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arXiv:0705.0788 · Physical Review A

Nonlinear coupling of continuous variables at the single quantum level

Christian F. Roos, Thomas Monz, Kihwan Kim, Mark Riebe, Hartmut Häffner, Daniel F. V. James, R. Blatt

Published Apr 29, 2008• 75 citations
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Abstract

We experimentally investigate nonlinear couplings between individual quanta of the vibrational modes of strings of cold ions stored in linear ion traps. The nonlinearity is caused by the ions' Coulomb interaction and gives rise to a Kerr-type interaction Hamiltonian $H=\ensuremath{\hbar}\ensuremath{\chi}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{r}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{s}$, where ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{r}$, ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{s}$ are phonon number operators of two interacting vibrational modes. We precisely measure the resulting oscillation frequency shift and observe a collapse and revival of the contrast in a Ramsey experiment over a range of trap frequencies. With $2.97\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}$ stretch mode and $3.61\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}$ rocking mode frequencies, we observed a coupling of $20.5\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}∕\mathrm{phonon}$. Implications for ion trap experiments aiming at high-fidelity quantum gate operations are discussed.

DOI
10.1103/physreva.77.040302

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We experimentally investigate nonlinear couplings between individual quanta of the vibrational modes of strings of cold ions stored in linear ion traps. The nonlinearity is caused by the ions' Coulomb interaction and gives rise to a Kerr-type interaction Hamiltonian H=\ensuremathℏ\ensuremathχn\ifmmode^\elseˆ\firn\ifmmode^\elseˆ\fisH=\ensuremath{\hbar}\ensuremath{\chi}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{r}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{s}H=\ensuremathℏ\ensuremathχn\ifmmode^\elseˆ\fir​n\ifmmode^\elseˆ\fis​, where n\ifmmode^\elseˆ\fir{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{r}n\ifmmode^\elseˆ\fir​, n\ifmmode^\elseˆ\fis{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{s}n\ifmmode^\elseˆ\fis​ are phonon number operators of two interacting vibrational modes. We precisely measure the resulting oscillation frequency shift and observe a collapse and revival of the contrast in a Ramsey experiment over a range of trap frequencies. With 2.97MHz2.97\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}2.97MHz stretch mode and 3.61MHz3.61\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}3.61MHz rocking mode frequencies, we observed a coupling of 20.5Hz∕phonon20.5\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}∕\mathrm{phonon}20.5Hz∕phonon. Implications for ion trap experiments aiming at high-fidelity quantum gate operations are discussed.
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 the presence of single phonons in the second mode and use it to reconstruct the phonon number distribution of the second mode.
A strong nonlinear coupling between harmonic oscillators is highly desirable for quantum information processing and quantum simulation, but is difficult to achieve in many physical systems. Here, we exploit the Coulomb interaction between two trapped ions to achieve strong nonlinear coupling between normal modes of motion at the single-phonon level. We experimentally demonstrate phonon up- and down-conversion and apply this coupling to directly measure the parity and Wigner functions of the ions' motional states. Our results represent the fully quantum operation of a degenerate parametric oscillator and hold promise for quantum computation schemes that involve continuous variables.
Laser cooled ions trapped in a linear Paul trap are long-standing ideal candidates for realizing quantum simulation, especially of many-body systems. The properties that contribute to this also provide the opportunity to demonstrate unexpected quantum phenomena in few-body systems. A pair of ions interacting in such traps exchange vibrational quanta through the Coulomb interaction. This linear interaction can be anharmonically modulated by an elementary coupling to the internal two-level structure of one of the ions. Driven by thermal energy in the passively coupled oscillators, which are themselves coupled to the internal ground states of the ions, the nonlinear interaction autonomously and unconditionally generates entanglement between the mechanical modes of the ions. We examine this counter-intuitive thermally induced entanglement for several experimentally feasible model systems and propose parameter regimes where state-of-the-art trapped ion systems can produce such phenomena. In addition, we demonstrate a multiqubit enhancement of such thermally induced entanglements.
