by Sara Marzban
In this post I’m going to briefly describe the solid state system, namely rare-earth crystalline material, that the Tittel lab at Qutech is using to do all sorts of cool quantum communication experiments, including building the hardware required for an elementary link of a quantum repeater.
In this era of online communications, the security of transmitted and received information is extremely important. Quantum communication is an absolutely secure method of communication between points A and B. However, building these secure quantum communication networks has proven to be difficult. In a quantum communication network, fragile quantum states are transmitted between a transmitter (point A) and a receiver (point B). During transmission, decoherence can be introduced into the system, either from losses in the transmission line, absorption in the system or from environmental contamination. Some of the losses occurring over the full length of the transmission line scale exponentially with distance and as a consequence, quantum communication is restricted to a range of about 200 km, beyond which quantum states can no longer be reliably measured [1, 2, 3].
In the classical communication network, the problem of loss in the transmission line is resolved by placing amplifiers along the transmission line. Due to the no-cloning theorem, which states that it is impossible to create identical copies of unknown quantum states , it is not possible to simply amplify quantum information. Therefore, a quantum repeater protocol is used which is able to establish entanglement between two quantum states that are spatially separated . The quantum repeater is an essential part of the quantum internet that part of QuTech is working to achieve.
Rare-earth ion doped crystals
An appealing system for constructing a complete quantum communication network can be ensembles of rare-earth ion optical centers in a transparent rare-earth crystal. These crystals have a unique property where light can be strongly coupled to densely packed ions in the structure, while the ions are isolated from environmental perturbations. This results in strong coupling to the ensemble and leads to long coherence times on the optical and hyperfine transitions, long storage times [2, 5] and low loss compared to other solid state platforms [6, 3].
In this post I’m going to briefly describe the solid state system, namely rare-earth crystalline material, that the Tittel lab at QuTech is using to do all sorts of cool quantum communication experiments, including building the hardware required for an elementary link of a quantum repeater. The name “rare-earth” elements is misleading as they are relatively abundant in the Earth’s crust. Cerium is actually more abundant on the earth than Copper. They got their mysterious name due to their geochemical properties, which makes these elements typically scattered and not concentrated in rare-earth minerals. In 1778, the first rare-earth mineral, Gadolinite, was extracted from a mine in the town of Ytterby in Sweden. These elements have quite similar chemical properties, which has made it difficult to separate them from each other . Figure 1 shows a Nd:YAG crystal that is often used as a laser material .
Figure 1: Nd:YAG, Yttrium Aluminum Garnet 
(Reproduced with permission from FLIR Scientific Materials)
In these elements, 14 electrons are added to the 4f electronic orbital, consecutively. The 4f electron shell is filled from cerium with 1 electron in this shell to lutetium with a full shell of 14 electrons . The two elements scandium and yttrium are also commonly included in the rare-earth group, due to their trivalent ions having filled electron shells. The rare earth’s oxidation state is usually dominated by the trivalent state. The trivalent structure of rare-earth ions is described by [Xe]4fN, with N increasing from 1 to 14. The interesting optical, magnetic and chemical properties of the rare-earths are due to their unique electronic structure.
The unique electronic structure of the rare-earth elements causes them to exhibit interesting optical properties. The approximate radial probability distributions of the electronic orbitals of rare-earth ion structures are shown in Figure 2. From Figure 2, it can be observed that the electrons in the 4f shell, are mostly within the filled 5s and 5p shells. The outer shells protect the 4f’s from environmental perturbations, forming a strong Faraday cage around the electrons in the 4f shell .
Figure 2: Radial probability distribution for the 4f, 5s, 5p and 6s electron subshells of Gd+.
Because of the strong shielding of the 4f electrons, their energy levels are similar to the free ion energy levels, making the crystal field act as a perturbation to the free-ion Hamiltonian. As an example, the energy level structure of Tm3+:𝑌3𝐺𝑎5𝑂12 (Tm:YGG), which is the crystal we use in the Tittel lab, is shown in Figure 3.
Figure 3: Energy levels of the valence (4f) electrons in Tm:YGG
The development of many applications of rare-earth doped crystals, like real-time high frequency signal processing, storage of information in computers or communications networks [10, 11, 12, 13, 14, 15, 16] or frequency stabilizing lasers for spectroscopy of materials or quantum information applications, require narrow homogeneous broadening and low spectral diffusion. As a result, there has been a significant effort over the last 30 years to understand and control mechanisms causing broadening on the optical transition .
