Ruprecht Karls Universit�t Heidelberg

Theory Group on Ultracold Atomic and Molecular Systems - Ultracold few-atom systems

Why ultracold few-atom systems?

Ever since the experimental realization of Bose-Einstein condensation (awarded with the Nobel prize in 2001; see http://cua.mit.edu/ketterle_group/research.htm for a nice website), ultracold atoms have been an enourmously popular research object. Using the atoms' interaction with electric (Laser) and magnetic fields, one can virtually `design' both their external trapping forces and their interactions. That makes them an extremely useful tool to study all kinds of fundamental physics (such as superconductivity/superfluidity). But there also applications exploiting their coherence: e.g., in atom interferometry or in quantum computation.

Typically, experiments with ultracold atoms are done with about $ 10^{3-5}$ atoms. In this limit, `individual' interactions between the particles become irrelevant, and their are replaced by a mean field (the so-called Gross-Pitaevskii description, where all atoms `Bose-condense' into the same single-particle state $ \Psi=\phi^{\otimes N}$). For few atoms, in turn, one can explore mutual correlations in detail, which promises much more thrilling effects. There are some other advantages:

  • Few-atom systems are amenable to exact calculations, allowing for a `bottom-up' approach to a better understanding of larger systems.
  • There is a much higher level of control than for a system with many atoms.

One example: 1-D Bose gas

One system that neatly lends itself as an object of study is the quasi-one-dimensional (1D) Bose gas. Here the transverse degrees of freedom (say, $ y,z$) have been `frozen' such that the system can effectively be described as one-dimensional. Then, it turns out, one can tune the effective interaction strength $ g_{\mathrm{1D}}$ at will by merely making the transverse confinement narrower. This allows us to explore the limits of strong correlations, this way testing the physics beyond the mean-field regime.

  • Surprisingly, the limit of (repulsive) ultrastrong interactions $ g_{\mathrm{1D}}\to\infty$ is known exactly in 1D: Such hard-core bosons have a 1-1 correspondence to non-interacting fermions. In particular, their ground state is simly given by the absolute value of the free fermionic wave function

    $\displaystyle \Psi_{\mathrm{Bose}}=\left\vert\Psi_{\mathrm{Fermi}}^{(0)}\right\vert,$

    i.e., the `Slater determinant'. This makes it tempting to think of the hard-core repulsion as mimicking the Pauli exclusion principle - hence this limit is termed `fermionization' (even though the bosons of course stay bosonic!).
  • The opposite case of weak interaction strengths is also known, of course: It is the regime of the Gross-Pitaevskii equation mentioned above, when Bose condensation into a single-particle orbital $ \phi_{0}$ occurs, $ \Psi=\phi_{0}^{\otimes N}$.
Relatively little in turn is known about the evolution between these two very different borderline cases. In fact, there is an analytic solution for the homogeneous case (without traps), but for realistic atom traps, one generally has to resort to numerical simulations.

Appetizer: Ground state of 1-D bosons in harmonic traps

The transition from a weakly to a strongly correlated ground state is nicely illustrated on the example of few interacting bosons in a harmonic potential $ U(x)=x^{2}/2$ (in dimensionless units). More concretely, we can model our system by the Hamiltonian

$\displaystyle H=\sum_{i=1}^{N}\left[\frac{1}{2}p_{i}^{2}+U(x_{i})\right]+\sum_{i<j}g_{\ mathrm{1D}}\delta(x_{i}-x_{j}).$

Figure 1: Density profile $ \rho(x)$ for a harmonic trap ($ N=5$ atoms) for different interactions $ g_{\mathrm{1D}}$. Note how the profile changes from a weakly interacting one ($ g=0.2$) to a flattened one due to fragmentation, and finally to a fermionized profile featuring $ N$ humps ($ g\ge15$).
\includegraphics[% width=8cm, keepaspectratio]{p5d1_DW.h0.eps}

If you look at the density profile $ \rho(x)$ - the probability density for having one atom at position $ x$ - in Fig. 1, you can see that for $ g\equiv g_{\mathrm{1D}}\simeq0$ the particles are all in the harmonic orbital $ \phi_{0}(x)\propto e^{-x^{2}/2}$. As we switch on interactions between the atoms, these try to repel each other and thus move further apart. This smears out the density (see $ g=4.7$). If we drive that to extermes, you can see that something happens: at $ g=15$, e.g., five peaks emerge (just as many as there are bosons). This is quite intuitive: When the particles repel each other very strongly, they tend to isolate, and some localization takes place.

