Interaction of Rydberg Excitons in Cuprous Oxide with Phonons and Photons: Optical Linewidth and Polariton Effect

We demonstrate that the optical linewidth of Rydberg excitons in Cu2O can be completely explained by scattering with acoustical and optical phonons, whereby the dominant contributions stems from the non-polar optical modes. The consequences for the observation of polariton effects are discussed. We find that an anti-crossing of photon and exciton dispersions exists only for states with main quantum numbers n>28, so polariton effects do not play any role in the experiments reported up to now.

continuum states, is deduced. In section 3, the theory is applied to those phonon processes, which contribute most to the linewidth and quantitative scattering rates are derived which are compared to experiment. In section 4, we discuss other scattering processes, like ionization into the continuum. In section 5, we derive the polariton dispersion curves for the multicomponent P excitons and discuss the possibility of a polariton splitting. The results are used to define a correct exciton radiative lifetime.

General considerations
According to the theory of Toyozawa, the linewidth is given by Fermi's golden rule and relates the full width at half maximum of the absorption band to the reciprocal of the total lifetime of the exciton state 1 / ii T  [6,11]. 1/ 1 is given by Fermi's golden rule as (2.1) Here denotes the possible final states of all scattering processes characterized by the transition matrix element M. This first approximation is quite accurate as long as the exciton bands do not overlap and includes also the radiative coupling.
Possible phonon scattering processes for excitons in a perfect crystal (see [12,13]) are: Since we are interested not only in P-excitons but in general angular momentum states, we cannot choose the coordinate system free as in [7]. Rather, we expand the plane wave in (2.3) into spherical harmonics where the first factor denotes the Clebsch-Gordan coefficient and the second denotes the reduced matrix element. Instead of using the Clebsch-Gordan coefficients, for numerical calculations one can use the integrals ( , ', '', , ') GC l l l m m directly. From inspection of the overlap integrals ( , ' ' ', ) W nlm n l m Q we note that all terms in the -expansion have the same ′′. This means that in calculating the scattering matrix elements we can take out from ( , ' ' ', ) W nlm n l m Q a common factor exp( '' ) Q im  which by taking the absolute square drops out. So we can state that the scattering matrix element depends on | ⃗ | and the angle of ⃗ to the -axis (denoted by cos Q x   ). Note that W in general is either real or imaginary.
Besides the available phase space which is restricted by energy and momentum conservation, the scattering probability between excitons with different quantum numbers , , is determined by the overlap functions. Here we assume that we can approximate the exciton states by hydrogen wave functions (scaled by an appropriate Bohr radius B a ). First we notice that scattering within the same exciton states (intraband) always has a term with ′′ = 0. Since 0 (0) 1 j  , ( , , ) W nlm nlm Q always starts at 1 for . This means that intraband phonon scattering rates at low T and small K are almost independent of n and l, as long as the phonon wave vectors are smaller than max Q . For different states (interband scattering) ( , ' ' ', ) W nlm n l m Q always starts at 0 for 0 Q  , so interband scattering requires a finite Q.
As an example of the scaling laws which can be derived from ( , ' ' ', ) W nlm n l m Q , we discuss the behavior of ( 10,100, ) W n Q which determines the scattering rate from a P-exciton with main quantum number n to the 1S state. In Fig. 1, we have plotted the dependence of | ( 10,100, ) | W n Q for -4-different n multiplied by a factor 52 / ( 1) nn  . We see that, except for 2 n  , all curves fall on the same dependence. Since the phase space for the scattering which depends on the energy difference between the initial state and the 1S state is almost the same for all 2 n  , we conclude that the linewidth of the P-states scales as the square of 2 25 ( 10,100, ) ( 1) / W n Q n n  which is just that what is observed experimentally [1]. The maximum of the scaled overlap integral at max ≈ 2/3 B −1 is the same for all n .

