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Implementation of Tavis Cummings model in solid-state defect qubits: Diamond nitrogen-vacancy center V. Roopini, R. Radhakrishnan ⇑ Department of Theoretical Physics, University of Madras, Chennai 600025, India

a r t i c l e

i n f o

Article history: Received 6 September 2019 Accepted 24 November 2019 Available online xxxx Keywords: NV center Vacancies Defects Diamond Qubit Tavis Cummings model

a b s t r a c t Nitrogen-vacancy color centers, which are formed in diamond by one substitutional nitrogen atom and an adjacent carbon vacancy, are among the most intensely studied lattice defects in recent years because of its applications in quantum information processing. In this paper, the negatively charged nitrogenvacancy center is modelled using the Tavis Cummings Hamiltonian representing two non-interacting identical qubits coupled to a cavity. The energy spectrum corresponding to the Hamiltonian matches with the spin triplet states of the color center. It is found that, by appropriately tunig the parameters such as the coupling constant, cavity and qubit frequencies this model can be used to simulate other solid-state defect qubits. The dynamical behaviour of the system has been studied by varying the coupling strength and the time evolution of the system has been systematically examined. The atom-cavity field interaction is investigated in the strong coupling regime under dissipation. Ó 2019 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of the scientific committee of the First International Conference on Recent Advances in Materials and Manufacturing 2019

1. Introduction Color centers in wide-bandgap semiconductors are of great interest for the past two decades and it is a viable resource for quantum information processing. These solid-state defects with nonzero spin ground states fulfills the DiVincenzo criteria as qubits and can be manipulated using various control mechanisms. Circuit quantum electrodynamics with spin qubits have been implemented in a wide variety of solid-state materials, such as semiconductors, superconductors, and ion-doped insulators [1–5]. In silicon, the electron/nuclear spins of individual donor phosphorus atoms was reported as a primary candidate for spin qubits [6,7]. Another promising group IV material, diamond with its wide band gap (5.5 eV) supports a large number of optically active point defects which are paramagnetic in nature and therefore serve as spin qubits. The nitrogen-vacancy (NV) center in diamond is reported to have remarkable coherence properties that persist up to room temperature due to diamond’s large band gap, weak spin–orbit interactions and extremely high Debye temperature. The interactions of the individual nuclear spins of the intrinsic nitrogen atom and

⇑ Corresponding author.

proximal 13 C nuclei with the electronic spin state makes each NV center a small ‘‘quantum register” consisting of several individually addressable quantum systems that can be initialized, manipulated, entangled and measured with high fidelity at room temperature [8–13]. The crystal structure of Nitrogen-vacancy (NV) center in diamond consisting of a substitutional nitrogen atom adjacent to a vacancy in diamond is shown in Fig. 1(a). The multiparticle states of the color center consists of six electrons of which five are contributed by the four atoms surrounding the vacancy, and one is captured from the bulk [15]. The electronic structure of NV center in diamond is shown in Fig. 1(b). The lowest energy states are the spin triplet states (3A2) with the spin sublevels ms = +1, 0, 1 which differs slightly in energy. The coherent rotations between the sublevels ms = 0 and 1 is induced by applying microwave radiation tuned to the energy splitting between them. The excited state triplet (3E) is 1.945 eV higher in energy than the ground state triplets. As shown in Fig. 1(b) there is a nonradiative transition from 3E to an intermediate spin singlet state (1A1). Through the spinconserving optical transition and fluorescence detection, these spin states can be optically initialized and measured [15,16]. It has been identified that the isotopic purification of spin-free 12 C diamond leads to ultra-long coherence times, up to several milliseconds at room temperature and more than one million coherent

E-mail address: [email protected] (R. Radhakrishnan). https://doi.org/10.1016/j.matpr.2019.11.266 2214-7853/Ó 2019 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of the scientific committee of the First International Conference on Recent Advances in Materials and Manufacturing 2019

Please cite this article as: V. Roopini and R. Radhakrishnan, Implementation of Tavis Cummings model in solid-state defect qubits: Diamond nitrogenvacancy center, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.11.266

