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.     

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.     

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.     

Coupled quantized mechanical oscillators

The harmonic oscillator is one of the simplest physical systems but also one of the most fundamental. It is ubiquitous in nature, often serving as an approximation for a more complicated system or as a building block in larger models. Realizations of harmonic oscillators in the quantum regime include electromagnetic fields in a cavity and the mechanical modes of a trapped atom or macroscopic solid. Quantized interaction between two motional modes of an individual trapped ion has been achieved by coupling through optical fields, and entangled motion of two ions in separate locations has been accomplished indirectly through their internal states. However, direct controllable coupling between quantized mechanical oscillators held in separate locations has not been realized previously. Here we implement such coupling through the mutual Coulomb interaction of two ions held in trapping potentials separated by 40?μm (similar work is reported in a related paper). By tuning the confining wells into resonance, energy is exchanged between the ions at the quantum level, establishing that direct coherent motional coupling is possible for separately trapped ions. The system demonstrates a building block for quantum information processing and quantum simulation. More broadly, this work is a natural precursor to experiments in hybrid quantum systems, such as coupling a trapped ion to a quantized macroscopic mechanical or electrical oscillator.     

Self-cooling of a micromirror by radiation pressure

Cooling of mechanical resonators is currently a popular topic in many fields of physics including ultra-high precision measurements, detection of gravitational waves and the study of the transition between classical and quantum behaviour of a mechanical system. Here we report the observation of self-cooling of a micromirror by radiation pressure inside a high-finesse optical cavity. In essence, changes in intensity in a detuned cavity, as caused by the thermal vibration of the mirror, provide the mechanism for entropy flow from the mirror's oscillatory motion to the low-entropy cavity field. The crucial coupling between radiation and mechanical motion was made possible by producing free-standing micromirrors of low mass (m approximately 400 ng), high reflectance (more than 99.6%) and high mechanical quality (Q approximately 10,000). We observe cooling of the mechanical oscillator by a factor of more than 30; that is, from room temperature to below 10 K. In addition to purely photothermal effects we identify radiation pressure as a relevant mechanism responsible for the cooling. In contrast with earlier experiments, our technique does not need any active feedback. We expect that improvements of our method will permit cooling ratios beyond 1,000 and will thus possibly enable cooling all the way down to the quantum mechanical ground state of the micromirror.     

Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip

An optical cavity enhances the interaction between atoms and light, and the rate of coherent atom-photon coupling can be made larger than all decoherence rates of the system. For single atoms, this 'strong coupling regime' of cavity quantum electrodynamics has been the subject of many experimental advances. Efforts have been made to control the coupling rate by trapping the atom and cooling it towards the motional ground state; the latter has been achieved in one dimension so far. For systems of many atoms, the three-dimensional ground state of motion is routinely achieved in atomic Bose-Einstein condensates (BECs). Although experiments combining BECs and optical cavities have been reported recently, coupling BECs to cavities that are in the strong-coupling regime for single atoms has remained an elusive goal. Here we report such an experiment, made possible by combining a fibre-based cavity with atom-chip technology. This enables single-atom cavity quantum electrodynamics experiments with a simplified set-up and realizes the situation of many atoms in a cavity, each of which is identically and strongly coupled to the cavity mode. Moreover, the BEC can be positioned deterministically anywhere within the cavity and localized entirely within a single antinode of the standing-wave cavity field; we demonstrate that this gives rise to a controlled, tunable coupling rate. We study the heating rate caused by a cavity transmission measurement as a function of the coupling rate and find no measurable heating for strongly coupled BECs. The spectrum of the coupled atoms-cavity system, which we map out over a wide range of atom numbers and cavity-atom detunings, shows vacuum Rabi splittings exceeding 20 gigahertz, as well as an unpredicted additional splitting, which we attribute to the atomic hyperfine structure. We anticipate that the system will be suitable as a light-matter quantum interface for quantum information.     

