Nonlinear optical properties and Kerr nonlinearity of Rydberg excitons
in Cu₂O quantum wells

Rydberg physics in semiconductors has started in 2014 by an observation of highly excited excitonic states with principal quantum numbers as high as $n$=25 in cuprous oxide, the material of a very large exciton binding energy [1]. This experiment revealed a plethora of Rydberg excitons’ unusual properties such as extraordinary large dimensions up to 1 $\mu$m, long life-times of order of nanosecond, vulnerability to interactions with external fields and restrictions of their coupling arising from Rydberg blockade, which precludes a simultaneous excitation of two Rydberg excitons that are separated by less then a blockade radius $r_{b}$. A lot of papers have been devoted to studies of spectroscopic characteristic of REs in natural and synthetic bulk systems of Cu₂O [2, 3, 4] (see more references therein). Simultaneously, the explorations of RE in the field of quantum optics have begun by demonstration of a generation and control of strong excitonic interactions with the help of two-color pump-probe technique [5], Rydberg exciton-assisted coupling between microwave and optical fields [6] and the experimental verification of the strong coupling of REs to cavity photons [7]. Moreover, some efforts have been made to investigate nonlinear interactions of REs with electromagnetic fields [8, 9]. The recent one-photon experiment has shown a giant nonlinear optical index in a bulk Cu₂O crystal, caused by sharp Rydberg resonances and revealed a Kerr phase-shift much larger than in typical nonlinear crystals [10]. Interesting, giant microscopic dimensions of Rydberg excitons together with an intrinsic Rydberg blockade effect in cuprous oxide cause enhanced nonlinearities at much smaller densities compared with other semiconductors [1, 11].

Those results indicate that Rydberg excitons are a unique platform for obtaining strong interactions in solid systems and allow one to hope for a realization, in a close future, of solid state masers [12, 13] and few-photon devices. The first step to achieve a scalable solid-state platform characterized by controlled interactions between Rydberg excitons and photons to realize such technologically demanding miniaturized systems, consisted in an investigation of REs’ properties in strongly confined systems such as quantum dots, wires or wells [14, 15, 16, 17]. The experiment, which has verified a change of oscillator strength due to quantum confinement of REs in a nanoscale system [17], is an important step towards exploiting their large nonlinearities for quantum applications. The recent progress in fabricating synthetic cuprous oxide elements has shown an enormous progress of their quality manifested by observations of high excitonic states [4, 18, 19] and now the natural direction of subsequent explorations seems to be the study of a nonlinear interaction between confined REs and light. A great challenge in quantum optics is an accomplishment of the gigant Kerr nonlinearities in solid-state low-dimensional media. This phenomenon was realized in semiconductor quantum wells mostly under the conditions of the ellectromagnetically induced transparency [20, 21, 22] or in ultrathin gold films [23]. In our paper we propose a realization of the Kerr nonlinearity in the Cu₂O quantum well with REs, taking advantage from the fact that confinement effects result in a significant optical Kerr susceptibility.

The theoretical tool which we use to calculate the optical functions for nonlinear interacion of electromagnetic radiation with Rydberg excitons in a quantum well is a mesoscopic method, called Real Density Matrix Approach [24, 25]. It allows for a calculation of transition probability amplitudes taking into account finite lifetimes, all mutual and external interactions and a system geometry. The detailed description of RDMA as well as a presentation of the iteration procedure, which allows one to obtain a nonlinear susceptibility for Rydberg excitons confined in a quantum well, is presented in Sections II and III. The effect of the Rydberg blockade is also included into our treatment and is considered in Sec. IV. Phase-sensitive Kerr nonlinearity appearing in discussed system is examined in Sec. V. Sec. VI contains the presentation of numerical results and their discussion, while the summary and conclusions of our paper are presented in the last Sec.VII.

The next application of the amplitude $Y_{12}^{(1)}$ is related to the iteration process. Inserting $Y_{12}^{(1)}$ in the source terms on the r.h.s. of Eqs. (II.1,II.1) and using appropriate expressions for the irreversible terms, one obtains the matrices $C^{(2)},D^{(2)}$, where the superscript indicates the order with respect to the electric field strength $E$. Substituting the matrices into the saturating terms on the r.h.s. of Eq. (II.1) one obtains the equation for the nonlinear amplitude $Y^{(3)}$ which, with respect to Eq. (II.1), defines the nonlinear susceptibility $\chi^{(3)}$. We obtain the following expression

where $\ell=0,1,2...$ The nonlinear oscillator strengths can be written as

where

and the derivation of constants $\mathcal{A}$, $\mathcal{B}$ is presented in Appendix A. The potentials $V_{N_{e}N_{h}}$ are given by

with the thermal lengths $\tilde{\lambda}_{\hbox{\tiny th e,h}}$ defined above in Eq. (31).

