## Exact quantum simulations of many-body Fermi systems

*Supervisors: Dr Joel Corney *

The simulation of many-body Fermi systems is a topic of widespread interest in many fields of physics, including condensed matter physics, astrophysics, and ultra-cold atomic and molecular physics. A Nobel prize was even awarded recently in this field. However, current techniques generally employ various approximations to avoid a problem known as the Fermi sign problem. We have recently developed a novel technique that employs a Gaussian operator expansion to address this problem, and have demonstrated its successful use in the well-known Hubbard lattice model [1,2].

The thesis topic will be to extend this new technique to treat other related problems, including the BEC-BCS crossover in ultracold fermions, the theory of strongly-coupled superconductivity, and simplified models of QCD.

[1] J. F. Corney and P. D. Drummond, 'Gaussian quantum Monte Carlo methods for fermions', Phys. Rev. Lett. **93**, 260401 (2004).

[2] P. D. Drummond, P. Deuar, and K. V. Kheruntsyan, 'Canonical Bose gas simulations with stochastic gauges', Phys. Rev. Lett. **92**, 40405 (2004).

## Quantum dynamics of a Bose-Einstein condensate in an optical lattice

*Supervisor: Dr Joel Corney*

Recent pioneering experiments in ultracold atoms have opened up experimental regimes in which many-body quantum physics can be investigated with unprecedented simplicity and precision. Fermionic atoms confined to an optical lattice provide a direct realisation of the Hubbard model, which has long been of interested in condensed matter physics. While solutions of the Hubbard model are known in 1D, first-principle results in higher dimensions have relied on Quantum Monte Carlo methods, which suffer from intractable "Fermi sign" problems.

A system of interacting bosons on a lattice is described by the bosonic version of the Hubbard model, which shows a quantum phase transition between Mott insulator and superfluid regimes. Even for bosonic systems, which do not suffer from sign problems, computing the physics of quantum many-body systems is a notoriously difficult endeavour, particularly when there is a phase transition present. In many cases such calculations are only possible for very small numbers of particles.

This project will make use of recently developed phase-space methods based on a Gaussian representation to perform exact calculations of lattice systems. Technically, one of the main challenges will be the development of `stochastic gauges' to reduce sampling error and improve stability for strongly interacting systems.

[1] J. F. Corney and P. D. Drummond, 'Gaussian quantum Monte Carlo methods for fermions', Phys. Rev. Lett. **93**, 260401 (2004).

[2] J. F. Corney and P. D. Drummond, `Gaussian quantum operator representations of bosons', Phys. Rev. A **68**, 063822 (2003).

## Thermal fluctuations in Bose-Einstein condensates and the atom laser

*Supervisor: A/Prof Matthew Davis *

Calculating the physics of quantum many-body systems, even for dilute gases, is a notoriously difficult problem. The lack of exact analytic solutions to the applicable quantum field equation means that the problems must usually be tackled computationally. Even then, the scope of the simulations is usually limited to calculating time-independent properties of equilibrium systems. Dynamical calculations are usually either restricted to a few particles with limited degrees of freedom or else involve simplifying, semi-classical assumptions.

This project will continue the development of a relatively new method for calculating the dynamics of BECs at finite temperature. This is based on a Wigner function representation of an approximate `high-temperature' master equation for the condensate and other low-lying excitations, collectively known as the condensate band. It has the advantage that it reduces to effectively solving classical field equations that are within the capacity of the computing resources of today. The most important restriction on the method is that the condensate band modes must be highly occupied, a condition that is satisfied in many experiments.

One of the most exciting prospects for applications of BECs is in the development of the atom laser that could potentially be used for high-precision measurements. The atom laser essentially consists of outcoupling atoms from a BEC, however there are many difficulties associated with doing this in a continuous manner. This project will study the atom laser from two different perspectives. The first will be using quantum kinetic theory to investigate what conditions must be satisfied to continuously replenish the condensate mode. A continuous wave atom laser will necessarily be at finite temperature, and therefore the output will not be ideal (T=0). This project will be use newly developed finite temperature simulation techniques to determine the linewidth and correlations that would be present in the atom laser beam.

Other topics of interest in this project are thes issue of phase fluctuations, quasi-condensates, and vortices, especially in reduced dimensional systems. These issues are very topical at the present time, with several groups planning experiments on these phenomena in coming months.

