GOVERNMENT INTEREST

This invention was made with government support under grant number 1729383 awarded by the National Science Foundation. The government has certain rights in the invention.

PRIORITY CLAIM

This application claims the benefit of U.S. Provisional Patent Application Serial No. 63/160,751 , filed March 13, 2021 , the disclosure of which is incorporated herein by reference in its entirety.

TECHNICAL FIELD

The subject matter described herein relates generally to quantum information processing. More particularly, the subject matter described herein relates to methods and systems for High temperature superradiant and superfluorescent states and photon sources in for quantum information processing and communication applications.

BACKGROUND

Quantum sensing, computation and communication applications require creation, manipulation and reading the quantum states of matter. Light matter interactions can be used to create tailored collective many body phases in solidsl ,2,3, which could be used for realization of these quantum applications. However, in most cases these collective quantum states are fragile, with a short decoherence and dephasing time, limiting their existence to precision tailored structures under delicate conditions such as cryogenic temperatures and/or high magnetic fields. SUMMARY

This document describes methods and systems for quantum information processing. In some examples, a system for producing a superfluorescence photon source includes thin films of hybrid and lead halide containing perovskites under laser excitation.

The subject matter described herein may be implemented in hardware, software, firmware, or any combination thereof. As such, the terms “function” or “node” as used herein refer to hardware, which may also include software and/or firmware components, for implementing the feature(s) being described. In some exemplary implementations, the subject matter described herein may be implemented using a computer readable medium having stored thereon computer executable instructions that when executed by the processor of a computer control the computer to perform steps. Exemplary computer readable media suitable for implementing the subject matter described herein include non-transitory computer readable media, such as disk memory devices, chip memory devices, programmable logic devices, and application specific integrated circuits. In addition, a computer readable medium that implements the subject matter described herein may be located on a single device or computing platform or may be distributed across multiple devices or computing platforms.

BRIEF DESCRIPTION OF DRAWINGS

Figure 1 is a diagram illustrating an example of generation of a qubit;

Figure 2 illustrates an example of manipulation of a qubit;

Figure 3 is a block diagram illustrating an example system for an operation of reading out of a qubit;

Figure 5A shows the absorption spectra of MAPbl3 at temperatures ranging from 78 K to room temperature;

Figure 5B shows the continuous wave photoluminescence (CW PL) spectra at 78 K;

Figure 5C shows the PL spectra when the sample is excited with 120 fs pulses at 400 nm; Figure 6A shows the transient PL kinetics measured at two different above threshold excitation fluences;

Figures 6A - 6C show the quantitative values of the delay time “ _D ”, characterizing the time in which spontaneous synchronization takes place, and the real width “x _R " of the SF burst, obtained from these fits;

Figures 6D - 6G show pump-probe traces;

Figure 6H shows how the electronic levels in the coherent macroscopic system form a Dicke-ladder;

Figure 7A shows an example detector setup with two orthogonal detectors;

Figure 7B shows a normalized intensity curve from the two detectors;

Figure 8A shows a particle in a box that simulates the electronic transition in which the superposition of the first two quantum levels forms the excited electronic polarization;

Figure 8B shows correlated coupling as a particular spring mode modulating the potential barrier of the particle in a box in Figure 8B;

Figure 8C shows an illustration of the vibrational isolation;

Figure 8D depicts multiple quantum wells connected to one single phonon mode that modulates their potential collectively;

Figures 9A and 9C show the progress of the PL spectra as the excitation fluence increased;

Figures 9B and 9D show the intensity as the excitation fluence is increased;

Figures 10A and 10D show the time resolved traces for the sharp PL feature;

Figures 10B-C and 10E-F show the time delay and the real widths of the SF bursts and the fits;

Figure 11A illustrates that, when a superradiant state is formed, it makes a giant dipole that can strongly interact with vacuum fields and other superradiant giant dipoles;

Figure 11 B shows the evolution of the PL spectra as the excitation fluence increased; Figures 11 C and 11 D show the peak shift and the broadening in the spectra;

Figures 12A - 12C are tables listing 3D perovskite materials;

Figure 13 is a table listing 2D perovskites;

Figure 14 is a list of metal ion dopants; and

Figure 15 is a list of small molecule dopants

DETAILED DESCRIPTION

Quantum sensing, computation and communication applications require creation, manipulation and reading the quantum states of matter. Light matter interactions can be used to create tailored collective many body phases in solids, which could be used for realization of these quantum applications. Because these macroscopic states are larger than a single defect site or single atoms, they can be physically addressed using external fields, forces enabling quantum application. However, in most cases these collective quantum states are fragile, with a short decoherence and dephasing time, limiting their existence to precision tailored structures under delicate conditions such as cryogenic temperatures and/or high magnetic fields.

In this specification, we show that lead-halide perovskites exhibit such a collective coherent quantum many-body phase, namely, superfluorescent state of phase-locked dipoles at high temperatures (beyond room temperature). Pulsed laser excitation first creates a population of high energy excitations, which quickly relax to lower energy domains and then develop a macroscopic quantum coherence through spontaneous synchronization. The temperature and excitation power dependence of the steady state and transient photoluminescence unambiguously confirm all the characteristics of superfluorescence. Creation and manipulation of collective coherent states in lead-halide perovskites can be used as the basic building blocks for quantum applications.

Figure 1 is a diagram illustrating an example of generation of a qubit. A first column 100 illustrates N individual quantum objects in a two level system. The N individual quantum objects are incoherent. A second column 104 shows a single quantum object in a two level system. The quantum object is coherent. For example, the single quantum object can be a giant dipole, formed by superradiance (SR) or superfluorescence (SF) 102. Unlike individual dipoles or two level systems, at atomic or microscope scale, SR and SF is a collective state. The physical size can be, e.g., much larger than a qubit in a point defect in a semiconductor, or in a spin state qubit in one of various vacancy centers.

Moreover, since the coherent system can be a giant dipole, the dipole moment can be accessible with external electric fields to perform quantum operations using Rabi oscillations. In some examples, the Rabi oscillations scale with d ^* E, where d is the dipole moment.

