our case indicates that different QDs effectively contribute to the lasing in different moments. Actually, since the value of Rabi splitting ΩRν varies with time in our case, the spectral position of QDs that satisfy the condition of maximal gain, ΩRν = Ωc, also changes in time, see white dashed curve in Figure 5b. This trajectory coincides with the region of DS inversion depletion. Thus, using pulses with varying intensity allows increasing the number of QDs involved in lasing in comparison with the case of quasi-CW driving field. It is important to emphasize that the discussed mechanism of THz radiation emission can be also described in terms of inelastic scattering of an optical photon accompanied by creation of a THz quantum. Actually, DS population inversion, Sz̅ ν > 0, implies positive detunings, Δν > 0, see eq 12. According to definitions 5a in the case of large Δν, the upper DS |1⟩ν represents a mixture of a major part of the QD ground state |g⟩ν and a tiny amount of the excited state |e⟩ν. Thus, in the absence of THz photons, these QDs are in a state close to the ground state |g⟩ν since they are far detuned from the driving field and can hardly absorb it. However, the increase of the THz cavity occupancy during the onset of lasing enables absorption of the driving field by QDs, which switch from the upper to the lower DS emitting THz photon. In the QD eigenbasis, the latter process corresponds to the excitation of the upper level |e⟩ν by absorption of the driving field. However, the frequency of the driving field exceeds the frequency of the interband transition of these QDs, Δν > 0. Hence the excess energy should be transferred to the THz mode, which is not empty now and amplifies such relaxation by the effect of bosonic stimulation. Note that this process becomes possible only due to the presence of the stationary dipoles of asymmetric QDs. The effect of such inelastic scattering of the driving field should always occur during interaction of light with a two-level system with broken inversion symmetry.90 However, in our case its efficiency is increased by the presence of the THz cavity. Influence of Phonon-Induced Dephasing. Since the primary goal of this paper is to propose a practical device, it is necessary to estimate an impact of dephasing processes on operation of the proposed laser. In semiconductor structures, the main cause of dephasing is an interaction with a phonon bath. In the case when it is strong enough, for instance, in the high temperature91 regime, the induced coherency in DS basis can be broken leading to the spoiling of lasing. Accounting for dephasing due to the interaction with phonons can be done within a Lindblad approach.92,93 It implies modification of the master eq 6 with extra terms responsible for phonon-mediated processes. In the absence of an intense driving field, the presence of phonons leads only to 2733


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Note that the impact of phonons should rise as the temperature increases during the action of the driving pulse since the occupation of phonon modes np(Ωc) grows. However, for the particular lasing frequency the value of np(Ωc) is less than 10 up to T = 103 K. Thus, heating of the structure cannot reduce the population inversion between DSs. The value of dephasing rate Γϕν, which affects the mutual

pure dephasing for off-diagonal elements of the density matrix written in the basis of bare states. It does not contribute to the populations of eigenstates of QDs since phonon frequency is sufficiently lower than the frequency of transition between bare states. However, this is not the case for dressed QDs. In the presence of driving field, phonons trigger both dephasing for DS polarization and transitions between DS levels. In the limit of strong light−matter coupling after the secularization procedure described in ref 92, the Lindblad terms correspondi n g t o t h e a f o r e m e n t i o n e d p r o c e s s e s re a d : 9 2 , 9 4 Γ↓ ν3[ςν−]ϱ + Γ↑ ν3[ςν+]ϱ + Γϕν3[ςνz]ϱ, where

coherence between DSs, is proportional to

(22a)

Γz̃ ν = Γzν + 2(Γ↑ ν + Γ↓ ν)

(22b)

Γs̃ ν = Γsν − 2(Γ↓ ν − Γ↑ ν)

