Storing Light in an Atomic Frequency Comb

ATOMIC FREQUENCY COMB STORAGE - THEORY

3.1 Storing Light in an Atomic Frequency Comb

C h a p t e r 3

a) b)

Figure 3.1: Schematic of dephasing due to inhomogeneous broadening on a Bloch sphere. a) Dephasing a short time after absorption. b) Dephasing at 𝑡 ∼ _Δ ¹

inhom. Details in main text.

frequencies that are multiples of the comb periodΔ:

𝜔_𝑗 =2𝜋Δ𝑛_𝑗, 𝑛_𝑗 ∈Z. (3.2)

In that case, the Dicke state |Ψi characterizing the ion ensemble after a photon is absorbed is given by Equation 3.3:

|Ψi=

𝑁

Õ

𝑗=1

𝑐_𝑗𝑒^{𝑖2𝜋Δ𝑛𝑗𝑡}𝑒^{−𝑖 𝑘®𝑟𝑗}

0...1𝑗...0𝑁

(3.3)

Looking at the phase evolution term, when Δ ×𝑡 = 𝑚 where 𝑚 is an integer, the term is equal to 𝑒^{𝑖2𝜋𝑚 𝑛𝑗} = 1 for all ions, indicating a rephasing event. This means that a chain of rephasing events will occur at𝑡 = ^𝑚_Δ,𝑚=1,2,3...where𝑡 =0 is the time when the photon was absorbed.

Figure 3.2 presents the dephasing and first rephasing of a photon absorbed by an atomic frequency comb (AFC). Each term in the sum in Equation 3.3 is represented as an arrow on the Bloch sphere. The inset shows a sketch of an AFC with 5 teeth.

Each tooth is not an ideal delta function, but rather a distribution of frequencies, as in a realistic AFC. Immediately after the photon is absorbed, the ions start to dephase, but the phase evolution is described by just 5 groups, one for each comb tooth. At𝑡 = ¹

4Δ, the phases of the 5 groups are maximally scrambled, but the phase evolution continues. By𝑡 = _Δ¹−𝜖, all groups are close to arriving at the same phase again: the phase of one group of ions (green) has stayed put, two groups (cyan and orange) have evolved by 𝜙 = ±2𝜋, and two groups (red and purple) have evolved

by 𝜙 = ±2×2𝜋, and all groups are rephasing. When all arrows point in the same direction again at𝑡 = _Δ¹, the ions will spontaneously emit a photon that is identical in frequency, polarization, and spatial distribution to the one that was absorbed at 𝑡 =0. This is the output pulse of the AFC memory. Note that the groups of arrows are now broader than when they started. This results from the finite width 𝛾of the comb, and leads to a limit on the efficiency of storage through the term𝜂_𝑑[4] (see Section 3.2).

a) b) c)

Figure 3.2: Sketch of atomic frequency comb evolution on a Bloch sphere. a) Dephasing a short time after absorption (𝑡 = 𝜖). Inset shows a schematic of an atomic frequency comb (optical depth versus frequency). (b) Dephasing at𝑡 = ¹

4Δ. c) Shortly before rephasing, at 𝑡 = _Δ¹ −𝜖. The diagram is in the rotating frame of the ions in the center of the comb (green). Details in main text.

Although the AFC protocol can in principle store any kind of photonic qubit, qubits encoded in the polarization degree of freedom of light are not compatible with our implementation. This is because the photonic crystal cavities only support one polarization of light. In this work, we store time-bin qubits for the form

|𝜓_ini =𝛼|earlyi +𝑒^{𝑖 𝜙}

√

1−𝛼²|latei. Since AFC is a first-in-first-out storage protocol [101], the |earlyi and |latei pulses will be emitted in the same order, leading to an output state identical to the input state in the case of an ideal protocol:

|𝜓_outi=𝛼|earlyi +𝑒^{𝑖 𝜙}

√

1−𝛼²|latei.

The atomic frequency comb is created using spectral holeburning. To create an atomic frequency comb in ¹⁶⁷Er³⁺:Y₂SiO₅ in our experiments, a set of periodic spectral holes is burned in the absorption line after hyperfine initialization (see Section 2.3). Each hole is created by pumping on the transitions at that optical frequency with a laser, until population is moved to other long-lived hyperfine levels. The teeth of the comb are the remaining absorption peaks between the

spectral holes. See Figure 5.2 for pulse sequence and comb scan. Side-holes and anti-holes from other hyperfine transitions are usually outside of the frequency window of the comb. Superhyperfine side-holes and anti-holes, however, affect the comb structure (see Section 2.4). Combs of arbitrary finesse 𝐹 = ^Δ

𝛾 are created by creating spectral trenches between the comb teeth. These trenches are created by scanning the laser in frequency as it pumped away population. See Figure 6.2 for pulse sequence, and Figure 6.3 for a scan of a high finesse comb.

In some cases, the AFC was created using the accumulated AFC method [34], where the entire frequency comb is created simultaneously by creating an optical frequency comb and imprinting that onto the ion absorption line. See Figure 5.3 for a pulse sequence used to generate a comb in this way. The finesse obtained with this method is always𝐹 ∼2.

AFC with spin-wave

Up to this point, we have described storing light on an optical transition using an atomic frequency comb. All quantum storage experiments in this thesis were done according to this procedure, which is usually referred to as "optical AFC". However, this is only part the AFC protocol as it was originally proposed in Reference [4].

The full protocol, called spin-wave AFC, requires two additional optical pulses and an empty hyperfine level to which population from the excited state can be coherently and reversibly transferred. Additionally, a microwave frequency oscillating magnetic field (∼ 850 MHz) would be needed to rephase the inhomogeneously broadened spins during storage.

In the AFC with spin-wave protocol, after an input pulse (or input qubit) is stored on the optical transition, an optical 𝜋 pulse coherently moves population from the optical excited state to a second long-lived spin state in the optical ground state manifold. The coherence is then stored between the two ground state levels for a time that is limited by the spin coherence time 𝑇_spin. This time can be much longer than the optical coherence time. For example, in¹⁶⁷Er³⁺:Y₂SiO₅, hyperfine coherence times longer than 1 second have been measured [86]. After some desired wait time𝑇_s < 𝑇_spin, another optical𝜋 pulse moves the population from the second spin state back to the excited state, and the phase evolution continues until𝑡 = _Δ¹+𝑇_s, when the ions rephase and the output pulse is emitted. During the storage on the spin state, microwave pulses are usually used to rephase the inhomogeneously broadened spin ensemble.