## Dynamics of quantum phase transitions with ultra-cold atoms

*Supervisor: A/Prof Matthew Davis*

A quantum phase transition (QPT) occurs at zero temperature as a single parameter of a Hamiltonian passes through a critical value, and there is a qualitative change in the nature of the ground state of the system [1]. Systems of ultra-cold atoms are ideal for the experimental investigation of QPTs, due to the well-controlled potentials that are achievable, as well as the ability to precisely control the strength of atomic interactions with the use of Feshbach resonances. This level of control is unacheivable in tranditional condensed-matter systems. Also, the dynamics of passing through the critical point can be investigated experimentally with ultra-cold atom systems. Quantum correlations become significant near the critical point and traditional calculation techniques become unfeasible in this regime.

A well known-example of a QPT is the Bose-Hubbard model of interacting bosons on a lattice. As the ratio of on-site interaction energy to tunnelling rate between sites is increased, the ground state makes a transition from a superfluid state to a insulating state. This phase transition has been observed in ultra-cold Bose gas systems loaded into an optical lattice [2]. Another quantum phase transition occurs in attractive bosons confined to a 1D ring geometry. Above a critical interaction strength the ground state changes from being entirely delocalized to forming a localised soliton – a symmetry breaking transition [3,4].

This project will study the boundary in quantum many-body physics between many-body wave functions and quantum phase space representations. In particular it focus on the dynamics of quantum phase transitions, initially starting with attractive bosons in a 1D ring geometry. This has the advantage that the many-body eigenstates can be calculated numerically (although this is still an enormous task!), and so quantum dynamical simulation techniques can be compared directly with exact results. Another topic of interest with be the superfluid-insulator transition of the Bose-Hubband model.

[1] Quantum Phase Transitions, Subir Sachdev (Cambridge, 1999).

[2] Greiner *et al *., Nature **415**, 39 (2002).

[3] Kanamoto *et al *., Phys. Rev. A **67**, 013608 (2003).

[4] Kanamoto *et al *., Phys. Rev. Lett. **94**, 090404 (2005).

## Quantum atom optics using molecule dissociation

*Supervisor: Dr Karen Kheruntsyan *

Advances in the experimental control of ultracold quantum gases have now reached the stage where atomic correlations and quantum statistics can be directly accessed via the measurement of atomic shot-noise. Recent breakthrough experiments demonstrating this include quantum correlation measurements on atoms created through dissociation of weakly-bound molecular dimers near a Feshbach resonance [1] and on atoms in a Mott insulator phase released from an optical lattice [2].

These remarkable experimental developments offer intriguing parallels with quantum optics with photons, including applications similar to those achieved with the well-known entangled photon pairs from optical parametric down-conversion. The aim of this research project is to explore these parallels in the matter-wave or atom optics analog of parametric down-conversion, which can be realized through dissociation of a Bose-Einstein condensate of molecular dimers into pairs of constituent atoms [1,3].

The most direct analogy with optics corresponds to dissociation into bosonic atoms [3,4,5]. The recent experiments at JILA [1], on the other hand, have an intriguing twist in that the statistics of the constituent atoms is fermionic.

This research project in quantum atom optics aims at understanding the many-body physics underlying atomic quantum correlations in molecule dissociation. The research will include the case of bosonic statistics of the atoms, extending the previous results [4,5] into two and three spatial dimensions, as well as the fermionic case, which can be thought of as a new paradigm of quantum atom optics with correlated fermions. The project will analyze Einstein-Podolsky-Rosen correlations and Bell-like inequalities with correlated atomic fragments. The significance of this study is in offering novel tools for coherent control and manipulation of ultracold quantum gases, including applications in precision measurements, fundamental tests of quantum mechanics in new macroscopic regimes, and possible developments into the next generation quantum technology.

The project will involve analytic approaches similar to those employed in quantum optics, as well as numerical simulations using the positive P-representation method for bosons [6] and the recently developed Gaussian quantum Monte Carlo method for fermions [7].

[1] M. Greiner, C. A. Regal, J. T. Stewart, and D. S. Jin, Phys. Rev Lett. **94**, 110401 (2005).

[2] S. Fölling, et al., Nature (London) **434**, 481 (2005).

[3] S. Dürr, S. T. Volz, and G. Rempe, Phys. Rev. A **70**, 031601(R) (2004).

[4] K. V. Kheruntsyan and P. D. Drummond, Phys. Rev. A **66**, 031602(R) (2002).

[5] K. V. Kheruntsyan, M. K. Olsen, and P. D. Drummond, arXiv:cond-mat/0407363 (to appear in Phys. Rev. Lett.)

[6] P. D. Drummond and C. W. Gardiner, J. Phys. A: Math & Gen. **13**, 2353 (1980).

[7] J. F. Corney and P. D. Drummond, Phys. Rev. Lett. **93**, 260401 (2004).