A string of cold ions confined in a linear trap represents a man-made quantum object with a broad range of applications in atomic and molecular spectroscopy, such as high-precision measurement of atomic properties. An efficient isolation from the environment guarantees excellent coherent properties of such systems and makes them suitable for practical realization of the quantum information processing. The pioneering theoretical [Cirac and Zoller] and experimental [Wineland and Monroe] work in 1990s resulted in the explosive expansion of this field during the last decade. In this dissertation an alternative new method for controlling the quantized motional/vibrational states of ions in a trap is explored theoretically. It is proposed to create small anharmonicity in the trapping potential which would modify the spectrum of states and allow addressing the state-to-state transitions selectively. In this approach all ions remain in the ground electronic state and their motion is controlled adiabatically and coherently by applying the optimally shaped electric fields (RF). The optimal control theory, accurate numerical calculation of the energies and wavefunctions, and numerical propagation of wave packets are employed. Two sources of vibrational anharmonicity are studied: the intrinsic Coulomb anharmonicity due to the ion-ion interactions and the external anharmonicity of the trapping potential. It is shown that the magnitude of Coulomb anharmonicity is insufficient for the control. In contrast, anharmonicity of the trapping potential allows controlling the motion of ions very accurately. It is demonstrated that one ion in a slightly anharmonic trap can be easily controlled and used to represent one qubit. A multi-qubit system can be created by employing a long progression of states of a single ion, or by trapping multiple ions and controlling several normal vibration modes of the ion string. Up to four qubits are modeled in this work and accurate pulses are optimized for a set of universal quantum gates: NOT, conditional NOT (CNOT) and Hadamard transformation. The control field for Shor's algorithm (quantum algorithm for factorization onto prime numbers) is also obtained. It is demonstrated that a careful choice of system properties allows achieving very high accuracy of qubit transformations, up to 0.999.
A long-lived coherent state and nonlinear interaction have been experimentally demonstrated for the vibrational mode of a trapped ion. We propose an implementation of quantum computation using coherent states of the vibrational modes of trapped ions. Differently from earlier experiments, we consider a far-off resonance for the interaction between external fields and the ion in a bidimensional trap. By appropriate choices of the detunings between the external fields, the adiabatic elimination of the ionic excited level from the Hamiltonian of the system allows for beam splitting between orthogonal vibrational modes, production of coherent states, and nonlinear interactions of various kinds. In particular, this model enables the generation of the four coherent Bell states. Furthermore, all the necessary operations for quantum computation, such as preparation of qubits and one-qubit and controlled two-qubit operations, are possible. The detection of the state of a vibrational mode in a Bell state is made possible by the combination of resonant and off-resonant interactions between the ion and some external fields. We show that our read-out scheme provides highly efficient discrimination between all the four Bell states. We extend this to a quantum register composed of many individually trapped ions. In this case, operations on two remote qubits are possible through a cavity mode. We emphasize that our remote-qubit operation scheme does not require a high-quality factor resonator: the cavity field acts as a catalyst for the gate operation.
Following the path of Bužek et al. in [Phys. Rev. A 56 2352 1997], we investigate the full intensity-dependent interaction between a motional two-level ion which is trapped in a f-deformed oscillator (so possessing vibrational modes too) with a single-mode nonlinear quantized field in the Lamb-Dicke limit. After finding the explicit form of the entire ion-field state vector in a general framework, we particularly restrict ourselves to an appropriate nonlinearity function (describing intensity-dependent interaction) corresponding to the center-of-mass motion of a trapped ion which explicitly depends on the Lamb-Dicke parameter as well as another well-known nonlinearity function in the form n. Then, we evaluate the effects of two different initial quantized field states as well as the considered nonlinearity function (via varying the Lamb-Dicke parameter) on a few physical properties of the system. For this purpose, we pay attention to the amount of entanglement, mean number of vibrational quanta (phonons), population inversion of the two-level ion and Q-distribution function. Finally, by comparing our results with the previously considered “linear” one which is pointed in the above Ref., we establish that, by tuning the Lamb-Dicke parameter, one can adjust the mentioned nonclassical properties of the system appropriately.