Measuring the phase variance of light

Abstract

The results of experiments designed to measure the operational phase cosine and sine variances of weak states of light disagree with the variances predicted by canonical phase formalisms. As these variances are fundamental manifestations of the quantum nature of phase, it is important to be able to measure the canonical variances also. A recent suggestion to do so, based on the use of a two-component probe, involves the difficult preparation of exotic states of light which have not yet been produced. In this paper we show how the variances can be measured with simple coherent state inputs. The retrodictive formalism of quantum mechanics provides useful insight into the physics involved.     

Optical state measurement by information transfer

Abstract

Mathematically, the simplest state of light containing phase information is a superposition of the vacuum and the one-photon state and, as we show in this paper, such a state is reasonably simple to measure. We investigate how the information contained in a more complicated pure state of light, in particular the ratio of successive number-state coefficients, can be transferred selectively to fields in this two-state superposition for subsequent measurement. By this means the number-state representation of the more complicated state can be ascertained, provided there are no gaps in the number state distribution. We also discuss how to correct for the effect of non-unit efficiencies of the photodetectors involved in the transferral process.     

Retrodictive quantum optical state engineering

Abstract

Although it has been known for some time that quantum mechanics can be formulated in a way that treats prediction and retrodiction on an equal footing, most attention in engineering quantum states has been devoted to predictive states, that is, states associated with a preparation event. Retrodictive states, which are associated with a measurement event and propagate backwards in time, are also useful, however. In this paper it is shown how any retrodictive state of light that can be written to a good approximation as a finite superposition of photon number states can be generated by an optical multiport device. The composition of the state is adjusted by controlling predictive coherent input states. It is shown how the probability of successful state generation can be optimized by adjusting the multiport device and also a versatile configuration that is useful for generating a range of states is examined.     

Hilbert-Schmidt distance and non-classicality of states in quantum optics

Abstract

The Hilbert—Schmidt distance between two arbitrary normalizable states is discussed as a measure of the similarity of the states. Unitary transformations of both states with the same unitary operator (e.g. the displacement of both states in the phase plane by the same displacement vector and squeezing of both states) do not change this distance. The nearest distance of a given state to the whole set of coherent states is proposed as a quantitative measure of non-classicality of the state which is identical when considering the coherent states as the most classical ones among pure states and the deviations from them as non-classicality. The connection to other definitions of the non-classicality of states is discussed. The notion of distance can also be used for the definition of a neighbourhood of considered states. Inequalities for the distance of states to Fock states are derived. For given neighbourhoods, they restrict common characteristics of the state as the dispersion of the number operator and the squared deviation of the mean values of the number operator for the considered state and the Fock state. Possible modifications in the definition of non-classicality for mixed states with dependence on the impurity parameter and by including the displaced thermal states as the most classical reference states are discussed.     

The information of ambiguity

The phase space characteristics of a quantum state are best captured by the Wigner distribution. This displays transparently the diagonality information of the density matrix. The complementary function offering transparently the off-diagonal elements is captured by a function called the S-function, or the ambiguity. In carrying the maximal information about the quantum coherences it represents the uncertainties or ambiguity of the diagonal information. Mathematically this is manifested in its role as the phase space moment generating function. Formally it complements the information in the Wigner function. These formal relations provide the starting point for the present investigations. As a measure of quantum uncertainties, ambiguity may be used to define a probability measure on the off-diagonality. The mathematical and physical consistency of this view is presented in this paper. For a pure state, we find the extraordinary result that such distributions are their own Fourier transforms. The physical interpretation of this distribution as a carrier of classical signal fuzziness suggests the introduction of heuristic approximations to the observational uncertainties. We illustrate the properties and interpretation of the ambiguity function by some specific examples. We find that for smooth, ‘Gaussian-like’ distributions, the heuristic considerations provide good approximations. On the other hand, representing quantum interferences, the ambiguity serves as the most positive probe for the ultimate quantum structures which have been called sub-Planckian. They are interesting because it has been argued that such structures are physically observable.     

Optical qubit generation by state truncation using an experimentally feasible scheme

Generation of arbitrary superposition of vacuum and one-photon states using a quantum scissors device (QSD) is studied. The device allows the preparation of states by truncating an input coherent light. Optimum values of the intensity of the coherent light for the generation of any desired state using the experimentally feasible QSD scheme are found.     

