Hao-Sheng Zeng, Le-Man Kuang, Ke-Lin Gao
The beam splitter and phase shifter, which are the key elements in the experiments of light interference, are realized in the motion of trapped ions. Some applications, such as the creation of quantum motional states and the realization of Mach-Zehnder interferometer, are illustrated. Several detection methods of motional states used in the interferometer are also discussed.
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We present a selective vibronic interaction for manipulating motional states in two trapped ions, acting resonantly on a previously chosen vibronic subspace and dispersively on all others. This is done respecting technical limitations on ionic laser individual addressing. We discuss the generation of Fock states and entanglement in the ionic collective motional degrees of freedom, among other applications.
We propose and demonstrate a scalable scheme for the simultaneous determination of internal and motional states in trapped ions with single-site resolution. The scheme is applied to the study of polaritonic excitations in the Jaynes- Cummings Hubbard model with trapped ions, in which the internal and motional states of the ions are strongly correlated. We observe quantum phase transitions of polaritonic excitations in two ions by directly evaluating their variances per ion site. Our work establishes an essential technological method for large-scale quantum simulations of polaritonic systems.
The decoherence of nonclassical motional states of a trapped $^9 {\rm Be^+}$ ion in a recent experiment is investigated theoretically. Sources of decoherence considered here destroy the characteristic coherent quantum dynamics of the system but do not cause energy dissipation. Here they are first introduced phenomenologically and then described using a microscopic Hamiltonian formulation. Theoretical predictions are compared to experimental results.
Individual electrodynamically trapped and laser cooled ions are addressed in frequency space using radio-frequency radiation in the presence of a static magnetic field gradient. In addition, an interaction between motional and spin states induced by an rf field is demonstrated employing rf-optical double resonance spectroscopy. These are two essential experimental steps towards realizing a novel concept for implementing quantum simulations and quantum computing with trapped ions.
Resonance fluorescence of a single trapped ion is spectrally analyzed using a heterodyne technique. Motional sidebands due to the oscillation of the ion in the harmonic trap potential are observed in the fluorescence spectrum. From the width of the sidebands the cooling rate is obtained and found to be in agreement with the theoretical prediction.
We present a scheme to generate an arbitrary two-dimensional quantum state of motion of a trapped ion. This proposal is based on a sequence of laser pulses, which are tuned appropriately to control transitions on the sidebands of two modes of vibration. Not more than $(M+1)(N+1)$ laser pulses are needed to generate a pure state with upper phonon number $M$ and $N$ in the $x$ and $y$ direction respectively.
We present two methods for characterization of motional-mode configurations that are generally applicable to the weak and strong-binding limit of single or multiple trapped atomic ions. Our methods are essential to realize control of the individual as well as the common motional degrees of freedom. In particular, when implementing scalable radio-frequency trap architectures with decreasing ion-electrode distances, local curvatures of electric potentials need to be measured and adjusted precisely, e.g., to tune phonon tunneling and control effective spin-spin interaction. We demonstrate both me…
Non-Gaussian states represent a powerful resource for quantum information protocols in the continuous variables regime. Cat states, in particular, have been produced in the motional degree of freedom of trapped ions by controlled displacements dependent on the ionic internal state. An alternative method harnesses the Kerr nonlinearity naturally existent in this kind of system. We present detailed calculations confirming its feasibility for typical experimental conditions. Additionally, this method permits the generation of complex non-Gaussian states with negative Wigner functions. Especially,…
To achieve high-fidelity operations on a large-scale quantum computer, the parameters of the physical system must be efficiently characterized with high accuracy. For trapped ions, the entanglement between qubits are mediated by the motional modes of the ion chain, and thus characterizing the motional-mode parameters becomes essential. In this paper, we develop and explore physical models that accurately predict both magnitude and sign of the Lamb-Dicke parameters when the modes are probed {\it in parallel}. We further devise an advanced characterization protocol that shortens the characteriza…
Generally noise-resilient quantum gates for trapped-ions
提出一种针对囚禁离子链的纠缠门方案，在多种误差源下利用激发态运动实现高保真度操作，包含所有运动模式并增强对加热效应和失谐误差的鲁棒性，助力构建可扩展量子计算机。
We present a method for the deterministic generation of all the electronic Bell states of two trapped ions. It involves the combination of a purely dispersive with a resonant laser excitation of vibronic transitions of the ions. In contrast to other methods presented up to now, our proposal does not require differential laser addressing of the individual ions and may be easily implemented with present available techniques. It is further shown that this excitation scheme is highly adequate for the complete determination of the motional state of the ions.
