We investigate the presence of spin- and planar- squeezing in generalized superpositions of atomic (or spin) coherent states (ACS). Spin-squeezing has been shown to be a useful tool in determining the presence of entanglement in multipartite systems, such as collections of two-level atoms, as well as being an indication of reduced projection noise and sub-shot-noise limited phase uncertainty in Ramsey spectroscopy, suitable for measuring phases $ϕ\sim 0$. On the other hand, planar-squeezed states display reduced projection noise in two directions simultaneously and have been shown to lead to enhanced metrological precision in measuring phases without the need for explicit prior knowledge of the phase value. In this paper, we show that the generalized superposition state can be parametrized to display both spin-squeezing along all orthogonal axes and planar-squeezing along all orthogonal planes for all values of $J>1/2$. We close with an application of the maximally spin- and planar-squeezed states to quantum metrology.
We consider the superpositions of spin coherent states and study the coherence properties and spin squeezing in these states. The spin squeezing is examined using a new version of spectroscopic squeezing criteria. The results show that the antibuching effect can be enhanced and spin squeezing can be generated in the superpositions of two spin coherent states.
In this study we investigate the spin squeezing in superposition of a Biaxial state [1,2] and Bell state. Numerical and analytical solutions for the length of mean spin, mean spin direction and spin squeezing are given. It is shown that both the mean spin direction and spin squeezing parameter are determined by the coefficients of superposition and the relative phase.
In the present paper, we study spin squeezing under decoherence in the superposition of tripartite maximally entangled GHZ and W states. Here we use amplitude damping, phase damping and depolarisation channel. We have investigated the dynamics of spin squeezing with the interplay of superposition and decoherence parameters with different directions of the mean spin vector. We have found the mixture of GHZ and W states is robust against spin squeezing generation for amplitude damping and phase damping channels for certain directions of the mean spin vector. However, the depolarisation channel p…
We analyze effects of squeezing upon superpositions of coherent states (SCSs) and entangled coherent states (ECSs) for Bell-inequality tests. We find that external squeezing can always increase the degrees of Bell violations, if the squeezing direction is properly chosen, for the case of photon parity measurements. On the other hand, when photon on/off measurements are used, the squeezing operation can enhance the degree of Bell violations only for moderate values of amplitudes and squeezing. We point out that a significant improvement is required over currently available squeezed SCSs in orde…
We propose an efficient approach to generate the superposed macroscopically squeezed states with enhanced squeezing in a two-mode optomechanical system. This can be achieved by introducing a sinusoidal modulation to either the cavity frequencies or the coupling strengths between two cavity modes. The squeezement of the oscillator can be significantly enhanced to 12.16 dB with single photon, once the relative ratio of coupling strength is optimized under proper conditions. Further enhanced squeezing can be obtained by carefully adjusting the system parameters. In terms of the Wigner quasi-proba…
Planar squeezed states, i.e. quantum states which are squeezed in two orthogonal spin components, have recently attracted attention due to their applications in atomic interferometry and quantum information [Q. Y. He et al, New J. Phys. 14, 093012 (2012)]. While canonical variables such as quadratures of the radiation field can be squeezed in at most one component, simultaneous squeezing in two orthogonal spin components can be achieved due to the angular momentum commutation relations. We present a novel scheme for planar squeezing via quantum non-demolition (QND) measurements in spin-1 syste…
Non-Gaussian states, and specifically the paradigmatic Schrödinger cat state, are well-known to be very sensitive to losses. When propagating through damping channels, these states quickly loose their non-classical features and the associated negative oscillations of their Wigner function. However, by squeezing the superposition states, the decoherence process can be qualitatively changed and substantially slowed down. Here, as a first example, we experimentally observe the reduced decoherence of squeezed optical coherent-state superpositions through a lossy channel. To quantify the robustness…
