Modified coherence of quantum spins in a damped pure-dephasing model

We consider a spin-$j$ particle coupled to a structured bath of bosonic modes that decay into thermal baths. We obtain an analytic expression for the reduced spin state and use it to investigate non-Markovian spin dynamics. In the heavily overdamped regime, spin coherences are preserved due to a quantum Zeno affect. We extend the solution to two spins and include coupling between the modes, which can be leveraged for preservation of the symmetric spin subspace. For many spins, we find that intermode coupling gives rise to a privileged symmetric mode gapped from the other modes. This provides a handle to selectively address that privileged mode for quantum control of the collective spin. Finally, we show that our solution applies to defects in solid-state systems, such as negatively charged nitrogen vacancy centers in diamond.

Figure 1

Dephasing factor ${\mathcal{I}}_{\text{Re}}(t;\omega ,\mathrm{\Gamma })$ and unitary phase factor ${\mathcal{I}}_{\text{Im}}(t;\omega ,\mathrm{\Gamma })$ in Eqs. (31a) and (31b) as a function of time for the (a) underdamped and (b) overdamped regimes. Parameters are $\eta /\omega =1, \beta \to \infty$ (zero temperature), and critical damping occurs at $\mathrm{\Gamma }/\omega =1$. The amplitude of a spin coherence between $m$ and $m^{\prime }$ is determined by ${\mathcal{I}}_{\text{Re}}$ [Eq. (30)], which in the underdamped regime increases as $\mathrm{\Gamma }/\omega$ does. However, in the overdamped regime, increasing $\mathrm{\Gamma }/\omega$ decreases the ${\mathcal{I}}_{\text{Re}}$, thus producing less spin dephasing. In this regime, the accumulated unitary phase between $m$ and $m^{\prime }$, determined by ${\mathcal{I}}_{\text{Im}}$, also decreases with $\mathrm{\Gamma }/\omega$; in the heavily overdamped regime, spin coherences are frozen and experience no dephasing or unitary-type evolution at all.

Figure 2

Decay of the off-diagonal coherences of a $j=\frac{1}{2}$ spin coupled to a single bosonic mode for varying decay rates. Plotted is the matrix element ${\rho }_{\frac{1}{2},\frac{1}{2}}\left(t\right)$ using Eq. (30) with ${\rho }_{\frac{1}{2},\frac{1}{2}}\left(0\right)=1$. Parameters are the same as in Fig. 1. (a) Underdamped ($\mathrm{\Gamma }<\omega$) and (b) overdamped ($\mathrm{\Gamma }>\omega$) regimes. The transition from underdamped, highly non-Markovian dynamics to simpler, overdamped dynamics is evident in the disappearance of oscillations. In the underdamped regime, the decay rate of the spin coherence increases with increasing decay rate $\mathrm{\Gamma }$, while in the overdamped regime the decay rate of the spin coherence decreases with increasing decay rate $\mathrm{\Gamma }$. This can be seen from Eq. (34), which shows that in the underdamped case the decay rate of the spin coherences scales proportional to $\mathrm{\Gamma }$, while in the overdamped case the decay rate scales proportional to ${\mathrm{\Gamma }}^{-1}$. This is a manifestation of the quantum Zeno effect; when the information the environment gains about the spin is lost fast enough, the spin state is frozen and does not decohere.

Figure 3

Decay of the symmetric two-spin state $|J=1,M=0\rangle$ as a function of time for various ${\mathrm{\Gamma }}_{-}$. The parameters are $\eta /{\omega }_0=1, \kappa =0$, and $\beta \to \infty$. The curves show population decaying out of the initial state. Lost population is incoherently pumped into the antisymmetric state $|J=0,M=0\rangle$. The quantum Zeno effect is apparent, as the population is preserved for longer times by increasing ${\mathrm{\Gamma }}_{-}$.

Figure 4

Influence of intermode coupling $\kappa$ on symmetric-state decay. Decay of the symmetric two-spin state $|J=1,M=0\rangle$ induced by coupling to the decaying bosonic modes. The parameters are $\eta /{\omega }_0=1, \mathrm{\Gamma }/{\omega }_0=1, {\mathrm{\Gamma }}_{\pm }/{\omega }_0={({\omega }_{\pm }/{\omega }_0)}^3$, and $\beta \to \infty$.

Figure 5

Depiction of multiple spins whose local modes are coupled together, illustrated in the context of two-level defects in a solid state (see Sec. 5). Each pseudospin couples in a state-dependent manner to local bosonic modes, e.g., a local vibronic mode, shown as a state-dependent displacement of the harmonic oscillator potential by an amount $x_0=2x_{\mathrm{rms}}\eta /\omega$, where $x_{\mathrm{rms}}$ is the ground-state width and $\omega$ is the energy of the mode. Furthermore, the modes are coupled to each other with potentially different strengths ${\kappa }_{ij}$ (blue dotted lines) and also experience decay into a bath of extended thermal modes, e.g., phonons, at rate $\mathrm{\Gamma }$ (red wavy lines).

Figure 6

Process fidelity (blue) $F_{\mathrm{pro}}(\mathcal{E},U)$ for a target spin-squeezing unitary $\widehat{U}=e^{-i\chi {\widehat{J}}_z^2}$ due to a collective spin with angular momentum $j=5$ evolving according to the damped, pure-dephasing model with a single mode ${\omega }_s$. The system parameters are in the underdamped regime, $\eta /{\omega }_s=0.1, \mathrm{\Gamma }/{\omega }_s={10}^{-3}$, and $\beta =10$. The accumulated coherent phase $\chi$ is plotted in red. At $t=157.08/{\omega }_s, \chi =\pi /2$ with $F_{\mathrm{pro}}=0.9703$.

Figure 7

Illustration of the separation of the symmetric mode from the other modes for a collection of $N$ randomly coupled modes. (a) Shown are the lowest-energy eigenmode, i.e., the symmetric mode ${\widehat{w}}_s$ (purple); the second-lowest-energy mode (orange); and the highest-energy mode (tan). Here, the bare mode frequency is ${\omega }_0$, and the mode couplings are chosen randomly and uniformly in the interval $[-\kappa ,0]$, where $\kappa /\omega ={10}^{-3}$. For each eigenmode and value of $N$, two points are plotted (nearly overlapping on the plot), indicating the mean $\pm 1$ standard deviation energy as calculated over ten realizations of random couplings. (b) Mean fidelity error of the lowest-energy eigenmode to the fully symmetric mode, where $F={\left(\frac{1}{\sqrt{N}}{\sum }_{j=1}^N\langle j|{\lambda }_1\rangle \right)}^2$.