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 system.
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.
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…
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…
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…
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, $τ$.…
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…
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）
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 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.
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 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.
We present squeezed light that is robust against spatial mode mismatch (beam displacement, tilt, and beam-size difference), which is generated from a self-imaging optical parametric oscillator below the threshold. We investigate the quantum properties of the generated light when the oscillator is detuned from the ideal self-imaging condition for stable operation. We find that the generated light is more robust to mode mismatch than single-mode squeezed light having the same squeezing level, and it even outperforms the single-mode infinitely squeezed light as the strength of mode mismatch incre…
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 …
Re-examination of the role of displacement and photon catalysis operation in continuous variable measurement device-independent quantum key distribution
We study continuous variable coherence of phase-dependent squeezed state based on an extended Hanbury Brown-Twiss scheme. High-order coherence is continuously varied by adjusting squeezing parameter $r$, displacement $α$, and squeezing phase $θ$. We also analyze effects of background noise $γ$ and detection efficiency $η$ on the measurements. As the squeezing phase shifts from 0 to $π$, the photon statistics of the squeezed state continuously change from the anti-bunching ($g^{(n)}<1$) to super-bunching ($g^{(n)}>n!$) which shows a transition from particle nature to wave nature. The experiment…
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…
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 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.
In this paper, we theoretically investigate the displacement and momentum fluctuations spectra of the movable mirror in a standard optomechanical system driven by a finite bandwidth squeezed vacuum light accompanying a coherent laser field. Two cases in which the squeezed vacuum is generated by degenerated and non-degenerate parametric oscillators (DPO and NDPO) are considered. We find that for the case of finite bandwidth squeezed vacuum injection, the two spectra exhibit unique features, which strongly differ from those of broadband squeezing excitation. In particular, the spectra exhibit a …
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…
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 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…
Laser cooling is a fundamental technique used in primary atomic frequency standards, quantum computers, quantum condensed matter physics and tests of fundamental physics, among other areas. It has been known since the early 1990s that laser cooling can, in principle, be improved by using squeezed light as an electromagnetic reservoir; while quantum feedback control using a squeezed light probe is also predicted to allow improved cooling. Here, we implement quantum feedback control of a micro-mechanical oscillator for the first time with a squeezed probe field. This allows quantum-enhanced feed…
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…
Quantum enhanced sensing is a powerful technique in which nonclassical states are used to improve the sensitivity of a measurement. For enhanced mechanical displacement sensing, squeezed states of light have been shown to reduce the photon counting noise that limits the measurement noise floor. It has long been predicted, however, that suppressing the noise floor with squeezed light should produce an unavoidable increase in radiation pressure noise that drives the mechanical system. Such nonclassical radiation pressure forces have thus far been hidden by insufficient measurement strengths and …
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.
The Standard Quantum Limit (SQL) in interferometric displacement measurement imposes a restriction on the precision of the measurement due to its quantum back-action noise. This limit can be surpassed over a broad frequency band by injecting squeezed vacuum states with a frequency dependent (FD) quadrature angle into the measurement system. Here, we demonstrate the generation of FD squeezed vacuum states by utilizing Einstein-Podolsky-Rosen (EPR) entangled states. The frequency dependence of the squeezed state can be tuned by conditionally filtering the measurement once the two entangled field…
Generalized squeezing rotating-wave approximation (GSRWA) is proposed by employing both the displacement and the squeezing transformations. A solvable Hamiltonian is reformulated in the same form as the ordinary RWA ones. For a qubit coupled to oscillators experiment, a well-defined Schrödinger-cat-like entangled state is given by the displaced-squeezed oscillator state instead of the original displaced state. For the isotropic Rabi case, the mean photon number and the ground-state energy are expressed analytically with additional squeezing terms, exhibiting a substantial improvement of the GS…
