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On the Hydrodynamic Canonical Formalism of the Gross-Pitaevskii Field

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On the Hydrodynamic Canonical Formalism of the Gross-Pitaevskii Field On the hydrodynamic canonical formalism of the Gross-Pitaevskii field Y. Buggy and P. Ohberg¨ SUPA, Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom We derive a canonical formalism for the hydrodynamic representation of the Gross-Pitaevskii field (nonlinear Schr¨odinger field), where the density and the phase of the condensate form a canonical pair of conjugate field variables. To do so, we treat the meanfield as a singular Lagrangian system and apply both the Dirac-Bergmann and Faddeev-Jackiw methods. The Faddeev-Jackiw method is found to be a more direct approach to the problem. I. INTRODUCTION II. THE GROSS-PITAEVSKII FIELD The Hamiltonian of a dilute cloud of N Bose atoms In the early twentieth century, it was discovered that with weak contact interactions, may be written as fluids at low temperatures acquire unusual properties, 2 such as flow without resistance. This phenomenon called pˆi Hˆ = + V (ri) + gδ (ri rj ) , (1) superfluidity, has since attracted great interest, both 2m − i i6=j from a fundamental and applications point of view. Su- X X 2 perfluidity was first discovered in Helium. In 1995, exper- where pˆi = i~∇r , g = 4π~ a/m, a is the s-wave − i imentalists managed to cool and trap alkali gases [1–3] scattering length and V is an external potential. The to such low temperatures that quantum effects started to Gross-Pitaevskii equation may be retrieved as a mean- play a major role. These ultracold bosonic gases formed field approximation to the true dynamics of the system, a Bose-Einstein condensate with exotic properties, such by treating the many-body wavefunction as a product of as superfluidity and other coherent many-body effects. identical single particle states: Nowadays, Bose-Einstein condensates are used as a tool N to study the properties of matter, by creating macro- Ψ (r1, r2, , rN )= φ (ri) , (2) scopic quantum states and using these for probing effects ··· i=1 such as quantum phase transitions and nonequilibrium ef- Y 2 fects in quantum gases. In contrast to superfluid 4He, the where d3r φ (r) = 1. To do so, it is convenient to atomic Bose-Einstein condensate is a weakly interacting introduce the| Lagrangian| of the many-body system [14– gas and therefore well described by a meanfield theory 18] R and the Gross-Pitaevskii equation [4]. The mechanics of N this complex scalar field are analogous to those of a classi- 3 ∗ L = d riΨ i~∂t Hˆ Ψ, (3) cal fluid [5], where the amplitude squared ρ and the phase − i=1 θ of the field may be identified with the density and the Z Y velocity potential of a fluid, respectively. This analogy whose variation with respect to Ψ∗ yields the many-body is well-known and made evident most frequently through Schr¨odinger equation. Inserting Eq. (2) for Ψ into Eq. the Hamilton-Jacobi or Lagrangian formalism. Perhaps (3), leads to the meanfield Lagrangian less known, is the canonical formalism for the hydrody- 3 ∗ namic representation of the field where ρ and θ form a LMF = d rψ i~∂t HˆMF ψ, (4) − canonically conjugate pair. In the condensed matter liter- Z ature (see [4, 6–10] for instance), this fact is often postu- where the meanfield Hamiltonian, HˆMF , acts on the arXiv:1908.07090v2 [cond-mat.quant-gas] 17 Mar 2020 lated seemingly a priori, where one observes that the cor- macroscopic wavefunction, ψ (r)= √Nφ (r), as rect wave equations for the field components emerge from ~2 the proposed canonical field equations. This presupposes 2 g 2 HˆMF ψ = + V + ψ ψ. (5) that ρ and θ are conjugate variables to be begin with. −2m∇ 2 | | The conjugate nature of the hydrodynamic variables has Since henceforth our concern lies only with the meanfield, been shown to result from general symmetry properties of we drop the subscript MF . In accordance with Eqs. (4) the system [11–13], where ρ and θ play the role of the and (5), the meanfield Lagrangian density, may be pre- momentum and coordinate− field variables, respectively. sented, as In this paper, we examine an alternative forward deriva- 2 tion of the problem, by treating the Gross-Pitaevskii field i~ ~ g 4 2 = ψ∗ψ˙ ψψ˙∗ ∇ψ ∇ψ∗ ψ V ψ , as a singular Lagrangian system and applying both the L 2 − − 2m · − 2 | | − | | Dirac-Bergmann and Faddeev-Jackiw methods. (6) 2 where we have denoted partial differentiation with re- Dynamical systems with this property are called singu- spect to time by a dot and performed the transforma- lar Lagrangian systems or constrained Hamiltonian sys- i~ ∗ tion 2 ∂t (ψ ψ), such that IR. Inserting tems [21, 22]. Two equivalent [23] methods have been de- the LagrangianL→L− density (6) into the Euler-LagrangeL ∈ field vised to eliminate redundant variables and construct a re- equation for ψ∗, duced phase space for such systems: the Dirac-Bergmann method [22, 24–32] and the Faddeev-Jackiw method δL ∂ δL =0, (7) [28, 33–36]. In the particular case of the Schr¨odinger field, δψ∗ − ∂t ˙∗ δψ an alternative route is made available by performing a yields the Gross-Pitaevskii equation suitable canonical transformation [20, 37], where one be- gins by decomposing the field into real and imaginary 2 ~ 2 2 parts and then supplements the resulting Lagrangian by i~∂tψ = ∇ + V + g ψ ψ, (8) −2m | | a total time derivative to obtain a single pair of real conju- gate variables. Several other field transformations yield- where δL/δψ = ∂ /∂ψ ∇ (∂ /∂ (∇ψ)) and δL/δψ˙ = ing Schr¨odinger’s equation from a canonical field equa- L − · L ∂ /∂ψ˙ are functional derivatives [19, 20]. Carrying out tion, can also be found in the literature [16, 38]. How- L the same procedure for ψ, yields the complex conjugate ever, these depend either on one or two complex pairs of of the Gross-Pitaevskii equation. conjugate variables, meaning redundant variables have not entirely been eliminated. For a review of these for- malisms, and an application of the Dirac-Bergmann and III. THE GROSS-PITAEVSKII FIELD AS A the Faddeev-Jackiw methods to the Schr¨odinger field, see SINGULAR LAGRANGIAN SYSTEM [39]. In this paper, we take a different approach and derive Notice how a complex field variable, ψ, automatically a canonical formalism in terms of the single pair of real requires that also depend on ψ∗ in order for the action variables (ρ,θ), namely the density and the phase of the L to be real [20]. Yet, the fact that the Gross-Pitaevskii matter-field ψ. These constitute the natural pair of vari- equation is first order in time, signifies that ψ, ψ∗, ψ˙ and ables connecting the field and fluid descriptions of a con- ψ˙ ∗ are not independent and that an excess of dynamical densate. Accordingly, we shall refer to this formalism as variables are contained in the Lagrangian [16, 20]. This the “hydrodynamic canonical formalism”. is invariably the situation when the Lagrangian density is linear in the time derivatives of the fields. To see this, let ˙ φα, ∇φα, φα be a multi-component Lagrangian den- IV. HYDRODYNAMIC FORMALISM FOR THE L sity which is linear in the time derivatives of the fields, GROSS-PITAEVSKII FIELD where φα φ1, , φn. In this situation, the total La- ≡ ··· grangian of the system, may be written as Following up on previous discussions, let us represent 3 the Gross-Pitaevskii field in the polar, or Madelung form L φα, φ˙α = d r i (φα, ∇φα) φ˙i U [φα] , (9) A − i i θ ∗ −i θ h i Z X ψ = √ρe ~ , ψ = √ρe ~ , (13) where U is an interaction functional of the field compo- nents: and treat (ρ,θ) as the independent variables subject to 3 the process of variation. From a dynamical perspective, U [φα]= d r (φα, ∇φα) . (10) U a clear role is played by the new field variables: the first Z defines the distribution or amplitude of the matter-field, When L is of the form (9), the canonical momenta, while the second dictates the flow of this distribution. πi = i (φα, ∇φα), can not be treated as independent This may be seen by substituting Eq. (13) into the cur- A dynamical variables. Indeed, upon constructing the to- rent density tal Hamiltonian ~ n i ∇ ∗ ∗∇ 3 ˙ ˙ J = (ψ ψ ψ ψ) , (14) H [φα, πα]= d r πiφi L φα, φα , (11) 2m − =1 − Z i h i X which yields the more perspicuous form the field velocities φ˙i no longer appear on the right hand side of Eq. (11) as they would typically. Hence, the ρ J = ∇θ, (15) Hamiltonian reduces to the interaction functional, H = m U, and it is not possible to invert φ˙i as a function of φα and πα, since and allows for the identification of the velocity field 2 ∂ ∂ i ∇ L = A =0. (12) θ ˙ ˙ ˙ v = . (16) ∂φj ∂φi ∂φj m 3 A. Lagrangian formalism according to ∂ ρθ Under substitution (13), the Lagrangian density (6), + (24) L → L ∂t 2 becomes 2 2 2 2 1 (∇θ) g ~ 2 (∇θ) g ~ 2 = θρ˙ ρθ˙ ρ + ρ + V (∇ρ) . = ρ θ˙ + + ρ + V (∇ρ) . (17) 2 − − 2m 2 − 8mρ L − 2m 2 − 8mρ ! The Euler-Lagrange field equations for θ and ρ, yield re- This will allow for the possibility of readily identifying spectively, the wave equations a canonical transformation (54), which separates out the physical pair of conjugate variables from the redundant ∂tρ + ∇ J =0, (18) pair of conjugate variables, the latter representing the · constraints of the theory. The first order nature of the 1 2 ∂tθ + mv + gρ + V + Q =0, (19) Lagrangian density (24), means that the canonical mo- 2 menta where v = v is the modulus of the superfluid flow from ∂ θ ∂ ρ Eq.
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