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Generalized Squeezing
Abstract
We consider a generalized form of parametric amplification which produces k-photon correlations. We show numerically that this process is well-defined quantum mechanically, and we explain the quantum phase-space structures produced by such parametric amplification.
... BG coherent states were intensively investigated in the 1980s as a possible alternative to N-boson squeezed-states e za † N −za N |0⟩. The latter are mathematically problematic [20,21], whereas e zA † N −zA N |0⟩ are as well-behaved as the ordinary coherent states in H. The N = 2 case of e zA † N −zA N |0⟩ indeed leads to squeezing of variances of position X ∼ a + a † and momentum P ∼ i(a † − a), although qualitatively different from the squeezing implied by e za †2 −za 2 |0⟩ [22]. ...
Any single system whose space of states is given by a separable Hilbert space is automatically equipped with infinitely many hidden tensor-like structures. This includes all quantum mechanical systems as well as classical field theories and classical signal analysis. Accordingly, systems as simple as a single one-dimensional harmonic oscillator, an infinite potential well, or a classical finite-amplitude signal of finite duration can be decomposed into an arbitrary number of subsystems. The resulting structure is rich enough to enable quantum computation, violation of Bell’s inequalities, and formulation of universal quantum gates. Less standard quantum applications involve a distinction between position and hidden position. The hidden position can be accompanied by a hidden spin, even if the particle is spinless. Hidden degrees of freedom are, in many respects, analogous to modular variables. Moreover, it is shown that these hidden structures are at the roots of some well-known theoretical constructions, such as the Brandt–Greenberg multi-boson representation of creation–annihilation operators, intensively investigated in the context of higher-order or fractional-order squeezing. In the context of classical signal analysis, the discussed structures explain why it is possible to emulate a quantum computer by classical analog circuit devices.
... This models an n-photon drive with n ≥ 1, where the detuning between the natural oscillator frequency ω 0 and the frequency of the driving force ω s is denoted by ∆ = ω 0 −ω s . This n-photon parametric process produces squeezing effects for n > 1 [38] and will be called the squeezing term in the following. The parameter η n controls the driving strength and ϕ represents its phase. ...
Quantum oscillators with nonlinear driving and dissipative terms have gained significant attention due to their ability to stabilize cat-states for universal quantum computation. Recently, supercon-ducting circuits have been employed to realize such long-lived qubits stored in coherent states. We present a generalization of these oscillators, which are not limited to coherent states, in the presence of different nonlinearities in driving and dissipation, exploring different degrees. Specifically, we present an extensive analysis of the asymptotic dynamical features and of the storage of squeezed states. We demonstrate that coherent superpositions of squeezed states are achievable in the presence of a strong symmetry, thereby allowing for the storage of squeezed cat-states. In the weak symmetry regime, accounting for linear dissipation, we investigate the potential application of these nonlinear driven-dissipative resonators for quantum computing and quantum associative memory and analyze the impact of squeezing on their performance.
... Unlike the displacement operator, the squeeze operator consists of two-photon creation and annihilation operators, in which the nonlinear optical processes are involved. However, numerical evaluations for unitary transformations of arbitrary quantum state based on the classical Taylor series expansion become cumbersome [5] [6] [7] because the system's Hamiltonian is time-independent and nonlinearly constructed by non-commuting operators. ...
In quantum optics, unitary transformations of arbitrary states are evaluated by using the Taylor series expansion. However, this traditional approach can become cumbersome for the transformations involving non-commuting operators. Addressing this issue, a nonstandard unitary transformation technique is highlighted here with new perspective. In a spirit of “quantum” series expansions, the transition probabilities between initial and final states, such as displaced, squeezed and other nonlinearly transformed coherent states are obtained both numerically and analytically. This paper concludes that, although this technique is novel, its implementations for more extended systems are needed.