Figure 2 illustrates an example of manipulation of a qubit, e.g., the second column 104 of Figure 1. In Figure 2, a single qubit rotation 200 is shown as a Rabi oscillation. Figure 2 shows an example field 202 to interrogate and manipulate a giant dipole oscillator. The polarization direction and the electric field strength are used to tune the Rabi oscillation for manipulation and interrogation of the oscillator. This external field can be, e.g., pulsed, and beam and pulse shaping can be configured on the operation to perform (interrogation or manipulation).

Figure 3 is a block diagram illustrating an example system 300 for an operation of reading out of a qubit. The system 300 can also be used to manipulate a qubit. Figure 3 does not depict all of the optics and delay and phase control that could be used in the system 300; however, a person of ordinary skill in the art would understand how to configure the system for reading out a qubit.

The system 300 includes a sample 302 configured such that a giant dipole qubit is in the sample 302. An initialization pulse 304 is incident on the sample 302. SF or SR state is a coherent emission.

A mirror 306 reflects the SF or SR pulse 308 towards a beam splitter 310. A first portion of the emission from the beam splitter can be used to determine a time domain signal 312. A second portion of the emission from the beam splitter can be used to determine the frequency domain response 314, e.g., using spectral interferometry. The system 300 can include a local oscillator 316 to provide interference, e.g., a local oscillator with a well defined phase.

The system 300 and other systems known to those of ordinary skill in the art can be used for interrogating or manipulating qubits. The composition of the sample 302 and the configuration of the sample 302 for creating qubits is described further below with reference to two papers, “High Temperature Superfluorescence in Methyl Ammonium Lead Iodide” and “Room Temperature Superfluorescence in Lead Halide Perovskites Through Polarons in Lead Halide Perovskites.” Further examples for using perovskites in quantum computing are further provided in a paper, “Room-Temperature Superfluorescence to Room-Temperature Superconductivity in Perovskites.”

High Temperature Superfluorescence in Methyl Ammonium Lead

Iodide

Spontaneous synchronization of oscillators is a fascinating process. While it is best visualized as the buildup of a collective phase in initially randomly oscillating metronomes coupled to the same medium, spontaneous synchronization is a universal phenomenon occurring in natural processes such as the initial ordering of the planetary orbits, frequency locking of triode generators, and signal synchronization of fireflies in the wild (1 ). This phenomenon prevails not only in macro and micro classical realms, spanning physical and biological systems, but it also leads to the manifestation of exotic collective quantum phenomena (2). In the quantum domain, systems are described by wave functions in which the “phase” plays a dominant role as it determines the waveform and its relation to other waves, along with their collective behavior under external stimuli. While an incoherent population of quantum objects has a random distribution of phases, spontaneous synchronization leads to symmetry breaking and the observation of exotic collective quantum phenomena including but not limited to, superconductivity, Bose-Einstein condensation, and the collective dynamics of Josephson junctions (3). A remarkable example of spontaneous synchronization is the superfluorescence of optically excited dipoles in a small volume. Figure 4 illustrates the process in which the initially excited population of dipoles has a random phase distribution. The vacuum field interactions spontaneously synchronize the phases of these oscillators. As a result, the system experiences a phase transition into the Dicke superradiant state (4-6). In this state, all the excitations interact with the radiation field collectively and coherently. Acting like a giant atom, they emit a high-intensity short burst of photons. This phase transition of an ensemble of incoherent dipoles into a coherent macroscopic quantum state and its collective radiation is called superfluorescence (SF).

The quantum phase transition that leads to SF depends on the dephasing time of the excitations. Macroscopic coherence can only build up if the dephasing is slower than spontaneous synchronization. Therefore, initial observation of SF has been primarily in gas-phase systems (7). Due to fast electronic dephasing in condensed matter, SF has only been observed at cryogenic temperatures (~10K) in a handful of solids. These include excitonic transitions in oxygen-doped KCI (8), localized atomic-like states, such as CuCI nanocrystals embedded into a NaCI matrix (9), and excitonic transitions in bulk ZnTe single crystals (10). Also, electron-hole plasma in InGaAs quantum wells exhibits SF under a high magnetic field (>10 Tesla) (11 ). More recently superfluorescence was observed in CsPbBr3 perovskite nanocrystals at 6 K (12). In contrast to previous solid-state systems where low cryogenic temperatures and/or high magnetic field are required to demonstrate superfluorescence, we demonstrate that hybrid perovskite MAPbl3 exhibits SF at temperatures achievable by liquid nitrogen, indicating that the versatile hybrid perovskite material system is an ideal platform to study SF and create collective coherent quantum states of matter suitable for quantum applications at elevated non-cryogenic temperatures.

Since the SF process requires spontaneous synchronization and the emission is from a macroscopic coherent state, it exhibits characteristic spectroscopic signatures that unambiguously help distinguish it from other collective radiation processes, such as amplified spontaneous emission (ASE). These spectroscopic signatures are measurable using steady-state photoluminescence (PL), time-resolved emission, and time-resolved absorption spectroscopies. First and foremost, the initial population in SF does not have macroscopic coherence, therefore there is a delay time during which spontaneous synchronization takes place, preceding the formation of a macroscopic coherent state (13). Secondly, this delay time reduces with an increase in the excitation density (13). Also, since the emission is from a Dicke superradiant state, the lifetime decreases with the density “N” of phase-locked indistinguishable quantum oscillators (13). Since all the excitations interact coherently with the radiating field, the maximum intensity of the SF pulse scales with N2, leading to a quadratic dependence on the excitation density. Moreover, SF emission exhibits interference and propagation effects, leading to oscillations called Burnham-Chiao ringing in the time evolution (14). Last but not least, the emission kinetics of SF is similar to the relaxation of an inverted pendulum (15); as a result, it has a very specific time-dependent functional form based on "sech ² ", unlike the exponential behavior of spontaneous recombination of individual excitations (16). Here we show that the hybrid perovskite MAPbl3 exhibits all of these characteristics of SF when excited at and above a certain threshold using ultrafast laser pulses (17).

