Superradiant Terahertz Emission by Dipolaritons
Dipolaritons are mixed light-matter quasiparticles formed in double quantum wells embedded in microcavities. Due to resonant coupling between direct and indirect excitons via electronic tunnelling, dipolaritons possess large dipole moments. Resonant excitation of the cavity mode by a short pulse of light induces oscillations of the indirect exciton density with a characteristic frequency of Rabi flopping. This results in oscillations of classical Hertz dipoles array which generate supperradiant emission on a terahertz (THz) frequency. Resulting THz signal may be enhanced using the supplementary THz cavity in the weak coupling regime.
Introduction.—The generation and frequency modulation of terahertz radiation are among the major technological challenges nowadays Siegel (). Applications of terahertz sources span from communication technologies to medicine and security. The existing THz emitters are based on a large variety of physical principles Gallerano (). One possible solution is a conventional solid state oscillator based on a high frequency Gunn or tunnelling diode Eisele (). The operating frequency of such oscillator is limited to the microwave region and the lower boundary of terahertz range. Second, quantum cascade lasers (QCL) allow for coherent emission of radiation in a terahertz range Faist (); Kohler (). Exploiting the multiple photon emission from intersubband transition of wide quantum wells Geiser (), QCL cover the upper boundary of THz range with relatively high power of emission (up to 50 mW). QCL operate at cryogenic temperatures which strongly limits their application area. The laser driven terahertz emitters form a third group of THz sources, where femtosecond optical pulse illuminating the semiconductor structure leads to oscillation of carrier density which generates terahertz radiation Auston (); Fattinger (); Heinz (); Johnston (). Finally, the wide group of THz emitters are free electron based sources like klystron or free electron laser. These are powerful but quite bulky sources. Optimization of spectral characteristics, size, efficiency, cost and operation temperature of terahertz devices is one of the priority objectives of modern optoelectronics. Here we propose a compact optical to terahertz radiation converter allowing for efficient frequency modulation and operating at high temperatures.
Theoretical proposals for THz sources in solid state and semiconductor physics present a rich diversity. One example is the use of carbon nanostructures, in particular, carbon nanotubes (CNT’s) Kibis (); Portnoi (), or Aharonov-Bohm quantum rings Alexeev (). Another area where a possibility of THz generation has been widely studied theoretically is polaritonics — the interdisciplinary research area at the boundary of solid state physics and quantum optics KavokinBook (); PolaritonDevices (). Polaritonic devices are based on the strong coupling between excitons in semiconductor quantum wells and confined photons. Proposals include the polariton based THz emitters where signal is generated from transitions between upper and lower polariton branches KVKavokin (); Savenko (); delValle () and transition between and exciton states, the latter one strongly coupled to the cavity mode Kavokin (). Recent theoretical studies also include the proposal for bosonic cascade lasers realized due to the multiple THz photon emission by excitons (exciton-polaritons) confined in a parabolic potential trap Liew (). Our present proposal is based on the recent experimental realization of dipolaritons — cavity exciton-polaritons characterized by large dipole moments Cristofolini ().
The mechanism for terahertz signal generation described in this Letter relies on the beats between spatially direct and indirect excitons. Spatially indirect excitons are composed of electrons and holes situated in separate quantum wells (QWs) Lozovik (). Coupled QW systems where separation of electrons and holes and energy splitting between direct and indirect excitons can be controlled by applied electric fields have been studied in the recent decades ButovPRL (); SnokeScience (); ButovNature (). The radiative lifetimes of indirect excitons are usually much longer than those of direct excitons due to the lower electron–hole overlap. Another important feature of indirect excitons is their large dipole moments in the normal to QW plane direction resulting in strong exciton-exciton interactions KyriienkoIndirect ().
Recently, it has been shown that exciton polaritons and spatially indirect excitons can be intermixed in the biased semiconductor microcavities with embedded coupled QWs ChristmannPRB (); ChristmannAPL (); Cristofolini (). In these structures new quasiparticles being linear superposition of cavity photon (C), direct exciton (DX) and indirect exciton (IX) appear. They form three exciton-polariton modes, namely, the upper polariton (UP), middle polariton (MP) and lower polariton (LP) modes. These modes may be characterized by large dipole moments which is why they are referred to as dipolaritons Cristofolini (). Here we study theoretically the effect of beats between dipolariton modes due to the tunnel coupling between direct and indirect excitons, which result into superradiant THz emission by array of Hertz dipoles. Its characteristics can be tuned by the applied bias and the calculated efficiency of proposed emitter is comparable with state-of-art generators.
