Polarized distances between quantum states and observables
The proposition 1 is incomplete. In some of the examples D(a,b) may not obey the triangle inequality. The paper is withdrawn for further elaboration.
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arXiv subjects
D. A. Trifonov
Publications and source records attributed to D. A. Trifonov.
Generalized uncertainty relations and coherent and squeezed states
Characteristic uncertainty relations and their related squeezed states are briefly reviewed and compared in accordance with the generalizations of three equivalent definitions of the canonical coherent states. The standard SU(1,1) coherent states are shown to be the unique states that minimize the Schroedinger uncertainty relation for every pair of the three generators and the Robertson relation for the three generators. The characteristic uncertainty inequalities are naturally extended to the case of several states. It is shown that these inequalities can be written in the equivalent complementary form.
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Diagonalization of Hamiltonians, uncertainty matrices and Robertson inequality
The problem of diagonalization of Hamiltonians of N-dimensional boson systems by means of time-dependent canonical transformations (CT) is considered, the case of quadratic Hamiltonians being treated in greater detail. The unitary generator of time-dependent CT which can transform any Hamiltonian to that of a system of uncoupled stationary oscillators is constructed. The close relationship between methods of canonical transformations, time-dependent integrals of motion and dynamical symmetry is noted. The diagonalization and symplectic properties of the uncertainty matrix for 2N canonical observables are studied. It is shown that the normalized uncertainty matrix is symplectic for the squeezed multimode Glauber coherent states and for the squeezed Fock states with equal photon numbers in each mode. The Robertson uncertainty relation for the dispersion matrix of canonical observables is shown to be minimized in squeezed coherent states only.
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Remarks on the Extended Characteristic Uncertainty Relations
Three remarks concerning the form and the range of validity of the state-extended characteristic uncertainty relations (URs) are presented. A more general definition of the uncertainty matrix for pure and mixed states is suggested. Some new URs are provided.
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The Uncertainty Way of Generalization of Coherent States
The three ways of generalization of canonical coherent states are briefly reviewed and compared with the emphasis laid on the (minimum) uncertainty way. The characteristic uncertainty relations, which include the Schroedinger and Robertson inequalities, are extended to the case of several states. It is shown that the standard SU(1,1) and SU(2) coherent states are the unique states which minimize the second order characteristic inequality for the three generators. A set of states which minimize the Schroedinger inequality for the Hermitian components of the su_q(1,1) ladder operator is also constructed. It is noted that the characteristic uncertainty relations can be written in the alternative complementary form.
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State Extended Uncertainty Relations
A scheme for construction of uncertainty relations (UR) for n observables and m states is presented. Several lowest order UR are displayed and briefly discussed. For two states |ψ> and |ϕ> and canonical observables the (entangled) extension of Heisenberg UR reads [Δp(ψ)]^2[Δq(ϕ)]^2 + [Δp(ϕ)]^2[Δq(ψ)]^2 \geq 1/2. Some possible applications of the new inequalities are noted.
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Generalized intelligent states and SU(1,1) and SU(2) squeezing
A sufficient condition for a state |ψ> to minimize the Robertson-Schrödinger uncertainty relation for two observables A and B is obtained which for A with no discrete spectrum is also a necessary one. Such states, called generalized intelligent states (GIS), exhibit arbitrarily strong squeezing (after Eberly) of A and B. Systems of GIS for the SU(1,1) and SU(2) groups are constructed and discussed. It is shown that SU(1,1) GIS contain all the Perelomov coherent states (CS) and the Barut and Girardello CS while the Bloch CS are subset of SU(2) GIS.
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Exact Solutions for the General Nonstationary Oscillator with a Singular Perturbation
Three linearly independent Hermitian invariants for the nonstationary generalized singular oscillator (SO) are constructed and their complex linear combination is diagonalized. The constructed family of eigenstates contains as subsets all previously obtained solutions for the SO and includes all Robertson and Schrödinger intelligent states for the three invariants. It is shown that the constructed analogues of the SU(1,1) group-related coherent states for the SO minimize the Robertson and Schrödinger relations for the three invariants and for every pair of them simultaneously. The squeezing properties of the new states are briefly discussed.
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Characteristic uncertainty relations
New uncertainty relations for n observables are established. The relations take the invariant form of inequalities between the characteristic coefficients of order r, r = 1,2,...,n, of the uncertainty matrix and the matrix of mean commutators of the observables. It is shown that the second and the third order characteristic inequalities for the three generators of SU(1,1) and SU(2) are minimized in the corresponding group-related coherent states with maximal symmetry.
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Barut-Girardello coherent states for u(p,q) and sp(N,R) and their macroscopic superpositions
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 optics (even and odd N-mode CS, pair CS) and provide an approach to quadrature squeezing, alternative to that of intelligent states. New subsets of weakly and strongly nonclassical states are pointed out and their statistical properties (first- and second-order squeezing, photon number distributions) are discussed. For specific values of the angle parameters and small amplitude of the canonical CS components these states approaches multimode Fock states with one, two or three bosons/photons. It is shown that eigenstates of a squared non-Hermitian operator A^2 (generalized cat states) can exhibit squeezing of the quadratures of A.
