Trapped-ion antennae for the transmission of quantum information

More than 100 years ago, Hertz succeeded in transmitting signals over a few metres to a receiving antenna using an electromagnetic oscillator, thus proving the electromagnetic theory developed by Maxwell. Since this seminal work, technology has developed, and various oscillators are now available at the quantum mechanical level. For quantized electromagnetic oscillations, atoms in cavities can be used to couple electric fields. However, a quantum mechanical link between two mechanical oscillators (such as cantilevers or the vibrational modes of trapped atoms or ions) has been rarely demonstrated and has been achieved only indirectly. Examples include the mechanical transport of atoms carrying quantum information or the use of spontaneously emitted photons. Here we achieve direct coupling between the motional dipoles of separately trapped ions over a distance of 54 micrometres, using the dipole-dipole interaction as a quantum mechanical transmission line. This interaction is small between single trapped ions, but the coupling is amplified by using additional trapped ions as antennae. With three ions in each well, the interaction is increased by a factor of seven compared to the single-ion case. This enhancement facilitates bridging of larger distances and relaxes the constraints on the miniaturization of trap electrodes. The system provides a building block for quantum computers and opportunities for coupling different types of quantum systems.     

Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode

Optical laser fields have been widely used to achieve quantum control over the motional and internal degrees of freedom of atoms and ions, molecules and atomic gases. A route to controlling the quantum states of macroscopic mechanical oscillators in a similar fashion is to exploit the parametric coupling between optical and mechanical degrees of freedom through radiation pressure in suitably engineered optical cavities. If the optomechanical coupling is 'quantum coherent'--that is, if the coherent coupling rate exceeds both the optical and the mechanical decoherence rate--quantum states are transferred from the optical field to the mechanical oscillator and vice versa. This transfer allows control of the mechanical oscillator state using the wide range of available quantum optical techniques. So far, however, quantum-coherent coupling of micromechanical oscillators has only been achieved using microwave fields at millikelvin temperatures. Optical experiments have not attained this regime owing to the large mechanical decoherence rates and the difficulty of overcoming optical dissipation. Here we achieve quantum-coherent coupling between optical photons and a micromechanical oscillator. Simultaneously, coupling to the cold photon bath cools the mechanical oscillator to an average occupancy of 1.7?±?0.1 motional quanta. Excitation with weak classical light pulses reveals the exchange of energy between the optical light field and the micromechanical oscillator in the time domain at the level of less than one quantum on average. This optomechanical system establishes an efficient quantum interface between mechanical oscillators and optical photons, which can provide decoherence-free transport of quantum states through optical fibres. Our results offer a route towards the use of mechanical oscillators as quantum transducers or in microwave-to-optical quantum links.     

Sideband cooling of micromechanical motion to the quantum ground state

The advent of laser cooling techniques revolutionized the study of many atomic-scale systems, fuelling progress towards quantum computing with trapped ions and generating new states of matter with Bose-Einstein condensates. Analogous cooling techniques can provide a general and flexible method of preparing macroscopic objects in their motional ground state. Cavity optomechanical or electromechanical systems achieve sideband cooling through the strong interaction between light and motion. However, entering the quantum regime--in which a system has less than a single quantum of motion--has been difficult because sideband cooling has not sufficiently overwhelmed the coupling of low-frequency mechanical systems to their hot environments. Here we demonstrate sideband cooling of an approximately 10-MHz micromechanical oscillator to the quantum ground state. This achievement required a large electromechanical interaction, which was obtained by embedding a micromechanical membrane into a superconducting microwave resonant circuit. To verify the cooling of the membrane motion to a phonon occupation of 0.34?±?0.05 phonons, we perform a near-Heisenberg-limited position measurement within (5.1?±?0.4)h/2π, where h is Planck's constant. Furthermore, our device exhibits strong coupling, allowing coherent exchange of microwave photons and mechanical phonons. Simultaneously achieving strong coupling, ground state preparation and efficient measurement sets the stage for rapid advances in the control and detection of non-classical states of motion, possibly even testing quantum theory itself in the unexplored region of larger size and mass. Because mechanical oscillators can couple to light of any frequency, they could also serve as a unique intermediary for transferring quantum information between microwave and optical domains.     

