The time evolution of even and odd squeezed states, as well as that of squeezed number states, has been given in simple, analytic form. This follows experimental work on trapped ions which has demonstrated even and odd coherent states, number states, and squeezed (but not displaced) ground states. We review this situation and consider the extension to even and odd squeezed number states. Questions of uncertainty relations are also discussed.
We derive a general expression of the quantum Fisher information for a Mach-Zehnder interferometer, with the port inputs of an \emph{arbitrary} pure state and a squeezed thermal state. We find that the standard quantum limit can be beaten, when even or odd states are applied to the pure-state port. In particular, when the squeezed thermal state becomes a thermal state, all the even or odd states have the same quantum Fisher information for given photon numbers. For a squeezed thermal state, optimal even or odd states are needed to approach the Heisenberg limit. As examples, we consider several…
A direct proof of the resolution of the identity in the odd sector of the Fock space in terms of squeezed number states $D(ξ)|2m+1>; D(ξ) = \exp((ξa^{\dagger2}-ξ^*{a^2})/2)$ is given. The proof entails evaluation of an integral involving Jacobi polynomials. This is achieved by the use of Racah identities.
A completely analytic description is given of the motion of a trapped ion which is in either an even or an odd squeezed state. Comparison is made to recent results on the even or odd coherent states, and possible experimental work is discussed.
Particle distributions in squeezed states, even and odd coherent states are given in terms of multivariable Hermite polynomials. The Q--function and Wigner function for nonclassical field states are discussed.
The two-mode even and odd coherent states and two-mode squeezed correlated state are discussed. Photon distribution functions, means, dispersions, Fano factor for even and odd coherent states and squeezed correlated state are calculated. The photon distribution function for two-mode squeezed correlated state is obtained. The tomograms of two-mode even and odd coherent states and squeezed correlated states are investigated within the framework of a symplectic tomography scheme. It is shown that the phenomenon of entanglement appears in the system. Two different measures of entanglement are empl…
Several teleportation protocols, namely those using entangled coherent states, entangled squeezed states, and the single-photon EPR state, are all shown to be particular instances of a more general scheme that relies on the detection of an odd number of photons.
We proposed and analyzed a scheme to generate large-size Schrödinger cat states based on linear operations of Fock states and squeezed vacuum states and conditional measurements. By conducting conditional measurements via photon number detectors, two unbalanced Schrödinger kitten states combined by a beam splitter can be amplified to a large-size cat state with the same parity. According to simulation results, two Schrödinger odd kitten states of $β=1.06$ and $β=1.11$ generated from one-photon-subtracted squeezed vacuum states of $-$3 dB, are amplified to an odd cat state of $β=1.73$ with a fi…
We investigate a single atom cavity-QED system directly driven by a broadband squeezed light. We demonstrate how the squeezed radiation can be used to sense the presence of a single atom in a cavity. This happens by transferring one of the photons from the field in a state with even number of photons to the atom and thereby populating odd number Fock states. Specifically, the presence of the atom is sensed by remarkable changing in the presence of one photon and the loss of squeezing of the cavity field. A complete study of quantum fluctuations and the excitation of multiphoton transitions is …
We explore the effects of quantum mechanical squeezing on the nonequilibrium fluctuations of bosonic transport between two squeezed harmonic reservoirs and a two level system. A standard full counting statistics technique based on a quantum master equation is employed. We derive a nonzero thermodynamic affinity under equal temperature setting of the two squeezed reservoirs. The odd cumulants are shown to be independent of squeezing under symmetric conditions, whereas the even cumulants depend nonlinearly on the squeezing parameters. The odd and even cumulants saturate at two different but uniq…
A superconducting qubit in the strong dispersive regime of a circuit quantum electrodynamics system is a powerful probe for microwave photons in a cavity mode. In this regime, a qubit spectrum is split into multiple peaks, with each peak corresponding to an individual photon number in the cavity (discrete ac Stark shift). Here, we measure the qubit spectrum in the cavity that is driven continuously with a squeezed vacuum field generated by a Josephson parametric amplifier. By fitting the qubit spectrum with a model which takes into account the finite qubit excitation power, the photon number d…
In this paper we discuss the quantum properties for superposition of squeezed displaced number states against multiphoton Jaynes-Cummings model (JCM). In particular, we investigate atomic inversion, photon-number distribution, purity, quadrature squeezing, Mandel $Q$ parameter and Wigner function. We show that the quadrature squeezing for three-photon absorption case can exhibit revivals and collapses typical to those occurring in the atomic inversion for one-photon absorption case. Also we prove that for odd number absorption parameter there is a connection between the evolution of the atomic…
We provide a scheme for the generation of controlled entangled number states of Bose-Einstein condensates in multiple wells, and also provide a novel method for the creation of squeezed states without severe adiabatic constraints on barrier heights. The condensate in a multiple well trap can be evolved, starting with a specific initial phase difference between the neighboring wells, to a tunable entangled state or a squeezed state. We propose a general formula for the initial phase difference between the neighboring wells that is valid for any number of wells, even and odd.