The model.—The structure we consider represents two QWs separated by a barrier which is sufficiently thin to allow for resonant electron tunnelling (Fig. 1). The electron wave function is shared between two QWs in this case. The optical microcavity is tuned to the wider QW exciton resonance (LQW in Fig. 1), while the right QW remains decoupled from the optical pump. Pulsed pumping creates electron-hole pairs which form direct excitons. The bias applied in growth direction induces mixing of direct and indirect exciton states. Coupling of the cavity photons to direct excitons resulting in appearance of two polariton modes has been extensively studied KavokinBook (); PolaritonDevices (). The coupling and anticrossing of DX-IX resonances tuned by electric field is also a well-known effect in GaAs/AlGaAs QW structures Kardiff (); Bayer (); Kyriienko2011 (). Both effects combined result in the appearance of dipolaritons studied recently by Cristofolini et al. Cristofolini ().
The correct treatment of a real dipolariton system involves both coherent and decoherent parts for the Hamiltonian, . The Hamiltonian corresponding to the coherent part reads
where , and are creation and annihilation operators for cavity photons, direct excitons and indirect excitons, respectively. Here , and denote cavity mode, direct exciton and indirect exciton energies, and first three terms of Hamiltonian describe energy of the bare modes. The fourth and fifth terms describe coupling between the modes, where denotes the coupling constant between photon and direct exciton, and the tunnelling rate corresponding to DX–IX coupling is . The last two terms in Hamiltonian describe the optical pumping of cavity mode with amplitude , where is a pumping frequency.
The decoherent part of dynamics of the system is mainly governed by the radiative lifetimes of the modes and phonon-scattering processes, which lead to population of thermalized exciton reservoirs SM (). They can be described by the exciton-phonon interaction Hamiltonian , where
correspond to processes with emission () and absorption () of phonon with in-plane momentum , respectively. Here and are creation and annihilation operators for direct and indirect excitons in reservoirs. denote exciton-photon interaction constants. These incoherent processes can be conveniently treated using master equation for the density matrix, , where corresponds to standard Lindblad superoperator describing the finite lifetime of the modes Savenko (); Kavokin (), and is responsible for phonon-assisted processes (see details of derivation in SM ()).
In the following we shall be interested in the large occupation numbers of the modes, and mean-field approximation can be applied. The dynamics of occupation numbers of the modes can be found from the master equation with operators treated as classical fields SM ()
where denotes averaging using the density matrix . Here we applied the rotating wave approximation , and introduced the photonic and excitonic detunings. The pumping amplitude reads as , where denotes detuning of the coherent optical pumping. The damping rates of the modes are defined by parameters , . Typical lifetimes of the modes are ps, ns and ns. The reservoir dynamics is governed by scattering rate , being proportional to a square of exciton-phonon interaction constant, and is a constant accounting for the energy conservation, which is taken to be inverse broadening of exciton states divided by square of Plank constant. For simplicity we consider , assume reservoir states, and use total scattering rate ps Savenko (). is Bose distribution of phonons.
Additionally, we derived dynamic equations for occupation numbers for the modes going to higher order of mean-field theory and verified validity of the system (S57)–(7) for chosen pumping conditions. The former model represents the general description of dipolariton system and allows for accounting of various decoherent processes including pure and phonon-assisted dephasing, which lead to faster decay of oscillations between modes (see SM (), section B). The effects of non-linearities introduced by exciton-exciton interactions are discussed in the section C of supplemental material.
Results and discussion.—We calculate numerically the dynamics of the system with coupled quasiparticles subjected to a picosecond pulsed optical pumping. The presence of mixing terms between different modes implies the oscillating behavior similar to Rabi flopping in the classical model of a two-level system subjected to a time-varying field. It is important to note that our system has several characteristic frequencies. They are governed by the exciton-photon coupling strength and detuning , the IX-DX coupling strengths and detuning . Additionally, the pumping frequency governs the efficiency of the pump. Varying these characteristic frequencies one can control the frequency, magnitude and damping rate of indirect exciton density oscillations.
