In quantum mechanics a geometric phase, also called Berry phase, is a phase difference that a given physical system acquires during a cycle in which the system itself is under the action of an adiabatic process. This phase is linked to the geometric properties of the system itself (which is a simplification, but for our purposes there is no need to go into too much detail).
It was discovered independently by Shivaramakrishnan Pancharatnam in 1956(1), Hugh Christopher Longuet-Higgins(2) in 1958 and subsequently generalized by Michael Berry(3) in 1984. This phase, although geometric, has measurable physical effects, for example in an interference experiment. An example of a geometric phase is Foucault's pendulum.
The most famous version of this experiment, designed by Léon Foucault, dates back to 1851 when the French physicist, with the aim of showing the rotation of the Earth around its axis, suspended a ball of 28 kilograms of lead coated with brass over a surface of sand using a 67 meter cable hooked to the top of the dome of the Panthéon in Paris. The plane of the pendulum was observed to rotate clockwise at approximately 11.3 degrees per hour, completing a full circle in 31.8 hours. A more refined examination shows that after 24 hours there is a difference between the initial and final orientation of the trace left on Earth which is equal to
Showing posts with label quantum mechanics. Show all posts
Showing posts with label quantum mechanics. Show all posts
Imaging quantum entanglement
The main object of a paper published a couple of days ago on Science is to find an answer to the following question:
The results is the production of some images that shots the Bell inequality violation, like the following image:
what kind of imaging process could reveal a Bell inequality?The experimental set-up used a $\beta$-Barium Borate crystal pumped by a (quasi-cotinuous) laser. The pairs of entagled photons generated are subsequently separated on a beam splitter and propagate into two distinct optical systems like LIGO interferometer.
The results is the production of some images that shots the Bell inequality violation, like the following image:
Moreover, our demonstration shows that one can detect the signature of a Bell-type behavior within a single image acquired by an imaging setup. By demonstrating that quantum imaging can generate high-dimensional images illustrating the presence of Bell-type entanglement, we benchmark quantum imaging techniques against the most fundamental test of quantum mechanics.
Moreau, P. A., Toninelli, E., Gregory, T., Aspden, R. S., Morris, P. A., & Padgett, M. J. (2019). Imaging Bell-type nonlocal behavior. Science Advances, 5(7), eaaw2563. doi:10.1126/sciadv.aaw2563
Wigner's theorem
The Wigner’s theorem was formulated and demonstrated for the first time by Eugene Paul Wigner on Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektrum(1). It states that for each symmetry transformation in Hilbert’s space there exists a unitary or anti-unitary operator, uniquely determined less than a phase factor.
For symmetry transformation, we intend a space transformation that preserved the characteristics of a given physical system. Asymmetry transformation implies also a change of reference system.
Invariants play a key role in physics, being the quantities that, in any reference system, are unchanged. With the advent of quantum physics, their importance increased, particularly in the formulation of a relativistic quantum field theory. One of the most important tools in the study of invariants is the Wigner’s theorem, an instrument of fundamental importance for all the development of quantum theory.
In particular, Wigner was interested in determining the properties of transformations that preserve the transition’s probability between two different quantum states. Given $\phi$ the wave function detected by the first observer, and $\bar {\phi}$ the wave function detected by the second observer, Wigner assumed that the equality \[|\langle \psi | \phi \rangle| = |\langle \bar \psi | \bar \phi \rangle|\] must be valid for all $\psi$ and $\phi$.
In the end, if we exclude time inversions, we find that the operator $\operatorname{O}_{R}$, such that $\bar{\phi} = \operatorname{O} _{R} \phi$, must be unitary and linear, but also anti-unitary and anti-linear. Consequence of this fact is that the two observers’ descriptions are equivalent. So the first observes $\phi$, the second $\bar{\phi}$, while the operator $\operatorname{H}$ for the first will be $\operatorname{O}_R \operatorname{H} \operatorname{O}_R^{-1}$ for the second.
For symmetry transformation, we intend a space transformation that preserved the characteristics of a given physical system. Asymmetry transformation implies also a change of reference system.
In particular, Wigner was interested in determining the properties of transformations that preserve the transition’s probability between two different quantum states. Given $\phi$ the wave function detected by the first observer, and $\bar {\phi}$ the wave function detected by the second observer, Wigner assumed that the equality \[|\langle \psi | \phi \rangle| = |\langle \bar \psi | \bar \phi \rangle|\] must be valid for all $\psi$ and $\phi$.
In the end, if we exclude time inversions, we find that the operator $\operatorname{O}_{R}$, such that $\bar{\phi} = \operatorname{O} _{R} \phi$, must be unitary and linear, but also anti-unitary and anti-linear. Consequence of this fact is that the two observers’ descriptions are equivalent. So the first observes $\phi$, the second $\bar{\phi}$, while the operator $\operatorname{H}$ for the first will be $\operatorname{O}_R \operatorname{H} \operatorname{O}_R^{-1}$ for the second.