... BG coherent states were intensively investigated in the 1980s as a possible alternative to N -boson squeezedstates e za † N −za N |0 . The latter are mathematically problematic [8,9], whereas e zA † N −zAN |0 are as well-behaved as the ordinary coherent states in H. The N = 2 case of e zA † N −zAN |0 indeed leads to squeezing of variances of position X ∼ a + a † and momenta P ∼ i(a † − a), although qualitatively different from the squeezing implied by e za † 2 −za 2 |0 [10]. ...
Standard architecture of quantum information processing is based on bottom-up design: One begins with a one-digit one-particle system, while multi-digit quantum registers demand multi-particle configurations, mathematically modeled by tensor products of single quantum digits. Here we show that any single quantum system is automatically equipped with hidden tensor structures that allow for single-particle top-down designs of quantum information processing. Hidden tensor structures imply that any quantum system, even as simple as a single one-dimensional harmonic oscillator, can be decomposed into an arbitrary number of subsystems. The resulting structure is rich enough to enable quantum computation, violation of Bell's inequalities, and formulation of universal quantum gates. In principle, a single-particle quantum computer is possible. Moreover, it is shown that these hidden structures are at the roots of some well known theoretical constructions, such as the Brandt-Greenberg multi-boson representation of creation-annihilation operators, intensively investigated in the context of higher-order or fractional-order squeezing. In effect, certain rather tedious standard proofs known from the literature can be simplified to literally one line. The general construction is illustrated by concrete examples.
... Although, the standard approach based on time-ordered exponentials is extremely useful [1,2,3], it may occasionally turn out to be challenging, particularly, in the case of revealing nonlinear quantum dynamics [4,5] that requires rigorous numerical simulations [6,7,8]. Quantum dynamics for arbitrary system are traditionally realized by time evolutions of wave functions in Hilbert space, which can also be expressed in terms of density operators in the Liouville space [2,3]. ...
Quantum dynamics for arbitrary system are traditionally realized by time evolutions of wave functions in Hilbert space and/or density operators in Liouville space. However, the traditional simulations may occasionally turn out to be challenging for the quantum dynamics, particularly those governed by the nonlinear Hamiltonians. In this letter, we introduce a nonstandard iterative technique where time interval is divided into a large number of discrete subintervals with an ultrashort duration; and the Liouville space is briefly expanded with an additional (virtual) space only within these subintervals. We choose two-state spin raising and lowering operators for virtual space operators because of their simple algebra. This tremendously reduces the cost of time-consuming calculations. We implement our technique for an example of a charged particle in both harmonic and anharmonic potentials. The temporal evolutions of the probability for the particle being in the ground state are obtained numerically and compared to the analytical solutions. We further discuss the physics insight of this technique based on a thought-experiment. Successive processes intrinsically 'hitchhiking' via virtual space in discrete ultrashort time duration, are the hallmark of our simple iterative technique. We believe that this novel technique has potential for solving numerous problems which often pose a challenge when using the traditional approach based on time-ordered exponentials.
Quanta splitting is an essential generator of Gaussian entanglement, exemplified by Einstein-Podolsky-Rosen states and apparently the most commonly occurring form of entanglement. In general, it results from the strong pumping of a nonlinear process with a highly coherent and low-noise external drive. In contrast, recent experiments involving efficient trilinear processes in trapped ions and superconducting circuits have opened the complementary possibility to test the splitting of a few thermal quanta. Stimulated by such small thermal energy, a strong degenerate trilinear coupling generates large amounts of nonclassicality, detectable by more than 3 dB of distillable quadrature squeezing. Substantial entanglement can be generated via frequent passive linear coupling to a third mode present in parallel with the trilinear coupling. This new form of entanglement, outside any Gaussian approximation, surprisingly grows with the mean number of split thermal quanta; a quality absent from Gaussian entanglement. Using distillable squeezing we shed light on this new entanglement mechanism for nonlinear bosonic systems.
