Cold Caesium atoms in a magneto optical trap are probed with off resonant laser light in one arm of a Mach-Zehnder interferometer. The interferometer measures the phase of the light, which records information of the atomic level population in a non-destructive way. Such a quantum non-demolition (QND) measurement reduces the uncertainty of the measured quantum variable, whereby an intrinsically non-classical (squeezed) atomic state is created. Implementing a quantum feedback protocol would allow a specific quantum state to be engineered. Non-classical states like this find several applications in the expanding field of quantum information theory and can improve the precision of atomic clocks. The construction and characterization of the interferometer has been the subject of my master thesis, and as a natural continuation, the Ph.D project would be to further develop and improve the experiment. The experiment has been a key ingredient in the European Union's CAUAC project and will be an important part of the COVAQUIAL project from Sep. 2004.

Quantum theory predicts several features that are distinctively non-classical, i.e. they cannot be described in any classical theory. One such quantum property is the uncertainty relation, which sets a lower limit to the uncertainty with which two properties, that are so-called conjugate, can be predicted. More precisely, the relation rules that the product of the uncertainties of two properties of a quantum system must be greater than a certain value. Another facet at the core of quantum theory is the notion of a measurement of a property of a quantum system. It has been the subject of intense discussion up till this day, but the Copenhagen interpretation proposed by Niels Bohr and physicists around him at the NBI, remains as one of the most widely accepted explanations for this feature.

The project that is the subject of this application combines the above stated facets of quantum theory. It involves using an optical interferometer to do a quantum non-demolition (QND) measurement of a particular property of a quantum system which in this case is a cloud of cold Caesium (Cs) atoms. Such a measurement will, if it is performed with sufficient sensitivity, yield a greater knowledge of the particular property of the system that has been measured. The point of the measurement being non-destructive is, that the property itself is unaffected by the measurement. For the system as a whole to adhere to the uncertainty principle, the increased knowledge, and thereby reduced uncertainty, of one property, must be accompanied by an increased uncertainty of another property.

The state that is created with an unequal distribution of uncertainties is distinctively quantum mechanical, in the sense that it does not have a classical counter-part. In the case, where the properties of interest are atomic spins or pseudo-spins, the states are called squeezed spin states (SSS). In applications in Cs atomic clocks and atom interferometry the reduced uncertainty of one the spin components in a SSS can be exploited for increased sensitivity. Moreover, the QND measurement gives a value for the measured property, which can be fed back to the quantum system and used to prepare it in a specific state. Naturally, states of a quantum system can be prepared in other ways e.g. by optical pumping, but the novelty of the state preparation with the QND measurement is, that the states that can be prepared are highly non-classical. Such, non-classical states are indispensable in quantum information theory e.g. as quantum memory devices, in teleportation protocols or for quantum cryptography.
The particular implementation with cold Cs atoms circumvents the Doppler broadening which allows all atoms in the ensemble to be addressed on an equal footing and also provides high atomic densities, resulting in larger interaction strengths between the atoms and the light in the interferometer.

During the past two years I have been responsible for the construction of the interferometric detection system, and in my Master work I have characterized two versions of the interferometer, namely one consisting of optical fibers and one in a free space configuration. The aim has been to stabilize the interferometer to the extent needed for doing a QND measurement, which was achieved in both interferometer configurations, as has been documented in a publication. The PhD project would thus come as a natural continuation of my previous work on the experiment.

The experiment outlined above has been a key ingredient in the research group's contribution to the European Union project CAUAC and will from September 2004 be an important part of the European Union COVAQUIAL project.

The properties of a quantum mechanical system are described by operators, and a measurement of a property corresponds to finding the expectation value of the related operator. The expectation value of the operator, in turn, depends on the state of the system, which is described by a state vector in Hilbert space. The degree to which one can predict the outcome of a measurement depends on the uncertainty of the state, i.e. how well defined is the state with respect to the property described by the operator. Unlike classical physics, quantum mechanics sets a limit to the precision with which we can know the state of a system. The limit is given by the Heisenberg uncertainty relation, which rules that the product of the uncertainty on two conjugate operators must be larger than a certain value, determined by their commutator. Herein also lies the key to, how the uncertainty limit on one operator can be overcome. The trick is that the uncertainty on one operator can be reduced as long as the uncertainty of the conjugate operator is increased sufficiently, so that the product of the uncertainties remains unchanged. The question arises, how this re-arrangement of uncertainties can be achieved in practice?

When a measurement is performed something peculiar happens to the state vector. According to the Copenhagen interpretation a measurement of an operator with a certain outcome will collapse the state vector to a new state that agrees with the measurement outcome. If the precision of the measurement is infinite or there is only a discrete set of measurement outcomes, then the updated state vector will also be exact. It is exact in the sense that another measurement of the same operator will give the same result as the first measurement.

There is a snag in this, because in many cases the interaction between the measured system and the measuring device will change the measured operator. In some sense the measurement perturbs or destroys the initial state, and therefore the subsequent measurement will not be correlated with the first measurement. An extra condition is needed, which demands that the measurement should not affect the measured property. If this condition is fulfilled the measurement is called a quantum non-demolition (QND) measurement.

In this experiment we will deal with variables that have a continuous spectrum of measurement outcomes, and the measurements, we will make, are of a certain finite precision. Therefore, the measurement does not leave the state fully specified, but just better specified. However, the reduction in the uncertainty of the measured value can be significant, so that a subsequent measurement will be strongly correlated with the first measurement. The state that is created is named a squeezed state, and it has already been implemented in a couple of physical systems ranging from laser beams to gas cells.