JMP #58, 4: path integrals and friends
Bernardo, R. C. S., & Esguerra, J. P. H. (2017). Euclidean path integral formalism in deformed space with minimum measurable length. Journal of Mathematical Physics, 58(4), 042103. doi:10.1063/1.4979797
We study time-evolution at the quantum level by developing the Euclidean path-integral approach for the general case where there exists a minimum measurable length. We derive an expression for the momentum-space propagator which turns out to be consistent with recently developed $\beta$-canonical transformation. We also construct the propagator for maximal localization which corresponds to the amplitude that a state which is maximally localized at location $\xi'$ propagates to a state which is maximally localized at location $\xi"$ in a given time. Our expression for the momentum-space propagator and the propagator for maximal localization is valid for any form of time-independent Hamiltonian. The nonrelativistic free particle, particle in a linear potential, and the harmonic oscillator are discussed as examples.Other papers from JMP #58, 4, follows:
The quantum Zeno paradox
The standard axioms of quantum mechanics imply that in the limit of continuous observation a quantum system cannot evolve.Initially known as Turing’s paradox, in honor of the mathematician who formulated it in the 1950s, was subsequently identified as quantum Zeno effect, resulting an advanced version of the famous Zeno’s arrow paradox, whose phylosophical result is the negation of motion. A first formulation and derivation of the effect is found in Does the lifetime of an unstable system depend on the measuring apparatus?(1), while George Sudarshan and Baidyanath Misra were the first to identify it as quantum Zeno paradox. The two theoretical physicists established that an unstable particle will not decay as long as it is kept under continuous observation(2). However they try to save goat and cabbage:
(Andrew Hodges in Alan Turing: the logical and physical basis of computing - pdf)
There is a fundamental principle in quantum theory that denies the possibility of continuous observation.(2)On the other hand, Ghirardi, Omero, Weber and Rimini show that:
if the uncertainty relations are properly taken into account the arguments leading to the paradox are not valid.(3)
JMP 58, 3: math paradoxes in quantum mechanics
Facchi, P., & Ligabò, M. (2017). Large-time limit of the quantum Zeno effect Journal of Mathematical Physics, 58 (3) DOI: 10.1063/1.4978851 (arXiv)
If very frequent periodic measurements ascertain whether a quantum system is still in its initial state, its evolution is hindered. This peculiar phenomenon is called quantum Zeno effect. We investigate the large-time limit of the survival probability as the total observation time scales as a power of the measurement frequency, $t \propto N^\alpha$. The limit survival probability exhibits a sudden jump from $1$ to $0$ at $\alpha = 1/2$, the threshold between the quantum Zeno effect and a diffusive behavior. Moreover, we show that for $\alpha \geq 1$, the limit probability becomes sensitive to the spectral properties of the initial state and to the arithmetic properties of the measurement periods.
Selvitella, A. (2017). The Simpson’s paradox in quantum mechanics Journal of Mathematical Physics, 58 (3) DOI: 10.1063/1.4977784 (sci-hub)
In probability and statistics, the Simpson’s paradox is a paradox in which a trend that appears in different groups of data disappears when these groups are combined, while the reverse trend appears for the aggregate data. In this paper, we give some results about the occurrence of the Simpson’s paradox in quantum mechanics. In particular, we prove that the Simpson’s paradox occurs for solutions of the quantum harmonic oscillator both in the stationary case and in the non-stationary case. In the non-stationary case, the Simpson’s paradox is persistent: if it occurs at any time $t=\tilde t$, then it occurs at any time $t\not= \tilde t$. Moreover, we prove that the Simpson’s paradox is not an isolated phenomenon, namely, that, close to initial data for which it occurs, there are lots of initial data (a open neighborhood), for which it still occurs. Differently from the case of the quantum harmonic oscillator, we also prove that the paradox appears (asymptotically) in the context of the nonlinear Schrödinger equation but at intermittent times.Read also: Two quantum Simpson's paradoxes
Meng, F., & Liu, C. (2017). Necessary and sufficient conditions for the existence of time-dependent global attractor and application Journal of Mathematical Physics, 58 (3) DOI: 10.1063/1.4978329 (sci-hub)
In this paper, we are concerned with infinite dimensional dynamical systems in time-dependent space. First, we characterize some necessary and sufficient conditions for the existence of the time-dependent global attractor by using a measure of noncompactness. Then, we give a new method to verify the sufficient condition. As a simple application, we prove the existence of the time-dependent global attractor for the damped equation in strong topological space.
Cen, J., Correa, F., & Fring, A. (2017). Time-delay and reality conditions for complex solitons Journal of Mathematical Physics, 58 (3) DOI: 10.1063/1.4978864 (arXiv)
We compute lateral displacements and time-delays for scattering processes of complex multi-soliton solutions of the Korteweg de-Vries equation. The resulting expressions are employed to explain the precise distinction between solutions obtained from different techniques, Hirota’s direct method and a superposition principle based on Bäcklund transformations. Moreover they explain the internal structures of degenerate compound multi-solitons previously constructed. Their individual one-soliton constituents are time-delayed when scattered amongst each other. We present generic formulae for these time-dependent displacements. By recalling Gardner’s transformation method for conserved charges, we argue that the structure of the asymptotic behaviour resulting from the integrability of the model together with its $PT$-symmetry ensures the reality of all of these charges, including in particular the mass, the momentum, and the energy.
