Two-photon Coherent States Research Articles

Optomechanical parametric oscillation of a quantum light-fluid lattice

Two-photon coherent states are one of the main building pillars of non-linear and quantum optics. It is the basis for the generation of minimum-uncertainty quantum states and entangled photon pairs, applications not obtainable from standard coherent states or one-photon lasers. Here we describe a fully-resonant optomechanical parametric amplifier involving a polariton condensate in a trap lattice quadratically coupled to mechanical modes. The quadratic coupling derives from non-resonant virtual transitions to extended discrete excited states induced by the optomechanical coupling. Non-resonant continuous wave (cw) laser excitation leads to striking experimental consequences, including the emergence of optomechanically induced inter-site parametric oscillations and inter-site tunneling of polaritons at discrete inter-trap detunings corresponding to sums of energies of the two involved mechanical oscillations (20 and 60 GHz confined vibrations). We show that the coherent mechanical oscillations correspond to parametric resonances with threshold condition different to that of standard linear optomechanical self-oscillation. The associated Arnold tongues display a complex scenario of states within the instability region. The observed new phenomena can have applications for the generation of entangled phonon pairs, squeezed mechanical states relevant in sensing and quantum computation, and for the bidirectional frequency conversion of signals in a technologically relevant range.

A rigorous uncertainty relation (Cauchy Inequality) for an electromagnetic field in quantum superpositions of coherent states and in a two-photon coherent state

It is shown that the Heisenberg uncertainty relation (or soft uncertainty relation) determined by the commutation properties of operators of electromagnetic field quadratures differs significantly from the Robertson–Schrodinger uncertainty relation (or rigorous uncertainty relation) determined by the quantum correlation properties of field quadratures. In the case of field quantum states, for which mutually noncommuting field operators are quantum-statistically independent or their quantum central correlation moment is zero, the rigorous uncertainty relation makes it possible to measure simultaneously and exactly the observables corresponding to both operators or measure exactly the observable of one of the operators at a finite measurement uncertainty for the other observable. The significant difference between the rigorous and soft uncertainty relations for quantum superpositions of coherent states and the two-photon coherent state of electromagnetic field (which is a state with minimum uncertainty, according to the rigorous uncertainty relation) is analyzed.

Nanoelectronic RF Josephson Devices

Superconducting RF nanoelectronic devices exhibit a considerable potential for application in future electronics. Josephson effect based devices allow generation, detection, mixing, and parametric amplification of high-frequency signals up into the terahertz region and exhibit high sensitivity, low energy consumption, and small size. Traveling-wave devices can be realized with distributed Josephson junctions. Nonlinear lumped-element circuits can be realized that are small enough so that the circuit dynamics are governed by quantum mechanics. This allows to generate two-photon coherent states and entangled states and will open the door for quantum information processing. Nanotechnological fabrication techniques and the availability of novel materials give a strong impact on the development of novel Josephson-effect-based devices and systems. An overview over the physical principles and the possible applications of nanoelectronic RF Josephson devices is presented.

POLARIZATION ENTANGLEMENT THROUGH NOISE

We show that a suitably engineered external non-Gaussian noise can entangle a class of initially separable, polarized two-photon coherent states.

Generalized coherent states and the quadratic Hamiltonian

The idea of coherent states has proved very useful in the study of the quantum nature of electromagnetic radiation. Coherent states are also interesting because their evolution in time comes closest to classical dynamics. So they are the best approximation in quantum mechanics to classical behaviour. In recent years, attempts have been made to generalize the concept of coherent states to systems other than those made up of the simple harmonic oscillator (such as light). In one such attempt, Klauder introduces a set of physical criteria that generalized coherent states should have (Klauder J R 1996 J. Phys. A: Math. Gen. 29 L293). These criteria have been applied to the hydrogen atom and several other systems. In this paper, we apply Klauder's criteria to identify generalized coherent states for the quadratic Hamiltonian H = ℏ(f1a†a + f*2a2 + f2a†2). We check if two-photon coherent states of light satisfy these criteria. We find that they do not, which is expected: their non-classical properties are well known. We then use Klauder's criteria to find a set of states that are associated with the quadratic Hamiltonian. We find that these states too display non-classical behaviour even for large values of the quantum numbers. This non-classical behaviour is revealed in the form of sub-Poisson photon statistics. We will also briefly consider the one-photon Hamiltonian, apply Klauder's criteria to it and comment on its statistics.

