Yusuke Morita, Assistant Professor, Department of Physics

Takataka Yoshioka, Associate Professor, Photon Science Center

Makoto Gokami, Professor, Department of Physics

Key Points of the Presentation

• We have succeeded in directly visualizing exciton Bose-Einstein condensates (BECs) in three-dimensional semiconductors, a macroscopic quantum phenomenon that has been a pending issue in physics for 60 years.
• By developing a new cryogenic experimental technique, mid-infrared spectroscopy at dilute refrigerator temperature, BECs due to excitons, which are compound Bose particles, were visualized for the first time. BECs of finite-lived quasiparticles exhibit features different from those of conventional condensed matter.
• Macroscopic quantum phenomena in systems in contact with the environment are also closely related to error control in quantum computers using artificial quantum systems such as superconducting circuits. In addition, various new observation techniques at dilute refrigerator temperatures that have been developed will be useful for future quantum technology development.

Summary of Presentations

Bose-Einstein condensation ^{(Note 1)} is a phenomenon in which a group of homogeneous Bose particles condense into a single state at low temperature and high density as a natural consequence of the quantum theory principle of quantum statistical properties ^{(Note 2)}, a phase transition occurring without interaction. Since the resulting condensate is macroscopic and exhibits quantum properties, it is important to artificially create and control it for applications in quantum technology, such as quantum computers. In particular, macroscopic quantum phenomena exhibited by electron systems in semiconductors and metals are important because of their high affinity with electronics technology.

Among them, Bose-Einstein condensation of excitons ^{(Note 3} ) in bulk semiconductors, which was theoretically predicted 60 years ago, has not been observed yet and has been a long-standing concern in modern physics.

In this study, Assistant Professor Yusuke Morita (Graduate School of Science, The University of Tokyo), Associate Professor Takataka Yoshioka (Photon Science Center, Graduate School of Engineering, The University of Tokyo), and Professor Makoto Goshin (Graduate School of Science, The University of Tokyo) have developed a new method of absorption imaging using mid-infrared light in an extreme environment near absolute zero (-273.09 degrees Celsius). and succeeded in directly visualizing and observing condensates formed in a three-dimensional trap of crystals held in a dilution refrigerator for the first time. The observed condensates were found to exhibit a peculiar behavior different from that predicted by conventional theories. This behavior is closely related to the fact that the quantum condensate is in contact with the environment of the crystal, and is a finding that will lead to new developments in the physics of quantum coherence in nonequilibrium open systems. At the same time, this finding is expected to be useful for research on error suppression and control, which is a central issue in the development of quantum computers using artificial quantum systems, which is currently the focus of active research around the world.

Contents of presentation

Purpose of Research
Bose-Einstein condensation (BEC), which was predicted at the dawn of quantum mechanics, is a phenomenon in which quantum mechanically identical Bose particles gather in three-dimensional space and condense into a single state at low temperatures as a natural consequence of quantum statistical properties. This is a purely quantum phase transition that occurs without interaction, whereas in ordinary phase transitions such as liquid solidification, interaction between particles is intrinsic. Therefore, as one of the distinguishing features of quantum mechanics, its experimental verification has been a goal of many researchers.

On the other hand, Bose-Einstein condensates are macroscopic objects that exhibit quantum properties, and their artificial creation and control is very important from the viewpoint of developing new technologies based on quantum mechanical principles that go beyond the principles of classical physics, such as quantum computers. In particular, macroscopic quantum phenomena exhibited by electron systems in semiconductors and metals are highly compatible with electronics technology, and there is a strong need to develop technologies that take advantage of such phenomena.

Bose-Einstein condensation is a central issue in modern physics, and its verification has been attempted in various systems. Exciton systems in semiconductors, the subject of this study, have also been the focus of attention. Especially in the late 1970s, with the invention of wavelength-tunable lasers, it became possible to generate excitons at high density in various semiconductors, and studies to verify the BEC of excitons were actively conducted. On the other hand, BEC was first observed in 1995 by combining laser cooling and evaporative cooling methods to trap a gas of neutral atoms interacting with each other at very low temperatures, and the Nobel Prize in Physics was awarded for this achievement in 2001, Since 1995, the focus of BEC research has shifted to cooled atomic systems and has made significant progress.