Quantum coherence is one of the clearest departures from classical physics, exhibited when a system is in a superposition of different basis states. Here the coherent superposition of three motional Fock states of a single trapped ion is experimentally certified, with a procedure provably robust against imperfect operation. As the motional state cannot be directly interrogated, our scheme uses an interference pattern generated by projective measurement of the coupled qubit state. The minimum number of coherently superposed states is inferred from a series of threshold values based on analysis …
Motional state analysis of a trapped ion by ultra-narrowband composite pulses
本文提出利用超窄带复合脉冲扫描蓝边带跃迁，实现对囚禁离子运动态的无先验测量，适用于Lamb-Dicke区内外，并演示了单离子实验验证。
Favored schemes for trapped-ion quantum logic gates use bichromatic laser fields to couple internal qubit states with external motion through a "spin-dependent force." We introduce a new degree of freedom in this coupling that reduces its sensitivity to phase decoherence. We demonstrate bichromatic spin-dependent forces on a single trapped $^{111}$Cd$^+$ ion, and show that phase coherence of the resulting "Schrödinger-cat" states of motion depends critically upon the spectral arrangement of the optical fields. This applies directly to the operation of entangling gates on multiple ions.
Entangling gates in trapped-ion quantum computing have primarily targeted stationary ions with initial motional distributions that are thermal and close to the ground state. However, future systems will likely incur significant non-thermal excitation due to, e.g., ion transport, longer operational times, and increased spatial extent of the trap array. In this paper, we analyze the impact of such coherent motional excitation on entangling-gate error by performing simulations of Molmer-Sorenson (MS) gates on a pair of trapped-ion qubits with both thermal and coherent excitation present in a shar…
Coherent coupling and non-destructive measurement of trapped-ion mechanical oscillators
本研究通过振荡电势实现了囚禁离子晶体中不同运动模式间单量子的相干交换，演示了高保真度量子态转移、分束器操作、模式纠缠及非破坏性投影测量，提升了其在连续变量量子计算与纠错中的适用性。
Directional Motional Control via Engineered Conical Intersections in Trapped Rydberg Ions
本文利用量子最优控制技术，通过在囚禁里德堡离子中设计锥形交叉来打破动力学对称性，实现了高保真度的方向性运动波包控制，为量子信息处理中的门操作提供了新资源。
We investigate theoretical decoherence effects of the motional degrees of freedom of a single trapped atomic/ionic electronically coded qubit. For single bit rotations from a resonant running wave laser field excitation, we found the achievable fidelity to be determined by a single parameter characterized by the motional states. Our quantitative results provide a useful realistic view for current experimental efforts in quantum information and computing.
A method for gaining information about the phonon-number moments and the generalized nonlinear and linear quadratures in the motion of trapped ions (in particular, position and momentum) is proposed, valid inside and outside the Lamb-Dicke regime. It is based on the measurement of first time derivatives of electronic populations, evaluated at the motion-probe interaction time t=0. In contrast to other state-reconstruction proposals, based on measuring Rabi oscillations or dispersive interactions, the present scheme can be performed resonantly at infinitesimal short motion-probe interaction tim…
In this paper we propose a scheme to model the phonon analog of optical elements, including a polarizing beam splitter, a half- wave plate, and a quarter- wave plate, as well as an implementation of CNOT and Pauli gates, by using two atoms confined in a 2D plane. The internal states of the atoms are taken to be Rydberg circular states. Using this model we can manipulate the motional state of the atom, with possible applications in optomechanical integrated circuits for quantum information processing and quantum simulation. Towards this aim, we study the interaction of two trapped atoms with tw…
Quantum manipulation of individual phonons could offer new resources for studying fundamental physics and creating an innovative platform in quantum information science. Here, we propose to generate quantum states of strongly correlated phonon bundles associated with the motion of a trapped atom. Our scheme operates in the atom-phonon resonance regime where the energy spectrum exhibits strong anharmonicity such that energy eigenstates with different phonon numbers can be well-resolved in the parameter space. Compared to earlier schemes operating in the far dispersive regime, the bundle states …
Nonlinear Coupling between Motional Modes in Trapped Ion Quantum Processors
本文针对离子阱量子处理器中运动模式间的非线性耦合问题，建立了包含三阶库仑项的通用模型，分析了其对Molmer-Sorensen门保真度的影响，并提出了通过失谐、调节阱各向异性和优化门波形来抑制误差的设计规则。
