Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

Comment:

      The Gross-Pitaevskii operator is simply a partial trace over the second particle:

(45)

      where Pθ(2) is the projection operator Pθ(2) of the state of particle 2 onto |θ〉:

(46)

To show this, let us compute the partial trace on the right-hand side of (45). To obtain this trace (Complement E_III, § 5-b), we choose for particle 2 a set of basis states {|θ_n〉} whose first vector |θ₁〉 coincides with |θ〉:

      (47)

Replacing Pθ(2) by its value (46) yields the product of δik (for the scalar product associated with particle 1) and δ_n1 (for the one associated with particle 2). This leads to:

      (48)

      which is simply the initial definition (40) of . Relation (45) is therefore another possible definition for the Gross-Pitaevskii potential.

4. Physical discussion

      We have established which conditions the variational wave function must obey to make the energy stationary, but we have yet to study the actual value of this energy. This will allow us to show that the parameter μ is in fact the chemical potential associated with the system of interacting bosons. We shall then introduce the concept of a relaxation (or “healing”) length, and discuss the effect, on the final energy, of the fragmentation of a single condensate into several condensates, associated with distinct individual quantum states.

4-a. Energy and chemical potential

Since the ket |φ〉 is normalized, multiplying (44) by the bra 〈φ| and by N, we get:

      (49)

We recognize the first two terms of the left-hand side as the average values of the kinetic energy and the external potential. As for the last term, using definition (41) for , we can write it as:

      (50)

      which is simply twice the potential interaction energy given in (33) when |θ₁〉 = |φ〉. This leads to:

(51)

      To find the energy , note that N μ/2 is the sum of and of half the kinetic and external potential energies. Adding the missing halves, we finally get for :

      (52)

      An advantage of this formula is to involve only one- (and not two-) particle operators, which simplifies the computations. The interaction energy is implicitly contained in the factor μ.

The quantity μ does not yield directly the average energy, but it is related to it, as we now show. Taking the derivative, with respect to N, of equation (34) written for |θ〉 = |φ〉, we get:

      (53)

      For large N, one can safely replace in this equation (N — 1/2) by (N — 1); after multiplication by N, we obtain a sum of average energies:

      (54)

Taking relation (51) into account, this leads to:

      (55)

      We know (Appendix VI, § 2-b) that in the grand canonical ensemble, and at zero temperature, the derivative of the energy with respect to the particle number (for a fixed volume) is equal to the chemical potential. The quantity μ introduced mathematically as a Lagrange multiplier, can therefore be simply interpreted as this chemical potential.

4-b. Healing length

      The “healing length” is an important concept that characterizes the way a solution of the time-independent Gross-Pitaevskii equation reacts to a spatial constraint (for example, the solution can be forced to be zero along a wall, or along the line of a vortex core). We now calculate an approximate order of magnitude for this length.

      Assuming the potential V₁ (r) to be zero in the region of interest, we divide equation (28) by φ(r) and get:

(56)

      Consequently, the left-hand side of this equation must be independent of r. Let us assume φ(r) is constant in an entire region of space where the density is n₀, independent of r:

      (57)

but constrained by the boundary conditions to be zero along its border. For the sake of simplicity, we shall treat the problem in one dimension, and assume φ(r) only depends on the first coordinate x of r; the wave function must then be zero along a plane (supposed to be at x = 0). We are looking for an order of magnitude of the distance ξ over which the