Sub-kelvin optical cooling of a micromechanical resonator

Micromechanical resonators, when cooled down to near their ground state, can be used to explore quantum effects such as superposition and entanglement at a macroscopic scale. Previously, it has been proposed to use electronic feedback to cool a high frequency (10 MHz) resonator to near its ground state. In other work, a low frequency resonator was cooled from room temperature to 18 K by passive optical feedback. Additionally, active optical feedback of atomic force microscope cantilevers has been used to modify their response characteristics, and cooling to approximately 2 K has been measured. Here we demonstrate active optical feedback cooling to 135 +/- 15 mK of a micromechanical resonator integrated with a high-quality optical resonator. Additionally, we show that the scheme should be applicable at cryogenic base temperatures, allowing cooling to near the ground state that is required for quantum experiments--near 100 nK for a kHz oscillator.     

Radiation-pressure cooling and optomechanical instability of a micromirror

Recent table-top optical interferometry experiments and advances in gravitational-wave detectors have demonstrated the capability of optical interferometry to detect displacements with high sensitivity. Operation at higher powers will be crucial for further sensitivity enhancement, but dynamical effects caused by radiation pressure on the interferometer mirrors must be taken into account, and the appearance of optomechanical instabilities may jeopardize the stable operation of the next generation of interferometers. These instabilities are the result of a nonlinear coupling between the motion of the mirrors and the optical field, which modifies the effective dynamics of the mirror. Such 'optical spring' effects have already been demonstrated for the mechanical damping of an electromagnetic waveguide with a moving wall, the resonance frequency of a specially designed flexure oscillator, and the optomechanical instability of a silica microtoroidal resonator. Here we present an experiment where a micromechanical resonator is used as a mirror in a very high-finesse optical cavity, and its displacements are monitored with unprecedented sensitivity. By detuning the laser frequency with respect to the cavity resonance, we have observed a drastic cooling of the microresonator by intracavity radiation pressure, down to an effective temperature of 10 kelvin. For opposite detuning, efficient heating is observed, as well as a radiation-pressure-induced instability of the resonator. Further experimental progress and cryogenic operation may lead to the experimental observation of the quantum ground state of a micromechanical resonator, either by passive or active cooling techniques.     

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.     

Low-temperature shear modulus changes in solid 4He and connection to supersolidity

Superfluidity--liquid flow without friction--is familiar in helium. The first evidence for 'supersolidity', its analogue in quantum solids, came from torsional oscillator measurements involving 4He. At temperatures below 200 mK, the torsional oscillator frequencies increased, suggesting that some of the solid decoupled from the oscillator. This behaviour has been replicated by several groups, but solid 4He does not respond to pressure differences, and persistent currents and other signatures of superflow have not been seen. Both experiments and theory indicate that defects are involved; these should also affect the solid's mechanical behaviour. Here we report a measurement of the shear modulus of solid 4He at low frequencies and strains. We observe large increases below 200 mK, with the same dependence on measurement amplitude, 3He impurity concentration and annealing as the decoupling seen in the torsional oscillator experiments. We explain this unusual elastic behaviour in terms of a dislocation network that is pinned by 3He at the lowest temperatures but becomes mobile above 100 mK. The frequency changes in the torsional oscillator experiments appear to be related to the motion of these dislocations, perhaps by disrupting a possible supersolid state.     

Energy spectra of quantum rings.

Quantum mechanical experiments in ring geometries have long fascinated physicists. Open rings connected to leads, for example, allow the observation of the Aharonov-Bohm effect, one of the best examples of quantum mechanical phase coherence. The phase coherence of electrons travelling through a quantum dot embedded in one arm of an open ring has also been demonstrated. The energy spectra of closed rings have only recently been studied by optical spectroscopy. The prediction that they allow persistent current has been explored in various experiments. Here we report magnetotransport experiments on closed rings in the Coulomb blockade regime. Our experiments show that a microscopic understanding of energy levels, so far limited to few-electron quantum dots, can be extended to a many-electron system. A semiclassical interpretation of our results indicates that electron motion in the rings is governed by regular rather than chaotic motion, an unexplored regime in many-electron quantum dots. This opens a way to experiments where even more complex structures can be investigated at a quantum mechanical level.     