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Fig. 1. (a) A schematic diagram of nitrogen vacancy center in diamond lattice. Carbons atoms are marked in red, nitrogen (N) atom in blue, and (V) is the vacancy. (b) The electronic structure of NV center in diamond [14]. The green arrows represent the spin-conserving (1.945 eV) optical transition to the excited state. The red arrows indicate the fluorescence emission. The intersystem crossing is shown by the gray arrrows and the non-radiative decay is indicated by brown arrows. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

operations can be performed within the NV center’s spin coherence time [17,18]. A demonstration of a quantum register at room temperature consisting of a single 13C nucleus weakly coupled to a NV center also reports a coherence time that exceeds 1s [19–22]. With the implementation of the color-centers in solid-state nanocavities, there is a great impact on the development of high-quality integrated sources of single photons, optical switches in GHz speed and qubit gates.

where the JC model works well. The generalized RWA [44] bridges the gap between the validity regime of the AA and JC model. A generalization of the JC model, called the Tavis Cummings (TC) model, consists of N two-level systems that are identically coupled to a cavity and the Hamiltonian of the system can be written as (for — h ¼ 1) [45] N X

H ¼ x0

riz þ xay a þ

i¼1

2. Tavis Cummings model The quantum Rabi model (QRM) governing the dynamics of two level systems interacting with a harmonic oscillator is used to implement many physical phenomena, such as nuclear spins interacting with magnetic field, atoms interacting with electromagnetic field, electrons coupled to a phonon mode of a crystal lattice, superconducting qubits interacting with a nano-mechanical resonator [23–28]. The model was initially presented in 1936 by Rabi and stands out to be a significant model in quantum optics. The QRM finds an important place in circuit quantum electrodynamics (QED) [29,30] and cavity QED [31,32]. It has been reported that the QRM can be used to realize several solid state qubits and implement quantum entanglement and quantum information processing [33–36]. The analytical solutions of QRM has been investigated and the eigenvalues and eigen-functions of the Rabi model was presented by Braak and Chen [37,38]. The QRM with two qubits has been analytically solved using the method of extended coherent states [39]. A number of approximations have been developed to solve QRM. Rotating wave approximation (RWA) [24,40] is one such approximation which works based on the assumption of near resonance and weak coupling between qubits and oscillator. The Jaynes Cummings (JC) model describes the physical situations where the qubits are nearly resonant with the oscillator and the qubit-oscillator interaction strengths are much smaller than the qubit and oscillator frequencies [24]. The ultrastrong coupling between qubits and oscillator has been achieved with the development of circuit QED and cavity QED. Irish et al., proposed the adiabatic approximation (AA) [41–43] which works best at Quasi-degenerate qubit and ultra-strong coupling. It is imperative to study the dynamical properties and the evolution of entanglement between two qubits in the ultra-strong coupling regime. The adiabatic approximation (AA) was shown to fail in the regime

N X g a þ ay riþ þ ri N i¼1

ð1Þ

where riz , rix , riþ and ri ði ¼ 1; 2Þ are the usual Pauli matrices in the Hilbert space of the i-th qubit, and ay and a refer to the creation and the annihilation operators of an interacting mode of a harmonic oscillator, x0 and x are the frequency of the qubit and the cavity respectively. The coupling strength between the qubit and the cavity is represented by g. The collective N-atom interaction strength is predicted to be pﬃﬃﬃﬃ GN ¼ g i N , where g i is the dipole coupling strength of each individual atom i. The TC model gathers profound interest as it can be used to implement quantum-information protocols [46] and study the multiqubit properties like quantum entanglement by employing various entanglement measures such as concurrence for mixed-state pairs of qubits, quantum negativity and Schmidt weights [47–49]. 2.1. The spectrum of model Hamiltonian (N = 2) For a system consisting of two identical qubits interacting with a cavity the Hamiltonian can be written as

H ¼ x0

g

r1z þ r2z þ xay a þ pﬃﬃﬃ a þ ay r1x þ r2x 2

ð2Þ

Writing the model Hamiltonian in terms of the collective angular momentum operators J z and J ,

k H ¼ x0 J z þ xay a þ pﬃﬃﬃﬃ a þ ay Jþ þ J N

ð3Þ

where,

Jz ¼

2 X i¼1

rðziÞ ; J ¼

2 X

rðiÞ

i¼1

The eigenstates j/i of the Hamiltonian is the product states that can be written as:

