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 − + ψ = 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 ψ 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 ˆ ˆ ˆ ˆ 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 ˆ 2 ! X Dp Classical methods were used previously to create a de- Hˆ = + q ϕ − µ σˆαBα 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 αβσˆασˆβ, (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- Hˆ = − εαβγ Ωα · rβDˆ γ + ~σˆα · 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 ˆ X µp αβγ α β ˆ γ physical space. To do so we should derive equations HSO = − ε σˆp · 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 ˆ γ particle quantum hydrodynamics (MPQHD) in the non- + ε σˆn · ∂p GpnDn mnc inertial frame. p6=n

N X µp II. CONSTRUCTION OF FUNDAMENTAL EQUATIONS + εαβγ εβµν ενijσˆα · Bµ rj Ωi Dˆ 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): ˆ ˆ qp particle Pauli-Schrodinger equation (MPSE). We receive Dp = −i~∇p − c Ap, where Aext, ϕp,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. It is important the charge of electrons qe = −e or for the charge of ions 3 qp = e, and ~ is the Planck constant, g ' 2.0023193, mp of particles in the neighborhood of r in a physical space. denotes the mass of particles, c is the speed of light in vac- If we define the concentration of particles as quantum uum. In this case the many-particle Pauli-Schrodinger average of the concentration operator in the coordinate equation for quantum particles motion in the external PN representationn ˆ = p δ(r − rp) we arrive at the conclu- fields in the rotating frame has the form sive definition for the concentration

∂Ψ(R, t) Z N i = Hˆ Ψ(R, t), (6) X + + ~ ∂t n(r, t) = dR δ(r − rp)Ψ (R, t)Ψ(R, t) = hΨ Ψi. p where R = (~r1, ..., ~rN ) and Ψ(R, t) is the rank−Nspinor. (7) The first term at the right-hand side of the (3) gives sum We will use the definition for the average value of the of kinetic energies of all particles, it contains long deriva- physical quantity f(R, t) of the particles tive including action of the vortex part of the external electromagnetic field on the particle charge. The second N Z X term in (3) describes potential energy of charges in the hfi = dR δ(r − r )f(R, t). (8) external electric field. The third term describes the Zee- p p man effect with an external magnetic field and represents Differentiation of n(r, t) with respect to time and ap- the Zeeman energy. The fourth term in (3) presents the plying of the Pauli-Schrodinger equation with Hamilto- Coulomb interaction between particles and the last term nian (2) leads to continuity equation in the rotating frame characterizes the spin-spin interactions. Green func- tions of the Coulomb and spin - spin interactions are αβ α β Gpn = 1/rpn,Fpn = 4πδαβδ(~rpn) + ∂p ∂n (1/rjk). ∂tn(r, t) + ∇j(r, t) = 0, (9) The Hamiltonian (4) characterizes the inertial effects. The first term in (4) describes the mechanical rotation where the current density vector takes a form of coupling with the angular momentum of particles rp ×Dˆ p and leads to the Euler, Coriolis and centrifugal forces in * !+ the equation of motion. The second term is the spin- 1 + + j(r, t) = (ˆjpΨ) Ψ + Ψ ˆjpΨ . (10) rotation coupling and results in the Barnett effect. The 2mp spin-rotation coupling can be unified with the Zeeman energy µpσˆp · (Bp + BΩ) using the effective Barnett field In the definition for the current density vector (10) we of the particle BΩ = cmpΩ/qp. use the current operator of i-th particle in the form The spin-orbit coupling is characterized by the terms of Hamiltonian (5). The first term on the right-hand side ˆ ˆ µp represents the interaction of internal angular momentum jp = Dp − mpΩ × rp − σˆp× (11) c or spin with the external electric field and the second term describes the effect of the Coulomb electric field on the spin. The third term in the Hamiltonian (5) describes ! N (Ω × rp) 1 X qp the influence of the effective electric field on the moving × E + ×B + µ εαβγ σˆβ∇γ G . p,ext c p 2 c p p p pn spin in the inertial frame. The effective spin-dependent p6=n electric field arises in the rotating frame. The first step in the construction of MQHD apparatus We derive the velocity of i-th particle vp is determined for the inertial frame is to determine the concentration by equation