Microwave quantum logic gates for trapped ions

Control over physical systems at the quantum level is important in fields as diverse as metrology, information processing, simulation and chemistry. For trapped atomic ions, the quantized motional and internal degrees of freedom can be coherently manipulated with laser light. Similar control is difficult to achieve with radio-frequency or microwave radiation: the essential coupling between internal degrees of freedom and motion requires significant field changes over the extent of the atoms' motion, but such changes are negligible at these frequencies for freely propagating fields. An exception is in the near field of microwave currents in structures smaller than the free-space wavelength, where stronger gradients can be generated. Here we first manipulate coherently (on timescales of 20 nanoseconds) the internal quantum states of ions held in a microfabricated trap. The controlling magnetic fields are generated by microwave currents in electrodes that are integrated into the trap structure. We also generate entanglement between the internal degrees of freedom of two atoms with a gate operation suitable for general quantum computation; the entangled state has a fidelity of 0.76(3), where the uncertainty denotes standard error of the mean. Our approach, which involves integrating the quantum control mechanism into the trapping device in a scalable manner, could be applied to quantum information processing, simulation and spectroscopy.     

Decoherence of quantum superpositions through coupling to engineered reservoirs

The theory of quantum mechanics applies to closed systems. In such ideal situations, a single atom can, for example, exist simultaneously in a superposition of two different spatial locations. In contrast, real systems always interact with their environment, with the consequence that macroscopic quantum superpositions (as illustrated by the 'Schrodinger's cat' thought-experiment) are not observed. Moreover, macroscopic superpositions decay so quickly that even the dynamics of decoherence cannot be observed. However, mesoscopic systems offer the possibility of observing the decoherence of such quantum superpositions. Here we present measurements of the decoherence of superposed motional states of a single trapped atom. Decoherence is induced by coupling the atom to engineered reservoirs, in which the coupling and state of the environment are controllable. We perform three experiments, finding that the decoherence rate scales with the square of a quantity describing the amplitude of the superposition state.     

介观互感电容耦合双谐振电路在压缩真空态下的量子涨落

对介观互感电容耦合电路作双模耦合谐振子处理,将其量子化.通过三次幺正变换,将体系的哈密顿量对角化,给出了体系的本征能谱,研究了压缩真空态下回路中电荷和电流的量子涨落.     

Laser cooling of a nanomechanical oscillator into its quantum ground state

The simple mechanical oscillator, canonically consisting of a coupled mass-spring system, is used in a wide variety of sensitive measurements, including the detection of weak forces and small masses. On the one hand, a classical oscillator has a well-defined amplitude of motion; a quantum oscillator, on the other hand, has a lowest-energy state, or ground state, with a finite-amplitude uncertainty corresponding to zero-point motion. On the macroscopic scale of our everyday experience, owing to interactions with its highly fluctuating thermal environment a mechanical oscillator is filled with many energy quanta and its quantum nature is all but hidden. Recently, in experiments performed at temperatures of a few hundredths of a kelvin, engineered nanomechanical resonators coupled to electrical circuits have been measured to be oscillating in their quantum ground state. These experiments, in addition to providing a glimpse into the underlying quantum behaviour of mesoscopic systems consisting of billions of atoms, represent the initial steps towards the use of mechanical devices as tools for quantum metrology or as a means of coupling hybrid quantum systems. Here we report the development of a coupled, nanoscale optical and mechanical resonator formed in a silicon microchip, in which radiation pressure from a laser is used to cool the mechanical motion down to its quantum ground state (reaching an average phonon occupancy number of 0.85 ± 0.08). This cooling is realized at an environmental temperature of 20?K, roughly one thousand times larger than in previous experiments and paves the way for optical control of mesoscale mechanical oscillators in the quantum regime.     

