Quantum-correlated two-photon transitions to excitons in semiconductor quantum wells

The dependence of the excitonic two-photon absorption on the quantum correlations (entanglement) of exciting biphotons by a semiconductor quantum well is studied. We show that entangled photon absorption can display very unusual features depending on space-time-polarization biphoton parameters and absorber density of states for both bound exciton states as well as for unbound electron-hole pairs. We report on the connection between biphoton entanglement, as quantified by the Schmidt number, and absorption by a semiconductor quantum well. Comparison between frequency-anti-correlated, unentangled and frequency-correlated biphoton absorption is addressed. We found that exciton oscillator strengths are highly increased when photons arrive almost simultaneously in an entangled state. Two-photon-absorption becomes a highly sensitive probe of photon quantum correlations when narrow semiconductor quantum wells are used as two-photon absorbers.


Introduction
Quantum correlations (entanglement) with no classic analogs are among the most surprising consequences of quantum mechanics. A multipartite quantum system in an entangled state must be described as a non-separable entity without any reference to well defined individual states. Although, the measurement of a local observable can yield to completely random values, there are strong correlations between different parts of the composite system. These non-classical correlations can usually appear in the form of energy, linear momentum or angular momentum conservation requirements. They are crucial for quantum-enhanced technologies [1] and would give access to the implementation of new quantum algorithms. Widely studied entangled systems are those formed by quantum-correlated photons which are produced by the socalled spontaneous-parametric-down-conversion (SPDC) mechanism [2] where a continuous or pulsed pump laser beam is sent to an optical nonlinear crystal. In a SPDC elemental process one laser pump photon is destroyed and two lower energy photons, signal and idler, are created within the material. A signal-idler pair (or biphoton) is a highly entangled system in variables such as energy, linear momentum, angular momentum and polarization [3]. In the last two decades, this process has been exploited in different quantum optics experimental scenarios especially designed for probing the foundations of quantum physics [4,5]. Recently, the interfacing of such quantum optical beams with solid-state quantum objets, like Cooper-pairs in a superconductor box [6] and many-body correlations in semiconductors [7], has proven to be an emerging fertile field worthwhile of investigation. In the present work the theoretical study of the absorption of quantum-correlated photons by electron-hole pairs in a semiconductor quantum well (QW) is addressed. Of special interest is the understanding of the interplay between field-correlation features and photon absorption. A well known result is that the two-photonabsorption (TPA) process provides information about higher-order correlation functions of the radiation field [8,9]. In particular, the photon first-order correlation function, although important in determining the spectrum of one-photon absorption response by a matter system, is of limited usefulness for describing multi-photon processes. Therefore, in order to properly describe the matter optical response to new light sources, including quantum-correlated photons, the inclusion of higher-order correlation functions is crucial to theoretically get the correct optical response. This is a first point of interest in the present work. On the other hand, from an experimental point of view, several procedures have been established to control the different entanglement features of signal/idler pairs. For instance, by inserting spectral filters in the paths of signal and idler photons allows for the production of narrow-band time-entangled biphotons. Additional control mechanisms affecting the biphoton wavefunction shape, including the engineering of the spatial profile of the pump laser and selecting the geometry of the nonlinear crystal [10][11][12], provide useful tools to manipulate the two-photon wavefunction or equivalently the photon temporal correlations. As a second goal of the present work, we perform a comparative study of the absorption response of a semiconductor QW for differently quantum-correlated signal/idler photon pairs.
Previous entangled-two-photon-absorption (ETPA) by specific matter systems include simple molecules [13], dendrimer molecular systems [14,15] , receptors. Semiconductor materials provide advantages for photon absorption over atomic systems due to the possibility of tailoring the band density of states and/or the density of carriers through the control of the confinement dimensionality and excitation mechanisms. Thus, they offer the opportunity of measuring photon temporal correlations in a femtosecond timescale by exploiting processes where the almost simultaneous arriving of two photons is required. It is therefore natural to ask whether more versatile large-bandwidth devices could be engineered by resorting to semiconductor nanostructures like a QW as the two-photon receptor. Large optical nonlinearities have been measured in such systems, with a wealth of new optical phenomena which are difficult to observe in bulk crystals. In particular, photon absorption in such nanostructures turns out to be a nonmonotonic function of the photon energy with steps occurring whenever transitions are available between new pairs of quantized subbands. In the quantum limit of strong confinement (L ∼ few nanometers) those spectra present a very rich structure showing new features associated to a transformed density of available states as well as to inter-particle interaction effects. Furthermore, two-photon emission from semiconductor QWs has been realized to be a source of high-intensity room-temperature entangled photons [24]. Since ETPA processes provide information about higher order correlation functions of the radiation field, QWs nanostructures may be therefore useful for in-situ detection of entangled photons produced by other on-chip semiconductor sources like QWs [24] or quantum dots [25].
