Discrete Time Crystals in Noninteracting Dissipative Systems

TL;DR

This work shows that a robust subharmonic response characteristic of discrete time crystals can emerge in a noninteracting open quantum system when dissipation is engineered to stabilize the dynamics. By pairing environment interaction for a duration $\tau$ with a brief $\theta$-pulse sequence, the authors derive analytical results for $\theta = \pi$ and $\theta = \pi + \delta$, revealing a $2T$-periodic response in the magnetization $M_z$ whose lifetime is set by the dissipative timescales $T_1$ and $T_2$ and is independent of system size. The experimental demonstration using Nuclear Magnetic Resonance in $\mathrm{D}_2\mathrm{O}$ confirms the EDTC phenomenon, showing rigidity under near-perfect pulses and controlled degradation under pulse errors, with the ability to mitigate errors by adjusting $\tau$. The findings establish environment-assisted DTC as a distinct dissipative mechanism for time-crystalline order in open quantum systems, broadening the scope beyond interacting or Floquet-prethermal regimes. Overall, the work highlights a practical route to stabilize DTC behavior via controlled dissipation, independent of initial conditions and system size, with potential implications for robust quantum dynamics in noisy environments.

Abstract

Many-body quantum systems, under suitable conditions, exhibit time-translation symmetry breaking and settle in a discrete time crystalline (DTC) phase -- an out-of-equilibrium quantum phase of matter. The defining feature of DTC is a robust subharmonic response. However, the DTC phase is fragile in the presence of environmental dissipation. Here, we propose and exemplify a DTC phase in a noninteracting system that owes its stability to environmental dissipation. The lifetime of this DTC is independent of initial conditions and the size of the system, though it depends on the frequency of the external driver. We experimentally demonstrate this realization of DTC using Nuclear Magnetic Resonance spectroscopy.

Figures (5)

• Figure 1: (color online) (a) Pulse sequence for environment-assisted DTC. Here, the system interacts with its environment for a $\tau$ time period (red, deeper grey) and is followed by a $\theta$-pulse about $y$-direction (green, lighter grey), and this sequence, having $T$ periodicity, is repeated many times. As $T_1 \gg T_2$, relaxation of $M_z$ is slower than the decay of $M_x$. Hence, the time delay $\tau$ acts similarly to a spin-locking pulse along $z$-direction Choi_2017Beatrez_2023. Whilst, the $\theta$-pulse, when close to $\pi$, provides the $2T$-periodicity for $M_z$. (b) The existence of DTC depends on the $\theta$-pulse. Here $M_z$ oscillates between positive and negative values in alternate cycles when $\theta$ is close to $\pm \pi$. (c) Spectrum of $M_z$ in the frequency domain. The Fourier spectrum shows peaks around $\nu = 0.5$ frequency, confirming the $2T$-periodicity of $M_z$ in the same regions of $\theta$. The parameters used to generate the plots of (b) and (c) are $T_1 = 100 T_2$, $\tau = 10 T_2$, $M_\circ = 0.8$, and $M_z(0) = -0.9M_\circ$.
• Figure 2: (color online) Plots of experimentally determined values of $M_z$ versus the number of cycles are shown in (a), (b), (c) and their corresponding Fourier transform, $S(M_z)$, versus frequency ($\nu$) are shown in the insets. (a) Perfectly calibrated $\pi$ pulses are used to get the DTC phase, which produces a sharp peak at $0.5$ in the frequency domain. Here $\tau = 25ms$. (b) In this case the pulses were with error of $\delta = 0.0674 \pi$, with $\tau = 25ms$, kills the DTC phase to produce a decaying beat pattern. This corresponds to peaks at $0.5 \left( 1 \pm \delta \right)$ in the frequency domain. (c) Here, the time delay has been increased, $\tau = 0.2s$, for the same imperfect pulse of (b) to get back DTC, showing the rigidity of such a phase. However, the lifetime of DTC has reduced significantly.
• Figure 3: (color online) (a) Crystalline fraction (f), defined by Choi et al.Choi_2017, dependence on the perturbation ($\delta$) in the $\pi$ rotation about $y$-direction and the ratio of the timescales $T_1$ and $T_2$. The Crystalline fraction, f, acts as a measure for the amount of DTC phase present in the system. To have DTC phase $\delta$ needs to be small, i.e. we need near perfect $\pi$ pulses and $T_2 \ll T_1$, as it follows from Eq. \ref{['eq:7']}. We have used $\tau = 5 T_2$ for the plot. (b) Crystalline fraction's dependence on $\delta$ and time interval $\tau$ (in the units of $T_2$). It shows $\delta$ needs to be small and $T_2 < \tau \ll T_1$ for having DTC phase, following Eq. \ref{['eq:7']}. Here, $T_1 = 1000 T_2$ is considered for the plot. The other parameters used to generate the plots, of (a) and (b), are $M_\circ = 0.8$, and $M_z(0) = -0.9M_\circ$. (c) Plot of the FWHM of the Fourier spectrum of $M_z$, i.e. $S(M_z)$, versus the error ($\delta$) in $\pi$ pulse (experimental data for $\tau = 25ms$). The data is fitted with $f(\delta) = a \delta ^\lambda + b$ for $\lambda = 2.24$. Hence, FWHM varies as $\delta^2$; therefore, the lifetime of the DTC phase decreases at the same rate.
• Figure 4: (color online) The variation of Crystalline fraction, f, with $\theta$ pulse is plotted, with different time delays $\tau$. It shows that we need a pulse very close to $\pi$, for a short $\tau$, to get the EDTC phase. Whilst EDTC persists for imperfect pulses when a longer $\tau$ is applied, showing the rigidity of the phase.
• Figure 5: (color online) The variation of $M_z$ with time (in the units of $T_1$) is shown for different time delays $\tau$'s. This shows the lifetime of this DTC increases with $\tau$. Hence, the lifetime decreases with increasing frequency of the pulse. The parameters used to generate the plot are $T_1 = 1000 T_2$, $\delta = 0.1 \pi$, $M_z(0) = -0.9M_\circ$, and $M_\circ = 0.8$.