Fock态下介观电容耦合阻尼双谐振RLC电路的量子涨落

将介观电容耦合阻尼电路作双模耦合阻尼谐振子处理,使其量子化,通过3次幺正变换,将体系的哈密顿量对角化,并给出了体系的本征能谱.研究了Fock态、真空态下回路中电荷和电流的量子涨落.     

压缩真空态的激发态下介观互感电容耦合双谐振电路的量子涨落

对介观互感电容耦合电路作双模耦合谐振子处理,将其量子化.通过三次幺正变换,将体系的哈密顿量对角化,给出了体系的本征能谱.研究了压缩真空的激发态下回路中电荷和电流的量子涨落.结果表明:这种量子涨落不仅与电路器件的参数、激发量子数、压缩参数有关,而且还和彼此的互感、电容耦合有关.     

压缩真空态的激发态下介观互感电容藕合双谐振电路的量子涨落

对介观互感电容耦合电路作双模耦合谐振子处理，将其量子化．通过三次幺正变换，将体系的哈密顿量对角化，给出了体系的本征能谱．研究了压缩真空的激发态下回路中电荷和电流的量子涨落．结果表明：这种量子涨落不仅与电路器件的参数、激发量子数、压缩参数有关，而且还和彼此的互感、电容耦合有关．     

Realization of the Cirac-Zoller controlled-NOT quantum gate

Quantum computers have the potential to perform certain computational tasks more efficiently than their classical counterparts. The Cirac-Zoller proposal for a scalable quantum computer is based on a string of trapped ions whose electronic states represent the quantum bits of information (or qubits). In this scheme, quantum logical gates involving any subset of ions are realized by coupling the ions through their collective quantized motion. The main experimental step towards realizing the scheme is to implement the controlled-NOT (CNOT) gate operation between two individual ions. The CNOT quantum logical gate corresponds to the XOR gate operation of classical logic that flips the state of a target bit conditioned on the state of a control bit. Here we implement a CNOT quantum gate according to the Cirac-Zoller proposal. In our experiment, two 40Ca+ ions are held in a linear Paul trap and are individually addressed using focused laser beams; the qubits are represented by superpositions of two long-lived electronic states. Our work relies on recently developed precise control of atomic phases and the application of composite pulse sequences adapted from nuclear magnetic resonance techniques.     

耦合谐振子的量子绝热捷径设计

量子绝热捷径技术旨在加快量子绝热慢过程, 已经被广泛用于原子的冷却、转移等量子信息处理过程. 研究耦合谐振子模型的量子绝热捷径设计及其热机应用, 基于耦合谐振子模型的量子不变量, 首先得到 Ermakov 方程, 然后反设计耦合谐振子频率, 最终加快绝热过程而不产生末态激发. 本工作为耦合谐振子的量子态操控及超绝热量子热机提供了新思路.     

具有坐标二阶和动量一阶耦合的两量子谐振子的演化

在Ｄａｎｔａｓ等人所作量子振子工作的基础上提出了二个相互耦合的量子谐振子的演化问题，通过引进几个算符和辅助函数，求出了问题的精确解。     

外加电场下谐振子量子线的电子拉曼散射

文章研究了外加电场下谐振子量子线的电子拉曼散射.利用有效质量近似,以GaAs材料为例,计算了微分散射截面,结果表明,谐振子频率和外加电场的强度对拉曼光谱均有影响.拉曼光谱的峰值随着外加电场强度的增大而增大;在恒定的外加电场下,与电场方向相同的谐振子频率只影响峰值的大小,而与电场方向垂直方向的谐振子频率还对峰值的位置有影...     