We limit ourselves to the case when one-photon absorption is excluded. This paper is organized as follows: In Section II we briefly review the main aspects of quantum-correlated photons and the general ETPA formalism for both non-interacting electron-hole pairs and excitons in QWs. In Section III results of ETPA by QWs are discussed. Finally, in Section IV the main conclusions of the present work are summarized.

Theoretical background
The ETPA probability by a semiconductor QW requires a full quantum description of both the radiation field and the matter system. The radiation-matter system of interest is outlined in Fig.  1. We first address the issue concerning the generation and description of quantum correlated biphotons. Secondly, the description of the absorber system, a semiconductor QW, is provided by focusing on the selection rules governing the interaction of radiation-matter of interest for the present work. Explicit expressions are finally given for the ETPA rate in both cases of no excitonic effects as well as when excitons are included.

Quantum-correlated biphoton wavefunctions
We start by describing the quantum-correlated photon wavefunctions in both time and frequency variables. For the purposes of concreteness we shall focus on realistic entangled photon sources based on type-II SPDC processes [26] for which there exists the possibility of experimentally tailoring the degree of quantum correlation among the photons, allowing even to pro-duce uncorrelated twin photons. In this way, it becomes possible to analyze, within the same theoretical framework, the novel aspects that photon quantum correlations bring to the basic process of absorption by a semiconductor nanostructure.The biphoton wavefunction is given by where |Ψ 2 is the radiation quantum state emerging from the type-II SPDC device whileâ 1 and a 2 denote annihilation operators for signal/idler photons with corresponding field amplitudes E 1 and E 2 , respectively (E i has dimensions of electric field times square root of time). . The Schmidt number for graphs (b) and (d) is the same, K = 3. In (b) τ T < 1 corresponding to a a FAC biphoton while in (d) τ T > 1 related to a FC biphoton. For graph (c) the Schmidt number is K = 1 corresponding to an unentangled biphoton state. Axes display the signal and idler frequencies in units of ω The Fourier transform of the type-II entangled biphoton wavefunction [2] has the form whereN is a normalization constant which ensures that dω 1 dω 2 |Ψ(ω 1 , ω 2 )| 2 = 1, ω (0) p = ω s + ω i is the central pump frequency, ω s (ω i ) the central signal (idler) frequency, T is the pump coherence time (inverse of pump bandwidth) and δ ω = ω s − ω i . In a great variety of cases, especially when the relation 2v −1 withN = 2 τT /π and the effective signal/idler quantum-correlation time is represented by τ. While different values of τ can be attained by modifying non-linear crystal properties such as its length, and group velocities of different pump/signal/idler photons, the values of T are essentially associated to coherence properties of the exciting laser pump, thus being affected by the pump laser pulse duration, among other factors. Thus, quantum-correlated photons produced by SPDC devices can be designed with a high spectral purity for sufficiently large values of T , and at the same time with a narrow timecoincidence window as characterized by small values of τ. This last feature marks a strong departure from the behavior of classical (laser) light.
The biphoton wavefunction given in Eq. (3) encloses different photon quantum correlations which can be controlled by changing the pump bandwidth (equivalently the pump pulse duration T ) and the correlation time τ. These two experimentally accessible control parameters lead directly to a measure of the amount of entanglement of two photon modes as quantified by the so-called Schmidt number, which turns out to be [28] It reaches the minimum value of 1 (unentangled biphoton state) when T = τ as shown in Fig.  2(a). Although, identical values of the Schmidt number can be obtained for two different ratios τ/T , the quantum correlation features can differ significantly. This point is illustrated in Fig.  2 where the biphoton joint spectral intensity is plotted for different pump pulse duration. Figs. 2(b) and 2(d) correspond to identical biphoton entanglement, K = 3. However, while in Fig. 2(b) a frequency-anti-correlated (FAC) biphoton is displayed for long pump pulses, τ/T < 1, in the opposite case, short pump pulses, τ/T > 1, the biphoton state as shown in Fig.  2(d) is frequency-correlated (FC). The uncorrelated biphoton joint intensity, K = 1, produced by a pump pulse with T = τ, is shown in Fig. 2(c).

