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- Springer Journals
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- Copyright © The Author(s) 2023
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- 10.1007/s43673-023-00081-8
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Abstract
We present a microscopic many‑body calculation of the nonlinear two ‑ dimensional coherent spectroscopy (2DCS) of trion‑polaritons and exciton‑polaritons in charge ‑tunable transition‑metal‑ dichalcogenides monolayers placed in an optical microcavity. The charge tunability leads to an electron gas with nonzero density that brings brightness to the trion — a polaron quasiparticle formed by an exciton with a nonzero residue bounded to the electron gas. As a result, a trion‑polariton is created under strong light ‑matter coupling, as observed in the recent experiment by Sidler et al. [Nat. Phys. 13, 255 (2017)]. We analyze in detail the structure of trion‑polaritons, by solving an extended Chevy ansatz for the trion quasiparticle wave‑function. We confirm that the effective light ‑matter coupling for trion‑polaritons is determined by the residue of the trion quasiparticle. The solution of the full many‑body polaron states within Chevy ansatz enables us to microscopically calculate the nonlinear 2DCS spectrum of both trion‑polaritons and exciton‑ polaritons. We predict the existence of three kinds of off‑ diagonal cross‑peaks in the 2DCS spectrum, as an indication of the coherence among the different branches of trion‑polaritons and exciton‑polaritons. Due to the sensitivity of 2DCS spectrum to quasiparticle interactions, our work provides a good starting point to explore the strong nonlinear‑ ity exhibited by trion‑polaritons in some recent exciton‑polariton experiments. [4–6], such as MoS , WS , M oSe , and WSe . In these two- 1 Introduction 2 2 2 2 dimensional materials, robust bright excitons of electrons Exciton-polaritons in microcavities are hybrid light-matter and holes with relatively large effective masses and large quasiparticles, formed due to strong coupling between exci- exciton binding energy dominate the optical response even tons and tightly confined optical modes [1 –3]. Owing to the at room temperature. As a result, TMD monolayers are half-matter, half-light nature, they open a research frontier promising candidates for ultrafast polariton-based nonlin- of polaritonics to explore novel nonlinear quantum phe- ear optical integrated devices, such as ultra-low threshold nomena that are impossible to observe in linear optical sys- lasers, fast and low-power switches, and all-optical inte- tems and are difficult to reach in pure matter systems. This grated quantum gates. For this purpose, strong polariton potential is further amplified by the recent manipulation nonlinearity is typically required. However, so far it remains of atomically thin transition metal dichalcogenides (TMD) a challenge to obtain strong exciton-exciton interaction and polariton-polariton interaction [7, 8]. *Correspondence: In this respect, the recent observation of trion-polari- Hui Hu tons in charge-tunable MoSe monolayers by Sidler et al. hhu@swin.edu.au 2 Centre for Quantum Technology Theory, Swinburne University received considerable interest [9]. At first glance, the exist - of Technology, Melbourne 3122, Australia ence of trion-polaritons is a surprise, since a trion is a © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Hu et al. AAPPS Bulletin (2023) 33:12 Page 2 of 13 fermionic three-particle bound-state of one hole and two we would like to restrict ourselves to the case of a single electrons and therefore in principle it should not be able to trion-polariton or exciton-polariton in the system [32]. couple with bosonic photon of light. But now, we under- This rules out the possibility of addressing the inter - stand that trions in charge-tunable monolayers are actually action effect between two trion-polaritons that is of the quasiparticles of Fermi polarons [10], which are exci- major interest. However, our calculation would capture tons (as impurities) dressed by the whole Fermi sea of an the basic features of the 2DCS spectrum, which could electron gas [11–15]. Except in the true trion limit (with then be used to discriminate the possible interaction vanishing electron gas density), where the three-particle effects between two trion-polaritons in future 2DCS bound-state is recovered, the trion is better viewed as a measurements. dressed exciton with a nonzero residue that characterizes The rest of the paper is organized as follows. In the next the free motion of the exciton [9, 10]. As a result, cavity section (Section 2), we outline the model Hamiltonian mode can indeed couple to the trion and lead to the for- for the Fermi-polaron-polaritons in TMD monolayers mation of trion-polaritons. The real surprise of trion-polar - and present the many-body solutions by using the Chevy itons comes with the observation that there seems to be a ansatz approximation. In Section 3, we discuss the struc- large nonlinearity in the optical response, as revealed by the tures of trion-polaritons and exciton-polaritons and the pump-probe measurement [16]. The understanding of such optical responses of both photons and excitons. In Sec- a large nonlinearity has been the focus of several theoreti- tion 4, we predict the the 2DCS spectroscopy and discuss cal analyses [17–19]. Further experimental investigations in detail the off-diagonal cross-peaks, which show the are definitely needed. In particular, a nonlinear four-wave- coherence between