Figure 5A shows the absorption spectra of MAPbl3 at temperatures ranging from 78 K to room temperature (RT). It is known that MAPbl3 is in a tetragonal phase (TP) at room temperature and it exhibits a transition to an orthorhombic phase (OP) at temperatures below 150 K. The structural phase transition leads to a blue-shift and a relatively sharp excitonic feature at the band edge of the absorption spectra (18). It is also well-known that this phase transition is incomplete with a small fraction of TP coexists in the OP domain dominant thin film (19). Figure 5B shows the continuous wave photoluminescence (CW PL) spectra at 78 K. The sample is excited above the bandgap of the OP domains. The CW PL emission has two distinct features; one at 1 .54 eV and the other at 1 .58 eV, both from the TP domains (18, 20) The absence of OP emission indicates the efficient transfer of excitons to the TP domains (19). Under CW excitation both features increase superlinearly with the fluence, i.e. / oc F ¹ · ^{410 1} . jhj _s j _s because the recombination kinetics is a bimolecular process (21 ).

Figure 5C shows the PL spectra when the sample is excited with 120 fs pulses at 400 nm, in which 1 .54 eV feature exhibits a threshold behavior at around 6.3 m/ cm ^~2 . Below the threshold, both peaks increase linearly. Beyond the threshold, the 1.54 eV feature exhibits a quadratic increase up to 22.8 m/ cm ^~2 , while the 1.58 eV feature saturates. The linear increase below the threshold, as opposed to the superlinear increase observed at CW excitation ( a = 1.4 ± 0.1), indicates the existence of excitation density- dependent nonradiative recombination kinetics such as Auger recombination (22). In contrast, beyond the threshold, the quadratic increase of 1.54 eV feature indicates that a radiative recombination process with a rate much faster than the nonradiative processes takes place. Below we show that this fast recombination process is SF.

We first characterize the time evolution of the sharp feature at 1 .54 eV to investigate whether it exhibits the SF dynamics. Figure 6A shows the transient PL kinetics measured at two different above threshold excitation fluences 10.9 m/ cm ^~2 and 135.8 m/ crrT ² . The transients in Figs. 6A have several characteristics that are unique to SF. First, for the range that the integrated PL increases quadratically, the peaks of the PL transients increase superlinearly with fluence (F) with a power-law given by, I _max = F ³ - ^9±0 · ⁷ (Figure 6A inset). This observation is consistent with SF because, in a bimolecular process, the rate of exciton formation increases quadratically with the excitation fluence, as a result, the SF peak is expected to increase with the 4th power of the fluence (17).

Next, we examine the rise and decay characteristics. At low fluence there is no measurable emission for the first 3 picoseconds, then the PL starts to rise slowly. However, as the excitation fluence is increased, the PL signal starts earlier and rises to its peak faster. This excitation density-dependent delay in PL rise time is a signature characteristic of SF. The decay also shows faster recombination kinetics as the excitation fluence increases (Figure 6A), hence the emission takes the shape of a burst with its temporal width (so- called real width) becomes narrower. While conventional exponential functions do not fit these transients, the SF theoretical models based on seek ² functions (Eqs. (S5) and (S8)) fit them (black dashed lines in Figure 6A) reasonably well (16, 17). The quantitative values of the delay time “x _D ", characterizing the time in which spontaneous synchronization takes place, and the real width “x _R " of the SF burst, obtained from these fits, are displayed in circles in Figs. 6B and 6C respectively. Finally, as fluence increases, the emission exhibits clear recurrences, known as Burnham-Chiao ringing behavior (14). All these observations, namely the fluence dependent maximum intensity, delay time, decay, and ringing behavior agree with SF emission.

The delay in emission and its density dependence is a distinctive property of SF and separates it from other collective recombination processes, such as ASE. To further investigate if the delayed emission is due to SF, we first need to show that dynamics related to exciton transfer from orthorhombic to tetragonal domains do not cause the fluence dependent delayed emission. To probe the time evolution of the population in the TP domains, we performed transient absorption spectroscopy.

Figures 6D - 6G shows the pump-probe traces measured at 1 .54 eV at 78 K. The results show that within the first 2 ps of the optical excitation a negative signal occurs and then recovers. This negative feature is a result of bandgap renormalization (BGR) (17, 23). The population kinetics show that even at the lowest excitation fluence within the first 2 ps after the optical excitation, the population growth in the TP domains is completed and remains steady (17). Flowever, as the excitation fluence increased beyond the threshold, the pump-probe traces show unusual behavior. For instance, at an excitation fluence of 21.5 m/ cm ^{~ 2} the traces exhibit similar steady behavior for about 8 ps. But after 8 ps the signal suddenly decays. Interestingly, this waiting period in which the population remains steady depends on the excitation fluence. In Figure 6B circles show the waiting times measured from the pump- probe experiments and they accurately follow the same trend as extracted x _D values from the SF model, further confirming that observed kinetics is due to SF. All these observations indicate that the population in the TP domains reaches its maximum value within 2 ps after excitation and the delay in the PL emission is not because excitons did not populate TP domains, but because they need time to establish macroscopic coherence.

According to the SF theory the delay time x _D scales with In N/N, and the real width x _R scales as 1/N, where N is the exciton density (13, 17). To do this analysis, one needs to determine the exciton density involved in SF. Exciton density in the TP domains depends both on the excitation fluence and the carrier relaxation dynamics. Since the exciton formation is predominantly bimolecular, the exciton density is proportional to the square of optically excited carrier density (i.e. electron and hole density n), i.e. N oc n ² . Using transient absorption spectroscopy, we determined the excitation fluence dependence of the TP population density right before the SF burst emitted. We found that in the fluence range that the PL behave quadratic, the pump- probe signal at 1.54 eV (TP population) increases linearly with the excitation fluence, i.e. n oc F (17). Therefore, for this range, exciton density N increases with F ² . The black solid lines in Figs. 6B and 6C are the In N/N and 1/N curves fitting x _D and x _R , for the fluence range that population increases linearly.

