Dynamics of the collective modes of an inhomogeneous spin ensemble in a cavity

We study the excitation dynamics of an inhomogeneously broadened spin ensemble coupled to a single cavity mode. The collective excitations of the spin ensemble can be described in terms of generalized spin waves, and, in the absence of the cavity, the free evolution of the spin ensemble can be described as a drift in the wavenumber without dispersion. In this article we show that the dynamics in the presence of coupling to the cavity mode can be described solely by a modified time evolution of the wavenumbers. In particular, we show that collective excitations with a well-defined wavenumber pass without dispersion from negative to positive-valued wavenumbers without populating the zero wavenumber spin wave mode. The results are relevant for multimode collective quantum memories where qubits are encoded in different spin waves.

Figure 1

(a) The model studied in this paper describes a set of $N+1$ quantum oscillators with different frequencies interacting in a star topology. We assume that the width of the distribution of the outer oscillator frequencies can be controlled by some external parameter. (b) A physical motivation for the model studied is a spin ensemble in a cavity exposed to a controllable linearly varying Zeeman shift provided by an adjustable magnetic field gradient [15].

Figure 2

Time evolution of a collective spin excitation expanded in the bare-time (wavenumber) basis. The figures show the overlap $|\langle \tau |\Psi (t)\rangle |{}^2$ of the time-evolving spin state with the bare-time basis states. Panels (a) and (c) show that the evolution under the free Hamiltonian ${\mathcal{H}}_0$ ($\Omega =0$) is a simple translation with time. Under the full Hamiltonian $\mathcal{H}$ ($\Omega =10\pi /T$), the strongly coupled state $|\tau =0\rangle$ is frozen (b), while a bare-time state with initial condition $|\Psi (t\to -\infty )\rangle \equiv |\tau =t\rangle$ passes the $\tau =0$ region without ever populating the strongly coupled $|\tau =0\rangle$ state (d). Snapshots of the expansion of the states in (c) and (d) are shown in Fig. 3.

Figure 3

Evolution of a wave packet in the bare-time (wavenumber) basis in the presence and absence of a strongly coupled cavity. The initial state (bottom figure) is the negative bare-time state $|\tau =-4T\rangle$ at $t=-4T$, and the figures show the overlaps $\langle \tau |\Psi (t)\rangle$ (solid) and $\langle \tau |{\Psi }_0(t)\rangle$ (dashed) at three different times during the evolution. With no cavity coupling, the spin wave packet evolution is a simple translation, while the cavity coupling freezes the superradiant state, and unitarity forces the evolving wave packet to maintain a vanishing overlap $\langle 0|\Psi (t)\rangle =0$ during the evolution. The net effect on the final state (upper figure) is a phase shift $\pi$ and an additional translation of the wave packet by $\Delta t~T$.

Figure 4

Examples of the phase $\varphi$ in the strong coupling limit for three different system geometries discussed in the text. For each part (a)–(c), the effective spin density ${\rho }_{\text{eff}}$ is illustrated by the inset, and the plot shows the value of ${\varphi }_{\infty }(\omega )$ as given by Eq. (11) (solid lines), together with the linear approximation (17) corresponding to the shear observed for fully delocalized wave packets [$\psi (t)=\delta (t)$] in the system as discussed in Sec. 3b and Fig. 6.

Figure 5

The shear approximation for a localized wave packet. (a) The value of ${\gamma }_{\text{c}}$ (filled) and ${\delta }_{\text{c}}$ (solid) as given by Eq. (10) for the spin system with uniform distribution of $\omega$, ${\rho }_{\text{eff}}(\omega )=1_{[-\pi /T,\pi /T]}(\omega )T/2\pi$, as also considered in Fig. 4a. (b) The phase $\varphi (\omega )$ given by Eq. (9) at ${\omega }_{\text{c}}=0$ and $\Omega =\{2\pi /3,\pi ,\infty \}/T$ (dotted) and at ${\omega }_{\text{c}}=-\pi /4T$, $\Omega =\pi /3T$ (solid). Dashed lines are linear approximations of $\varphi (\omega )$ according to Eq. (17) in two different frequency domains around ${\omega }_0$, where the wave packets illustrated in (c) have significant value. The slope of the lines is given by the $\Delta t$ that maximizes the fidelity (19). (c) The frequency domain wave packets $\mathrm{\psi ̃}(\omega )$ and the shear fidelity $F$ for these wave packets.

Figure 6

Shear fidelity for delocalized wave packets. Plot shows the overlap of $|\tau +t\rangle$ with (i) the unshifted state $|t\rangle$ (dashed), (ii) the state $|t+\Delta t\rangle$ corresponding to the shear approximation (dotted), and (iii) the actual asymptotic state compensated for ${\varphi }_0$, $e^{i{\varphi }_0}|\Psi (t)\rangle$ (solid), for the same three spin systems (a)–(c) as described in Fig. 4 in the limit of $t\to \infty$. For all three systems the wave packet traverses the $\tau \approx 0$ region with little dispersion as discussed in Sec. 3b and a time shift $\Delta t$ in good agreement with the results of Sec. 3 as illustrated by Fig. 4.