The theory of quantum propagator and time--dependent integrals of motion in quantum optics is reviewed as well as the properties of Wigner function, Q--function, and coherent state representation. Propagators and wave functions of a free particle, harmonic oscillator, and the oscillator with varying frequency are studied using time--dependent linear in position and momentum integrals of motion. Such nonclassical states of light (of quantum systems) as squeezed states, correlated states, even and odd coherent states (Schrödinger cat states) are considered. Photon distribution functions of Schrö…
A new class of states of light is introduced that is complementary to the well-known squeezed states. The construction is based on the general solution of the three-term recurrence relation that arises from the saturation of the Schrödinger inequality for the quadratures of a single-mode quantized electromagnetic field. The new squeezed states are found to be linear superpositions of the photon-number states whose coefficients are determined by the associated Hermite polynomials. These results do not seem to have been noticed before in the literature. As an example, the new class of squeezed s…
We investigate the cavity excitation spectrum and the photon number distribution in a cavity QED system driven by a broadband squeezed vacuum. In an empty cavity, we show that only states with even number of photons can be measured under resonant condition since the squeezed vacuum consists of states with even number of photons only. When a single atom is trapped in the cavity, the strong coupling between the atom and cavity results in energy splittings of the system, and there exist two peaks in the cavity excitation spectrum at two-photon transition frequencies. At the central frequency, how…
Improved entanglement indicators for optical fields and its application in the event-ready experiment for bright squeezed vacuum with induced non-gaussianity
We report on the experimental observation of quantum-network-compatible light described by a non-positive Wigner function. The state is generated by photon subtraction from a squeezed vacuum state produced by a continuous wave optical parametric amplifier. Ideally, the state is a coherent superposition of odd photon number states, closely resembling a superposition of weak coherent states (a Schroedinger cat), with the leading contribution from a single photon state in the low parametric gain limit. Light is generated in a nearly perfect spatial mode with a Fourier-limited frequency bandwidth …
We revisit the dissipative approach to producing and stabilizing spin-squeezed states of an ensemble of $N$ two-level systems, providing a detailed analysis of two surprising yet generic features of such protocols. The first is a macroscopic sensitivity of the steady state to whether $N$ is even or odd. We discuss how this effect can be avoided (if the goal is parity-insensitive squeezing), or could be exploited as a new kind of sensing modality to detect the addition or removal of a single spin. The second effect is an anomalous emergent long timescale and a "prethermalized" regime that occur…
Theoretical analysis is given for a two-photon subtraction from a continuous wave (CW) squeezed vacuum with finite time separation between two detection events. In the CW photon subtraction process, the generated states are inevitably described by temporal mutimode states. Our approach is based on Schrödinger picture which provides mathematically simple forms and an intuitive understanding of its multimode structure. We show that, in our process, the photon subtracted squeezed vacuum is generated in two temporal modes and one of these modes acts as an ancillary mode to make the other one a lar…
Quantum information processing using photons has recently been stimulated by the suggestion to use linear optics, single photon sources and detectors. The recent work by Knill has also shown that errors in photon detectors leads to a high error rate threshold (around 29%). An important missing element are good single photon sources. In this paper we show how to make a single photon source using squeezed states, linear optics and conditional measurement. We use degenerate squeezed vacuum states, in contrast to the normal non-degenerate squeezed vacuum states used for single photon production. W…
Photon counting measurement has been regarded as the optimal measurement scheme for phase estimation in the squeezed-state interferometry, since the classical Fisher information equals to the quantum Fisher information and scales as $\bar{n}^2$ for given input number of photons $\bar{n}$. However, it requires photon-number-resolving detectors with a large enough resolution threshold. Here we show that a collection of $N$-photon detection events for $N$ up to the resolution threshold $\sim \bar{n}$ can result in the ultimate estimation precision beyond the shot-noise limit. An analytical formul…