The population relaxation kinetics measured by the pump-probe experiments are also consistent with the SF kinetics and distinct from the dynamics expected from the ASE process. At very high excitation fluences the pump-probe signal rapidly drops (Figs. 6F and 6G). This abrupt drop is similar to that observed in the pump-probe measurements of SF in InGaAs quantum wells studied earlier (11 ). The electronic levels in the coherent macroscopic system form a Dicke-ladder as shown in Figure 6H. In this representation, N number of excitations coherently exist in the same state. At the top of the Dicke-ladder, all the dipoles are excited and at the bottom, all of them are in the ground state. The decay rate, G, is proportional to N, at the top and the bottom, while at the mid-levels it is proportional to N ² . This unique fluence dependence of decay rate causes a sudden population drop, which becomes as fast as 750 fs at 128.9 m/ cm ^{~ 2} .

One of the important differences in SF and ASE is that in SF all the coherent population relaxes to the bottom of the Dicke ladder (11), whereas in ASE the recombination only takes place up until population inversion is lost. As a result, in SF one can observe complete population depletion in contrast to ASE. In our system, the complete depletion of the population in TP domains is hard to observe since there is always a transfer of excess population from the OP domains. As a result, population depletion due to SF and refilling due to transfer from the OP domains compete. A complete population depletion in the TP domains can only be observed when the SF recombination rate exceeds the exciton capture rate by the TP domains significantly. Strikingly, at the highest excitation fluence, we observe that the sudden drop (within 750 fs) in the TP domain population leads to almost complete depletion and then the population rises in a few ps time scale, indicating the cooperative emission is significantly faster than the exciton capture from OP domains, again confirming the observed population kinetics is due to SF.

Flaving established that the 1.54 eV emission is superfluorescence from TP domains through analysis of the PL and absorption kinetics, we also studied emission directionality to investigate the role of TP domain distribution in SF emission. In extended systems, SF exhibits strong directionality determined with the excitation profile (24). In MAPbl3 TP domains have random sizes (100 nm-500 nm) and shapes (17). If multiple TP domains establish a macroscopic coherence, then the directionality should be determined by the excitation profile. In contrast, if SF is originated from individual domains, then the domain sizes and shapes should determine the SF direction. While in the former case SF and ASE should have the same directionality, in the latter case SF direction becomes independent of the excitation profile hence differs from ASE. Our studies show that SF emission from MAPbl3 exhibits a preferentially stronger emission in the forward direction compared to any other direction. This is true regardless of the size and the shape of the excitation profile (17). For instance, in Figure 6A, we show that even when the sample is excited using a stripe-shaped beam with a spot size of 4 mm x 15 mpi, the SF emission at the forward direction is stronger compared to the emission at the edge, which is the optical gain direction. This observation further proves that the observed signal is not ASE, but it is SF radiated from an individual or small clusters of TP domains.

Figure 7A shows an example detector setup with two orthogonal detectors. Figure 7B shows a normalized intensity curve from the two detectors.

The observation of superfluorescence in solid-state systems has been extremely rare due to stringent requirements to support macroscopic coherence (8, 10, 11 ). Recently, perovskite nanocrystal superlattices based on CsPbBr3 have also been shown to exhibit SF at 6 K (12). Our observation of high-temperature SF in MAPbl3 thin films suggests that these materials are intrinsically suitable for maintaining quantum coherence. Interesting property in these materials is that optical excitations form strongly bound exciton- polarons. Polaron formation has been known to protect electronic coherence. For instance, in light-harvesting systems, similar polaron formation is found to be a primary reason for sustaining electronic coherence and promoting wave like energy transfer (25). In the current example of hybrid perovskites, polaron formation can be the reason for extended electronic coherence and observation of macroscopic quantum state. Observation of SF in these versatile materials can make them an ideal platform for quantum applications. We anticipate that these materials can further be used in various microcavities (19) as building blocks for complex systems where quantum information can be stored, processed, and read out by manipulation and interrogation of giant dipoles (26-28). Finally, from a fundamental point of view, the electronic properties of hybrid perovskites such as the nature of the electronic states, their localization, coupling to phonons modes (29), and dephasing kinetics, have to be further studied to fully understand the dynamics leading to superfluorescence. Methods

Synthesis of MAPbh Thin Films: Perovskite precursor solution was prepared by dissolving Pbb, MAI, and DMSO (molar ratio is 1 : 1 : 1 ) in DMF with a concentration of 1 M. The precursor solution was spin coated at 3000 rpm on glass substrate, during which 100 pL of toluene was dropped on the sample at the 8th second. Thickness of the film was measured to be 300 nm. The film was then annealed at 100 °C for 10 min to complete the crystallization.

PL experiments: A Mightex spectrometer is used to measure the PL. Sample is excited using 400 nm pulses obtained by frequency doubling of a Ti-Sapphire amplifier with a repetition rate of 1kHz and 800 nm fundamental wavelength.

Kerr-Gate Experiment. Time-resolved SF was measured with a homebuilt Kerr-gate set up using a 1 kHz amplified Integra-C Ti:Sapphire laser with 120 fs pulsed output at 800 nm. The laser output beam was split into two paths; one for the optical Kerr gate pulse and the other for the excitation beam, which will be converted to 400 nm through second harmonic generation using a BBO crystal. The collected PL and the gate pulse were focused on CS2, which was used as the Kerr medium. The measurements were performed in a Janis continuous flow liquid-ISh cryostat. The time-resolved photoluminescence was collected with a Hamamatsu photomultiplier tube (H10721 -20) attached to a monochromator.

