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arXiv:1205.4524 · Physical Review A

Homodyne detection of matter-wave fields

Stefan Rist, Giovanna Morigi

Published May 24, 2012• 6 citations
arXivHTMLPDF

Abstract

A scheme is discussed that allows for performing homodyne detection of the matter-wave field of ultracold bosonic atoms. It is based on a pump-probe laser setup, which both illuminates a Bose-Einstein condensate, acting as reference system, and a second ultracold gas, composed by the same atoms but in a quantum phase to determine. Photon scattering outcouples atoms from both systems, which then propagate freely. Under appropriate conditions, when the same photon can either be scattered by the Bose-Einstein condensate or by the other quantum gas, both flux of outcoupled atoms and scattered photons exhibit oscillations, the amplitude of which is proportional to the condensate fraction of the quantum gas. The setup can be extended to measure the first-order correlation function of a quantum gas. The dynamics discussed here makes use of the entanglement between atoms and photons, which is established by the scattering process in order to access detailed information on the quantum state of matter.

DOI
10.1103/physreva.85.053635

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A scheme is discussed that allows for performing homodyne detection of the matter-wave field of ultracold bosonic atoms. It is based on a pump-probe laser setup, which both illuminates a Bose-Einstein condensate, acting as reference system, and a second ultracold gas, composed by the same atoms but in a quantum phase to determine. Photon scattering outcouples atoms from both systems, which then propagate freely. Under appropriate conditions, when the same photon can either be scattered by the Bose-Einstein condensate or by the other quantum gas, both flux of outcoupled atoms and scattered photons exhibit oscillations, the amplitude of which is proportional to the condensate fraction of the quantum gas. The setup can be extended to measure the first-order correlation function of a quantum gas. The dynamics discussed here makes use of the entanglement between atoms and photons, which is established by the scattering process in order to access detailed information on the quantum state of matter.
A scheme is proposed, that allows one for performing homodyne detection of the matter-wave field of ultracold bosonic atoms. It is based on a pump-probe lasers setup, that both illuminates a Bose-Einstein condensate, acting as reference system, and a second ultracold gas, composed by the same atoms but in a quantum phase to determine. Photon scattering outcouples atoms from both systems, which then propagate freely. Under appropriate conditions, when the same photon can either be scattered by the Bose-Einstein condensate or by the other quantum gas, both flux of outcoupled atoms and scattered photons exhibit oscillations, whose amplitude is proportional to the condensate fraction of the quantum gas. This setup allows one, for instance, to perform thermometry of a condensate or to monitor the Mott-insulator/superfluid phase transition in optical lattices, and can be extended in order to measure the first-order correlation function of a quantum gas. The dynamics here discussed make use of the entanglement between atoms and photons, which is established by the scattering process, in order to access detailed information on the quantum state of matter.
The field of atom optics has progressed rapidly over the past 20 years since the realisation of Bose-Einstein condensation, such that the wave behaviour of atomic gases is now routinely demonstrated. Furthermore, the study of quantum atom optics, which goes beyond a ‘mean-field’ description of quantum systems to consider the behaviour of single particles, has demonstrated both the similarities between photons and massive species, and their differences as a result of the internal structure and external interactions of atoms. An important class of observable quantities which allow such effects to be measured are nth order correlation functions, which can be interpreted as a result of either particle or wave behaviour. These functions provide a statistical description of fluctuations in n-tuples of particles in a source, which rigorously defines concepts such as coherence. The quantum statistics of a Bose-Einstein condensate should be the same as that for an optical laser, while an ideal thermal Bose gas matches the behaviour of incoherent light. However, correlation measurements can also be used to quantify the influence of interactions, dimensionality, confining potentials and waveguides, and the difference in quantum statistics between fermions and bosons, which illustrates the rich range of behaviour exhibited by atomic gases. In this thesis, several aspects of quantum atom optics are explored with experiments using ultracold metastable helium, a species with the unique advantage of facilitating simple single-atom detection with high resolution, while still allowing Bose-Einstein condensates to be formed. The coherence of atomic systems is shown to be maintained when outcoupled as pulsed atom lasers, and the long-range order characteristic of Bose-Einstein