A Mach-Zender interferometer with a gaussian number-difference squeezed input state can exhibit sub-shot-noise phase resolution over a large phase-interval. We obtain the optimal level of squeezing for a given phase-interval $Δθ_0$ and particle number $N$, with the resulting phase-estimation uncertainty smoothly approaching $3.5/N$ as $Δθ_0$ approaches 10/N, achieved with highly squeezed states near the Fock regime. We then analyze an adaptive measurement scheme which allows any phase on $(-π/2,π/2)$ to be measured with a precision of $3.5/N$ requiring only a few measurements, even for very la…
We study the nonclassicality of the evolution of a superposition of an arbitrary number of photon-added squeezed coherent Schrödinger cat states in a nonlinear Kerr medium. The nonlinearity of the medium gives rise to the periodicities of the quantities such as the Wehrl entropy $S_{Q}$ and the negativity $δ_{W}$ of the $W$-distribution, and a series of local minima of these quantities arise at the rational submultiples of the said period. At these local minima the evolving state coincides with the transient Yurke-Stoler type of photon-added squeezed kitten states, which, for the choice of the…
We report electromagnetically induced transparency using quantized fields in optomechanical systems. The weak probe field is a narrow band squeezed field. We present a homodyne detection of EIT in the output quantum field. We find that the EIT dip exists even though the photon number in the squeezed vacuum is at the single photon level. The EIT with quantized fields can be seen even at temperatures of the order of 100 mK paving the way for using optomechanical systems as memory elements.
We construct even and odd nonlinear coherent states of a parametric oscillator and examine their nonclassical properties.It has been shown that these superpositions exhibit squeezing and photon antibunching which change with time.
A complete set of solutions |z,u,v>_{sa} of the eigenvalue equation (ua^2+va^{dagger 2})|z,u,v> = z|z,u,v> ([a,a^{dagger}]=1) are constructed and discussed. These and only these states minimize the Schrödinger uncertainty inequality for the squared amplitude (s.a.) quadratures. Some general properties of Schrödinger intelligent states (SIS) |z,u,v> for any two observables X, Y are discussed, the sets of even and odd s.a. SIS |z,u,v;+,-> being studied in greater detail. The set of s.a. SIS contain all even and odd coherent states (CS) of Dodonov, Malkin and Man'ko, the Perelomov SU(1,1) CS and …
We address the possibility of using even/odd states of orbital angular momentum (OAM) of photons for the quantum information tasks. Single photon qubit states and two photon entangled states in even/odd basis of OAM are considered. We present a method for the tomography and general projective measurement in even/odd basis. With the general projective measurement, we show the Bell violation and quantum cryptography with Bell's inequality. We also describe hyper and hybrid entanglement of even/odd OAM states along with polarization, which can be applied in the implementation of quantum protocols…
Explicit expressions for optical tomograms of the photon-added coherent states, even/odd photon-added coherent states and photon-added thermal states are given in terms of Hermite polynomials. Suggestions for experimental homodyne detection of the considered photon states are presented.
This paper has been withdrawn. It is based on numerical results limited by computing resources to N=3000 atoms. Using a newly understood geometric method we find that the observed scaling with N saturates at around N=7000 or even higher. In light of this new finding we withdraw the paper and will submit a revised manuscript reflecting our new understanding.
The use of artificial atoms as an active lasing medium opens a way to construct novel sources of nonclassical radiation. An example is the creation of photon-number squeezed light. Here we present a design of a laser consisting of multiple Cooper-pair transistors coupled to a microwave resonator. Over a broad range of experimentally realizable parameters, this laser creates photon-number squeezed microwave radiation, characterized by a Fano factor $F \ll 1$, at a very high resonator photon number. We investigate the impact of gate-charge disorder in a Cooper-pair transistor and show that the s…
We report demonstrations of both quadrature squeezed vacuum and photon number difference squeezing generated in an integrated nanophotonic device. Squeezed light is generated via strongly driven spontaneous four-wave mixing below threshold in silicon nitride microring resonators. The generated light is characterized with both homodyne detection and direct measurements of photon statistics using photon number-resolving transition edge sensors. We measure $1.0(1)$~dB of broadband quadrature squeezing (${\sim}4$~dB inferred on-chip) and $1.5(3)$~dB of photon number difference squeezing (${\sim}7$…