Time Resolved Pump-probe experiment: The pump-probe measurements were performed in a commercial Helios system, with pump and probe originating from amplified Ti:Sapphire laser system producing a femtosecond beam with 100 fs pulses at 800 nm. Part of the fundamental beam is frequency doubled using a Coherent OPerA Solo OPA to produce a 400 nm excitation light. The rest is used to generate the UV-stable white light continuum probe with a CaF2 crystal. References:

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Room Temperature Superfluorescence in Lead Halide Perovskites Through Polarons in Lead Halide Perovskites

The formation of tailored coherent macroscopic states and the control of their entanglement using external stimuli is essential for emerging quantum applications. However, the observation of these many-body states is rare due to fast dephasing of quasiparticles. As a result, Bose-Einstein condensation, superconductivity, superradiance and similar collective coherent electronic phenomena are only observed in a handful of solid-state systems at very low temperatures, at which dephasing due to random thermal events is highly suppressed. Here we propose that the strongly coupled polaron formation in lead halide perovskites increases the electronic dephasing time of oscillating dipoles by increasing their immunity from random scattering events while also promoting the synchronization between them. Consequently, these materials can host superradiant phase transitions at high temperatures and form a “giant dipole”. This crucial role of polarons in achieving coherent states is explained using a quantum analogue of a vibration isolation system. Experimental results show that for the quasi-2D perovskite PEA:CsPbBr3, these giant superradiant dipoles survive beyond room temperature and exhibit spectral signatures of dipole-dipole interactions. The entanglement of multiple superradiant ensembles in lead halide perovskites provides a platform for the creation and manipulation of quantum information.

Superfluorescence (SF) is a prominent example of macroscopic quantum coherence of optical excitations. In this process, a laser pulse creates above bandgap photoexcitations, which later relax to the band edge and form an incoherent ensemble. If the electronic phases of these incoherent excitations spontaneously synchronize, an ensemble of macroscopically coherent dipole oscillator forms. This superradiant phase transition is a symmetry breaking phenomena, i.e a second order quantum phase transition. The resulting Dicke superradiant state acts like a “giant atom”(7) , and relaxes to ground state by collectively emitting a burst of photons in a time duration that is orders of magnitude faster than the emission of individual emitters. In solids, fast electronic dephasing caused by scattering events impedes superradiant phase transition. Therefore up until recently, solid-state SF has been only observed in a few systems at stringent conditions such as cryogenic temperatures (~10K) and/or high magnetic fields (>10T)(2).

Flere we demonstrate room temperature SF in a quasi-2D PEA:CsPbBr3 perovskite and propose that this unusually high temperature SF in both this system and in MAPB films is observed due to the large polaron nature of their fundamental excitations. In other words, the excited electron- hole pair couples to a particular lattice distortion mode and aids the process for SF. It has been shown that carriers in perovskites form large polarons within a picosecond after excitation (3). This is due to their unusually high dielectric responsiveness (6), which is comparable to ionic liquids (4,5). The role of large polarons in the protection of carriers against random scattering events with other phonons and carriers has been discussed in earlier studies ( 6 , 7). Flere we show that this strong coupling of electronic dipoles to lattice distortion not only protects the electronic coherence, but also facilitates formation of coherent macroscopic phenomena. Since the nature of polarons is similar in all lead-halide perovskites, we predict that most of these materials can exhibit SF at unusually high temperatures. A quasi-2D material based on PEA:CsPbBr3 exhibits strongly bound polarons (8). As a result, in this material SF survives even at room temperature.

SF requires electronic dephasing time to be longer compared to spontaneous synchronization. In a typical electronic oscillator, dephasing time depends on the fluctuations in the electronic energy levels. At high temperatures, scattering with phonons is the primary dephasing mechanism in solid materials. In Figure 8A, the particle in a box simulates the electronic transition in which the superposition of the first two quantum levels forms the excited electronic polarization. This polarization oscillates within the box potential at a frequency determined by the difference of the two electronic energy levels “o _eg ". Scattering with phonon modes will induce changes in the box potential, which will lead to electronic dephasing.

In perovskites, this picture is modified due to large polaron formation. The lattice distortion, which couples to the electronic dipole, relaxes the potential that the electronic dipole is confined in. Unlike polarons observed in other solids, electronic dipoles in perovskites strongly correlate with the lattice distortion mode, which is evidenced by coherent phonon oscillations and spectral phonon shifts in their optical spectra (9). In our straightforward model, we illustrate this correlated coupling as a particular spring mode modulating the potential barrier of the particle in a box in Figure 8B. As the spring oscillates, the “w _b3 ” will be modulated. This strong coupling of an electronic transition with a particular lattice distortion mode plays an important role for electronic coherence dynamics. Since phonons in the crystal are also mechanical in nature, i.e they are atomic displacements, their interaction with electronic dipoles will be mediated by the lattice distortion mode of the polaron. This configuration of a high frequency electronic oscillator, which is connected with the rest of the material through a low frequency lattice distortion mode forms a quantum analogue of a vibration isolation system for the electronic dipole of interest. The optical response and dephasing dynamics of this coupled system depends on the interaction of these two oscillators, i.e lattice distortion and the electronic dipole, with the vacuum fields and lattice scattering events.

Vibrational isolation is a common way to reduce the random environmental noise in mechanical systems. The most straightforward way is to use a low frequency oscillator between the system of interest and its environment. In this classical representation, the transmissibility, T, of any random impact from an environmental effect has a frequency dependence given by {10):

In this equation, the displacements in the system of interest “x , due to oscillations in the environment “x ₂ ”, depends on the w _h -the natural frequency of the spring vibrational mode that is coupled to the system, w -the frequency of the oscillations in the ambient and x -the damping factor of the spring. Figure 8C shows an illustration of the vibrational isolation with a well- known plot of transmissibility as a function of — . It is clear that higher w _h frequency noise is filtered from the system while the lower frequencies are transmitted. An underdamped vibrational isolator filters high frequency noise very effectively. In the lead halide platform, the system is the electronic dipole and the spring is the particular lattice distortion mode that forms the polaron. Consequently, the low energy lattice distortion mode serves the purpose of a high frequency noise filter.

This strong phonon-electron dipole correlation in the polaron mode leads to extended high temperature electronic coherence and SF through multiple means. First, the lattice distortion mode that couples to form a polaron is a low energy Pb-halide distortion mode ( 3 , 11), therefore it impedes many higher energy phonon modes from interacting with the electronic oscillation. As a result, the intrinsic electronic coherence time will be significantly enhanced. Secondly, at higher excitation fluences these polarons will be closely packed. Although phonons with lower energies will be able to impart random phase fluctuations on the electronic oscillation, their larger sizes will allow them to interact with the closely packed polarons coherently. Therefore, at high excitation densities, the lower energy phonon scattering will have a limited impact on electronic dephasing.

