Auxiliary-state facilitated phase synchronization phenomena in isolated spin systems

TL;DR

The paper presents a theoretical framework for quantum phase synchronization of an effective spin-1 system engineered by coupling three ground states to finite-lifetime auxiliary states. By integrating out the excited states, the authors derive an effective Hamiltonian with both coherent and incoherent contributions, whose dissipators carry complex phases that enable new synchronization pathways. They show that phase synchronization can be tuned via the phase parameter $\alpha$, and that a synchronization blockade persists for certain parameter choices, while a dissipation-driven, purely incoherent mechanism emerges at zero Zeeman splitting when $|\Omega'|$ is finite. The results, supported by perturbation theory and full master-equation benchmarks in $^{87}$Rb, illuminate dissipation engineering as a robust route to control quantum synchronization with potential relevance for quantum technologies.

Abstract

Extending classical synchronization to the quantum domain is of great interest both from the fundamental physics point of view and with a view toward quantum technology applications. This work characterizes phase synchronization of an effective spin-1 system, which is realized by coupling three quantum states with infinite lifetime to auxiliary excited states that have a finite lifetime. Integrating out the excited states, the effective spin-1 model features coherent and incoherent effective couplings. Our key findings are: (i) Phase synchronization can be controlled by adjusting the phases of the couplings to the excited states. (ii) Unlike in the paradigmatic spin-1 system studied in the literature, where the dissipative couplings describe decay into the limit cycle state, the effective spin-1 model investigated in this work is governed by a competition between dissipative decay into and out of the limit cycle state, with the dissipative decay out of the limit cycle state playing a critical role. (iii) We identify a parameter regime where phase synchronization of the effective spin-1 system is -- in the absence of coherent effective couplings -- governed entirely by the effective dissipators. The effective spin-1 model is benchmarked through comparisons with master equation calculations for the full Hilbert space. Physical insights are gained through analytical perturbation theory calculations. Our findings, which are expected to hold for a broad class of energy level and coupling schemes, are demonstrated using hyperfine states of $^{87}$Rb.

Figures (7)