More detailed calculations of $\chi^{(3)}$ are presented in Appendix B and the table of material parameters is included in Appendix C. The long-wave approximation we have used in above calculations limits our results, which are appropriate for quantum wells thicknesses $L$ significantly bellow $1\mu m$.

One of the important characteristics of the theoretical approach described above is the fact that it is derived under the assumption of a relatively low power level, when the medium is not saturated with excitons. Thus, the so-called Rydberg blockade [1] is not inherently present in the calculations and its effects have to be taken into account in a separate step. This has been done in [13, 10] and the description outlined below is an extension of the approaches presented in the cited works.

For an exciton with principal quantum number $(j+1)$, the blockade volume is given by [1]

Similarly to the recent experiments [10], we assume that the laser beam illuminating the sample has a circular beam spot of area $S$ of 0.1 mm² and the sample length is $L$; the volume , where the light can be absorbed and an exciton created is $V=LS$. Within this volume, a new exciton can be formed only when its location is outside of the blockade volume of existing excitons. Thus, assuming that the blockade volume is spherical, the upper limit of exciton density is the perfect sphere packing, where approximately $74\%$ of the volume is occupied, e. g. for the number of excitons $N_{e}$, $N_{e}\frac{V_{B}}{V}\approx 0.74$. However, the positions of the excitons formed within the laser beam are random and thus highly unlikely to form a perfect sphere packing. To estimate the practical upper limit of exciton density imposed by Rydberg blockade, a Monte Carlo simulation has been performed; within given volume $V$, excitons with their associated blockade volumes are added at random positions and the number of attempts to place an exciton in a free space (not occupied by blockade volume) is counted. Then, the probability of excitation (inverse of the number of attempts) is calculated. The results are shown on the Fig. 1.

One can see that the system is effectively saturated when the fraction of occupied volume approaches 0.2. An exponential function can be fitted to the data (dashed line), providing a simple model of saturation; the probability of excitation is

When calculating the susceptibility from Eq. (II.2) and Eq. (37), one has to multiply the oscillator strengths $F$ by the above probability. This is a similar approach to that one used in [10] and [13], where also an exponential function $\exp(-AN_{e}V_{B}/V)$ with one fitted constant $A$ was used.

Finally, to calculate the number of excitons (and thus the blocked volume), one can consider the power to sustain a single exciton

where $E_{j}$ and $\tau_{j}$ are the energy and lifetime of excitonic state. The number of excitons $N_{e}$ is

where $P_{A}$ is the absorbed laser power; for a sufficiently thick sample, it is equal to the total laser power.

As a first verification of the presented theoretical description, one can examine the results obtained in the asymptotic limit of a very large thickness, e.g. a bulk crystal. The results of such a comparison are presented on the Fig.2. Specifically, Fig. 2 a) depicts the calculated optical density spectrum in the region of $j=6-11$ excitonic states, for two illumination powers. One can notice a quick decrease of absorption in the high power regime, approximately proportional to blockade volume $\sim j^{7}$. This is the so-called optical bleaching [1]. The same result can be seen on the Fig. 2 b), where calculations are compared to the experimental data from [1]. A very good agreement obtained in a wide range of powers and across multiple excitonic states indicates that the saturation model in Eq. (42) is sufficiently precise.

In the self Kerr effect the refractive index is changed due to the response of the incoming field itself, in other words it consists in the change of the refractive index of the medium with a variation of the propagating light intensity. The third-order nonlinear susceptibility is the basis of theoretical description of this phenomenon. The nonlinear optical response is conveniently described in terms of a field-dependent index $n(E)$ defined as

The real part of the nonlinear susceptibility defines the nonlinear index of refraction, which characterizes so-called Kerr media, $n_{2}=\frac{Re\chi^{(3)}}{c\epsilon_{0}n_{0}^{2}}$, with $n_{0}^{2}=1+\chi^{(1)}$.
The self-Kerr interaction is an optical nonlinearity that produces a phase shift proportional to the square of the field intensity (or a number of photons in the field). In the Kerr medium the phase of an electromagnetic wave propagating at the distance $L$ increases and the increment in phase due to intensity-dependent term is proportional to the distance and to the square of the electric field strength, which is called self-phase modulation. The phase shift is calculated from

The considerable nonlinear susceptibility of Rydberg excitonic system, further amplified in a thin quantum well, is expected to cause a noticeable phase shift even for small $L\sim 100$ nm. The confinement states, even when not directly visible, still contribute to the total height of the excitonic line, increasing $\chi^{(3)}$ and phase shift.