Lastly, at the natural frequency of the polaron lattice distortion mode "w _h ", the interaction with the ambient is amplified. Consequently, when there is a significant density of excitations, all the polarons can interact cooperatively through their lattice distortion mode as they synchronize. The modulation of the electronic dipole oscillation by these cooperative polarons is an ideal configuration for boosting synchronization between electronic dipoles and building a macroscopic coherence. This can be depicted as multiple quantum wells connected to one single phonon mode that modulates their potential collectively, as seen in Figure 8D. In this scheme, the critical parameters are polaron binding energy and the damping of the lattice distortion mode. At temperatures higher than the polaron binding energy, electronic dipole oscillation will be free from a particular lattice mode and will dephase very quickly. Similarly, when the phonon mode is overdamped, then the filter will leak higher energy phonon modes, which will also lead to electronic dephasing. Since phonons intrinsically have a longer dephasing time compared to electronic transitions, SF can be observed in many different lead halide perovskites at temperatures below polaron binding energy.

Accordingly, hybrid perovskites with strong polaron binding energy can potentially also exhibit high temperature SF. As an example, we studied a quasi-2D PEA:CsPbBr3 perovskite film. This material is ideal for the observation of SF at high temperatures for several reasons. First, the electron- phonon coupling in Cs-based systems is well-known to be above thermal energy at room temperature (12-14). Second, in the quasi-2D thin film, the coexistence of high energy 2D domains and low energy 3D domains creates a high density of photoexcitations at a moderate laser fluence. Moreover, time resolved population kinetics measured by transient absorption experiments shows the fast energy funneling to the low energy 3D domains within the first ps after excitation ( 15)(16 ). In order to show SF in quasi-2D PEA:CsPbBr3 perovskite film we performed steady state and time resolved PL spectroscopy at different temperatures. Figures 9A and 9C show the progress of the PL spectra as the excitation fluence increased measured at 78K, and 300K, respectively. At both temperatures, the PL exhibits qualitatively similar evolution. At the lowest fluence there is a single broad PL feature located at about 2.4 eV. However, at a threshold fluence, a sharp feature starts to rise at the lower energy tail. The SF excitation density threshold is 0.26 x 10 ¹⁸ cm ^~3 at 78 K and 1.5 x 10 ¹⁸ cm ^~3 at 300K. Figures 9B and 9D show the intensity as the excitation fluence is increased, where it increases superlinearly as N ⁵ · ⁶ at 78 K and N ⁵³ at room temperature. Compared to the expected quadratic dependence for SF {17), this increase is much larger. While the mechanism for such a high nonlinear behavior is not clear, it is not an unseen behavior for SF emission. For instance, a similar superlinear response has been observed in an ensemble of CuCI nanocrystals embedded in a NaCI matrix. {18).

In order to confirm that this sharp feature is indeed SF, we measured the time evolution of the PL using Kerr gate experiments at various temperatures. The key signature for SF is observation of a delay in the emission. This delay is associated with the time that it takes for the synchronization of the oscillators and it supposed to decrease with the increased excitation density. {19, 20) Figures 10A and 10D show the time resolved traces for the sharp PL feature at 78 K, and 300L\ In Figure 10A near the threshold fluence, 0.59 x 10 ¹⁸ cm ^~3 , the PL emission starts to rise about 10 ps after the excitation. Even with a slight increase in the excitation fluence, the PL starts rising and decaying faster. In order to quantitatively analyze the excitation dependence of the delay time, we fit the PL transients. According to the SF theory the PL decay should exhibit a unique functional form that depends on seek function {21). In Figure 10A and 10D the black lines are the resulting fits, which show that each spectrum indeed exhibit SF decay characteristics. Moreover, it is also expected that in SF the delay and time width (i..e. real width) of the transients should depend on the excitation fluence by In - and - respectively {21). Figures 10B-C and 10E-F show the time delay and the real widths of the SF bursts and the fits, which agrees well with the expected theoretical behavior.

This sharp low energy tail in the PL spectra has been commonly explained as amplified spontaneous emission (ASE) in perovskites. However, according to our findings, this behavior has been misinterpreted in many cases. The superlinear increase in intensity along with the delay time and real width trends strongly supports this is an emission caused by the coherence of electronic dipoles.

We further performed time resolved absorption experiments to solidify this claim. In SF, intensity amplification is due to macroscopic coherence whereas in ASE, amplification is due to the stimulated emission of an inverted population. Therefore, if the observed superlinear increase in the emission was due to ASE, we would observe it in the transient absorption spectra as well whereas if it was due to SF, we would not observe the same amplification as suggested in the original paper by Dicke (22). During the regime that the PL feature is increasing superlinearly, the stimulated emission peak increases sublinearly with the excitation fluence. Therefore, the observed enhancement in the PL cannot be ASE, but it is indeed due to cooperative emission due to SF.

We further studied the behavior of the SF feature at higher excitation fluences. Since the perovskite thin film is an extended object, laser excitation using a large spot size creates multiple separated superradiant states. When a superradiant state is formed, it makes a giant dipole that can strongly interact with vacuum fields and other superradiant giant dipoles. Figure 11 A illustrates this interaction. The detailed microscopic theory of multi-superradiant states has not been developed yet, but a theoretical analysis of such states has been presented by Haude and Rajabi recently (23). Their analysis revealed that dipole fields of superrandiant systems dress Dicke ladder states and lead to entanglement. The analysis predicts that shifts in the energy levels are proportional to /N. Figure 11 B shows the evolution of the PL spectra as the excitation fluence increased to 3.7 x 10 ¹⁸ cm ^~3 at 78 K. As the excitation fluence is increased, the spectra substantially redshifts and broadens to take an asymmetric line shape. We analyzed the peak shift and the broadening in the spectra in Figures 11 C and 11 D, which quantitatively agrees with the prediction in reference (23). These observations indicate that within the excited volume, the resulting superradiant states are not independently radiating, but they exhibit strong coupling to each other, i.e they are entangled.