! 1 qp µp i~ + vp = ∇pS − Ap − Ω × rp − σˆp × Ep,eff − φ ∇pφ, (12) mp c cm mp

where the second term in the definition (12) is the ro- and can be displayed in the large SOI systems. In general tational velocity characterizing the effect of mechanical case vp(R, t) depends on the coordinate of all particles rotation. The effective electric field Eeff consists of the of the system R, where R is the totality of 3N coordi- external electric field, the Coulomb internal field and the nates of N particles of the system R = (r1, ..., rN ). The internal electric field originating from the mechanical ro- S(R, t) value in the formula (12) represents the phase of tation. It was shown that this field effects due to the the wave function spin-orbit interaction SOI with the mechanical rotation 4

velocity v(r, t) in equations of continuity (9) and of the ! momentum balance (15). For that we substituted the iS(R, t) wave function (13) in the definition of the basic hydro- Ψ(R, t) = a(R, t) exp φ(R, t), (13) ~ dynamical quantities. The momentum current density tensor (34) will has the new form of where φ, normalized such that φ+φ = 1, is the new spinor. Velocity field v(r, t) is the velocity of the local centre of mass in the rotating frame, is measured in the Πβ(r, t) = mn(r, t)v(r, t)vβ(r, t) inertial frame and indicates a quantity measured in the rotating frame β β +pthermal(r, t)+Υquantum(r, t). (17)

j(r, t) = n(r, t)v(r, t), (14) As we can see, the kinetic pressure tensor and quantum pressure tensor appear in the definition for the momen- where the quantum equivalent of the thermal speed is tum current density tensor. determined as up(r, R, t) = vp(R, t) − v(r, t). We differ- entiate the momentum density (10) with respect to time and apply the many-particle Pauli-Schrodinger equation * + to time derivatives of the wave functions Ψ(R, t). As a β 2 β pthermal(r, t) = a mp · upup (18) result, a momentum balance equation can be obtained in the form is the tensor of kinetic pressure and

1 β ∂tj(r, t) + ∂βΠ (r, t) = F, (15) m * 2 2 2 2 + β 2 ~ ∂ ln a ~ a β Υquantum(r, t) = −a β + ∇sp · ∇ sp , where 2mp ∂rp∂rp 4mp * !+ (19) β 1 ˆ +ˆβ + ˆ ˆβ where the first term is the Madelung quantum potential. Π (r, t) = (jpΨ) jp Ψ + Ψ (jpjp Ψ) + c.c. 4mp This tensor is proportional to ~2, has a purely quantum (16) origin and can therefore be interpreted as an additional represents the momentum current density tensor and F quantum pressure. The second term characterizes the represents a force field including the Coriolis, centrifugal force produced by the self-interactions of the spins sp. and Euler forces in the rotating frame. We introduce the Taken in the approximation of self-consistent field, the separation of particles thermal movement with velocities continuity equation and momentum balance equation in up(r, R, t) and the collective movement of particles with the rotating frame have a form

∂tn(r, t) + ∇(nv)(r, t) = 0, (20)

e mn(r, t)(∂ + v∇)v(r, t) = en(r, t)E(r, t) + n(r, t)v(r, t) × B(r, t) − ∇ pβ(r, t) t c β

2 p ! 2 ! ~ 4 n(r, t) β ~ β − n(r, t)∇ + µn(r, t)sβ(r, t)∇B (r, t) − ∂β n∇s(r, t) · ∂ s(r, t) 2m pn(r, t) 4m