We introduce an entanglement-depth criterion optimized for planar quantum squeezed (PQS) states. It is connected with the sensitivity of such states for estimating an arbitrary, not necessarily small phase. We compare numerically our criterion with the well-known extreme spin squeezing condition of Sørensen and Mølmer [Phys. Rev. Lett. 86, 4431 (2001)] and show that our condition detects a higher depth of entanglement when both planar spin variances are squeezed below the standard quantum limit. We employ our theory to monitor the entanglement dynamics in a PQS state produced via quantum non-d…
We obtain a lower bound on the sum of two orthogonal spin component variances in a plane. This gives a novel planar uncertainty relation which holds even when the Heisenberg relation is not useful. We investigate the asymptotic, large $J$ limit, and derive the properties of the planar quantum squeezed states that saturate this uncertainty relation. These states extend the concept of spin squeezing to any two conjugate spin directions. We show that planar quantum squeezing can be achieved experimentally as the ground state of a Bose-Einstein condensate in two coupled potential wells with a crit…
We show how one can prepare and detect entanglement and Einstein-Podolsky-Rosen (EPR) steering between two distinguishable groups (modes) of atoms in a Bose-Einstein condensate (BEC) atom interferometer. Our paper extends previous work that developed criteria for two-mode entanglement and EPR steering based on the reduced variances of two spins defined in a plane. Observation of planar spin squeezing will imply entanglement, and sufficient planar spin squeezing implies EPR steering, between the two groups of atoms. By using a two-mode dynamical model to describe BEC interferometry experiments,…
Based on N different coherent states with equal weights and phase-space rotation symmetry, we introduce N-headed incoherent superposition states (NHICSSs) and N-headed coherent superposition states (NHCSSs). These N coherent states are associated with N-order roots of the same complex number. We study and compare properties of NHICSSs and NHCSSs, including average photon number, Mandel Q parameter, quadrature squeezing, Fock matrix elements and Wigner function. Among all these states, only 2HCSS (i.e., Schrodinger cat state) presents quadrature-squeezing effect. Our theoretical results can be …
In this paper, we study some quantum properties of a superposition of displaced squeezed two-mode vacuum and single-photon states, such as the second-order correlation function, the Cauchy-Schwartz inequality, quadrature squeezing, quasiprobability distribution functions and purity. This type of states includes two mechanisms, namely, interference in phase space and entanglement. We show that these states can exhibit sub-Piossonian statistics, squeezing and deviate from the classical Cauchy-Schwartz inequality. Moreover, the amount of entanglement in the system can be increased by increasing t…
We discuss the general matrix elements of the squeezing operator between number eigenstates of a harmonic oscillator (which may also represent a quantized mode of the electromagnetic radiation). These matrix elements have first been used by Popov and Perelomov (1969) long ago, in their thorough analysis of the parametric excitation of harmonic oscillators. They expressed the matrix elements in terms of transcendental functions, the associated Legendre functions. In the present paper we will show that these matrix elements can also be expressed by the classical Gegenbauer polynomials. This new …
Considering the concept of "{\it nonlinear coherent states}", we will study the interference effects by introducing the {\it "superposition of two classes of nonlinear coherent states"} which are $\fracπ{2}$ out of phase. The formalism has then been applied to a few physical systems as "harmonious states", "SU(1,1) coherent states" and "the center of mass motion of trapped ion". Finally, the nonclassical properties such as sub-Poissonian statistics, quadrature squeezing, amplitude-squared squeezing and Wigner distribution function of the superposed states have been investigated, numerically. E…
We study the relations between spin squeezing and concurrence, and find that they are qualitatively equivalent for an ensemble of spin-1/2 particles with exchange symmetry and parity, if we adopt the spin squeezing criterion given by the recent work (G. Toth et al. Phys. Rev. Lett. 99, 250405 (2007)). This suggests that the spin squeezing has more intimate relations with pairwise entanglement. We exemplify the result by considering a superposition of two Dicke states.
