arXiv subjects

Koichiro Umetsu

Publications and source records attributed to Koichiro Umetsu.

15 recordsLinked to original sources

arXiv2023

Berry's phase and quantum mechanical formulation of anomalous Hall effect

The canonical commutation relations in quantum mechanics are not maintained in the anomalous Hall effect described by Berry's phase in the presence of the electromagnetic vector potential. To define quantum mechanical formulation, one may avoid the electromagnetic vector potential but then the anomalous Nernst effect induced by Berry's curvature is not described using a modified phase space volume, although the anomalous Nernst effect by itself may be generated by the adiabatic Berry's curvature. We also comment on the Born-Oppenheimer approximation if it describes the anomalous Hall effect in the absence of the electromagnetic vector potential in quantum mechanics. An alternative view of the Bjorken-Johnson-Low prescription which is consistent with the principle of quantum mechanics and the existing Berry's phase theory of the anomalous Hall effect is also mentioned.

cond-mat.str-el↗

arXiv2022

Berry's phase and chiral anomalies

The basic materials of Berry's phase and chiral anomalies are presented to appreciate the phenomena related to those notions. As for Berry's phase, a general survey of the subject is presented using both Lagrangian and Hamiltonian formalisms. The canonical Hamiltonian formalism of the Born-Oppenheimer approximation, when applied to the anomalous Hall effect, can incorporate the gauge symmetry of Berry's connection but unable to incorporate the electromagnetic vector potential simultaneously. Transformed to the Lagrangian formalism with a time-derivative term allowed, the Born-Oppenheimer approximation can incorporate the electromagnetic vector potential simultaneously with Berry's connection, but the consistent canonical property is lost and thus becomes classical. The Lagrangian formalism can thus incorporate both gauge symmetries simultaneously but spoils the basic quantum symmetries, and thus results in classical anomalous Poisson brackets and the classical Nernst effect as in the conventional formalism. As for chiral anomalies, we present basic materials by the path integral formulation with an emphasis on fermions on the lattice. A chiral fermion defined by $γ_{5}$ on the lattice does not contain the chiral anomaly for the non-vanishing lattice spacing $a\neq0$. The idea of a spectral flow on the lattice does not lead to an anomaly for each species doubler separately but rather to a pair production in a general sense. We also mention that a specific construction called the Ginsparg-Wilson fermion, which is free of species doublers, may practically be useful. We discuss the representative applications of Berry's phase and chiral anomalies in nuclear physics and related fields to illustrate the use of these two basic notions.

nucl-th↗

arXiv2022

A path integral derivation of the equations of anomalous Hall effect

A path integral (Lagrangian formalism) is used to derive the effective equations of motion of the anomalous Hall effect with Berry's phase on the basis of the adiabatic condition $|E_{n\pm1}-E_{n}|\gg 2π\hbar/T$, where $T$ is the typical time scale of the slower system and $E_{n}$ is the energy level of the fast system. In the conventional definition of the adiabatic condition with $T\rightarrow {\rm large}$ and fixed energy eigenvalues, no commutation relations are defined for slower variables by the Bjorken-Johnson-Low prescription except for the starting canonical commutators. On the other hand, in a singular limit $|E_{n\pm1}-E_{n}|\rightarrow \infty$ with specific $E_{n}$ kept fixed for which any motions of the slower variables $X_{k}$ can be treated to be adiabatic, the non-canonical dynamical system with deformed commutators and the Nernst effect appear. In the Born-Oppenheimer approximation based on the canonical commutation relations, the equations of motion of the anomalous Hall effect is obtained if one uses an auxiliary variable $X_{k}^{(n)}=X_{k}+{\cal A}^{(n)}_{k}$ with Berry's connection ${\cal A}^{(n)}_{k}$ in the absence of the electromagnetic vector potential $eA_{k}(X)$ and thus without the Nernst effect. It is shown that the gauge symmetries associated with Berry's connection and the electromagnetic vector potential $eA_{k}(X)$ are incompatible in the canonical Hamiltonian formalism. The appearance of the non-canonical dynamical system with the Nernst effect is a consequence of the deformation of the quantum principle to incorporate the two incompatible gauge symmetries.

cond-mat.str-el↗

arXiv2021

Lensing of Dirac monopole in Berry's phase

Berry's phase, which is associated with the slow cyclic motion with a finite period, looks like a Dirac monopole when seen from far away but smoothly changes to a dipole near the level crossing point in the parameter space in an exactly solvable model. This topology change of Berry's phase is visualized as a result of lensing effect; the monopole supposed to be located at the level crossing point appears at the displaced point when the variables of the model deviate from the precisely adiabatic movement. The effective magnetic field generated by Berry's phase is determined by a simple geometrical consideration of the magnetic flux coming from the displaced Dirac monopole.

hep-th↗

arXiv2019

A classical limit of Grover's algorithm induced by dephasing: Coherence vs entanglement