• Figure 1: Illustrations of (a/b) the full $(3+3)$-level system and (c/d) the effective 3-level system. The black horizontal lines show the energy levels. While the (3+3)-system consists of the ground states $|1\rangle$, $|2\rangle$, and $|3\rangle$ and the auxiliary states $|4\rangle$, $|5\rangle$, and $|6\rangle$, the effective 3-level system only contains the ground states. Gray dashed lines show the energy of the $F=1$ ground state manifold and the $F'=1$ excited state manifold (only the $m_{F'}=\pm1$ states are shown) for vanishing magnetic field. Throughout, we assume $\Delta_\pi"=\Delta_\sigma"=\Delta_\pi'=0$ (these detunings are defined in Appendix \ref{['sec_appendix0']}) and $|\Omega_{+1}|=|\Omega_{-1}|$. (a) Arrows indicate coherent couplings between ground and auxiliary states. Levels $|1\rangle$, $|2\rangle$, and $|3\rangle$ are coupled to $|4\rangle$ with Rabi frequencies $\Omega_{+1}$, $\Omega_{0}$, and $\Omega_{-1}$, respectively. Level $|1\rangle$ is coupled to $|5\rangle$ and level $|3\rangle$ is coupled to $|6\rangle$ with Rabi frequency $\Omega'$. The Zeeman shifts for the $F=1$ ground state manifold and the $F'=1$ excited state manifold are $\Delta_B$ and $\Delta_B'$, respectively. For finite magnetic field, i.e., for non-zero $\Delta_B$ and non-zero $\Delta_B'$, the "decay" beams ($\Omega'$) and the "control" beams ($\Omega_{\pm1}$) are off-resonant while the "probe" beam ($\Omega_{0}$) is on resonance. The phases of the coherent couplings, which play an important role in our analysis, are not shown. (b) Wiggly lines show the decay paths of the auxiliary states. The auxiliary state $|4\rangle$ decays into each of $|1\rangle$, $|2\rangle$, and $|3\rangle$ at rate $\Gamma_{\text{aux}}"/3$. The auxiliary state $|5\rangle$ decays into each of $|1\rangle$ and $|2\rangle$ at rate $\Gamma_{\text{aux}}'/2$. The auxiliary state $|6\rangle$ decays into each of $|2\rangle$ and $|3\rangle$ at rate $\Gamma_{\text{aux}}'/2$. (c) The two arrows show the coherent effective couplings. Throughout, we work in a parameter regime where the coherent coupling between states $|1\rangle$ and $|3\rangle$ is zero. The effective detuning $\Delta_{\text{eff}}$ comes from the Zeeman shift of the ground state manifold and the light shifts. (d) The wiggly lines show dissipative effective decay. The control beams ($\Omega_{\pm1}$) mediate effective dissipation from state $|1\rangle$ to states $|2\rangle$ and $|3\rangle$ and from state $|3\rangle$ to states $|1\rangle$ and $|2\rangle$ with rate $\Gamma_{\text{control}}$. The probe beam ($\Omega_{0}$) mediates effective dissipation from state $|2\rangle$ to states $|1\rangle$ and $|3\rangle$ with rate $\Gamma_{\text{probe}}$. The decay beams ($\Omega'$) mediate effective dissipation from states $|1\rangle$ and $|3\rangle$ to state $|2\rangle$ with rate $\Gamma_{\text{decay}}$. To keep the schematic readable, dissipative effective self-dephasing processes ("effective self-dissipation") are not shown (see Appendix \ref{['sec_appendixA']}).
• Figure 2: (a) Steady-state synchronization $S_q$ as a function of the phase angle $\alpha$ for $\Delta_B=2\pi \times 0.4$ MHz, $\phi_{\pm1}=\phi_0=\phi'=0$, $|\Omega_{\pm1}|=2\pi \times 9.5$ MHz, $|\Omega_{0}|=2\pi \times 1.0$ MHz, and $|\Omega'|=2\pi \times 3.0$ MHz (the decay rates $\Gamma_{\text{aux}}'$ and $\Gamma_{\text{aux}}"$ are those for $^{87}$Rb; see Appendix \ref{['sec_appendix0']}). The black dashed, blue solid, and red dotted lines are for the full $(3+3)$-level system, the effective 3-level model, and the perturbative treatment of the effective 3-level system, respectively. The three curves nearly coincide for all $\alpha$. Husimi-$Q$ distributions for points B ($\alpha=0$) and C ($\alpha=\pi$) as functions of $\phi$ (horizontal axis) and $\theta$ (vertical axis) are shown in (b) and (c), respectively; the color scheme is the same as that used in Figs. \ref{['rewrite_fig_beta']} and \ref{['rewrite_fig_HusimiQ']}. The Husimi-$Q$ function for $\alpha=0$ shows phase localization while that for $\alpha=\pi$ shows a small deformation of the limit cycle state but no phase localization.
• Figure 3: Phase angle $\phi_{\text{max}}$ at which the Husimi-$Q$ function takes its maximum as a function of $\Delta_B$ for $\phi_0=0$, $\phi'=0$, $|\Omega_{\pm1}|=2\pi \times 9.5$ MHz, $|\Omega_{0}|=2\pi \times 1.0$ MHz, and $|\Omega'|=2\pi \times 3.0$ MHz (the decay rates $\Gamma_{\text{aux}}'$ and $\Gamma_{\text{aux}}"$ are those for $^{87}$Rb; see Appendix \ref{['sec_appendix0']}). Two different parameter combinations of the control laser phases are considered: The dark lines are for $\phi_{\pm 1} = 0$ while the faded lines are for $\phi_{\pm 1} = \pm \pi/2$. The dark/faded black dashed, dark/faded blue solid, and dark/faded red dotted lines show results for the full $(3+3)$-level system, the effective 3-level system, and the perturbative treatment of the effective 3-level system, respectively.
• Figure 4: The green dashed and blue solid lines in panels (d) and (e) show the synchronization $S_q$, calculated by solving the master equation numerically for the effective 3-level system with scaled dissipative rates (see text for details), as a function of $\beta$ for approach (1) and approach (2), respectively. Panel (d) is for $\Delta_B=2\pi \times 0.4$ MHz and panel (e) for $\Delta_B=0$; the other parameters are the same as in Fig. \ref{['rewrite_fig_alpha']}, except for $\phi_{-1}$, which is chosen such that $\phi_{\text{eff}}=0$ holds. For comparison, the red dotted and orange dash-dotted lines show the corresponding perturbative expressions $S_{q,a1}$ [Eq. (\ref{['eq_sync_pt_finite_beta1']})] and $S_{q,a2}$ [Eq. (\ref{['eq_sync_pt_finite_beta2']})], respectively. Husimi-$Q$ distributions for point A ($\beta=0$), point B ($\beta=1/2$), and point C ($\beta=1$) are included above panel (d) for $\Delta_B=2 \pi \times 0.4$ MHz while Husimi-$Q$ distributions for point F ($\beta=0$), point G ($\beta=1/2$), and point H ($\beta=1$) are included below panel (e) for $\Delta_B=0$; the color scheme is the same as that used in Figs. \ref{['rewrite_fig_alpha']} and \ref{['rewrite_fig_HusimiQ']}. The green and blue diamonds at $\beta=0$ show $S_q$ for the expanded spin-1 model using $\gamma_g=\gamma_d=\Gamma_{\text{decay}}$ and $\gamma_g=\gamma_d=\Gamma_{\text{decay}}+\Gamma_{\text{control}}$, respectively. The parameters for the expanded spin-1 system follow from Eqs. (\ref{['eq_Delta_eff']}),(\ref{['eq_Omega_eff']}),(\ref{['eq_phi_eff']}), and (\ref{['steady_states_eq_lindblad_eff_gamma']}).
• Figure 5: The black dashed, blue solid, and red dotted lines show the synchronization $S_q$ as a function of the Zeeman splitting $\Delta_B$ for the full $(3+3)$-level system, the effective 3-level model, and the perturbative treatment of the effective 3-level model, respectively, for (a) $|\Omega'|=2\pi \times 3.0$ MHz and (b) $|\Omega'|=2\pi \times 6.0$ MHz. The other parameters are $\phi_{\pm1}=\phi_0=\phi'=0$, $|\Omega_{\pm1}|=2\pi \times 9.5$ MHz, and $|\Omega_{0}|=2\pi \times 1.0$ MHz (the decay rates $\Gamma_{\text{aux}}'$ and $\Gamma_{\text{aux}}"$ are those for $^{87}$Rb; see Appendix \ref{['sec_appendix0']}).