Interaction strengths
The scattering strengths , cv   depend on the mechanism and are given for deformation potential scattering by [11,13] ,, 0 identity. The first order 2   scattering is forbidden for the yellow states since 4 Information about the magnitude of the corresponding scattering strengths with the first order deformation potentials 1 , cv D  can be obtained from the resonant Raman studies in [18], where the observation of both scattering processes has been reported. (see Ref. [18], Fig. 7). Since the cross sections at the same relative energy should depend only on the absolute square of the scattering strengths, we can obtain from the known strength of the LO 1 process (given by Eq. Finally, we have to discuss carefully the consequences of the energy conservation in the different type of scattering processes. The scattering rate between the initial state | ; > and all final states | ′ ′ ′ > is given by While the  -integration is trivial, the integration over (set cos x   ) requires to determine the zeros of the argument of the delta function Here we have to distinguish between 1. Stokes (phonon emission) and 2. Anti-Stokes (phonon absorption) processes. The energies of initial and final states (argument of the delta function in Eq. (2.13) ) are We see that they depend on the angle  between ⃗ ⃗ and ⃗ and on the magnitude of ⃗ . Note that we are not allowed to set K=0, as the excitons are excited at the finite optical wave vector (see Appendix A). The functions are also slightly different between acoustical and optical phonon scattering. While in the latter processes is approximately constant, in the former the phonon energy scales linearly with Q.

Scattering by acoustical Phonons
For scattering by LA phonons we have where we introduced the mass factor ' x is the zero of ( , , ) as a function of , limited between -1 and +1, the result of theintegration is then simply We have to distinguish two situations: Either For case a) we define (2.17) The integration limits for Q are then obtained as follows: For the case 1a) , the lower limit is while the upper limit is The case a 0 K  (degenerate intraband scattering) gives 1 0 Q  , but 3 0 Q  as long as This shows that Stokes scattering is suppressed for excitons at small wave vectors.
In case 2a) we obtain as limits for the Q integration In case of 0 0 K  it gives 13 which agrees with previous results [19].
For the case of up scattering (b) we can set ' 1 nn   as here only the equal mass case is relevant (the 1S state is so far away from the others that up scattering occurs only at very high temperatures T>300K).
This results for Stokes scattering (case 1b) the integration limits are This shows that Stokes scattering only occurs if b 2 KK   . In case of Anti-Stokes scattering (case 2b) we have as limits for the Q integration

Scattering by Optical Phonons
Due to the low temperature, we have here to consider only case 1a. We assume that the phonon the integration limits are For the 3   and 5   optical phonons with a first order deformation potential we have the same expressions as for the LO phonons (replace the phonon energy in Eq. (1.47)). Note that in the Qintegral there is an additional factor 2 Q due to the matrix element. ). Furthermore, the effective mass is also much larger than the sum of electron and hole masses. We will take these effects into account by using in the overlap integral the smaller Bohr radius and in the energy conservation the larger exciton mass, but neglect the effect on e  and h  .

Scattering rates and linewidth
The interaction strength of LA scattering depends on the deformation potentials. In Ref. [7] the authors used eh 2.4 eV and 2.2 eV DD  taken from Ref. [20]. However, we think that these values are highly questionable.  [21], strain measurements [22] and paraexcitons propagation beats [23].
Obviously, assuming eh 0.2 eV DD  contradicts these results.
2. Measurements of Hall mobility gave ℎ = 0.7 eV [24], which would fit exactly. 3. However, a closer inspection of the theory used in Ref. 20 shows, that their interpretation of the experimental results is incorrect. To derive the deformation potentials from the measured T-dependence of the cyclotron resonance linewidth, they use a theory which assumes sufficiently high temperature, so that equipartition of the phonon modes occurs (see [12]). This is not valid for Cu 2 O due to the high masses compared to usual semiconductors. Therefore, one has to use the exact expression for the scattering rate at low temperatures (see [25]). A re-evaluation of their results with the correct theory gives as deformation potentials = 3.5 eV and ℎ = 1.8 eV, the difference of which would exactly give the right exciton 1S deformation potential of 1.7 eV.