+ ~n(r, t)s (r, t)∇Ωβ(r, t) + F (r, t) + F (r, t), (21) 2 β inertial SO

where n, m and v denote the density, mass and fluid ve- lar momentum, where ξp is the thermal fluctuations of locity, E and B are the electric and magnetic fields in the spin about the macroscopic average. The first two the rotating frame, pβ is the thermal pressure tensor, terms describe the interaction with external electromag- 2 2 s = 1− < a ξp · ξp > /n is the macroscopic spin angu- netic field. As we can see, the transformations to the ro- 5 tating frame of reference does not change the continuity where the displacement vector equation (20). The first term at the right side of the equa- * + tion (21) represents the effect of external electric field on + the charge density and the second term represents the P(r, t) = Ψ rpΨ (23) Lorentz force in the rotating frame. The fourth term at the right side of the equation (21) is a quantum force pro- and duced by quantum Madelung potential. The fifth term * !+ is the effect of non-uniform magnetic field on the mag- 1 Λβ(r, t) = r (ˆjβΨ)+Ψ + Ψ+ˆjβΨ . netic moment. The fifth term appears in the equation of 2m p p p motion (21) through the magnetization energy and repre- p sents the Stern-Gerlach force due to the coupling between The force field (22) depends on the mechanical rotation magnetic moment and magnetic field. The sixth term at velocity Ω and leads from the coupling between the me- the right side of the equation (21) is the spin force, pro- chanical rotation and angular momentum (4). The first duced by the spin stress38. The spin stress in the one- term at the right side of the (22) is the Coriolis force particle model was derived by Takabayasi39. This devel- density, the third term represents the centrifugal force opment was performed using the postulate that a cor- density, the second and forth terms form the Euler force puscle of mass is embedded in the spinor wave. Spinor, field density. The main features of the Coriolis force that in that way, is represented in terms of the spacetime- this force cannot do work on the fluid and strives to de- dependent Euler angles which define the orientation of flect a fluid particle in a direction perpendicular to its the triad relative to the fixed set of Cartesian axes40,41. instantaneous velocity. The last term on the right side of In the context of this representation the spinor wave must equation (21) is the spin-orbit force field density. constitute a new form of physical field that affects on the corpuscle of mass moving within it. A. Equation for the evolution of displacement vector The seventh term on the right side of equation (21) represents the spin-rotation coupling. The eighth term is the inertial force density in the rotating frame To close the QHD equations set (20), (21) we derive equation for the displacement evolution. If we differen- tiate the definition for displacement (23) with respect to time and apply the Schrodinger equation, the required equation for the displacement evolution can be obtained ∂Ω Finertial(r, t) = −2mΩ × n(r, t)v(r, t) − m × P(r, t) β ∂t ∂tP(r, t) + ∂βΛ (r, t) = 0, (24)

We have two ways to close the QHD equations set. The first one is to express Λβ in terms of n, υ and P using additional assumptions or experimental data. The other way is to derive the equation for evolution Λβ in the same fashion it was accomplished previously for other material fields. Now the evolution equation Λβ occurs in the form β −mΩ×(Ω×P(r, t))−m∇βΩ×Λ (r, t), (22) of

1 e e ∂ Λβ(r, t) + ∂ Λβγ (r, t) = P(r, t)Eβ (r, t) + βγδΛ (r, t)Bδ(r, t) − 2Ω × Λ(r, t) t m γ m ext mc β

P α(r, t) P α(r, t) −Ω × (Ω × P(r, t) · ) − ∂ Ω × P(r, t) (25) n(r, t) t n(r, t)

The first two terms describe the interaction of parti- B. Equation for the spin evolution cles with external electromagnetic field. The next terms represent the inertial forces. To close the equations set (20) and (21) we derive equa- tion for the spin evolution. If we differentiate the defini- tion for spin-polarization 6

with respect to time and apply the Pauli-Schrodinger equation with the Hamiltonian in the rotating frame, the * + required equation for the spin evolution s without ther- + s(r, t) = φ σˆpφ (26) mal effects can be obtained

2µ (∂t + v∇)s(r, t) = s(r, t) × B(r, t) + s(r, t) × Ω(r, t) ~

~ β 2µ αβγ µ γν + s(r, t) × ∂β(n(r, t) · ∂ s(r, t)) − ε εβµν Eeff (r, t)JM (r, t) (27) 2mn(r, t) ~cn(r, t)

The first term at the right side of the equation (27) C. The evolution of polarization in the rotating frame represents the torque caused by the interaction with the external magnetic field and the magnetic field of the spin- The evolution of polarization for the particles with spin interparticle interactions in the rotating frame. The the electric dipole moment (EDM) can be derived us- second term is the spin rotation coupling term in the ro- ing a method of quantum hydrodynamics. We receive tating frame and can be interpreted as the torque caused the equations for the system of charged particles with by the interaction with effective magnetic field BΩ in- EDM. We research the fluid of EDM-having polar par- troducing the Barnett effect to the right side of equation ticles (polar molecules). Obtaining equations could be 39 (27). The third term is the spin torque . The fourth used for neutral particles with the EDM as well. Method term characterizes the torque resulting from spin-orbit of quantum hydrodynamics allows to present dynamic coupling in the effective electric field. The magnetic mo- of system of interacting quantum particles in terms of ment flux tensor occurs in this equation (27) in the form functions defined in 3D physical space in the rotating reference frame. The many-particle Hamiltonian for the * !+ EDM-having particles in the rotating reference frame has β 1 + ˆβ ˆβ + JM (r, t) = Ψ σˆpjp Ψ + (ˆσpjp Ψ) Ψ . (28) the form 4mp