While the coupling constants and are dependent on the geometry of the structure and can hardly be tuned for a given sample, the detunings between modes and are strongly sensitive to the applied electric field and the incidence angle of the cavity pump, respectively Cristofolini (). Tuning these parameters one can bring the oscillating dipole system to different regimes. If the cavity mode is far-detuned from IX-DX anti-crossing the light-exciton coupling is weak, which will be referred to as the regime I. If the detuning is small the strong intermixing of IX, DX, and C modes takes place, which corresponds to the regime II.
First, we assume low temperature of the system, when phonon processes are suppressed. The behavior of the system in the regime I is shown in Fig. 2(a) for the detuning meV. We observe antiphase oscillations of IX and DX densities with decaying amplitudes. The inset in Fig. 2(a) shows a zoom of long-standing phase-locked oscillations which last for several tens of picoseconds. For the electric field corresponding to the resonance between IX and DX modes, , the frequency of oscillations is given by THz. The electric field at resonance is equal to , where is exciton detuning at zero applied field, and is a separation between QWs centers. Considering InGaAsGaAsInGaAs (10 nm/4 nm/10 nm) structure studied in Ref. ChristmannAPL (), where meV and nm, it can be calculated as kV/cm.
The time dependence of indirect exciton density oscillations , calculated numerically from the Eqns. (S57)–(S59), can be fitted with the analytical function , where is the frequency of oscillations, denotes the magnitude of oscillations which decreases in time with the damping rate . Tuning the electric field, the frequency of generated oscillations changes in the range of several THz due to its dependence on IX-DX detuning [Fig. 3(a)], given as ButovJPCM (). Other important parameters of the system such as the amplitude of oscillations and dimensionless oscillation quality factor defined as a ratio of magnitude to the decrement of oscillations can be tuned by variation of pumping conditions, as well as by the applied electric field SM ().
Rising the temperature of the system, the phonon scattering and subsequent population of the reservoir states lead to increase of characteristic decay rate of the oscillations [Fig. 3(b)]. However, the beats of excitons density are still observable for comparably high frequency, and can be exploited in the pulsed regime.
Reducing the detuning between exciton and photon modes one can bring the system into the regime II. In this regime, the IX density oscillations are observed as well, while their quality factor is different. Strong interactions of the cavity mode with the IX-DX resonance result in a higher magnitude of oscillations than in the regime I [Fig. 2(b)]. The damping rate of these oscillations is also larger. Clearly, the regime of strong coupling between all modes is advantageous for the pulsed pumping regime and it allows for high power generation. On the other hand, the regime I is preferential for the long-standing signal generation providing higher quality factor , while lower amplitude of the emission.
We have shown that due to the coupling between modes IX and DX densities oscillate with a THz frequency. This infers that a dipolariton is an oscillating dipole, with a dipole moment in direction changing periodically from value of IX to corresponding to DX. One can define the total dipole moment of the system as , where is the number of indirect excitons within the area illuminated by the pumping light. Since we have shown before that after initial transient regime the density of indirect excitons is a decaying harmonic function of time, the total dipole moment of the system can be found as . Here is the maximum density of indirect excitons.
The total intensity of the far field radiation emitted by a classical Hertz dipole can be found as Landau (), where is the total dipole moment of the dipolariton array, is vacuum permittivity and is the speed of light. For the particular case of an array of harmonic dipole oscillations the intensity is , where is a dipole moment of indirect exciton. Here for simplicity we did not consider the damping part of oscillations. It will result in damping of the total power of emission, which is why to achieve stable cw radiation one therefore needs to using a sequence of pump pulses. Similarly to the conventional case of an elementary dipole emitter, the polar pattern is given by , where is an angle between direction of radiation and the growth axis of the structure SM ().
It is important to note that the total emitted intensity is proportional to the square of the density of indirect excitons and is dependent on the pump intensity in a non-linear way. This is a manifestation of the superradiance Dicke (); Bohnet (); Yukalov () effect: due to the interference of coherent in-phase oscillations of elementary dipoles the output power is enhanced. This effect is sensitive to the quality factor of the cavity: the longer Rabi oscillations persist the stronger the amplification effect is. Superradiance is a specific feature of the dipolariton THz emitter, which makes it more efficient than any existing laser-to-THz converter.