Wilming, H., Kastoryano, M., Werner, A., & Eisert, J. (2017). Emergence of spontaneous symmetry breaking in dissipative lattice systems Journal of Mathematical Physics, 58 (3) DOI: 10.1063/1.4978328 (arXiv)
A cornerstone of the theory of phase transitions is the observation that many-body systems exhibiting a spontaneous symmetry breaking in the thermodynamic limit generally show extensive fluctuations of an order parameter in large but finite systems. In this work, we introduce the dynamical analog of such a theory. Specifically, we consider local dissipative dynamics preparing an equilibrium steady-state of quantum spins on a lattice exhibiting a discrete or continuous symmetry but with extensive fluctuations in a local order parameter. We show that for all such processes, there exist asymptotically stationary symmetry-breaking states, i.e., states that become stationary in the thermodynamic limit and give a finite value to the order parameter. We give results both for discrete and continuous symmetries and explicitly show how to construct the symmetry-breaking states. Our results show in a simple way that, in large systems, local dissipative dynamics satisfying detailed balance cannot uniquely and efficiently prepare states with extensive fluctuations with respect to local operators. We discuss the implications of our results for quantum simulators and dissipative state preparation.
JMP 58, 2: quantum abstract
Alhaidari, A., & Taiwo, T. (2017). Wilson-Racah quantum system Journal of Mathematical Physics, 58 (2) DOI: 10.1063/1.4975138 (arXiv)
Using a recent formulation of quantum mechanics without a potential function, we present a four-parameter system associated with the Wilson and Racah polynomials. The continuum scattering states are written in terms of the Wilson polynomials whose asymptotics give the scattering amplitude and phase shift. On the other hand, the finite number of discrete bound states are associated with the Racah polynomials.
Dorsch, F. (2017). Accumulation rate of bound states of dipoles generated by point charges in strained graphene Journal of Mathematical Physics, 58 (2) DOI: 10.1063/1.4976201 (arXiv)
We consider strained graphene, modelled by the two-dimensional massive Dirac operator, with potentials corresponding to charge distributions with vanishing total charge, non-vanishing dipole moment and finitely many point charges of subcritical coupling constants located in the graphene sheet. We show that the bound state energies accumulate exponentially fast at the edges of the spectral gap by determining the leading order of the accumulation rate.
Feynman in comics
Italian version written with Maria-Angela Silleni.
Telling a man’s life is always an arduous and difficult task. So it is necessary to make choices, often focusing on successes and leaving aside failures, especially if you talk about a physicist who has been interested in the most disparate fields within his specialization.
This is the some idea that Lawrence Krauss has dealt with the life of Richard Feynman in Quantum Man and the same spirit seems to animate Feynman, the graphic novel by Jim Ottaviani and Leland Myrick. After the series of Introducing, a hybrid of illustrated and comic books about science, one of the most effective comics dedicated to science, brings the signature of Ottaviani, which, on the pages of Suspended In Language, with the contribution of the comic artist Leland Purvis, told the life of Niels Bohr, the master of the Copenhagen School, whose interpretation of quantum mechanics dominated during the first steps of this new approach of physics to nature.
Ottaviani knows very well the risks in comic book science genre: in particular, falling into teaching and slamming the reader into boredom is always around the corner; so thanks to an episode narrative (which is also a limit, as we shall see below), accompanied by synthetic drawings, the volume reaches the disclosure purpose without betraying the aspect of entertainment.
A great role in the success of the graphic novel is dued by the subject: Feynman, Nobel Prize in Physics in 1965, was an eccentric character with a strong sense of humor and impatient with social conventions. Thanks to these features he often found himself in embarrassing situations with unexpected consequences. A passionate bongo player and amateur sketcher of naked women, picked up his two-volume adventures, Surely You’re Joking, Mr. Feynman! and What Do You Care What Other People Think?, which wrote to seem not overly serious: and most of the episodes narrated in the comic are getting from his autobiographical stories.
Feynman by Ottaviani and Myrick, reported by Maria Popova as one of the top 11 scientific popular books of 2012, is a splendid example of how to make interesting science. Even with the comics.
This is the some idea that Lawrence Krauss has dealt with the life of Richard Feynman in Quantum Man and the same spirit seems to animate Feynman, the graphic novel by Jim Ottaviani and Leland Myrick. After the series of Introducing, a hybrid of illustrated and comic books about science, one of the most effective comics dedicated to science, brings the signature of Ottaviani, which, on the pages of Suspended In Language, with the contribution of the comic artist Leland Purvis, told the life of Niels Bohr, the master of the Copenhagen School, whose interpretation of quantum mechanics dominated during the first steps of this new approach of physics to nature.