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。     

简谐振子波函数的代数解及Hermite多项式的递推

简谐振子模型是量子力学中极其简单又重要的模型,其物理思想在其他相关的学科中都有着广泛的应用,通过多种途径去深入理解简谐振子模型,对理解量子力学的实质和运用量子力学作为工具去研究微观物理模型都有重要的意义;另一方面在实际工作中应用代数方法去求解力学量的本征值和波函数是研究量子力学的主要手段.以简谐振子为例,运用代数方法,先给出一维简谐振子的波函数,从而推广到多维简谐振子,并结合相应算符的对易关系给出Hermite多项式及其递推关系,回避了通过级数展开去求解Hermite方程的过程;同时指出<厄米本征值问题的探究>一文中的不足之处.     

Laser cooling and real-time measurement of the nuclear spin environment of a solid-state qubit

Control over quantum dynamics of open systems is one of the central challenges in quantum science and engineering. Coherent optical techniques, such as coherent population trapping involving dark resonances, are widely used to control quantum states of isolated atoms and ions. In conjunction with spontaneous emission, they allow for laser cooling of atomic motion, preparation and manipulation of atomic states, and rapid quantum optical measurements that are essential for applications in metrology. Here we show that these techniques can be applied to monitor and control individual atom-like impurities, and their local environment, in the solid state. Using all-optical manipulation of the electronic spin of an individual nitrogen-vacancy colour centre in diamond, we demonstrate optical cooling, real-time measurement and conditional preparation of its nuclear spin environment by post-selection. These methods offer potential applications ranging from all-optical nanomagnetometry to quantum feedback control of solid-state qubits, and may lead to new approaches for quantum information storage and processing.     

Generation of Fock states in a superconducting quantum circuit

Spin systems and harmonic oscillators comprise two archetypes in quantum mechanics. The spin-1/2 system, with two quantum energy levels, is essentially the most nonlinear system found in nature, whereas the harmonic oscillator represents the most linear, with an infinite number of evenly spaced quantum levels. A significant difference between these systems is that a two-level spin can be prepared in an arbitrary quantum state using classical excitations, whereas classical excitations applied to an oscillator generate a coherent state, nearly indistinguishable from a classical state. Quantum behaviour in an oscillator is most obvious in Fock states, which are states with specific numbers of energy quanta, but such states are hard to create. Here we demonstrate the controlled generation of multi-photon Fock states in a solid-state system. We use a superconducting phase qubit, which is a close approximation to a two-level spin system, coupled to a microwave resonator, which acts as a harmonic oscillator, to prepare and analyse pure Fock states with up to six photons. We contrast the Fock states with coherent states generated using classical pulses applied directly to the resonator.     

Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system

The field of cavity quantum electrodynamics (QED), traditionally studied in atomic systems, has gained new momentum by recent reports of quantum optical experiments with solid-state semiconducting and superconducting systems. In cavity QED, the observation of the vacuum Rabi mode splitting is used to investigate the nature of matter-light interaction at a quantum-mechanical level. However, this effect can, at least in principle, be explained classically as the normal mode splitting of two coupled linear oscillators. It has been suggested that an observation of the scaling of the resonant atom-photon coupling strength in the Jaynes-Cummings energy ladder with the square root of photon number n is sufficient to prove that the system is quantum mechanical in nature. Here we report a direct spectroscopic observation of this characteristic quantum nonlinearity. Measuring the photonic degree of freedom of the coupled system, our measurements provide unambiguous spectroscopic evidence for the quantum nature of the resonant atom-field interaction in cavity QED. We explore atom-photon superposition states involving up to two photons, using a spectroscopic pump and probe technique. The experiments have been performed in a circuit QED set-up, in which very strong coupling is realized by the large dipole coupling strength and the long coherence time of a superconducting qubit embedded in a high-quality on-chip microwave cavity. Circuit QED systems also provide a natural quantum interface between flying qubits (photons) and stationary qubits for applications in quantum information processing and communication.     