A new class of quantum states generated by m -fold application of Bogoliubov's transformation

By applying higher powers of the Bogoliubov transformation operator bdagger = v*a + mu*adagger to the two-photon coherent states (or minimum uncertainty squeezed states) we construct a new type of quantum state which we call the generalized excited two-photon coherent states. Analytic expressions for the quantum statistical properties are derived, and through numerical computation the phase space quasi-probability distributions are found. These states can exhibit highly nonclassical behaviour depending on the degree of excitation m and other parameters. For particular values of two parameters lambda and rho, these generalized states reduce to other classes of coherent states formerly reported. Our theory thus presents a much broader approach to these types of quantum states.

Photon-bunching, photon-antibunching and the nonclassical photon statistics of coherent light coupled to a driven harmonic oscillator of time dependent mass and frequency

The analytical solution of the quantum driven harmonic oscillator of time dependent mass and frequency (DHTDMF) is exploited to obtain the photon-bunching, photon-antibunching and the nonclassical photon statistics of the input coherent light coupled to the oscillator. We observe that the model Hamiltonian of a DHTDMF coincides with those of the model Hamiltonian of a two-photon coherent state due to H.P. Yuen [Phys. Rev. A 13 (1976) 226]. To substantiate our analytical results, we give a special case corresponding to a damped harmonic oscillator subject to an external periodic force. The quantum statistical properties of the radiation field are discussed for under-, over-, and for critically-damped situations of the oscillator coupled to an initial coherent field. The off-resonance and the near- or on-resonance of the oscillator with respect to the frequency of the external force are discussed. In the determination of quantum statistical properties of the radiation field, we observe that the effects of the driving force are important only when the frequency of the oscillator is in near- or in on-resonance condition with that of the frequency of the external periodic force.

Victor Kozlov's reply

We prove formal identity of the dispersed states and the two-photon coherent states. As the last fall in the class of squeezed states and are non-classical, so are the dispersed states. A state which exhibits squeezing and/or non-classicality should not be called coherent as suggested in the Comment.

Statistical Properties and Algebraic Characteristics of Quantum Superpositions of Negative Binomial States**The project supported in part by National Natural Science Foundation of China under Grant No. 19875008

We introduce new kinds of states of quantized radiation fields, which are the superpositions of negative binomial states. They exhibit remarkable nonclassical properties and reduce to Schrödinger cat states in a certain limit. The algebras involved in the even and odd negative binomial states turn out to be generally deformed oscillator algebras. It is found that the even and odd negative binomial states satisfy the same eigenvalue equation with the same eigenvalue and they can be viewed as two-photon nonlinear coherent states. Two methods of generating such the states are proposed.

Photon-added one-photon and two-photon nonlinear coherent states

From the photon-added one-photon nonlinear coherent states a†m|α, ƒ〉, we introduce a new type of nonlinear coherent states with negative values of m. The nonlinear coherent states corresponding to the positive and negative values of m are shown to be the result of nonunitarily deforming the number states |m〉 and |0〉, respectively. As an example, we study the sub-Poissonian statistics and squeezing effects of the photon-added geometric states with negative values of m in detail. Finally, we investigate the photon-added two-photon nonlinear coherent states and find that they are still the two-photon nonlinear coherent states with certain nonlinear functions. PACS Nos.: 42.50Dv, 03.65Fd

Superpositions of negative binomial states

The states formed by a superposition of the negative binomial states are introduced. The sub-Poissonian statistics and squeezing properties of the superposition states are studied in detail. Moreover, we obtain the ladder operator formalism of the superposition state and find that this state is a type of two-photon nonlinear coherent state. A scheme for the generation of the superposition state is also discussed.