We introduce the thermal fluctuation wp(r, R, t) = Hˆ = Hˆ0 + Hˆrotor (31) sp(R, t) − s(r, t) of the spin about the macroscopic av- erage s(r, t) which determined in the neighborhood of r in a physical space. Using the Madelung decomposition N ! of the N-particle wave function (13) we have obtained Dˆ 2 ˆ X p the definition for the magnetic moment flux tensor in the H0 = + qpϕp,ext − dpEp,ext 2m rotating frame p=1 p

β β β N N JM (r, t) = s(r, t)n(r, t)v + jthermal(r, t) (29) 1 X 1 X + q q G − Gαβdˆαdˆβ, (32) 2 p n pn 2 pn p n p6=n p6=n,n − ~ n(r, t)s(r, t) × ∂ s(r, t) 2m β where we introduce the definition for the thermal flux N ! X αβγ α β γ ~ α α density tensor Hˆ = − ε Ω · r Dˆ + σˆ · Ω . (33) rotor p p p 2 p p p=1 * + β The new third term in (32) is considered in the Hamil- j (r, t) = a2 · s uβ . (30) thermal p p tonian of particles through the dipole energy in the ex- ternal electric field. The fourth term in (32) presents We have two ways to close the equations set. The the Coulomb interaction between charged particles and β first one is to express JM (r, t) in terms of n(r, t), v(r, t) the last term characterizes the dipole-dipole interactions αβ α β and s(r, t) using additional assumptions or experimen- between dipoles, where Gpn = ∂p ∂n /rpn. tal data. The other way is to derive the equation for From the many-particle Hamiltonian (31) the required β evolution JM in the same fashion it was accomplished equation for the polarization evolution can be obtained previously for other material fields. as36 7

70 β 60 ∂tD(r, t) + ∂βR (r, t) = 0, (34) 50 where the polarization vector field of the EDM-having 40 particles has the form Β 30 + D(r, t) = hΨ dpΨi, (35) 20 10 and a polarization current can be derived in the form 0 0 2 ´ 107 4 ´ 107 6 ´ 107 8 ´ 107 1 ´ 108 * + k d β p ˆ+β + +ˆβ (a) R (r, t) = (jp Ψ Ψ + Ψ jp Ψ) . (36) 2mp

60 III. WAVE OF POLARIZATION 50 40 A. The inertial frame Β 30 We investigate the system of neutral particles resides 20 in a uniform electromagnetic field. It is also assumed that interactions make the largest contribution into the 10 changes in Rβ36. If so then we use the equation (34) 0 7 7 7 7 8 and the equation for the polarization current density 0 2 ´ 10 4 ´ 10 6 ´ 10 8 ´ 10 1 ´ 10 evolution36 k (b)

γ β D(r, t)D (r, t) FIG. 1. Color online: The figure (a) presents of the variable ∂tR (r, t) = σ × mn(r, t) β1(k) on the wave vector k. Ionic or molecular radius r0 are assumed to equal 0.1 nm. The figure (b) presents the dependence of the variable β(k) on the wave vector k. Ionic Z 0 γµ 0 0 or molecular radius r0 is assumed to equal 0.1 nm. × ∇β dr G (r, r )Dµ(r , t) (37)

The term on the right hand side of (37) characterizes where the dipole-dipole interactions between the particles. This Z ∞ allows the analysis of polarization waves in a system of cos(r) β1(k) = 2 dr 3 . (41) neutral particles. If we derive a solution for eigenwaves ξ r in a 2D system the dispersion equation has a form of The quantity (41) is presented on Figs. (1(a)). The dis- persion relations (38) and (40) characterize the waves of s polarization in the system of neutral particles with dipole β(k) 3/2 ω = σ | κ | E0k , (38) moments. This waves exist on a level with the acoustic mn 0 waves. Equations of continuity (20) and of the momen- where β(k) is defined by the relation tum balance (21) herein describe the dynamics of the acoustic wave. Dispersion branches of a novel type that Z ∞ occurs due to the polarization dynamics were discovered J0(r) β(k) = 2π dr , (39) 36 r2 in various physical systems . The waves of electric po- ξ larization we discovered possess the following feature - their frequencies ω tends to zero provided that k → 0. here ξ = r0k, r0 there is an ionic or molecular radius q 2 2 and k = kx + ky is a modulus of the wave vector. As