Please cite this article as: V. Roopini and R. Radhakrishnan, Implementation of Tavis Cummings model in solid-state defect qubits: Diamond nitrogenvacancy center, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.11.266

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j/i ¼ jjmijwm i

ð4Þ

Here jj; mi are the eigenstates of J x and jwm i are the oscillator eigenstates. When the system is subjected to dissipation the model Hamiltonian must be expanded to an open system treatment. The dynamics of the system is studied using the Lindblad master equation for the density operatorq

k 2

q_ ¼ i½qðtÞ; H þ L½aqðtÞ þ

N X C L½J qðtÞ 2 z i¼1

ð5Þ

where k is the cavity-field decay rate, C represents the atomic spontaneous decay and the super-operator L for any operator O is given by

1 1 L½Oq ¼ OqOy Oy Oq qOy O 2 2

ð6Þ

3. Numerical analysis of the atom-cavity field interaction The eigenvalue spectrum of the TC Hamiltonian has been systematically studied by numerically solving the Schrödinger Equation using QuTiP (Quantum Toolbox in Python) [50]. It is found that coherent interactions in the system give rise to a new set of eigenfrequencies that forms the energy spectrum of the Hamiltonian in Eq. (2). For a system with N quibits interacting with a cavity, each rung consists of N þ 1 states. Thus, for a system with two non-interacting identical qubits equally coupled to a cavity, the energy spectrum of the corresponding Hamiltonian matches with the spin triplet states of the color center as given in Fig. 1(b). The energy difference between the rungs can be controlled by the cavity frequncy x and difference between the triplet states are tuned by the quibit frequncy x0 . The energy spectrum of the modelled Hamiltonian is shown in Fig. 2(a) for x ¼ 1:945 and x0 ¼ 0:0001x. By tuning the qubit frequency as x0 ¼ 0:05 x, it is found that the energy splitting between the triplet states can be controlled as shown in Fig. 2(b). The energy spectrum for the system when x ¼ 1:945 and x0 ¼ 0:05x for coupling strength g ¼ 0:42 is show in Fig. 2(c). It is observed that in the strong coupling regime ð0:5 < g < 1Þ two of the eigenstates become nearly degenerate and its energy difference with the third eigenstate increases. The modelled Hamiltonian can therefore be used to realize a wide range of solid-state defect qubits.

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3.1. Occupation probability The time evolution of the cavity occupation probability for the systems with and without dissipation is studied by numerically solving the Lindblad master equation. QuTiP code is used for solving the master equation for the steady-state density operator in Eq. (5) [50]. For x ¼ 1:945 and x0 ¼ 0:00001x by choosing a coherent state, the time evolution of the cavity occupation probability without dissipation is presented in Fig. 3(a). The effect of dissipation is studied by including decay rates k ¼ 0:05 and C ¼ 0:15 in Eq. (5) and the time evolution is presented in Fig. 3(b). Simialrly the time evolution of the atom occupation probability is plotted for the closed system in Fig. 4(a) and for the open quantum system in Fig. 4(b). The resonator probability ha þ ay i are plotted in Fig. 5(a) and (b) for the closed and open system respectively. It is seen that, in all the three cases the occupation probability for the closed system follows an oscillatory pattern and when the system is subjected to dissipation the pattern collapses and the system decays with the respective rates k and C as time evolves. 3.2. Wigner function The Wigner function is given as

W ðaÞ ¼

i y 2 h y tr D ðaÞqDðaÞð1Þa a

p

ð7Þ

where DðaÞ ¼ expðaay a aÞ is the displacement operator, pﬃﬃﬃ a ¼ ðx þ iyÞ= 2 and q is the reduced density operator for the cavity field. The Wigner function W ðaÞ has been calculated for the modelled Hamiltonian in Eq. (2) for the cavity field being in the ground state. For frequencies x ¼ 1:945 and x0 ¼ 0:00001x, Fig. 6 shows the contour plots of the Wigner finction and the fock state distribution for various values of coupling strengths g ¼ ð0; 0:5; 1; 1:5; 2Þ. With increase in the qubit-cavity interaction strength, W ðaÞ evolves into two well-separated peaks. For g ¼ 2 the separation of the field amplitude that corresponds to the two peaks of the Wigner function is greater. When the initial state of the cavity field is considered to be a coherent state, the time evolution of the Wigner function W ðaÞ and the corresponding fock state distribution is shown in Fig. 7 for t ¼ ð0; 20; 40; 60; 80Þ. Initially the cavity state shows classical behaviour and as time evolves the coherent state superpositions and the nonclassical states are observed. Under dissipation with

