Quantum state engineering using conditional measurement on a beam splitter

State preparation via conditional output measurement on a beam splitter is studied, assuming the signal mode is mixed with a mode prepared in a Fock state and photon numbers are measured in one of the output channels. It is shown that the mode in the other output channel is prepared in either a photon-subtracted or a photon-added Jacobi polynomial state, depending upon the difference between the number of photons in the input Fock state and the number of photons in the output Fock state onto which it is projected. The properties of the conditional output states are studied for coherent and squeezed input states, and the probabilities of generating the states are calculated. Relations to other states, such as near-photon-number states and squeezed-state-excitations, are given and proposals are made for generating them by combining the scheme with others. Finally, effects of realistic photocounting and Fock-state preparation are discussed. Received: 17 March 1998 / Revised and Accepted: 8 May 1998     

Preparation of multi-photon Fock states and quantum entanglement properties in circuit QED

We demonstrate the controllable generation of multi-photon Fock states in circuit quantum electrodynamics （circuit QED）. The external bias flux regulated by a counter can effectively adjust the bias time on each superconducting flux qubit so that each flux qubit can pass in turn through the circuit cavity and thereby avoid the effect of decoherence. We further investigate the quantum correlation dynamics of coupling superconducting qubits in a Fock state. The results reveal that the lower the photon number of the light field in the number state, the stronger the interaction between qubits is, then the more beneficial to maintaining entanglement between qubits it will be.     

Reduction to the Fock state via a degenerate four-wave mixing

A scheme to generate the Fock state via a degenerate four-wave mixing and partial measurement is proposed. A conditional state of the signal mode of a degenerate four-wave mixer will be shown to reduce to the Fock state when the outcome of the measurement on the reference mode satisfies a certain condition. The photon number of the Fock state is distributed when the measurement is repeated many times. Conditioning for the quantum state of the signal mode is carried out by the photon counting measurement on the reference mode whose input state is assumed to be the vacuum. The phase coherence of the conditional signal state is also considered. Furthermore conditioning for the signal mode by means of heterodyne and homodyne detections are briefly considered.     

探测器对量子增强马赫-曾德尔干涉仪相位测量灵敏度的影响

研究了强度差测量方案下,探测器量子效率对光子数态、关联数态、压缩真空态三种量子光源注入的马赫-曾德尔干涉仪相位测量灵敏度的影响.获得了相位测量灵敏度与效率的定量关系,比较了探测效率对不同量子态注入的干涉仪相位灵敏度的影响.研究表明：光子数态注入时,相位测量灵敏度始终不能超越标准量子极限;关联数态注入时,无论多大的光子数,要获得相位测量的量子增强,探测效率不得小于75%;对于压缩真空态,只要有压缩存在就可以获得一定的相位测量的量子增强;关联数态、压缩真空态的注入,相位灵敏度皆随探测效率的增大而不同程度的提高,且压缩真空态比关联数态具有更好的量子增强效果.给出了在量子增强的精密测量实验中对探测效率的要求,并结合实际应用说明了探测效率的提高有助于提高干涉仪探测的灵敏度.     

Generation of Entanglement Between Two Noninteracting Qubits via Interaction with Nonclassical Radiation Field

We study the interaction between a single-mode quantized field and a quantum system composed of two qubits. We suppose that two qubits initially be prepared in the mixed and separable state, and study how entanglement between two qubits arises in the course of evolution according to the Jaynes-Cummings type interaction with nonclassical radiation field. We also investigate the relation between entanglement and purity of qubit subsystem. We show that different photon statistics have different effects on the dynamical behavior of the qubit subsystem. When the qubits are initially prepared in the maximally mixed and separable state, for coherent state field we cannot find entanglement between two qubits; for squeezed state field entanglement between two qubits exists in several short period of time; for even and odd coherent state fields of large photon number, the dynamical behavior of the entanglement between two qubits shows collapse and revival phenomenon. For odd coherent state field of small photon number, the entanglement with both qubits initially prepared in maximally mixed state can be stronger than that with both qubits initially prepared in pure states. For fields of small photon number, the entanglement strongly depends on the states they are initially prepared in. For coherent state field, and odd and even coherent state fields of large photon number, the entanglement only depends on the purity of the initial state of qubit subsystem. We also show that during the evolution the unentangled state may be purer than the entangled state, and the maximum degree of entanglement may not occur at the time when the qubit subsystem is in the purist state.     