Ottaviani knows very well the risks in comic book science genre: in particular, falling into teaching and slamming the reader into boredom is always around the corner; so thanks to an episode narrative (which is also a limit, as we shall see below), accompanied by synthetic drawings, the volume reaches the disclosure purpose without betraying the aspect of entertainment.
A great role in the success of the graphic novel is dued by the subject: Feynman, Nobel Prize in Physics in 1965, was an eccentric character with a strong sense of humor and impatient with social conventions. Thanks to these features he often found himself in embarrassing situations with unexpected consequences. A passionate bongo player and amateur sketcher of naked women, picked up his two-volume adventures, Surely You’re Joking, Mr. Feynman! and What Do You Care What Other People Think?, which wrote to seem not overly serious: and most of the episodes narrated in the comic are getting from his autobiographical stories.
JMP 58, 1: magnetic monopoles, spacetime and gravity
Fine, D., & Sawin, S. (2017). Path integrals, supersymmetric quantum mechanics, and the Atiyah-Singer index theorem for twisted Dirac Journal of Mathematical Physics, 58 (1) DOI: 10.1063/1.4973368
Feynman’s time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ensure that these products converge. The limit defines a path integral which agrees pointwise with the heat kernel for a generalized Laplacian. The result is a rigorous construction of the propagator for supersymmetric quantum mechanics, with potential, as a path integral. Further, the class of Laplacians includes the square of the twisted Dirac operator, which corresponds to an extension of $N = 1/2$ supersymmetric quantum mechanics. General results on the rate of convergence of the approximate path integrals suffice in this case to derive the local version of the Atiyah-Singer index theorem.
Kováčik, S., & Prešnajder, P. (2017). Magnetic monopoles in noncommutative quantum mechanics Journal of Mathematical Physics, 58 (1) DOI: 10.1063/1.4973503
We discuss a certain generalization of the Hilbert space of states in noncommutative quantum mechanics that, as we show, introduces magnetic monopoles into the theory. Such generalization arises very naturally in the considered model, but can be easily reproduced in ordinary quantum mechanics as well. This approach offers a different viewpoint on the Dirac quantization condition and other important relations for magnetic monopoles. We focus mostly on the kinematic structure of the theory, but investigate also a dynamical problem (with the Coulomb potential).
JMP 57, 12: quantum mechanics for the universe
Andersson, A. (2016). Electromagnetism in terms of quantum measurements Journal of Mathematical Physics, 57 (12) DOI: 10.1063/1.4972287
We consider the question whether electromagnetism can be derived from the theory of quantum measurements. It turns out that this is possible, both for quantum and classical electromagnetism, if we use more recent innovations such as smearing of observables and simultaneous measurability. In this way, we justify the use of von Neumann-type measurement models for physical processes. We apply the operational quantum measurement theory to gain insight into fundamental aspects of quantum physics. Interactions of von Neumann type make the Heisenberg evolution of observables describable using explicit operator deformations. In this way, one can obtain quantized electromagnetism as a measurement of a system by another. The relevant deformations (Rieffel deformations) have a mathematically well-defined "classical" limit which is indeed classical electromagnetism for our choice of interaction.
Aerts, D., & Sassoli de Bianchi, M. (2016). The extended Bloch representation of quantum mechanics: Explaining superposition, interference, and entanglement Journal of Mathematical Physics, 57 (12) DOI: 10.1063/1.4973356
An extended Bloch representation of quantum mechanics was recently derived to offer a possible (hidden-measurements) solution to the measurement problem. In this article we use this representation to investigate the geometry of superposition and entangled states, explaining interference effects and entanglement correlations in terms of the different orientations a state-vector can take within the generalized Bloch sphere. We also introduce a tensorial determination of the generators of $SU(N)$, which we show to be particularly suitable for the description of multipartite systems, from the viewpoint of the sub-entities. We then use it to show that non-product states admit a general description where sub-entities can remain in well-defined states, even when entangled. This means that the completed version of quantum mechanics provided by the extended Bloch representation, where density operators are also considered to be representative of genuine states (providing a complete description), not only offers a plausible solution to the measurement problem but also to the lesser-known entanglement problem. This is because we no longer need to give up the general physical principle saying that a composite entity exists and therefore is in a well-defined state, if and only if its components also exist and therefore are also in well-defined states.
How quantum mechanics explains global warming
So, according to NASA scientists, if all the ice in 14 million sq km Antarctica melts, sea levels will rise more than 200 feet. Greenland alone has another huge chunk of the Earth’s water tied up in ice; some scientists say that its ice sheet has passed a tipping point and will be gone in the next centuries, raising ocean levels by 24 feet. These are scary amounts of sea level rise that put huge areas of population centers (New York, Boston, Miami, San Francisco, etc.) under water.Well, it's really interesting, about the climate change, to see the following Ted-Ed lesson:
In the end, one can deny climate change (although I’d not recommend it), but one cannot deny math.
You've probably heard that carbon dioxide is warming the Earth. But how exactly is it doing it? Lieven Scheire uses a rainbow, a light bulb and a bit of quantum physics to describe the science behind global warming.
