In recent years substantial efforts have been expended in extending thermodynamics to single quantum systems. Quantum effects have emerged as a resource that can improve the performance of heat machines. However in the fully quantum regime their implementation still remains a challenge. Here, we report an experimental realization of a quantum absorption refrigerator in a system of three trapped ions, with three of its normal modes of motion coupled by a trilinear Hamiltonian such that heat transfer between two modes refrigerates the third. We investigate the dynamics and steady-state properties of the refrigerator and compare its cooling capability when only thermal states are involved to the case when squeezing is employed as a quantum resource. We also study the performance of such a refrigerator in the single shot regime made possible by coherence and demonstrate cooling below both the steady-state energy and a benchmark set by classical thermodynamics.

A system of harmonic oscillators coupled via nonlinear interaction is a fundamental model in many branches of physics, from biophysics to electronics and condensed matter physics. In quantum optics, weak nonlinear interaction between light modes has enabled, for example, the preparation of squeezed states of light and generation of entangled photon pairs [1, 2]. While strong nonlinear interaction between the modes has been realized in circuit QED systems [3,4], achieving significant interaction strength on the level of single quanta in other physical systems remains a challenge [5,6]. Here we experimentally demonstrate such interaction that is equivalent to photon up-and downconversion using normal modes of motion in a system of two Yb ions [7,8]. The nonlinearity is induced by the intrinsic anharmonicity of the Coulomb interaction between the ions and can be used to simulate fully quantum operation of a degenerate optical parametric oscillator [9]. We exploit this interaction to directly measure the parity and Wigner functions of ion motional states. The nonlinear coupling, combined with near perfect control of internal and motional states of trapped ions, can be applied to quantum computing [6,10], quantum thermodynamics [11,12], and even shed some light on the quantum information aspects of Hawking radiation [13].Ions in a Paul trap experience a pseudopotential that is harmonic to a high degree and their motion is usually approximated by a set of normal modes that do not interact with each other. Coulomb interaction between the ions is, however, nonlinear and can introduce coupling between the modes and anharmonicity to the ion motion. The linear coupling of motional modes due to mutual Coulomb repulsion of ions was previously demonstrated in quantum regime [14,15], where the ions were trapped in independent potential wells. The higher order terms in the Coulomb interaction lead, for example, to cross Kerr-type nonlinear coupling that results in a shift of the normal mode frequencies [7,8] which has been experimentally observed [16].In this Letter, we engineer the nonlinear interaction between modes of motion similar to a degenerate parametric oscillator at the single-phonon level. We consider a system of two ions with the same mass m and charge e in a linear Paul trap that is characterized by the singleion secular frequencies ω x , ω y , ω z [7,8]. The potential energy of the system has the formwhere ǫ 0 is the permittivity of free space, X, Y, Z are the center-of-mass coordinates, and x, y, z are half the separation between the ions along the direction of principle trap axes. When ω z < (ω x , ω y ), the ions crystallize along the axial (z) direction at an equilibrium distance z 0 from the trap center. According to Eq. 1, the motion of the centerof-mass modes is harmonic, but the out-of-phase modes are coupled to each other by the Coulomb interaction. For small axial displacement u = z − z 0 and keeping only terms up to the third order that contribute to the coupling between the x and z modes, the potential...

State measurement of a quantum harmonic oscillator is essential in quantum optics and quantum information processing. In a system of trapped ions, we experimentally demonstrate the projective measurement of the state of the ions' motional mode via an effective cross-Kerr coupling to another motional mode. This coupling is induced by the intrinsic nonlinearity of the Coulomb interaction between the ions. We spectroscopically resolve the frequency shift of the motional sideband of the first mode due to presence of single phonons in the second mode and use it to reconstruct the phonon number distribution of the second mode.The quantum harmonic oscillator is one of the foundational models in physics which describes, among many other systems, the mode of the electromagnetic field and the motion of trapped particles. A rich toolbox of methods exists to characterize its quantum state. In optics such methods include homodyning [1], photon counting [2] and photon number resolving detection [3,4]. In a trapped-ion system the motion of ions is usually probed by coupling it to the ion's internal state via a motional sideband transition, which enables reconstruction of phonon number distribution [5][6][7][8][9] or measurement of the parity of the motional state [10]. However most of the methods to determine the motional state are destructive in nature and the state of the oscillator after measurement does not correspond to its outcome.An ideal projective measurement should leave the quantum system immediately after measurement in the state defined by the measurement outcome. Such measurements have been performed by utilizing the nonlinear dispersive interaction between the oscillator and another quantum system such as Rydberg atoms [11,12], superconducting circuits [13,14] or electron motion in a Penning trap [15]. An example of such an interaction in the context of a projective measurement of the Fock state [16][17][18] is the cross-Kerr nonlinear coupling between two different quantum oscillators. This coupling is described by the HamiltonianĤ kerr ∼ χn anb , wherê However, several difficulties arise in the practical implementation of the cross-Kerr nonlinearity. In optics, the interaction between the photons is usually weak [22] and difficult to control. In addition, locality and causality arguments preclude large conditional phase shifts for traveling single photon wave packets [23][24][25][26]. One way of overcoming these limitations was recently demonstrated by mapping one of the photons to an atomic excitation [27][28][29]. Other similar schemes have also been proposed [30].In our approach we simulate the cross-Kerr interaction in quantum optics in a system of trapped ions. The anharmonicity of the Coulomb interaction induces a strong nonlinear interaction between the motional modes [31]. In the dispersive regime, where no energy exchange between the modes is possible, this coupling manifests itself as a shift of the frequency of the motional mode that is proportional to the number of phonons in another motional mode ...