A new approach to the classical limit of Grover's algorithm is discussed by assuming a very rapid dephasing of a system between consecutive Grover's unitary operations, which drives pure quantum states to decohered mixed states. One can identify a specific element among $N$ unsorted elements by a probability of the order of unity after $k\sim N$ steps of classical amplification, which is realized by a combination of Grover's unitary operation and rapid dephasing, in contrast to $k\sim π\sqrt{N}/4$ steps in quantum mechanical amplification. The initial two-state system with enormously unbalanced existence probabilities, which is realized by a chosen specific state and a superposition of all the rest of states among $N$ unsorted states, is crucial in the present analysis of classical amplification. This analysis illustrates Grover's algorithm in extremely noisy circumstances. A similar increase from $k\sim \sqrt{N}$ to $k\sim N$ steps due to the loss of quantum coherence takes place in the {\em analog} model of Farhi and Gutmann where the entanglement does not play an obvious role. This supports a view that entanglement is crucial in quantum computation to describe quantum states by a set of qubits, but the actual speedup of the quantum computation is based on quantum coherence.

quant-ph↗

arXiv2016

Separability criteria with angular and Hilbert space averages

The practically useful criteria of separable states $ρ=\sum_{k}w_{k}ρ_{k}$ in $d=2\times2$ are discussed. The equality $G({\bf a},{\bf b})= 4[\langle ψ|P({\bf a})\otimes P({\bf b})|ψ\rangle-\langle ψ|P({\bf a})\otimes{\bf 1}|ψ\rangle\langle ψ|{\bf 1}\otimes P({\bf b})|ψ\rangle]=0$ for any two projection operators $P({\bf a})$ and $P({\bf b})$ provides a necessary and sufficient separability criterion in the case of a separable pure state $ρ=|ψ\rangle\langleψ|$. We propose the separability criteria of mixed states, which are given by ${\rm Tr}ρ\{{\bf a}\cdot {\bf σ}\otimes {\bf b}\cdot {\bf σ}\}=(1/3)C\cosφ$ for two spin $1/2$ systems and $4{\rm Tr}ρ\{P({\bf a})\otimes P({\bf b})\}=1+(1/2)C\cos2φ$ for two photon systems, respectively, after taking a geometrical angular average of ${\bf a}$ and ${\bf b}$ with fixed $\cosφ={\bf a}\cdot{\bf b}$. Here $-1\leq C\leq 1$, and the difference in the numerical coefficients $1/2$ and $1/3$ arises from the different rotational properties of the spinor and the transverse photon. If one instead takes an average over the states in the $d=2$ Hilbert space, the criterion for two photon systems is replaced by $4{\rm Tr}ρ\{P({\bf a})\otimes P({\bf b})\}=1+(1/3)C\cos2φ$. Those separability criteria are shown to be very efficient using the existing experimental data of Aspect et al. in 1981 and Sakai et al. in 2006. When the Werner state is applied to two photon systems, it is shown that the Hilbert space average can judge its inseparability but not the geometrical angular average.

quant-ph↗

arXiv2014

A hidden-variables version of Gisin's theorem

It is generally assumed that {\em local realism} represented by a noncontextual and local hidden-variables model in $d=4$ such as the one used by Bell always gives rise to CHSH inequality $|\langle B\rangle|\leq 2$. On the other hand, the contraposition of Gisin's theorem states that the inequality $|\langle B\rangle|\leq 2$ for arbitrary parameters implies (pure) separable quantum states. The fact that local realism can describe only pure separable quantum states is naturally established in hidden-variables models, and it is quantified by $G({\bf a},{\bf b})= 4[\langle ψ|P({\bf a})\otimes P({\bf b})|ψ\rangle-\langle ψ|P({\bf a})\otimes{\bf 1}|ψ\rangle\langle ψ|{\bf 1}\otimes P({\bf b})|ψ\rangle]=0$ for any two projection operators $P({\bf a})$ and $P({\bf b})$. The test of local realism by the deviation of $G({\bf a},{\bf b})$ from $G({\bf a},{\bf b})=0$ is shown to be very efficient using the past experimental setup of Aspect and his collaborators in 1981.

quant-ph↗

arXiv2012

Aspects of universally valid Heisenberg uncertainty relation

A numerical illustration of a universally valid Heisenberg uncertainty relation, which was proposed recently, is presented by using the experimental data on spin-measurements by J. Erhart, et al.[ Nature Phys. {\bf 8}, 185 (2012)]. This uncertainty relation is closely related to a modified form of the Arthurs-Kelly uncertainty relation which is also tested by the spin-measurements. The universally valid Heisenberg uncertainty relation always holds, but both the modified Arthurs-Kelly uncertainty relation and Heisenberg's error-disturbance relation proposed by Ozawa, which was analyzed in the original experiment, fail in the present context of spin-measurements, and the cause of their failure is identified with the assumptions of unbiased measurement and disturbance. It is also shown that all the universally valid uncertainty relations are derived from Robertson's relation and thus the essence of the uncertainty relation is exhausted by Robertson's relation as is widely accepted.