We investigate how the entanglement properties of a two-mode state can be improved by performing a coherent superposition operation of photon subtraction and addition, proposed by Lee and Nha [Phys. Rev. A 82, 053812 (2010)], on each mode. We show that the degree of entanglement, the EPR-type correlation, and the performance of quantum teleportation can be all enhanced for the output state when the coherent operation is applied to a two-mode squeezed state. The effects of the coherent operation are more prominent than those of the mere photon subtraction and the addition particularly in the sm…
Quantum non-Gaussian gate is a missing piece to the realization of continuous-variable universal quantum operations in the optical system. In a measurement-based implementation of the cubic phase gate, a lowest-order non-Gaussian gate, non-Gaussian ancillary states that has a property we call nonlinear squeezing are required. This property, however, has never been experimentally verified. In this paper, we generate a superposition between a vacuum state and a single photon state whose nonlinear squeezing are maximized by the optimization of the superposition coefficients. The nonlinear squeezi…
We derive a phase-entanglement criterion for two bosonic modes which is immune to number fluc- tuations, using the generalized Moore-Penrose inverse to normalize the phase-quadrature operator. We also obtain a phase-squeezing criterion that is immune to number fluctuations using similar techniques. These are utilized to obtain an operational definition of relative phase-measurement sensitivity, via analysis of phase measurement in interferometry. We show that these measures are proportional to enhanced phase-measurement sensitivity. The phase-entanglement criterion is a hallmark for a new type…
The manuscript investigates the entropy squeezing of a qubit (two-level atom) interacting with the cavity field in the Schrödinger cat states. The effects of the Schrödinger cat states of the field on the squeezing are examined. Our results suggest that entropy squeezing is not sensitive to the superposition features of two coherent states of the Schrödinger cat states. Furthermore, it is found that entropy squeezing occurs as the qubit initially close to its maximally coherent state, and the qubit initially in the maximally coherent state can generate optimal squeezing. This also demonstrates…
Recent years have witnessed revolutionary improvement in the production, manipulation, characterization and quantification of multiatom (multiqubit) states - because of their promising applications in high precision atomic clocks, atomic interferrometry, quantum metrology and quantum information protocols. In the characterization and quantification of non-classical atomic correlations, two concepts namely, spin squeezing and quantum entanglement emerge. While spin squeezing originates from the uncertainty relation between atomic collective angular momentum operators, entanglement is a peculiar…
Nonlinear coherent states (CSs) and their {\it dual families} were introduced recently. In this paper we want to obtain their superposition and investigate their non-classical properties such as antibunching effect, quadrature squeezing and amplitude squared squeezing. For this purpose two types of superposition are considered. In the first type we neglect the normalization factors of the two components of the dual pair, superpose them and then we normalize the obtained states, while in the second type we superpose the two normalized components and then again normalize the resultant states. As…
We have analyzed the quadrature squeezing in a cavity mode driven by coherent light and interacting with a two-level atom. We have found that the cavity mode is in a squeezed state, with the maximum quadrature squeezing being $50\%$ below the vacuum-state level. We have also considered the superposition of a pair of cavity modes, each driven by coherent light and interacting with a two-level atom. We have found that the superposed cavity modes are in a squeezed state and the squeezing occurs in both quadratures. We have established that the uncertainty relation perfectly holds for this case. I…
In this paper we explore the quantum behaviour of a SQUID ring which has a significant Josephson coupling energy. We show that that the eigenfunctions of the Hamiltonian for the ring can be used to create macroscopic quantum superposition states of the ring. We also show that the ring potential may be utilised to squeeze coherent states. With the SQUID ring as a strong contender as a device for manipulating quantum information, such properties may be of great utility in the future. However, as with all candidate systems for quantum technologies, decoherence is a fundamental problem. In this pa…
We implement the squeezing operation as a genuine quantum gate, deterministically and reversibly acting `online' upon an input state no longer restricted to the set of Gaussian states. More specifically, by applying an efficient and robust squeezing operation for the first time to non-Gaussian states, we demonstrate a two-way conversion between a particle-like single-photon state and a wave-like superposition of coherent states. Our squeezing gate is reliable enough to preserve the negativities of the corresponding Wigner functions. This demonstration represents an important and necessary step…
We make a comparative study of quadrature squeezing, photon-number distribution and Wigner function in a decayed quantum system. Specifically, for a field mode prepared initially in cat states interacting with a zero-temperature environment, we show that the rate of reduction of the nonclassical effects in this system is proportional to the occurrence of the decoherence process.
We investigate how entangled coherent states and superpositions of low intensity coherent states of non-Gaussian nature can be generated via non-resonant interaction between either two linearly or circularly polarized field modes and an ensemble of X-like four-level atoms placed in an optical cavity. We compare our results to recent experimental observations and argue that the non-Gaussian structure of the field states may be present in those systems.
We analyze an experimental method for creating interesting nonclassical states by processing the entanglement generated when two large coherent states interact in a cross-Kerr medium. We specifically investigate the effects of loss and noise in every mode of the experiment, as well as the effect of "binning" the post-selection outcomes. Even with these imperfections, we find an optimal set of currently-achievable parameters which would allow a proof-of-principle demonstration of number squeezing in states with large mean photon number. We discuss other useful states which can be generated with…
In this doctoral thesis we have studied the quantum properties of several models which have been classified as statical and dynamical systems. The first part has been devoted to investigate the properties of the statical models including the superposition of squeezed displaced number states with and without thermal noise. Also we have developed a new type of multidimensional squeeze operator including two different squeezing mechanisms. In the second part the dynamical models were given to show the interaction between modes in the nonlinear optical coupler.