On the other hand, the question of whether an exciton can transition to the BEC state in thermal equilibrium with a solid crystal is similar to the assumption of Einstein's original paper. In order to explore the nature of BEC as a quantum phase transition not due to interactions, a verification of BEC for excitons, where cooling takes place without interaction between Bose particles, has long been awaited. However, excitons have remained an experimental concern because they are quasiparticles with finite lifetimes, and it is difficult to quantitatively evaluate their temperature and density. From a theoretical point of view, exciton systems have the feature of being open systems in which the number of particles is not conserved, and this is an unexplored field as a new quantum many-body system beyond Einstein's assumptions. The purpose of this report is to solve this problem experimentally and to contribute to the theoretical verification.

Background of Research
When a semiconductor cooled to low temperature absorbs light, electrons in the valence band are excited to the conduction band. The excited electrons combine with holes, which are loopholes in the valence band, to form virtual atoms such as hydrogen atoms. This is called an exciton. Excitons are classified as atoms that contain particles other than electrons, protons, and neutrons (exotic atoms: ^{(Note 4} )). Excitons are composite particles consisting of a pair of Fermi particles, an electron and a hole, bound together by the Coulomb force, and a quasiparticle with a finite lifetime, excited from the ground state of a multi-electron system in a semiconductor. If sufficiently dilute, they are expected to behave as Bose particles, and BECs are expected in a state where the entire system can be regarded as being in approximate thermal equilibrium (quasi-thermal equilibrium) due to contact with the environment within their lifetime. However, it is by no means obvious that this "virtual atom" with a limited lifetime will exhibit the same quantum properties as a real cooled atom.

About 60 years ago, a theoretical proposal for BEC verification by excitons was made, and many studies have been conducted since then. Theoretical and experimental studies have been conducted in various parts of the world prior to the aforementioned experiments on cooled atom systems. In particular, excitons formed in a semiconductor called copper suboxide ( 1s para exciton) are characterized by an extremely low probability of photon emission and annihilation. The exciton lifetime of ordinary semiconductors is about nanoseconds, but copper suboxide has been observed to have a lifetime as long as microseconds. As a result, it is possible to cool the system to sufficiently low temperatures within the lifetime while still being a virtual atom, and it has been considered a strong candidate for observing exciton BEC. Studies have been conducted around the world using this system to observe BECs by cooling them to about 2 Kelvin (-271 degrees Celsius), which can be reached with a liquid helium refrigerator. However, experimental observations have been inconclusive, and the observation of Bose-Einstein condensation of excitons has remained an important unsolved problem in solid state physics and quantum statistical mechanics.

The group reported here has been working for more than 20 years to solve this unsolved problem. First, they quantitatively investigated the reason why BEC could not be observed in the search at 2 Kelvin. In the course of the investigation, we have quantitatively evaluated with high precision the basic parameters that determine whether or not exciton BEC is observed, such as the lifetime of para-excitons and the rate of exciton annihilation due to two-body collisions. We have developed a method for quantitative evaluation of exciton density by using the internal transition absorption of excitons in the mid-infrared region (exciton Lyman spectroscopy: ^{(Note 5)} ). As a result, it was found that the reason is that the process of annihilation of excitons by two-body collision occurs at a high frequency as the density of para-excitons increases.

Based on the results of the study, we aimed to induce BEC at lower densities. Specifically, they switched to a refrigerator using liquid helium 3 as the refrigerant and cooled the excitons to a lower temperature (0.8 Kelvin) than 2 Kelvin to lower the density at which BEC occurs. By introducing a three-dimensional potential (trap potential) that traps excitons in a small space in the semiconductor, and by also devising the excitation conditions for the laser light used to create the excitons, the incident power was minimized to prevent heating of the crystal. As a result, we observed that when the conditions for the exciton-BEC transition are met, a phenomenon called "relaxation explosion" ^{(Note 6),} which is caused by the spontaneous formation of high-density states in the trap, occurs. The observation of this phenomenon confirmed that the BEC transition is indeed captured. However, since the condensate immediately undergoes a relaxation explosion, direct detection of the condensate itself remains an issue.

