The dynamics of a qubit reveals its coupling to a N level system

doi:10.1016/j.aop.2015.08.006

Annals of Physics

Volume 362, November 2015, Pages 287-297

https://doi.org/10.1016/j.aop.2015.08.006 Get rights and content

Abstract

Modeling the environment of a single qubit as a $N$ dimensional quantum system, we show that the dynamics of the qubit alone, if measured in sufficient detail, can reveal the parameters of the qubit–environment coupling Hamiltonian. We show that data from quantum process tomography experiments can be used to get information about the environment that can be used to minimize its deleterious effects on the state of the qubit.

Introduction

The dynamics of an open quantum system can by itself be a probe of the nature and dynamics of its immediate surroundings. Reconstructing open quantum dynamics through quantum state tomography and quantum process tomography [1], [2], [3], [4] protocols have become standard tools in the characterization and development of quantum information processing devices. In particular, detailed state and process tomography on quantum bits implemented in different types of physical systems are routinely done and the data are used to verify the fidelity of initialization procedures, gate operations, readout schemes, etc. For instance in [5], state tomography of solid state superconducting phase qubits using single shot measurements is done while in [6] a nuclear spin qubit in a high finesse optical cavity is studied and its states are mapped out. These are just a couple of examples of quantum state tomography on qubits picked from the available literature. It is worth noting that the evolution of the state of the superconducting qubit is measured in [5] at high enough data rates that the time dependence of the components of the Bloch vector representing the state of the qubit can be tracked almost continuously.

In this paper we point out that the wealth of data [5], [6], [7], [8], [9], [10] available on quantum processes can not only be used to do more than just characterize the open dynamics but also to learn about the coupling between the system of interest and its immediate environment. Reconstruction of the system–environment interaction can lead to the identification of the sources of decoherence and dephasing on the system as well as to the design of better quantum control [11] and isolation [12] protocols. The information can also be used to test our understanding of decoherence phenomena, pointer states and emergence of classicality in the classical world [13]. The reconstruction can also be useful in exploring molecular energy transfer mechanisms in chemical and biological processes where the environment is believed to be playing an enabling role in making the process robust and efficient even in vivo or in solution [14], [15].

This question of reconstructing the parameters of the Hamiltonian coupling between the system and its environment was addressed in a very limited context in [16] where the system, as well as its environment, was assumed to be single qubits. Here, we remove the restriction that the environment is a single qubit and let its Hilbert space be $N$ dimensional. It can be conceived as either an $N$ level quantum system or a collection of lower dimensional quantum systems with a tensor product Hilbert space of dimension $N$ . Since quite a few experiments involving individual quantum systems have qubits as the system of interest, we also let the system in our analysis to be a qubit. We show how, in principle, the parameters of the coupling Hamiltonian can be extracted by making sufficiently detailed observations on the system qubit and its dynamics. It is possible to extract partial information about the state of the environment also from the system dynamics but we defer that question to a later time and focus on the coupling Hamiltonian. We give a detailed example showing the reconstruction of the parameters of the Hamiltonian starting from simulated measurement data of the system qubit.

Section snippets

Single qubit coupled to $N$ dimensional environment: SU(2) $\times$ SU(N)

The state of the system of interest–the qubit–is written in terms of the three Pauli matrices, which are also SU(2), generators, denoted by $\vec{Σ} = (Σ_{1}, Σ_{2}, Σ_{3})$ . We often refer to the system qubit as the $Σ$ -system from here on. The $Σ_{i}$ operators satisfy the commutation relations, $[Σ_{i}, Σ_{j}] = 2 i ϵ_{ijk} Σ_{k} .$ The environment is assumed to be a general $N$ level quantum system with its state written in terms of the $N^{2} - 1$ generators of SU(N) denoted by $\vec{Λ}$ [17] (see Appendix A). These generators satisfy the commutation

Numerical example

In the absence of actual experimental data, to do a numerical example, we start by constructing a Hamiltonian by assigning the following values to the parameters $α_{j}$ , $β_{j}$ and the $γ_{j k}$ : $α_{1} = 1$ , $α_{2} = 2$ , $α_{3} = 3$ , $β_{1} = 1$ , $β_{2} = 2$ , $β_{3} = 1$ , $β_{4} = 1$ , $β_{5} = 1$ , $β_{6} = 1$ , $β_{7} = 1$ , $β_{8} = 0.1$ , $γ_{11} = 1$ , $γ_{22} = 1$ , $γ_{33} = 1$ . All the rest were set to zero. Assuming again that the initial state of the environment is fully mixed, we evolved the combined system numerically for the three prototypical initial states for the system qubit namely, $ρ_{0}^{(1)}$ , $ρ_{0}^{(2}$

Conclusion

To summarize, we have shown that the parameters of the Hamiltonian of a qubit interacting with an $N$ dimensional quantum system can be obtained, in principle, from the time dependence of the qubit alone. The methods presented here can be generalized in a straightforward manner to situations where the system is higher dimensional as well. It is worth noting that the magnitudes of the coupling of the system qubit to the environment given by ${| {\tilde{γ}}_{j} |}^{2}$ is obtained at the level of ${\ddot{a}}_{j}^{(k)}$ . The dot

Acknowledgments

V.J. thanks CSIR for an SRF. A.S. acknowledges the support of the Department of Science and Technology, Government of India, through the Fast-Track Scheme for Young Scientists (SERC Sl. No. 2786), and the Ramanujan Fellowship programme.

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The dynamics of a qubit reveals its coupling to a N level system

Abstract

Introduction

Section snippets

Single qubit coupled to N dimensional environment: SU(2) × SU(N)

Numerical example

Conclusion

Acknowledgments

J. Modern Opt.

Quantum Computation and Quantum Information

Phys.~Rev.~Lett.

Science

Phys.~Rev.~Lett.

Phys.~Rev.~A

New J. Phys.

Europhys. Lett.

The dynamics of a qubit reveals its coupling to a $N$ level system

Single qubit coupled to $N$ dimensional environment: SU(2) $\times$ SU(N)