Quantum Coherence Studies
Since an ion cooled to its ground state in a trap is such a well-controlled and isolated quantum mechanical system, it can show excellent coherence properties.
Rabi oscillations and optical coherence
In order to demonstrate the coherence properties of our system, we need to first prepare the ion in its ground state of motion, i.e. with motional quantum number n=0, as described in the sideband cooling page. A laser tuned exactly to the frequency corresponding to a transition from the electronic ground state of the ion (|g›) to a long-lived excited state (|e›) will drive the ion coherently between these two states. This is effectively a two-level atom and the system displays “Rabi oscillations” as it moves coherently from the ground state to the excited state and back in a series of oscillations, as shown in Figure 1. The ion stays in the motional ground state n=0 throughout. The number of oscillations seen is limited only by decoherence effects in the system such as laser intensity or frequency noise, environmental perturbations (e.g. electrical or magnetic field noise) or heating of the ion.
In our calcium system the optical coherence time is of the order of 1 ms and is probably limited by the linewidth of the highly-stabilised laser at 729 nm that we use to drive the Rabi oscillations. This laser linewidth is around 1 kHz.
We can also drive Rabi oscillations between the motional ground state (n=0) and the first excited state (n=1) by tuning the laser to the first blue sideband of the optical transition. The transition then couples the states |g,0› and |e,1› where we are now specifying both the electronic state (|g› or |e›) and the motional state (n=0 or 1). In this case the Rabi oscillations are slower because the strength of the sideband transition is weaker than the carrier.
Another way to measure the optical coherence time is to create a quantum mechanical superposition of the ground and excited electronic states, while the ion is in the n=0 motional state. Starting from the state |g,0>, we first drive a “π/2-pulse” on the carrier transition, which creates an equal superposition of the states |g,0> and |e,0>. If we reverse the pulse used to prepare this state after a delay time T we will find interference fringes when we scan the frequency of the laser around the resonance. A set of such fringes is shown in Figure 2 taken from . The change of visibility of these fringes as the delay time T is changed gives information on the coherence time which again is measured to be around 1 ms. It can be extended using spin-echo techniques (see for example ).
The motional coherence time for our system is much longer than the optical coherence time, because it does not depend on the presence of an ultra-stable laser. In order to measure the motional coherence time, we have to create a quantum mechanical superposition of the ground and first excited motional states, i.e. n=0 and 1, using the ultrastable laser. Starting from the state |g,0>, we first drive a “π/2-pulse” on the carrier transition, which creates an equal superposition of the states |g,0> and |e,0>. Then we use a “π-pulse” on the first red sideband to transfer that part of the population that is in the state |e,0> to the state |g,1>. We now have an equal superposition of |g,0> and |g,1> and if we reverse the two pulses used to prepare this state after a delay time T we will find interference fringes when we vary the delay time T. A set of such fringes is shown in Figure 3. The change of visibility of these fringes as the delay time T is changed gives information on the motional coherence time which in this case is approaching 1 s.
Coherence of highly-excited motional states
The sideband cooling technique can be used to gradually reduce the motional quantum number towards n=0, the ground state. If we reverse this process by tuning to the first blue sideband rather than the first red sideband, we drive a process of “sideband heating”. As explained in ref. , this process stops around a narrow range of n where the strength of the sideband approaches zero. In this way we can prepare the ion in a fairly well-defined highly-excited motional state. We can then probe the coherence properties of this state using the same techniques as described above (for details see ref. ). The plot of fringe visibility against delay time (Figure 4) shows three main regions: there is nearly full visibility for delays up to the optical coherence time (around 0.7 ms); then there is roughly 40% visibility for longer delays than the optical coherence time. In this region there is no optical coherence between the electronic states, but there is still motional coherence within each electronic state. When the motional coherence time is reached (at around 160 ms), the fringe visibility falls to zero because all coherence is lost.
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