We propose a displacement-operator approach to some aspects of squeezed states for general multiphoton systems. The explicit displacement-operators of the squeezed vacuum and the coherent states are achieved and expresses as the ordinary exponential form. As a byproduct the coherent states of the $q$-oscillator are obtained by the {\it usual exponential} displacement-operator.
We introduce nonlinear canonical transformations that yield effective Hamiltonians of multiphoton down conversion processes, and we define the associated non-Gaussian multiphoton squeezed states as the coherent states of the multiphoton Hamiltonians. We study in detail the four-photon processes and the associated non-Gaussian four-photon squeezed states. The realization of squeezing, the behavior of the field statistics, and the structure of the phase space distributions show that these states realize a natural four-photon generalization of the two-photon squeezed states.
Multiphoton heralding generates large-amplitude squeezed Schrödinger cat states and parity-selective Fock superpositions from squeezed vacuum via an OPA
In terms of the characteristic functions of the quantum states, we present a complete operator description of a lossy photon-subtraction scheme. Feeding a single-mode squeezed vacuum into a variable beam splitter and counting the photons in one of the output channels, a broad class of multiphoton-subtracted squeezed vacuum states (MSSVSs) can be generated in other channel. Here the losses are considered in the beginning and the end channels in the circuit. Indeed, this scheme has been discussed in Ref. [Phys. Rev. A 100, 022341 (2019)]. However, different from the above work, we give all the 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 introduce a linear, canonical transformation of the fundamental single--mode field operators $a$ and $a^{\dagger}$ that generalizes the linear Bogoliubov transformation familiar in the construction of the harmonic oscillator squeezed states. This generalization is obtained by adding to the linear transformation a nonlinear function of any of the fundamental quadrature operators $X_{1}$ and $X_{2}$, making the original Bogoliubov transformation quadrature--dependent. Remarkably, the conditions of canonicity do not impose any constraint on the form of the nonlinear function, and lead to a set…
Extending the scheme developed for a single mode of the electromagnetic field in the preceding paper ``Structure of multiphoton quantum optics. I. Canonical formalism and homodyne squeezed states'', we introduce two-mode nonlinear canonical transformations depending on two heterodyne mixing angles. They are defined in terms of hermitian nonlinear functions that realize heterodyne superpositions of conjugate quadratures of bipartite systems. The canonical transformations diagonalize a class of Hamiltonians describing non degenerate and degenerate multiphoton processes. We determine the coherent…
We introduce a formalism of nonlinear canonical transformations for general systems of multiphoton quantum optics. For single-mode systems the transformations depend on a tunable free parameter, the homodyne local oscillator angle; for n-mode systems they depend on n heterodyne mixing angles. The canonical formalism realizes nontrivial mixings of pairs of conjugate quadratures of the electromagnetic field in terms of homodyne variables for single-mode systems; and in terms of heterodyne variables for multimode systems. In the first instance the transformations yield nonquadratic model Hamilton…
A simple technique is used to obtain a general formula for the Berry phase (and the corresponding Hannay angle) for an arbitrary Hamiltonian with an equally-spaced spectrum and appropriate ladder operators connecting the eigenstates. The formalism is first applied to a general deformation of the oscillator involving both squeezing and displacement. Earlier results are shown to emerge as special cases. The analysis is then extended to multiphoton squeezed coherent states and the corresponding anholonomies deduced.