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On the squeezed states for n observables
Three basic properties (eigenstate, orbit and intelligence) of the canonical squeezed states (SS) are extended to the case of arbitrary n observables. The SS for n observables X_i can be constructed as eigenstates of their linear complex combinations or as states which minimize the Robertson uncertainty relation. When X_i close a Lie algebra L the generalized SS could also be introduced as orbit of Aut(L^C). It is shown that for the nilpotent algebra h_N the three generalizations are equivalent. For the simple su(1,1) the family of eigenstates of uK_- + vK_+ (K_\pm being lowering and raising operators) is a family of ideal K_1-K_2 SS, but it cannot be represented as an Aut(su^C(1,1)) orbit although the SU(1,1) group related coherent states (CS) with symmetry are contained in it. Eigenstates |z,u,v,w;k> of general combination uK_- + vK_+ + wK_3 of the three generators K_j of SU(1,1) in the representations with Bargman index k = 1/2,1, ..., and k = 1/4,3/4 are constructed and discussed in greater detail. These are ideal SS for K_{1,2,3}. In the case of the one mode realization of su(1,1) the nonclassical properties (sub-Poissonian statistics, quadrature squeezing) of the generalized even CS |z,u,v;+> are demonstrated. The states |z,u,v,w;k=1/4,3/4> can exhibit strong both linear and quadratic squeezing.
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Canonical equivalence of quantum systems, multimode squeezed states and Robertson relation
It is shown that any two Hamiltonians H(t) and H'(t) of N dimensional quantum systems can be related by means of time-dependent canonical transformations (CT). The dynamical symmetry group of system with Hamiltonian H(t) coincides with the invariance group of H(t). Quadratic Hamiltonians can be diagonalized by means of linear time-dependent CT. The diagonalization can be explicitly carried out in the case of stationary and some nonstationary quadratic H. Linear CT can diagonalize the uncertainty matrix σ(ρ) for canonical variables p_k, q_j in any state ρ, i.e., σ(ρ) is symplectically congruent to a diagonal uncertainty matrix. For multimode squeezed canonical coherent states (CCS) and squeezed Fock states with equal photon numbers in each mode σis symplectic itself. It is proved that the multimode Robertson uncertainty relation is minimized only in squeezed CCS.
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Robertson Intelligent States
Diagonalization of uncertainty matrix and minimization of Robertson inequality for n observables are considered. It is proved that for even n this relation is minimized in states which are eigenstates of n/2 independent complex linear combinations of the observables. In case of canonical observables this eigenvalue condition is also necessary. Such minimizing states are called Robertson intelligent states (RIS). The group related coherent states (CS) with maximal symmetry (for semisimple Lie groups) are particular case of RIS for the quadratures of Weyl generators. Explicit constructions of RIS are considered for operators of su(1,1), su(2), h_N and sp(N,R) algebras. Unlike the group related CS, RIS can exhibit strong squeezing of group generators. Multimode squared amplitude squeezed states are naturally introduced as sp(N,R) RIS. It is shown that the uncertainty matrices for quadratures of q-deformed boson operators a_{q,j} (q > 0) and of any k power of a_j = a_{1,j} are positive definite and can be diagonalized by symplectic linear transformations. PACS numbers: 03.65.Fd, 42.50.Dv
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Barut-Girardello coherent states for sp(N,C) and multimode Schrödinger cat states
Overcomplete families of states of the type of Barut-Girardello coherent states (BG CS) are constructed for noncompact algebras $u(p,q)$ and $sp(N,C)$ in quadratic bosonic representation. The $sp(N,C)$ BG CS are obtained in the form of multimode ordinary Schrödinger cat states. A set of such macroscopic superpositions is pointed out which is overcomplete in the whole $N$ mode Hilbert space (while the associated $sp(N,C)$ representation is reducible). The multimode squared amplitude Schrödinger cat states are introduced as macroscopic superpositions of the obtained $sp(N,C)$ BG CS.}
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Algebraic Coherent States and Squeezing
Eigenstates of general complex linear combination of SU(1,1) generators (su^c(1,1) algebraic coherent states (ACS)) are constructed and discussed. In case of quadratic boson representation ACS can exhibit strong both linear and quadratic amplitude squeezing. ACS for a given Lie group algebra contain the corresponding Perelomov CS with maximal symmetry.
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Schrödinger Intelligent States and Linear and Quadratic Amplitude Squeezing
A complete set of solutions |z,u,v>_{sa} of the eigenvalue equation (ua^2+va^{dagger 2})|z,u,v> = z|z,u,v> ([a,a^{dagger}]=1) are constructed and discussed. These and only these states minimize the Schrödinger uncertainty inequality for the squared amplitude (s.a.) quadratures. Some general properties of Schrödinger intelligent states (SIS) |z,u,v> for any two observables X, Y are discussed, the sets of even and odd s.a. SIS |z,u,v;+,-> being studied in greater detail. The set of s.a. SIS contain all even and odd coherent states (CS) of Dodonov, Malkin and Man'ko, the Perelomov SU(1,1) CS and the squeezed Hermite polynomial states of Bergou, Hillery and Yu. The even and odd SIS can exhibit very strong both linear and quadratic squeezing (even simultaneously) and super- and subpoissonian statistics as well. A simple sufficient condition for superpoissonian statistics is obtained and the diagonalization of the amplitude and s.a. uncertainty matrices in any pure or mixed state by linear canonical transformations is proven.
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