Fig. 2. Energy spectrum of the Hamiltonian in Eq. (2) as a function of the coupling strength (a) x ¼ 1:945, x0 ¼ 0:00001x (b) x ¼ 1:945, x0 ¼ 0:05 x and (c) Forg ¼ 0:42

Please cite this article as: V. Roopini and R. Radhakrishnan, Implementation of Tavis Cummings model in solid-state defect qubits: Diamond nitrogenvacancy center, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.11.266

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Fig. 3. Time evolution of the cavity occupation probability hay ai for the (a) closed and (b) open system by varying the coupling strengths.

Fig. 4. Time evolution of the probability of the atomic inversion hJz i for the (a) closed and (b) open system by varying the coupling strengths.

Fig. 5. Time evolution of the resonator probability ha þ ay i for the (a) closed and (b) open system by varying the coupling strengths.

the decay rates k ¼ 0:05 and C ¼ 0:15, the negativity of the Wigner function W ðaÞ that corresponds to the non-classical character of the cavity state disappears as shown in Fig. 8 and its fock state distribution for t ¼ ð0; 10; 20; 30; 40Þ is studied.

The time evolution of the Wigner function W ðaÞ and the correspondiong fock state distribution when the intial state of the cavity field is a fock state is shown in Fig. 9. It is observed that the initial fock state undergoes a full revival through non-classical states and

Please cite this article as: V. Roopini and R. Radhakrishnan, Implementation of Tavis Cummings model in solid-state defect qubits: Diamond nitrogenvacancy center, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.11.266

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Fig. 6. Contour plots of the Wigner function W ðaÞ and the fock state distribution, showing snapshots cavity field being in the groundstate. Parameters x ¼ 1:945 and x0 ¼ 0:00001x. Starting from left to right, the coupling strengths for the snapshots are g ¼ ð0; 0:5; 1; 1:5; 2Þ.

Fig. 7. Time evolution of Wigner function W ðaÞ and the fock state distribution for an initial state that is chosen to be a coherent state. For g ¼ 1 starting from left to right, the time for the snapshots are t ¼ ð0; 20; 40; 60; 80Þ.

Fig. 8. Time evolution of Wigner function W ðaÞ with dissipation and the fock state distribution for an initial state that is chosen to be a coherent state. For g ¼ 1 starting from left to right, the time for the snapshots are t ¼ ð0; 10; 20; 30; 40Þ.

coherent superpositions as time evolves. When the system is subjected to dissipation with k ¼ 0:05 and C ¼ 0:15, the time evolution of W ðaÞ and the correspondiong fock state distribution is

presented in Fig. 10. It is noted that at for t ¼ 0, the cavity is in its intial fock state with W ðaÞ having negative values. This indicates a truly quantum mechanical state. As time evolves

Please cite this article as: V. Roopini and R. Radhakrishnan, Implementation of Tavis Cummings model in solid-state defect qubits: Diamond nitrogenvacancy center, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.11.266

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Fig. 9. Time evolution of Wigner function W ðaÞ and the fock state distribution for an initial state that is chosen to be a fock state. For g ¼ 1 starting from left to right, the time for the snapshots are t ¼ ð0; 20; 40; 60; 80Þ.

Fig. 10. Time evolution of Wigner function W ðaÞ with dissipation and the fock state distribution for an initial state that is chosen to be a fock state. For g ¼ 1 starting from left to right, the time for the snapshots are t ¼ ð0; 20; 40; 60; 80Þ:

ðt ¼ 20; 40; 60; 80Þ, the Wigner function has no negative values and is therefore considered to be a classical state. 3.3. Entropy and correlation measures For the modelled system described by the density operator q, the von Neumann entropy is given as

SðqÞ ¼ Tr½q logq

ð8Þ

The time evolution of the von Neumann entropy SðqÞ is obtained by numerically solving the Lindblad master equation without dissipation, for frequencies x ¼ 1:945 and x0 ¼ 0:00001x. The total von Neumann entropy as a function of time for various values of coupling strengths ðg ¼ 0:25; 0:5; 0:75; 1Þ is shown in Fig. 11. It is observed that for values of time t > 5, the entropy reaches a maximum value and saturates. On increasing the coupling strength from 0 to 0:5, it is noted that the entropy increases and remains nearly equal with further increase in coupling strength values.