We demonstrate the possibility of surpassing the quantum noise limit for simultaneous multi-axis spatial displacement measurements that have zero mean values. The requisite resources for these measurements are squeezed light beams with exotic transverse mode profiles. We show that, in principle, lossless combination of these modes can be achieved using the non-degenerate Gouy phase shift of optical resonators. When the combined squeezed beams are measured with quadrant detectors, we experimentally demonstrate a simultaneous reduction in the transverse x- and y- displacement fluctuations of 2.2…
We report on an orbital-angular-momentum-enhanced scheme for angular displacement estimation based on two-mode squeezed vacuum and parity detection. The sub-Heisenberg-limited sensitivity for angular displacement estimation is obtained in an ideal situation. Several realistic factors are also considered, including photon loss, dark counts, response-time delay, and thermal photon noise. Our results indicate that the effects of the realistic factors on the sensitivity can be offset by raising orbital angular momentum quantum number $\ell$. This reflects that the robustness and the practicability…
In this article, results from the previous paper (I) are applied to calculations of squeezed states for such well-known systems as the harmonic oscillator, free particle, linear potential, oscillator with a uniform driving force, and repulsive oscillator. For each example, expressions for the expectation values of position and momentum are derived in terms of the initial position and momentum, as well as in the $(α,z)$- and in the $(z,α)$-representations described in I. The dependence of the squeezed-state uncertainty products on the time and on the squeezing parameters are determined for each…
We demonstrate an angular momentum-enhanced protocol that permits an angular displacement estimation by using tunable squeezed Bell state and parity detection. We consider the resolution and the sensitivity, super-resolution is presented along with Heisenberg scaling sensitivity for arbitrary tunable factor, the tunable factor which can optimize the sensitivity is also discussed. Additionally, we analyze the advantages of using angular momentum via considering and comparing simulation results. Under the situation of the optimal tunable factor, the Heisenberg-limited sensitivity and $2\left(\el…
It has been shown that entanglement distillation of Gaussian entangled states by means of local photon subtraction can be improved by local Gaussian transformations. Here we show that a similar effect can be expected for the distillation of an asymmetric Gaussian entangled state that is produced by a single squeezed beam. We show that for low initial entanglement, our largely simplified protocol generates more entanglement than previous proposed protocols. Furthermore, we show that the distillation scheme also works efficiently on decohered entangled states as well as with a practical photon s…
We study entanglement concentration of continuous variable Gaussian states by local photon subtractions enhanced by coherent displacements. Instead of the previously considered symmetric two-mode squeezed vacuum states, we investigate the protocol for input states in the form of split single-mode squeezed vacuum, i.e., states obtained by mixing a single-mode squeezed vacuum with a vacuum state on a beam splitter, which is an experimentally highly relevant configuration. We analyze two scenarios in which the displacement-enhanced photon subtraction is performed either only on one, or on both of…
We present a review of theoretical and experimental aspects of multiphoton quantum optics. Multiphoton processes occur and are important for many aspects of matter-radiation interactions that include the efficient ionization of atoms and molecules, and, more generally, atomic transition mechanisms; system-environment couplings and dissipative quantum dynamics; laser physics, optical parametric processes, and interferometry. A single review cannot account for all aspects of such an enormously vast subject. Here we choose to concentrate our attention on parametric processes in nonlinear media, w…
In this paper we use the Lie algebra of space-time symmetries to construct states which are solutions to the time-dependent Schrödinger equation for systems with potentials $V(x,τ)=g^{(2)}(τ)x^2+g^{(1)}(τ)x +g^{(0)}(τ)$. We describe a set of number-operator eigenstates states, $\{Ψ_n(x,τ)\}$, that form a complete set of states but which, however, are usually not energy eigenstates. From the extremal state, $Ψ_0$, and a displacement squeeze operator derived using the Lie symmetries, we construct squeezed states and compute expectation values for position and momentum as a function of time, $τ$.…
Optimal Displacement Sensing with Spin-Dependent Squeezed States
University of Amsterdam（阿姆斯特丹大学）；QuSoft；University of Sydney（悉尼大学）；Sydney Nano Institute, University of Sydney（悉尼大学悉尼纳米研究所）；Indian Institute of Technology Madras（马德拉斯印度理工学院）；Center for Quantum Information, Communication and Computing, Indian Institute of Technology Madras（马德拉斯印度理工学院量子信息、通信与计算中心）
We report on a hitherto unexplored application of squeezed light: for quantum-enhancement of mechanical transduction sensitivity in microcavity optomechanics. Using a toroidal silica microcavity, we experimentally demonstrate measurement of the transduced phase modulation signal with a sensitivity $-0.72(\pm 0.01)$\,dB below the shot noise level. This is achieved for resonant probing in the highly under-coupled regime, by preparing the probe in a weak coherent state with phase squeezed vacuum states at sideband frequencies.
Displacement of propagating quantum states of light is a fundamental operation for quantum communication. It enables fundamental studies on macroscopic quantum coherence and plays an important role in quantum teleportation protocols with continuous variables. In our experiments we have successfully implemented this operation for propagating squeezed microwave states. We demonstrate that, even for strong displacement amplitudes, there is no degradation of the squeezing level in the reconstructed quantum states. Furthermore, we confirm that path entanglement generated by using displaced squeezed…
We consider the phase space for a system of $n$ identical qudits (each one of dimension $d$, with $d$ a primer number) as a grid of $d^{n} \times d^{n}$ points and use the finite field $GF(d^{n})$ to label the corresponding axes. The associated displacement operators permit to define $s$-parametrized quasidistribution functions in this grid, with properties analogous to their continuous counterparts. These displacements allow also for the construction of finite coherent states, once a fiducial state is fixed. We take this reference as one eigenstate of the discrete Fourier transform and study …
Gaussian boson sampling (GBS) is quantum sampling task in which one has to draw samples from the photon-number distribution of a large-dimensional nonclassical squeezed state of light. In an effort to make this task intractable for a classical computer, experiments building GBS machines have mainly focused on increasing the dimensionality and squeezing strength of the nonclassical light. However, no experiment has yet demonstrated the ability to displace the squeezed state in phase-space, which is generally required for practical applications of GBS. In this work, we build a GBS machine which …
As an aid to understanding the {\it displacement operator} definition of squeezed states for arbitrary systems, we investigate the properties of systems where there is a Holstein-Primakoff or Bogoliubov transformation. In these cases the {\it ladder-operator or minimum-uncertainty} definitions of squeezed states are equivalent to an extent displacement-operator definition. We exemplify this in a setting where there are operators satisfying $[A, A^{\dagger}] = 1$, but the $A$'s are not necessarily the Fock space $a$'s; the multiboson system. It has been previously observed that the ground state…
We propose a method of an improving quality of a ring cavity which is imperfect due to non-unit mirror reflectivity. The method is based on using squeezed states of light pulses illuminating the mirror and gradual homodyne detection of a radiation escaping from the cavity followed by single displacement and single squeezing operation performed on the released state. We discuss contribution of this method in process of storing unknown coherent and known squeezed state and generation of squeezing in the optical ring cavities.
