On the origin of Stark effect of rotons In what follows from: (i) a proposal of London1,9 that the Bose–Einstein condensate (BEC) (also known as in He-II and the existence of p = 0 p = 0 condensate) is responsible for superfluidity and condensate related aspects of He-II (superfluid phase of LHe-4), and (ii) the conclusion of a microscopic model of weakly 10 Y. S. Jain 1, *, L. Chhangte1, S. Chutia 1,2 and S. Dey 1,3 interacting bosons developed by Bogoliubov that inter- 1 particle repulsion pushes a fraction of particles to the Department of Physics, North-Eastern Hill University, states of p ≠ 0, most people working in the field believe Shillong 793 022, India 2 Department of Physics, St Anthony’s College, Shillong 793 001, India that a SIB in its superfluid phase has depleted value of 7,11 3 Don-Bosco College of Engineering and Technoogy, Assam Don-Bosco p = 0 condensate. The He-II is reported to have a University, Guwahati 781 017, India maximum of about 10% atoms in the p = 0 state and the BEC state of trapped dilute gases (TDG) about 60% (ref. Linear Stark effect of roton transition, experimentally 4). London’s proposal was questioned by Landau12 as observed through microwave absorption in He-II 4 soon as it was advanced. Landau rightly argued that LHe- (superfluid He) in the presence of varying external 4 is not a system of non-interacting bosons (SNIB) for electric field, is critically analysed. We find that: (i) which Einstein13 concluded the existence of BEC below The effect cannot be explained in terms of conven- 2 2/3 tional microscopic theory (CMT) of He-II which pre- T = TBEC = (h /2 π mkB)(N/2.61V) , where N is the total sumes the existence of p = 0 condensate and concludes number of particles and V the volume of the system. The that 4 He atoms even at T = 0 have random motions issue was debated by Landau and London for a long time and mutual collisions which do not support the basic until it was settled in favour of London’s proposal with factor (viz. an ordered arrangement of atomic electric the publication of Bogoliubov’s model 7. dipoles) needed for its occurrence. (ii) The desired The conventional microscopic theories (CMT) of a SIB order is concluded, rather, by a non-conventional micro- use single particle basis (SPB) by identifying a single scopic theory (NCMT) as an intrinsic property of particle as the basic unit of the fluid and by assuming tha t He-II. Accordingly, all atoms in He-II define a close the particles occupy states of a single particle confined to packed arrangement of their wave packets (CPA-WP) volume V. However, SPB has a missing link with two with identically equal nearest neighbour distance (d), 2 2 important realities of a SIB: (a) particles interact with per particle zero-point energy (ε0 = h /8md ) and equivalent momentum, h/2d. (iii) The CPA-WP pre- two-body interactions, indicating that a pair of particles vent atoms from having relative motions and mutual should form the basic unit of the system, and (b) on cool- collisions capable of disturbing any order of atomic ing the system to T at which λT becomes of the order of d, dipoles. As such the NCMT and the observed Stark the particles assume a state of their wave super-position effect have strong mutual support; whereas the for which no particle can be described as independent former concludes CPA-WP necessary for the occur- represented by a plane wave; one at least needs two parti- rence of the effect, the latter strengthens the experi- cles with their representative plane waves to find the mental support for the former, which means that the wave function resulting from their wave super-position observation does not support the presence of p = 0 and the way it affects the physical behaviour of particles condensate in He-II. occupying such a state. This is particularly important Keywords: Bosons, microwave absorption, roton tran- because superfluidity is undoubtedly a consequence of the sition, Stark-effect. wave nature which assumes dominance at LT. In order to avoid the said missing link, Jain’s non-conventional LIQUID helium-4 (LHe-4), a system of interacting bosons 8 1–8 microscopic theory (NCMT) , rightly uses pair of parti- (SIB), has been a subject of extensive research , for its cles basis (PPB), and it is for this reason that: (i) an unique behaviour, such as superfluidity (flow of the fluid apparently simple difference of SPB and PPB renders without viscosity) and related properties which arise significantly different G-states (symbolized as G- when