In conventional wisdom, producing curved space requires distortions, such as bending or stretching flat space. A team of researchers from Purdue University has discovered a new method for creating curved spaces that also solves a mystery in physics. Without any physical distortion of physical systems, the team devised a scheme using non-hermiticity, which exists in all systems coupled with environments, to create a hyperbolic surface and a variety of other prototypical curved spaces.

“Our work may revolutionize the general public’s understanding of curvatures and distance,” says Qi Zhou, professor of physics and astronomy. “He also answered long-standing questions about non-Hermitian quantum mechanics by linking non-Hermitian physics and curved spaces. These two topics were supposed to be completely disconnected. The extraordinary behaviors of non-Hermitian systems, which puzzled physicists for decades. , only become more mysterious if we recognize that space has been curved. In other words, non-hermiticity and curved spaces are dual to each other, being the two faces of the same room.

The team recently published their findings in *Nature Communication*. Of the team members, most work at Purdue University’s West Lafayette campus. Graduate student Chenwei Lv is the lead author, and fellow Purdue team members include Professor Qi Zhou and postdoctoral fellow Zhengzheng Zhai. Co-first author Professor Ren Zhang of Xi’an Jiaotong University was a visiting scholar at Purdue when the project was launched.

In order to understand how this discovery works, one must first understand the difference between Hermitian and non-Hermitian systems in physics. Zhou explains it using an example in which a quantum particle can “hop” between different sites on a lattice. If the probability for a quantum particle to jump in the right direction is the same as the probability of jumping in the left direction, then the Hamiltonian is Hermitian. If these two probabilities are different, the Hamiltonian is non-Hermitian. This is why Chenwei and Ren Zhang used arrows of different sizes and thicknesses to indicate the probabilities of jumping in opposite directions in their plot.

“Typical quantum mechanical textbooks focus primarily on systems governed by Hamiltonians who are Hermitian,” says Lv. “A quantum particle moving through a lattice must have an equal probability of tunneling in left and right directions. While Hermitian Hamiltonians are well-established frameworks for studying isolated systems, couplings with the environment inevitably lead to dissipations in open systems, which can give rise to Hamiltonians that are no longer Hermitian For example, tunneling amplitudes in a lattice are no longer equal in opposite directions, a phenomenon called non-reciprocal tunneling. In such non-Hermitian systems, the familiar textbook results no longer apply and some may even seem completely opposite to those of Hermitian systems For example, the eigenstates of non-Hermitian systems are no longer orthogonal, contrary to what we learned in first class of an undergraduate quantum mechanics course.These extraordinary behaviors of systems Non-Hermitian phenomena have puzzled physicists for decades, but many unanswered questions remain open.”

He further explains that their work provides an unprecedented explanation of non-Hermitian fundamental quantum phenomena. They discovered that a non-Hermitian Hamiltonian bent the space where a quantum particle resides. For example, a quantum particle in a non-reciprocal tunneling lattice actually moves on a curved surface. The ratio of tunneling amplitudes in one direction to that in the opposite direction controls the size of the curved surface. In such curved spaces, all strange non-Hermitian phenomena, some of which may even appear unphysical, immediately become natural. It is finite curvature that requires orthonormal conditions distinct from their counterparts in flat spaces. As such, the eigenstates would not appear orthogonal if we used the derived theoretical formula for flat spaces. It is also finite curvature that gives rise to the extraordinary non-Hermitian skin effect that all eigenstates concentrate near one edge of the system.

“This research is of fundamental importance and its implications are twofold,” says Zhang. “On the one hand, it establishes nonhermiticity as a unique tool for simulating intriguing quantum systems in curved spaces,” he explains. “Most quantum systems available in laboratories are flat and it often requires significant effort to access quantum systems in curved spaces. Our results show that non-hermiticity provides experimenters with an extra button to access and manipulate curved spaces. An example is that a hyperbolic surface could be created and then be strung by a magnetic field.This could allow experimenters to explore the responses of quantum Hall states to finite curvatures, an open question in condensed matter physics. On the other hand, duality allows experimenters to use curved spaces to explore non-Hermitian physics. For example, our results offer experimenters a new approach to access exceptional points using curved spaces and improve the accuracy of quantum sensors without resorting to dissipations.

Now that the team has published its findings, it expects it to spread in several directions for further study. Physicists studying curved spaces could implement their devices to answer difficult questions in non-Hermitian physics. Additionally, physicists working on non-Hermitian systems could adapt dissipations to access non-trivial curved spaces that cannot be easily obtained by conventional means. The Zhou research group will continue to theoretically explore more connections between non-Hermitian physics and curved spaces. They also hope to help bridge the gap between these two physics topics and bring these two different communities together for future research.

Researchers discover a new way to achieve quantum loss control

**More information:**

Chenwei Lv et al, Curving the space by non-Hermiticity,

*Nature Communication*(2022). DOI: 10.1038/s41467-022-29774-8

Provided by Purdue University

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