L. -L. Yan

Nonadiabatic holonomic quantum computation (NHQC) offers intrinsic resilience to certain control imperfections. However, conventional nonadiabatic holonomic protocols are constrained by the fixed-pulse-area condition, which limits flexibility and prolongs duration of small-angle gates. Here we experimentally demonstrate a universal brachistochrone nonadiabatic holonomic quantum gate scheme in a trapped 40Ca+ ion, and realized the construction of pX gate under the conventional NHQC, brachistochrone NHQC (BNHQC) and composite BNHQC (CBNHQC) protocols. By characterizing the performance of gate performance in the presence of dissipation, Rabi-frequency errors and detuning errors, we show that BNHQC and CBNHQC outperform conventional NHQC, and BNHQC can offer a favorable balance between operation speed and robustness. It further shows that keeping high fidelity and strong robustness need decrease the accumulated population of excited state in the evolution process. These results highlight nonadiabatic holonomic computation as a practical route toward fast and robust quantum gates in trapped-ion platforms.

Time crystals constitute a novel phase of matter defined by the spontaneous breaking of timetranslation symmetry. Here we present a scheme to realize a continuous-time crystal of the vibrational phonon in the normal mode of two coupled ultra-cold ions. By utilizing two addressable standing-wave lasers and adiabatic elimination method, we generate a controllable nonlinear phonon mode with the well-designed efficient linear gain and nonlinear damping. By controlling these parameters to satisfy the phase transition conditions of Hopf bifurcation and limit cycle phase, it behaves as a stable dissipative dynamics over timescales significantly longer than the oscillation period, indicating the emergence of discrete time-translation symmetry breaking in the phonon mode, i.e., a phonon time crystal. We further numerically simulate this phonon time crystal by using accessible experimental parameters and also demonstrate a robustness to the initial thermal state and thermalization of phonon mode, spin dephasing, and the control errors of Rabi frequencies. These results provide a practical scheme for observing a time crystal in a nonlinear phonon mode and will advance the research of time crystals.

We investigate the thermodynamics and fluctuations of a finite-time quantum Otto engine alternatively driven by a hot squeezed and a cold thermal reservoir. We show that reservoir squeezing significantly enhances the performance by increasing the thermodynamic efficiency and the power, and enables higher stability by decreasing the relative power fluctuations and speeding up the convergence of quantum efficiency to its most probable value. These results are explained by our theoretical analysis that incorporates the effect of reservoir squeezing on the irreversibility associated with quantum friction and coherence due to finite time. An experimental scheme for realizing this quantum heat engine is proposed using a single-electron spin pertaining to a trapped 40Ca$^+$ ion. We provide a general framework for reliably studying the finite-time quantum heat engine and derive important insights into the novel thermodynamic behaviors beyond the classical thermal machines.

Quantum gates induced by geometric phases are intrinsically robust against noise due to their global properties of the evolution paths. Compared to conventional nonadiabatic geometric quantum computation (NGQC), the recently proposed nonadiabatic noncyclic geometric quantum computation (NNGQC) works in a faster fashion, while still remaining the robust feature of the geometric operations. Here, we experimentally implement the NNGQC in a single trapped ultracold $^{40}$Ca$^{+}$ ion for verifying the noise-resilient and fast feature. By performing unitary operations under imperfect conditions, we witness the advantages of the NNGQC with measured fidelities by quantum process tomography in comparison with other two quantum gates by conventional NGQC and by straightforwardly dynamical evolution. Our results provide the first evidence confirming the possibility of accelerated quantum information processing with limited systematic errors even in the imperfect situation.

Quantitative measure of disorder or randomness based on the entropy production characterizes thermodynamical irreversibility, which is relevant to the conventional second law of thermodynamics. Here we report, in a quantum mechanical fashion, the first theoretical prediction and experimental exploration of an information-theoretical bound on the entropy production. Our theoretical model consists of a simplest two-level dissipative system driven by a purely classical field, and under the Markovian dissipation, we find that such an information-theoretical bound, not fully validating quantum relaxation processes, strongly depends on the drive-to-decay ratio and the initial state. Furthermore, we carry out experimental verification of this information-theoretical bound by means of a single spin embedded in an ultracold trapped $^{40}$Ca$^{+}$ ion. Our finding, based on a two-level model, is fundamental to any quantum thermodynamical process and indicates much difference and complexity in quantum thermodynamics with respect to the conventionally classical counterpart.

We demonstrate for the first time a linear-zigzag phase transition induced solely by temperature of the $^{40}$Ca$^{+}$ ion crystals in a surface-electrode trap. In contrast to the previously observed counterparts based on change of the mechanical equilibrium conditions of the ions, our presented structural phase transition occurs due to controllable influence of thermal fluctuation. The ions' temperature is well controlled by tuning the cooling laser and the experimental observation could be fully understood by classical Langevin equation in addition to the effects from thermal fluctuation. Our experimental investigation indicates the fantastic role of thermal fluctuation played in the thermodynamic process at atomic level, which might bridge the thermodynamics from the macroscopic domain to the quantum regime.

Most non-equilibrium processes in thermodynamics are quantified only by inequalities, however the Jarzynski relation presents a remarkably simple and general equality relating non-equilibrium quantities with the equilibrium free energy, and this equality holds in both classical and quantum regimes. We report a single-spin test and confirmation of the Jarzynski relation in quantum regime using a single ultracold $^{40}Ca^{+}$ ion trapped in a harmonic potential, based on a general information-theoretic equality for a temporal evolution of the system sandwiched between two projective measurements. By considering both initially pure and mixed states, respectively, we verify, in an exact and fundamental fashion, the non-equilibrium quantum thermodynamics relevant to the mutual information and Jarzynski equality.

One of the outstanding challenges to information processing is the eloquent suppression of energy consumption in execution of logic operations. Landauer principle sets an energy constraint in deletion of a classical bit of information. Although some attempts have been paid to experimentally approach the fundamental limit restricted by this principle, exploring Landauer principle in a purely quantum mechanical fashion is still an open question. Employing a trapped ultracold ion, we experimentally demonstrate a quantum version of Landauer principle, i.e., an equality associated with energy cost of information erasure in conjunction with entropy change of the associated quantized environment. Our experimental investigation substantiates an intimate link between information thermodynamics and quantum candidate systems for information processing.

Here we respond to a comment [arXiv:1802.01382] submitted recently on 'Experimental Verification of a Jarzynski-Related Information-Theoretic Equality by a Single Trapped Ion' PRL 120 010601 (2018). We consider that the argument is the thermalization in a quantum system, which is different from the counterpart in conventional thermodynamics.

The uncertainty relations, pioneered by Werner Heisenberg nearly 90 years ago, set a fundamental limitation on the joint measurability of complementary observables. This limitation has long been a subject of debate, which has been reignited recently due to new proposed forms of measurement uncertainty relations. The present work is associated with a new error trade-off relation for compatible observables approximating two incompatible observables, in keeping with the spirit of Heisenberg's original ideas of 1927. We report the first \textsl{direct} test and confirmation of the tight bounds prescribed by such an error trade-off relation, based on an experimental realisation of optimal joint measurements of complementary observables using a single ultracold $^{40}Ca^{+}$ ion trapped in a harmonic potential. Our work provides a prototypical determination of ultimate joint measurement error bounds with potential applications in quantum information science for high-precision measurement and information security.