For meV and the typical distance between QWs of nm, the power of THz emission of an elementary dipole formed by a dipolariton is W aW. The total power emitted is given by this quantity multiplied by the square number of elementary dipoles . For the typical concentration of indirect excitons achievable experimentally of and a m diameter of the pumping spot one obtains W. By embedding a stack of double quantum wells in each microcavity and using several sets of cavities on one chip one can obtain the output power similar to one of quantum cascade lasers. Furthermore, to improve the efficiency of THz radiation, the system can be placed in an external cavity tuned to the THz frequency Geiser (); Walther (), with efficiency of emission being increased by the Purcell factor of the external cavity (see SM (), Fig. S5, for the sketch of setup).
Conclusions.—We have shown that the system of dipolaritons can provide an efficient tunable source of THz radiation. In this system, due to the direct-indirect exciton coupling, the optically excited dipolariton system exhibits density oscillations with a subpicosecond period. The dipole oscillations lead to the superradiant emission of radiation in the THz frequency range. The spectral properties as well as the power output of this system are expected to be strongly improved with respect to the existing laser induced THz emitters. Additionally, we propose a way to enhance the radiation efficiency using a supplementary THz cavity in the weak coupling regime.
We thank Jeremy J. Baumberg and Sven Höfling for useful discussions on the subject. This work has been supported by FP7 IRSES projects “POLATER” and “POLAPHEN”. O. K. acknowledges the support from Eimskip Fund. A. K. acknowledges support from Russian ministry of Science and Education. I. A. S. acknowledge the support of Tier1 project “Novel polaritonic devices”.
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Supplemental Material: Superradiant terahertz emission by dipolaritons
.1 A: Derivation of dynamic equations (lowest order of mean-field theory)
Here we present the derivation of Eqns. (4)–(8) of the main text, which describe the dynamics of dipolariton system in the lowest order of mean field theory.
Coherent part. In order to find the macroscopic occupation of the modes one can treat fields as classical and calculate the dynamics of the system using mean values of their creation (or annihilation) operators. Using the Hamiltonian (1) and the master equation, one can write the equation of motion for the mean value of operator as
where we used the cyclic properties of the trace. The commutator in the RHS may be written using explicit form of the Hamiltonian as
and it can be calculated part by part. Naturally, only the first, fourth and penultimate terms contribute to the equation. Using standard commutation relations for bosons, one obtains , , and . The coherent part of the equations for and can be written in the same way.
The first term gives , where we used the relation . The second term reads as , and third term yields . The dissipative part of dynamics is described by a kinetic equation
The lifetime of and modes can be introduced in the same fashion.
Interaction with phonons. Next, we want to account for interaction of indirect and direct excitons with a thermal reservoir of acoustic phonons in the system. In general, it requires the accounting of full momentum space of exciton-photon system and represents a formidable task for a two-dimensional system. In the following we shall assume pumping at a normal incidence angle, so that the optically generated excitons are having zero in-plane momentum. The emission or absorption of acoustic phonon therefore leads to transition of particles to the thermalized reservoir states.
The generic Hamiltonian of exciton-phonon interaction can be written as
where and are creation and annihilation operators for acoustic phonon with momentum . Here are direct (indirect) exciton-phonon interaction constants, and are energies of direct and indirect excitons at . Additionally, we introduced operators for particles in the reservoir, and , with energies given by and .
Processes of exciton-phonon scattering contribute to the incoherent part of the dynamics of the system and can be described using the Liouville-von Neumann equation written in an integro-differential form Savenko ()
where corresponds to generic exciton-phonon interaction Hamiltonian and by index we denote the part of dynamics coming from the thermal bath interaction. Applying Born-Markov approximation we obtain
where is a constant accounting for the energy conservation and is taken to be inverse broadening of exciton states divided by square of Plank constant. One can see that the time-dependence of the Hamiltonian disappears because of the energy conservation. Here and denote exciton-phonon interaction Hamiltonians corresponding emission and absorption of phonon, which can be written as , and
In the following Hamiltonian we accounted only for the energy conserving terms.