(22c)

which is

not negligible in general, see ref 93. However, according to eq 22a the presence of Γϕν causes increasing the width of homogeneous line Lν(ω), see eq 15. As a result the number of QDs effectively amplifying the THz cavity field, Neff, grows and the total efficiency of the laser does not go down but improves. This counterintuitive result is a direct consequence of the large inhomogeneous broadening (in comparison with the spectral width of the gain), which excludes the main part of QDs from lasing. In this case, the loss of coherence between DSs caused by dephasing is compensated by the increase of number of emitters participating in the gain. In order to illustrate this effect, we introduce a factor f, determining Γ̃dν = fΓdν. For simplicity, we suppose that f is identical for all QDs and neglect phonon-induced transitions between DSs. Then in the case of quasi-CW excitation, the steady-state number of THz photons calculated from eq 19 grows as factor f increases, see Figure 6.

3[A]ϱ ≡ A ϱA† − (A†A ϱ + ϱA†A)/2. The rate Γ↓ν corresponds to transitions from the upper to the lower DS with an emission of phonon, while Γ↑ν is responsible for the inverse processes, and Γϕν is the dephasing acting in the DS basis. The presence of a phonon bath leads to renormalization of the dissipative terms in Maxwell−Bloch eqs 9:

Γd̃ ν = Γdν + 2Γϕν + (Γ↓ ν + Γ↑ ν)

+(ω) , ω ω= 0

The values of phonon-mediated transition rates are determined by the specific coupling mechanism of QDs to the phonon bath. For GaN/AlN QDs, an interaction with acoustic phonons is governed by both deformation potential coupling and piezoelectric coupling.95 Without loss of generality, we write Γ↓ = (Ω/2Ω R )2 +(Ω R )(n p(ΩR ) + 1), Γ↑ = (Ω/2Ω R )2 +(Ω R )n p(Ω R )

(23)

where +(ω) is the effective spectral density, which accounts simultaneously for all phonon-coupling mechanisms, and np(ω) = (eℏω/(kbT) − 1)−1 is the thermal occupation of phonon modes. According to experimental data, the spectral density is not uniform but has a high-frequency cutoff, ωc, that is, we take 2

Figure 6. Dephasing strength dependence of THz photon number calculated by eq 19 in the case of quasi-CW regime. For both lines, ℏΩ = 4 meV. Other values are the same as for Figure 4.

2

+(ω) ∝ e−ω / ωc . Since we are particularly interested in dephasing for those QDs that participate in lasing, we take ω = Ωc. In the case of coupling due to deformation potential, which is the strongest (although not the dominant) mechanism for GaN/AlN QDs,95 the value of cutoff can be estimated as ωc = 2 vs/rQD, where vs is a sound velocity (∼11 × 103 m/s for AlN) and rQD is the electron and hole confinement length, which can be estimated as QD transverse size (we take rQD = 5 nm). For these parameters, ωc ≈ 3.11 ps−1, which is four times smaller than the lasing frequency, Ωc = 2·2π ps−1. Thus, the rate of phonon induced relaxation of DS inversion is sufficiently suppressed in the spectral range under consideration. Definitions 23 also contain the prefactor (Ω/2ΩR)2 arising when writing phononic Lindblad terms in the basis of DSs.92 The presence of this coefficient reflects the fact that phonons interact only with excited QDs. Thus, the efficiency of this interaction is proportional to the fraction of matter component in the Rabi level, which is initial during the phonon-assisted transition. So, if the condition Ω2 ≪ ΩR2 is satisfied (this is the case shown in Figure 5a,b), the effect of phonon-mediated relaxation of DSs are additionally suppressed by the photonic character of the upper dressed state.

Note that in the case of homogeneous ensemble (ξ = 0), the dephasing has only destructive character and leads to the decrease of lasing intensity. However, this effect is rather small and is not visible on the scale of Figure 6. Summarizing all the discussion presented above, the influence of dephasing due to interaction with a phonon bath can be effectively suppressed by the proper choice of system parameters. The main restriction is applied to the lasing frequency, which should be on the order of units of terahertz. Reduction of Ωc down to sub-terahertz range leads to strong increase of the density of states +(Ωc) and the phonon occupation np(Ωc) triggering phonon-mediated transitions between DS, which spoil lasing.