quant-ph↗

arXiv2011

Hawking radiation as tunneling from squashed Kaluza-Klein black hole

We discuss Hawking radiation from a five-dimensional squashed Kaluza-Klein black hole on the basis of the tunneling mechanism. A simple manner, which was recently suggested by Umetsu, is possible to extend the original derivation by Parikh and Wilczek to various black holes. That is, we use the two-dimensional effective metric, which is obtained by the dimensional reduction near the horizon, as the background metric. By using same manner, we derive both the desired result of the Hawking temperature and the effect of the back reaction associated with the radiation in the squashed Kaluza-Klein black hole background.

hep-th↗

arXiv2010

Uncertainty relation and probability: Numerical illustration

The uncertainty relation and the probability interpretation of quantum mechanics are intrinsically connected, as is evidenced by the evaluation of standard deviations. It is thus natural to ask if one can associate a very small uncertainty product of suitably sampled events with a very small probability. We have shown elsewhere that some examples of the evasion of the uncertainty relation noted in the past are in fact understood in this way. We here numerically illustrate that a very small uncertainty product is realized if one performs a suitable sampling of measured data which occur with a very small probability. It is also shown that our analysis is consistent with the Landau-Pollak type uncertainty relation. It is suggested that the present analysis may help reconcile the contradicting views about the "standard quantum limit" in the detection of gravitational waves.

quant-ph↗

arXiv2010

Tunneling Mechanism in Kerr-Newman Black Hole and Dimensional Reduction near the Horizon

It is shown that the derivation of the Hawking radiation from a rotating black hole on the basis of the tunneling mechanism is greatly simplified by using the technique of the dimensional reduction near the horizon. This technique is illustrated for the original derivation by Parikh and Wilczek, but it is readily applied to a variant of the method such as suggested by Banerjee and Majhi.

hep-th↗

arXiv2010

Recent Attempts in the Analysis of Black Hole Radiation

In this thesis, we first present a brief review of black hole radiation which is commonly called Hawking radiation. The existence of Hawking radiation by itself is well established by now because the same result is derived by several different methods. On the other hand, there remain several aspects of the effect which have yet to be clarified. We clarify some arguments in previous works on the subject and then attempt to present the more satisfactory derivations of Hawking radiation. To be specific, we examine the analyses in the two recent derivations of Hawking radiation which are based on anomalies and tunneling; both of these derivations were initiated by Wilczek and his collaborators. We then present a simple derivation based on anomalies by emphasizing a systematic use of covariant currents and covariant anomalies combined with boundary conditions which have clear physical meaning. We also extend a variant of the tunneling method proposed by Banerjee and Majhi to a Kerr-Newman black hole by using the technique of the dimensional reduction near the horizon. We directly derive the black body spectrum for a Kerr-Newman black hole on the basis of the tunneling mechanism.We directly derive the black body spectrum for a Kerr-Newman black hole on the basis of the tunneling mechanism.

hep-th↗

arXiv2009

Hawking Radiation from Kerr-Newman Black Hole and Tunneling Mechanism

We present the derivation of Hawking radiation by using the tunneling mechanism in a rotating and charged black hole background. We show that the 4-dimensional Kerr-Newman metric, which has a spherically nonsymmetric geometry, becomes an effectively 2-dimensional spherically symmetric metric by using the technique of the dimensional reduction near the horizon. We can thus readily apply the tunneling mechanism to the nonspherical Kerr and Kerr-Newman metric.

hep-th↗

arXiv2008

Clear evasion of the uncertainty relation with very small probability

We entertain the idea that the uncertainty relation is not a principle, but rather it is a consequence of quantum mechanics. The uncertainty relation is then a probabilistic statement and can be clearly evaded in processes which occur with a very small probability in a tiny sector of the phase space. This clear evasion is typically realized when one utilizes indirect measurements, and some examples of the clear evasion appear in the system with entanglement though the entanglement by itself is not essential for the evasion. The standard Kennard's relation and its interpretation remain intact in our analysis. As an explicit example, we show that the clear evasion of the uncertainty relation for coordinate and momentum in the diffraction process discussed by Ballentine is realized in a tiny sector of the phase space with a very small probability. We also examine the uncertainty relation for a two-spin system with the EPR entanglement and show that no clear evasion takes place in this system with the finite discrete degrees of freedom.

quant-ph↗

arXiv2008

Ward Identities in the Derivation of Hawking Radiation from Anomalies

Robinson and Wilczek suggested a new method of deriving Hawking radiation by the consideration of anomalies. The basic idea of their approach is that the flux of Hawking radiation is determined by anomaly cancellation conditions in the Schwarzschild black hole (BH) background. Iso et al. extended the method to a charged Reissner-Nordstroem BH and a rotating Kerr BH, and they showed that the flux of Hawking radiation can also be determined by anomaly cancellation conditions and regularity conditions of currents at the horizon. Their formulation gives the correct Hawking flux for all the cases at infinity and thus provides a new attractive method of understanding Hawking radiation. We present some arguments clarifying for this derivation. We show that the Ward identities and boundary conditions for covariant currents without referring to the Wess-Zumino terms and the effective action are sufficient to derive Hawking radiation. Our method, which does not use step functions, thus simplifies some of the technical aspects of the original formulation.

hep-th↗