Contents of Research
In this report, based on the knowledge obtained in the previous studies, we identified the conditions for stable observation of the condensate and redesigned the experimental method. First, we switched from a helium-3 refrigerator to a dilution refrigerator ^{(Note 7)} to conduct the experiment under cryogenic conditions. In order to quantitatively measure the exciton density and temperature, we used internal transitions of excitons with resonances in the mid-infrared region in addition to conventional luminescence measurements (Figures 1 and 2).

Figure 1: Schematic of exciton formation by laser light and mid-infrared optical absorption imaging
Non-uniform strain is applied to a copper suboxide crystal (red rectangular body) through a pressurized lens under the crystal. When the crystal is irradiated with a laser beam at the point where this distortion occurs, excitons are locally captured and a high density of excitons can be achieved. This "cloud" of dense excitons is observed by absorption imaging of mid-infrared light. The position-controllable lens is placed inside the dilution refrigerator, just after the crystals, to achieve high position resolution imaging.

Figure 2: Schematic of the experimental apparatus
In this report, a dilution refrigerator (blue cylindrical device) is used to create the cryogenic state of excitons. The dilution refrigerator is maintenance-free and can be operated for long periods of time (several months). In the experiments in this report, precise measurements were achieved through long-time measurements over many days, so the feature of being able to operate for long periods of time was very important.
A semiconductor single-crystal of copper suboxide (in the figure: red cube) was mounted in the center of the sample stage in the dilution refrigerator, and mid-infrared beams emitted from an excitation laser (in the figure: orange line, wavelength 606.1 nm) for making excitons from the outside and from a quantum cascade laser for absorption imaging of the excitons (wavelength: 9.8 μm) for exciton absorption imaging. To introduce the two beams, a window is installed in the heat shield of the dilution refrigerator. When installing the window, we carefully designed its size and material to minimize the inflow of heat. As a result, the temperature rise was suppressed to 0.02 Kelvin.
In this report, an optical system with a mirror inside the dilution refrigerator was assembled to combine the conventional orthodox measurement method of observing emission from excitons and absorption imaging in the mid-infrared region.

Furthermore, using this system, we have succeeded in directly visualizing and observing condensed matter by realizing mid-infrared imaging. Specifically, we observed that a large change in the distribution of excitons captured in the microscopic space appears when the exciton temperature becomes high density under cryogenic conditions below 0.4 Kelvin (Figures 3 and 4).

Figure 3: Observation of Bose-Einstein condensates of excitons
In this report, mid-infrared absorption imaging is used to measure the density distribution of the para-exciton population captured in a three-dimensional potential in a semiconductor crystal. (left) Exciton density distribution at relatively high temperature (0.5 Kelvin) where the conditions for condensate formation are not met (right) Exciton density distribution at low temperature (0.1 Kelvin) where the conditions for condensate formation are met.
A spatially localized high-density region (red peak in the figure), which was not seen under relatively high temperature conditions, appears at the center of the potential; this signal, which appears only at low temperatures and high densities that satisfy the BEC conditions, is the Bose-Einstein condensate of excitons.

Figure 4: Change in exciton density distribution due to the formation of Bose-Einstein condensates
Exciton density distribution measured while changing the peak power of the laser used to generate excitons. At peak powers below 940 μW, the exciton density distribution has only a thermal component fitted by a Gaussian distribution, but at peak powers above 1.6 mW, a characteristic density distribution appears with a local high-density region at the center of the trap. This local high-density signal is the condensate.

A large number of excitons are distributed in an extremely small region (<10 μm), even smaller than the spatial extent of the original excitons. This signal is the exciton condensate that appears only when the BEC condition is satisfied.

Using this experimental system, we systematically evaluated the properties of the condensate while controlling exciton density and temperature. As a result, we found that the condensates in the para-exciton system can be treated as weakly interacting bosons as in the atomic system, and that there is weak repulsion between the excitons. On the other hand, we found that the fraction of condensates in the entire exciton system behaves very differently from what is predicted by conventional theories describing Bose-Einstein condensation. The essential difference from atomic systems that leads to this behavior is that in the case of exciton systems, the quasiparticles are in contact with a macroscopic heat bath, the lattice system of a crystal. This is an important finding from the viewpoint of exploring the mechanism of coherence and decoherence of quantum systems prepared in the real world.