The Barut-Girardello coherent states (BG CS) representation is extended to the noncompact algebras u(p,q) and sp(N,R) in (reducible) quadratic boson realizations. The sp(N,R) BG CS take the form of multimode ordinary Schrödinger cat states. Macroscopic superpositions of 2^{n-1} sp(N,R) CS (2^n canonical CS, n=1,2,...) are pointed out which are overcomplete in the N-mode Hilbert space and the relation between the canonical CS and the u(p,q) BG-type CS representations is established. The sets of u(p,q) and sp(N,R) BG CS and their discrete superpositions contain many states studied in quantum opt…
Effect of biased noise fluctuations on the degree of squeezing as well as the intensity of a radiation generated by a one-photon coherent beat laser is presented. It turns out that the radiation exhibits squeezing inside and outside the cavity under certain conditions. The degree of squeezing is enhanced by the biased noise input significantly in both regions. Despite the presence of the biased environment modes outside the cavity, the degree of squeezing outside the cavity can be greater than or equal to or even less than the cavity radiation depending on the initial preparation of the atomic…
We report the suitability of an Einstein-Podolsky-Rosen (EPR) entanglement source for gaussian continuous-variable quantum key distribution at 1550 nm. Our source is based on a single continuous-wave squeezed vacuum mode combined with a vacuum mode at a balanced beam splitter. Extending a recent security proof, we characterize the source by quantifying the extractable length of a composable secure key from a finite number of samples under the assumption of collective attacks. We show that distances in the order of 10 km are achievable with this source for a reasonable sample size despite the f…
Generalized cat states represent arbitrary superpositions of coherent states, which are of great importance in various quantum information processing protocols. Here we demonstrate a versatile approach to creating generalized itinerant cat states in the microwave domain, by reflecting coherent state photons from a microwave cavity containing a superconducting qubit. We show that, with a coherent control of the qubit state, a full control over the coherent state superposition can be realized. The prepared cat states are verified through quantum state tomography of the qubit state dependent refl…
It is shown that a linear superposition of two macroscopically distinguishable optical coherent states can be generated using a single photon source and simple all-optical operations. Weak squeezing on a single photon, beam mixing with an auxiliary coherent state, and photon detecting with imperfect threshold detectors are enough to generate a coherent state superposition in a free propagating optical field with a large coherent amplitude ($α>2$) and high fidelity ($F>0.99$). In contrast to all previous schemes to generate such a state, our scheme does not need photon number resolving measurem…
In a two-mode Bose Josephson junction the dynamics induced by a sudden quench of the tunnel amplitude leads to the periodic formation of entangled states. For instance, squeezed states are formed at short times and macroscopic superpositions of phase states at later times. The two modes of the junction can be viewed as the two arms of an interferometer; use of entangled states allows to perform atom interferometry beyond the classical limit. Decoherence due to the presence of noise degrades the quantum correlations between the atoms, thus reducing phase sensitivity of the interferometer. We co…
Spin squeezing - a central resource for quantum metrology - can be generated via the non-linear, entangling evolution of an initially factorized spin state. Here we show that robust (i.e. persistent) squeezing dynamics is generated by a very large class of $S=1/2$ spin Hamiltonians with axial symmetry, in relationship with the existence of a peculiar structure of the low-lying Hamiltonian eigenstates - the so-called Anderson's tower of states. Such states are fundamentally related to the appearance of spontaneous symmetry breaking in quantum systems; and, for models with sufficiently high conn…
The dissipative dynamics of Gaussian squeezed states (GSS) and coherent superposition states (CSS) are analytically obtained and compared. Time scales for sustaining different quantum properties such as squeezing, negativity of the Wigner function or photon number distribution are calculated. Some of these characteristic times also depend on initial conditions. For example, in the particular case of squeezing, we find that while the squeezing of CSS is only visible for small enough values of the field intensity, in GSS it is independent of this quantity, which may be experimentally advantageou…
We consider macroscopic, mesoscopic and "S-scopic" quantum superpositions of eigenstates of an observable, and develop some signatures for their existence. We define the extent, or size $S$ of a superposition, with respect to an observable \hat{x}, as being the range of outcomes of \hat{x} predicted by that superposition. Such superpositions are referred to as generalized $S$-scopic superpositions to distinguish them from the extreme superpositions that superpose only the two states that have a difference $S$ in their prediction for the observable. We also consider generalized $S$-scopic super…
