Light scattering is a prominent effect of atomic physics, described in section 1.2. It can indeed be applied to quantum memories if we consider this process as a reversible quantum-state transfer from light to the atoms: all the information from impinging light is converted via inelastic scattering into information encoded in the atomic state, and
vice versa. Raman scattering is the exemplary form of inelastic scattering with photons of low energy compared to the ionisation energy[1]. Let us give a quick reminder on the
Raman scattering process and then look into the details of how it can be applied to a memory scheme.
Reminder on Raman scattering In a Λ configuration, a light field at frequency ω can lead to the emission of an anti-Stokes photon at frequency ω0 which matches the energy difference between states |gi and |si, and which transfers one atom to
state |si, as shown on figure 4.1[2]. The process involves a virtual level of energy
~(ω3+ ∆) =~(ω1+ω) =~(ω2+ω0), where ∆ is the frequency difference between the
excited state|ei and this virtual level.
Figure 4.1: The absorption of a pump photon ω and emission of an anti-Stokes photon at frequency ω0. This operation transfers one atom from state |gi to|si.
The emission of a photon at frequency ω0 can be strongly enhanced with stimulated emission: applying a strong driving field at this frequency results instimulated Raman scattering.
Another form of scattering is coherent anti-Stokes Raman scattering (CARS), which corresponds to the simultaneous generation of a Stokes and anti-Stokes photons from two pump photons. As it involves four electromagnetic waves, the process is often dubbed four-wave mixing (FWM)[3].
[1]
Photons with energies much larger than the ionisation energy lead to Compton scattering.
[2]In the case of state|gilying energetically below|si, it is a less energetic Stokes photon that is emitted,
verifying~ω0=~ω−∆E12.
4.1 Theoretical basis for a Raman memory 95
Figure 4.2: The signal ˆE detuned by ∆ from the |gi → |eitransition is mapped to the
spin-wave ˆS between|giand |si by the two-photon resonant control Ω.
Spin wave The Raman scattering with the emission of an anti-Stokes photon leaves
the atomic ensemble with a collective excitation: one of all the atoms from the ensemble has been transferred to state|si.
Raman memory scheme In the context of optical quantum memories, it is customary
to define the light field to be stored as the signal and the auxiliary field stimulating the scattering as a control or coupling beam. The Raman memory scheme in Λ-type atoms consists of the absorption and subsequent retrieval of a signal detuned by ∆ from the transition between ground state|gi and excited state|ei, stored in a spin wave between
states |giand |si (see figure 4.2).
Although the applicable theory of stimulated Raman scattering has been known for much longer [Raymer1985, Raymer1981], and similar equations were laid out during the first demonstration of a quantum memory [Kozhekin2000], I use as a reference the 2007 article series by Alexey Gorshkov, which is the most comprehensive in the analytical treat- ment of Λ-type atomic ensembles [Gorshkov2007a, Gorshkov2007b, Gorshkov2007c]. The thorough description of this type of system is aimed at the storage of light, which is especially relevant to our approach.
As a reminder of chapter 1.2, Λ-type three-level atoms are governed by the set of Maxwell-Bloch equations 1.5. On resonance, and additionally ignoring the Stark shift and dispersion terms, these equations simplify to:
(∂t+c∂z) ˆE = i √ dΩ(t) ∆ Sˆ(z, t) ∂tSˆ= i √ dΓΩ(t) ∆ Eˆ(z, t). (4.1)
b)
a)
Figure 4.3: Symmetric configuration for backward-retrieval Raman memories: a) The spin-wave is orthogonal to the signal axis, as the angle θbetween the signal and control matches their energy mismatch. The spin-wave displays a k-vector mismatch ∆k. b) The spin-wave is not orthogonal to the signal axis. Consequently, the retrieved signal is not collinear with the input. Notice that the k-vector mismatch ∆k0 is smaller than ∆kin the previous configuration.
Efficient retrieval Upon retrieval, the choice can be made to retrieve the signal in
the backward direction and indeed time-reversal of the absorption process in the memory medium implies re-emission in this direction [Moiseev2011]. To this end, the mirror image of the control beam needs to be applied. This backward-retrieval configuration enables fundamentally-higher conversion efficiencies to and from any memory based on three-level atoms than in the forward direction [Gorshkov2007b, Nunn2007, LeGouet2009], an argument I will now explain.
First, a necessary extension to the theoretical model is the introduction of the backward versions of the signal and control, noted with a minus subscript, and for clarity a plus subscript is added to the forward-propagating quantities. The propagation in the backward direction modifies the set of equations as follows:
(∂t+c∂z) ˆE+= +i √ dΩ+ ∆− ˆ S (∂t−c∂z) ˆE−=−i √ dΩ− ∆+ ˆ S ∂tSˆ= i √ dΓ Ω + ∆+ ˆ E++ Ω− ∆− ˆ E− . (4.2)
The zand tdependence have been dropped for clarity. Note the minus sign of the left- hand side of the second equation, describing the propagation in the−z direction. Most notably, the spin-wave, which can now be converted into either forward or backward light through the controls Ω+ and Ω−, is governed by the third equation.
As we want to perform backward retrieval, special care has to be taken of the phase matching of the storage-and-retrieval process, as previously shown in [Surmacz2008]. Choosing the storage state energetically below the initial state, enables the configuration depicted in figure 4.3(a), with an angleθ between each signal-and-control pair, in which the spin wave is orthogonal to the signal axis, which conveniently makes the input signal
4.1 Theoretical basis for a Raman memory 97
and its read-out collinear. Figure 4.3(b) shows an alternative symmetric configuration where the signal and control make an angle θ0 < θinstead, which makes the effective k-vector mismatch smaller[4]. Only the control beams, not the signals, can be made collinear if the storage state lies energetically above the initial state.
Moreover, as pointed out in [Moiseev2011], the time-reversal of the storage process requires the detuning of the control beam to be inverted for read-out, i.e. ∆+ =−∆−. As will be explained in the description of the experiment, with available light resonant on |si → |ei transition, the forward and backward control can have opposite detuning
by respectively picking the positive and negative orders of the AOM on each beam path.