This report is a milestone achievement that solves a long-standing unsolved problem. At the same time, it is a discovery of a new aspect of coherence in non-equilibrium open quantum systems, which is directly related to the development of techniques for controlling and suppressing errors, which is a central issue in the development of quantum computers using artificial quantum systems that are currently being actively studied around the world.

The following is a supplemental description of the features and novelty of the present experiment.
In this report, single crystals of copper suboxide were cooled to 0.06 Kelvin (-272.9 degrees Celsius) using a dilution refrigerator, an ultra-low-temperature cooling device. The temperature of the exciton in quasi-thermal equilibrium was found to be 0.17 Kelvin. This exciton temperature is close to the world's lowest exciton temperature (0.09 Kelvin) that we have achieved. Cooling to this temperature suppresses the relaxation explosion when BEC is realized.

The research group focused on the fact that para-excitons absorb a specific wavelength of mid-infrared light (9.8 μm) and conducted an experiment to determine the density from the amount of absorption. By measuring this amount of absorption at each point in the space where the exciton exists (imaging), the density distribution of the exciton can be obtained. The features of this experimental technique are that the density can be determined extremely accurately and that the formed condensates can also be detected. The orthodox measurement method used in experiments for exciton BEC observation has been to observe the light emitted from the excitons. However, our report shows that once a condensate is formed, it is theoretically and experimentally confirmed that the condensate cannot emit light. As a result, a new method beyond the conventional one is necessary to observe condensates, and this group has tried this new method.

On the other hand, the introduction of mid-infrared light in a cryogenic experimental environment using a dilution refrigerator is an extremely challenging experimental problem. In order to irradiate samples with mid-infrared light, a window must be installed in the dilution refrigerator to allow optical access (Figure 2). However, the window allows heat to flow in from the outside. At low temperatures, the heat capacity of the material is small and the temperature rises easily due to light absorption, so it is important to design an experimental system that minimizes heat inflow. We designed the material of the window to minimize the inflow of thermal radiation as much as possible, and furthermore, we carefully designed the size of the window to be as small as possible in conjunction with the precise control of the laser beam, and succeeded in suppressing the temperature increase to a mere 0.02 Kelvin.

Future Prospects
Researchers around the world are currently competing to develop quantum computers using artificial quantum systems such as superconducting quantum circuits and semiconductor quantum dots. In the development of such systems, it is impossible to eliminate the non-equilibrium open system nature of the artificial quantum system because it is in contact with the external environment. The present discovery is an important discovery because it confirms experimentally the existence of macroscopic quantum state formation by quantum condensation transition at cryogenic temperatures, as predicted by Einstein, even in such a system. The condensate discovered at the same time was found to be strongly affected by the interaction with the environment and to exhibit qualitatively different aspects from the ideal model systems that have been explored so far. This report is expected to provide an important opportunity to pursue new aspects of quantum physics. At the same time, it is expected to provide important insights into the development of techniques for error handling and control, which are indispensable for the advancement of quantum computers in the real world.

The experimental techniques developed in this study can be described as advanced experiments such as visible and mid-infrared precise laser spectroscopic imaging performed remotely from the outside world in a laboratory under cryogenic conditions built into a dilute refrigerator temperature environment. It is a treasure trove of new technologies that can be utilized in various aspects of future quantum technology development.

This research was supported by JSPS KAKENHI (Grant Numbers JP20104002, JP26247049, JP25707024, JP15H06131, JP17H06205); by the Photon Frontier Network Program, Quantum Leap Flagship Program (Q-LEAP) Grant No. JPMXS0118067246 of MEXT; and by the JSPS through its FIRST Program.

Journals

Journal name	Nature Communications
Title of paper	Observation of Bose-Einstein condensates of excitons in a bulk semiconductor
Author(s)	Yusuke Morita, Kosuke Yoshioka*, Makoto Kuwata-Gonokami*, and Yusuke Morita
DOI Number	10.1038/s41467-022-33103-4