Let us now derive dynamic equation with additional treatment of temperature effects and reservoir states. The corresponding contribution to the time evolution of the mean value of any operator reads as
We proceed with calculation of phonon-assisted part of dynamics for DX and IX modes. For instance, the phonon contribution to dynamics of mean value of IX field yields
The calculation of commutators is straightforward and gives , , where and we account only for diagonal terms of phonon density matrix. Similar expression can be found for direct exciton field dynamics.
In order to close the set of equations we need to derive the expression for dynamical equations of reservoirs. For instance, the DX and IX reservoir states equations read as
Finally, merging all contribution to dynamics of the fields and reservoir states, we come to the system of five coupled equations (4)–(8) in the main text.
.2 B: The full model accounting for phonons and dephasing
In the previous section we derived equations which describe the dipolariton system in the lowest order of the mean-field theory. Here we generalize it to the next order and derive equation for the mean values of occupation numbers of the modes and their correlators.
Coherent part. We start from the Hamiltonian given by Eq. (1). The time dependence of the occupation numbers and correlators can be found using relation
where represents operator of different type (). The equation can be rewritten in form
and simplified using the explicit expression for coherent part of the Hamiltonian. To describe the dynamics we need to write a system of six coupled equations for occupation numbers of the modes , , and correlators , and . Using the mean-field approximation, we derive equation for each type of occupation number and correlator. For instance, the occupation number of indirect excitons can be written as
The first four terms of commutator are zero, while fifth term gives
with final equation being
where we used relation . One can write equations for and in the same fashion.
Derivation of dynamic equations for correlators require more efforts, but can be done straightforwardly. For example, the equation for correlator reads as
Considering terms one by one, the first gives and second term yields . Naturally, third and forth terms give zero, while the fifth term is equal to . The sixth term reads , and seventh term from correlator in Eq. (S27) is zero. This leads to dynamic equation for the correlator
Equations for and can be derived analogously.
Finally, assuming the presence of coherent pump, we need to add three equation for , and given by Eqns. (4)–(6) in the main text (excluding thermal part). This leads to complete system of Eqns. (S51)–(S59) for dynamics of the dipolariton system written at the end of the section.
Decoherent part. The master equation for density matrix is a powerful tool which allows one to account for various incoherent processes. They can be introduced in the model using Lindblad superoperator formalism Savenko (); Kavokin (). First, let us account for dissipation processes coming from the leakage of cavity mode out of the mirrors, as well as radiative and non-radiative recombination of excitons.
Dissipation processes can be modelled with the Lindblad superoperator written for any operator of type in the form
where represent the decay rate of the mode, . Reminding the master equation, , one can write the incoherent part of dynamic equation for occupation numbers and correlators. For instance, in the case of cavity photon occupation number it reads
where to write the latter expression we used the cyclic properties of the trace. Applying the mean-field approximation, and , it can be rewritten as
The same considerations are valid for , and correlators.
The radiative decay of the mode is not the only cause which affects the incoherent dynamics of the system. Indeed, the coherence in the system can be spoiled due to dephasing even for infinitely long-lived atomic systems. There the main effect of loosing coherence can be introduced by a pure decoherence term, which acts solely on off-diagonal terms or “coherences”, while live diagonal terms are unaffected. Usually written for two-level system (atom, qubit etc), the Lindblad operator for pure coherence in general form reads Schlosshauer ()
where , is a fermionic creation operator, and denotes the rate of pure dephasing. In particular, for two-level system it thus can be rewritten as Laussy (). In our system we want to account for the decay of coherence of the interaction between different modes which have bosonic statistics. Therefore, we can map the angular momentum operators into bosonic operators using Holstein-Primakoff transformations HP (), , where is spin of the particle. Then Lindblad operator (S32) can be rewritten opening the commutator and reads
Plugging (S33) into equation for occupation number
one can check that it gives zero contribution to the lifetime of the mode. Here we used definition . Performing the same analysis for correlator (e. g. ) we straightforwardly derive
Thus, coherence between mode is destroyed for large damping rates , and index corresponds to the type of correlator we are interested in. Similarly, one can introduce pure decoherence for and correlators.
Thermal part. Following the same procedure for accounting of exciton-phonon interaction, which we introduced in the previous section, we derive thermal part of dynamics for occupation numbers and correlators. For example, the thermal contribution to dynamics of indirect exciton occupation number is given by
and commutators inside can be calculated as