## Final Report Summary - SQOD (Solid-state Quantum Optical Devices)

The SQOD (Solid-state Quantum Optical Devices) project aimed at integrating a theory fellow into an experimental group in the field of semiconductor based quantum optics. Research was conducted in the host group of Prof. J. J. Finley at the Walter Schottky Institut (WSI) of the Technical University of Munich in Garching bei München, in Germany with the main objectives,

1. To observe or evidence the Jaynes-Cummings nonlinearities in solid state cavity QED systems based on semiconductor quantum dots embedded within photonic crystal defect nano cavities.

2. To develop the theory of coloured (frequency resolved) photon-counting and account quantitatively for relevant quantum correlations in the experiment.

3. To develop the theory to describe the quantum dynamics of strongly-coupled semiconductor quantum dots under incoherent pumping, taking into account the spin degree of freedom and Coulomb interaction of carriers, and match it to (or guide it with) the experiments.

4. To model theoretically and assist the experimental realisation of elementary but working and usable quantum devices, such as single-photon sources and entangled-photons pair emitter.

as listed in Annex 1 of the project

The completion of these objectives are summarised below:

1.1 Observe or evidence quantum nonlinearities

The quantum-nonlinear regime is difficult to achieve in semiconductors because of the large dissipation in these systems. To better understand the interplay of the coherent and incoherent dynamics, I have refined the theoretical understanding of light-matter coupling under incoherent pumping in the nonlinear regime, obtaining semi-analytical solutions allowing to conveniently characterize various regimes of operations for a given system. According to these results, the figures of merits of the samples then available at the host institution did not allow an explicit observation of quantum nonlinearities. This situation changed only recently with new samples under production and characterization. In the meantime, the fellow has proposed a configuration allowing to explicitly demonstrate such nonlinearities even in strongly dissipative systems, such as those of the WSI, relying on a particular configuration, namely, coherent excitation of the quantum emitter detuned from the cavity mode and measurement of the cavity-photons statistics. The fellow has shown how strong, clear and robust (to dephasing, lifetime, etc.) resonances appear at precisely the anharmonic frequencies characteristic of this system. The scheme presents a few technical difficulties for an experimental implementation, as quantum statistics is difficult to acquire and the coherent excitation of the emitter is not standard. The theory has been, however, as successful as could have been by identifying a clear manifestation of quantum nonlinearities even in systems with very strong dissipation. The theoretical investigations also led to a new definition of lasing in strong coupling by identifying a regime where an ideal thresholdless laser with no threshold in the characteristic curve for the intensity or the statistics of the emitted light is almost realized as a function of the pumping power, modulo a deviation caused by a fundamental property of Bose particles: the indistinguishability of two photons. Apart from this interesting twist (due to quantum nonlinearities), the project motivated why the single-atom laser is the one that maximises strong-coupling rather than the one that minimises spontaneous emission of the emitter.

1.2 Theory of colored photon-counting

One of the ambitious goals of the project was to develop a theory of frequency-filtered photon correlations. Quantum optics is mostly concerned with measuring correlations between photons, typically in their arrival time: natural light exhibits bunching (photons arrive together), lasers emit uncorrelated photons while quantum sources emit a given number of photons at a given time. While the theoretical formalism to compute such correlations is well known since the work of Glauber (that earned him the Nobel prize in 2005), experimentalists increasingly supplement the time information with the energy of the photons. In this case, however, the theory needs to be revisited at a fundamental level since frequency and time are conjugate variables. A formalism has been successfully developped in the late seventies but consisted in an extremely awkward integral formulation with an exponential complexity of computation. This made actual computations impossible but for the simplest cases. With theorist co-workers from a nearby (Physik Department in Garching) and my present (Universidad Auton´oma de Madrid) institution, we have developped a formalism able to compute such time and frequency-resolved correlations for arbitrary systems and for any number of photons. While all previous works since the 1970s have been limited to two-photon correlations for a single atom, we computed up to four-photon correlations for a vastly more complicated system: an atom in a cavity (the Jaynes-Cummings model). This is a tremendous progress on the state of the art related to an important and increasingly popular experimental scheme. The results now available have already identified new types of emission in cavity QED (e.g. so-called “leap-frog processes” whereby the system emits simultaneously pairs of photons by jumping over intermediate rungs of the ladder of dressed states, resulting in strongly correlated emission at energies specified by our theory). The difficulty of the task made this result arrive too late to be implemented experimentally by the host, at least during the project itself, but setting up such a theory that is both efficient and practical is a complete and likely to be resounding success of this project.