Tavis-Cummings模型中的量子态平均保真度

运用全量子理论,研究了Tavis-Cummings模型中量子态平均保真度的演化特性,对原子和光场的初态以及两原子的关联程度对平均保真度的影响进行了研究.结果表明,原子、光场和系统的平均保真度依赖于初态时2个原子处在基态的几率以及光场处于真空态的几率;初态两原子在基态无关联,光场处于真空态,体系不失真.     

激子极化激元光子学研究进展

研究光与物质相互作用是腔量子电动力学的一个重要方向.早在20世纪50年代,黄昆先生就提出了固体环境中的光子与晶格连续作用的时间演化图像,并指出光子-声子时间上连续不断的相互转化会在物质中形成声子极化激元波,从理论上计算了声子极化激元波的色散关系.Hopfield把这种图像推广到半导体环境中的光子-激子作用上.随后人们在微腔中实现了单原子、单量子点激子的真空拉比振荡.随着半导体微腔生长和微纳加工工艺的提高,激子极化激元的凝聚、超流、涡旋等宏观量子态被实验证明.通过控制微腔结构和光场调控的手段,人们进一步实现了对宏观量子态的相干调控.有机半导体、钙钛矿、二维半导体等新材料体系展现了极大的激子束缚能,有望实现室温量子器件的制备.微腔激子极化激元的研究进入了黄金时代.本文首先从激子极化激元的基本图像入手,详细介绍激子极化激元的概念、色散关系以及常见的激子极化激元体系.其次,总结了研究微腔激子极化激元的材料体系和实验方法,详细介绍了平板微腔和微纳材料自构型微腔的工作原理和具体实例,以及共焦显微荧光光谱和角分辨荧光光谱.第三,对激子极化激元的量子调控进行了总结.详细介绍了激子极化激元的重要宏观量子态以及通过微纳加工和光场调控的方式对宏观量子态的操控.具体分析了两个量子态操控的实例,包括氧化锌超晶格中多重量子态的制备以及凝聚体的参量散射过程.第四,对新型材料中激子极化激元的研究进行了总结,包括二维半导体、有机半导体和钙钛矿.最后,对本文进行总结,并且从理论、实验的角度分别预测了该领域的发展趋势.     

Controlling cavity reflectivity with a single quantum dot

Solid-state cavity quantum electrodynamics (QED) systems offer a robust and scalable platform for quantum optics experiments and the development of quantum information processing devices. In particular, systems based on photonic crystal nanocavities and semiconductor quantum dots have seen rapid progress. Recent experiments have allowed the observation of weak and strong coupling regimes of interaction between the photonic crystal cavity and a single quantum dot in photoluminescence. In the weak coupling regime, the quantum dot radiative lifetime is modified; in the strong coupling regime, the coupled quantum dot also modifies the cavity spectrum. Several proposals for scalable quantum information networks and quantum computation rely on direct probing of the cavity-quantum dot coupling, by means of resonant light scattering from strongly or weakly coupled quantum dots. Such experiments have recently been performed in atomic systems and superconducting circuit QED systems, but not in solid-state quantum dot-cavity QED systems. Here we present experimental evidence that this interaction can be probed in solid-state systems, and show that, as expected from theory, the quantum dot strongly modifies the cavity transmission and reflection spectra. We show that when the quantum dot is coupled to the cavity, photons that are resonant with its transition are prohibited from entering the cavity. We observe this effect as the quantum dot is tuned through the cavity and the coupling strength between them changes. At high intensity of the probe beam, we observe rapid saturation of the transmission dip. These measurements provide both a method for probing the cavity-quantum dot system and a step towards the realization of quantum devices based on coherent light scattering and large optical nonlinearities from quantum dots in photonic crystal cavities.