Figure 2: Two-body correlation function $ \rho_{2}(x_{1},x_{2})$ for $ N=5$ bosons in a harmonic trap; shown are the interaction strengths $ g=0.4, 4.7, 10$ from left to right.
\includegraphics[% width=5cm, keepaspectratio]{p5d1_DW.1.h0.eps}\includegraphics[% width=5cm, keepaspectratio]{p5d1_DW.2.h0.eps}\includegraphics[% width=5cm, keepaspectratio]{p5d1_DW.201.h0.eps}

You can also look at this from the two-body perspective. Figure 2 shows the two-body correlation function $ \rho_{2}(x_{1},x_{2})$, giving the distribution for finding one atom at $ x_{1}$ and any second one at $ x_{2}$. In the left picture ($ g=0.4$), the atoms are almost uncorrelated and $ \rho_{2}=\rho\otimes\rho$ factorizes. As the two-body forces get stronger, it will cost the particles more and more energy to be at the same spot, $ x_{1}=x_{2}$, and they will avoid that. This is seen in the black diagonal in the central plot - the correlation hole. When fermionization sets in, $ g=10$, there is a striking checkerboard pattern. This signifies that, if we find the `first' atom at, say, $ x_{1}\simeq2$, then finding the second one at the same spot will be excluded. Rather, the other four atoms will be found at four discrete spots, marked by four density maxima.

Figure: Momentum density $ \tilde{\rho}(k)$ for five bosons in a harmonic trap; shown are the interaction strengths $ g=0.4, 4.7, 15$.
\includegraphics[% width=8cm, keepaspectratio]{p5d1_DW.h0_k.eps}

An interesting view on the transition is afforded by the momentum distribution for one atom, $ \tilde{\rho}(k)$ (Fig. 3). Of course, the momentum and the spatial distribution for $ g\to0$ are both simply Gaussian. (Without a trap, this would mean that the bosons condense into the $ k=0$ ground state). When interactions are added, the atoms are driven out of the condensate and the momenta are redistributed to higher $ k$. One can show that, in the fermionization limit $ g\to\infty$, the asymptotic behavior is $ \tilde{\rho}(k)=O(k^{-4})$ as $ k\to\infty$.

Perspectives:

This is only an appetizer. There are many intriguing ways to extend the above studies:

  • Other traps: What happens if we run up a potential-energy barrier in the middle so as to turn the harmonic trap into a double well? This is interesting in its own right; for it is a paradigm system for fundamental effetcs such as interference or tunneling, see [1,2]. Along this line, one can also study multiple wells, such as lattices.
  • Exotic interactions: We have also started the route to inhomogeneous interactions, where the coupling $ g$ depends on the position of the collision. This way single atoms might be extracted from an ensemble. [1]
  • Excited states & Dynamics: Many interesting problems are inherently time dependent, such as tunneling or the expansion of an ensemble followed by interference.

What methods do we use?

We basically rely on two different approaches.

  • Exact diagonalization: The idea here is simply to expand the (exact) many-body wave function $ \Psi$ in terms of suitable (i.e., easily treatable) basis function - for instance occupation-number states $ \vert\boldsymbol{n}\rangle\equiv\vert n_{0},n_{1},\dots\rangle$, and diagonalize the corresponding Hamiltonian matrix $ (\langle\boldsymbol{n}'\vert H\vert\boldsymbol{n}\rangle)$.
    This effort is still in progress, and programming-type contributions are welcome.
  • Wave-packet dynamics: We exploit a method developed in the Theoretical Chemistry group in Heidelberg. The key is to solve the time-dependent Schrödinger equation $ i\dot{\Psi}=H\Psi$ via expansion in terms of direct (or Hartree) products $ \Phi_{J}\equiv\varphi_{j_{1}}^{(1)}\otimes\cdots\otimes\varphi_{j_{N}}^{(N)}$< /SPAN>:

    $\displaystyle \Psi(Q,t)=\sum_{J}A_{J}(t)\Phi_{J}(Q,t),$ (1)

    hence the full name Multi-Configuration Time-Dependent Hartree method (MCTDH). Note that both the coefficients $ A_{J}$ and the single-particle functions $ \varphi_{j}$ have a time dependence. This is determined by some variational principle; in this sense, the basis function $ \Phi_{J}$ are optimal at each time step, which makes MCTDH rather efficient.


Sounds interesting?
If this has sparked your interest in doing research on cold atoms, please don't hesitate to contact Prof. Peter Schmelcher

Please also feel free to explore some of the material given in the Bibliography below.
[Download PDF version of this website]

Bibliography

1
S. Zöllner, H.-D. Meyer, and P. Schmelcher, Phys. Rev. A 74, 053612 (2006).

2
S. Zöllner, H.-D. Meyer, and P. Schmelcher, Phys. Rev. A 74, 063611 (2006).


Last update: Sascha Zoellner 2007-01-23
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