Anthony E. Miller, Gopal Sarma, Orion Crisafulli, Andrew Silberfarb, and Hideo Mabuchi
Former project members: Ramon van Handel, John K. Stockton, JM Geremia, Andrew C. Doherty

Introduction

We are working to experimentally demonstrate magnetometer with sub Standard Quantum Limited (SQL) sensitivity and spin-squeezing of an enseble of cold Cesium atoms. The experiment is shown schematically in Figure 1. More than 100 million Cesium atoms are initially collected in a Magneto-Optical Trap. After the atoms are collected, the trap is switched off, and the atomic spins are aligned by optical pumping. After pumping, the atoms begin to fall under the influence of gravity and thermally diffuse. Before the atoms diffuse too much, a linearly polarized laser beam (with frequency detuned from the atomic resonance) traverses the atomic ensemble and the optical plane of polarization rotates in a way that depends on the collective atomic spin state (see schematic below). This effect is Faraday rotation, and we can observe dynamics such as Larmor precession of the spins (due to a magnetic field) via a polarimeter measurement of the probe polarization.

The following paragraphs contain a brief introduction to a number of topics including weak continuous measurement, entangled state preparation via measurement and feedback, quantum parameter extimation and magnetometry, and entanglement in symmetric ensembles. A more complete discussion of these topics can be found on the following pages:

• Magnetometry with Atomic Ensenbles
• Collective Entanglement of Ensembles

Figure 1: A: A cartoon representation of collective atomic spin states. The initial coherent spin state (CSS) is transformed into the spin squeezed state (SSS) via the measurement process. The uncertainty in the z direction is squeezed at the expense of the uncertainty in the y direction. B: The schematic of the experiment described in the text.

Weak Continuous Measurement

Any physical measurement is performed by interacting a probe system (the laser beam) with system to be measured (the atom cloud). The interaction leads to a joint evolution of the two systems and after the interaction both systems carry information about the other. In our experiment, and in fact whenever useful information is carried byt the probe, the interaction engangles the two systems. Because of this correlation, a measurement of the state of the probe system teaches us something about the state of atomic system. However, the coupling between the atoms and the light is weak in comparison to the quantum noise sources in the experiment (optical shot noise) and so only a small amount of information about the atoms is gained in any finite time interval. This means that instead of canonical projective measurements, our measurement, (and all real measurements) is continuous and the state of the atoms gradually diffuses towards an eigenstate of the measurement operator. (Note: We assume that the optical detector is "classical." One can play the game of endlessly coupling to larger systems and retaining keeping everything quantum. However, the actual measurement record from our detectors is classical; we will leave the question there.)

There are several approaches to understanding how exactly the state of the atoms evolves under this continuous measurement. By measuring the polarization state of probe laser, we learn something about the atoms. However, optical shot noise makes it impossible to measure the polarization state of the light perfectly, which is why we acquire information about the atoms slowly. If the optical density of the atoms is large enough and we have suppressed technical noise sources, we can resolve the noise contribution of the atoms to the shotnoise. We use this random signal to change our description of the atoms (the state). This technique, known as conditioning, is a method to generate the best possible estimate of the atomic state using the measurement record based upon a detailed model of the system. The formulation of the formulation of the filter for our system can be found in reference 5.

When our experiment is perfectly configured, it is in principle impossible to know which of the atoms interacted with (rotated the polarization of) the laser light. Consequently, when we condition upon the measurement record, we condition the collective state of the system and not each individual state of each atom. This means that individual members of the atomic ensemble can be correlated as a result of the measurement. The quantum measurement is intrinsically random. Because the measurement is weak and continuous, however, we can apply feedback control (mapping the continuous measurement results to a control parameter like a magnetic field) to deterministically and robustly steer the system to a target state. We take advantage of the weak measurements and guide the state diffusion to a non-random goal. The collective states that we prepare in this way can be useful in a variety of metrology applications such as quantum noise limited magnetometry, atomic time standards, and spin gyroscopes.

Below we attempt to put our various research articles in context and give the reader a short guide through our contributions to this field. Of course, many other groups are highly active in these subject areas and we refer the reader to the references below for more on their work. We somewhat arbitrarily divide our work into three inter-related topics: state preparation, metrology, and entanglement characterization.

Entangled State Preparation via Measurement and Feedback

The experiment described above is one of the more simple experimental demonstrations of the Heisenberg uncertainty principle. The initial collective state is a coherent spin state with all spins aligned along one direction. Yet the component of the spins in the other two perpendicular directions are not perfectly well known and there exists an uncertainty relation involving those variables. As one perpendicular direction is measured and becomes more certain, the trade-off is that the other direction must become less certain (to maintain the uncertainty relation). This results in what is called a spin-squeezed state (see Figure 1A).

Without feedback, the direction of the spin-squeezed state is random and different on every trial. By adding a feedback magnetic field to correct this randomness, the spin-squeezed state production can be made both deterministic and robust. The detailed theory of the atom-light interaction process (including the non-trivial effect of the atom being multi-level) is discussed in reference 1.