Generation of Entanglement Between Two Noninteracting Qubits via Interaction with Nonclassical Radiation Field

We study the interaction between a single-mode quantized field and a quantum system composed of two qubits. We suppose that two qubits initially be prepared in the mixed and separable state, and study how entanglement between two qubits arises in the course of evolution according to the Jaynes-Cummings type interaction with nonclassical radiation field. We also investigate the relation between entanglement and purity of qubit subsystem. We show that different photon statistics have different effects on the dynamical behavior of the qubit subsystem. When the qubits are initially prepared in the maximally mixed and separable state, for coherent state field we cannot find entanglement between two qubits; for squeezed state field entanglement between two qubits exists in several short period of time; for even and odd coherent state fields of large photon number, the dynamical behavior of the entanglement between two qubits shows collapse and revival phenomenon. For odd coherent state field of small photon number, the entanglement with both qubits initially prepared in maximally mixed state can be stronger than that with both qubits initially prepared in pure states. For fields of small photon number, the entanglement strongly depends on the states they are initially prepared in. For coherent state field, and odd and even coherent state fields of large photon number, the entanglement only depends on the purity of the initial state of qubit subsystem. We also show that during the evolution the unentangled state may be purer than the entangled state, and the maximum degree of entanglement may not occur at the time when the qubit subsystem is in the purist state.     

Optimal quantum estimation of the coupling constant of Jaynes-Cummings interaction

We address the estimation of the coupling constant of the Jaynes-Cummings Hamiltonian for a coupled qubit-oscillator system. We evaluate the quantum Fisher Information (QFI) for the system undergone the Jaynes-Cummings evolution, considering that the probe initial state is prepared in a Fock state for the oscillator and in a generic pure state for the qubit; we obtain that the QFI is exactly equal to the number of excitations present in the probe state. We then focus on the two subsystems, namely the qubit and the oscillator alone, deriving the two QFIs of the two reduced states, and comparing them with the previous result. Next we focus on possible measurements on the system, and we find out that if population measurement on the qubit and Fock number measurement on the oscillator are performed together, the Cramer-Rao bound is saturated, that is the corresponding Fisher Information (FI) is always equal to the QFI. We compare also the performances of these energy measurements performed alone, that is when one of the two subsystem is ignored. We show that, when the qubit is prepared in either the ground or the excited state, the local measurements are still optimal. Finally we investigate the case when the harmonic oscillator is prepared in a thermal state and observe how, particularly for small values of the coupling constant, the QFI increases with the average number of thermal photons of the initial state.     

Generation of a displaced qubit and entangled displaced photon state via conditional measurement and their properties

We study optical schemes for generating both a displaced photon and a displaced qubit via conditional measurement. Combining one mode prepared in different microscopic states (one-mode qubit, single photon, vacuum state) and another mode in macroscopic states (coherent state, single photon added coherent state), a conditional state in the other output mode exhibits properties of a superposition of the displaced vacuum and a single photon. We propose to use the displaced qubit and entangled states composed of the displaced photon as components for quantum information processing. Basic states of such a qubit are distinguishable from each other with high fidelity. We show that the qubit reveals both microscopic and macroscopic properties. Entangled displaced states with a coherent phase as an additional degree of freedom are introduced. We show that additional degree of freedom enables to implement complete Bell state measurement of the entangled displaced photon states.     