Circuit cavity electromechanics in the strong-coupling regime

Demonstrating and exploiting the quantum nature of macroscopic mechanical objects would help us to investigate directly the limitations of quantum-based measurements and quantum information protocols, as well as to test long-standing questions about macroscopic quantum coherence. Central to this effort is the necessity of long-lived mechanical states. Previous efforts have witnessed quantum behaviour, but for a low-quality-factor mechanical system. The field of cavity optomechanics and electromechanics, in which a high-quality-factor mechanical oscillator is parametrically coupled to an electromagnetic cavity resonance, provides a practical architecture for cooling, manipulation and detection of motion at the quantum level. One requirement is strong coupling, in which the interaction between the two systems is faster than the dissipation of energy from either system. Here, by incorporating a free-standing, flexible aluminium membrane into a lumped-element superconducting resonant cavity, we have increased the single-photon coupling strength between these two systems by more than two orders of magnitude, compared to previously obtained coupling strengths. A parametric drive tone at the difference frequency between the mechanical oscillator and the cavity resonance dramatically increases the overall coupling strength, allowing us to completely enter the quantum-enabled, strong-coupling regime. This is evidenced by a maximum normal-mode splitting of nearly six bare cavity linewidths. Spectroscopic measurements of these 'dressed states' are in excellent quantitative agreement with recent theoretical predictions. The basic circuit architecture presented here provides a feasible path to ground-state cooling and subsequent coherent control and measurement of long-lived quantum states of mechanical motion.     

有限自由度谐振子系统的力学问题

研究了有限自由度的谐振子系统，先给出了它的运动方程，并对其求解，又导出了它的运动方程的拉格朗日形式和哈密顿形式，将谐振子系统量子化后，由海森伯方程也可得出其运动方程。最后把经典力学和量子力学同时运用到谐振子系统的多种状态并且把它们推广到n维，从而验证了海森伯方程和哈密顿正则方程对谐振子系统的等价性。     

三维各向同性谐振子的升、降算子

本文利用在粒子数表象中三维各向同性谐振子的升、降算子、推导出在坐标表象中三维各向同性谐振子的升、降算子并求出对三维各向性谐振的三个量子数（N,l,m)的升、降算子。     

具有q^4的非谐振子与谐振子的坐标二阶耦合系统的演化

在Ｄａｎｔａｓｗ．Ｃａｒｕｓｏｔｔｏ等人单一非谐振子演化研究的基础上提出了两个量子振子的耦合问题,并讨论了具有ｑ＾４的非谐振子与谐振子的坐标二阶耦合系统的演化。通过引进适当的算符和辅助函数,求出了问题的精确解。     

Nanometre-scale displacement sensing using a single electron transistor

It has been a long-standing goal to detect the effects of quantum mechanics on a macroscopic mechanical oscillator. Position measurements of an oscillator are ultimately limited by quantum mechanics, where 'zero-point motion' fluctuations in the quantum ground state combine with the uncertainty relation to yield a lower limit on the measured average displacement. Development of a position transducer, integrated with a mechanical resonator, that can approach this limit could have important applications in the detection of very weak forces, for example in magnetic resonance force microscopy and a variety of other precision experiments. One implementation that might allow near quantum-limited sensitivity is to use a single electron transistor (SET) as a displacement sensor: the exquisite charge sensitivity of the SET at cryogenic temperatures is exploited to measure motion by capacitively coupling it to the mechanical resonator. Here we present the experimental realization of such a device, yielding an unequalled displacement sensitivity of 2 x 10(-15) m x Hz(-1/2) for a 116-MHz mechanical oscillator at a temperature of 30 mK-a sensitivity roughly a factor of 100 larger than the quantum limit for this oscillator.     

介观阻尼LC并联电路的量子化

借鉴阻尼谐振子的量子力学处理的研究思想，将介观阻尼LC并联电路量子化，在此基础上研究了真空态下，各支路电流和电压的量子涨落．结果表明，各支路电流电压的量子涨落均与电路器件的参数有关，且随时间衰减．