Quantum oscillators with nonlinear driving and dissipative terms have gained significant attention due to their ability to stabilize cat states for universal quantum computation. Recently, superconducting circuits have been employed to realize such long-lived qubits stored in coherent states. We present a generalization of these oscillators, which are not limited to coherent states. The key ingredient lies in the presence of different nonlinearities in driving and dissipation, beyond the quadratic one. Through an extensive analysis of the asymptotic dynamical features for different nonlinearities, we identify the conditions for the storage and retrieval of quantum states, such as squeezed states, in both coherent and incoherent superpositions. We explore their applications in quantum computing, where squeezing prolongs the lifetime of memory storage for qubits encoded in the superposition of two symmetric squeezed states, and in quantum associative memory, which has so far been limited to the storage of classical patterns.
Algorithms for associative memory typically rely on a network of many connected units. The prototypical example is the Hopfield model, whose generalizations to the quantum realm are mainly based on open quantum Ising models. We propose a realization of associative memory with a single driven-dissipative quantum oscillator exploiting its infinite degrees of freedom in phase space. The model can improve the storage capacity of discrete neuron-based systems in a large regime and we prove successful state discrimination between n coherent states, which represent the stored patterns of the system. These can be tuned continuously by modifying the driving strength, constituting a modified learning rule. We show that the associative-memory capability is inherently related to the existence of a spectral separation in the Liouvillian superoperator, which results in a long timescale separation in the dynamics corresponding to a metastable phase.
With the emerging possibility to obtain emissions of triples of Greenberger-Horne-Zeilinger–entangled photons via direct parametric generation, we study here bright emissions of this kind which involve higher-order emissions of two triples, three triples, etc. Such states would constitute a natural generalization of the four-mode (two beams plus polarization) squeezed vacuum. We show how to avoid technical difficulties related to straight-ahead generalization of the usual approximate description of parametric down-conversion which uses a classical pump field. Using Padé approximation, we turn the first terms of the nonconverging perturbation expansion into elements of a converging series. This allows us to study nonclassicality of the new bright generalized Greenberger-Horne-Zeilinger states. We present both violations of local realism by such a radiation and simple entanglement indicators tailored for this case.
A precise theory of the simultaneous meaSllrement of a pair of conjugate observables is necessary for obtaining the classical limit from the quantum theory, for determining the limitations of coherent quantum mechanical amplifiers, etc. The uncertainty principle, of course, does not directly address this problem, since it is a statement about the variances of two hypothetical ideal measurements. We will adopt the approach that there exist instantaneous inexplicable ideal measurements of a single observable. Just as von Neumann1 uses an ideal measurements together with all interaction to explain an indirect observation, we use ideal measurements together with iuteractions to explain the simultaneous measurements of an observable and its conjugate.
The high rates of reservoir construction throughout the world give evidence of the fact that at the present stage of economic development reservoirs are objectively necessary for solving water resources problems both in industrially-developed and in developing countries.
The great achievements in the field of hydraulic construction make it possible to regulate the flow of the largest rivers in the world and to create reservoirs with parameters exceeding those of many large natural lakes. The high scales achieved in reservoir construction during the past two decades, and the many-sided effects of reservoirs on the natural surroundings and on the economy call for an intensification of scientific investigations and for the unification of the efforts of scientists in different countries.
One of most fruitful devices for scientific research, widely applied also in reservoir design, is the analogy method. Its utilization and development call for the preparation of different reservoir classifications and typologies, which in turn generates the need for an adequate approach to the use and publication of data on reservoirs. A list of the most desirable reservoir indices should be an object for discussion by specialists in different countries.