-8-
In our calculations we therefore use for the deformation potentials = 3.5 eV and ℎ =  [27] (note that we use the same value for the ortho-and paraexciton mass, neglecting the anisotropy of the orthoexciton mass due to the complex valence band structure [28]). To make the calculations more simple, we further combine the 3   and 5   processes in one with an effective D of 38 eV.

nP states
Here we first show the results for the contributions of the different scattering processes for the nP  Obviously, the dominant contribution comes from the LO2 and the 35 /   processes, whereby the latter contributes about 2/3 to the total linewidth. In all previous calculations, it was just this process which was neglected explaining their failure.
Since the optical phonon energies involved in the scattering are all above 10 meV, the temperature dependence of the scattering rates which is given approximately by B 1 2 ( ) n  is negligible up to 50 K. The acoustical modes which would already increase strongly above 10 K, contribute less than 1% to the linewidth, so their influence can be neglected. Up to 50 K we, therefore, predict the linewidth to be temperature independent.
In conclusion, we can explain the linewidth of al P exciton states by the following scattering processes: 1. Scattering by LO1 and LO2 phonons via the Fröhlich mechanism 2. Scattering by 3,5   phonons by a first order deformation potential scattering. In the continuum the exciton states are determined besides the angular momentum , lm by a continuous quantum number k and by the center of mass momentum K . The energy is given by 2  The matrix elements have to be calculated with the continuum wave functions , , GegC are the Gegenbauer polynomials [31].

Thermal ionization of Rydberg excitons by phonons
In the following we only consider the case of Anti-Stokes LA scattering (due to the low temperature optical phonon modes are not occupied) and restrict to the case with 21<n<70.
We start off with the integration over the angles of Q.
with B () nE being the Bose function (Anti-Stokes scattering!).
The results of the calculation for 0  are shown in Fig. 3.
The red curve for T=1.35K shows that ionization is not important for the observation limits for higher n states, as the rate becomes larger than the lifetime only for n > 30. For 20K, thermal ionization sets in for n > 24, for very low T=100mK it is completely negligible.