quantum nature of the atoms dominates its low state(SPB) and G-state(PPB)) which are depicted by the temperature (LT) behaviour at macroscopic scale. Among momentum distribution (Np) of particles in Figure 1 a and all the liquids in nature, only LHe-4 assumes superfluid- b respectively, and the respective position distribution in T T 4 ity at < λ = 2.17 K, because He atoms can have the Figure 1 c and d, and (ii) only G-state(PPB) helps in find- largest ratio of the thermal de Broglie wavelength ing a clear understanding of superfluidity and related (λ = h/ 2π mk T , h k 8 T B with being the Planck constant, B properties of He-II . m the Boltzmann constant, the mass of helium atom) to A critical analysis of N (Figure 1 a) of G-state(SPB) d p the inter-atomic separation . The motivation for its stud- (as discussed briefly in Appendix 1, and in detail in ies also lies with the fact that the microscopic understand- Jain14), unequivocally establishes that the corresponding ing of the said behaviour is still unclear. energy E0(SPB) does not assume the least possible value as expected. This remained unnoticed for more than six *For correspondence. (e-mail: [email protected]) decades, possibly due to the strong bias for the existence of p = 0 condensate. Evidently, G-state(SPB) does not only explains the properties of LHe-4 at quantitative describe the true G-state of a SIB. The analysis further scale but also concludes: (i) the absence of p = 0 conden- concludes (Appendix 1) that the true G-state is described sate and (ii) true G-state which agrees with G-state(PPB) by G-state(PPB) having no particle with p < h/2d, which and related inferences by Jain 14 and (iii) the real nature of establishes the absence of p = 0 condensate in He-II. It is, BEC that exists in He-II. The absence of the p = 0 con- obviously, not surprising that a number of recent studies densate is also supported strongly25,26 by the physical re- of LHe-4 or smaller systems of 4 He atoms (published ality of the existence of electron bubble in LHe-4 (ref. over the last 15 years) report several interesting new 27) and the experimentally observed spectroscopy of em- results, viz. the loss of viscosity for the rotation of mole- bedded molecules15–18. cules embedded in microscopic clusters and droplets15–18, Next we analyse the observed effect19–21 not only to linear Stark effect in roton mode observed in the absorp- understand its origin but also to identify it as another ex- tion of microwaves19–21 , etc. which question the existence perimental evidence for the absence of p = 0 condensate of the p = 0 condensate. in He-II; to this effect we summarize relevant important as - Neutron inelastic scattering experiments, supposed to pects of a SIB and roton excitation in He-II in Appendix 1. render a direct proof for the existence of the p = 0 con- In what follows from Rybalko et al.19–21 we note that: densate, have been performed using neutron beams of dif- (a) Microwave absorption in LHe-4 is found to peak at ferent energies and their results are reviewed in several the frequency (corresponding to the energy of its well- reports 3,11,22. While different reports analysing these known quantum quasi-particle excitation known as roton ) experiments conclude different values of p = 0 conden- which increases smoothly from ≈ 5.2 K (≡ 125.0 GHz) at sate ranging from 0% to about 20% (refs 3, 11, 22), a T = Tλ to ≈ 8.65 K (≡ 180.3 GHz) at T = 0, with its width value around 10% has been accepted for the strong bias decreasing from ≈ 400 KHz at 2.2 K to ≈ 40 KHz at of people in favour of its existence, otherwise several ex- 1.6 K. 23 perts raise doubts on its existence . However, the said (b) Under the influence of external electric field, Eex , the bias still exists (possibly for reasons similar to those said peak splits into two Stark components separated by 24 counted by Hirsch for the strong bias in favour of BCS pE E theory of superconductivity). Consequently, new ideas ∆f = ex ≈ 2.8×10-34 ex , (1 ) and approaches to understand the LT behaviour of a SIB h h –34 are not finding their due place, although a microscopic showing its linear dependence on Eex with p ≈ 2.8 × 10 theory that explains the experimental properties of LHe-4 SI (here and henceforth SI refers to the SI units of the at quantitative scale is still awaited. In addition, we also physical quantity) being the experimental value of th e lack right understanding of: (i) superfluidity, (ii) the true electric dipole moment, presumably, of a 4He atom.