Terminology

1 Bose-Einstein condensation

When particles are cooled, their properties as matter waves become apparent. Particularly for Bose particles, multiple particles can adopt the same state, and when the distance between particles, which is determined by the spread as a matter wave and the particle density, reaches the same level, the particles begin to become indistinguishable from each other. As a result, the distribution of their kinetic states as a group changes. When certain conditions of extremely low temperature and high density are met, a macroscopic number of particles occupy the ground state of the system and form a condensate. As a result of this condensate formation, the phases of the wave functions of the particles constituting the system are aligned, resulting in a phase transition with spontaneous gauge symmetry breaking. Since this phase transition is not caused by the interaction between particles but by the characteristics of Bose particles, this phenomenon is a remarkable reflection of the quantum nature of matter. This phase transition based on quantum statistical mechanics predicted by Einstein was experimentally demonstrated by American researchers in 1995, and the Nobel Prize in Physics was awarded in 2001. ↑up

Note 2 Quantum statistical properties

Particles are classified by quantum statistical properties into Bose particles (bosons), which follow Bose-Einstein statistics, and Fermi particles (fermions), which follow Fermi-Dirac statistics. Bose particles have the property that multiple particles can assume the same state. Fermi particles can only occupy one particle in the same state due to Pauli's exclusion law, and all particles must take different states. At low temperatures and high densities, these differences due to quantum statistical properties become important. ↑up

Note 3 Exciton

When a semiconductor absorbs light, electrons in the valence band transition to the conduction band. The hole created in the valence band by the transition of electrons to the conduction band is called a hole. Since electrons and holes have negative and positive charges, they are attracted to each other by Coulomb attraction (electrical interaction). The attraction between electrons and holes creates a stable bound state, similar to a hydrogen atom with electrons bound around a proton. This is called an exciton. While excitons exhibit an energy level structure similar to that of hydrogen atoms, the finite lifetime effect and the shielding of the Coulomb attraction by solids make them very different from real particles. Being a particle due to electron-hole pairs that obey Fermi statistics, the exciton is considered a Bose particle that obeys Bose statistics. ↑up

Note 4 Exotic atoms

Normally, atoms are formed by the attraction of electrons and nuclei (protons and neutrons). On the other hand, the state in which particles other than electrons, protons, and neutrons electrically attract and combine with each other is called an exotic atom. The exciton discussed in this study is a bound state of electrons and holes. In addition to excitons, there are positronium and muonium, in which electrons and positrons combine to form a bound state like a hydrogen atom. Research on these has been intensively pursued in recent years. ↑up

Note 5 Exciton Lyman spectroscopy

Excitons have an energy level structure similar to that of hydrogen atoms. In each energy level, the relative motions of electrons and holes are represented by wave functions similar to those of hydrogen atoms, such as 1s, 2s, 2p, and 3s or bitals. The para-exciton under study in this report is in the ground state and therefore in the bound state of the 1s orbital. The exciton absorbs the incident light whose energy corresponds to the difference between the energy level of the 1s orbital state and that of the 2p orbital state. As a result, a transition of the internal state from the 1s orbital state to the 2p orbital state occurs. Exciton Lyman spectroscopy is a technique to observe the exciton state using the optical absorption process associated with this internal state transition. In this report, the density of excitons is precisely calculated from the absorption of mid-infrared light using exciton Lyman spectroscopy. ↑up

Note 6: Relaxation Explosion

In the case of an ideal BEC, when a quantum phase transition occurs, the macroscopic quantities of particles constituting the system assume only one lowest energy state (ground state). In the case of Bose particles held in a trapping potential, the ground state of the system is located in a very small spatial region of the trapping center. Therefore, when a large number of particles flow into this ground state as a result of the BEC transition, the system becomes very dense. In a system in which particles are lost from the trap due to collisions between particles, the increase in density results in a rapid outflow of particles from the center of the trap. This is called a relaxation explosion, which was theoretically predicted to occur during the BEC transition of hydrogen atoms. For exciton systems, a threshold increase in both spatial and energy broadening was observed, which may be due to this phenomenon. ↑up

Note 7 Dilution refrigerator

This is a device that uses the ^3He-4He dilution refrigeration method, which is the only technology capable of continuously cooling solids down to 0.1 Kelvin or less. The isotopes ^{3He and4He} are liquefied respectively, and cooling is performed using the entropy difference generated by pouring the ^3He liquid phase into the ^4He liquid phase. In principle, there is no lower limit to the temperature that can be reached, but in practice, the minimum temperature is determined by the inflow of heat from the outside, and cryogenic temperatures down to about 0.01 Kelvin can be achieved. ↑up