The following is a complete list of articles published or submitted in conjunction with the authors PhD. The research described in articles 1 − 3 and 5− 6 is presented in this thesis. Papers are sorted chronologically by the date of first submission.
1. “Collapsing Bose-Einstein Condensates beyond the Gross-Pitaevskii Ap- proximation”; S. W¨uster, J.J. Hope and C.M. Savage; Phys. Rev. A 71, 033605 (2005).
2. “Numerical Study of the stability of Skyrmions in a Bose-Einstein Conden- sate”; S. W¨uster, T.E. Argue and C.M. Savage; Phys. Rev. A 72, 043616 (2005).
3. “Supersonic optical tunnels for Bose-Einstein condensates”; S. W¨uster and B.J. D¸abrowska-W¨uster; cond-mat/0602108 (2006).
4. “Quantum effects in the dynamical localization of Bose-Einstein conden- sates in optical lattices”; B.J. D¸abrowska-W¨uster, S. W¨uster, A.S. Bradley, M.J. Davis and E.A. Ostrovskaya; cond-mat/0607332 (2006).
5. “Quantum depletion of collapsing Bose-Einstein condensates”; S. W¨uster, B.J. D¸abrowska-W¨uster, A.S. Bradley, M.J. Davis, P.B. Blakie, J.J. Hope and C.M. Savage; cond-mat/0609417; Phys. Rev. A in press (2007). 6. “Limits to the analogue Hawking temperature in a Bose-Einstein conden-
sate”; S. W¨uster and C.M. Savage; in preparation (2007).
The contents of articles 1 and 5 are presented in chapter 3. Article 2 is included in chapter 4, which also contains contributions from article 3. The analogue gravity chapter 5 contains research from papers 3 and 6.
Theory
This chapter briefly but systematically introduces the theoretical tools which are required in order to answer the questions posed in this thesis. In the process, we define our mathematical notation and highlight the relations between the approaches employed.
The most basic description of a Bose-Einstein condensate, on the mean-field level, is conceptually only slightly more involved than Schr¨odinger’s equation. However the BEC’s equation of motion is nonlinear, giving rise to a multitude of new phenomena [35]. Additionally, many of the topics covered in this thesis require more sophisticated tools: to deal with atom losses, supersonic flow, multi- component condensates, rotating traps and quantum corrections.
In our overview of the theory, we start from the many-body Hamiltonian, derive various incarnations of the Gross-Pitaevskii equation and finally include quantum corrections using two very different schemes: time dependent Hartree- Fock Bogoliobov theory and the truncated Wigner approach.
Our primary objective is a self-contained thesis. We also wish to offer practical guidance, regarding how and when subtleties in the theories have to be carefully considered and where they can be safely ignored.
This thesis is not primarily concerned with theory development. Some ele- ments of the presented methods were however worked out (or re-developed) within the author’s PhD, such as:
• The renormalisation of the HFB equations using Feynman diagrams (ap- pendix A).
• The implementation of the time dependent Hartree-Fock Bogoliubov (HFB) equations in a spherical harmonic trap (appendix B).
• The sketch of the inclusion of three-body loss processes, also for the uncon- densed component, in the HFB equations (section 2.4.5).
2.1
Microscopic Description
of an Interacting Bose-Gas
After second quantisation [31], a system of interacting Bosons possesses the many- body Hamiltonian: ˆ H = d3xΨˆ†(x) ˆH0(x) ˆΨ(x) +1 2 d3xd3xΨˆ†(x) ˆΨ†(x)Vint(x−x) ˆΨ(x) ˆΨ(x), (2.1) where ˆH0(x) = − 2 2m∇ 2
x +V(x) is the single particle Hamiltonian. Here ˆΨ(x)
denotes the field operator in the Heisenberg picture that annihilates atoms of mass m at positionx. The fields obey the canonical commutation relations
ˆ
Ψ(x),Ψ(ˆ x) = 0, Ψ(ˆ x),Ψˆ†(x) =δ(3)(x−x). (2.2)
V(x) is a potential, which the atoms experience, for example, due to interaction with external electromagnetic fields (section 1.2). A common example regularly encountered in this thesis is a cylindrically symmetric harmonic trap:
V(x) = 1 2m
ω⊥2x2+ω⊥2y2+ωz2z2. (2.3)
Vint(x−x) parametrises the interaction between atoms at positions x and x.
