Quantop > Quantum Optics Lab > Research > Cs Cell Experiment > Scientific Topics > Spin State

During our work with entangled spin samples we have developed some tools to characterize the properties of the ground state spin in Cesium. These tools rely on the well known magneto-optical resonance effect - when an RF-magnetic field is tuned into resonance with magnetic transitions in the ground state, the transitions can be probed optically by laser beams.

The principle of the experimental setup is shown below. The general setting with lasers and atoms is very similar to other experiments we have performed. Hence, the spin state characterization can be performed under "real experimental conditions".

The experimental setup for the magneto-optical resonance method. The atomic sample is polarized along the x-axis parallel to a bias magnetic field. The bias field causes Larmor precession. An applied RF-magnetic field modulates the transverse spin state when the RF-field and the Larmor frequency meet resonance conditions. The transverse spin components are measured by the probe laser.

The static magnetic field B_bias causes the atomic spins to precess at the Larmor frequency corresponding to the energy splitting in the magnetic ground state. The splitting between two magnetic sublevels m+1 and m corresponds to the frequency ω_m+1,m. When the frequency ω of the RF-magnetic field B_RF approaches ω_m+1,m, the spins will respond resonantly with a Lorentzian profile. The width Γ_m+1,m of this response characterizes the decay rate of the coherence between the levels m+1 and m.

Now, our cesium atoms have hyperfine split ground state and we consider the F=4 atoms. The 2F+1 = 9 levels give rise to eight possible transitions which all respond with a Lorentzian shape. The overall response which is read out on the laser beam and measured by photo detectors follows the equation below. This is what we call the Magneto-Optical Resonance Signal (MORS).

We see that this is just the coherent sum of 2F+1 Lorentzians with their own resonance frequency ω_m+1,m and line width Γ_m+1,m. Each signal is also proportional to the population differences σ_m+1,m+1 - σ_m,m, and the algebra of angular momentum also contributes with different weights F(F+1)-m(m+1). The total number of atoms in the sample is denoted N. Given a MORS spectrum we can tell much about the atomic ground state spin. Examples are shown below:

Two examples of magneto-optical resonance signals (MORS). On the abscissa is the frequency of the RF-magnetic field. On the ordinate we plot the response of the transverse spin given by the equation above. (a) The many peaks show that many magnetic sub-levels are populated with a resulting low orientation p = 82%. The line width characterizes the coherence time T₂ of the transverse spin components. (b) Extra lasers provide higher orientation (almost only one peak visible) by optical pumping processes. Scattered pumping photons result in a slightly higher line width.

In the above examples the static magnetic field is close to one Gauss, the Larmor frequencies are close to 325kHz. The quadratic Zeeman effect makes a small contribution, the eight resonance lines are split by 23Hz from this effect. We see that this really enables us to resolve the different lines and characterize the populations among the ground state sub-levels.

With the magneto-optical resonance method we are able to measure coherence time of spins, populations of different sub-levels and hence the degree of spin polarization, and the number of atoms in the sample (on a relative scale). To fit the complicated spectra to the theoretical value we need to employ useful models for populations and line widths. This has been done with great success, see the reference below for details. Obtained orientations exceed 98% for quite dense samples. An important factor is the use of 894nm light on the D1 transitions where the outermost level with m=4 is in a dark state.

• B. Julsgaard, J. Sherson, J. L. Sørensen, and E. S. Polzik,
  Characterizing the spin state of an atomic ensemble using the magneto-optical resonance method,
  J. Opt. B: Quantum Semiclass. Opt. 6, 5 (2004).
  
  
• B. Julsgaard,
  Entanglement and Quantum Interactions with Macroscopic Gas Samples,
  PhD-thesis, University of Aarhus (2003).