condensates is demonstrated to third order for the first time. Conversely, thermal bunching is observed for a variety of atomic systems, including the measurement of correlation functions up to sixth order with near-ideal interference contrast. These results clearly demonstrate the correspondence between the quantum statistics of photons and atoms as was formalised by Glauber, as well as confirming the validity of applying Wick’s theorem to simplify the statistics of atomic gases. Correlation functions are also shown to be an ideal tool to probe the quantum state of an ultracold gas, and were used to observe the phenomenon of transverse condensation in an elongated Bose gas, as well as characterise the mode occupancy of matter waves guided by an optical potential. Ultracold metastable helium is also suitable for exploring other fundamental topics in quantum optics such particle/wave duality. The notion of complementarity stimulated long running philosophical discussions about how apparently mutually exclusive behaviours can coexist, which culminated in Wheeler devising his famous ‘delayed choice’ gedankenexperiment. A proposed experimental method to realise Wheeler’s experiment with ultracold atoms is discussed, and preliminary measurements presented which indicate that the completion of this experiment could be achieved in the near future. Not only is this of interest in its own right, but the implementation of this experiment has also developed techniques which may enable further studies in quantum atom optics such as investigations of the Hong-Ou-Mandel effect and quantum entanglement with massive particles.
In the present thesis we study several schemes where the detection of the emitted light from an atomic ensemble gives information about the state of the atoms. In chapter 1 we review the basic theory for the light matter interaction in second quantized form. In chapter 2 we show how conservative potentials for the atoms, such as dispersive optical lattices, can be realized experimentally due to the dispersive interaction with a far detuned laser We then review some basic properties of the physics of a particle in a periodic potential. We show that the ground state and the lowest lying excitations of a Bose-Einstein condensate in a dispersive optical lattice can be described by the Bose-Hubbard Hamiltonian. In chapter 3 we study the photonic bandstructure of a chain of fixed pointlike atoms in a biperiodic configuration. We calculate the photonic spectra and the probe transmission if the atoms are in free space and inside a standing wave optical resonator. In chapter 4 we consider light scattering from ultracold atoms in an optical lattice where we calculate the photonic scattering cross section as a function of energy and direction of emission along the Mott-insulator-superfluid phase transition. We take into account the finite tunneling rate for the atoms, when evaluating the scattering cross section of the photons and the effect of photon recoil onto the atoms. In chapter 5 we consider a gas of ultracold atoms trapped at two spatially separated regions in space. Using a two Raman lasers to couple atoms out of the systems we show that it is possible to measure the mean value of the atomic field operator of the two systems by measuring the scattered light intensity of one of the Raman lasers. We show how this setup might be used to measure the temperature of a Bose-Einstein condensate and to monitor the superfluid to Mott-insulator phase transition for a gas of ultracold atoms in an optical lattice.
Using single atom detection offered by metastable helium atoms (He*), we are able to measure correlations between atoms. With this we have demonstrated second-order coherence of matter waves that emerge from amplified four-wave mixing (FWM). Our results provide the first second-order, or quantum test of coherence for amplified matter waves. Using the same detection scheme, we have investigated the onset of coherence during the formation of a Bose-Einstein condensate (BEC), attempting to reveal how the long-range order of a BEC is established. Our initial results support the existence of quasi-condensates.
One of the seminal advances in quantum optics was the understanding that a quantised description of ensembles of photons is best characterised by correlation functions. Correlations are also a fundamental property of matter waves, and the single wavefunction that describes a Bose-Einstein condensate (BEC) is in principle characterised by long range coherence to all orders (i.e. a universal correlation value of unity). Here we measure the higher order correlation properties of ultra-cold matter waves and use them to probe the coherence of the gas.
One of the seminal advances in quantum optics was the understanding that a quantised description of ensembles of photons is best characterised by correlation functions. Correlations are also a fundamental property of matter waves, and the single wavefunction that describes a Bose-Einstein condensate (BEC) is in principle characterised by long range coherence to all orders (i.e. a universal correlation value of unity). Here we measure the higher order correlation properties of ultra-cold matter waves and use them to probe the coherence of the gas.