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 paper we apply an open systems approach to model the effect of coupling a quantum mechanical SQUID ring to a thermal bath. We use this model to demonstrate the manner in which decoherence affects the quantum states of the ring.
We study a system of two nanomechanical resonators embedded in a dc SQUID. We show that the inductively-coupled resonators can be treated as two entangled quantum memory elements with states that can be read from, or written on by employing the SQUID as a displacement detector or switching additional external magnetic fields, respectively. We present a scheme to squeeze the even modeof the state of the resonators and consequently reduce the noise in the measurement of the magnetic flux threading the SQUID. Wefinally analyze the effect of dissipation on the squeezing using the quantum master eq…
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…
Exploiting multiple modes in a quantum acoustic device could enable applications in quantum information in a hardware-efficient setup, including quantum simulation in a synthetic dimension and continuous-variable quantum computing with cluster states.We develop a multimode surface acoustic wave (SAW) resonator with a superconducting quantum interference device (SQUID) integrated in one of the Bragg reflectors. The interaction with the SQUID-shunted mirror gives rise to coupling between the more than 20 accessible resonator modes. We exploit this coupling to demonstrate two-mode squeezing of SA…
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 numerically integrate the time-dependent Schrodinger equation in a single-degree-of-freedom model of SQUID with a variable potential barrier between the basis flux states. We find that linear superpositions of the basis states, with relatively little residual excitation, can be formed by pulsed modulations of the barrier, provided the pulse duration exceeds the period of small oscillations of the flux. Two pulses applied in sequence exhibit strong interference effects, which we propose to use for an experimental determination of the decoherence time in SQUIDs.
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 e…
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…
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…
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 …
Quantum computers take advantage of the superpositional logic of quantum mechanics to allow for dramatic increases in computational efficiency. rf-SQUIDs show potential for quantum computing applications by forming the qubit component of a quantum computer, through simply treating the direction of current - clockwise or counterclockwise - as the value of the bit. rf-SQUIDs present a major advantage over atomic-scale qubit systems - they are sensitive to parameters that can be engineered. Flux qubits are linked through controlled inductive coupling: the magnetic field of each junction affects t…
We study cooling and squeezing the fluctuations of a nanomechanical beam using quantum feedback control. In our model, the nanomechanical beam is coupled to a transmission line resonator via a superconducting quantum interference device (SQUID). The leakage of the electromagnetic field from the transmission line resonator is measured using homodyne detection. This measured signal is then used to design a quantum-feedback-control signal to drive the electromagnetic field in the transmission line resonator. Although the control is imposed on the transmission line resonator, this quantum-feedback…
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 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…
We introduce a technique to control the macroscopic quantum state of an rf SQUID qubit. We propose to employ a stream of single flux quantum (SFQ) pulses magnetically coupled to the qubit junction to momentarily suppresses its critical current. This effectively lowers the barrier in the double-well rf-SQUID potential thereby increasing the tunneling oscillation frequency between the wells. By carefully choosing the time interval between SFQ pulses one may accelerate the interwell tunneling rate. Thus it is possible to place the qubit into a chosen superposition of flux states and then effectiv…
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…
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…
Based on the interaction between the radiation field and a superconductor, we propose a way to engineer quantum states using a SQUID charge qubit inside a microcavity. This device can act as a deterministic single photon source as well as generate any Fock states and an arbitrary superposition of Fock states for the cavity field. The controllable interaction between the cavity field and the qubit can be realized by the tunable gate voltage and classical magnetic field applied to the SQUID.