We describe and analyse numerically schemes (i) for entangling orthogonal motional modes of one or a few harmonically-trapped atoms or ions, and (ii) for transferring the entanglement from one of these local modes to a distant trapped atom (or atoms) via a light-mediated quantum state transfer procedure proposed in previous work [A.S. Parkins and H.J. Kimble, J. Opt. B: Quantum Semiclass. Opt. 1, 496 (1999)]. Possibilities arising from these schemes include the generation of an Einstein-Podolsky-Rosen state in the positions and momenta of distantly-separated trapped atoms and the preparation o…
We have demonstrated transport of Be+ ions through a 2D Paul-trap array that incorporates an X-junction, while maintaining the ions near the motional ground-state of the confining potential well. We expand on the first report of the experiment [1], including a detailed discussion of how the transport potentials were calculated. Two main mechanisms that caused motional excitation during transport are explained, along with the methods used to mitigate such excitation. We reduced the motional excitation below the results in Ref. [1] by a factor of approximately 50. The effect of a mu-metal shield…
A simple technique for robust generation of vibrational Fock states in a chain of trapped ions is proposed. The method is fast and easy to implement, since only a single chirped laser pulse, simultaneously addressing all of the ions, is required. Furthermore, because the approach uses collective adiabatic passage, significant fluctuations in the intensity or frequency of the laser pulse can be tolerated, and the technique performs well even on the border of the Lamb-Dicke regime. We also demonstrate how this technique may be extended in order to create non-classical superposition states of the…
First order Doppler effects are usually ignored in laser driven trapped ions when the recoil frequency is much smaller than the trapping frequency (Lamb-Dicke regime). This means that the central, carrier excitation band is supposed to be unaffected by vibronic transitions in which the vibrational number changes. While this is strictly true in the Lamb-Dicke limit (infinitely tight confinement), the vibronic transitions do play a role in the Lamb-Dicke regime. In this paper we quantify the asymptotic behaviour of their effect with respect to the Lamb-Dicke parameter. In particular, we give ana…
Remote Gottesman-Kitaev-Preskill Entanglement Generation between Trapped Ions
该研究提出一种预示式协议，利用囚禁离子的运动模式与光子接口，实现远程GKP量子比特间的纠缠生成，在实际场景中可实现超95%的逻辑贝尔对保真度，结合了已验证的实验操作。
Robust Nonperturbative Trapped-Ion Quantum Logic
研究囚禁离子量子逻辑中因弱耦合和谐波动力学对门时间有限制的问题，通过设计最优驱动方案，利用非谐动力学实现快速高保真纠缠门，并增强对系统缺陷的抗性。
Characterization and cancellation of power-line-induced motional-mode frequency noise in a trapped-ion system
研究60 Hz电源线噪声对离子阱量子计算中运动模式频率的影响，通过自旋回波拉姆齐光谱表征并利用补偿信号实现近完全抑制，将相干时间从10 ms延长至35 ms。
High-fidelity two-qubit quantum logic gates in a trapped-ion chain using axial motional modes
本研究通过优化轴向运动模式，实现了高保真度的双量子比特逻辑门，提升了离子阱量子处理器的可扩展性和应用潜力。
Efficient preparation and detection of the motional state of trapped ions is important in many experiments ranging from quantum computation to precision spectroscopy. We investigate the stimulated Raman adiabatic passage (STIRAP) technique for the manipulation of motional states in a trapped ion system. The presented technique uses a Raman coupling between two hyperfine ground states in $^{25}$Mg$^+$, implemented with delayed pulses, which removes a single phonon independent of the initial motional state. We show that for a thermal state the STIRAP population transfer is more efficient than a …
Trapped atomic ions are a versatile platform for studying interactions between spins and bosons by coupling the internal states of the ions to their motion. Measurement of complex motional states with multiple modes is challenging, because all motional state populations can only be measured indirectly through the spin state of ions. Here we present a general method to determine the Fock state distributions and to reconstruct the density matrix of an arbitrary multi-mode motional state. We experimentally verify the method using different entangled states of multiple radial modes in a 5-ion chai…
We consider a system of laser-cooled ions in a linear harmonic trap and study the phenomenon of squeezing exchange between their internal and motional degrees of freedom. An interesting relation between the quantum noise reduction (squeezing) and the dynamical evolution is found when the internal and motional subsystems are prepared in properly squeezed (intelligent) states. Specifically, the evolution of the system is fully governed by the relative strengths of spectroscopic and motional squeezing, including the phenomenon of total cancellation of the interaction when the initial squeezing pa…
We present a new method of spin-motion coupling for trapped ions using microwaves and a magnetic field gradient oscillating close to the ions' motional frequency. We demonstrate and characterize this coupling experimentally using a single ion in a surface-electrode trap that incorporates current-carrying electrodes to generate the microwave field and the oscillating magnetic field gradient. Using this method, we perform resolved-sideband cooling of a single motional mode to its ground state.
