The Jarzynski equality relates the free-energy di erence between two equilibrium states to the work done on a system through far-from-equilibrium processes-a milestone that builds on the pioneering work of Clausius and Kelvin. Although experimental tests of the equality have been performed in the classical regime, the quantum Jarzynski equality has not yet been fully verified owing to experimental challenges in measuring work and work distributions in a quantum system. Here, we report an experimental test of the quantum Jarzynski equality with a single 171 Yb + ion trapped in a harmonic potential. We perform projective measurements to obtain phonon distributions of the initial thermal state. We then apply a laser-induced force to the projected energy eigenstate and find transition probabilities to final energy eigenstates after the work is done. By varying the speed with which we apply the force from the equilibrium to the far-from-equilibrium regime, we verify the quantum Jarzynski equality in an isolated system. T here is increasing interest in non-equilibrium dynamics at the microscopic scale, crossing over quantum physics, thermodynamics and information theory as the experimental control and technology at such a scale have been developing rapidly. Most of the principles in non-equilibrium processes are represented in the form of inequalities, as seen in the example of the maximum work principle, W − F ≥ 0, where the average work W is equal to the free-energy difference F only in the case of the equilibrium process. In close-to-equilibrium processes, the fluctuation-dissipation theorem is valid and connects the average dissipated energy W diss ≡ W − F and the fluctuation of the system σ 2 /2k B T . Here σ is the standard deviation of the work distribution, T is the initial temperature of the system in thermal equilibrium and k B is the Boltzmann constant. Beyond the nearequilibrium regime, no exact results were known until Jarzynski found a remarkable equality 1 that relates the free-energy difference to the exponential average of the work done on the system:The Jarzynski equality (1) is satisfied irrespective of the protocols of varying parameters of the system even when the driving is arbitrarily far from equilibrium. The relation enables us to experimentally determine F of a system by repeatedly performing work at any speed. Experimental tests of the classical Jarzynski equality and its relation to the Crooks fluctuation theorem 2 have been successfully performed in various systems 3-12 .In classical systems, work can be obtained by measuring the force and the displacement, and then integrating the force over the displacement during the driving process. In the quantum regime, however, as a result of Heisenberg's uncertainty principle, we cannot determine the position and the momentum simultaneously-thus invalidating the concepts of force and displacement. Instead of measuring these classical observables, it is necessary to carry out projective measurements over the energy eigenstates to determine the work d...

The quantum Rabi model, involving a two-level system and a bosonic field mode, is arguably the simplest and most fundamental model describing quantum light-matter interactions. Historically, due to the restricted parameter regimes of natural light-matter processes, the richness of this model has been elusive in the lab. Here, we experimentally realize a quantum simulation of the quantum Rabi model in a single trapped ion, where the coupling strength between the simulated light mode and atom can be tuned at will. The versatility of the demonstrated quantum simulator enables us to experimentally explore the quantum Rabi model in detail, including a wide range of otherwise unaccessible phenomena, as those happening in the ultrastrong and deep strong-coupling regimes. In this sense, we are able to adiabatically generate the ground state of the quantum Rabi model in the deep strong-coupling regime, where we are able to detect the nontrivial entanglement between the bosonic field mode and the two-level system. Moreover, we observe the breakdown of the rotating-wave approximation when the coupling strength is increased, and the generation of phonon wave packets that bounce back and forth when the coupling reaches the deep strongcoupling regime. Finally, we also measure the energy spectrum of the quantum Rabi model in the ultrastrong-coupling regime.

The application of adiabatic protocols in quantum technologies is severely limited by environmental sources of noise and decoherence. Shortcuts to adiabaticity by counterdiabatic driving constitute a powerful alternative that speed up time-evolution while mimicking adiabatic dynamics. Here we report the experimental implementation of counterdiabatic driving in a continuous variable system, a shortcut to the adiabatic transport of a trapped ion in phase space. The resulting dynamics is equivalent to a ‘fast-motion video' of the adiabatic trajectory. The robustness of this protocol is shown to surpass that of competing schemes based on classical local controls and Fourier optimization methods. Our results demonstrate that shortcuts to adiabaticity provide a robust speedup of quantum protocols of wide applicability in quantum technologies.

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Single-quantum level operations are important tools to manipulate a quantum state. Annihilation or creation of single particles translates a quantum state to another by adding or subtracting a particle, depending on how many are already in the given state. The operations are probabilistic and the success rate has yet been low in their experimental realization. Here we experimentally demonstrate (near) deterministic addition and subtraction of a bosonic particle, in particular a phonon of ionic motion in a harmonic potential. We realize the operations by coupling phonons to an auxiliary two-level system and applying transitionless adiabatic passage. We show handy repetition of the operations on various initial states and demonstrate by the reconstruction of the density matrices that the operations preserve coherences. We observe the transformation of a classical state to a highly non-classical one and a Gaussian state to a non-Gaussian one by applying a sequence of operations deterministically.

