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Quantum Hydrodynamics in the Rotating Reference Frame Mariya Iv

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Quantum Hydrodynamics in the Rotating Reference Frame Mariya Iv Quantum hydrodynamics in the rotating reference frame Mariya Iv. Trukhanovaa) (Dated: 5 November 2018) In this paper we apply quantum hydrodynamics (QHD) to study the quantum evolution of a system of spinning particles and particles that have the electric dipole moments EDM in the rotating reference frame. The method presented is based on the many-particle microscopic Schrodinger equation in the rotating reference frame. Fundamental QHD equations for charged or neutral spinning and EDM-bearing particles were shaped due to this method and contain the spin-dependent inertial force field. The polarization dynamics in systems of neutral particles in the rotating frame is shown to cause formation of a new type of waves, the dipole-inertial waves. We have analyzed elementary excitations in a system of neutral polarized fluids placed into an external electric field in 2D and 3D cases. We predict the novel type of 2D dipole-inertial wave and 3D - polarization wave modified by rotation in systems of particles with dipole-dipole interactions. PACS numbers: 47.35.+i, 52.65.Kj, 77.22.Ej Keywords: inertial forces, quantum hydrodynamic, dipole wave, inertial wave to the circular motions of fluid elements. Large-scale I. INTRODUCTION geophysical and astrophysical flows are populated by a great variety of internal waves, some maintained by den- Over many years scientists paid attention to the ef- sity stratification (internal gravity waves), some by the fects occurring in the non-inertial frames. The influence background planetary or stellar rotation (inertial waves), of the inertial effects on electrons have always been the and yet others by the large-scale magnetic fields which main focus of most studies, starting with those that be- thread through interplanetary space and are generated long to Barnett and Einstein - de Haas1,2. Recently the in the interiors of planets and stars (Alfven waves)8. influence of rotation in spintronic applications has been The investigation of inertial waves had been imple- noticed. The spin-dependent inertial force in an acceler- mented in various geometries. The viscous flow inside ating system under the presence of electromagnetic fields a closed rotating cylinder of gas subjected to periodic have been derived from the generally covariant Dirac axial compression had been investigated numerically17. equation3. It was shown that mechanical vibration in a An experimental study of inertial waves in a closed cone high frequency resonator can create a spin current via the had been presented18, in which the inertial waves are ex- spin-orbit interaction augmented by the linear accelera- cited by a slight periodic oscillation superimposed on the tion. The enhancement of the spin-rotation coupling due cone's basic rotation rate. 4 to the interband mixing was predicted in Ref. The the- The experimental visualization of inertial waves had oretical investigation of the inertial effects’ influence on been realized using particle image velocimetry19. The the spin dependent transport of conduction electrons in author presented direct visualization of the velocity and 5 a semiconductor was reviewed in Ref. Equations of mo- vorticity fields in a plane normal to the rotation axis and tion for a wavepacket of electrons in the two-dimensional determined the characteristic wavelength. The dynamics planes subject to the spin-orbit interaction were derived of the anisotropy of grid-generated decaying turbulence 6 in , using the inertial effects due to the mechanical ro- in a rotating frame had been experimentally investigated tation. The author obtained a superposition of two cy- by means of particle image velocimetry on the large-scale clotron motions with different frequencies and a circular "Coriolis" platform20,21. spin current. The dipole-dipole interactions are the longest-range A homogeneous fluid rotating with constant angular interactions possible between two neutral atoms or velocity leads to the emergence of an unusual class of iner- molecules and occur in many systems in nature. This tial waves spreading in the interior of the fluid or inertial interaction is one of the most important interactions be- waves7 -11. Inertial waves, caused by fluid rotation, are tween atoms or molecules. In recent years much attention circularly polarized waves with a sense of rotation fixed arXiv:1610.09848v1 [physics.plasm-ph] 31 Oct 2016 is paid to the effect of the intrinsic electric dipole moment by their helicity. Inertial waves have many important (EDM) on the characteristics of charged and neutral par- features and play an important role in Geophysics11,13, ticle systems. The propagation of perturbation of EDM in evolution of liquid planet core14,15, rotating stars16. does not require much energy as it