Dynamics of dispersive photon-number QND measurements in a micromaser

A numerical analysis of dispersive quantum nondemolition measurement of the photon number of a microwave cavity field is presented. Simulations show that a key property of the dispersive atom-field interaction used in Ramsey interferometry is the extremely high sensitivity of the dynamics of atomic and field states to basic parameters of the system. When a monokinetic atomic beam is sent through a microwave cavity, a qualitative change in the field state can be caused by an uncontrollably small deviation of parameters (such as atom path length through the cavity, atom velocity, cavity mode frequency detuning, or atom-field coupling constants). The resulting cavity field can be either in a Fock state or in a super-Poissonian state (characterized by a large photon-number variance). When the atoms have a random velocity spread, the field is squeezed to a Fock state for arbitrary values of the system’s parameters. However, this makes detection of Ramsey fringes impossible, because the probability of detecting an atom in the upper or lower electronic state becomes a random quantity almost uniformly distributed over the interval between zero and unity, irrespective of the cavity photon number.     

基于量子测量的Fock态制备和自相位调制的影响

提出一种制备Fock态的新方案研究在包含有自相位调制的Kerr介质中两束光的相互作用,信号光场和探测光场初始处于相干态,经由非线性相互作用后演化成为纠缠态.若对探测光场的正交位相分量实行第一类量子测量,信号光场的光子数分布会受到调制.重复上述过程,发现信号光场最终演化成为一个纯的Fock态.这种制备Fock态的原理是基于互相位调制,而自相位调制则起着阻碍Fock态形成的作用.     

远程控制纠缠原子的偶极压缩

考虑初始处于EPR态的两个二能级原子A、B,将B原子注入处于真空态和单光子态的叠加态的腔中,演化一段时间后,对B原子进行选择性测量,通过选择合适的腔场初始叠加状态和演化时间,可控制原子A的偶极矩压缩效应.     

Entanglement generation by Fock-state filtration

We demonstrate a Fock-state filter which is capable of preferentially blocking single photons over photon pairs. The large conditional nonlinearities are based on higher-order quantum interference, using linear optics, an ancilla photon, and measurement. We demonstrate that the filter acts coherently by using it to convert unentangled photon pairs to a path-entangled state. We quantify the degree of entanglement by transforming the path information to polarization information; applying quantum state tomography we measure a tangle of T=(20+/-9)%.     

The quantum computational speed of a single Cooper-pair box

The speed of computations is investigated by means of the orthogonality speed for a charged qubit interacting with a single cavity field prepared initially in a Fock state or Binomial state. We observe that the rate of the computational speed is related to the number of photons inside the cavity. Moreover, we show that the qubit–field coupling plays an opposite role, where the speed of computations is decreased as the coupling is increased. We suggest using the number of photons in the field as a control parameter to improve the speed of computations.     

Efficient scheme for realizing a multiplex-controlled phase gate with photonic qubits in circuit quantum electrodynamics

We propose an efficient scheme to implement a multiplex-controlled phase gate with multiple photonic qubits simultaneously controlling one target photonic qubit based on circuit quantum electrodynamics (QED). For convenience, we denote this multiqubit gate as MCP gate. The gate is realized by using a two-level coupler to couple multiple cavities. The coupler here is a superconducting qubit. This scheme is simple because the gate implementation requires only one step of operation. In addition, this scheme is quite general because the two logic states of each photonic qubit can be encoded with a vacuum state and an arbitrary non-vacuum state |φ> (e.g., a Fock state, a superposition of Fock states, a cat state, or a coherent state, etc.) which is orthogonal or quasi-orthogonal to the vacuum state. The scheme has some additional advantages: because only two levels of the coupler are used, i.e., no auxiliary levels are utilized, decoherence from higher energy levels of the coupler is avoided; the gate operation time does not depend on the number of qubits; and the gate is implemented deterministically because no measurement is applied. As an example, we numerically analyze the circuit-QED based experimental feasibility of implementing a three-qubit MCP gate with photonic qubits each encoded via a vacuum state and a cat state. The scheme can be applied to accomplish the same task in a wide range of physical system, which consists of multiple microwave or optical cavities coupled to a two-level coupler such as a natural or artificial atom.     

