Problems on Non-Equilibrium Statistical Physics

Four problems in non-equilibrium statistical physics are investigated: 1. The thermodynamics of single-photon gas; 2. Energy of the ground state in Multi-electron atoms; 3. Energy state of the H2 molecule; and 4. The Condensation behavior in N weakly interacting Boson gas. In the single-photon heat engine, we have derived the equation of state similar to that in classical ideal gas and applied it to construct the Carnot cycle with a single photon, and showed the Carnot efficiency in this single-photon heat engine. The energies of the ground state of multi-electron atoms are calculated using the modi ed Bohr model with a shell structure of the bound electrons. The di erential Schrodinger equation is simpli ed into the minimization problem of a simple energy functional, similar to the problem in dimensional scaling in the H-atom. For the C-atom, we got the ground state energy -37:82 eV with a relative error less than 6 %. The simplest molecular ion, H+ 2 , has been investigated by the quasi-classical method and two-center molecular orbit. Using the two-center molecular orbit derived from the exact treatment of the H+ 2 molecular ion problem, we can reduce the number of terms in wavefunction to get the binding energy of the H2 molecule, without using the conventional wavefunction with over-thousand terms. We get the binding energy for the H2 with Hylleraas correlation factor 1 + kr12 as 4:7eV, which is comparable to the experimental value of 4:74 eV. Condensation in the ground state of a weakly interacting Bose gas in equilibrium is investigated using a partial partition function in canonical ensemble. The recursive relation for the partition function developed for an ideal gas has been modi ed to be applicable in the interacting case, and the statistics of the occupation number in condensate states was examined. The well-known behavior of the Bose-Einstein Condensate for a weakly interacting Bose Gas are shown: Depletion of the condensate state, even at zero temperature, and a maximum uctuation near transition temperature. Furthermore, the use of the partition function in canonical ensemble leads to the smooth cross-over between low temperatures and higher temperatures, which has enlarged the applicable range of the Bogoliubov transformation. During the calculation, we also developed the formula to calculate the correlations among the excited states

On the connection between quantum nonlocality and phase sensitivity of two-mode entangled Fock state superpositions

In two-mode interferometry, for a given total photon number $N$, entangled Fock state superpositions of the form $(|N-m\rangle_a|m\rangle_b+e^{i (N-2m)\phi}|m\rangle_a|N-m\rangle_b)/\sqrt{2}$ have been considered for phase estimation. Indeed all such states are maximally mode-entangled and violate a Clauser-Horne-Shimony-Holt (CHSH) inequality. However, they differ in their optimal phase estimation capabilities as given by their quantum Fisher informations. The quantum Fisher information is the largest for the $N00N$ state $(|N\rangle_a|0\rangle_b+e^{i N\phi}|0\rangle_a|N\rangle_b)/\sqrt{2}$ and decreases for the other states with decreasing photon number difference between the two modes. We ask the question whether for any particular Clauser-Horne (CH) (or CHSH) inequality, the maximal values of the CH (or the CHSH) functional for the states of the above type follow the same trend as their quantum Fisher informations, while also violating the classical bound whenever the states are capable of sub-shot-noise phase estimation, so that the violation can be used to quantify sub-shot-noise sensitivity. We explore CH and CHSH inequalities in a homodyne setup. Our results show that the amount of violation in those nonlocality tests may not be used to quantify sub-shot-noise sensitivity of the above states.Comment: Published online in Quantum Information Processin

Inefficiency of classically simulating linear optical quantum computing with Fock-state inputs

Aaronson and Arkhipov recently used computational complexity theory to argue that classical computers very likely cannot efficiently simulate linear, multimode, quantum-optical interferometers with arbitrary Fock-state inputs [Aaronson and Arkhipov, Theory Comput. 9, 143 (2013)]. Here we present an elementary argument that utilizes only techniques from quantum optics. We explicitly construct the Hilbert space for such an interferometer and show that its dimension scales exponentially with all the physical resources. We also show in a simple example just how the Schr\"odinger and Heisenberg pictures of quantum theory, while mathematically equivalent, are not in general computationally equivalent. Finally, we conclude our argument by comparing the symmetry requirements of multiparticle bosonic to fermionic interferometers and, using simple physical reasoning, connect the nonsimulatability of the bosonic device to the complexity of computing the permanent of a large matrix.Comment: 7 pages, 1 figure Published in PRA Phys. Rev. A 89, 022328 (2014

The two electron molecular bond revisited: from Bohr orbits to two-center orbitals

In this review we first discuss extension of Bohr's 1913 molecular model and show that it corresponds to the large-D limit of a dimensional scaling (D-scaling) analysis, as developed by Herschbach and coworkers. In a separate but synergetic approach to the two-electron problem, we summarize recent advances in constructing analytical models for describing the two-electron bond. The emphasis here is not maximally attainable numerical accuracy, but beyond textbook accuracy as informed by physical insights. We demonstrate how the interplay of the cusp condition, the asymptotic condition, the electron-correlation, configuration interaction, and the exact one electron two-center orbitals, can produce energy results approaching chemical accuracy. Reviews of more traditional calculational approaches, such as Hartree-Fock, are also given. The inclusion of electron correlation via Hylleraas type functions is well known to be important, but difficult to implement for more than two electrons. The use of the D-scaled Bohr model offers the tantalizing possibility of obtaining electron correlation energy in a non-traditional way.Comment: 99 pages, 29 figures, review article, to appear in Advances in Atomic, Molecular and Optical Physic