ETPA: no-excitonic effects
Now, we describe how quantum-correlated photons interact with a semiconductor nanostructure. We restrict ourselves to consider a semiconductor infinite square well potential of thickness L in direction z. The initial state corresponds to the QW in the ground state |G and the radiation field with two quantum-correlated photons described by the state |Ψ 2 as produced by a SPDC device, while the final state corresponds to the QW in any excited state |n c , n v , k with an electron in conduction sub-band n c , a hole in the valence sub-band n v (identical twodimensional k wavevectors for electron and hole, as discussed below) and the photon field in the vacuum state, |0 . The ETPA amplitude [29] is given by α 2 (n c , n v , k) = n c , n v , k, 0|Û 2 |G, Ψ 2 where the evolution operator is evaluated up to second order in the radiation-matter coupling HamiltonianĤ I (t) (in the interaction picture), corresponding toÛ 2 The ETPA probability amplitude turns out to be where the biphoton wavefunction in time domain, ψ(t 1 ,t 2 ), is given by Eq. (1), the interband transition energy corresponds tō and the intraband transition energy can be one of the followinḡ In the last two equations, the first one describes an intraband conduction transition whereas the second one is associated to an intraband valence transition. In Eqs. (5)-(7) m 0 is the free electron mass, m * c (m * v ) the effective electron (hole) mass, µ denotes the reduced electron-hole mass and E g is the bulk semiconductor energy gap. The r-gauge for the dipole transition elements have been adopted, which read as n c , k, n v , k |û ε · eˆ r|G = − ieh m 0 E g (û ε · P cv )δ n c ,n v δ k, k for direct interband transitions with P cv the interband Kane transition element, and n ν , k|û z · eˆ r|ñ ν , k = − 8eL ν ) 2 δ k, k for intraband transitions within the ν = c conduction band or the ν = v valence band. For intraband transitions a selection rule, imposing a change of parity, implies the condition |ñ ν − n ν | = 1, 3, 5... should hold [30,31]. In these expressionsû ε denotes an unitary vector pointing in directions x, y or z and the allowed transistions are schematically depicted in Fig. 3. Quantum correlated photons emitted by a type-II SPDC crystal have orthogonal polarizations (see Fig. 1). Consequently, we assume the inter-band transition is induced by an on-plane polarized photon, i.e.û ε =û x , while intra-band transitions respond to perpendicularly polarized photons, i.e.û ε =û z . In this way, TPA from a heavy-hole in the initial ground state is possible when the interband transition occurs first excited by theû x polarized photon followed by the intra-band transition induced by theû z polarized photon. Thus, a heavy-hole TPA, which is a forbidden transition [31] under laser excitation with identicalû z photon polarizations, is now permitted for type-II SPDC generated biphotons. Therefore, inter-band transitions involving heavy holes are the only ones we shall consider below.
Finally, the ETPA probability is the sum over final states as P 2 (ω) = ∑ n c ∑ n v ∑ k |α 2 (n c , n v , k)| 2 , which for the present case turns out to be Dimensionless variables have been used: energies in units of the bulk semiconductor gap E g , frequencies in units of 2π , being A QW the QW transversal area. In case the biphoton wavefunction is unentangled (see Fig. 2(c)) the classical laser absorption results [30,31] are easily recovered.

ETPA: excitonic effects
The above results, can be easily extended to the case where excitonic effects are taken into account. The interband transition occurs now between the semiconductor ground state |G and an exciton state denoted by |n c , n v , β , m where n c and n v represent as before sub-band indices for electron and hole, respectively, while the relative exciton bound state is marked by the principal quantum number, β , and the angular momentum projection number in direction z is m. The matrix element for this transition becomes now [32] n c , n v , β , m|û ε · eˆ r|G = − ieh m 0 E g (û ε · P cv ) A QW Φ β ,m ( r = 0)δ n c ,n v δ m,0 where a B denotes the 3D Bohr radius and Φ β ,m ( r) is the exciton wavefunction. As indicated by this selection rule, only interband transitions producing m = 0 or S-excitons are allowed. On the other hand, the intra-band matrix element becomes now [32] n c , n v , β , m |û ε · eˆ r|n c , n v , β , m = − 8eL π 2 a B L 2 n ν n ν (n 2 ν −n 2 ν ) 2 δ β ,β δ m,m for intraband transitions within the ν = c conduction band or the ν = v valence band.