We experimentally measure the Q-function and reconstruct the Wigner function of the phononic state of the vibrational motion for an 171 Yb + ion resonantly interacting with its internal energy states. The scheme for the Q-function is based on highly efficient vacuum phonon detection using the transitionless adiabatic passage on the external motion-internal states interaction. The JaynesCummings dynamics of entanglement between the external and internal states has been known to bring the state of the field to a superposition of two composite states, which are nearly coherent states. The measured Q-function clearly shows two peaks in phase space at a certain interaction time and the quantum superposition of this state is confirmed by the reconstruction of the Wigner function with negativities manifesting the quantum interferences between the two composite states. The scheme can be applied to other physical setups including circuit-QED and opto-mechanical systems.PACS numbers: 03.65. Ta, 37.10.Ty, 42.50.Dv, 42.50.Md One of the most fundamental interaction models in quantum mechanics is the so-called Jaynes-Cummings model (JCM) [1], where a single two-level atom resonantly interacts with a single-mode field. The JCM has enabled the theoretical and experimental investigations of the basic properties of quantum electrodynamics such as Rabi oscillations of the energy transfer between the two subsystems and collapses and revivals of the oscillations [2]. More recently, the model has been widely studied for its rich properties of quantum control, coherent superposition and entanglement which are closely related to the current development of quantum technology. In order to see the nonclassical effects due to quantum interaction, the JCM is often studied with the state initially prepared in a coherent field, which is normally considered as the most classical state among all the pure quantum states and the atom in its energy eigenstate. It has been shown that the field and the atom are entangled [3] as soon as the interaction starts, but at a certain time they are nearly disentangled to bring the field into a superposition of two coherent states of π phase difference [4,5]. Earlier, Eiselt and Risken [6,7] showed that the Gaussian probability distribution of the initial coherent state in phase space breathe at the very initial points of interaction, reflecting the Rabi oscillations. Then the Gaussian peak bifurcates to travel around a circle in opposite direction in phase space. The bifurcation is a consequence of quantum nature of interaction and was experimentally probed through the measurement of field phase distribution [8][9][10]. However, the full reconstruction of the dynamics of the JCM field has not been experimentally demonstrated.A phase-space reconstruction of the density operator of a field can be performed by the measurement of quasiprobabilities, such as the Husimi Q and the Wigner functions [11]. There have been many developments of reconstructing the quantum state of a field in various systems and situa...

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We theoretically propose and experimentally demonstrate a nonclassicality test of single-mode field in phase space, which has an analogy with the nonlocality test proposed by Wódkiewicz [Phys. Rev. Lett. 82, 2009 (1999)]. Our approach to deriving the classical bound draws on the fact that the Wigner function of a coherent state is a product of two independent distributions as if the orthogonal quadratures (position and momentum) in phase space behave as local realistic variables. Our method detects every pure nonclassical Gaussian state, which can also be extended to mixed states. Furthermore, it sets a bound for all Gaussian states and their mixtures, thereby providing a criterion to detect a genuine quantum non-Gaussian state. Remarkably, our phase-space approach with invariance under Gaussian unitary operations leads to an optimized test for a given non-Gaussian state. We experimentally show how this enhanced method can manifest quantum non-Gaussianity of a state by simply choosing phase-space points appropriately, which is essentially equivalent to implementing a squeezing operation on a given state.PACS numbers: 03.65. Ta, 42.50.Dv, 42.50.Ar Introduction-Nonclassicality of a quantum state is a topic of crucial importance that has attracted a lot of theoretical and experimental efforts for long. It provides not only a profound conceptual framework to distinguish quantum phenomena from classical ones, but also an essential practical basis for numerous applications, e.g. in quantum information processing. An important approach to studying quantum mechanics in comparison with classical mechanics is to adopt a phase-space description of a quantum state [1]. A wide variety of quantum systems of continuous variables (CVs) can be addressed in phase space, including quadrature amplitudes of light fields, collective spin states of atomic ensembles, and motional states of trapped ions, Bose-Einstein condensate, or mechanical oscillators, etc. [2]. Investigating quantum dynamics in phase space has yielded a great deal of intuition to quantum-to-classical transition [3]. It also offers a powerful tool to treat problems in quantum optics [4] and CV quantum informatics [5].A clear signature of nonclassicality is the negativity of phase-space distribution, which does not exist in classical probability distributions. However, its demonstration typically requires a full reconstruction of Wigner function [6] and it is of fundamental and practical significance to have a simpler set of measurements manifesting nonclassicality [7,8], desirably even when the Wigner function is non-negative. For instance, every Gaussian state possesses a positive-definite Wigner function, which restricts a possible set of nonclassicality tests. To manifest the Bell nonlocality, e.g., by employing homodyne detections, a Gaussian state must be transformed to a non-Gaussian state having a non-positive Wigner function to rule out hidden-variable models [9,10]. Banaszek and Wódkiewicz (BW) introduced a different seminal approach to manifesting CV ...