λmin = 2π/kmax > 2r0 then ξ ⊂ (0, π). The dispersion B. Dipole-inertial wave dependence (38) is presented on fig. (1(a)). In 1D case ω(k) occurs as 1. 2D-eigenwaves s σβ1(k) 2 In this section we consider the fluid of neutral parti- ω = | κ | E0k , (40) mn0 cles in the rotating reference frame. We derive disper- 8

1 ´ 107 for eigenwaves in a 2D system the dispersion equation has a form of 8 ´ 106

6 ´ 106 s Ω β(k) ω = σ | κ |2 E2k3 + 4Ω2, (43) 6 0 4 ´ 10 mn0

2 ´ 106 the dispersion relation characterizes the 2D-polarization wave modified by rotation (fig. 2(b)), where the z axis is 0 0 5.0 ´ 107 1.0 ´ 108 1.5 ´ 108 directed along the rotation axis Ω = Ωz. k (a) 2. 3D-wave 5 To analyze 3D systems we use the set of equations (34) 4 and the polarization current density equation. The equa-

3 tion for the polarization current density in the rotating Ω frame and in the approximation of the self-consistent field 2 has the form of

1 α γ α D D γ α ∂tR = σ × ∇E − 2Ω × R , (44) 0 mn 0 20 000 40 000 60 000 80 000 100 000 k where the total electric field E leads from a field equa- (b) tion ∇ · E = −4π∇ · D. Here z axis is directed along the rotation axis, D0 = Dkz and two-dimensional position FIG. 2. Color online: The dependence of frequency ω on vector r⊥ is directed in the xy plane normal to the ro- the wave vector k is displayed for the case of two-dimension tation axis, a wave number kk is the component of the polarization mode which dispersion characteristic is defined three-dimension wave vector along the rotation axis and by the equation (38) for the figure (a) and of two-dimension k⊥ is the wave vector in the xy plane. These equations polarization mode modified by rotation for the figure (b). The give the dispersion low dashed branch describes the polarization wave modified by rotation or dipole-inertial 2D-mode (43). The radius r0 is supposed to be 0.1 nm. Equilibrium polarization has form Ω2 + 4Ω2 ω2 = d ± (45) D0 = κE0. Static electric permeability κ is defined by the 2 2 equation κ = n0d0/(3kB T ), where d0 - is a dipole moment of an molecule, T - temperature of the medium, kB - Boltzmann s constant. System parameters are assumed to be as follows: Ω2 2 n = 1012sm−2, d ' 0.16D, T = 100K, E ' 102V/m and d 2 2 2 2 0 0 0 ± + 2Ω − 4ΩdΩ cos θ. m = 10−23g. 2

For a fluid at rest (Ω = 0) this relation gives the fre- sion characteristics of eigenwaves in 2D systems of neu- quencies the zero-frequency transverse mode of a fluid tral particles that take account of the EDM collective at rest ω = 0 and ω = Ωd corresponding to the trans- dynamics. The particles are assumed here to localize on verse polarization wave. It seems interesting to analyze a xy - plane and reside in a uniform electromagnetic field the effect of rotation on the wave propagation in differ- orthogonal to the plane xy. For simplicity the centrifugal ent limits. Obviously, in the approximation of large wave force is neglected and only the effects of the Coriolis force 2 2 vectors kk → ∞ and Ωd >> Ω one branch of dispersion are taken into consideration |Ω × (Ω × P)| << 2|nΩ × v| yields the inertial wave the equation (37) transforms into k2 ω2 = 4Ω2 k , (46) Dα(r, t)Dγ (r, t) k2 ∂ Rα(r, t) = σ × t mn(r, t) and another represents the polarization wave modified by Z rotation 0 µγ 0 0 α × ∇ dr G (r, r )Dµ(r , t) − 2Ω × R , (42)

