Degeneracy produces robust protocols
Elementary quantum mechanics deals predominantly with closed quantum systems, which are described by Hermitian Hamiltonians. The Hermiticity of the Hamiltonian dictates many of the postulates of quantum mechanics, including the fact that expectation values are real. However, most realistic systems are open in the sense that they can exchange energy with their environment, like atoms in solids that interact with photons and phonons in the lattice. Open systems are conveniently described with non-Hermitian operators. As such, their energy spectrum is complex, and their eigenvectors are not necessarily orthogonal [1]. A unique feature of non-Hermitian operators is that they can have exceptional points – where several eigenvectors of the Hamiltonian coalesce – in striking contrast to Hermitian Hamiltonians. It implies that the eigenvectors do not form a complete basis of the Hilbert space and traditional perturbation theory breaks down. For this reason, exceptional points can lead to a variety of counterintuitive effects [2], such as loss-induced transmission and giant response to small perturbations. An example of an exceptional point in an NV center (a substitutional nitrogen atom and a vacancy in the diamond lattice) is shown below.
Exceptional points in the energy spectrum of the NV center (defect in diamond) (a) Triplet energy levels of the NV center, including 2 driving fields (red/blue) and incoherent jumps (green). (b) Resonant frequencies of the NV center as a function of the fields’ parameters. The energy sheets cross at a singular point called an “exceptional point.”
It was recently discovered that exceptional points can be used to develop noise-resilient protocols for quantum-state manipulation [3]. Such protocols make use of cyclic adiabatic processes -- where the parameters of a system are varied slowly along closed loops that encircle exceptional points in its parameter space. In traditional (Hermitian) quantum mechanics, when a system is subjected to a cyclic adiabatic process, its quantum state acquires a geometric phase [4]. However, in non-Hermitian quantum mechanics, when the loop encircles an exceptional point, its quantum state may change entirely. Surprisingly, the final state is dictated by the direction in which the exceptional point is encircled but is independent of the details of the loop. We utilize this effect to develop applications for quantum information processing. We have recently proposed protocols for robust mode conversion in NV centers [5] and asymmetric energy flow in elastodynamic systems. We explore ways to optimize our protocols and, together with our experimentalist collaborators, realize these theoretical predictions in the lab. An example of a cyclic adiabatic protocol that produces a robust geometric phase gate is shown below.
[1] N. Moiseyev, Non-Hermitian Quantum Mechanics, Cambridge University (2011)
[2] M.A. Miri and A. Alu. "Exceptional points in optics and photonics." Science 363, 6422 (2019)
[3] R. Uzdin, A. A. Mailybaev, and N. Moiseyev. "On the observability and asymmetry of adiabatic state
flips generated by exceptional points." J. Phys. A, 44, 435302 (2011).
[4] M. V. Berry. "Quantal phase factors accompanying adiabatic changes." Proc. R. Soc. Lond., A Math. Phys. Sci. 392, 1802, 45-57 (1984)
[5] A. Pick, S. Silberstein, N. Moiseyev, and N. Bar-Gill. "Robust mode conversion in NV centers using
exceptional points." Phys. Rev. Research 1, 013015 (2019).
Singularity enhances light emission
Radiation rates from excited particles are determined by the number of available electromagnetic modes that a particle can emit into, formally called the local the density of states (LDOS). The LDOS can be controlled by introducing dielectric or metallic structures with associated resonant modes. Specifically,
LDOS enhancement near resonators with non-degenerate (spectrally separated) modes is given by the Purcell factor, which is proportional to the ratio of the resonator’s quality factor and mode volume [1]. Although the common trend for enhancing emission rates is to optimize this ratio, one may ask whether this approach is optimal? Is it possible to achieve greater enhancements with degenerate resonances and exceptional points (EPs)? While early work suggested that dramatic noise enhancements should occur at EPs [2], as of a few years ago, a comprehensive analysis of this effect seemed to be missing.
In a series of studies, we developed a framework for analyzing and utilizing EPs for controlling light emission [3-6]. We demonstrated that constructive interference of the coalescing modes at EPs produces qualitative changes in the emission spectrum (see figure). Specifically, the emission peak near an n-fold degeneracy changes from an ordinary Lorentzian to a Lorentzian to the nth power. Energy conservation dictates that the enhancement in passive systems (without pumping energy) is bounded, but dramatic enhancements can be achieved with gain. These predictions have attracted the attention of the research community and initiated collaborations with leading experimental groups around the world (including the Soljacic group at MIT [7] and the Nguyen group at Ecole Central de Lyon). We developed these ideas
further and showed that EPs in laser-riven atoms and molecules can lead to a variety of exciting phenomena [8]. We found that “dressing fields” can create EPs of resonant orbitals and, in this way, shape the photoionization spectrum. Future research directions include (i) using “dressing fields” to shape high-harmonic pulses generated in strong fields, (ii), study collective-emission effects (e.g., superradiance) at EPs, and (iii) revisit remaining open questions regarding laser noise and stability at EPs.
