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Physical Review A

Nonclassical maximum-entropy states

Ulrike Herzog, János A. Bergou

Published Aug 1, 1997• 1 citation
Publisher

Abstract

We investigate specific maximum-entropy states of a single harmonic-oscillator mode that arise when only number states (Fock states) differing by a multiple of a certain integer $k$ $(k>~1)$ are allowed to be occupied. For $k=2$ the number-probability distribution of the even-number maximum-entropy state has a close resemblance to that of a squeezed-vacuum state. These maximum-entropy states can be obtained as the stationary solutions of a master equation which takes into account $k$-quantum absorption as well as $k$-quantum emission processes only. The steady-state solution of this master equation depends on the initial conditions. For the vibrational motion of a trapped ion such nonclassical maximum-entropy states could be produced with the help of the recently proposed method of laser-assisted quantum reservoir engineering.

DOI
10.1103/physreva.56.1658

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We investigate specific maximum-entropy states of a single harmonic-oscillator mode that arise when only number states (Fock states) differing by a multiple of a certain integer kkk (k> 1)(k>~1)(k> 1) are allowed to be occupied. For k=2k=2k=2 the number-probability distribution of the even-number maximum-entropy state has a close resemblance to that of a squeezed-vacuum state. These maximum-entropy states can be obtained as the stationary solutions of a master equation which takes into account kkk-quantum absorption as well as kkk-quantum emission processes only. The steady-state solution of this master equation depends on the initial conditions. For the vibrational motion of a trapped ion such nonclassical maximum-entropy states could be produced with the help of the recently proposed method of laser-assisted quantum reservoir engineering.
The density matrix in the position and momentum representation, the position-momentum uncertainty product, the Wigner and Q functions, and thermal properties of the family of the nonclassical maximum-entropy states of a single harmonic oscillator are determined. Any such state, having the mean number of quanta n , has the uncertainty product x p = (2 n +1)( /2), and this product attains its minimum value for the temperature parameter 0. Generically, the Wigner function has alternating sign so that the underlying maximum-entropy state is truly nonclassical. The von Neumann entropy and the heat capacity, expressed via the temperature parameter , coincide with the corresponding quantities for the thermal state. The properties discussed here are of interest for the description and analysis of the vibrational motion of a trapped ion in a harmonic-oscillator potential since the equilibrium states which result under certain conditions are the maximum-entropy states.
The robust generation of quantum states in the presence of decoherence is a primary challenge for explorations of quantum mechanics at larger scales. Using the mechanical motion of a single trapped ion, we utilize reservoir engineering to generate squeezed, coherent, and displaced-squeezed states as steady states in the presence of noise. We verify the created state by generating two-state correlated spin-motion Rabi oscillations, resulting in high-contrast measurements. For both cooling and measurement, we use spin-oscillator couplings that provide transitions between oscillator states in an engineered Fock state basis. Our approach should facilitate studies of entanglement, quantum computation, and open-system quantum simulations in a wide range of physical systems.
We present a generic quantum master equation whose dissipative dynamics autonomously stabilizes a harmonic oscillator in the n=1n=1n=1 Fock state. A multimode optomechanical system is analyzed and shown to be an example of a physical system obeying this model. We show that the optomechanical setup enables preparation of a mechanical oscillator in a nonclassical steady state, and that this state indeed approaches a single phonon Fock state in the ideal parameter regime. The generic model may be useful in other settings, such as cavity or circuit quantum electrodynamics or trapped ion physics.
We report the creation of thermal, Fock, coherent, and squeezed states of motion of a harmonically bound 9 Be + ion. The last three states are coherently prepared from an ion which has been initially laser-cooled to the zero-point of motion. The ion is trapped in the regime where the coupling between its motional and internal states, due to applied (classical) radiation, can be described by a Jaynes-Cummings-type interaction. With this coupling, the evolution of the internal atomic state provides a signature of the number state distribution of the motion. PACS numbers:42.50.Vk, 32.80.Pj, 32.80.Qk Typeset using REVT E X Nonclassical states of the harmonic oscillator associated with a single mode of the radiation field (for example, squeezed states) have been a subject of considerable interest. One method for analyzing these states has been through the dynamics of a single, two-level atom which radiatively couples to the single mode radiation field. This system, described by the Ja...