Portrait of an atom
The study of the structure of the atom is long story, and it begins with Democritus, or from the point of view of the modern science, with John Dalton in 1808: indeed he tried to fix in scientific terms the ideas of the greek philosopher and naturalist.
Dalton's theory was based on five fundamental points:
A few years later, however, in 1911, Rutherford devised and conducted an important experiment(1) in which he sent a beam of alpha particles against gold nuclei. The cross section observed, i.e. the surface on which the scattered particles resulting bump, it was too large to be compatible with the Thomson's hypothesis, but was compatible with that of Rutherford, namely that the atom was made up of a positive nucleus and by a number of electrons that revolved around the core itself at a large distance (compared to nuclear ones, of course).
However, this is not the last step: in 1913 Niels Bohr refined the Rutherford's model(2). Accepted the planetary structure of the atom, Bohr suggested that the electrons in their rotational motion, could not occupy orbits at their leisure, but they must themselves at very specific distances from the nucleus: this is the dawn of quantum mechanics, that further refine the atomic model thanks to the famous Schroedinger's equation.
The atom, now, was intended as a postive nucleus consists of protons and neutrons, with a little cloud of electrons that moved around, and not on an orbital but in a sort of spherical cap. And these caps at different energy was recently directly observed by Aneta Stodolna's team(3, 4):
via phys.org, io9
(1) Rutherfor E. (1911). The scattering of α and β particles by matter and the structure of the atom, Philosophical Magazine Series 6, 21 (125) 669-688. DOI: 10.1080/14786440508637080
(2) Bohr N. (1913). On the constitution of atoms and molecules, Philosophical Magazine Series 6, 26 (151) 1-25. DOI: 10.1080/14786441308634955
(3) Stodolna, A., Rouzée, A., Lépine, F., Cohen, S., Robicheaux, F., Gijsbertsen, A., Jungmann, J., Bordas, C., & Vrakking, M. (2013). Hydrogen Atoms under Magnification: Direct Observation of the Nodal Structure of Stark States Physical Review Letters, 110 (21) DOI: 10.1103/PhysRevLett.110.213001
(4) Smeenk, C. (2013). A New Look at the Hydrogen Wave Function Physics, 6 (58) DOI: 10.1103/Physics.6.58
Dalton's theory was based on five fundamental points:
- matter is made of tiny building blocks called atoms, which are indivisible and indestructible;
- atoms of the same element are all equal to each other;
- the atoms of different elements combine with each other (via chemical reactions) in ratios of whole numbers and generally small, thus giving rise to compounds;
- atoms can be neither created nor destroyed;
- atoms of an element can not be converted into atoms of other elements.
A few years later, however, in 1911, Rutherford devised and conducted an important experiment(1) in which he sent a beam of alpha particles against gold nuclei. The cross section observed, i.e. the surface on which the scattered particles resulting bump, it was too large to be compatible with the Thomson's hypothesis, but was compatible with that of Rutherford, namely that the atom was made up of a positive nucleus and by a number of electrons that revolved around the core itself at a large distance (compared to nuclear ones, of course).
(...) an experimental method was proposed about thirty years ago, when it was suggested that experiments ought to be performed projecting low-energy photoelectrons resulting from the ionization of hydrogen atoms onto a position-sensitive two-dimensional detector placed perpendicularly to the static electric field, thereby allowing the experimental measurement of interference patterns directly reflecting the nodal structure of the quasibound atomic wave function.(3)
via phys.org, io9
(1) Rutherfor E. (1911). The scattering of α and β particles by matter and the structure of the atom, Philosophical Magazine Series 6, 21 (125) 669-688. DOI: 10.1080/14786440508637080
(2) Bohr N. (1913). On the constitution of atoms and molecules, Philosophical Magazine Series 6, 26 (151) 1-25. DOI: 10.1080/14786441308634955
(3) Stodolna, A., Rouzée, A., Lépine, F., Cohen, S., Robicheaux, F., Gijsbertsen, A., Jungmann, J., Bordas, C., & Vrakking, M. (2013). Hydrogen Atoms under Magnification: Direct Observation of the Nodal Structure of Stark States Physical Review Letters, 110 (21) DOI: 10.1103/PhysRevLett.110.213001
(4) Smeenk, C. (2013). A New Look at the Hydrogen Wave Function Physics, 6 (58) DOI: 10.1103/Physics.6.58
Triphoton Wigner quasi-probability distributions on the Poincaré sphere
Quantum mechanics places a fundamental limit on the accuracy of measurements. In most circumstances, the measurement uncertainty is distributed equally between pairs of complementary properties; this leads to the 'standard quantumlimit' for measurement resolution. Using a technique known as 'squeezing', it is possible to reduce the uncertainty of one desired property below the standard quantumlimit at the expense of increasing that of the complementary one. Squeezing is already being used to enhance the sensitivity of gravity-wave detectors and may play a critical role in other high precision applications, such as atomic clocks and optical communications. Spin squeezing (the squeezing of angular momentum variables) is a powerful tool, particularly in the context of quantum light–matter interfaces. Although impressive gains in squeezing have beenmade, optical spin-squeezed systems are stillmany orders of magnitude away from the maximum possible squeezing, known as the Heisenberg uncertainty limit. Here we demonstrate how an optical system can be squeezed essentially all the way to this fundamental bound. We construct spin-squeezed states by overlapping three indistinguishable photons in an optical fibre and manipulating their polarization (spin), resulting in the formation of a squeezed composite particle known as a 'triphoton'. The symmetry properties of polarization imply that the measured triphoton states can be most naturally represented by quasi-probability distributions on the surface of a sphere. In this work we show that the spherical topology of polarization imposes a limit on how much squeezing can occur, leading to the quasi-probability distributions wrapping around the sphere — a phenomenon we term 'oversqueezing'. Our observations of spin-squeezing in the few-photon regime could lead to new quantum resources for enhanced measurement, lithography and information processing that can be precisely engineered photon-by-photon.In optics, the Poincaré sphere is a graphical tool for visualizing different types of polarized light, and was introduced by Henri Poincaré in Theorie mathematique de la lumiere (archive.org).