Error propagation in polarimetric demodulation

The polarization analysis of light is typically carried out using modulation schemes. The light of an unknown polarization state is passed through a set of known modulation optics, and a detector is used to measure the total intensity passing the system. The modulation optics is modified several times, and, with the aid of several such measurements, the unknown polarization state of the light can be inferred. How to find the optimal demodulation process has been investigated in the past. However, since the modulation matrix has to be measured for a given instrument and the optical elements can present problems of repeatability, some uncertainty is present in the elements of the modulation matrix or covariances between these elements. We analyze in detail this issue, presenting analytical formulas for calculating the covariance matrix produced by the propagation of such uncertainties on the demodulation matrix, on the inferred Stokes parameters, and on the efficiency of the modulation process. We demonstrate that even if the covariance matrix of the modulation matrix is diagonal, the covariance matrix of the demodulation matrix is in general nondiagonal because matrix inversion is a nonlinear operation. This propagates through the demodulation process and induces correlations on the inferred Stokes parameters.     

Homodyne tomography of classical and non-classical light

Abstract

States with explicit quantum character, such as squeezed vacuum and bright squeezed light, as well as coherent states and incoherent superpositions of coherent states were generated and analysed by tomographical methods. Wigner functions, photon-number distributions, density matrices and phase distributions were reconstructed with high accuracy. Features such as photon number oscillations, sub-Poissonian and super-Poissonian photon statistics, bifurcations of the phase distribution, and loss of coherence were observed, demonstrating the usefulness of quantum state reconstruction as an analysing tool in quantum optics experiments.     

Superposition states of light in a three-photon absorbing dissipative medium

We consider the process of simultaneous absorption of three photons in a medium in the presence of weak one-photon absorption. We show that in such a system stationary three-component superposition states of light may be formed in the range of small values of state amplitude (weak perturbation). This circumstance is associated with the fact that in this range of interaction the field spends more time in one of the three types of superposition states of light (constituting an ensemble of quantum trajectories of the system) than in two other types. We also show that in the range of large values of state amplitude it is possible to obtain three types of non-stationary superposition states of light. By using numerical simulation of quantum trajectories of the system we study the dynamics of the quantum entropy of the field. We calculate the Wigner functions of the field states. We also obtain analytical results for the density matrix of the steady state of the system.     

The effect of probe coherence and atomic coherence on the response of an open lambda-type pump–probe system

The response of a coherently prepared four-level λ-type system, interacting with two electromagnetic fields in a Doppler-free pump–probe configuration is analytically formulated. Under density matrix formalism the probe coherence (field-dependent phase) and atomic coherence (field-independent phase) are introduced through the off-diagonal matrix elements. The coherent effects on probe response is investigated for probe coherence, as well as for the driving contribution (coherence) of the pump field. We show how probe coherence can modify the Rabi splitting and two-photon absorption, through shifting and broadening of spectral lines for on- and off-resonance pumping, respectively. In addition, we report on the enhancement of absorptionless dispersion (i.e. refractive index for on and around probe resonance) and two-photon absorption, via coherent control of the driving contribution of the pump field.     

Dissipation and Amplification of Jaynes-Cummings Superposition States

Abstract

There have recently been several proposals for generation of optical superposition states in the resonant atom-field interaction and more practically in microwave cavities. In the present paper we study the influence of the vacuum reservoir on properties of the near-superposition state of the cavity field which is described by the Jaynes-Cummings model at one-half of the revival time. Instead of introducing the cavity loss from the first instance of the atom-field interaction we consider the cavity loss only after the near-superposition state is produced and after the atom leaves the cavity. We solve the corresponding master equation with the initial condition being the Jaynes-Cummings field at one-half of the revival time. We find that under the influence of the vacuum reservoir the photon number distribution of the superposition state we study exhibits certain asymmetry around the mean photon induced by the decay process. We show that an analogous effect can be seen when the Jaynes-Cummings superposition state is amplified. For a basic test of our approach we study the dissipation and amplification of Fock states.     

Phase control of six-wave mixing from circularly polarized light

We investigate the phase control of six-wave mixing (SWM) in atomic system with multi-Zeeman levels theoretically and experimentally. With the relative phase varying, the switch between bright and dark state can appear in probe transmission signal. Then we demonstrate the evolution of six-wave mixing generated in bright and dark states by scanning the frequency detuning of the dressing field at different polarized probe field. Meanwhile, by utilizing the strong dressing effect of circular polarized light, we observe pure dark state switched to pure bright state in terms of energy level splitting, and compare different phases under different detuning of circularly polarized light. Theoretical calculations are in well agreement with the experimental observations.     