Due to the limited amount of experimental data regarding nonlinear properties of Cu₂O quantum wells, as a first step we verify our calculations with a comparison to a bulk medium. It should be stressed that while the calculated spectra approach the bulk ones as $L\rightarrow\infty$, the presented method is derived under the assumptions of strong confinement and long-wave approximation, so it yields fully correct results only for well thickness significantly below $1\mu$m. In contrast to [14], where weak confinement regime is studied, the confinement affects the relative motion of the electron-hole pair.

As mentioned above, one can make a rough comparison with experimental results in bulk medium by assuming a large value of $L$, skipping the wide quantum well regime at moderate $L\sim 1$ $\mu$m. The linear and nonlinear parts of susceptibility have been calculated from Eqs. (II.2) and (37), in a wide range of laser powers and for a thick crystal $L=50$ $\mu$m. The results are shown on the Fig. 3.

The calculated spectra are in the range from $j=3$ exciton resonance (2.161 meV) to the band gap (2.172 meV). The effect of Rydberg blockade is included in calculations by multiplying the obtained susceptibility by the factor $p_{e}$, Eq.(42); as the power increases, the density of excitons reaches saturation and $p_{e}\rightarrow 0$. In such a way one is able to control whether one is still in the regime, in which additional effects due to Rydberg blockade preventing the transmission are absent, and do not influence the excitons-light interaction. One can see that the overall amplitude of the linear susceptibility changes in the order of $10^{-4}$ and the nonlinear part is approximately 3 orders of magnitude lower. As expected, the number of observed resonances is strongly dependent on the power $P=\frac{1}{2}cS_{b}\epsilon|E|^{2}$, where $S_{b}=0.1$ mm² is the beam area; for $P=1$ W, the bleaching is considerable even for $j=3$. The results are consistent with our previous calculations in a bulk medium [10, 8], as well as experimental observations [30] and indicate that the model in Eq. (42) is correct.

As the next step, let us consider a thin $L=100$ nm quantum well. In such a system, one can expect that the absorption spectrum will contain multiple confinement states corresponding to the quantum number $N=1,2,3...$. This is the case shown in Fig. 4. Although there is no strict upper limit on the confinement state number $N_{max}$, in practice only a few lowest confinement states are observable and thus in calculations one can assume $N_{max}=10$. The linear part of susceptibility is consistent with the results presented in [15] for the case of a quantum well. Specifically, one can see a series of secondary peaks originating from every excitonic line, which shift towards high energy as $L$ becomes very small. It should be stressed that these lines corresponding to confinement states are only detectable in the case of a very thin quantum well; in the micrometer-sized nanoparticles, the energy spacing between these lines is small enough that they completely overlap, forming a single, broadened excitonic line [17]. Moreover, in this size range, one cannot observe oscillations of the absorption coefficient caused by the spatial matching between the center-of-mass exciton motion and light waves [18]. Naturally, a very small absorption of a thin sample makes a direct observation of confinement states challenging. Moreover, just like in the case of large quantum dots [17], the oscillator strength of excitonic states decreases faster than $j^{-3}$; this is also consistent with the observations in [14]. This effect, in addition to the broadening and chaotic ,,background” formed by multiple confinement lines, puts an upper limit on the maximum principal number of the observable state.

The nonlinear susceptibility shown in the Fig. 4 b) is apparently similar to the bulk case in the Fig. 3. The influence of the confinement on the nonlinear part $\chi^{(3)}$ is complex. One can see from Eq. (III) that oscillating terms of $V_{N_{e},N_{h}}$ interplay with slowly varying factors $\lambda_{the,h}$, describing plasma effects, resulting in an absorption attenuation. Namely, the confinement lines are much less pronounced and only $N=1$ line is readily visible. This effect follows from Eq. (III).

For low-dimensional systems the nonlinear optical effects depend strongly on the shape of the confinement potentials. For the above-used no-escape boundary conditions we obtained the expressions $V_{N_{e},N_{h}}$ decaying as $N^{-1}$. The physics behind is that the rapid motion of electrons and holes in the confinement in $z$ direction, especially for states with higher $N$, hinders the creation of plasma which is responsible for the reduction of the absorption while the linear absorption does not depend on $N$. For low-dimensional systems the nonlinear optical effects depend strongly on the shape of the confinement potentials. In the considered quantum well with no-escape boundary conditions, the overall amplitude of the nonlinear part of the susceptibility is enhanced as compared to bulk system. The influence of the confinement on the nonlinear part of $\chi^{(3)}$ is more complex. One can see that oscillating functions $V_{{N_{e}}{N_{h}}}$, characteristic for low dimensional confined systems interplay with relatively slowly varying exponential functions due to plasmonic terms, which results in increasing of the nonlinear absorption.