We study quantum mechanics in the phase space associated with the coherent state (CS) manifold of Lie groups. Eigenstates of generators of the group are constructed as one dimensional integral superpositions of CS along their orbits. We distinguish certain privileged orbits where the superposition is in phase. Interestingly, for closed in phase orbits, the geometric phase must be quantized to $2π\mathbb{Z}$, else the superposition vanishes. This corresponds to exact Bohr-Sommerfeld quantization. The maximum of the Husimi-Kano $Q$ quasiprobability distribution is used to diagnose where in phase…
Role of quantum interference in the origin of higher-order nonclassical characteristics of radiation field has been probed vis-a-vis a discrete and a continuous superposition of coherent states. Specifically, the possibilities of observing higher-order nonclassical properties (e.g., higher-order antibunching (HOA), higher-order sub-Poissonian photon statistics (HOSPS), higher-order squeezing (HOS) of Hong-Mandel type and Hillery type) have been investigated using a shifted symmetric cat state that reduces to Yurke-Stoler, even and odd coherent states at various limits. This shifted symmetric c…
We present and analyze a proposal for a macroscopic quantum delayed-choice experiment with massive mechanical resonators. In our approach, the electronic spin of a single nitrogen-vacancy impurity is employed to control the coherent coupling between the mechanical modes of two carbon nanotubes. We demonstrate that a mechanical phonon can be in a coherent superposition of wave and particle, thus exhibiting both behaviors at the same time. We also discuss the mechanical noise tolerable in our proposal and predict a critical temperature below which the morphing between wave and particle states ca…
Two-mode nonlinear interaction (second-harmonic and second-subharmonic generation) in a planar waveguide with a small periodic corrugation at the surface is studied. Scattering of the interacting fields on the corrugation leads to constructive interference that enhances the nonlinear process provided that all the interactions are phase matched. Conditions for the overall phase matching are found. Compared with a perfectly quasi-phase-matched waveguide, better values of squeezing as well as higher intensities are reached under these conditions. Procedure for finding optimum values of parameters…
Evolutions of quantum noise, characterized by quadrature squeezing parameter and Fano factor, and of mixedness, quantified by quantum von Neumann and linear entropies, of a pumped dissipative non-linear oscillator are studied. The model can describe a signal mode interacting with a thermal reservoir in a parametrically pumped cavity with a Kerr non-linearity. It is discussed that the initial pure states, including coherent states, Fock states, and finite superpositions of coherent states evolve into the same steady mixed state as verified by the quantum relative entropy and the Bures metric. I…
We investigate the generation of entanglement (spin squeezing) in an optical-transition atomic clock through the coupling to a vacuum electromagnetic field that is enhanced by an optical cavity. We show that if each atom is prepared in a superposition of the ground state and a long-lived electronic excited state, and viewed as a spin-1/2 system, then the collective vacuum light shift entangles the atoms, resulting in a squeezed distribution of the ensemble collective spin. This scheme reveals that even a vacuum field can be a useful resource for entanglement and quantum manipulation. The metho…
We propose a theoretical scheme to show the possibility of generating motional nonlinear coherent states and their superposition for an undamped vibrating micromechanical membrane inside an optical cavity. The scheme is based on an intensity-dependent coupling of the membrane to the radiation pressure field. We show that if the cavity field is initially prepared in a Fock state, the motional state of the membrane may evolve from vacuum state to a special type of nonlinear coherent states. By examining the nonclassical properties of the generated state of the membrane, including the quadrature …
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 …
We present a novel proposal for generating quantum superpositions of macroscopically distinct states of a bulk mechanical oscillator, compatible with existing optomechanical devices operating in the readily achievable bad-cavity limit. The scheme is based on a pulsed cavity optomechanical quantum non-demolition (QND) interaction, driven by displaced non-Gaussian states, and measurement-induced feedback, avoiding the need for strong single-photon optomechanical coupling. Furthermore, we show that single-quadrature cooling of the mechanical oscillator is sufficient for efficient state preparatio…