exciton-polaritons and trion-polar- mixing measurement, such as the two-dimensional itons. Finally, Section 5 is devoted to conclusions and coherent spectroscopy (2DCS) [20–24] would be ideally outlooks. suitable to quantitatively characterize the large nonlinearity of trion-polaritons. The purpose of this work is two-fold. On the one hand, 2 Model Hamiltonian and the Chevy ansatz we wish to clarify the nature of trion-polaritons by care- solution fully examining the full many-body Fermi polaron wave- We consider the systems of TMD monolayers explored functions of either exciton-polariton or trion-polariton, experimentally in Refs. [9, 23, 24] and theoretically in Refs. with the use of the Chevy ansatz that describes the one-par- [6, 10, 28]. As discussed in great detail in Ref. [28] (see, i.e., ticle-hole excitations of the Fermi sea [25]. The variational Fig. 1 of Ref. [28] on the band structure and optical transi- Chevy ansatz [25] (or equivalently the many-body T-matrix tions of TMD monolayers), in charge-tunable TMD mon- theory [26]) has been previously used to determine the olayers tightly bound excitons formed by electrons and self-energy and the spectral function of trion-polaritons holes near the K (or K ) valley move in the Fermi sea of an [9, 18]. However, a detail analysis of the many-body wave- electron gas in other valley with a nonzero electron density functions is of lack. Here, our strategy is to follow the recent that corresponds to a Fermi energy at about ε ∼ 10 meV. theoretical study of the wave-functions of the three-par- Electrons in the electron gas have opposite spin with respect ticle trion bound state [27, 28], where a single excess elec- to the electron inside excitons. By solving the three-body tron is approximately used to simulate the whole Fermi sea problem with two unlike electrons (i.e., with opposite spin) through the k-space discretization. Our calculation is free and one hole, in the presence of Coulomb-like interactions, from such a k-space approximation. A trade-off, however, is it was found numerically by Fey and collaborators [33] that the ignorance of the internal degree of freedom of the exci- an exciton experiences an effective short-range interac - ton wave-function. This ignorance is fully justified by the tion with the background Fermi sea. Therefore, we model large exciton binding energy (∼ 500 meV), which is at least the electron-exciton interaction by a contact interaction ten times larger than the trion binding energy in TMD mon- with strength U < 0 , following the theoretical treatment olayers (∼ 30 meV) [5]. The internal structure of excitons in Ref. [9]. The value of the interaction strength U is tuned then should only bring negligible effects on the low-energy to yield the trion binding energy E ∼ 30 meV [9, 28]. The properties of trion-polaritons. TMD monolayers can be placed in the antinode of a pla- On the other hand, the full many-body Fermi polaron nar photonic microcavity, with cavity photon mode being wave-functions obtained within the Chevy ansatz tuned near resonance with the excitonic and trionic optical approximation allow us to microscopically calculate transitions. the 2DCS spectrum of trion-polaritons, in addition to that of exciton-polaritons. The microscopic determi - 2.1 Model Hamiltonian nation of the 2DCS spectrum of an interacting many- We denote the cavity photon mode and the exciton by body system is highly non-trivial [29–32]. Therefore, the creation (or annihilation) field operators a ( a ) k Hu et al. AAPPS Bulletin (2023) 33:12 Page 3 of 13 Fig. 1 Two‑ dimensional contour plots of the zero‑ momentum spectral function of the exciton (a) and of the photon (b), as a function of the photon energy ω = δ + ω at the electron Fermi energy ε = 7.8 meV. The two black horizontal dot‑ dashed lines show the energies of the X F ph exciton (i.e., the repulsive polaron branch with ε ≃ 2004.4 meV ) and the trion (i.e., the attractive polaron branch with ε ≃ 1963.5 meV ), in the X T absence of the cavity photon field. The diagonal white dotted line indicates the cavity photon energy ω = δ + ω . Two avoided crossings at ph ω = ε and ω = ε are clearly visible. The spectral functions are measured in arbitrary units and are plotted in a linear scale X T † † X X and ( ), respectively. The electrons in the electron k (a a + X X ) ≤ 1, k k k k k (3) gas are described by the creation and annihilation field c c operators and . The polariton system under con - which realizes the Fermi polaron limit. In contrast, the sideration therefore can be well described by a Fermi density of the electrons ( n = c c ) is tunable, by = 1 k k polaron model Hamiltonian ( ) [9], adjusting the Fermi energy ε through gate voltage in the (0) † † † experiments [9, 16]. H =H + ǫ c c + U X c c X , k k q−p p aX k k q−k (1) In the absence of the electron gas, the strong light-matter k qkp coupling leads to the well-defined two branches of exciton- polaritons: the lower polariton and upper polariton [1–3]. (0) † † † H = ω a a + ǫ X X + a X + h.c. With the electron gas, one may naively anticipate the effec - k k k k aX k k k k k tive interactions between lower (upper) polarities and the (2) electron gas, and hence the formation of two separate lower 2 2 Here, ǫ = k /(2m ) , ω = k /(2m ) + δ , and and upper branches of Fermi polarons. However, the cor- k e k ph X 2 ǫ = k /(2m ) are the single-particle energy disper- rect physical picture turns out to be the formation of attrac- sion relation of electrons, cavity photons and exci- tive and repulsive Fermi polarons of dressed excitons in the