In this project the quantum mechanical system that we wish to manipulate, is a cloud of cold Caesium (Cs) atoms. These are cooled in a magneto optical trap and in the future will be loaded in to a dipole force trap. The atomic operator that we wish to measure is the difference in the populations of the two hyper-fine ground levels of atomic Cs. The well known correspondence between a two level system and a spin ½ system allows the operator to be expressed as the z component of a pseudo-spin, i.e. an operator which has commutators like those of an angular momentum. The x and y components of the pseudo-spin also have meanings, in terms of the coherences between the two ground levels.

The QND measurement is done with laser light passing through a Mach-Zehnder interferometer, one arm of which interacts with the atoms, while another arm serves as a reference arm. To put the light and the atoms on an equal footing, one can describe the light in the interferometer with spin operators.

The interferometer measures the phase-shift caused by different optical path-lengths in the two arms of the interferometer, which depends on the physical path-length as well as the index of refraction along the beam path. Since only the light in one arm passes the atoms the interferometer can measure any change in the index of refraction caused by the atomic sample. If now the frequency of the light is tuned, so that it corresponds to the transition from the Cs ground levels to an excited level, one observes a large dependence of the refractive index on the frequency. As a matter of fact light, that is tuned to a frequency which lies in-between the transition frequencies from each of the ground level to an excited level, will receive a negative phase-shift from atoms that populate the lower ground level and a positive phase-shift from atoms which populate the upper ground level. In this way the light can measure the difference in the ground state populations expressed by the z-component of the pseudo-spin.
Naturally, the atoms also absorb light, which is a destructive interaction. However, absorption can be avoided by going off resonance. The reason why this improves the quality of the measurement is, that the phase-shift of light caused by the atoms scales as the inverse detuning from the atomic transition, whereas the absorption scales as the inverse detuning squared. In other words, the absorption drops off faster with the detuning than the phase-shift does.

With off resonant light the interaction of the light with the atoms can be reduced to a very simple form, where the Hamiltonian of the interaction contains only the z components of the atomic and light pseudo-spins. This form is well known to allow a QND measurement of the spin components in the Hamiltonian and hence induce squeezing of the pseudo-spins.

It is important to realize, that the interaction itself merely serves to entangle (correlate) the atomic level population difference with the phase of the light field. Unless we perform the measurement of the light phase, we will not learn about the population difference, and thus the atomic state will not become squeezed. When we do measure the phase of the light with the interferometer we create a squeezed atomic state, and a subsequent measurement will with a higher probability give a result similar to the first one. But the outcome of the first measurement is a random value within the uncertainty of the initial state. Thus, the QND measurement does not create a predetermined state. However, for applications of the state in atomic clocks and quantum information, one wants a particular predefined state. Fortunately, the measured value exactly tells one how the state should be corrected to obtain the desired state. In this experiment the quantum feedback, that ensures that a specific squeezed state is created, is done by applying a microwave field, which transfers population in-between the hyperfine ground states, as it is also done in atomic clock operation.

To sum up the discussion, the QND measurement of the atomic pseudo-spin together a feedback protocol provides a technique to engineer a squeezed atomic state. The squeezed state is a intrinsically quantum mechanical state in the sense that its cannot be explained in a classical theory. The reason is that the reduction in uncertainty involves correlations in the form of entanglement between the atoms, which are a consequence of the non-locality of quantum theory. Such a state would find numerous applications in the novel field of quantum information theory e.g. as a quantum memory device and in entanglement and teleportation experiments. Because the experiment uses Cs atoms, the states also find applications in atomic clocks.

Experimentally, the status of the project is as follows. The interferometer is currently in a free space configuration, but previously a fiber optic interferometer was also tested. I have been responsible for constructing and testing the two interferometers, and based on the results, the final choice has been the free space interferometer. The tests of this interferometer have shown that it is at the fundamental quantum limit of sensitivity given by the so-called shot noise of light. This is important for the QND measurement, since the sensitivity of the detection determines how well we determine the atomic spin state and thus the degree of squeezing obtained. In the process of reaching the shot noise sensitivity we have developed a number of interesting techniques to suppress residual noise contributions.

The cold Cs atoms are obtained in a magneto optical trap (MOT). The first step towards creating a spin squeezed state is to observe the correct quantum fluctuations of the measured population number difference coming from the uncertainty relations. This noise is often referred to as projection noise as it comes from the uncertainty of which the final state the initial atomic state collapses into or alternatively is projected onto.

The future work on the experiment, which would be the subject of this PhD project, is related both to the atomic sample and the interferometric detection. What regards the interferometer the next step would be to implement a two-color detection. The two colors refer to the frequency of the laser light, and while the current setup involves only one probe laser beam, the improved setup would involve two laser beams each detuned from a ground level to hyperfine level transition. We have calculated that this scheme would increase the sensitivity of the measurement. It is relatively simple to implement in practice, and on top of the improved sensitivity our calculations suggest, that if we also alter the alignment of the interferometer it will additionally suppress residual noise of the detection.

As for the atomic sample, the intention is to load it into a far-off-resonant dipole force trap (FORT), which uses the conservative potential caused by the Stark effect from a tightly focused laser beam. The high power laser and the optics for the FORT are in place, and at the moment we are attempting to trap the Cs atoms in it.

Further tasks are to implement and optical pumping scheme, which would serve to create an appropriate (polarized) initial state of the atoms, in order for the subsequently engineered squeezed state to be usable in the suggested applications. To verify the quality of the optical pumping we need to characterize the level populations, which will be done by magnetic resonance spectroscopy. The basic parts of the setup for this characterization are already at hand. Beyond this lies a multitude of interesting possible improvements that would involve the use of novel techniques such as photonic integrated circuits.