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 nondegenerate parametric oscillations in a multimode superconducting microwave resonator that is terminated by a SQUID. The parametric effect is achieved by modulating magnetic flux through the SQUID at a frequency close to the sum of two resonator-mode frequencies. For modulation amplitudes exceeding an instability threshold, self-sustained oscillations are observed in both modes. The amplitudes of these oscillations show good quantitative agreement with a theoretical model. The oscillation phases are found to be correlated and exhibit strong fluctuations which broaden the osci…
We investigate the Nth order amplitude squeezing in the fan-state |ξ;2k,f>_F which is a linear superposition of the 2k-quantum nonlinear coherent states. Unlike in usual states where an ellipse is the symbol of squeezing, a 4k-winged flower results in the fan state. We first derive the analytical expression of squeezing for arbitrary k, N, f and then study in detail the case of a laser-driven trapped ion characterized by a specific form of the nonlinear function f. We show that the lowest order in which squeezing may appear and the number of directions along which the amplitude may be squeezed…
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…
We show that squeezing is an irreducible resource which remains invariant under transformations by linear optical elements. In particular, we give a decomposition of any optical circuit with linear input-output relations into a linear multiport interferometer followed by a unique set of single mode squeezers and then another multiport interferometer. Using this decomposition we derive a no-go theorem for superpositions of macroscopically distinct states from single-photon detection. Further, we demonstrate the equivalence between several schemes for randomly creating polarization-entangled sta…
We describe a possible implementation of the nanomechanical quantum superposition generation and detection scheme described in the preceding, companion paper [Armour A D and Blencowe M P 2008 New. J. Phys. Submitted]. The implementation is based on the circuit quantum electrodynamics (QED) set-up, with the addition of a mechanical degree of freedom formed out of a suspended, doubly-clamped segment of the superconducting loop of a dc SQUID located directly opposite the centre conductor of a coplanar waveguide (CPW). The relative merits of two SQUID based qubit realizations are addressed, in par…
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.
The generation of nonclassical states of a radiation field has become increasingly important in the past years given its various applications in quantum communication. The feasibility of generating such nonclassical states has been established in several branches of physics such as cavity electrodynamics, trapped ions, quantum dots, atoms inside cavities and so on. In this sense, we will discuss the issue of the generation of nonclassical states in the context of a superconducting qubit in a microcavity. It has been recently proposed a way to engineer quantum states using a SQUID charge qubit …
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 quantum states of flux qubit is suggested to observe with the help of a new device, the superconducting differential double contour interferometer (DDCI). The flux qubit and the superconducting quantum interference device (DC-SQUID) are connected in the DDCI through the phase of the wave function rather than through magnetic flux. The critical current of DC-SQUID should change to the maximum value at the change of the flux qubit state thanks to this phase coupling. A large jump in the critical current and voltage enables to observe continuously the change in time the state of the flux qubi…
The principle of linear superposition is a hallmark of quantum theory. It has been confirmed experimentally for photons, electrons, neutrons, atoms, for molecules having masses up to ten thousand amu, and also in collective states such as SQUIDs and Bose-Einstein condensates. However, the principle does not seem to hold for positions of large objects! Why for instance, a table is never found to be in two places at the same time? One possible explanation for the absence of macroscopic superpositions is that quantum theory is an approximation to a stochastic nonlinear theory. This hypothesis may…
Time-dependent linear coupling between macroscopic quantum resonator modes generates both a parametric amplification also known as a {}"squeezing operation" and a beam splitter operation, analogous to quantum optical systems. These operations, when applied properly, can robustly generate entanglement and squeezing for the quantum resonator modes. Here, we present such coupling schemes between a nanomechanical resonator and a superconducting electrical resonator using applied microwave voltages as well as between two superconducting lumped-element electrical resonators using a r.f. SQUID-mediat…
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…
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 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 …
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…
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…
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 propose a technique to couple the position operator of a nano mechanical resonator to a SQUID device by modulating its magnetic flux bias. By tuning the magnetic field properly, either linear or quadratic couplings can be realized, with a discretely adjustable coupling strength. This provides a way to realize coherent nonlinear effects in a nano mechanical resonator by coupling it to a Josephson quantum circuit. As an example, we show how squeezing of the nano mechanical resonator state can be realized with this technique. We also propose a simple method to measure the uncertainty in the po…
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…