Photon-added one-photon and two-photon nonlinear coherent states

From the photon-added one-photon nonlinear coherent states $a^{\dagger m}|\alpha,f>$, we introduce a new type of nonlinear coherent states with negative values of $m.$ The nonlinear coherent states corresponding to the positive and negative values of $m$ are shown to be the result of nonunitarily deforming the number states $|m>$ and $|0>$, respectively. As an example, we study the sub-Poissonian statistics and squeezing effects of the photon-added geometric states with negative values of $m$ in detail. Finally we investigate the photon-added two-photon nonlinear coherent states and find that they are still the two-photon nonlinear coherent states with certain nonlinear functions.

Temperature dependence of four-wave-mixing spectra in ZnO thin films on sapphire substrates grown with laser MBE

We investigated the optical nonlinearity around the excitonic resonance region of ZnO thin films with thickness of about 200 nm grown with laser molecular beam epitaxy (MBE) method by using four-wave-mixing (FWM) spectroscopy. It is found that the FWM signal can be confirmed in the temperature range (10–294 K) and that its spectra significantly depends on polarization of exciting pulses. It is suggested, through the dependence, that the induced annihilation of a two-photon coherent state of biexcitons plays a negligibly small role in the FWM spectra compared with the excitonic population-grating effect.

Quantum correlation coefficient for position and momentum

We study the general statistical measures of covariance and correlation coefficient for noncommuting quantum observables as exemplified by position and momentum operators in quantum optics. High values of the position-momentum quantum correlation coefficient are achievable with two-photon coherent states and Schrödinger cat states of the electromagnetic field for parametric values which renounce quadrature squeezing. We discuss quantum-optical paradigms to detect these measures.

Excited two-photon coherent state of the radiation field

A non-classical state of the radiation field, which we call the excited two-photon coherent state, is defined by m-times applying the photon creation operator on Yuen's two-photon coherent state . Its squeezing and sub-Poissonian character are demonstrated. Its generalized quasiprobability distribution functions in the representation are obtained and compared with those of the state and the Agarwal - Tara excited coherent state . It is observed that the exhibits remarkable non-classical properties differing from those of and .

Even and odd two-photon coherent states of the radiation field.

Even and odd two-photon coherent states of the radiation field.

Very small optimized phase uncertainties of two-photon coherent states.

Considering intense two-photon coherent states with minimum phase dispersion, the width of the phase probability distributions based on the Pegg-Barnett formalism, the Wigner function, and the Q function is discussed. Remarkable differences are found in certain regions depending on the mean photon number. These results are compared with the phase uncertainty of phase optimized states.

Realistic phase distributions derived from the Wigner function

The authors present the radius integrated Wigner function for a general pure state and discuss the coefficients of the different Fourier components. A fast converging expansion of these coefficients is given and its usefulness is shown for coherent light. Remarkable and measurable differences to the Pegg-Barnett formalism appear for weakly excited coherent states while for large excitations both dispersions agree in their leading terms. Large deviations are expected for specially designed two-photon coherent states.

General criterion for non-classical radiation fields

A general criterion for non-classical radiation fields is given which can answer whether a given state of a radiation field is non-classical or not and what kind of non-classical field it is if the field is non-classical. The authors have used the criterion to discuss photon number state, two-photon coherent state and number-phase squeezed state, and to prove that the radiation field in the thermal k-photon Jaynes-Cummings model, is always classical.

Nonclassical states generated via four-wave mixing

The nonclassical properties of quantum fields generated by four- or three-wave mixing are investigated on the basis of assumptions of two-photon coherent (squeezed) input states and the approximation of strong pumping. The influence of losses is shown supposing the model of mutually independent phonon reservoirs. The possibility of producing squeezed and sub-Poisson light outputs is demonstrated in single or superposed modes using computer solutions.