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CONCLUSIONS In conclusion, we present the theoretical study of an ensemble of asymmetric QDs embedded into a terahertz cavity and dressed by external electromagnetic field of optical frequency. Using mean-field approximation, we model the system by a set of Maxwell−Bloch equations for the dressed states taking into 2734


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(6) Diao, Z.; Bonzon, C.; Scalari, G.; Beck, M.; Faist, J.; Houdre, R. Continuous-wave vertically emitting photonic crystal terahertz laser. Laser & Photonics Reviews 2013, 7, L45. (7) Kohler, R.; Tredicucci, A.; Beltram, F.; Beere, H.; Linfield, E.; Davies, A.; Ritchie, D.; Iotti, R.; Rossi, F. Terahertz semiconductorheterostructure laser. Nature 2002, 417, 156−159. (8) Williams, B. Terahertz quantum-cascade lasers. Nat. Photonics 2007, 1, 517−525. (9) Yampolsky, N. A.; Fraiman, G. M. Conversion of laser radiation to terahertz frequency waves in plasma. Phys. Plasmas 2006, 13, 113108. (10) Kavokin, A. V.; Baumberg, J. J.; Laussy, F. P.; Malpuech, G. Microcavities; Oxford University Press, 2007. (11) Kavokin, A.; Liew, T. C. H.; Schneider, C.; Hofling, S. Bosonic lasers: The state of the art. Low Temp. Phys. 2016, 42, 323. (12) Christopoulos, S.; von Hogersthal, G. B. H.; Grundy, A. J. D.; Lagoudakis, P. G.; Kavokin, A. V.; Baumberg, J. J.; Christmann, G.; Butte, R.; Feltin, E.; Carlin, J.-F.; Grandjean, N. Room-Temperature Polariton Lasing in Semiconductor Microcavities. Phys. Rev. Lett. 2007, 98, 126405. (13) Daskalakis, K. S.; Eldridge, P. S.; Christmann, G.; Trichas, E.; Murray, R.; Iliopoulos, E.; Monroy, E.; Pelekanos, N. T.; Baumberg, J. J.; Savvidis, P. G. All-dielectric GaN microcavity: Strong coupling and lasing at room temperature. Appl. Phys. Lett. 2013, 102, 101113. (14) Zhang, B.; Wang, Z.; Brodbeck, S.; Schneider, C.; Kamp, M.; Hofling, S.; Deng, H. Zero-dimensional polariton laser in a subwavelength grating-based vertical microcavity. Light: Sci. Appl. 2014, 3, e135. (15) Bajoni, D.; Senellart, P.; Wertz, E.; Sagnes, I.; Miard, A.; Lemaitre, A.; Bloch, J. Polariton Laser Using Single Micropillar GaAsGaAlAs Semiconductor Cavities. Phys. Rev. Lett. 2008, 100, 047401. (16) Azzini, S.; Gerace, D.; Galli, M.; Sagnes, I.; Braive, R.; Lemaitre, A.; Bloch, J.; Bajoni, D. Ultra-low threshold polariton lasing in photonic crystal cavities. Appl. Phys. Lett. 2011, 99, 111106. (17) Duan, Q.; Xu, D.; Liu, W.; Lu, J.; Zhang, L.; Wang, J.; Wang, Y.; Gu, J.; Hu, T.; Xie, W.; Shen, X.; Chen, Z. Polariton lasing of quasiwhispering gallery modes in a ZnO microwire. Appl. Phys. Lett. 2013, 103, 022103. (18) Li, F.; et al. From Excitonic to Photonic Polariton Condensate in a ZnO-Based Microcavity. Phys. Rev. Lett. 2013, 110, 196406. (19) Xie, W.; Dong, H.; Zhang, S.; Sun, L.; Zhou, W.; Ling, Y.; Lu, J.; Shen, X.; Chen, Z. Room-Temperature Polariton Parametric Scattering Driven by a One-Dimensional Polariton Condensate. Phys. Rev. Lett. 2012, 108, 166401. (20) De Liberato, S.; Ciuti, C.; Phillips, C. C. Terahertz lasing from intersubband polariton-polariton scattering in asymmetric quantum wells. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87, 241304. (21) Liew, T. C. H.; Glazov, M. M.; Kavokin, K. V.; Shelykh, I. A.; Kaliteevski, M. A.; Kavokin, A. V. Proposal for a Bosonic Cascade Laser. Phys. Rev. Lett. 2013, 110, 047402. (22) Huppert, S.; Lafont, O.; Baudin, E.; Tignon, J.; Ferreira, R. Terahertz emission from multiple-microcavity exciton-polariton lasers. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 241302. (23) Kavokin, K. V.; Kaliteevski, M. A.; Abram, R. A.; Kavokin, A. V.; Sharkova, S.; Shelykh, I. A. Stimulated emission of terahertz radiation by exciton-polariton lasers. Appl. Phys. Lett. 2010, 97, 201111. (24) Kavokin, A. V.; Shelykh, I. A.; Taylor, T.; Glazov, M. M. Vertical Cavity Surface Emitting Terahertz Laser. Phys. Rev. Lett. 2012, 108, 197401. (25) Savenko, I. G.; Shelykh, I. A.; Kaliteevski, M. A. Nonlinear Terahertz Emission in Semiconductor Microcavities. Phys. Rev. Lett. 2011, 107, 027401. (26) Cohen-Tannoudji, C.; Dupont-Roc, J.; Grynberg, G. AtomPhoton Interactions: Basic Processes and Applications; Wiley, Chichester, 1998. (27) Scully, M. O.; Zubairy, M. S. Quantum Optics; Cambridge University Press, Cambridge, 2001. (28) Autler, S. H.; Townes, C. H. Stark Effect in Rapidly Varying Fields. Phys. Rev. 1955, 100, 703.