Optical phase estimation is a vital measurement primitive that is used to perform accurate measurements of various physical quantities like length, velocity and displacements. The precision of such measurements can be largely enhanced by the use of entangled or squeezed states of light as demonstrated in a variety of different optical systems. Most of these accounts however deal with the measurement of a very small shift of an already known phase, which is in stark contrast to ab-initio phase estimation where the initial phase is unknown. Here we report on the realization of a quantum enhanced…
We implement direct readout of the symmetric characteristic function of quantum states of the motional oscillation of a trapped calcium ion. Using suitably chosen internal electronic state-dependent displacements based on bi-chromatic laser fields we map the expectation value of the real or imaginary part of the displacement operator to the internal states, which are subsequently read out using fluorescence detection. Combining the two readout results provides full information about the symmetric characteristic function. We characterize the performance of the technique by applying it to a rang…
Quantum vacuum fluctuations impose strict limits on precision displacement measurements, those of interferometric gravitational-wave detectors among them. Introducing squeezed states into an interferometer's readout port can improve the sensitivity of the instrument, leading to richer astrophysical observations. However, optomechanical interactions dictate that the vacuum's squeezed quadrature must rotate by 90 degrees around 50Hz. Here we use a 2-m-long, high-finesse optical resonator to produce frequency-dependent rotation around 1.2kHz. This demonstration of audio-band frequency-dependent s…
Vacuum radiation has been the subject of theoretical study in both cosmology and condensed matter physics for many decades. Recently there has been impressive progress in experimental realizations as well. Here we study vacuum radiation when a field mode is driven both parametrically and by a classical source. We find that in the Heisenberg picture the field operators of the mode undergo a Bogolyubov transformation combined with a displacement, in the Schrödinger picture the oscillator evolves from the vacuum to a squeezed coherent state. Whereas the Bogolyubov transformation is the same as wo…
We show that the motion of a cold trapped ion can be squeezed by modulating the intensity of a phase-stable optical lattice placed inside the trap. As this method is reversible and state selective it effectively implements a controlled-squeeze gate. We show how to use this resource, that can be useful for quantum information processing with continuous variables, in order to prepare coherent superpositions of states which are squeezed along complementary quadratures. We show that these states, which we denote as "${\mathcal X}$-states", exhibit high sensitivity to small displacements along two …
Quantum sensing and quantum information processing use quantum advantages such as squeezed states that encode a quantity of interest with higher precision and generate quantum correlations to outperform classical methods. In harmonic oscillators, the rate of generating squeezing is set by a quantum speed limit. Therefore, the degree to which a quantum advantage can be used in practice is limited by the time needed to create the state relative to the rate of unavoidable decoherence. Alternatively, a sudden change of harmonic oscillator's frequency projects a ground state into a squeezed state w…
Reservoir engineering enables the robust preparation of pure quantum states in noisy environments. We show how a new family of quantum states of a mechanical oscillator can be stabilized in a cavity that is parametrically coupled to both the mechanical displacement and the displacement squared. The cavity is driven with three tones, on the red sideband, on the cavity resonance and on the second blue sideband. The states so stabilized are (squeezed and displaced) superpositions of a finite number of phonons. They show the unique feature of encompassing two prototypes of nonclassicality for boso…
The measurement of weak continuous forces exerted on a mechanical oscillator is a fundamental problem in various physical experiments. It is fundamentally impeded by quantum back-action from the meter used to sense the displacement of the oscillator. In the context of interferometric displacement measurements, we here propose and demonstrate the working principle of a scheme for coherent back-action cancellation. By measuring the amplitude quadrature of the light reflected from a detuned optomechanical cavity inside which a stiff optical spring is generated, back-action can be cancelled in a n…
The non-commutativity of position and momentum observables is a hallmark feature of quantum physics. However this incompatibility does not extend to observables which are periodic in these base variables. Such modular-variable observables have been suggested as tools for fault-tolerant quantum computing and enhanced quantum sensing. Here we implement sequential measurements of modular variables in the oscillatory motion of a single trapped ion, using state-dependent displacements and a heralded non-destructive readout. We investigate the commutative nature of modular variable observables by de…
