Performing Continuous Measurements on Coherent Spin States

This is a Perspective on "How to perform the coherent measurement of a curved phase space by continuous isotropic measurement. I. Spin and the Kraus-operator geometry of $\mathrm{SL}(2,\mathbb{C})$" by Christopher S. Jackson and Carlton M. Caves, published in Quantum 7, 1085 (2023).

By Robert Mann (University of Waterloo, Ontario, Canada N2L 3G1).

Coherent states have long played an important role in physics, ever since Glauber pointed out their utility in quantum optics. They are the unique eigenstates of the annihilation operator of a bosonic quantum field (such as the electromagnetic field), which means that they remain unchanged by the annihilation of a field excitation (or in other words, a particle). They are regarded as the “most classical” of all possible quantum states of a given Hilbert space, and have a number of interesting and useful properties, most notable of which is that they are states of minimum uncertainty. The balance of uncertainty between the conjugate quantum variables does not have to be equal (there can be less uncertainty in the position-type variable and more in the momentum one for example), in which case the state is said to be a squeezed coherent state. The most straightforward way of realizing a coherent state is with the quantum harmonic oscillator, constructing an eigenstate of the operator $\frac{\hat{Q}+ i \hat{P}}{\sqrt{2}}$, where $\hat{Q}$ and $\hat{P}$ are its respective position and momentum operators. A physical example is that of the laser, in which light is emitted in a coherent state of photons.

Since their introduction over 60 years ago, the concept of coherent state has been generalized to quantum systems having more complicated phase spaces than that of electromagnetism. Examples such as Generalized Coherent States (GCSs) include Spin-Coherent States (realized by a dipole in a magnetic field), multiqubit coherent states (realized by products of qubit states), and fermionic coherent states (realized by BCS superconductors). Each of these examples has a curved phase space, unlike the flat phase space of the originally conceived coherent states. The properties of these GCSs remain the subject of active research and are still being unravelled.

Perhaps the least well-understood aspect of GCSs is the notion of how they can be measured. Unlike the original bosonic coherent states, in which heterodyne measurement is the experimental way of implementing a Positive-Operator-Valued-Measurement (POVM), there is a disconnect between theoretical POVMs for GCSs, and the kinds of measurements that an experimentalist might consider doing. POVMs are essential for understanding how a distribution of experimental measurement outcomes can define a state on the phase space.

Jackson and Caves intend to address this issue in a series of papers of which this is the first. Concentrating solely on Spin-Coherent States (SCSs), this paper is a tour-de-force showing how an SCS POVM can be carried out via continuous isotropic measurement of the three spin components. Reading more like a textbook than a research paper, they provide a full geometric description of the symmetric space of POVM elements and of the curved phase space on the boundary of the symmetric space. Rich in detail and pedagogic in style, this paper will be foundational in advancing our understanding of how to perform continuous measurements not only for SCSs, but also for more general GCSs.

Turning to the main physical results of the paper, the key result is that a continuous isotropic measurement almost always collapses to an SCS POVM very rapidly. The collapse time — exponentially fast and governed by the measurement rate of the continuous measurements — is necessarily non-vanishing due to the curvature of the phase space. This in turn has its origins in the spin observables being “more noncommutative” than quadrature observables associated with the more familiar coherent states are. Starting at the centre of the sphere of SCSs, the continuous isotropic measurement outcomes spontaneously pick a direction after a few collapses, after which the POVM element moves exponentially rapidly in that direction to the surface of the sphere.

This is a paper that has the virtue of high accessibility to beginners in the subject whilst being rich in detail for experts. I look forward to see future papers along these lines by the authors.

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