The excitonic ETPA probability becomes where Φ 1S ( r) and Φ e−h ( κ, r) represent the bound 1S exciton wavefunction and the unbound, but Coulomb interacting, electron-hole pair wavefunction, respectively. We limit ourselves to consider only the exciton ground-state, or 1S-state, with dimensionless binding exciton energy denoted as α 1S = E 1S /E g . Excitonic effects in a finite thickness QW can be described by relying upon a variational treatment [33] with a trial bound exciton wavefunction for the transverse relative motion such as Φ 1S ( r) = 2 π 1 λ e − r λ , a hydrogen-like wavefunction with λ the variational parameter, which reproduces well the 2D limit corresponding to an infinitely narrow QW exciton. In this limit, Note that 1 ≤ S(k) ≤ 2 accounts for the effects of the electron-hole Coulomb interaction in the unbound states, taking its maximum value, S(k) = 2 at the bottom edge of sub-bands, i.e. at k = 0. In the limit of no Coulomb interaction between electrons and holes, i.e. λ → ∞, results following from Eq. (10) are obtained again. In case the biphoton wavefunction is unentangled (see Fig. 2(c)) and T → ∞, the classical laser absorption results [35] are also recovered.

Results
In order to illustrate the results about ETPA by a QW we rely on recent experimental data where the quantum-correlated photons are produced using non-linear materials like periodically poled First, we discuss ETPA results when Coulomb effects are ignored. As can be seen in Fig. 4, for a narrow QW with L = a B /2, the main effects of photon quantum correlations are evident in variations of the absorption edges. Steps in this plot occur when the degenerate signal/idler biphoton is on resonance with the transitions (from low to high energy): HH2-C1, HH1-C2 and HH3-C2, respectively, where HH denotes a heavy-hole sub-band while C denotes a conduction sub-band. This type of TPA transitions is forbidden when both photons are z-polarized, but here these transitions are allowed by the fact that type-II SPDC produces cross-polarized photons, breaking that selection rule. ETPA for two extreme values of the signal/idler correlation time τ, with the same entanglement Schmidt measure K = 1.3, are shown in Fig. 4 (for comparison, it is also depicted the unentangled case, K = 1). For highly quantum correlated photons, main graph in Fig. 4, ω g τ = 50, changes in the absorption edge for each allowed two-photon transition are different depending on the frequency-anti-correlated or frequency-correlated biphoton feature. These differences are better observed in the bottom inset where differential-ETPA (d-ETPA, defined as ET PA(K = 1)-ET PA(K = 1)) is shown. As compared with the unentangled signal,  Fig. 4, where a weakly quantum correlated biphoton is assumed, ω g τ = 500. In this last case, the ETPA dependence on the biphoton entanglement amount, as given by K > 1, is practically irrelevant, as ETPA curves for all cases (FAC, unentangled and FC biphotons) are hard to distinguish in that plot. This behavior indicates that quantum correlated photons arriving almost simultaneously, small τ limit, should produce two-photon absorption changes near the sub-band edges, leaving unaltered the TPA values well inside the continuous bands. By contrast, when the quantum correlation window broads, large τ, the classical TPA behavior is quickly recovered. Although beyond the scope of the present work, we mention that by including a time delay between signal/idler photons a further enhancement of entanglement induced transparency should be possible as well as the exploration of virtual-state spectroscopy in a semiconductor nanostructure like a QW. Biphoton quantum correlations become more prominent when Coulomb-excitonic effects are taken into account. Fig. 5 displays results where both excitonic and continuous electronhole contributions are included. Highly quantum correlated biphotons, ω g τ = 100, produce strong excitonic peaks when the photons are frequency-anti-correlated whereas in the case of frequency-correlated photons, with the same amount of entanglement, the excitonic peaks practically vanish. This fact can be simply understood by resorting to the photon joint distribution as seen in Fig. 2(b) and Fig. 2(d), where the squeezing in both cases is evident but the spreading is higher (the maximum height is lower) in the frequency-correlated case. Although matterradiation resonance conditions are met for different electron-hole transitions (bound and unbound) using FAC or FC biphotons, the photon joint density is larger in the former case as compared to the latter case. As a consequence, the exciton oscillator strengths are highly increased when the photons arrive almost simultaneously as can be checked by Fourier transforming the information given in Fig. 2(b).