Read also: spie.org | buffalo.edu | ipr.res.in
Some quotations about the amplituhedron
However, the calculation using the shape in an infinite-dimensional space, the amplituhedron, should provide us with completely new perspectives how to look at the dynamics – perspective that is timeless, obscures the location of objects and events in the space and time, and obscures the unitarity (the requirement that the total quantum-calculated probability of all possibilities remains 100%), but it unmasks some other key structures that dictate what the probabilities should be, structures we were largely ignorant about.(Luboš Motl on physics.stackexchange.com)
If the new picture becomes sufficiently generalized, you could perhaps throw away the old books because you will get an entirely new framework to compute these things and to think about all these things. But once again, you don't have to throw them away because the physical results are the same.
The amplituhedron is not built out of space-time and probabilities; these properties merely arise as consequences of the jewel’s geometry. The usual picture of space and time, and particles moving around in them, is a construct.(from Quanta Magazine via Marginal Revolution)
This rich structure is also completely new to the mathematicians.(Jaroslav Trnka - pdf)
Neutrinos: between Pontecorvo and Majorana
Neutrinos are the most elusive elementary particles in the whole zoo. The reasons are simple: first of all neutrinos don't have electric charge, so physicists cannot use electromagnetic experiments in order to detect them, and they must design indirect measures; furthermore they interact with other particles only with weak interaction. At the other hand, neutrino is, in Standard Model, massless, while from an experimental point of view, he has a really small mass: at the beginning of 2000, Mainz and Troitsk experiment measured a maximum value at 2.2 eV, that is about 4 milion less that the electron mass!
Carlo Franzinetti (left) and Bruno Pontecorvo (Right)
The idea of neutrino's mass is dued by Bruno Pontecorvo that introduced in 1957 the so called neutrino's oscillations(1, 2): in this model is expected the existence of three type of neutrinos that, combining with each other, giving rise to neutrinos usually observed in experiments. The thoery was further developed in 1962 by Ziro Maki, Masami Nakagawa and Shoici Sakata(3):
\[\begin{pmatrix} \nu_e \\ \nu_\mu \\ \nu_\tau \end{pmatrix} = \begin{pmatrix} U_{e_1} & U_{e_2} & U_{e_3} \\ U_{\mu_1} & U_{\mu_2} & U_{\mu_3} \\ U_{\tau_1} & U_{\tau_2} & U_{\tau_3} \end{pmatrix} \begin{pmatrix} \nu_1 \\ \nu_2 \\ \nu_3 \end{pmatrix}\]
where $e$, $\mu$, $\tau$ indicate the three different leptons (electron, muon and tau), $\nu$ are neutrinos, with $\nu_i$, where $i = 1,2,3$, the fundamental neutrinos.
But, if the Pontecorvo–Maki–Nakagawa–Sakata matrix describes neutrinos' oscillations, we could describe the neutrino also using a particular equation: the Majorana equation(4): \[i \gamma^\mu \partial_\mu \psi - m \psi_c = 0\] where \[\psi_c = \gamma^2 \psi^*\] is the so called conjugated charge.
Now, if a wave function $\psi$ respects the Majorana equation, then $m$ is called Majorana mass; if $\psi$ coincides with $\psi_c$, then $\psi$ is said Majorana spinor; finally, if there is a particle that can be described with the Majorana equation, then this is called a Majorana particle, i.e. a particle that coincides with its antiparticle. The leading candidate to be a Majorana particle is, look at the case, the neutrino, whose mass is probably not so important with regard to the ultimate fate of the universe. In fact, the astronomical data suggest a flat universe, where flat universe means a substantial balance between gravitational attraction and expansion of spacetime.
Carlo Franzinetti (left) and Bruno Pontecorvo (Right)
But, if the Pontecorvo–Maki–Nakagawa–Sakata matrix describes neutrinos' oscillations, we could describe the neutrino also using a particular equation: the Majorana equation(4): \[i \gamma^\mu \partial_\mu \psi - m \psi_c = 0\] where \[\psi_c = \gamma^2 \psi^*\] is the so called conjugated charge.
