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a)Experimental quantum computing with ultracold atoms
Currently quantum computing is being investigated in a variety of different systems, ranging from nonlinear optics, ion traps, superconducting qubits, cold atoms in optical lattices, NV centers in diamond, and semiconductor quantum dots. The range of different systems that are being investigated reflects the early stage technologically that the field is in, where it is still unclear what materials are best in order to make a quantum computer. As is well known, the main difficulty in making a quantum computer is overcoming decoherence where the environment causes what is essentially noise on the quantum state of the qubit register. Although error correcting methods have advanced to a level that gate errors in the vicinity of 1% can be tolerated using topological error correction, the field is still not advanced to a level that a scalable quantum computer can be built.

In this project you will help with our experimental effort of building a quantum processor chip using Bose-Einstein condensates (BECs) with atom chips. We are in collaboration with the State Key Laboratory of Precision Spectroscopy at East China Normal University developing atom chips to be used for various quantum information tasks. One of the key ideas is to perform some of the theory ideas developed in our group in the lab. Such coherent manipulation of two component BEC qubits have been realized on atom chip systems. Experiments to realize arbitrary manipulation of single BEC qubits have been achieved, with arbitrary control of the state on the Bloch sphere. Currently the primary application of such two component BECs is viewed in the community to be towards quantum metrology. However, we believe that such BECs have possibilities that extend beyond such applications.
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b)Investigations of topics in quantum information theory
Quantum information is a mathematical theory used to describe the nature of information, when quantum mechanics is involved. It is used to describe the nature of quantum states and all their properties. Examples including understanding and quantifying the nature of entanglement, information, and correlations. Coherence has long been one of central concepts of quantum physics and hence its detection and quantification is a fundamental task. The distinction between classical (coherence in the absence of quantum fluctuations e.g. bright coherent light) and quantum coherence is has been made traditionally using phase space distributions and higher order correlation functions. Though these methods give a distinction between the classical and quantum forms of coherence they do not give us any procedure to measure the amount of coherence in the system. One of the new entrants in the field of quantum correlations is quantum coherence. A scheme for measuring coherence using the framework of quantum information theory was proposed by Baumgratz, Cramer, and Plenio [1] recently. In this work, the conceptual steps were taken and the definition of incoherent states, incoherent operations and maximally coherent states were defined. Further the list of properties a measure should satisfy in order to be classified as a coherence measure were also proposed. Developments have been made towards understanding quantum coherence and using it as resource in information theory.

In this project the student will use mathematical and numerical techniques to investigate various topics in quantum information theory. In our recent work [2] we have shown how to decompose coherence further, and precisely locate in what way and where the coherence is located in the quantum system. This can be done because quantum coherence is a manifestation of the superposition which can occur either between the qubits or between the levels of the qubit. An illustration of the two different kinds of coherence is given in the Figure where we show a simple four particle system with each qubit shown by a black colored filled circle. One kind of coherence arises due to the correlations between the qubits which is shown through the red colored arrow between the qubits represented by the black dots. This coherence is the inter-qubit or Intrinsic coherence. An example of this kind of coherence is the one in a Bell state. The second form of coherence arises purely due to the superposition between the levels of the quantum system. In the figure this is shown by magnifying the qubit to exhibit the two levels which are superposed. Since this intra-qubit coherence is localized within the qubit we refer to it as local coherence. The quantum state |0>+|1> is an example of this form of coherence. These two forms of coherence are complementary to each other and we proposed a scheme to isolate and estimate these different forms of coherence.

[1] T. Baumgratz, M. Cramer and M.B. Plenio, Phys. Rev. Lett. 113, 140401 (2014).
[2] R. Chandrashekar, P. Manikandan, J. Segar, Tim Byrnes, Phys. Rev. Lett. 116, 150504 (2016).
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c)Theory of quantum computing with Bose-Einstein condensates
One of the fundamental difficulties with quantum computing is that quantum states are fundamentally very sensitive to their surroundings. This is typically because the quantum world is associated with the microscopic world, where we must deal with extremely small length scales to observe quantum phenomena. Thus typically to control the states in the qubit register we require control over extremely small objects: atoms, quantum dots, atomic vacancies in crystals, etc. Although classical digital circuitry uses discrete variables in order to perform logic operations, single atoms are of course not used in computers, macroscopic currents are used to encode the logical information. A macroscopic current is first discretized to store the logical information.

A question then arises of whether such a similar approach is possible for quantum computing. Usually this possibility is excluded because quantum effects tend to disappear for macroscopic objects. Typically the quantum effects become smeared out into their classical averages, and thus quantum computing also becomes impossible. There are however exceptions to this general rule. Bose-Einstein condensates (BECs) are one exception where quantum effects are visible on a macroscopic level. For example, in a BEC all the bosons in the system occupy the ground state of the trap. This similar structure causes the macroscopic wavefunction obey all the same properties as microscopic quantum particles such as superposition and tunneling, but on a macroscopic scale. How is quantum information then encoded on such BECs? This problem has been investigated by us previously, with a theoretical framework outlined in Tim Byrnes, Kai Wen, Yoshihisa Yamamoto, Phys. Rev A 85 040306(R) (2012). The basic strategy is the use of two component Bose-Einstein condensates. For atomic systems, this corresponds to the use of hyperfine levels of the atoms forming the BECs. Such two component BECs can be put in an arbitrary superposition of the two states, in the same way as qubits, and have similar manipulation properties as standard qubits. Importantly, entanglement can be generated between two such qubits, and quantum algorithms can be executed on such qubits.
There are several interesting directions of research we are carrying out in this direction, listed as below
– Quantum algorithm mapping between qubits and BEC qubits
– Decoherence minimizing schemes from creating entanglement between BECs
– Entanglement detection for spin ensembles and BECs
This will involve mainly theoretical work and numerical simulations to analyze methods of doing quantum computing in this way.
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