1.3 Cavity QED theory and experiments

One chief goal of the project was to interact strongly with experimentalist working on semiconductor based cavity QED. Results were mostly purely theoretical, with investigations of coupling with more than oen emitter or of the effect of Coulomb interaction as well as study of alternative schemes of excitation, such as pump-and-probe experiments. A better collaboration with the experiments was obtained with the work of N. Hauke et al., who studied the emission of Germanium islands in photonic crystals, using the latter to enhance the emission efficiency (by the Purcell effect that accelerates the emission rate and collection and redirection of the photons by the cavity). They observed an a-priori unexpected result that better cavities (higher Q factor) resulted in smaller intensity of emission; exactly the opposite trend as the one motivating their experiment. The fellow showed how an interesting regime of cavity QED precisely results in the observed phenomenology. n The best results in connection with experimental observations have been obtained in another type of experiments, with no cavity. In the coherent excitation of a quantum dot (resp. of a quantum dot molecule), V. Jovanov (resp. K. Müuller) observed an unexpected sharp and intense luminescence line, that was initially supposed to be a coherent feature arising from an elastic scattering. It took us a large chunk of the project, combining theory and experimental efforts, to identify its true nature: a Fluctuation Induced Luminescence (FIL) due—not to a coherent response of the system—but to fluctuations. The difficulty in such an identification came from the nature of such fluctuations, namely, scalefree fluctuations, characterised by arbitrarily large deviations from the median that, combined with the large absorption of their system, realise so-called Black Swan events: extremely rare occurences that have a very strong impact (a new luminescence line in their case). Beside reporting a new and peculiar type of luminescence, the observation could have applications in the sampling of fat-tail distributions, a problem of notorious difficulty.

1.4 Design of working and usable quantum devices

The project concluded with two proposals for quantum devices, both photon sources, one as a two-photon emitter, the other as a more versatile N-photon emitter, N any integer (starting from 1) (this latter work is not yet submitted). The former relies on a bi-exciton in a microcavity, with twice the cavity-photon energy matching the bi-exciton energy but the single cavity-photon being detuned from the single-exciton thanks to the bi-exciton binding energy. The latter is a variation of the scheme to evidence Jaynes–Cummings nonlinearities in a case of large detuning and weak-coupling of the bottom rungs of the ladder of dressed states. Through the theoretical analysis of such cavity QED dynamics with dissipation, the fellow came to the realization that an efficient quantum emission can be obtained through a strong Purcell enhancement of the quantum dynamics, that is, one first has to isolate in a pure Hamiltonian picture a process with a large quantum amplitude, and place it in a strongly dissipative environment. This has several advantages: first is that Hamiltonian dynamics is easier to deal with, theoretically. Second is that in this world view, dissipation suddenly becomes an asset whereas, when we started, it was regarded as a disadvantage spoiling the quantum features. What happens is that the system does not, indeed, develop a quantum dynamics of its own. It directly outputs quantum states to the outside world. The N-photon emitter relies on a system in which special quantum states that are superpositions of N photons and the excited-state of the emitter with no-photon and the groudn state of the emitter. These are extremely peculiar states (only the case N = 1 is familiar, namely, the so-called “polaritons”) that can be locked in the system by a driving laser at particular energies in a detuned system which the theory provides. When the Q factor is now reduced from infinity (lossless cavity) to a number small enough, the system does not Rabi-oscillate but emits bursts of N photons. The experimental realization of such a behaviour, even at the proof-of-principle level, would represent a tremendous advance both in the fundamental physics of light-matter interactions and for applications.

In conclusion, the project was very successful from the theory point of view, which was its main angle, fulfilling completely its objectives in particular on the difficult question of the time- and frequency-resolved photon correlations, where it surpassed expectations as only two-photon correlations were discussed in the proposal and the work turned out to be general in the number of photons. The project has also been mainly successful in proposing experiments as it resulted in various schemes to bring to the laboratory, including two proposal for quantum devices. Their experimental confirmation or infirmation is only a question of time. The project resulted in a total of 18 manuscripts, 14 published, 2 in press, 1 under review and 1 still under preparation at the time of writing. This includes 2 papers in Phys. Rev. Lett., 4 in high impact-factor (> 3) journals (Phys. Rev. or New J. Phys.), 2 chapters in books and a revised and extended edition of the book “Microcavities” (Oxford University Press).