tons, respectively, with electron mass m , photon mass first place, and then the coupling of Fermi polarons to the −5 m ∼ 10 m and exciton mass m ≃ 2m in 2D TMD light. For this reason, the trion-polaritons is better viewed as e X e ph materials [5]; δ is the photon detuning measured in rela- Fermi-polaron-polaritons [9], where the treatment of a trion tive to the exciton energy level; and finally, is the light- as an attractive Fermi polaron is explicitly emphasized. matter coupling (i.e., Rabi coupling). We do not explicitly consider the direct Coulomb interactions between the 2.2 The Chevy ansatz solution electrons in the background Fermi sea. According to the To solve the model Hamiltonian in the case of one exci- Fermi liquid theory, the effects of these direct interac - ton-polariton, let us take the following Chevy ansatz, tions can be formally taken into account by considering weakly interacting quasiparticles with effective mass and † † † † ˜ (residual) renormalized Landau interaction parameters. |P� = φ X + φ a + φ X c c 0 0 k k k 0 0 p h −k +k k h p h p These quasiparticles are precisely the electrons that we k k p h (4) are referring to in the model Hamiltonian. We will restrict ourselves to the case that the maximum † † ˜ + φ a c c |FS�, k k k p h −k +k k h p p number of exciton-polaritons is one, i.e., k k h Hu et al. AAPPS Bulletin (2023) 33:12 Page 4 of 13 for the Fermi-polaron-polariton states with zero total 4πN ε ≃ 4πnt = t . (9) F c c momentum K = 0 . Here, the Fermi sea at zero tempera- ture |FS� is obtained by filling the single-particle energy It is also easy to see the relations t /t = m /m ∼ 10 a c e ph level ǫ with N electrons, from the bottom of the energy and t /t = m /m ≃ 1/2 . We assume the periodic X c e X band up to the energy ε . The hole momentum k and the F h boundary condition, so the momentum k on the lattice particle momentum k satisfy the constraints ǫ ε p k F takes the values, and ǫ > ε , respectively. The energy of the whole Fermi k F sea is denoted as E . FS 2πn 2πn x y k , k = , , The ansatz involves the free motions of excitons and x y (10) L L photons with the amplitudes φ and φ , respectively. It 0 0 also describes the one-particle-hole excitations of the with the integers n , n =−L/2 + 1, ··· − 1, 0, 1, ··· L/2. x y Fermi sea due to the inter-particle interaction of excitons On the square lattice, we may identify that the Hil- and electrons, with the amplitude φ . Although there k k p bert space of the model Hamiltonian involves four dif- is no direct interaction between photons and electrons, ferent types of expansion basis states (at zero polaron † † for completeness we include the terms a c c |FS� momentum), −k +k k h p h p with the amplitude φ . These terms actually do not k k |1� =X |FS�, (11) contribute to the ansatz due to the negligible photon mass, since the related energy would be extremely large (i.e., ω becomes very significant for nonzero k =0). |2� =a |FS�, (12) Unlike the previous works that only minimize the ground-state energy of the Chevy ansatz for the variational † † |3� =X c c |FS�, ˜ ˜ k k k (13) p −k +k k h parameters ( φ , φ , φ , and φ ) or the self-energy of h p p 0 0 k k k k h p p h h polaritons [9, 16], here we are interested in solving all the many-body Fermi-polaron-polariton states, by using an † † |4� =a c c |FS�, k k k (14) p −k +k k h p h p alternative exact diagonalization approach. To this aim, we put the system — consisting of N electrons and a single It is straightforward to see that the dimension of the Hil- exciton-polariton — onto a two-dimensional square lattice 2 bert space is D = 2 + 2N (L − N ) . By using the expan- with L × L sites. The electron density then takes the value sion basis states, the Fermi-polaron-polariton model Hamiltonian then is casted into a D × D Hermitian n = , (5) matrix, with the following matrix elements ( H = H ), 2 ji ij (La) �1|H|1� =E + nU, FS (15) where a is the lattice spacing and unless specified other - wise is set to be unity ( a = 1 ). We consider that the pho- ton, exciton, and electrons hop on the lattice only to the �1|H|2� = , (16) nearest neighbor with strengths t , t and t , resp e ctively . a X c Their single-particle energy dispersion relations are then given by ( ω˜ = ω − δ), k k �1|H|3� ′ ′ = , (17) k k p h 2 2 k + k x y ω˜ = − 2t cos (k ) + cos k + 4t ≃ , k a x y a 2m ph �1|H|4� ′ ′ =0, k k (18) (6) 2 2 and k + k x y ǫ = − 2t cos (k ) + cos k + 4t ≃ , X x y X � | | � 2m 2 H 2 =E + δ, X FS (19) (7) 2 2 �1|H|3� ′ ′ =0, k + k k k (20) x y p ǫ =− 2t cos (k ) + cos k + 4t ≃ , c x y c 2m (8) �2|H|4� ′ ′ =0, k k (21) p h 2 2 where m ≡ 1/(2t a ) , m ≡ 1/(2t a ) , and ph a X X m ≡ 1/(2t a ) in the dilute limit ( n → 0 ) that is of inter- and e c est. In the same limit, we have the relation Hu et al. AAPPS Bulletin (2023) 33:12 Page 5 of 13 � � X probably due to a nearly perfect destructive interference ⟨3�H�3⟩ � � = E + − + + nU � � k k k k FS k k k k k k −k +k h p p p h p p h h p h h of the high-order contributions with more than one par- � � + � − � , ticle-hole pairs [36]. The validity of Chevy ansatz has also k k k k 2 h h p p been examined by using numerically exact Monte Carlo (22) simulations and an excellent agreement was found [37]. ′ ′ ′ �3|H|4� ′ = δ δ , (23) k k k k p k k k k h p p p h h h 3 Trion polaritons and the one‑dimensional � � optical response � � � � ⟨4�H�4⟩ = E + − + . (24) k k k k FS k k −k +k k k k k h p p h p h p h p h p h In our numerical calculations, we consider a square lat- tice of L = 16 . We set the hopping strength t = 10 meV and then determine t = t (m /m ) = 10 meV and