Now, if a wave function $\psi$ respects the Majorana equation, then $m$ is called Majorana mass; if $\psi$ coincides with $\psi_c$, then $\psi$ is said Majorana spinor; finally, if there is a particle that can be described with the Majorana equation, then this is called a Majorana particle, i.e. a particle that coincides with its antiparticle. The leading candidate to be a Majorana particle is, look at the case, the neutrino, whose mass is probably not so important with regard to the ultimate fate of the universe. In fact, the astronomical data suggest a flat universe, where flat universe means a substantial balance between gravitational attraction and expansion of spacetime.
Conclusion: the importance of neutrino oscillations are related to the property to possess a mass: experiments confirmed that property, owned by all three neutrinos in the game. The astronomical data, however, assign this property a minor role for the ultimate fate of the universe, while its mass shows instead of the Standard Model, at present, still does not understand much of the physics of our universe. Among the facts not included in the Standard Model are the Majorana particles: in particular, the neutrino could be one of them and if this is confirmed, then we would have a great step in order to know the symmetry breaking between matter and antimatter.
The B mesons and the new physics
The search about B mesons decays has a great importance in physics for the possible clues of new physics that could be discovered. So theoretical phisicists have developed some new observables in order to test this possibility. LHCb produced new results about these new parameters, in particular the so called $P_5'$.
The image shows the distribution of the $P_5'$ observable as a function of the $\mu^+ \mu^-$ invariant mass squared $q^2$. The black data points are compared with the Standard Model prediction.
According to Joaquim Matias from Universitat Autonoma de Barcelona and colleagues the deviation in $P_5'$ and small discrepancies in the other angular observables for this decay, follow a pattern. In a recent paper the authors claim that a global analysis of the LHCb data, together with previous measurements, show a deviation of $4.5 \sigma$ with respect to Standard Model expectations, which can be explained with the same mechanism. This demands further investigation, in particular to re-evaluate all the sources of theoretical uncertainty, and to understand the effects of correlations between the experimental measurements.(via LHCb)
The image shows the distribution of the $P_5'$ observable as a function of the $\mu^+ \mu^-$ invariant mass squared $q^2$. The black data points are compared with the Standard Model prediction.
The circle of life
Until the next waltz.
The image (via Quantum Diaries) represent a Feynman's diagram about the electron-positron annihilation and a next pair production. In the middle of the diagram, there is the auto energy of the photon, and it is named loop, and the diagram, loop diagram.
Read also: Let’s draw Feynman diagrams! | Feynman diagrams (pdf)
Higgs at the Tevatron
This is the week of the Higgs. Indeed, wednesday, at CERN, ATLAS and CMS announced the results of the elaboration of the data collected in the first part of 2012... and a lot of journalists write about the probable discover of the Higgs boson. Indeed the two collaborations are disegned in order to discover the boson related to the mechanism that provides the mass to the other particles. Waiting for the conference, today CDF and DZero, the two collaborations of Tevatron, publicize in two conferences the first elaboration of the complete set of data about Higgs research. Their result was summarize by the following plot:
In the image there is the combination of the final results from the two collaborations. The two experiments combined detecte an excess in signals around 125 GeV with a 2.5 sigma. It is not the discover of the Higgs boson, but it could be a good clue for the existence of the boson. So I don't know if ATLAS and CMS will confirm or update this result in their next conferences, but in every case I must remmber to the readers that with a mass of 125 GeV we have need of physics beyond Standard Model, because the only SM is not sufficient to explain our universe. In order to explain better, I reprint here some considerations that I just published for the previous Higgs day:
0.0909090909: Shroedinger's cat
Following Uri Geller (via Peter Woit), today it could be open a portal to another universe!
Geller support his original idea with a mix of string theory and numerology. At the other hand, this mixture of science and superstition is perfect for a fiction, for example The Invisibles, the cult comics written by Grant Morrison in the past century. The comic book series, behind the entertainment pourposes, hides a political purpose, to tell the world and the use of mass media in order to control people. But I don't write about this subject, but about the scientific background, in particular I would start from the many worlds hypothesis(1).
In The Invisibles there is, indeed, a war against alien from another universe. The scientific basis of the comics is the string theory: for example the simbols of early cristians is interpreted like a part of the infinity symbol, that it also represents the connection between the two universes, like two universal strings or two membranes (or branes). But the story of the scientific multiversity began woth the famous Schroedinger's cat: The thought experiment was proposed by Schroedinger in 1935 in order to proof the limits of quantum mechanics and Copenaghen interpretation, but Erwind could not know that his paradox would be generate a lot of intriguing theoretical science. One consequence is the many worlds interpretation, but another research line is the construction of a real Schroedinger's cat!
