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

Introduction

We are working to experimentally demonstrate spin-squeezing of an enseble of cold Cesium atoms. Spin-squeezing will allow us to implement a magnetometer with a sensitivity below the standard quantum limit for our sensor. The experiment is shown schematically in Figure 1. We use Faraday rotation to probe an ensemble of more than a billion laser-cooled Cesium atoms confined to a sphere several millimeters in diameter. The atoms are initially collected in a Magneto-Optical Trap. After the atoms are collected, the trap is turned off, the atomic spins are aligned by optical pumping, and the atoms then slowly start to fall 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 simply 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 the system to be measured (the atom cloud). The interaction imprints information about each system on the other so that after the interaction, both systems have changed. In fact, the atomic and optical systems become entangled through the interaction. By measuring the optical state of the system we learn about the 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. Our measurement, like all real measurements and unlike measurements in quantum mechanics texts, is continuous and the state 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.)

By measuring the polarization of the optical state, we learn something about the atoms. This measurement, however, is intrinsically random due to optical shot noise. If the optical density of the atoms is large enough and our detectors are quiet enough, we can resolve the contribution of the atoms to this intrinsic noise. Through a process known as conditioning, we use this random signal to change our description of the atoms (the state). Conditioning is a technique to generate the best estimate of the system using the measurement record based upon a model of the system.

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 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 becomes 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. This principles of this process were demonstrated experimentally with cold atoms in reference 1. The detailed theory of the atom-light interaction process (including the non-trivial effect of the atom being multi-level) is discussed in reference 2.

A more rigorous mathematical approach to this process is taken in references 3-5, 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 reference 3, 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 4 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 5 gives numerical evidence that this is also true for a large number of atoms with intuitively chosen control laws.

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. Reference 8 demonstrates these principles experimentally.

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 9, 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, "Real-Time Quantum Feedback Control of Atomic Spin-Squeezing," Science, 304, 270 (2004).	online
2.	JM Geremia, John K. Stockton and Hideo Mabuchi, "Continuous quantum measurement and conditional spin-squeezing in Alkali atoms," Phys. Rev. A, accepted, (2005).	PDF quant-ph/0501033
3.	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
4.	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
5.	John K. Stockton, Ramon van Handel and Hideo Mabuchi, "Deterministic Dicke state preparation with continuous measurement and control," Phys. Rev. A 70, 022106 (2004).	PDF quant-ph/0402137 online
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.	JM Geremia, John K. Stockton and Hideo Mabuchi, "Suppression of Spin-Projection Noise in Broadband Atomic Magnetometry," Phys. Rev. Lett. 94, 203002 (2005).	PDF quant-ph/0401107 online
9.	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 27 July 2005 by John K. Stockton)