The problem of expanding a density operator rho in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function P(alpha) of the P representation, the Wigner distribution W(alpha), and the function , where |alpha> is a coherent state, are discussed from a unified point of view. Each of these quasiprobability distributions is examined as the expectation value of a Hermitian operator, as the weight function of an integral representation for the density operator and as the function associated with the density operator by one of the operator-function correspondences defined in the preceding paper. The weight function P(alpha) of the P representation is shown to be the expectation value of a Hermitian operator all of whose eigenvalues are infinite. The existence of the function P(alpha) as an infinitely differentiable function is found to be equivalent to the existence of a well-defined antinormally ordered series expansion for the density operator in powers of the annihilation and creation operators a and a†. The Wigner distribution W(alpha) is shown to be a continuous, uniformly bounded, square-integrable weight function for an integral expansion of the density operator and to be the function associated with the symmetrically ordered power-series expansion of the density operator. The function , which is infinitely differentiable, corresponds to the normally ordered form of the density operator. Its use as a weight function in an integral expansion of the density operator is shown to involve singularities that are closely related to those which occur in the P representation. A parametrized integral expansion of the density operator is introduced in which the weight function W(alpha,s) may be identified with the weight function P(alpha) of the P representation, with the Wigner distribution W(alpha), and with the function when the order parameter s assumes the values s=+1, 0, -1, respectively. The function W(alpha,s) is shown to be the expectation value of the ordered operator analog of the delta function defined in the preceding paper. This operator is in the trace class for Res
Pertinent properties of the unitary operators that create coherent states and squeezed coherent states are discussed. We show that certain generalizations of squeezed coherent states do not exist. This is accomplished by demonstrating that for the generalized squeeze operators ${U}_{k}=\mathrm{exp}(i{A}_{k})=\mathrm{exp}[{z}_{k}{({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{k}+i{h}_{k$-${}1}$-${}{({z}_{k})}^{*}{a}^{k}]$, $〈0|{U}_{k}|0〉$ diverges, $k>2$. This implies that $|0〉$ is not an analytic vector of ${A}_{k}$ for all $k>2$, where ${h}_{k$-${}1}$ is a Hermitian polynomial in $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ up to powers of ($k$-${}1$).
The Hausdorff similarity transformation technique is used to perform complete disentangling of the exponential operators exp[-lambda(Â+B)2k], exp[lambda(Âk+B)] and exp[lambda(ÂBk+Bl)] where [Â, B]- = c. These disentanglings seem to be new.
The properties of squeezed states produced by higher order subharmonic generation processes are examined formally. A small-time approximation is applied to modeling the n-th subharmonic generation process of photons produced by a pump beam intersecting a nonlinear medium. It is shown that a squeezed state can be obtained for subharmonic levels greater than 2, i.e., the squeezing then becomes proportional to the signal mode. It is further demonstrated that the vacuum state is not squeezed in higher order nonlinear optical processes.
The performance of phase-sensing interferometers employing squeezed states and homodyne detection is analyzed and compared to the performance of systems employing direct detection. Standard differenced direct-detection Michelson and Mach-Zehnder interferometers are shown to be suboptimal in the sense that an observation/measurement-noise coupling occurs, which can degrade performance. Homodyne-detection interferometers in which the phase shift in one arm is the conjugate of that in the other arm do not suffer from the preceding drawback. Overall, however, the performance of differenced direct-detection and homodyne-detection interferometers is similar in single-frequency operation. In particular, both detection schemes reach the standard quantum limit on position-measurement sensitivity in single-frequency interferometric gravity-wave detectors at roughly the same average photon number. This limit arises from back action in the form of radiation pressure fluctuations entering through the energy-phase uncertainty principle. Multifrequency devices can circumvent this uncertainty principle, as illustrated by the conceptual design given for a two-frequency interferometer which can greatly surpass the standard quantum limit on position sensing. This configuration assumes that ideal photodetectors respond to photon flux rather than energy flux.
The Feynman sum represents a convenient formulation of quantum mechanics for Bose fields, but, to secure a similar formulation applicable to Fermion fields, it has been necessary to use “anticommuting c-number” field histories to insure the anticommutivity of the quantum field operators. Here, a method is presented to sum over histories for spinor fields which (1) employs the familiar classical c-number expression for the action, (2) predicts anti-commutation rules and Fermi statistics, and (3) retains the invariance of the theory under a change in phase of the complex ψ field.
The properties of a unique set of quantum states of the electromagnetic field are reviewed. These 'squeezed states' have less uncertainty in one quadrature than a coherent state. Proposed schemes for the generation and detection of squeezed states as well as potential applications are discussed.