occurs without mass The restoring force for such kind of waves in the rotat- transfer. This process may be used in the information ing frame of reference is the Coriolis force, which leads transfer. In biological systems, for example, polarization processes, i.e. EDM propagation, are the predominant way of signal transfer22,23. a)Faculty of physics, Lomonosov Moscow State University, Moscow, Dipole-dipole interactions, are studied and manipu- Russian Federation; Electronic mail: [email protected] lated in Rydberg atoms, provide a strong coupling be- 2 tween atoms in an ultracold Rydberg gas, because these at investigation of wave process, which take place in 3D atoms have high principal quantum number and have physical space. large electric transition dipole moments compared to For the beginning we present the one-particle Hamil- ground state atoms24,25. This interaction can be tuned tonian in the rotating frame which can be derived from into resonance with a small electric field. This feature the fundamental equation for spinning particle in a curve of Rydberg atoms can be used in quantum computing26, space-time3 in the quantum information processing with cold neu- 27 tral atoms . Generation of entanglement between two " # ! 87 individual Rydberg Rb atoms in hyperfine ground µ iqAµ mc γ @µ − Γµ − + = 0; (1) states had been observed in26 using Rydberg blockade ef- ~ ~ fect. Dipole-dipole interactions between Rydberg atoms had been first observed by Raimond using spectral line where c; ~ and q are the speed of light, the Planck con- 28 broadening . In the magneto-optical trap ground-state stant and charge of an electron respectively, Γµ is the spin atoms are cooled down to 100K by laser cooling29. Re- connection3. The 4−spinor wave function contains the cently it has been shown that Rydberg excitation den- spin-up and spin-down components. The Hamiltonian for sities in a magneto-optical trap are limited by these the system of charged particles contains the terms of the interactions30. Another way is deceleration and trap- induced electric and magnetic fields due to inertial effect ping of Rydberg atoms by static electric fields31. Reso- and can be obtained from the one-particle Dirac equa- nant electric dipole-dipole interactions between cold Ry- tion (1). The many-particle Hamiltonian in the rotating dberg atoms had been observed using the microwave frame can be obtained as spectroscopy32. The attraction of Rydberg atoms is that it is possible to tune the Rydberg energy levels through ^ ^ ^ ^ resonance for the dipoledipole energy transfer. The in- H = H0 + Hrotor + HSO (2) vestigation of the properties of a cold ( 100K) and dense ( ∼ 1010cm−3) atomic Rydberg Cs gas had been studied 33 in using a "frozen Rydberg gas" model. N ^ 2 ! X Dp Classical methods were used previously to create a de- H^ = + q ' − µ σ^αBα 0 2m p p;ext p p p;ext scription of the collective dynamics of particles in the ro- p=1 p tating frame that takes into account the inertial effects7 -10. We will use a quantum mechanics description for sys- N N tems of N interacting particles is based upon the many- 1 X 1 X + q q G − µ2F αβσ^ασ^β; (3) particle Schrodinger equation (MPSE) that specifies a 2 p n pn 2 p pn p n wave function in a 3N-dimensional configuration space p6=n p6=n;n of inertial frame. The many-particle quantum hydrody- namics in the inertial frame was investigated in Refs.34 N ! X -38. As wave processes, processes of information trans- H^ = − "αβγ Ωα · rβD^ γ + ~σ^α · Ωα ; (4) rotor p p p 2 p p fer, diffusion and other transport processes occur in the p=1 three-dimensional physical space of rotating frame, a need arises to turn to a mathematical method of phys- ically observable values which are determined in a 3D N ^ X µp αβγ α β ^ γ physical space. To do so we should derive equations HSO = − " σ^p · Ep;extDp mpc those determine dynamics of functions of three variables, p=1 starting from MPSE in the rotating frame. This problem has been solved with the creation of a method of many- N X qpµn αβγ α β ^ γ particle quantum hydrodynamics (MPQHD) in the non- + " σ^n · @p GpnDn mnc inertial frame. p6=n N X µp II. CONSTRUCTION OF FUNDAMENTAL EQUATIONS + "αβγ "βµν "νijσ^α · Bµ rj Ωi D^ γ : (5) AND THE MODEL ACCEPTED m c p p;ext p p p p=1 p In this section we derive the many-particle quan- We consider a system of N interacting particles. The tum hydrodynamics (MPQHG) equations from many- following designations are used in the Hamiltonian (2): ^ ^ qp particle Pauli-Schrodinger equation (MPSE). We receive Dp = −i~rp − c Ap; where Aext;'p;ext - are the vector the equations for the system of charged particles with and scalar potentials of external electromagnetic field, spins. Method of MPQHG allows to present dynamic µp = gµB=2, µp - is the electron magnetic moment and of system of interacting quantum particles in terms of µpB = qp~=2mpc is the Bohr magneton, qp stands for functions defined in 3D physical space.
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