Mechanisms causing broadening in rare-earth ion doped crystalline structures are typically categorized into three groups based on the time scale of the broadening. Static disorders cause a broadening called “inhomogeneous broadening”, which results in a large spread of the ions’ center frequencies over the ensemble. “Homogeneous broadening” is due to dynamic coupling to the environment that occurs on a time scale that is short compared to the coherence time (T2). “Spectral diffusion”, is due to dynamic coupling to the environment occurring on a time scale that is long compared to T2 .
Variations in the crystal lattice and inhomogeneous broadening effects result in the ensemble being broadened by orders of magnitude compared to the homogeneous broadening observed in these structures. On the optical transitions, this broadening mechanism could result in a crystal linewidth on the order of 0.1 GHz to 100 GHz. For example, for the crystal Tm3+:𝑌3𝐺𝑎5𝑂12 which is used in the Tittel lab, the inhomogeneous linewidth has been reported to be about 60 GHz [17, 3].
The spectrum in Figure 4 illustrates an overall and common image of the inhomogeneous line of the rare-earth crystals. Although each ion has a narrow linewidth, the optical transitions of each ion in the crystal happens at a slightly detuned frequency, due to variations in the crystalline structure. This causes the spectrum of the crystal as a whole to look much broader .
Doping the host crystalline structure with the dopant ions results in inhomogeneous broadening. This is due to the size mismatch between the host and dopant ions resulting in increased strain in the crystalline lattice. By lowering the doping concentration the inhomogeneous linewidth can be reduced .
Figure 4: Inhomogeneous broadening on an optical transition caused by random defects in the crystal structure, (a) Inhomogeneous spectrum showing packets of homogeneous lines much smaller than the inhomogeneous broadening, (b) Randomly distributed defects in the crystalline structure.
Broadening mechanisms influencing the homogeneous linewidth include broadening due to the lifetime of the excited ions, magnetic and phonon broadening, broadening due to interaction with Two-Level Systems and other effects due to the crystalline environment .
Spectral diffusion is caused by dynamic coupling to the environment that happens on a long time scale compared to the coherence time (T2). As time passes, the frequency shifts of the ions produce small random walks, or “diffusion”, in frequency space across the optical spectrum. The importance of the decoherence which results from spectral diffusion has been investigated in the field of optical spectroscopy of solids [18, 19, 20, 21, 22, 3].
Spectral diffusion in a rare-earth crystalline material happens when an interaction causing a big frequency shift and a mechanism which modulates the magnitude of the interaction occurs in the structure. A significant mechanism causing spectral diffusion in rare-earth ion doped crystalline systems is due to the magnetic dipole–dipole interactions of the active ions in the crystal and the electronic nuclear flips of the host material .
Spectroscopic technique for studying broadening
Now that I have briefly described the different broadening mechanisms in a rare-earth ion dopes crystal, I’ll describe one of the tools that we can use to study these spectroscopic properties, namely “Spectral Holeburning”. It is also used to tailor the absorption line and prepare groups of ions to perform certain operations. To optically select one pack of ions with a certain frequency, a laser beam tuned to that particular frequency is applied to the crystal. The laser excites the ions, therefore pumping them from their ground state to the excited state. After which, the excited ions spontaneously emit a photon and fall back down to any ground state, depending on the oscillator strength of that transition. If the laser is left on for long enough, all the ions are pumped out of that particular ground state, and hence a “hole” is created. A simple spectral holeburning scheme is illustrated in Figure 5 .
Figure 5: Spectral holeburning, (a) Energy level diagram of spectral holeburning, (b) Inhomogeneous line spectrum with burnt hole.
Spectral holeburning can also be used to tailor the inhomogeneous linewidth which is the basis for the Atomic Frequency Comb (AFC) memory schemes which is the scheme used to create quantum memories in the Tittel lab  .
I have tried to give you an extremely brief overview of the spectroscopy of rare-earth ion doped crystals and their broadening mechanisms. This is definitely not complete, if you are interested to know more about these interesting crystals please feel free to contact me.
Cover picture from Wikipedia (Creative Commons).
About the author
Hi, I’m Sara, a Postdoc working with Professor Tittel on quantum memories using rare-earth crystals in the Tittel lab. I’m interested in broadening my social network and I really like coffee and chocolate!
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