a c e ph We diagonalize the D × D Hermitian matrix to obtain all t = t (m /m ) = 5 meV. At these parameters, the spec- (n) X c e X the eigenvalues E and eigenstates, from which we extract tral broadening factor �E = 4t /L = 2.5 meV, which qual- (n) (n) c the Fermi-polaron-polariton energies E = E − E , FS (n) (n)∗ (n) itatively agrees the homogeneous broadening observed in the residue of excitons Z ≡ φ φ and the residue of X 0 0 (n) (n)∗ (n) the optical response of exciton-polaritons [5]. We take ˜ ˜ photons Z ≡ φ φ . Furthermore, we directly calcu- ph 0 0 an attractive interaction strength U = −8t =−80 meV, late the retarded Green functions of excitons and photons, which leads to a trion energy at about −3.2t =−32 meV (n) in the dilute limit (i.e., n → 0 or N = 1 at L = 16 ), in rea- G k = 0, ω = , ( ) (25) sonable agreement with the trion binding energy E ∼ 30 (n) + ω − E + i0 meV found in 2D TMD materials [5]. Most of our calculations are carried out for a num- (n) ber of electrons N = 16 , which corresponds to a Fermi ph G (k = 0, ω) = . (26) energy ε ≃ 4πNt /L ≃ 7.8 meV [9, 16]. At this num- ph F c (n) + ω − E + i0 ber of electrons, we find the attractive polaron energy E ≃−3.65t = −36.5 meV and the repulsive polaron A c and the associated spectral functions energy E ≃+0.44t = 4.4 meV, without the cavity field. R c 1 Measured from the top of the valence band, these values A (k = 0, ω) = − ImG (k = 0, ω), (27) X X give rise to the trion energy ε = E + ω = 1963.5 meV T A X and the exciton energy ε = E + ω = 2004.4 meV. X R X For convenience, we will also measure the pho- A (k = 0, ω) = − ImG (k = 0, ω). (28) ph ph ton energy ω with respect to the top of the valence ph band, which leads to ω = δ + ω . For the light-mat- ph X Here, since we use a finite-size square lattice, the level ter coupling, we always fix the Rabi frequency to be spacing in the single-particle dispersion relation is about = 2t = 20 meV. �E = 4t /L . We will use E to replace the infinitesi - In Fig. 1a and b, we report the zero-momentum spec- mal 0 in the spectral function and to eliminate the dis- tral functions A (k = 0, ω) and A (k = 0, ω) at the typi- X ph creteness of the single-particle energy levels. To make cal experimental Fermi energy ε = 7.8 meV for excitons connection with the experimental measurement, we and photons, respectively, in the form of the two-dimen- measure the energy ω in the spectral function from the sional contour plot with a linear scale (as indicated on the top of the valence band by adding a constant energy shift top of the figure). Both spectral functions clearly show an ω = E − E = 2 eV [5, 27], where E and E are the X g X g X avoided crossing at the energy close to ω = 2 eV. The two band gap and the binding energy of excitons, respectively. branches can be well-understood as the upper and lower (0) To close this subsection, let us briefly comment on the polaritons given by the model Hamiltonian H , which aX usefulness of Chevy ansatz. On lattice, this variational exist even in the absence of the electron gas. This is evi - approach with the inclusion of one-particle-hole excita- dent if we compare Fig. 1 with Fig. 9 in Appendix A, where tions was extensively used to qualitatively understand the latter figure reports the results at a much smaller Fermi the stability of a ferromagnetic phase in two-dimen- energy ε = 1 meV. For the upper and lower polariton sional Hubbard model [34, 35]. In the dilute limit, where branches, we find that the existence of the electron gas will the density or the filling factor n → 0 as adopted in this slightly shift the position of the avoided crossing (i.e., from work, it was used to describe the Fermi polaron in ultra- ω = 2000 meV to ε ≃ 2004.4 meV), due to the exciton- X X cold atomic systems [25]. It turns out that this approxi- electron interaction that becomes effectively repulsive for mation works quantitatively well in the dilute limit, the excited state of repulsive polarons. Hu et al. AAPPS Bulletin (2023) 33:12 Page 6 of 13 The main effect of the electron gas to the spectral func - tions is the appearance of an additional avoided crossing, the trion-polariton, at the trion energy ω = ε . At the Fermi energy ε = 7.8 meV in Fig. 1, this avoided crossing has an energy splitting smaller than but comparable to the Rabi coupling = 20 meV for the exciton-polariton. The shape of the avoided crossing is apparently asymmetric in the exciton spectrum. At the much smaller Fermi energy ε = 1 meV in Fig. 9, the avoided crossing can hardly be identified in both exciton spectrum and photon spectrum, which unambiguously indicates that the existence of a Fermi sea is the key source for the trion-polariton. To better understand the two avoided crossings for exciton-polaritons and trion-polaritons, we show in Figs. 2 and 3 the residues (upper panel) and spectral functions (lower panel) of excitons and photons, at the photon energy ω = ε and ω = ε , resp e ctively . ph X ph T Let us first focus on the avoided crossing for exciton- polaritons at ω = ε ≃ 2004.4 meV in Fig. 2. The com - ph X position of the different branches might be seen from the exciton and photon residues. The upper branch (or the rightest branch) locates at the energy ∼ 2.013 eV and consists of a number of many-body energy levels that Fig. 3 a Residues of the exciton (black solid circles) and the photon (red empty square) for each many‑body state that is arranged with increasing energy. b The spectral function of the exciton (black solid line) and the photon (red dotted line), shown in arbitrary units. Here, we take a photon