In conclusion, the strong correlation of lattice distortion and electric dipole oscillation in hybrid perovskites facilitates the formation of macroscopic collective electronic coherence and consequently, SF emission. The low frequency lattice distortion mode of the polaron effectively filters high energy phonon modes, preventing their interaction with the electronic dipole and amplifying the interaction with different polarons at its own frequency, promoting a collective modulation of electronic dipole oscillations. As a result, the system acts like a quantum analogue of vibration isolation system to reduce dephasing of electronic dipoles. The experimental results of PEA:CsPbBr3 samples clearly show SF even at room temperature. The experimental studies also exhibit significant broadening and shift in the SF peak at higher excitation fluences. This observation agrees with the long- range dipole coupling and entanglement of multiple superradiant states.

REFERENCES

Each of the following references is hereby incorporated by reference in its entirety:

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2. G. Timothy Noe li et al., Giant superfluorescent bursts from a semiconductor magneto-plasma. Nature Physics 8, 219-224 (2012).

3. K. Miyata et al., Large polarons in lead halide perovskites. Science Advances 3 e1701217 (2017).

4. Y. Guo et al., Dynamic emission Stokes shift and liquid-like dielectric solvation of band edge carriers in lead-halide perovskites. Nature Communications 10, 1175 (2019). 5. K. Miyata, T. L. Atallah, X.-Y. Zhu, Lead halide perovskites: Crystal-liquid duality, phonon glass electron crystals, and large polaron formation. Science Advances 3, e1701469 (2017).

6. X. Y. Zhu, V. Podzorov, Charge Carriers in Hybrid Organic- Inorganic Lead Halide Perovskites Might Be Protected as Large Polarons. The Journal of Physical Chemistry Letters 6, 4758-4761 (2015).

7. H. Zhu et al., Screening in crystalline liquids protects energetic carriers in hybrid perovskites. Science 353, 1409-1413 (2016).

8. H. Long et al., Exciton-phonon interaction in quasi-two dimensional layered (PEA)2(CsPbBr3)n-1 PbBr4 perovskite. Nanoscale 11, 21867-21871 (2019).

9. S. Neutzner et al., Exciton-polaron spectral structures in two- dimensional hybrid lead-halide perovskites. Physical Review Materials 2 064605 (2018).

10. J. Shah, Ultrafast spectroscopy of semiconductors and semiconductor nanostructures. (Springer Science & Business Media, 2013), vol. 115.

11. W. Chu, Q. Zheng, O. V. Prezhdo, J. Zhao, W. A. Saidi, Low- frequency lattice phonons in halide perovskites explain high defect tolerance toward electron-hole recombination. Science Advances 6, eaaw7453 (2020).

12. C. M. laru, J. J. Geuchies, P. M. Koenraad, D. I. Vanmaekelbergh, A. Y. Silov, Strong carrier-phonon coupling in lead halide perovskite nanocrystals. ACS nano 11, 11024-11030 (2017).

13. J. Ramade et al., Exciton-phonon coupling in a CsPbBr3 single nanocrystal. Applied Physics Letters 112, 072104 (2018).

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15. L. Lei etal., Efficient Energy Funneling in Quasi-2D Perovskites: From Light Emission to Lasing. Advanced Materials, 1906571 (2020).

16. See supplementary materials.

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C101 (2016).

20. J.-H. Kim etal., Fermi-edge superfluorescence from a quantum- degenerate electron-hole gas. Scientific reports 3, 3283 (2013).

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22. R. H. Dicke, Coherence in spontaneous radiation processes. Physical review 93, 99 (1954).

23. M. Houde, F. Rajabi, Interacting superradiance samples: modified intensities and timescales, and frequency shifts. Journal of Physics Communications 2, 075015 (2018).

24. Y. F. Ng, S. A. Kulkarni, S. Parida, N. F. Jamaludin, N. Yantara, A. Bruno, C. Soci, S. Mhaisalkar, N. Mathews, Highly efficient Cs-based perovskite light-emitting diodes enabled by energy funnelling, Chem. Commun. 53, 12004-12006 (2017). 25. Y. Liu, J. Ciu, K. Du, H. Tian, Z. He, Q. Zhou, Z. Yang, Y. Deng,

D. Chen, X. Zuo, Y. Ren, L. Wang, H. Zhu, B. Zhao, D. Di, J. Wang, R. H. Friend, Y. Jin, Efficient blue light-emitting diodes based on quantum-confined bromide perovskite nanostructures, Nature Photonics 13, 760-764(2019).

Room-Temperature Superfluorescence to Room-Temperature Superconductivity in Perovskites

Metal Halide Perovskites are an emerging class of semiconducting materials with a general chemical formula of ABX3, in which the [BCb] ⁴ octahedra are corner-shared in a three-dimensional (3D) framework with A- site cations occupying the vacancy. Here, the A-site is typically occupied by the monovalent cations (for example, methylammonium (MA ⁺ ) and formamidinium (FA ⁺ ) and Cs ⁺ ), the B-site is the divalent metal (for example, Pb ²⁺ and Sn ²⁺ ) and the X-site is the halide anion (for example, Cl , Br and I ). Except for the 3D perovskite structure, the low-dimensional perovskites are also introduced by inserting the large interlayer organic cations between the [BCb] ⁴ octahedra which provided the structure flexibility as well as tunability of the optical and electrical properties. Owing to the superior properties such as direct bandgap, high absorption coefficient, low defect density, high photoluminescence quantum yield, and high gain value, metal halide perovskites have exhibited unprecedented advancement in a wide range of applications such as solar cells, photodetectors, light-emitting diodes, and lasers.