1. To observe or evidence the Jaynes-Cummings nonlinearities in solid state cavity QED systems based on semiconductor quantum dots embedded within photonic crystal defect nano cavities.

2. To develop the theory of coloured (frequency resolved) photon-counting and account quantitatively for relevant quantum correlations in the experiment.

3. To develop the theory to describe the quantum dynamics of strongly-coupled semiconductor quantum dots under incoherent pumping, taking into account the spin degree of freedom and Coulomb interaction of carriers, and match it to (or guide it with) the experiments.

4. To model theoretically and assist the experimental realisation of elementary but working and usable quantum devices, such as single-photon sources and entangled-photons pair emitter.

as listed in Annex 1 of the project

The completion of these objectives are summarised below:

1.1 Observe or evidence quantum nonlinearities

The quantum-nonlinear regime is difficult to achieve in semiconductors because of the large dissipation in these systems. To better understand the interplay of the coherent and incoherent dynamics, I have refined the theoretical understanding of light-matter coupling under incoherent pumping in the nonlinear regime, obtaining semi-analytical solutions allowing to conveniently characterize various regimes of operations for a given system. According to these results, the figures of merits of the samples then available at the host institution did not allow an explicit observation of quantum nonlinearities. This situation changed only recently with new samples under production and characterization. In the meantime, the fellow has proposed a configuration allowing to explicitly demonstrate such nonlinearities even in strongly dissipative systems, such as those of the WSI, relying on a particular configuration, namely, coherent excitation of the quantum emitter detuned from the cavity mode and measurement of the cavity-photons statistics. The fellow has shown how strong, clear and robust (to dephasing, lifetime, etc.) resonances appear at precisely the anharmonic frequencies characteristic of this system. The scheme presents a few technical difficulties for an experimental implementation, as quantum statistics is difficult to acquire and the coherent excitation of the emitter is not standard. The theory has been, however, as successful as could have been by identifying a clear manifestation of quantum nonlinearities even in systems with very strong dissipation. The theoretical investigations also led to a new definition of lasing in strong coupling by identifying a regime where an ideal thresholdless laser with no threshold in the characteristic curve for the intensity or the statistics of the emitted light is almost realized as a function of the pumping power, modulo a deviation caused by a fundamental property of Bose particles: the indistinguishability of two photons. Apart from this interesting twist (due to quantum nonlinearities), the project motivated why the single-atom laser is the one that maximises strong-coupling rather than the one that minimises spontaneous emission of the emitter.

1.2 Theory of colored photon-counting

One of the ambitious goals of the project was to develop a theory of frequency-filtered photon correlations. Quantum optics is mostly concerned with measuring correlations between photons, typically in their arrival time: natural light exhibits bunching (photons arrive together), lasers emit uncorrelated photons while quantum sources emit a given number of photons at a given time. While the theoretical formalism to compute such correlations is well known since the work of Glauber (that earned him the Nobel prize in 2005), experimentalists increasingly supplement the time information with the energy of the photons. In this case, however, the theory needs to be revisited at a fundamental level since frequency and time are conjugate variables. A formalism has been successfully developped in the late seventies but consisted in an extremely awkward integral formulation with an exponential complexity of computation. This made actual computations impossible but for the simplest cases. With theorist co-workers from a nearby (Physik Department in Garching) and my present (Universidad Auton´oma de Madrid) institution, we have developped a formalism able to compute such time and frequency-resolved correlations for arbitrary systems and for any number of photons. While all previous works since the 1970s have been limited to two-photon correlations for a single atom, we computed up to four-photon correlations for a vastly more complicated system: an atom in a cavity (the Jaynes-Cummings model). This is a tremendous progress on the state of the art related to an important and increasingly popular experimental scheme. The results now available have already identified new types of emission in cavity QED (e.g. so-called “leap-frog processes” whereby the system emits simultaneously pairs of photons by jumping over intermediate rungs of the ladder of dressed states, resulting in strongly correlated emission at energies specified by our theory). The difficulty of the task made this result arrive too late to be implemented experimentally by the host, at least during the project itself, but setting up such a theory that is both efficient and practical is a complete and likely to be resounding success of this project.