energy ω = δ + ω = ε ≃ 1963.5 meV. The ph X T electron Fermi energy is ε = 7.8 meV distribute nearby with notable exciton and photon resi- dues. For this upper branch, due to its collective nature, it seems difficult to find a Hopfield coefficient that clearly defines the contributions or components from cavity pho - tons and excitons, as in the case of conventional exciton- polaritons. In contrast, for the lower branch (or the middle branch in the range of the whole plot, which is referred to as middle polariton in the literature) located at the energy ∼ 1.996 eV, we find that it is only contributed by one dominated state. All other nearby many-body states have residues much less than 1% . This branch seems to decou - ple from the particle-hole excitations of the Fermi sea and therefore retains the characteristic of the exciton-polari- ton without the electron gas. We note that, the energy splitting between the upper and lower branches is given by 2.013 − 1.996 = 0.017 eV or 17 meV, which is slightly smaller than the Rabi coupling = 20 meV. We attribute this slight difference to the transfer of the residue or the Fig. 2 a Residues of the exciton (black solid circles) and the photon oscillator strength to the third branch (the lowest-energy (red empty square) for each many‑body state that is arranged with branch) in the exciton spectrum, as shown in Fig. 2b. increasing energy. b The spectral function of the exciton (black solid line) and the photon (red dotted line), shown in arbitrary units. Here, The situation for the avoided crossing of trion-polaritons we take a cavity photon energy ω = δ + ω = ε ≃ 2004.4 meV. ph X X at ω = ε is very similar. As can be seen from Fig. 3, ph The electron Fermi energy is ε = 7.8 meV the upper branch of this avoided crossing near the energy Hu et al. AAPPS Bulletin (2023) 33:12 Page 7 of 13 ∼ 1.967 eV is formed by a bundle of many-body states In Fig. 4, we show the ground-state energy of the trion- with significant residues. The lower branch is instead polariton (a) and its excitonic and photonic residues contributed by one state only at the energy ∼ 1.958 eV. (b), as a function of the photon energy ω = δ + ω . ph X The energy splitting of the two branches is about 9 meV The excitonic residue does not change significant when and is less than the Rabi coupling = 20 meV. The small ω ε . In particular, at the avoided crossing of ph X energy splitting is again attributed to the reduced oscillator ω = ε , the excitonic residue Z ∼ 0.25 , which implies ph X X strength, which we now turn to discuss in greater detail. an effective Rabi coupling ≃ Z = 10 meV, eff X As we mentioned earlier, a plausible picture for the for- which is very close to the observed value of 9 meV. The mation of trion-polaritons is the strong effective light- slightly reduced Rabi coupling of 17 meV at the avoided matter coupling between a photons and an attractive crossing of the exciton-polariton might be understand Fermi polaron of the exciton impurity. It is clear that only in a similar way. We may identify that the excitonic the free part of the attractive polaron (as characterized by residue of the repulsive polaron at ω = ε is about ph X φ ) contribute to the light-matter coupling, in the form of Z ∼ 0.7 . Therefore the effective Rabi coupling is given 0 X the term (�/2)[a (φ X ) + h.c.] at zero momentum. In by ≃ Z = 16.7 meV, in agreement with our 0 0 eff X other words, the effective Rabi coupling would be given by finding. � ≃ �φ = � Z , (29) eff 0 X 4 Two‑dimensional coherent spectroscopy Let us now consider the 2DCS spectroscopy, which is to which is reduced by the square root of the excitonic resi- be implemented in future experiments on studying the due. This expression of the effective Rabi coupling would exciton-polariton physics in TMD materials. In 2DCS, also work well for the repulsive polaron (i.e., the exciton- three excitation pulses with momentum k , k and k are 1 2 3 polariton with the electron gas). applied to the system under study at times τ , τ and τ , 1 2 3 separated by an evolution time delay t = τ − τ and a 1 2 1 mixing time delay t = τ − τ , as illustrated in the left 2 3 2 part of Fig. 5. These pulses generate a signal with momen - tum k , as a result of the nonlinear third-order process of the many-body interaction effect. The signal can then be measured after an emission time delay t by using the fre- quency-domain heterodyne detection. During the excitation period, each excitation pulse cre- ates or annihilates an exciton. As the photon momentum of the excitation pulses is negligible, the exciton has the Fig. 5 Two double‑sided Feynman diagrams that represent the two contributions to the standard rephasing 2D coherent spectra under the phase‑match condition k =−k + k + k , with s 1 2 3 the time ordering of excitation pulses indicated on the left. The Fig. 4 a The ground‑state energy of Fermi‑polaron‑polaritons as a evolution, mixing, and emission time delays are labeled as t , t , 1 2 function of the photon energy ω = δ + ω , where ω = 2 eV. The and t , respec tively. a shows the process of excited‑state emission ph X X 3 red dotted line and the black dot‑ dashed line show the cavity photon (ESE), R (t , t , t ) . b corresponds to the ground‑state bleaching 2 1 2 3 detuning and the trion energy without cavity field ε ≃ 1963.5 meV. (GSB), R (t , t , t ) . In the diagrams, we use |g� to denote the Fermi T 3 1 2 3 b Residues of the exciton (black solid circles) and the photon (red sea and |e� to label the many‑body states with an exciton‑polariton, empty squares) of the ground‑state as a