318 research outputs found

### Single Cooper pair tunneling induced by non-classical microwaves

A mesoscopic Josephson junction interacting with a mode of non-classical
microwaves with frequency $\omega$ is considered. Squeezing of the
electromagnetic field drastically affects the dynamics of Cooper tunneling. In
particular, Bloch steps can be observed even when the microwaves are in the
squeezed vacuum state with {\em zero} average amplitude of the field $\langle
E(t) \rangle = 0$. The interval between these steps is double in size in
comparison to the conventional Bloch steps.Comment: 8 pages, 2 figures are available upon request to:
[email protected]

### Correlation properties of interfering electrons in a mesoscopic ring under nonclassical microwave radiation

Original paper can be found at: http://eproceedings.worldscinet.com/ Copyright World Scientific Publishing Co. DOI: 10.1142/9789812704474_0009Interfering electrons in a mesoscopic ring are irradiated with both classical and nonclassical microwaves. The average intensity of the charges is calculated as a function of time and it is found that it depends on the nature of the irradiating electromagnetic field. For various quantum states of the microwaves, the electron autocorrelation function is calculated and it shows that the quantum noise of the external field affects the interference of the charges. Two-mode entangled microwaves are also considered and the results for electron average intensity and autocorrelation are compared with those of the corresponding separable state. In both cases, the results depend on whether the ratio of the two frequencies is rational or irrational.Peer reviewe

### Symmetries of the finite Heisenberg group for composite systems

Symmetries of the finite Heisenberg group represent an important tool for the
study of deeper structure of finite-dimensional quantum mechanics. As is well
known, these symmetries are properly expressed in terms of certain normalizer.
This paper extends previous investigations to composite quantum systems
consisting of two subsystems - qudits - with arbitrary dimensions n and m. In
this paper we present detailed descriptions - in the group of inner
automorphisms of GL(nm,C) - of the normalizer of the Abelian subgroup generated
by tensor products of generalized Pauli matrices of orders n and m. The
symmetry group is then given by the quotient group of the normalizer.Comment: Submitted to J. Phys. A: Math. Theo

### The Frobenius formalism in Galois quantum systems

Quantum systems in which the position and momentum take values in the ring
${\cal Z}_d$ and which are described with $d$-dimensional Hilbert space, are
considered. When $d$ is the power of a prime, the position and momentum take
values in the Galois field $GF(p^ \ell)$, the position-momentum phase space is
a finite geometry and the corresponding `Galois quantum systems' have stronger
properties. The study of these systems uses ideas from the subject of field
extension in the context of quantum mechanics. The Frobenius automorphism in
Galois fields leads to Frobenius subspaces and Frobenius transformations in
Galois quantum systems. Links between the Frobenius formalism and Riemann
surfaces, are discussed

### Ultra-quantum coherent states in a single finite quantum system

A set of $n$ coherent states is introduced in a quantum system with
$d$-dimensional Hilbert space $H(d)$. It is shown that they resolve the
identity, and also have a discrete isotropy property. A finite cyclic group
acts on the set of these coherent states, and partitions it into orbits. A
$n$-tuple representation of arbitrary states in $H(d)$, analogous to the
Bargmann representation, is defined. There are two other important properties
of these coherent states which make them `ultra-quantum'. The first property is
related to the Grothendieck formalism which studies the `edge' of the Hilbert
space and quantum formalisms. Roughly speaking the Grothendieck theorem
considers a `classical' quadratic form ${\mathfrak C}$ that uses complex
numbers in the unit disc, and a `quantum' quadratic form ${\mathfrak Q}$ that
uses vectors in the unit ball of the Hilbert space. It shows that if
${\mathfrak C}\le 1$, the corresponding ${\mathfrak Q}$ might take values
greater than $1$, up to the complex Grothendieck constant $k_G$. ${\mathfrak
Q}$ related to these coherent states is shown to take values in the
`Grothendieck region' $(1,k_G)$, which is classically forbidden in the sense
that ${\mathfrak C}$ does not take values in it. The second property
complements this, showing that these coherent states violate logical Bell-like
inequalities (which for a single quantum system are quantum versions of the
Frechet probabilistic inequalities). In this sense also, our coherent states
are deep into the quantum region

### Weak mutually unbiased bases

Quantum systems with variables in ${\mathbb Z}(d)$ are considered. The
properties of lines in the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space of
these systems, are studied. Weak mutually unbiased bases in these systems are
defined as bases for which the overlap of any two vectors in two different
bases, is equal to $d^{-1/2}$ or alternatively to one of the $d_i^{-1/2},0$
(where $d_i$ is a divisor of $d$ apart from $d,1$). They are designed for the
geometry of the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space, in the sense
that there is a duality between the weak mutually unbiased bases and the
maximal lines through the origin. In the special case of prime $d$, there are
no divisors of $d$ apart from $1,d$ and the weak mutually unbiased bases are
mutually unbiased bases

### Generalised squeezing and information theory approach to quantum entanglement

It is shown that the usual one- and two-mode squeezing are based on reducible representations of the SU(1,1) group. Generalized squeezing is introduced with the use of different SU(1,1) rotations on each irreducible sector. Two-mode squeezing entangles the modes and information theory methods are used to study this entanglement. The entanglement of three modes is also studied with the use of the strong subadditivity property of the entropy

### Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions

Mutually unbiased bases in Hilbert spaces of finite dimensions are closely
related to the quantal notion of complementarity. An alternative proof of
existence of a maximal collection of N+1 mutually unbiased bases in Hilbert
spaces of prime dimension N is given by exploiting the finite Heisenberg group
(also called the Pauli group) and the action of SL(2,Z_N) on finite phase space
Z_N x Z_N implemented by unitary operators in the Hilbert space. Crucial for
the proof is that, for prime N, Z_N is also a finite field.Comment: 13 pages; accepted in J. Phys. A: Math. Theo

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