4πD2 k2 and to close the equation (42) we use the equation for ω2 = σ 0 k2 − 4Ω2 k → Ω2. (47) the polarization evolution (34). If we derive a solution mn k k2 d 9

C. Conclusions 15H. Bondi and R. A. Lyttleton, On the dynamical theory of the rotation of the earth. The effect of precession on the motion of the liquid core, Proceedings of the Cambridge Philosophical Society, In this paper the method of QHD was developed for 49(3):498, (1953) spin-bearing and EDM-bearing particles in the rotating 16Wen Fu and Dong Lai, Corotational instability, magnetic res- frame. QHD equations are a consequence of MPSE in onances and global inertial-acoustic oscillations in magnetized which particles interaction is directly taken into account. black hole accretion discs, Mon. Not. R. Astron. Soc. 410, 399 (2011) The system of QHD equations we constructed is com- 17Duguet Y., Oscillatory jets and instabilities in a rotating cylin- prised by equations of continuity (20), of the momentum der, Phys Fluids 18, 104104 (2006) balance (21) and of the spin evolution equation (27). Us- 18Beardsley RC., An experimental study of inertial waves in a ing the developed method we derived the inertial force closed cone, Stud Appl Math XLIX(2):187196 (1970) 19 field (22) which consists of the Coriolis force density, the L. Messio, C. Morize, M. Rabaud and F. Moisy, Experimental observation using particle image velocimetry of inertial waves in centrifugal force density and of the Euler force field den- a rotating fluid, Exp. in Fluids 44, 519 (2008) sity. We close our system of equations by the displace- 20C. Morize, F. Moisy, M. Rabaud and J. Sommeria, Dynamics ment evolution equation (II A). In our studies of wave of the anisotropy in decaying rotating turbulence with confine- processes we used a self-consistent field approximation ment effects, 18 - eme Congres Francais de Mecanique, Grenoble of the QHD equations. In this paper we also analyzed (2007). 21F. Moisy, C. Morize, M. Rabaud, J. Sommeria, Decay laws, wave excitations caused by EDM dynamics in systems of anisotropy and cyclone-anticyclone asymmetry in decaying ro- charged and neutral particles in the rotating frame. For tating turbulence J. Fluid Mech. 666, 5 (2011) this purpose we derived the polarization evolution equa- 22Gregory S. Engel, Tessa R. Calhoun, Elizabeth L. Read, Tae-Kyu tion (34) and the polarization current for 2D systems Ahn, Tomas Mancal, Yuan-Chung Cheng, Robert E. Blanken- (42) and 3D fluids (44). Waves in a 2D fluid of EDM- ship, and Graham R. Fleming, Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems, bearing neutral particles in various physical dimensions Nature 446 (2007), 782. were considered. The effect of mechanical rotation on the 23Hohjai Lee, Yuan-Chung Cheng, and Graham R. Fleming, Co- polarization dynamics was derived. Dispersion branches herence Dynamics in Photosynthesis: Protein Protection of Ex- of a novel type that occurs due to polarization dynamics citonic Coherence, Science 316 (2007), 1462. 24Kourosh Afrousheh, Observation of resonant electric Dipole- in the rotating frame were discovered for 2D (43) and Dipole interactions between cold Rydberg atoms using microwave 3D (45) physical systems with dipole-dipole interactions. spectroscopy, Waterloo, Ontario, Canada (2006) Transfer of polarization disturbances plays a major role 25H. J. Metcalf, and P. van der Straten, Laser cooling and trapping, in the information transfer in biological systems. Such Springer (1999). 26 processes do not require particles of the medium to pos- M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 87, 037901 (2001) sess EDM as the dynamics of a system of charged parti- 27M. S. Safronova, C. J. Williams, and C.W. Clark, Phys. Rev. A cles leads to collective polarization. 