function of the photon respectively. There are infinitely many many‑body (Fermi polaron) energy. Here, we take the electron Fermi energy ε = 7.8 meV states |e� , as indicated by different colors F Hu et al. AAPPS Bulletin (2023) 33:12 Page 8 of 13 zero momentum. Therefore, each pulse can be described states induced by the three excitation pulses and the by the interaction operator V, signal. For example, the transfer of the first pulse at (n) momentum k brings a factor of φ , while the trans- V ∝ X + X . (30) fer of the second pulse at momentum k comes with (m) a factor of [φ ] , and so on. When we combine all Following the standard nonlinear response theory [22], the four factors for the four transitions, we obtain the the signal is given by the third-order nonlinear response (n) (m) weight Z Z . On the other hand, the three dynami- X X function, cal (time-evolution) phase factors arise from the (3) phases accumulated during the time delays t , t , and 1 2 R ∝ � V t + t + t , V t + t , V t , V �, [[[ ( ) ( )] ( )] ] 1 2 3 1 2 1 t , respectively. The GSB process can be analyzed in an (31) exactly same way. The only difference is the absence of where the time-dependent interaction operator the mixing time ( t ) dependence in the expression. This iHt −iHt 2 V (t) ≡ e Ve , and �··· � stands for the quantum is easy to understand from Fig. 5b: between the second average over the initial many-body configuration of the and third pulses the system returns to the ground state system without excitation pulses, which at zero tem- of a Ferm sea, so there is no phase accumulation during perature is given by the ground state. By expanding the the mixing time delay. three bosonic commutators, we find four distinct corre - By taking a double Fourier transformation for t and t 1 3 lation functions and their complex conjugates [22]. For in R (t , t , t ) and R (t , t , t ) , we obtain the 2DCS spec- 2 1 2 3 3 1 2 3 the rephasing mode that is of major experimental inter- trum [32], est, t > 0 and k =−k + k + k . For this case, only 1 s 1 2 3 two contributions are relevant if we consider at most (n) (m) (n) (m) i E −E t Z Z 1 + e X X one excitonic excitation in the system: the process of so- S , t , = , (36) 1 2 3 − (n) + (m) − − E − E nm 1 called excited-state emission (ESE) [22, 23] , − + where (−ω ) ≡−ω − i0 , and ω and ω are the exci- 1 1 1 3 R = �VV (t + t )V (t + t + t )V (t )�, (32) 2 1 2 1 2 3 1 tation energy and emission energy, respectively. and the process of ground-state bleaching (GSB) [22, 23], 4.1 Zero mixing time delay t = 0 R = �VV (t )V (t + t + t )V (t + t )�. 3 1 1 2 3 1 2 (33) Let us first focus on the case of zero mixing time delay These two processes can be visualized by using double- t = 0 , where sided Feynman diagrams, as given in Fig. 5a and Fig. 5b, (n) (m) Z Z respectively. It is worth noting that we do not include X X S(ω , 0, ω ) = 2 , 1 3 − + ∗ (n) (m) the third process R (t t , t ) of excited-state absorption (−ω ) − E ω − E 1, 2 3 1 3 nm (ESA), which involves the many-body states of two exci- (37) tons and becomes important at large exciton density. and consider the dependence of the 2DCS spectrum For an exciton system, a microscopic calculation of the |S(ω , 0, ω )| on the photon energy ω = δ + ω , a s 1 3 X ph 2DCS spectrum has been recently carried out [32]. Here, shown in Fig. 6. By changing the photon detuning δ from we extend such a microscopic calculation to the exciton- the blue shift above the exciton-polariton crossing (a) to polariton system. After some straightforward algebra fol- the red shift below the trion-polariton crossing (f ), we lowing the line of Ref. [32], we obtain the ESE and GSB typically find three diagonal peaks located at the diagonal contributions, line ω =−ω (see the white dashed lines) and six off- 3 1 (n) (n) (m) (m) diagonal cross-peaks located symmetrically with respect (n) (m) iE t i E −E t −iE t 1 [ ] 2 3 R = Z Z e e e , X X (34) to the diagonal line. nm These peaks arise from the three branches of excita - tions, as we already seen in Fig. 1. Formally, with decreas- (n) (m) (n) (m) iE t −iE t 1 3 R = Z Z e e , ing energy the many Fermi-polaron-polariton states have X X (35) nm been grouped into the upper polariton, middle polari- ton, and lower polariton branches, as often referred to where the indices n and m run over the whole many-body in the literature [9, 16–18, 27]. Therefore, we can roughly polaron states. understand the Fermi-polaron-polariton as a three- These two expressions can be easily understood from (n) (m) energy-level system, with the energies E , E ∼ E , UP the double-sided Feynman diagrams. For the ESE pro- (n) (m) E , and E that are tunable by the cavity photon detun- MP LP cess illustrated in Fig. 5a, the weight Z Z meas- X X ing. The corresponding excitonic weights are given by the ures the transfer rates between different many-body (UP) (MP) (LP) excitonic residues Z , Z , and Z . Hence, from X X X Hu et al. AAPPS Bulletin (2023) 33:12 Page 9 of 13 Fig. 6 The simulated rephasing 2D coherent spectra (amplitude) at various photon energies ω = δ + ω at and at zero mixing time decays ph t = 0 . The photon energy decreases from ω = 2020 meV to from ω = 1950 meV in a–f. We typically find three peaks appearing on the ph ph diagonal dashed line. The red color illustrates the maximum amplitude, as indicated in the colormap above each subplot. The electron Fermi energy is set to be ε = 7.8 meV is useful to characterize the main component of the Eq. (37) we can easily identify that the diagonal peaks ground-state of the trion-polariton (or the lower occur