We present a scheme for achieving amplification of the displacement of the mirror in optomechanical cavity using single-photon postselection where the mirror is initially prepared in squeezed coherent state. The amplification depends on the enhanced fluctuations of the squeezed coherent state, and it is is caused by the noncommutativity of quantum mechanics relying on the squeezed coherent state, which can not be explained by the standard weak measurement [1,25].
A closed form expression for the higher-power coherent states (eigenstates of $a^{j}$) is given. The cases j=3,4 are discussed in detail, including the time-evolution of the probability densities. These are compared to the case j=2, the even- and odd-coherent states. We give the extensions to the "effective" displacement-operator, higher-power squeezed states and to the ladder-operator/minimum-uncertainty, higher-power squeezed states. The properties of all these states are discussed.
In this work, we have applied the integrals of motion method in a nonunitary approach and so obtained the time-dependent displacement and squeezed parameters of the coherent squeezed states (CSS). On its turn, CSS for one-dimensional systems with general time-dependent quadratic Hamiltonian are constructed. We discuss the properties of these states, in particular, minimization of uncertainty relation and transition probabilities. As an application, we calculate the CSS of an oscillator with a time-dependent frequency and shown that the solution can be obtained from these well-known Mathieu's e…
Using continuous wave superposition of spatial modes, we demonstrate experimentally displacement measurement of a light beam below the standard quantum limit. Multimode squeezed light is obtained by mixing a vacuum squeezed beam and a coherent beam that are spatially orthogonal. Although the resultant beam is not squeezed, it is shown to have strong internal spatial correlations. We show that the position of such a light beam can be measured using a split detector with an increased precision compared to a classical beam. This method can be used to improve the sensitivity of small displacement …
Using the transformations from paper I, we show that the Schrödinger equations for: (1)systems described by quadratic Hamiltonians, (2) systems with time-varying mass, and (3) time-dependent oscillators, all have isomorphic Lie space-time symmetry algebras. The generators of the symmetry algebras are obtained explicitly for each case and sets of number-operator states are constructed. The algebras and the states are used to compute displacement-operator coherent and squeezed states. Some properties of the coherent and squeezed states are calculated. The classical motion of these states is deom…
Re-examination of the role of displacement and photon catalysis operation in continuous variable measurement device-independent quantum key distribution
The generally deformed oscillator (GDO) and its multiphoton realization as well as the coherent and squeezed vacuum states are studied. We discuss, in particular, the GDO depending on a complex parameter q (therefore we call it q-GDO) together with the finite dimensional cyclic representations. As a realistic physical system of GDO the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator. It is pointed out that starting from the q-GDO with q root of unity one can define the hermitian phase operators…
It is shown that the maximal phase sensitivity of a two-path interferometer with high-intensity coherent light and squeezed vacuum in the input ports can be achieved by photon-number-resolving detection of only a small number of photons in a dark output port. It is then possible to achieve the quantum Cramér-Rao bound of the two-path interferometer using only the field displacement dependence of the photon number statistics in the single mode output of the dark port represented by a field-displaced squeezed vacuum state. We find that, at small field displacements, it is not sufficient to use t…
Bifractional displacement operators, are introduced by performing two fractional Fourier transforms on displacement operators. They are shown to be special cases of elements of the group G, that contains both displacements and squeezing transformations. Acting with them on the vacuum we get various classes of coherent states, which we call bifractional coherent states. They are special classes of squeezed states which can be used for interpolation between various quantities in phase space methods. Using them we introduce bifractional Wigner functions A(?, ?; ??, ??), which are a two-dimensiona…
We generalized the squeeze and displacement operators of the one-dimensional harmonic oscillator to the three-dimensional case and based on these operators we construct the corresponding coherent and squeezed states. We have also calculated the Wigner function for the three-dimensional harmonic oscillator and from the analysis of time evolution of this function, the quantum Liouville equation is also presented. Further properties of the quantum states including Mandel's Q and quadrature squeezing parameters are discussed as well.