Very recently, the room temperature superfluorescence has been discovered in perovskites, indicating they are promising for the realization of emerging quantum applications. Macroscopic quantum phenomena such as superconductivity, superfluorescence, and Bose-Einstein condensation can function as the building blocks of future quantum technology. However, almost all these macroscopic quantum phenomena are only observed at cryogenic temperature or under extreme pressure limiting their practical utilization. To have stable coherent quantum states at elevated temperatures, the dephasing time of an ensemble of quantum oscillators (i.e. a Cooper-pair, an exciton, an electronic dipole, or a spin) is required to be longer than the synchronization time. In practical temperatures, the electronic dephasing times are within 10s of femtoseconds for typical quantum systems, and hence it is almost impossible to achieve a synchronized coherent state. Therefore, the dephasing time needs to be extended. In perovskites, on the other hand, the formation of polarons through the coupling of the quantum oscillator with the lattice modes (i.e. thermal phonons) is prevalent due to the soft nature of the crystal lattice. And these polarons are known to protect quantum coherence through the quantum oscillators bounded with the lattice modes which filters out the disturbances and extends the dephasing time. This is the key to observation of the high temperature superfluorescence in metal hybrid perovskites. A similar analog can be applied to superconductivity. In this case, the macroscopic quantum phenomena at elevated temperature are possible as long as the quantum oscillators are bound to a lattice mode forming a polaron which is stable at high temperature and the lattice mode preferably involves the lowest possible energy mode that can be coupled to. Therefore, metal halide perovskites might fulfill the fundamental requirements to protect the quantum coherence and hold the promise to realize the high-temperature superconductivity.

To make conductors out of perovskites, we can introduce either p-type or n-type dopants in the perovskite lattice. Because the polarons can protect the carriers from the dopants in perovskites, carriers will not be scattered by phonons and results in superconductivity. In the table provided, we include the perovskite hosts along with the p- and n-dopants we use.

Description of perovskite superconductors

1- Macroscopic quantum state formation depends on the competition of the two different mechanisms. One of them is coherence build up time, the other one is the dephasing time. If dephasing time is slower than coherence build-up time then macroscopic quantum coherence forms.

2- In order to observe macroscopic quantum states of matter such as superconductivity, superfluorescence in condensed matter at high temperatures. One needs to bind the quantum particles that undergo phase transition to a low energy lattice mode and form a polaron.

3- The polaron will act as a quantum analog of vibrational isolator. In other words, the thermal phonons that lead to dephasing of the quantum oscillator will be isolated from the quantum oscillator

4- The characteristics of the polaron determines the critical conditions under which the quantum phase transition takes place. The primary characteristics are: a- The polaron binding energy is the strength of the bound between the vibrational lattice mode and the quantum oscillator. In order to have a macroscopic quantum phase at a particular temperature the polarons needs to be stable at that temperature, i.e the binding energy needs to be larger than kT. Polaron stability also depends on the density of the quantum oscillators. As the density increase the polarons can get screened. Therefore, the polaron can only provide protection for a certain range of densities of oscillators b- The energy of the lattice mode is another important factor. The lattice mode acts as a low pass filter. As a result, it filters the interaction of phonons with higher energy. Therefore, the lower the energy of the lattice mode the better the QAVI process.

5- The density of the quantum oscillator determines the time that is takes the transition of an incoherent ensemble to a “phase- synchronized” coherent ensemble of quantum oscillators. Unless the density-induced dephasing processes increase the dephasing rate, increased density leads to a faster coherence build-up process. Therefore, macroscopic state formation is a competition between the dephasing and the synchronization. The critical values of the density of oscillators are the lower bound and the upper bound values of the density at which a macroscopic quantum state exists. This consideration and also polaron stability consideration gives the conditions for upper and lower critical values of the quantum oscillator density for observing the macroscopic phenomena. For superconductivity that is the doping density and for superfluorescence that is the excitation density. a- The lower critical density is the threshold oscillator density that is required for phase synchronization is faster than dephasing time at a certain temperature. As the temperature increase the lower critical density has to increase because dephasing gets faster with temperature. b- The upper critical density is the density beyond which macroscopic quantum state can’t survive, because destabilization of the QAVI protection due to screening effect. The upper critical density decreases as the temperature increased. Because polarons also become less stable due to thermal screening c- Because the with temperature lower critical density increase and upper critical density decrease, the temperature and density phase diagram should make a dome like structure. The peak of the dome is the critical temperature beyond which superconductivity or superfluorescence or any other macroscopic quantum phenomena can’t take place.

These considerations lead to an equation for a polaron binding energy:

Eft > kT + E c _ouiom ft -screen

In other words, the polaron should be bound with a stronger binding energy compared to total of the kT and a density dependent screening energy given

We claim that for a superconductor the range of doping density should be material dependent because of all the arguments in the previous articles. But it should be within the limit of 0.5 per 10 unit cells to 3 per 10 unit cell. And it should depend also in the type of doping. As n-type and p-type doping will have different polaron characteristics in terms of binding energy and also the lattice mode they bind to.

Perovskite superconductors

Figures 12A - 12C are tables listing 3D Perovskite Materials: ABX3, A = Methylammonium (MA), Formamidinium (FA), Cs; B = Pb, Sn; X = Cl, Br, I. Figure 13 is a table listing 2D perovskites: (RNFl3)An-iBnX3n _+i , n = 1 , 2,

3...00; A = Methylammonium (MA), Formamidinium (FA), Cs; B = Pb, Sn; X = Cl, Br, I. The combination of A, B, X in Figures 12A - 12C are also applied here.

Figure 14 is a list of metal ion dopants. Figure 15 is a list of small molecule dopants.

Accordingly, while the methods, systems, and computer readable media have been described herein in reference to specific embodiments, features, and illustrative embodiments, it will be appreciated that the utility of the subject matter is not thus limited, but rather extends to and encompasses numerous other variations, modifications and alternative embodiments, as will suggest themselves to those of ordinary skill in the field of the present subject matter, based on the disclosure herein.

Various combinations and sub-combinations of the structures and features described herein are contemplated and will be apparent to a skilled person having knowledge of this disclosure. Any of the various features and elements as disclosed herein may be combined with one or more other disclosed features and elements unless indicated to the contrary herein. Correspondingly, the subject matter as hereinafter claimed is intended to be broadly construed and interpreted, as including all such variations, modifications and alternative embodiments, within its scope and including equivalents of the claims. It is understood that various details of the presently disclosed subject matter may be changed without departing from the scope of the presently disclosed subject matter. Furthermore, the foregoing description is for the purpose of illustration only, and not for the purpose of limitation.