The next Fig. 5 shows the susceptibility spectra calculated for low a laser power and various values of thickness.

One can see that both confinement lines and the main excitonic lines are blueshifted in the limit of small $L$; as noted in [14], the confined exciton gains additional energy and this energy shift is most pronounced for $L<4a_{B}$, which is approximately 100 nm for $n=10$ exciton. On the other hand, it is known that excitons cannot form in quantum dots when the dot size $r<0.4a_{B}$ [14] which indicates the lower limit of applicability of our theoretical description. As before, the lines corresponding to the confinement states are mostly invisible in the nonlinear susceptibility spectrum. In the linear part, one can see that peaks due to those states, located closely to those due to main excitonic states at $L=100$ nm , shift quickly towards higher energy for smaller $L$ because of the changing proportion between confinement energy and excitonic state energy. Due to this divergence, a considerable mixing of states occurs and also many lines can be visible in the energy region above the band gap. As mentioned before, the nonlinear part of susceptibility is enhanced in a thin quantum well; on the Fig. 5 b) one can observe that absorption peaks become higher as $L$ decreases. The confinement of electrons and holes in a QW results in, illustratively speaking, ”squeezing” of excitons, which increases the binding energy and the oscillator strength of excitons, thus leading to an enhancement of the absorption.

Finally, we can explore the real part of susceptibility and the associated Kerr shift. Naturally, as follows from the Kramers-Kronig relations, each peak in the absorption spectrum corresponds to a region of anomalous dispersion, where $Re$ $\chi$ and thus also phase shift changes sign. This is visible in the Fig. 6 a). As mentioned before, the confinement states are barely visible in the nonlinear part of susceptibility and thus the spectrum is dominated by lines corresponding to excitonic states $j=0,1,2...$. Again, we see a divergence towards higher energy as $L$ decreases and also a reduction of phase shift in the limit of small $L$ due to the reduced optical length $nL$ in Eq. 46. Even for a relatively low thickness $L<100$ nm, one can observe a phase shift on the order of 50 mrad. The dependence of phase shift on the laser power is shown on the Fig. 6 b). Overall, the lower excitonic states provide a larger phase shift due to their larger oscillator strengths. The shift increases with power but is limited by optical bleaching caused by Rydberg blockade; one can see that the influence of higher states vanishes at high power. On the other hand, in the relatively lower power regime, the stronger nonlinear properties of upper states result in a considerable phase shift.

An useful measure of the nonlinearity is the maximum phase shift that can be obtained throughout the whole spectrum. The results calculated for a range of input powers are shown on the Fig. 7 a).

As expected, the power dependence is linear due to the $|E|^{2}\sim P$ factor in Eq. (46). However, the dependence on thickness, shown on the Fig. 7 b), is more complicated. Initially, as $L$ increases, the phase shift is also rapidly increasing, starting from $\Delta\Phi(L\rightarrow 0)=0$. However, at some point, the increase of the optical length is compensated by the decrease of $\chi^{(3)}$, which is enhanced in very thin wells. Thus, the phase shift reaches a local maximum and then starts decreasing with increasing $L$. Eventually, in the region of $L\sim 300$ nm, the value of $\chi^{(3)}$ stabilizes on the same level as in bulk medium and the phase shift again becomes linearly dependent on optical length. One can also notice slight oscillations on the Fig. 7 b) in the region $L\sim 100$ nm. In this regime, the confinement states are mostly visible; the thickness-dependent overlapping of multiple states slightly affects the maximum value of the susceptibility and thus the phase shift. In conclusion, the choice of quantum well thickness, input power and specific excitonic state to realize a self-Kerr shift is highly nontrivial, with multiple tradeoffs influenced by amplification of nonlinear properties, overlap of confinement states and Rydberg blockade.

In summary, we have studied the nonlinear interaction between an electromagnetic wave and Rydberg excitons in Cu₂O quantum well using the Real Density Matrix Approach, incorporating the control of the Rydberg blockade. Our theoretical, analytical results for linear and nonlinear absorption are illustrated by numerical calculations and indicate the potential experimental conditions for the best observation of confinement states in linear and nonlinear optical spectra. We show that a clear separation of confinement states and an amplification of nonlinear properties of the system are possible in sufficiently thin ($L<100$ nm) quantum wells. The interplay between nonlinearity enhancement and optical length of the system is discussed. We theoretically demonstrate that the Kerr nonlinearity and significant self-phase modulation are accomplished in a semiconductor quantum well with REs.
In short, our work provides insights into the nonlinear interactions of RE with photons in quantum-confined systems, opening interesting opportunities to explore Rydberg excitons for future opto-electronic nanoscale applications. We hope that our results might be useful for future direct integration of Rydberg confined states with nanophotonic devices.