Similarly, the von Neumann entropy SðqÞ for the cavity and atom is plotted as a function of time for various g values and is presented in Fig. 12(a) and (b) respectively. In Fig. 12(a) it is shown that there is an increase in entropy when the coupling strength is increased from 0:25 to 0:5 and in Fig. 12(b) SðqÞ remains nearly equal for values of g > 0:25. In both the cases, SðqÞ as a function of time reaches a maximum value and gets saturated. The dynamics of quantum correlation for the open and the closed system is studied by evolving it over time. In Fig. 13(a) the effect of atom-cavity interaction strength ðg Þ on the quantum correlation of the closed system is shown. The modelled system with frequencies x ¼ 1:945 and x0 ¼ 0:00001x is subjected to dissipation. With the cavity-field ðk ¼ 0:05Þ and atomic ðC ¼ 0:15Þ decay rates, Eq. (5) is numerically solved to obtain the time evolution of quantum correlation for the open system dynamics and is presented in Fig. 13(b). It is observed that for the closed system the quantum correlation follows an oscillatory pattern and when the system is subjected to dissipation the pattern collapses and the system decays with the respective rates k and C as time evolves.

Please cite this article as: V. Roopini and R. Radhakrishnan, Implementation of Tavis Cummings model in solid-state defect qubits: Diamond nitrogenvacancy center, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.11.266

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Fig. 11. The von Neumann entropy SðqÞ as a function of time for various coupling strengths g.

Fig. 12. The von Neumann entropy SðqÞ of the (a) cavity and (b) atom as a function of time for various coupling strengths g.

Fig. 13. Time evolution of quantum correlation for the (a) closed and (b) open system for various coupling strengths g.

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4. Conclusion In this paper, the possibility of color centers as controllable quantum devices is theoretically investigated by modelling NV centers in the framework of cavity QED. This work offers an alternative approach to model and simulate solid state qubits, in particular NV color centers, for potential applications in quantum information sciences. In this proposed model consisting of two identical qubits interacting with a single cavity, the spin triplet states coming in the diamond NV center is successfully simulated. The purpose of initialization and measurement of arbitrary quantum states can be investigated in a fully controlled manner by varying the coupling strengths, cavity and qubit frequencies. The dynamics of the modelled system is extensively studied with the aid of occupation probability, Wigner function, entropy and correlation measures for a wide range of coupling strength. When system interacts with the environment, the nature of states is analyzed using the open system version of the modelled Hamiltonian. This model enables us to search for maximally entangled states, from the numerous schemes proposed for entanglementby-measurement of solid-state qubits. It can also be used for constructing gates and registers using NV center. This scheme of modelling NV center qubit, greatly simplifies the experimental realization of scalable quantum computers in solid-state qubits. References [1] R. Hanson et al., Nature 453 (2008) 1043–1049. [2] M.H. Devoret et al., Quantum Inf. Process. 3 (2004) 163–203. [3] S. Bertaina et al., Rare-earth solid-state qubits, Nat. Nanotechnol. 2 (2007) 39– 42. [4] K.D. Petersson et al., Nature 490 (2012) 380. [5] L. DiCarlo et al., Nature 467 (2010) 574. [6] B.E. Kane, Nature 393 (1998) 133. [7] J.J.L. Morton, D.R. McCamey, M.A. Eriksson, S.A. Lyon, Nature 479 (2011) 345. [8] M. H. Devoret, S. Reynaud, E. Giacobino, J. Zinn-Justin, Eds. Elsevier Science, Amsterdam, (1997). [9] V. Bouchiat, D. Vion, P. Joyez, D. Esteve, M.H. Devoret, Phys. Scr. 1998 (1998) 165.

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