Transport, separation, and merging of trapped ion crystals are essential operations for most large-scale quantum computing architectures. In this work, we develop a theoretical framework that describes the dynamics of ions in time-varying potentials with a motional squeeze operator, followed by a motional displacement operator. Using this framework, we develop a new, general protocol for trapped ion transport, separation, and merging. We show that motional squeezing can prepare an ion wave packet to enable transfer from the ground state of one trapping potential to another. The framework and p…
We present a study of the coherence properties of a variety of motional states of a single ion confined in a Penning ion trap. We demonstrate that the motion of the ion has a coherence time of the order of one second, using Ramsey interferometry. We introduce a technique for preparing the ion in an incoherent superposition of highly-excited motional states using a simple modification of optical sideband cooling. Coherent manipulation of these states allow measurements of optical and motional coherence to be carried out. We show that these highly-excited motional state superpositions have long …
We present a family of generalized unitary transformations that simplifies the Hamiltonian for a harmonically trapped two level atom (or ion) interacting with a plane wave laser field. Novel near resonant single as well as double vibrational phonon dynamical regimes are found. The validity condition of the often used motional rotating wave approximation (MRWA) is examined both numerically and analytically. Large errors are found within typical regimes of MRWA with respect to the motional degrees of freedom. The effects of MRWA in trapped ion systems are shown to be opposite to that of the rota…
Modular mixed-species ion-trap networks are a promising framework for scalable quantum information processing, where one species acts as a memory qubit and another as a communication qubit. This architecture requires high-fidelity mixed-species entangling gates to transfer information from communication to memory qubits through their collective motion. We investigate the character of the motional modes of a mixed-species ion chain for entangling operations and also sympathetic cooling. We find that the laser power required for high-fidelity entangling gates based on transverse modes is at leas…
Adiabatically controlled motional states of a ground-state cooled CaO$^{+}$ and Ca$^{+}$ trapped ion chain
本研究针对囚禁分子离子外部自由度调控需求，利用共囚禁Ca⁺协同边带冷却实现CaO⁺轴向运动模近基态冷却，验证绝热调谐模式频率时声子态可保持，为分子偶极-声子相互作用研究奠定基础。
Interfacing ion qubits in separate traps is among the challenges towards scaling up ion quantum computing. This theoretical study focuses on using a conducting wire to couple the motional quantum states of ions in separate planar traps. This approach of interfacing ion traps provides an alternative to coupling distant qubits with lasers. We include the effects of $1/f^{\tildeα}$ (Anomalous) surface heating noise, using aggregate and recent experimental findings as the basis for an analytical model of the motional state decoherence time $t_{\mathrm{deco.}}$. Our optimized design for the couplin…
We present a reliable scheme for engineering arbitrary motional ionic states through an adaptation of the projection synthesis technique for trapped-ion phenomena. Starting from a prepared coherent motional state, the Wigner function of the desired state is thus sculpted from a Gaussian distribution. The engineering process has also been developed to take into account the errors arising from intensity fluctuations in the exciting-laser pulses required for manipulating the electronic and vibrational states of the trapped ion. To this end, a recently developed phenomenological-operator approach …
In this work, we derive analytic formulae that determine the effect of error mechanisms on one- and two-qubit gates in trapped ions and electrons. First, we analyze, and derive expressions for, the effect of driving field inhomogeneities on one-qubit gate fidelities. Second, we derive expressions for two-qubit gate errors, including static motional frequency shifts, trap anharmonicities, field inhomogeneities, heating, and motional dephasing. We show that, for small errors, each of our expressions for infidelity converges to its respective numerical simulation; this shows our formulae are suff…
Pulse optimization for high-precision motional-mode characterization in trapped-ion quantum computers
针对离子阱量子计算机中运动模式表征受交叉模式耦合限制的问题，提出利用脉冲设计自由度主动抑制该效应，并设计稳定化方法准确表征Lamb-Dicke参数，在三离子链模拟中验证了塑形脉冲的优越性。
We implement faster-than-adiabatic two-qubit phase gates using smooth state-dependent forces. The forces are designed to leave no final motional excitation, independently of the initial motional state in the harmonic, small-oscillations limit. They are simple, explicit functions of time and the desired logical phase of the gate, and are based on quadratic invariants of motion and Lewis-Riesenfeld phases of the normal modes.