A more rigorous mathematical approach to this process is taken in references 2-5, albeit with some physical simplifications (e.g., ignoring spontaneous emission). The conditioning equation (stochastic master equation) which updates the quantum state given the random measurement results is derived in references 2 and 5, paying particular attention to the subtleties of stochastic calculus. When this equation is taken to its long time limit the squeezing effect eventually prepares a collective eigenstate of the measured spin component. Reference 3 studies this equation for a small number of atoms and shows analtyically that the eigenstates can be deterministically prepared with intelligently chosen control laws. Reference 4 gives numerical evidence that this is also true for a large number of atoms with intuitively chosen control laws. Reference 5 demonstrates a more rigorously mathematical derivation of the scattering interaction and the filter derived from it.

Quantum Parameter Estimation and Magnetometry

In quantum parameter estimation problems, everything is the same as above yet we add uncertainty about a certain system parameter. We assume that we know the type of the parameter (and what it does to the system) but not its value. For example, in our system imagine an unknown magnetic field perpendicular to the spins and the probe direction. We can measure the Larmor spin rotation due to the field (via the Faraday rotation of the optical beam) and then infer what the field is by observing how much the spins rotated in a given time. Normally in physics we try to give the probability of a measurement outcome given a known system, but here we are discussing the reverse: given some measurement results, how likely is it that the system parameter had a certain value?

In the system described above, this process is tricky because we have to evolve the distribution for the field and the distribution for the atomic spins (the state) simultaneously. In fact both distributions are squeezed in an interdependent way as we gain information: just as the field uncertainty degrades the spin-squeezing, the atomic uncertainty degrades the field measurement. The theory of this process is detailed in references 6 and 7 where a Kalman filter approach is given. In the latter, it is shown how this process is made robust (to other system uncertainties, like atom number) with the use of feedback control.

In general, this research has far reaching implications in quantum metrology as it addresses fundamental issues about how sensors work at fundamental quantum limits. Similar types of procedures can be applied to other metrology contexts including, but not limited to, atomic inertial/gravity sensors and atomic clocks.

Further information on magnetometry is given here.

Entanglement in Symmetric Spin Ensembles

Finally, we are interested in how concepts of quantum entanglement can be used to characterize our system. In the system described above, there are two kinds of entanglement: that between the atoms and the light (after interaction) and that between the atoms within the cloud (after conditioning). In the latter case, the many-particle entanglement is due to the indistinguishability of the measurement and is synonymous with the spin-squeezing. In the ideal case where the atoms are completely indistinguishable, this leads to a completely symmetric entangled state of the atomic cloud (invariant to particle index exchange).

For general multi-particle entanglement, it is very computationally intensive to calculate entanglement measures because the size of the state grows exponentially with particle number (e.g., the state vector for N spin-1/2 particles is of size 2^N). However if one restricts the atomic states to a certain subspace then the computations can become much more efficient. If we restrict ourselves to the symmetric subspace (for the physical reasons mentioned) then the size of the space is linear in particle number. The trick is then to restrict the entanglement operations to the same space so that the computer never needs to work in the unreasonably large space. In reference 8, we show how to restrict such operations in the case of the symmetric space. As a result we can compute various entanglement measures for hundreds of spins which would have been impossible without the symmetrization. Because our computations can explore this asymptotically large regimes, we were able to observe and subsequently prove certain statements about `maximally entangled' symmetric states.

Further information on symmetric state entanglement is given here.

References

1.	JM Geremia, John K. Stockton and Hideo Mabuchi, "Tensor polarizability and dispersive quantum measurement of multilevel atoms," Phys. Rev. A 73, 042112, (2006).	PDF quant-ph/0501033 online BibTeX
2.	Ramon van Handel, John K. Stockton and Hideo Mabuchi, "Modeling and feedback control design for quantum state preparation," J. Opt. B (Special Issue: Quantum Control), accepted (2005).	PDF
3.	Ramon van Handel, John K. Stockton and Hideo Mabuchi, "Feedback control of quantum state reduction," IEEE Trans. Automat. Control 50, 768-780 (2005).	PDF quant-ph/0402136 online
4.	John K. Stockton, Ramon van Handel and Hideo Mabuchi, "Deterministic Dicke state preparation with continuous measurement and control," Phys. Rev. A 70, 022106 (2004).	quant-ph/0402137 online
5.	Luc Bouten, John Stockton, Gopal Sarma and Hideo Mabuchi, "Scattering of polarized laser light by an atomic gas in free space: A quantum stochastic differential equation approach," Phys. Rev. A 75, 052111, (2007).	PDF quant-ph/0701224 online BibTeX
6.	JM Geremia, John K. Stockton, Andrew C. Doherty, and Hideo Mabuchi, "Quantum Kalman filtering and the Heisenberg Limit in atomic magnetometry," Phys. Rev. Lett., 91, 250801 (2003).	PDF quant-ph/0306192 online additional info
7.	John K. Stockton, JM Geremia, Andrew C. Doherty and Hideo Mabuchi, "Robust quantum parameter estimation: Coherent magnetometry with feedback," Phys. Rev. A, 69, 032109 (2004).	PDF quant-ph/0309101 online
8.	John K. Stockton, JM Geremia, Andrew C. Doherty, and Hideo Mabuchi, "Characterizing the entanglement of symmetric multi-particle spin-1/2 systems," Phys. Rev. A, 67, 022112 (2003).	PDF quant-ph/0210117 online additional info

(last updated 23 March 2009 by Anthony E. Miller)