Entangling Two Multi-atomic Clusters Through a Cavity Field

Two atomic clusters, which have NA and Ns two-level atoms, respectively, are placed in a cavity but separated spatially. There is no direct interaction between the atoms. All the atoms interact with a single-mode of the cavity field. Quantum entanglement between the two atomic clusters is investigated for various initial states of the two atomic clusters and the field. When the cavity field is initially in a Fock state, we find that the time evolution of entanglement quasi-periodically oscillates regardless of the initial states of atoms. The oscillation period increases as the initial photon number increases. When all the atoms in both of the atomic clusters are initially in the excited state, we show that there is no entanglement between the atomic clusters with NA = NB = 1 regardless the initial state of the cavity field. However, when either NA or NB is larger than one, we find that the entanglement always exists even for a strong thermal field. In cases with different initial states of the atomic clusters, we notice that the entanglement becomes stronger as number of the atoms increases. When all the atoms in both of the clusters in the ground state, we also find that the entanglement can be enhanced even by a thermal field. We also notice that a single qubit can be entangled with multi-atoms which are initially in the ground state by the cavity field initially being in vacuum, thermal, coherent, and squeezed states.     

三个两能级原子与数态场相互作用的纠缠特性

通过计算并发度和线性熵研究了初始处于GHZ态的三个两能级原子与数态场相互作用系统的纠缠动力学特性,讨论了场的初始光子数对并发度和线性熵的影响.结果表明,腔内两原子之间的纠缠出现突然产生现象,而且可以通过改变场的初始光子数来控制产生纠缠的阈值时间和纠缠的最大值.对腔外原子的探测导致了并发度随时间的演化呈现周期性规律,场的光子数的增加不但减小了并发度的最大值,而且缩短了并发度的演化周期.     

三个两能级原子与数态场相互作用的纠缠特性

通过计算并发度和线性熵研究了初始处于GHZ态的三个两能级原子与数态场相互作用系统的纠缠动力学特性,讨论了场的初始光子数对并发度和线性熵的影响.结果表明,腔内两原子之间的纠缠出现突然产生现象,而且可以通过改变场的初始光子数来控制产生纠缠的阈值时间和纠缠的最大值.对腔外原子的探测导致了并发度随时间的演化呈现周期性规律,场的光子数的增加不但减小了并发度的最大值,而且缩短了并发度的演化周期.     

Entanglement dynamics of two atoms successively passing a cavity

By means of concurrence, we investigate the dynamics of entanglement between two initially separate atoms in succession passing through a cavity and their interaction with a Fock state field. We then analyze the effects of the atomic coherence, photon number, and atomic motion on the time evolution of atom-atom entanglement. The results show that there can be entanglement between two separate atoms, and that the threshold time for the creation of the entanglement is controllable by the photon number, atomic motion, and field-mode structure.     

Quantum teleportation through an entangled state composed of displaced vacuum and single-photon states

We study a teleportation protocol of an unknown macroscopic qubit by means of a quantum channel composed of the displaced vacuum and single-photon states. The scheme is based on linear optical devices such as a beam splitter and photon number resolving detectors. A method based on conditional measurement is used to generate both the macroscopic qubit and entangled state composed from displaced vacuum and single-photon states. We show that such a qubit has both macroscopic and microscopic properties. In particular, we investigate a quantum teleportation protocol from a macroscopic object to a microscopic state. The text was submitted by the author in English.     

腔QED中超导量子干涉仪概率克隆机的物理实现

提出一个量子概率克隆机的物理实现方案,该方案首先将高Q腔中的两个超导量子干涉仪分别作为初始比特和目标比特,腔模作为测量比特,通过腔模和经典微波脉冲与超导量子干涉仪的多种相互作用实现量子概率克隆机的幺正演化;然后将腔模态映射到另一个超导量子干涉仪上,通过对该超导量子干涉仪磁通量的测量完成状态坍缩,从而以最优的成功概率实现量子态的精确克隆.本方案采用双光子拉曼共振过程加快单比特门的操作速率,并且总操作时间远小于自发辐射和腔模衰变时间,因而在实验上是可行的.