The dominant error sources for state-of-the-art laser-free trapped-ion entangling gates are decoherence of the qubit state and the ion motion. The effect of these decoherence mechanisms can be suppressed with additional control fields, or with techniques that have the disadvantage of reducing gate speed. Here, we propose using a near-motional-frequency magnetic field gradient to implement a laser-free gate that is simultaneously resilient to both types of decoherence, does not require additional control fields, and has a relatively small cost in gate speed.
We propose a generation method of Bell-type states involving light and the vibrational motion of a single trapped ion. The trap itself is supposed to be placed inside a high-$Q$ cavity sustaining a single mode, quantized electromagnetic field. Entangled light-motional states may be readily generated if a conditional measurement of the ion's internal electronic state is made after an appropriate interaction time and a suitable preparation of the initial state. We show that all four Bell states may be generated using different motional sidebands (either blue or red), as well as adequate ionic re…
Electric noise from metallic surfaces is a major obstacle towards quantum applications with trapped ions due to motional heating of the ions. Here, we discuss how the same noise source can also lead to pure dephasing of motional quantum states. The mechanism is particularly relevant at small ion-surface distances, thus imposing a new constraint on trap miniaturization. By means of a free induction decay experiment, we measure the dephasing time of the motion of a single ion trapped 50~$μ$m above a Cu-Al surface. From the dephasing times we extract the integrated noise below the secular frequen…
We study cooling of the collective vibrational motion of two 138Ba+ ions confined in an electrodynamic trap and irradiated with laser light close to the resonances S_1/2-P_1/2 (493 nm) and P_1/2-D_3/2 (650 nm). The motional state of the ions is monitored by a spatially resolving photo multiplier. Depending on detuning and intensity of the cooling lasers, macroscopically different motional states corresponding to different ion temperatures are observed. We also derive the ions' temperature from detailed analytical calculations of laser cooling taking into account the Zeeman structure of the ene…
In this work, we design and implement frequency-robust Molmer-Sorensen gates on a linear chain of trapped ions, using Gaussian amplitude modulation and a constant laser frequency. We select this frequency to balance the entanglement accumulation of all motional modes during the gate to produce a strong robustness to frequency error, even for long ion chains. We demonstrate this technique on a three-ion chain, achieving $<\,1\%$ reduction from peak fidelity over a $20\,$kHz range of frequency offset, and we analyze the performance of this gate design through numerical simulations on chains of t…
We propose a new scheme for supplying voltages to the electrodes of microfabricated ion traps, enabling access to a regime in which changes to the trapping potential are made on timescales much shorter than the period of the secular oscillation frequencies of the trapped ions. This opens up possibilities for speeding up the transport of ions in segmented ion traps and also provides access to control of multiple ions in a string faster than the Coulomb interaction between them. We perform a theoretical study of ion transport using these methods in a surface-electrode trap, characterizing the pr…
A protocol is provided to reconstruct the Wigner function for the motional state of a trapped ion via fluorescence detection on another ion in the same trap. This "sympathetic tomography" of a dark ion without optical transitions suitable for state measurements is based on the mapping of its motional state onto one of the collective modes of the ion pair. The quantum state of this vibrational eigenmode is subsequently measured through sideband excitation of the bright ion. Physical processes to implement the desired state transfer and read-out are derived, and the accomplishment of the scheme …
We have measured motional heating rates of trapped atomic ions, a factor that can influence multi-ion quantum logic gate fidelities. Two simplified techniques were developed for this purpose: one relies on Raman sideband detection implemented with a single laser source, while the second is even simpler and is based on time-resolved fluorescence detection during Doppler recooling. We applied these methods to determine heating rates in a microfrabricated surface-electrode trap made of gold on fused quartz, which traps ions 40 microns above its surface. Heating rates obtained from the two techniq…