1.3 Cavity QED theory and experiments

One chief goal of the project was to interact strongly with experimentalist working on semiconductor based cavity QED. Results were mostly purely theoretical, with investigations of coupling with more than oen emitter or of the effect of Coulomb interaction as well as study of alternative schemes of excitation, such as pump-and-probe experiments. A better collaboration with the experiments was obtained with the work of N. Hauke et al., who studied the emission of Germanium islands in photonic crystals, using the latter to enhance the emission efficiency (by the Purcell effect that accelerates the emission rate and collection and redirection of the photons by the cavity). They observed an a-priori unexpected result that better cavities (higher Q factor) resulted in smaller intensity of emission; exactly the opposite trend as the one motivating their experiment. The fellow showed how an interesting regime of cavity QED precisely results in the observed phenomenology. n The best results in connection with experimental observations have been obtained in another type of experiments, with no cavity. In the coherent excitation of a quantum dot (resp. of a quantum dot molecule), V. Jovanov (resp. K. Müuller) observed an unexpected sharp and intense luminescence line, that was initially supposed to be a coherent feature arising from an elastic scattering. It took us a large chunk of the project, combining theory and experimental efforts, to identify its true nature: a Fluctuation Induced Luminescence (FIL) due—not to a coherent response of the system—but to fluctuations. The difficulty in such an identification came from the nature of such fluctuations, namely, scalefree fluctuations, characterised by arbitrarily large deviations from the median that, combined with the large absorption of their system, realise so-called Black Swan events: extremely rare occurences that have a very strong impact (a new luminescence line in their case). Beside reporting a new and peculiar type of luminescence, the observation could have applications in the sampling of fat-tail distributions, a problem of notorious difficulty.

1.4 Design of working and usable quantum devices

The project concluded with two proposals for quantum devices, both photon sources, one as a two-photon emitter, the other as a more versatile N-photon emitter, N any integer (starting from 1) (this latter work is not yet submitted). The former relies on a bi-exciton in a microcavity, with twice the cavity-photon energy matching the bi-exciton energy but the single cavity-photon being detuned from the single-exciton thanks to the bi-exciton binding energy. The latter is a variation of the scheme to evidence Jaynes–Cummings nonlinearities in a case of large detuning and weak-coupling of the bottom rungs of the ladder of dressed states. Through the theoretical analysis of such cavity QED dynamics with dissipation, the fellow came to the realization that an efficient quantum emission can be obtained through a strong Purcell enhancement of the quantum dynamics, that is, one first has to isolate in a pure Hamiltonian picture a process with a large quantum amplitude, and place it in a strongly dissipative environment. This has several advantages: first is that Hamiltonian dynamics is easier to deal with, theoretically. Second is that in this world view, dissipation suddenly becomes an asset whereas, when we started, it was regarded as a disadvantage spoiling the quantum features. What happens is that the system does not, indeed, develop a quantum dynamics of its own. It directly outputs quantum states to the outside world. The N-photon emitter relies on a system in which special quantum states that are superpositions of N photons and the excited-state of the emitter with no-photon and the groudn state of the emitter. These are extremely peculiar states (only the case N = 1 is familiar, namely, the so-called “polaritons”) that can be locked in the system by a driving laser at particular energies in a detuned system which the theory provides. When the Q factor is now reduced from infinity (lossless cavity) to a number small enough, the system does not Rabi-oscillate but emits bursts of N photons. The experimental realization of such a behaviour, even at the proof-of-principle level, would represent a tremendous advance both in the fundamental physics of light-matter interactions and for applications.

In conclusion, the project was very successful from the theory point of view, which was its main angle, fulfilling completely its objectives in particular on the difficult question of the time- and frequency-resolved photon correlations, where it surpassed expectations as only two-photon correlations were discussed in the proposal and the work turned out to be general in the number of photons. The project has also been mainly successful in proposing experiments as it resulted in various schemes to bring to the laboratory, including two proposal for quantum devices. Their experimental confirmation or infirmation is only a question of time. The project resulted in a total of 18 manuscripts, 14 published, 2 in press, 1 under review and 1 still under preparation at the time of writing. This includes 2 papers in Phys. Rev. Lett., 4 in high impact-factor (> 3) journals (Phys. Rev. or New J. Phys.), 2 chapters in books and a revised and extended edition of the book “Microcavities” (Oxford University Press).