Generation of amplitude squeezed states of light from phase squeezed coherent states

Abstract

We propose a scheme to generate a displaced macroscopic superposition of phase squeezed coherent states by incorporating an optical parametric oscillator (OPO) and a nonlinear Kerr medium in a Mach-Zhender interferometer configuration. We show that the phase squeezed coherent state generated by the OPO evolves into a macroscopic superposition of phase squeezed coherent states on spending a specific amount of time inside a Kerr medium. The phase space displacement is achieved by the interference of the intense reference beam with the signal in the final beam splitter of the interferometer. The noise of the reference beam contaminating the signal is prevented by making this beam splitter highly reflective. We have calculated the photon number uncertainty of the outcoming beam and have shown that it is smaller than the usual amplitude squeezed coherent state's value.     

Nonlinear Dissipative Oscillator with Displaced Number States

Abstract

We study the interaction of a Kerr-like medium with light initially prepared in a displaced number state. We analyse squeezing properties and photon statistics at the output of a Kerr-like medium. We show that under certain conditions the superposition of two displaced number states can be created. We study the influence of dissipation on the formation of the superposition state.     

Phase reconstruction from undersampled intensity patterns

We demonstrate the uniqueness and convergence of phase recovery from high-spatial-frequency and undersampled intensity data. Furthermore, this is accomplished without the ambiguities that arise in phase unwrapping and without the need to employ a priori information. The method incorporates the technique of line integration of the phase gradient to find the first approximation to the phase and the algorithm of synthetic interferograms to find the unknown phase with high accuracy. The method may be used with any experimental method that at a certain data processing step obtains generalized sine and cosine intensity functions.     

Off-diagonal impedance in amorphous wires and its application to linear magnetic sensors

We investigated the magnetic-field behavior of the off-diagonal impedance in Co-based amorphous wires under sinusoidal (50 MHz) and pulsed (5 ns rise time) current excitations. For comparison, we measured the field characteristics of the diagonal impedance as well. In general, when an alternating current is applied to a magnetic wire, the voltage signal is generated not only across the wire but also in a pickup coil wound on it. These voltages are related to the diagonal and off-diagonal impedances, respectively. We demonstrate that these impedances have a different behavior as functions of axial magnetic field: the diagonal impedance is symmetrical, whereas the off-diagonal one is antisymmetrical with a near-linear portion within a certain field interval. For the off-diagonal response, the dc bias current is necessary to eliminate circular domains. In the case of the sinusoidal excitation without a dc bias current, the off-diagonal response is very small and irregular. In contrast, the pulsed excitation, combining both high- and low-frequency harmonics, produces the off-diagonal voltage response without additional biasing. This behavior is ideal for a practical sensor circuit design. We discuss the principles of operation of a linear magnetic sensor based on a complementary metal-oxide-semiconductor transistor circuit.     

Construction of Differential Material Matrices for the Orthogonal Finite-Integration Technique With Nonlinear Materials

The linearization of an electromagnetic formulation by the Newton method can be expressed similarly as for the linear case, by introducing differential material matrices. For the case of the finite-integration technique applied to an orthogonal grid, the chord material matrix is diagonal whereas the differential material matrices includes off-diagonal bands, representing the cross-directional coupling introduced by the nonlinearity. An approximative Newton method based on a unidirectional differential material matrix yields a diagonal matrix, which has a higher computational efficiency but may lead to a degenerated convergence.     

关联无序材料中电子态的研究方法

按照一般的无序理论,在非关联一维无序材料中不存在扩展态,在绝对零度下,电子的波函数是局域的,系统表现为绝缘体.在关联无序系统中,格点能量之间的长程关联,即对角关联,能导致扩展的波函数并由此导致系统的导电性.当关联强度于某一阈值,可以发现在热力学极限之下有存在于较宽能带范围内的扩展态,此阈值即为体系的金属-绝缘体转变临界点.考虑格点之间的长程相互作用势,即非对角关联时,在一维系统中通过使用传输矩阵方法,可以获得电子态的一些本质效应,长程跳跃可以改变系统的有效维度并能导出一维体系中存在有依赖于关联强度的金属-绝缘体转变.     

Squeezing in Finite Superpositions of Photon-number States without the Vacuum

Abstract

We show that a finite superposition of photon-number states without the vacuum state can lead to squeezed fluctuations. The properties of these states are discussed.     

Reconstructing the Wavefunction in Quantum Optics

Abstract

The problem of reconstructing a wavefunction from probability distributions is re-examined in the context of whether a pure state vector of a single-mode optical field can be reconstructed from the photon number and phase probability distributions. An analytical solution is given for the case where the state of the mode is a superposition of a finite number of Fock states.