account inhomogeneous broadening in the QD ensemble, incoherent losses, and thermal noise existing in the system. Analyzing the stationary solutions of the obtained equations, we obtain the conditions for the amplification of the THz cavity mode. The numerical simulation of the obtained system of stochastic equations allows us to describe the generation of terahertz pulses and estimate the power of the lasing and its quantum efficiency. We also provide several simple arguments that prove that interaction with a phonon bath cannot affect considerably the properties of the proposed laser. The most significant advantage of the proposed THz laser consists of the ability to smoothly and easily tune the lasing frequency by adjusting the intensity of the driving field. The frequency range available for tuning is practically limited by the width of a stop band of a photonic crystal cavity. In contrast, for the majority of THz lasing devices and prototypes of various types, the lasing frequency is fixed by the properties of a structure or a medium.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Igor Yu. Chestnov: 0000-0002-3949-5421 Vanik A. Shahnazaryan: 0000-0001-7892-0550 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS I.Yu.Ch. acknowledges funding from RFBR Grant nos. 16-3260102-mol_a_dk and 15-59-30406-rt-omi, by the President of Russian Federation for state support of young Russian scientists, Grant MK-2988.2017.2. Numerical simulation of laser dynamics was supported by Ministry of Education and Science of the Russian Federation, project no. 16.1123.2017/ 4.6. V.A.Sh. and I.A.Sh. acknowledge funding from mega-grant no. 14.Y26.31.0015 of the Ministry of Education and Science of Russian Federation. Determination of the laser properties was supported by the project 3.2614.2017/4.6 of the Ministry of Education and Science of Russian Federation. The work was also supported by FP7 ITN NOTEDEV, Horizon 2020 project CoExAN, Icelandic Research Fund, Grant No. 163082-051. I.Yu.Ch. also thanks the University of Iceland for the hospitality.

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REFERENCES

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