A first right attempt was made by Jonathan Friedman and collegues(2) (read physicsworld). The team realized a superconducting quantum interference device (SQUID):
They used two junctions in their experimental setup, and so they realized a superposition between two different states:
In the plot it is showed the probability to realize a transition like function of the flux $\phi_x$. The curves, plotted for different potentials, are shifted upwards in order to clarify the shapes. The quantum behaviour of the macroscopic system is argued by the existence of two peaks, that decreasing potential separate one from each other and reach the same amplitude. The two peaks correspond to two distinct macroscopic fluxes and we can conclude that they realize a macroscopic Schroedinger's cat.
A most recent attempt has made this year (via tumblr) by a team of China's researchers leaded by Xing-Can Yao(6). The team, using the following experimental setup:
Geller support his original idea with a mix of string theory and numerology. At the other hand, this mixture of science and superstition is perfect for a fiction, for example The Invisibles, the cult comics written by Grant Morrison in the past century. The comic book series, behind the entertainment pourposes, hides a political purpose, to tell the world and the use of mass media in order to control people. But I don't write about this subject, but about the scientific background, in particular I would start from the many worlds hypothesis(1).
In The Invisibles there is, indeed, a war against alien from another universe. The scientific basis of the comics is the string theory: for example the simbols of early cristians is interpreted like a part of the infinity symbol, that it also represents the connection between the two universes, like two universal strings or two membranes (or branes). But the story of the scientific multiversity began woth the famous Schroedinger's cat: The thought experiment was proposed by Schroedinger in 1935 in order to proof the limits of quantum mechanics and Copenaghen interpretation, but Erwind could not know that his paradox would be generate a lot of intriguing theoretical science. One consequence is the many worlds interpretation, but another research line is the construction of a real Schroedinger's cat!
A first right attempt was made by Jonathan Friedman and collegues(2) (read physicsworld). The team realized a superconducting quantum interference device (SQUID):
The simplest SQUID is a superconducting loop of inductance $L$ broken by a Josephson tunnel junction with capacitance $C$ and critical current $I_c$. In equilibrium, a dissipationless supercurrent can flow around this loop, driven by the difference between the flux that threads the loops and the external flux $\phi_x$ applied to the loop.
Such a superposition would manifest itself in an anticrossing, where the energy-level diagram of two levels of different fluxoid states (labelled $| 0 >$ and $| 1 >$) is shown in the neighbourhood in which they would become degenerate without coherent interaction (dashed lines). Coherent tunnelling lifts the degeneracy (solid lines) so that at the degeneracy point the energy eigenstates are \[\frac{1}{2} \left ( | 0 > + | 1 > \right )\] and \[\frac{1}{2} \left ( | 0 > - | 1 > \right ) \, ,\] the symmetric and anti-symmetric superpositions. The energy difference $E$ between the two states is given approximately by $E = \epsilon^2 + \Delta^2$, where $\Delta$ is known as the tunnel splitting.In order to proof the existence of the splitting, a necessary condition is that:
(...) the experimental linewidth of the states be smaller than $\Delta$(3). The SQUID is extremely sensitive to external noise and dissipation (including that due to the measurement of ), both of which broaden the linewidth. Thus, the experimental challenges to observing coherent tunnelling are severe. The measurement apparatus must be weakly coupled to the system to preserve coherence, while the signal strength must be sufficiently large to resolve the closely spaced levels. In addition, the system must be well shielded from external noise. These challenges have frustrated previous attempts5, 6 to observe coherence in SQUIDs.But the observation presents some difficulties, like the SQUID's sensibility to the noise, which must be shielded, and to the dissipation; the device must also preserve the coherence,
(...) while the signal strength must be sufficiently large to resolve the closely spaced levels.All of these problems influenced previous attempts(4, 5), but they found an answer thanks Friedman's team:
Probably not
A group velocity faster than $c$ does not mean that photons or neutrinos are moving faster thsn the speed of light.This is the conclusion of Fast light, fast neutrinos? by Kevin Cahill(12). He start his briefly analisys from some experimental observations of superluminal group velocity. In these experiments researchers measure a speed of light faster and slower than $c$ in vacuum. The first observation was occured in 1982(1), but an interesting collection of work in this subject is in Bigelow(7) and Gehring(11). Experimentally when some pulses journey into a highly dispersive media occur some exotic effects. One of these is the observation of a negative group velocity, that coincides with a superluminal speed.
In Bigelow's and Gehring's works wasn't a really theoretical explenation. For example Bigelow proposed the following explaination:
(...) as the combination of different absorption cross sections and lifetimes for Cr3+ ions at either mirror or inversion sites within the BeAl2O4 crystal lattice. The superluminal wave propagation is produced by a narrow “antihole” [612 Hz half width at half maximum (HWHM)] in the absorption spectrum of Cr3+ ions at the mirror sites of the alexandrite crystal lattice, and the slow light originates from an even narrower hole (8.4 Hz) in the absorption spectrum of Cr3+ ions at the inversion sites.They also considered
(...) the influence of ions both at the inversion sites and at the mirror sites. In addition, the absorption cross sections are assumed to be different at different wavelengths.
The arrows indicate the locations of ion sites that have mirror or inversion symmetry. On the right, the corresponding energy-level diagrams for Cr3+ ions at the different sites are shown.
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