## Quantum dynamics and thermodynamics of 1D Bose gases

*Supervisor: Dr Karen Kheruntsyan and A/Prof Matthew Davis*

An intriguing area of studies of Bose-Einstein condensates (BEC) at the moment is the theory of reduced spatial dimensions, namely degenerate Bose gases in geometries having one or two spatial dimensions. These are interesting for several reasons. For example, some aspects in the 1D case are exactly soluble, though usually only for a few of the characteristic observable properties. However, this does increase the power of theoretical tools available.

Also, it is known from the exact solutions that unusual and interesting features occur in the 1D case. For example, it is possible to have excitations with a fermionic (spin-half) character even when the underlying atomic gas is bosonic (integer-spin). This is an important fundamental topic, since the distinction between fermions and bosons is usually thought to be a fundamental one, that can't be changed by interactions.

The interesting question is how to observe this effect? We have recently made use of the known exact solutions to the uniform 1D Bose gas problem to calculate the exact local second-order correlation function at all densities and interaction strengths [1]. This is the most direct indication of `fermionic' behaviour, since the pair correlation is strongly reduced at low density and strong coupling – similar to the case of fermions, where it vanishes exactly, due to the Pauli exclusion principle. The first experimental evidence of reduced or `antibunched' correlations in a 1D Bose gas has recently been demonstrated in the Nobel Prize winning laboratory of B. Phillips at NIST [2], and in D. Weiss group at Pennsylvania State University [3].

This research project will further investigate the many-body physics of 1D trapped Bose gases. The research program will include the study of atomic second- and higher-order correlations as well as thermodynamic properties of trapped 1D Bose gas, and the question of how to dynamically excite the fermionic degrees of freedom. The theoretical modeling will closely follow the experimental procedures employed at Penn State University and will address an important question of adiabatic transfer of the Bose gas from a 3D to a 1D regime. One of the limitations of the current experiments is the absence of the precise knowledge of the temperature of the gas, as well as the knowledge of regime changes of the gas at constant entropy and varying densities. The expected outcomes of the project will provide detailed quantitative understanding of different regimes of the gas accessible in practice.

[1] K.V. Kheruntsyan, D.M. Gangardt, P.D. Drummond, and G.V. Shlyapnikov, Phys. Rev. Lett. **91**, 040403 (2003); Phys. Rev. A 71 , 053615 (2005).

[2] B. L. Tolra, K. M. O'Hara, J. H. Huckans, W. D. Phillips, S. L. Rolston, and J. V. Porto, Phys. Rev. Lett. **92**, 190401 (2004).

[3] T. Kinoshita, T. R. Wenger and D. S. Weiss, Science **305**, 1125 (2004); D. W. Weiss, Bulletin of the American Physical Society **50 ** (No. 3), 56 (2005); APS DAMOP 2005 Annual Meeting, Lincoln, Nebraska, USA (May 17-21, 2005).

[4] P.D. Drummond, P. Deuar, and K.V. Kheruntsyan, Phys. Rev. Lett. **92**, 40405 (2004).

## Atom-light entanglement

*Supervisor: Dr Murray Olsen *

Atom light entanglement is the study of how to obtain quantum-limited coupling between photonic (massless) and atomic (massive) degrees of freedom. This is interesting in that it has many possible applications, ranging from teleportation to gravity-wave detection.

The thesis topic will be to investigate a range of possible experimental schemes to determine their utility as atom-light entanglers. The emphasis will be on using the simplest possible schemes that can be analysed, with losses and nonlinearities included.

Two possible examples of this are the intra-cavity entanglement of the center-of-mass motion of a BEC with a cavity light-field, and the coupling of photonic propagation with EIT transitions in a three-level atomic vapor.

Experiments are underway in both these areas, both in Australia and elsewhere, leading to the possibility of future experimental tests of the predictions.

## Quantum cooling of mechanical oscillators

*Supervisors: Dr Murray Olsen*

Ultra-cold mechanical oscillators, especially mirrors, have been proposed for the measurement of extremely small displacements, as would be necessary for the detection of gravity waves. While dilution cooling techniques and optical feedback can lower temperatures dramatically, they do not generally cool quantum fluctuations, but leave the oscillator in thermal equilibrium at a lower temperature. The fluctuations remain huge, which is important if we wish to infer properties of the oscillator through its interactions with the electromagnetic field.

What we need is a cooling mechanism which will operate to "eat" quantum fluctuations. By changing the quantum state of the oscillator, rather than cooling the classical motion, the fluctuations could be reduced drastically without the need to cool the mean motion to such low temperatures. An electromagnetic example of this is the laser field, which effectively is at high temperature but can have fluctuations at near the standard quantum limit.