Throughout this thesis we assume this interaction potential to be local:
V(x−x) = U0δ(3)(x−x), (2.4)
with interaction strength U0. We justified this step physically in section 1.2.
Finally, we note that the Hamiltonian (2.1) is globally U(1) symmetric, i.e. it is invariant under phase rotations of the field operator: ˆΨ(x) → Ψ(ˆ x) exp (iϕ), exp (iϕ)∈U(1).
We can now derive the Heisenberg equation for the operator ˆΨ(x) from Eq. (2.1), using the above described contact potential:
i∂ ∂t
ˆ
The macroscopic occupation of a single mode of the interacting field is incorpo- rated through the assumption that the atom field operator acquires an expecta- tion value, which is the condensate wave function: φ(x) =Ψ(ˆ x). Equivalently, we can choose a particular many-body quantum state with a non-vanishing ex- pectation value, such as a coherent state. The appearance of the condensate wave function breaks the U(1) symmetry of the Hamiltonian, as condensates with dif- ferent phase are macroscopically distinct objects.
The decomposition of ˆΨ(x) into its mean and fluctuations ˆ
Ψ(x) =φ(x) + ˆχ(x), (2.6)
defines the fluctuation operator ˆχ(x). Inserting Eq. (2.6) into Eq. (2.5) and taking the expectation value yields:
i∂ ∂t ˆ Ψ(x)=i∂ ∂tφ(x) = ˆH0(x)φ(x) +U0|φ(x)| 2 φ(x) + 2U0χˆ†(x) ˆχ(x)φ(x) +U0χˆ(x) ˆχ(x)φ(x) +U0χˆ†(x) ˆχ(x) ˆχ(x). (2.7)
This equation forms our point of origin for the exploration of a couple of approx- imations to the full quantum evolution.
2.1.1
Loss Processes and the Master Equation
The Hermitian Hamiltonian in the previous section describes a system in which the number of particles is conserved. Atom losses can indeed often be neglected, as experiments are usually done in regimes where losses remain small. There are however cases where losses play a crucial role in the condensate dynamics. Two of these are covered in this thesis: (i) collapsing condensates and (ii) high density condensates for the creation of analogue Hawking radiation.
Due to the Hermiticity of any Hamiltonian, the loss has to be phenomenolog- ically modelled at the level of the equation of motion. To do so, we represent our quantum system by its density operator ˆρ. The evolution equation for ˆρ, termed amaster equation, in the presence of three-body losses with loss-constant K3 has
been derived by M. Jack in Ref. [52]:
dρˆ dt =− i [ ˆH,ρˆ] + K3 6 d3x2 ˆΨ(x)3ρˆΨˆ†(x)3−Ψˆ†(x)3Ψ(ˆ x)3ρˆ−ρˆΨˆ†(x)3Ψ(ˆ x)3. (2.8) We can similarly include two-body (one-body) losses by replacing ˆΨ(x)3 with
ˆ
formalism, we have Aˆ= Tr[ˆρAˆ] for an operator ˆA. From Eq. (2.8) we can then derive:1 dAˆ dt =− i [ ˆA,Hˆ]+ K3 6 d3xΨˆ†(x)3,Aˆ Ψ(ˆ x)3+Ψˆ†(x)3A,ˆ Ψ(ˆ x)3 . (2.9) We will make use of this result to include loss processes in the HFB formalism in sections 2.2.5 and 2.4.5. What we have presented so far applies to all theories of BEC contained in this chapter. As the next step, we focus our attention on the most basic and most popular one of these: the Gross-Pitaevskii theory.