Note that these results are obtained without any assumed line-shape function for the exciton. Thus, the variation in exciton peak heights and full widths at half maximum are intrinsic and only determined by the laser pump coherence associated to time T , or equivalently to K values, and to signal/idler temporal quantum correlations as specified by time τ. The bound-exciton ETPA signal shows a highly selective and enhanced excitation efficiency under quantum-correlated illumination, somewhat similar to recently reported theoretical results for a three-level atomic system [37]. However, the differences with the present QW case, are that selectivity and efficiency for the excitonic ETPA depend also on the specific sub-band transitions, giving higher values for the lowest-energy transitions involved. Thus, by contrast with laser induced TPA, ETPA allows the control of two-photon transitions to excitons by tailoring the frequency correlation between signal and idler photons produced by SPDC-II processes. The interplay between carrier spatial confinement in a QW and temporal photon correlations in ETPA features is shown in Fig. 6 for FAC biphotons (see Fig. 2(b)). Excitonic ETPA signals (left column) are separately displayed from continuous electron-hole ETPA signals (right column) for three QW thicknesses L = a B /2, two-dimensional QW limit, L = a B and L = 2a B . In Fig. 6 ETPA from highly quantum correlated biphotons with K = 1.3 and ω g τ = 50, are depicted in red curves. In order to compare with the ETPA signal from unentangled biphotons, results for K = 1 are also shown in black lines in Fig. 6, and they allow us to conclude that a strong 2D confinement enhances substantially the ETPA response from highly quantum correlated biphotons. Furthermore, Coulomb effects, enhanced by carrier confinement, also impact electron-hole or continuous ETPA results as can be seen by comparing Fig. 4, no Coulomb interaction, with Fig. 6 (right column), for a narrow QW, L = a B /2. Most importantly, exciton ETPA signals at exciton resonances are clearly resolved (second and third row, left column) when entangled photons are used. According to well known results, increasing the QW thickness yields to a dense packing of excitonic peaks. However, our results predict that an entangled photon source can be used to gain in spectroscopic resolution. Additionally, the variations of the three lowest in energy excitonic ETPA peaks, as a function of the pump laser coherence time T , are plotted in Fig. 7 for narrow QWs, L = a B /2, a B , and biphotons with ω g τ = 50, 100 (K > 1). In Fig. 7, full ETPA signals at the resonant two-photon exciton frequencies, including the sum of the exciton and electron-hole contributions, are shown. ETPA becomes a highly sensitive probe of photon correlations when narrow semiconductor QWs are used as two-photon absorbers. Thus, we find that one should be able to harness the non-trivial cross effects provided by carrier confinement and temporal photon quantum correlations to detect the latter ones.
Clearly, the main differences between classical TPA and ETPA results arise when strongly quantum correlated biphotons arrive to the QW absorber, i.e. small τ. This is due to the fact that high temporal correlation, or equivalently time coincidence, between the photons implies the photon pair is absorbed as a whole entity and not in a sequential way of individual photon processes through an intermediate virtual state. Indeed, in the latter case classical TPA occurs. One major source of losing spectroscopic information in TPA experiments is the dephasing associated with exciton quasi-particles. However, the high sensitivity of ETPA to small quantum correlation times, τ usually in the femtosecond range, justifies that dephasing excitonic effects in the entangled-photon absorption process can be safely ignored, as it has been the case in the present work.

Conclusions
Two-photon transitions to excitons in semiconductor QWs by absorbing an entangled biphoton have been addressed. Analytical expressions for the ETPA rate including proper second-order correlation functions describing the differently quantum-correlated biphotons were obtained. We have demonstrated that optical non-linear effects at the level of a single couple of photons can be observed by playing with the correlation time window of cross-polarized photons emitted by a type-II SPDC process. In particular, we found that a high ETPA rate occurs when the sum of frequencies of the signal/idler photon pair is on resonance with the discrete QW transitions even though each photon can have a wide frequency range. This effect, specially strong for narrow QWs and highly frequency-anti-correlated biphotons, amounts to a new resource for ultrafast two-photon counting with semiconductor devices [23]. Furthermore, semiconductor nanostructures like QWs provide new large-bandwidth multiple-photon detectors where both discrete and continuous carrier states play important roles in discriminating the degree of photon entanglement.
It should be further mentioned that the remarkable point is that the effects here discussed does not rely on any change in the matter itself but on the quantum correlations of the exciting light. In this way a break up, or at least a weakening, of the usual optical selection rules can be tailored. We also believe that our results on excitonic ETPA can be exploited by integration with other semiconductor nanostructures, like quantum dots, in order to probe photon entanglement produced by on-chip sources. These effects can be observed in semiconductor QWs samples where the homogeneous excitonic linewidth is smaller than the entanglement induced broadening of the exciton absorption. We also suggest that cross effects from spatial carrier confinement and temporal photon quantum correlations can be exploited in various optoelectronic solid-state based entanglement set-ups providing new ways of addressing controllable quantum correlations in condensed-matter environments.