67, 040303 (2003). 28J. M. Raimond, G. Vitrant, and S. Haroche, J. Phys. B 14, L655 1 S. J. Barnett, Phys. Rev. 6, 239 (1915). (1981). 2 A. Einstein and W. J. de Haas, Verh. Dtsch. Phys. Ges. 17, 152 29I. Mourachko, D. Comparat, F. de Tomasi, A. Fioretti, P. Nos- (1915). baum, V. M. Akulin, and P. Pillet, Many-Body Effects in a 3 Mamoru Matsuo, Junichi Ieda, Eiji Saitoh, and Sadamichi Frozen Rydberg Gas, Phys. Rev. Lett. 80 (1998), 253. Maekawa, Spin-dependent inertial force and spin current in ac- 30K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and celerating systems, Phys. Rev. B. 84 104410 (2011) M. Weidemuller, Phys. Rev. Lett. 93, 163001 (2004). 4 Mamoru Matsuo, Jun’ichi Ieda, and Sadamichi Maekawa Phys. 31E. Vliegen, H. J. Worner, T. P. Softley, and F. Merkt, Nonhy- Rev. B 87, 115301 (2013) drogenic Effects in the Deceleration of Rydberg Atoms in Inho- 5 B. Basu, Debashree Chowdhury, Inertial effect on spin orbit cou- mogeneous Electric Fields, Phys. Rev. Lett. 92 (2004), 033005. pling and spin transport, Annals of Phys, 335, 47 (2013); Spin 32K. Afrousheh, P. Bohlouli-Zanjani, D. Vagale, A. Mugford, M. Transport in non-inertial frame, Physica B, 448, 155 (2014). Fedorov, and J. D. D. Martin, PRL 93, 233001 (2004) 6 Mamoru Matsuo, Junichi Ieda, Eiji Saitoh, Sadamichi Maekawa, 33I. Mourachko, D. Comparat, F. de Tomasi, A. Fioretti, P. Nos- Phys. Rev. Lett. 106, 076601 (2011) baum, V. M. Akulin, and P. Pillet Phys. Rev. Lett. 80, 253 7 Edouard B. Sonin, Dynamics of Quantised Vortices in Superflu- (1998) ids, Cambridge University Press, London, (2016) 34L. S. Kuzmenkov and S. G. Maksimov, Theoretical and Mathe- 8 P. A. Davidson, Turbulence in Rotating, Stratified and Electri- matical Physics, 118, 227 (1999) cally Conducting Fluids, Cambridge University Press, London, 35L. S. Kuzmenkov, S. G. Maksimov, and V. V. Fedoseev, Theo- (2013) retical and Mathematical Physics, 126, 110 (2001) 9 Greenspan H., The theory of rotating fluids, Cambridge Univer- 36P. A. Andreev, L. S. Kuzmenkov, M. I. Trukhanova, Phys.Rev.B., sity Press, London, (1968) A quantum hydrodynamics approach to the formation of new 10 Lighthill J., Waves in fluids, Cambridge University Press, Lon- types of waves in polarized two-dimension systems of charged don, (1978) and neutral particles, 84, 245401 (2011) 11 Cushman-Roisin B., Introduction to geophysical fluid dynamics, 37P. A. Andreev, M. I. Trukhanova, Russian Physics Journal 53, Prentice-Hall, Englewood Cliffs, (1994) 1196 (2011) 12 J. Boisson, D. Cebron, F. Moisy and P. P. Cortet, Earth rotation 38Mariya Iv. Trukhanova, Prog. Theor. Exp. Phys., Quantum Hy- prevents exact solid body rotation of fluids in the laboratory, drodynamics Approach to The Research of Quantum Effects and EPL, 98, 59002 (2012) Vorticity Evolution in Spin Quantum Plasmas, 2013, 111I01 13 Pedlosky J., Geophysical fluid dynamics, Springer, Heidelberg (2013) (1987) 39T. Takabayasi, Prog. Theor. Phys. 14, 283 (1955); 14 K. D. Aldridge and L. I. Lumb., Inertial waves identified in the T. Takabayasi, J. P. Vigier, Prog. Theor. Phys. 18, 573 (1957); earths fluid outer core. Nature, 325(6103):421 (1987) 10

T. Takabayasi, Prog. Theor. Phys. 70, 1 (1983); P. R. Holland, Phys. Lett. A. 128, 9 (1988); P. R. Holland, Phys. T. Takabayasi, Prog. Theor. Phys. Suppl. 4, 1 (1957); Rep. 169, 293 (1988) 40P. R. Holland, P. N. Kyprianidis, Ann. Inst. Henri Poincare. 49, 41H. A. Kramers, Quantentheorie des Electrons and des Strablung 325 (1988); (Leipzig, 1938) 259 P. R. Holland, J. P. Vigier, Found. Phys. 18, 741 (1988); Peter R. Holland, The Quantum Theory of Motion, Cambridge University Press, (1993);