when ω =−E and ω = E with peak amplitude 1 α 3 α (α) 2 polariton branch). As the photon energy decreases (Z ) ( α = UP,MP,LP ), while the off-diagonal peaks across the avoided crossing for trion-polaritons, we appear when ω =−E and ω = E with peak amplitude 1 α 3 β (α) (β) find that the brightness of the diagonal trion-polariton Z Z ( α =β = UP,MP,LP). X X peak becomes much weaker. The experimental measurement of diagonal peaks Although the upper, middle and lower polariton and crossover peaks at zero mixing time delay t = 0 branches can also be conveniently measured by using then provides us the information of both the energies (α) one-dimensional optical response, such as the reflec - E and the residues Z . In particular, when the pho- tance spectroscopy and photoluminescence spectros- ton energy ω = δ + ω is near the two avoided cross- ph copy [9], the application of 2DCS spectroscopy has ings (as shown in Fig. 6b and e, respectively), we may unique features to discriminate the intrinsic homoge- easily identify the effective Rabi coupling from the cor - neous line-with of the resonance peaks [23] and the responding energy splitting. Near the avoided crossing interaction effects [21, 24]. Unfortunately, both effects for the trion-polariton, the asymmetry of the crossing (i.e., the disorder potential for excitons and the exci- can also be clearly seen. ton-exciton interaction) are not included in our model We note that, when the photon energy is tuned to Hamiltonian. Nevertheless, our results in Fig. 6 provide roughly the half-way between the two avoided cross- the essential qualitative features of the 2DCS spectrum, ing, the middle polariton disappears in the 2DCS which is to be measured in future exciton-polariton spectrum (see Fig. 6d). This is simply because, at this experiments. In addition, the appearance of the off-diag - photon energy the middle polariton is of photonic in onal peaks and their evolution as a function of the mix- characteristics, with negligible excitonic component. ing time decay t are useful to characterize the quantum Therefore, it can not be seen from the 2DCS, which coherences among the different branches of polaritons, probes the excitonic part instead of the photonic part which we now turn to discuss. of the system. This feature of the 2DCS spectrum Hu et al. AAPPS Bulletin (2023) 33:12 Page 10 of 13 4.2 Q uantum coherence of the cross‑peaks The 2DCS spectra in Fig. 7 are shown in 40 fs increments. In Fig. 7, we present the simulated rephasing 2D coherent We can clearly identify that the brightness of each cross- spectra |S(ω , t , ω )| with increasing mixing time decays peak oscillates with the mixing time delay t , revealing the 1 2 3 2 t . We choose a photon energy ω = δ + ω = 2004 coherent coupling among different branches of exciton- 2 ph X meV at the avoided crossing for exciton-polaritons, where polariton and trion-polaritons. In comparison with the zero all the three polariton branches are clearly visible at mixing time delay 2DCS in Fig. 6b, we find that the HPC1 t = 0 . We label the three off-peaks at the top-right corner cross-peak nearly recovers its full brightness at t = 120 2 2 of the figure as HCP1, HCP2, and HCP3 [23], respectively. fs and 240 fs, confirming that its periodicity is close to the As can be seen from Eq. (36), the time t -dependence anticipated value T ≃ 124.4 fs. For the HCP2 cross- 2 HCP1 of the 2D spectrum S(ω , t , ω ) comes in through the peak, we see similarly that it nearly disappears at t = 40 1 2 3 2 (n) (m) i E −E t [ ] 2 term . As we interpret the Fermi-polaron- e fs, 120 fs, and 200 fs and fully recovers at t t = 80 fs, 160 polariton as a three-level system, where the energy lev- fs, and 240 fs, in agreement with our anticipation that (n) (m) els E , E are to be replaced by E , E , and E , T ≃ 81.6 fs. In addition, the HPC3 cross-peak only UP MP LP HCP2 it is readily seen that the t -term gives rise to quantum returns to the its full brightness at T ≃ 240 fs. 2 HCP3 oscillations with three different periods: 2π/|E − E | To better characterize the quantum oscillations, we MP LP for the HCP1 cross-peak, 2π/|E − E | for report in Fig. 8 the simulated rephasing 2D signal at the UP LP HCP2, and 2π/|E − E | for HCP3. At the pho- crosspeaks as a function of the mixing time t , both in UP MP 2 ton energy ω = δ + ω = 2004 meV, we find that the form of its amplitude (the upper panel) and in its real ph X E − E ≃ 33.2 meV, E − E ≃ 50.5 meV, and part (the lower panel). The oscillations do not take the MP LP UP LP E − E ≃ 17.3 meV. Therefore, the periodicities exact form of 1 + cos(ωt ) , as one may naively antici- UP MP 2 −10 of the cross-peaks are at the order of 10 s or 100 fs, pate from Eq. (36). This is partly due to the existence and are given by T ≃ 124.4 fs, T ≃ 81.6 fs, and and competition of three different periods in the oscil - HCP1 HCP2 T ≃ 239.5 fs. lations, which may bring a slight irregular structure. On HCP3 Fig. 7 The simulated rephasing 2D coherent spectra (amplitude) at the photon energy ω = δ + ω = 2004 meV with increasing mixing time ph X decays t from a to f. In a, the three higher‑ cross‑peak (HCP) are indicated. The three peaks appearing on the diagonal dashed line essentially do not change. However, the higher‑ cross‑peaks and lower ‑ cross peaks oscillate as a function of t , revealing the quantum coherence among different quasiparticles. The red color illustrates the maximum amplitude, as indicated in the colormap above each subplot. The electron Fermi energy is set to be ε = 7.8 meV F Hu et al. AAPPS Bulletin (2023) 33:12 Page 11 