We show that the motion of a cold trapped ion can be squeezed by modulating the intensity of a phase-stable optical lattice placed inside the trap. As this method is reversible and state selective it effectively implements a controlled-squeeze gate. We show how to use this resource, that can be useful for quantum information processing with continuous variables, in order to prepare coherent superpositions of states which are squeezed along complementary quadratures. We show that these states, which we denote as "${\mathcal X}$-states", exhibit high sensitivity to small displacements along two …
We propose the implementation of selective interactions of atom-motion subspaces in trapped ions. These interactions yield resonant exchange of population inside a selected subspace, leaving the others in a highly dispersive regime. Selectivity allows us to generate motional Fock (and other nonclassical) states with high purity out of a wide class of initial states, and becomes an unconventional cooling mechanism when the ground state is chosen. Individual population of number states can be distinctively measured, as well as the motional Wigner function. Furthermore, a protocol for implementin…
We present a method for measuring the polarizability of Rydberg ions confined in the harmonic potential of a Paul trap. For a highly excited electronic state, the coupling between the electronic wave function and the trapping field modifies the excitation probability depending on the motional state of the ion. This interaction strongly depends on the polarizability of the excited state and manifests itself in the state-dependent secular frequencies of the ion. We initialize a single trapped $^{40}$Ca$^+$ ion from the motional ground state into coherent states with $|α|$ up to 12 using electric…
Ground state cooling and coherent manipulation of ions in an rf-(Paul) trap is the prerequisite for quantum information experiments with trapped ions. With resolved sideband cooling on the optical S1/2 - D5/2 quadrupole transition we have cooled one and two 40Ca+ ions to the ground state of vibration with up to 99.9% probability. With a novel cooling scheme utilizing electromagnetically induced transparency on the S1/2 - P1/2 manifold we have achieved simultaneous ground state cooling of two motional sidebands 1.7 MHz apart. Starting from the motional ground state we have demonstrated coherent…
Individual addressing of trapped ion qubits with geometric phase gates
提出一种基于运动频率选择囚禁离子量子比特的单独寻址新方案，利用几何相位门结合自旋无关力与自旋相关力执行单量子比特旋转，提高门速度并降低退相干敏感性，在无激光实现中串扰误差低于$10^{-6}$。
Electric field noise is a major source of motional heating in trapped ion quantum computation. While the influence of trap electrode geometries on electric field noise has been studied in patch potential and surface adsorbate models, only smooth surfaces are accounted for by current theory. The effects of roughness, a ubiquitous feature of surface electrodes, are poorly understood. We investigate its impact on electric field noise by deriving a rough-surface Green's function and evaluating its effects on adsorbate-surface binding energies. At cryogenic temperatures, heating rate contributions …
We study the entangling power of a nanoelectromechanical system (NEMS) simultaneously interacting with two separately trapped ions. To highlight this entangling capability, we consider a special regime where the ion-ion coupling does not generate entanglement in the system, and any resulting entanglement will be the result of the NEMS acting as an entangling device. We study the dynamical behavior of the bipartite NEMS-induced ion-ion entanglement as well as the tripartite entanglement of the whole system (ions+NEMS). We found some quite remarkable phenomena in this hybrid system. For instance…
We theoretically propose and experimentally demonstrate a parallel-electromagnetically-induced transparency (parallel-EIT) cooling technique for ion crystals in the Paul trap. It has less stringent requirements on the cooling resonance condition than the standard electromagnetically-induced transparency (EIT) cooling, thus allowing, in principle, to simultaneously cool the motional mode spectrum with an arbitrary range. A proof-of-principle validation for this cooling scheme is experimentally demonstrated with up to 4 trapped 40Ca+ ions. We observe simultaneous near-ground-state cooling for al…
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