We propose an approach that enables to use it for construction of rotations of coherent states, in particular, it gives a possibility to construct Hadamard gate for the coherent states. Our approach is based on representation of arbitrary one-mode pure state in free-travelling fields, in particular superposition of coherent states (SCSs), in terms of displaced number states with arbitrary amplitude of displacement. Studied optical scheme is based on alternation of photon additions and displacement operators (in general case, photon additions and displacements are required) with seed coherent s…
While the objective of conventional quantum key distribution (QKD) is to secretly generate and share the classical bits concealed in the form of maximally mixed quantum states, that of private quantum channel (PQC) is to secretly transmit individual quantum states concealed in the form of maximally mixed states using shared one-time pad and it is called Gaussian private quantum channel (GPQC) when the scheme is in the regime of continuous variables. We propose a GPQC enhanced with squeezed coherent states (GPQCwSC), which is a generalization of GPQC with coherent states only (GPQCo) [Phys. Rev…
Mesoscopic superpositions of distinguishable coherent states provide an analog to the Schrödinger's cat thought experiment. For mechanical oscillators these have primarily been realised using coherent wavepackets, for which the distinguishability arises due to the spatial separation of the superposed states. Here, we demonstrate superpositions composed of squeezed wavepackets, which we generate by applying an internal-state dependent force to a single trapped ion initialized in a squeezed vacuum state with 9 dB reduction in the quadrature variance. This allows us to characterise the initial sq…
We consider the distinguishability of Gaussian states from the view point of continuous-variable quantum cryptography using post-selection. Specifically, we use the probability of error to distinguish between two pure coherent (squeezed) states and two particular mixed symmetric coherent (squeezed) states where each mixed state is an incoherent mixture of two pure coherent (squeezed) states with equal and opposite displacements in the conjugate quadrature. We show that the two mixed symmetric Gaussian states (where the various components have the same real part) never give an eavesdropper more…
The fundamental properties of recently introduced stretched coherent states are investigated. It has been shown that stretched coherent states retain the fundamental properties of standard coherent states and generalize the resolution of unity, or completeness condition, and the probability distribution that $n$ photons are in a stretched coherent state. The stretched displacement and stretched squeezing operators are introduced and the multiplication law for stretched displacement operator is established. The results of the action of the stretched displacement and stretched squeezing operator…
We show that the binomial states (BS) of Stoler {\it et al.} admit the ladder and displacement operator formalism. By generalizing the ladder operator formalism we propose an eigenvalue equation which possesses the number and the squeezed states as its limiting solutions. The explicit forms of the solutions, to be referred to as the {\it generalized binomial states} (GBS), are given. Corresponding to the wide range of the eigenvalue spectrum these GBS have as widely different properties. Their limits to number and {\it squeezed} states are investigated in detail. The time evolution of BS is ob…
We introduce a general formalism, based on the stochastic formulation of quantum mechanics, to obtain localized quasi-classical wave packets as dynamically controlled systems, for arbitrary anharmonic potentials. The control is in general linear, and it amounts to introduce additional quadratic and linear time-dependent terms to the given potential. In this way one can construct for general systems either coherent packets moving with constant dispersion, or dynamically squeezed packets whose spreading remains bounded for all times. In the standard operatorial framework our scheme corresponds t…
In a recent paper [Nieto M M 1996 Quantum and Semiclassical Optics, 8 1061; quant-ph/9605032], the one dimensional squeezed and harmonic oscillator time-displacement operators were reordered in coordinate-momentum space. In this paper, we give a general approach for reordering multi-dimensional exponential quadratic operator(EQO) in coordinate-momentum space. An explicit computational formula is provided and applied to the single mode and double-mode EQO through the squeezed operator and the time displacement operator of the harmonic oscillator.
We extend the definition of generalized coherent states to include the case of time-dependent dispersion. We introduce a suitable operator providing displacement and dynamical rescaling from an arbitrary ground state. As a consequence, squeezing is naturally embedded in this framework, and its dynamics is ruled by the evolution equation for the dispersion. Our construction provides a displacement-operator method to obtain the squeezed states of arbitrary systems.
Teleportation and storage of continuous variable states of light and atoms are essential building blocks for the realization of large scale quantum networks. Rigorous validation of these implementations require identifying, and surpassing, benchmarks set by the most effective strategies attainable without the use of quantum resources. Such benchmarks have been established for special families of input states, like coherent states and particular subclasses of squeezed states. Here we solve the longstanding problem of defining quantum benchmarks for general pure Gaussian single-mode states with …