of 13 Fig. 8 The simulated amplitude a and real part b of the rephasing 2D signal at the three cross‑peaks as a function of the mixing time delays t . Note the different periodicity at different crosspeaks. We choose the photon energy ω = δ + ω = 2004 meV as in Fig. 6 and take the electron Fermi ph X energy ε = 7.8 meV the other hand, we find that the oscillations at HCP2 and many-body Fermi polaron states. Instead, the lower- HCP3 typically exhibit a decay. These dampings should polariton branch at the trion-polariton avoided cross- be related to the many-body nature of the upper-polar- ing involves only one Fermi polaron state. The situation iton branch, i.e., it is formed by a bundle of many-body for the middle-polariton branch varies, depending on states as we discussed in Fig. 2. Therefore, the upper whether it is close to the exciton-polariton crossing polariton has an intrinsic spectral broadening, which or close to the trion-polariton crossing. In the former eventually causes the damping in the quantum oscilla- case, the middle-polariton is also dominated by a single tion of the cross-peaks HCP2 and HCP3. In contrast, Fermi polaron state. both the lower-polariton and middle-polariton at the As there are three polariton branches [9, 17, 27], in photon energy ω = δ + ω = 2004 meV are domi- the 2D coherent spectroscopy, we have found three ph X nated by a single Fermi polaron state, and do not expe- diagonal peaks and six off-diagonal cross-peaks. From rience the intrinsic spectral broadening. As a result, the these peaks measured in future experiments, in princi- quantum oscillation at HCP1 is long-lived, if we do not ple we should be able to extract the excitonic residues take into account the lifetimes of excitons (due to the of different polariton branches. We have predicted the natural radiative decay) and of photons (due to the qual- existence of quantum oscillations in the 2D spectra as a ity of the cavity). function of the mixing time delay t , as the evidence for the quantum coherence among the different polariton branches [23]. 5 Conclusions and outlooks Although in the present study we have not considered In conclusions, based on the Fermi polaron description the effects of the disorder potential on excitons and the of an exciton-polariton immersed in an electron gas, we inter-exciton interaction, our results would provide a good have analyzed the structure of exciton-polaritons and starting point to understand the 2D coherent spectroscopy trion-polaritons in monolayer transition metal dichal- on exciton-polaritons to be experimentally measured in cogenides and have predicted their 2D coherent spec- the near future. Theoretically, the inclusions of the disor - troscopy for on-going experimental explorations in the der effect and interaction effect would be extremely chal - near future. lenging in numerics, since the dimension of the Hilbert From the structure analysis, we have found that space of the model Hamiltonian will increase dramatically. the upper-polariton branch at the exciton-polariton We will address these effects in future publications. avoided crossing typically consists of a number of Hu et al. AAPPS Bulletin (2023) 33:12 Page 12 of 13 Appendix A: Exciton‑polaritons level is basically given by the trion binding energy of and trion‑polaritons at small electron density E ≃ 32 meV. We can hardly identify the existence of In Fig. 9, we report the zero-momentum spectral func- the trion-polariton from the excitonic spectrum. Nei- tions of excitons and of photons for a small electron ther, the trion-polariton can barely be seen from the density with Fermi energy ε = 1.0 meV. At this den- photonic spectrum. Both spectra are very similar to the sity, the exciton energy level is barely affected by the spectrum of exciton-polaritons in the absence of the scattering with the electron gas and the trion energy electron gas. Fig. 9 Two‑ dimensional contour plots of the zero‑momentum spectral functions of the exciton (a) and of the photon (b), as a function of the photon energy ω = δ + ω at a small electron Fermi energy ε = 1.0 meV. The two black horizontal dot‑ dashed lines show the energies of the ph X F exciton (i.e., the repulsive polaron branch with ε ≃ 2000.4 meV ) and the trion (i.e., the attractive polaron branch with ε ≃ 1967.4 meV ), in the X T absence of the cavity photon field. The diagonal white dotted line indicates the excitation energy ω = ω . At this low electron density, the avoided ph crossing at ω = E is insignificant. The spectral functions are measured in arbitrary units and are plotted in a linear scale Acknowledgements Competing interests See funding support. The authors declare that they have no competing interests. Authors’ contributions All the authors equally contributed to all aspects of the manuscript. All the Received: 12 November 2022 Accepted: 31 March 2023 authors read and approved the final manuscript. Funding This research was supported by the Australian Research Council’s (ARC) Discovery Program, Grants No. DE180100592 and No. DP190100815 (J.W.), and References Grant No. DP180102018 (X.‑ J.L). 1. H. Deng, H. Haug, Y. Yamamoto, Exciton‑polariton Bose ‑Einstein conden‑ sation. Rev. Mod. Phys. 82, 1489 (2010) Availability of data and materials 2. I. Carusotto, C. Ciuti, Quantum fluids of light. Rev. Mod. Phys. 85, 299 The data generated during the current study are available from the contribut‑ (2013) ing author upon reasonable request. 3. T. Byrnes, N.Y. Kim, Y. 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AAPPS Bulletin – Springer Journals
Published: May 17, 2023