Oscillator strength, radiative lifetime and polariton effect for the P states in Cu 2 O
The question, whether the P excitons do form polaritons is of central importance for the properties of Rydberg states. In our discussion we start from the classical exciton-polariton (see e.g. [32]).
Consider first a single resonant exciton state with energy where M is the mass and 1 ( ) Here () X r  is the exciton envelope function (usually assumed as hydrogen-like) and is a degeneracy factor taking the singlet part of the ortho states into account. For second class transitions the dipole moment may depend on the direction of the light propagation (like for a quadrupole transition).
Note that for excitons the oscillator strength itself (and all derived quantities like transition dipole moment and radiative lifetime) is meaningless since it is proportional to the crystal volume and thus diverges in the limit V  .
The contribution of the exciton state to the dielectric function is given by the dispersion relation As the higher P lines are lying very close together, one has to take their total contribution to the dielectric function into account. Generalizing (5.7) to the case of many resonances, we have -13- which allow to reproduce the measured energies with an error of less than 0.5  µeV. Note that the value of the band gap differs from that in [1] as one has to subtract from the resonance energies the energy of the excitons at the crossing point with the light dispersion E=26.96 µeV (which is almost constant for n>10).
As we have shown in Section 2, the damping of the polaritons originates from phonon scattering (for the influence of radiative coupling see below). One can use for our purpose the approximate formula Now we are able to calculate the polariton dispersions for all Rydberg states. For the calculation we use the method of Cho [35]. The results are shown in Fig. 4 for the real part and Fig. 5 for the imaginary part.  -15-Obviously, the photon line crosses all exciton dispersions without any disturbance: this means that the polariton character as a propagating wave is completely absent. The reason for this can be seen in the imaginary part of the wave vector, which is the inverse damping length of the polariton wave. In the resonance region (where the photon damping has its maximum and directly gives the absorption coefficient (multiplied by 4 2 10  to give the value in 1/cm, the peak heights are fully in agreement with experiment), the damping of the polariton waves corresponding to the excitons is about 100/µm. This means that these waves are damped out within 10 nm! Consequently we have no polariton splitting by avoided crossing.
This picture is completely different from that in Ref. [8], where typical polariton dispersion relations for the P excitons have been claimed. These results have to be considered doubtfull for several reasons: One is that the strength of the light-matter coupling is strongly overestimated. This is related to the wrong value of the LT splitting that is used by the authors. Instead of their value 10 µeV it is actually only 1.25 µeV for the n=2 P state. Second, the exciton mass used by these authors is only about 0.015 m 0 , compared to the real mass of about 1.6m 0 . Third, unfortunately the authors do not specify the damping constants, but they seem to be much too small.
Still the question remains, whether a polariton splitting occurs for larger quantum numbers. Detailed calculations, which are reproduced in Fig. 6 show that for n=30 and beyond an extremely small polariton splitting is visible. Obviously, any experimental verification of these splittings by e.g. polariton beats [23,35] will be a challenging task! Despite the lack of anti-crossing, the polariton concept has important consequences for the exciton dynamics, as it allows one to define unambiguously the coherence volume of the excitons [32]. As one sees from Eq. (5.2), the oscillator strength depends on the volume of the sample, which taking literally, would make the concepts of atomic physics obsolete. This, however, requires that the exciton translational motion is coherent over the total crystal volume. As the excitons are scattered by -16-phonons, they lose their coherence over the mean free path due to the scattering processes, which is given by the ratio of group velocity over scattering rate. In the polariton picture, one should use for consistency the polariton group velocity gr V , which can be determined from the dispersion curves at the resonance frequency. The values obtained from the above dispersion curves are shown in Fig. 7 (red diamonds). While it decreases with n for n<8, it stays constant above. Therefore, the mean free path increases with n according to the decrease of the lifetime. . The blue triangles show the n dependence of the spatial coherence critical linewidth Figure 7: Polariton group velocities (red diamonds) and coherence length (blue diamonds) of the P excitons for n=2 to 40. Up to n=9 the coherence length is determined solely by the group velocity and non-radiative damping, for higher n by both radiative and nonradiative decay.
-17-These two equations have to be solved self-consistently. The results for the coherence length are shown in Fig. 7 (blue diamonds), for the radiative rate in Fig. 8 (red diamonds). We see that the radiative broadening never gets larger than 1 µeV, but for n>26 determines the width of the resonances. So we expect polariton effects to become important. Indeed the dispersion relations for n>28 do show a (very) small polariton splitting (see Fig. 6).
Actually, our results agree with the well-known criteria [32,36] for the existence of the polariton effect. Here we have to distinguish between two situations: (i) the quasi-particle picture, where one creates polaritons with well-defined wave vector and (ii) the forced harmonic situation where polaritons with well-defined frequency are created by an external harmonic driving source, e.g., an electromagnetic wave. The latter is appropriate for transmission experiments, while the first applies, e.g., to two-photon absorption.
The existence of polaritons in the first case is governed by the condition that the (polariton) Rabi frequency is larger than the damping  (temporal coherence). In the second case, polaritons exists (in the sense that one can observe a non-crossing of the dispersion relations) if the following criterion is fulfilled (spatial coherence) A plot of these quantities (Fig. 8) shows that forced harmonic polaritons would exist for nP>28, which is in nice agreement with our direct calculation of the dispersion curves. In contrast, quasi-particle polaritons would already show up for P excitons with quantum number n>3, which has important consequences for the excitation of these states via a two-step process involving the yellow orthoexciton and a suitable mid-infrared laser supplying the energy of the 1S to nP transition [38]. In such an experiment we expect not only the observation of pronounced interference fringes similar to those in two-photon excitation of the blue exciton states in Cu 2 O [39] but complete new aspects in the physics of Rydberg excitons, like a polariton-polariton blockade.

Conclusions
We have presented detailed calculations for the total linewidth of exciton states in Cu 2 O by considering the interactions with phonons and photons. Taking not only the well-known scattering with acoustical and polar optical phonons but also with non-polar optical phonons into account, we are able to deduce the linewidth of the P exciton states with angular momentum = 1 in almost quantitative agreement with experiments. We further show that for main quantum numbers n<28 the polariton effect does not lead to a splitting of the dispersion relations, in contrast to previous studies [8]. We further exploit the concept of exciton coherence length to obtain the radiative linewidth of the excitons. Only for n>28 it becomes larger than the linewidth due to phonon scattering, so that for these, up to now not observed, exciton states the radiative coupling, i.e., the polariton character, dominates. Our results should clarify the roles of phonon and photon coupling for Rydberg excitons and open the way to more advanced experiments.