Spontaneous emission at exceptional points (EPs). (a) Emission rates are modified by placing the emitter near two structures, both having resonant modes with frequency ω0 (a loss-free one and a leaky one with loss rate κ), with coupling strength Ω. (b) Eigenenergies (in arbitrary units) of resonant modes vs. Ω/κ. An EP forms when κ =2Ω. (c) The linewidth at the EP is a squared Lorentzian and its peak is 4 times larger than the Lorentzian peak at κ ≪ Ω.
[1] E. M. Purcell, "Spontaneous emission probabilities at radio frequencies." In Confined Electrons and Photons, pp. 839-839. Springer, Boston, MA, (1995).
[2] M. V. Berry, "Mode degeneracies and the Petermann excess-noise factor for unstable lasers." J. Mod. Opt. 50, 63-81 (2003).
[3] A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljačić, and S. G. Johnson. "General theory of spontaneous emission near exceptional points." Opt. Express 25, 12325-12348 (2017).
[4] Z. Lin, A. Pick, M. Lončar, and A.W. Rodriguez. "Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals." Phys. Rev. Lett. 117, 107402 (2016).
[5] F. Hernández, A. Pick, and S. G. Johnson. "Scalable computation of Jordan chains," arXiv:1704.05837 (2017).
[6] A. Pick, Z. Lin, W. Jin, and A. W. Rodriguez. "Enhanced nonlinear frequency conversion and Purcell enhancement at exceptional points." Phys. Rev. B 96, 224303 (2017).
[7] B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić. "Spawning rings of exceptional points out of Dirac cones." Nature 525, 354-358 (2015)
[8] A. Pick, P. R. Kaprálová-Žďánská, and N. Moiseyev. "Ab-initio theory of photoionization via resonances." J. Chem. Phys. 150, 204111 (2019).
Atoms in photonic quantum computation
Quantum information can be encoded in electronic degrees of freedom (for example, using NV centers, as in the previous section) or in photons – energy quanta in modes of the electromagnetic field. In contrast to electrons, photons essentially do not interact. This property makes photons nearly immune to dephasing and decoherence but, on the other hand, it presents a challenge for achieving photon-photon gates, which are essential for universal quantum computation [1]. A popular approach for mediating effective interactions between photons is to use linear optical elements and measurements. In this approach, one obtains probabilistic photonic gates, requiring multiple repetitions of the protocol for achieving high success rates. Due to their probabilistic nature, large-scale linear-optics computation schemes are associated with a huge overhead in resources [2]. Although ideally, atoms can provide strong
nonlinear light-matter interaction and deterministically entangle photons [3], it is challenging to achieve strong nonlinearity in a scalable manner. Finding protocols for efficient photon-photon gates is a central goal in photonic quantum computation.
In our recent work [4], we presented a new paradigm for photonic quantum computation, which surprisingly was never previously explored. We proposed utilizing moderate nonlinearity to boost photonic quantum-computation protocols (see figure). The key element in our scheme is a nonlinear router that preferentially directs photonic wavepackets to different ports depending on the number of photons in the incoming pulse. Routing is achieved by exploiting intensity-dependent phase shifts provided by a nonlinear atomic medium. We presented protocols for efficient Bell measurement (BM) and Greenberger–Horne–Zeilinger (GHZ) state preparation – both key elements in photonic quantum computation [5], as well as for the CNOT gate and quantum factorization [9]. As photonic quantum computation requires many elementary operations, a modest increase in the success probability of each operation is translated into a dramatic reduction in the required resources. For example, a conditional phase shift of π/3, which in our scheme increases the success probability of ancilla-assisted BM from 75% to 86%, is translated into two-orders-of-magnitude reduction in resources after 35 entangling operations and three-orders-of-magnitude improvement at 50 gates. With the recent developments in integrated photonics and microfabricated atomic vapor cells, few-photon nonlinearty on chip-scale devices is becoming experimentally feasible, making our protocols a promising new platform for photonic quantum information processing.
Moderate nonlinearity for efficient photonic gates. (a) Photon-atom nonlinearity mediates effective interaction between photons. (b) A nonlinear router uses atoms to direct pulses to ports I or II depending on their intensity. It utilizes a nonlinear intensity dependent phase shift ±φ. (c) Proposed setup for nonlinear Bell measurements (BMs). (d) Success probability of two nonlinear BM protocols as a function of φ.
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[2] Y. Li, P. C. Humphreys, G. J. Mendoza, and S. C. Benjamin. "Resource costs for fault-tolerant linear optical quantum computing." Phys. Rev. X, 5, 041007 (2015).
[3] L.-M. Duan, and H. J. Kimble. "Scalable photonic quantum computation through cavity-assisted interactions." Phys. Rev. Lett. 92, 127902 (2004).
[4] A. Pick, E. S. Matekole, Z. Aqua, G. Guendelman, O. Firstenberg, J. P. Dowling, and B. Dayan, “Boosting photonic quantum computation with moderate nonlinearity,” Phys. Rev. Appl., 15, 054054, (2021)
[5] D. E. Browne and T. Rudolph, "Resource-efficient linear optical quantum computation," Phys. Rev. Lett. 95, 010501 (2005).