Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms

Open physical systems with balanced loss and gain, described by non-Hermitian parity-time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {{\cal P}{\cal T}} \right)$$\end{document}PT reflection symmetric Hamiltonians, exhibit a transition which could engender modes that exponentially decay or grow with time, and thus spontaneously breaks the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetry. Such \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetry-breaking transitions have attracted many interests because of their extraordinary behaviors and functionalities absent in closed systems. Here we report on the observation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetry-breaking transitions by engineering time-periodic dissipation and coupling, which are realized through state-dependent atom loss in an optical dipole trap of ultracold 6Li atoms. Comparing with a single transition appearing for static dissipation, the time-periodic counterpart undergoes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetry breaking and restoring transitions at vanishingly small dissipation strength in both single and multiphoton transition domains, revealing rich phase structures associated to a Floquet open system. The results enable ultracold atoms to be a versatile tool for studying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetric quantum systems.

The corresponding scaled atom numbers are given by n σ (t) = e Γ0t n σ (t). The passive PT -symmetric breaking transition occurs at Γ 0 = 2J. When Γ 0 < 2J, λ is real and n σ (t) oscillates with a period of π/λ which increases as Γ 0 increases. When Γ 0 > 2J, λ becomes purely imaginary, leading to the two eigenmodes with different decay rates, one of which decreases as (2J) 2 /Γ 0 . So when Γ 0 increase, the total unscaled atom number n (t) decays slower. Correspondingly, the scaled atom number n(t) = n ↑ (t) + n ↓ (t) increases exponentially with time. Supplementary Figure 1a shows the total atom number n (t) for various static dissipation. For smaller dissipation in the PTS phase, the atom number n (t) decays faster as Γ 0 is increased (blue circles and diamonds); this trend is reversed in the PTSB phase, that is Γ 0 > 2J (red squares). Thus, the passive PT -symmetry breaking transition occurs when n (t) vanishes most rapidly. Supplementary Figure 1b shows the same data, plotted in terms of the scaled atom number n(t). As is expected, n(t) shows oscillatory behavior with increasing amplitude and period in the PTSB phase (blue circles and diamonds). Such behavior gives way to exponential-in-time in the PTSB phase (red squares).

n(t)
Supplementary Figure 1: Observation of the passive PT transition with static dissipation. a n (t) shows faster decay as the loss strength is increased from Γ 0 = 0.28J (blue circles) to Γ 0 = 0.9J (blue diamonds) in the PTS phase. This trend is reversed at large Γ 0 = 2.57J (red squares) indicating a PTSB phase. Here J = π × 2.15 kHz. b n(t), obtained from the same data, shows oscillations with increasing amplitude and period in the PTS phase and exponential rise (right vertical axis) in the PTSB phase. Each data point denotes average over 6 single-shot measurements and the error bars are the standard deviation of the measurements.
To determine the passive PT phase transition threshold, we track the individual-level atom numbers n σ (Γ 0 ) and their sum n (Γ 0 ) at a fixed time t m = π/2J as a function of dissipation strength in Supplementary Figure 2. n (Γ 0 ) first decreases and then increases, due to the emergence of the long-lived mode in the PTSB phase. We fit these fixedtime data n σ (Γ 0 ) to supplementary Eq.(1), with a fitting parameter for the PT -threshold, that is λ = Γ 2 exp − Γ 2 0 /2, and extract a threshold value Γ exp = (1.92 ± 0.09)J, which matches the theoretical value Γ exp = 2.0J well shown in the inset in Supplementary where The half-width δΩ n (Γ 0 ) decreases as the power-law with the order of the resonances. It is difficult to implement the sinusoidal modulation precisely in the experiment because the nonlinear relation between the intensity of the resonant beam and the atom loss strength Γ(t), so we use a square-wave loss profile with period 2τ = 2π/Ω as, The non-unitary time evolution operator can be analytically calculated for square-wave modulation, and we can use it to determine the phase diagram. The numerical PT phase diagram is shown in Supplementary Figure 3 as plotted in the (Γ 0 , Ω) plane. In the weak-loss limit, the eigenvalues of G P T (2π/Ω) are given by 1 is the dimensionless loss amplitude and ω = Ω/J is the dimensionless frequency. The system is in the PTSB phase when the two eigenvalues have different magnitudes, which happens when Ω ∈ [Ω n − ∆Ω n , Ω n + ∆Ω n ] where In comparison with the sinusoidal case [1], the half-width ∆Ω n (Γ 0 ) of the square-wave modulation remains linearly proportional to the loss amplitude. Supplementary Figure 3b shows the vicinity of first three resonances Ω n = {2J/5, 2J/3, 2J} from the numerical simulation as well as the analytical calculation of the phase boundaries (white lines)from supplementary Eq.(6). As n increases, both Ω n and the half-width of the PTSB phase ∆Ω n decrease. Note that the linear-phase-boundary is a better approximation for smaller loss amplitude. PT transition phase diagram with time-periodic coupling. We also apply the square-wave modulation on the coupling strength of two spin states. For time-periodic coupling experiments, we apply constant dissipation Γ 0 , but modulate J(t) with a square-wave. Applying the similar method to get the analytical eigenvalues G P T (2π/Ω) in the weak-loss limit, we have Following supplementary Eq.(7), the system is in the PTSB when the two eigenvalues have different magnitudes, which happens when Ω ∈ [Ω n − ∆Ω n , Ω n + ∆Ω n ] where Supplementary Figure 4b shows the numerical simulation the vicinity of four resonances Ω n = {J, J/2, J/3, J/4} in the (Γ, Ω) plane. The ∆Ω n (Γ 0 ) is linearly proportional to the loss amplitude in the weak-loss limit. As n increases, both Ω n and ∆Ω n decrease. We emphasize that the ∆Ω n (Γ 0 ) has larger values in the time-periodic coupling (supplementary Eq. 8) comparing with the time-periodic dissipation (supplementary Eq. 6).

Supplementary Note 3: Effective Floquet Hamiltonian
The long-term dynamic behaviors of the system can be studied by the effective Floquet Hamiltonian H F in a stroboscopic fashion with steps of the driving period, which is defined by Floquet Hamiltonian of time-periodic dissipation. In the time-periodic dissipation PT system, we derive H F under the weak dissipation limitation, which is expressed as where c n is a time-independent coefficients and = J 2 − γ 2 . It is very obvious we can derive out the PT phase diagram by diagonalizing H F . Eigenvalues become complex when 2 − [ cos( τ ) cos(Jτ ) − J sin( τ ) sin(Jτ ) < 0, which comes with the PTSB phase. Floquet Hamiltonian of time-periodic coupling. Similar as the above, the effective Hamiltonian under the weak dissipation limitation is given by c n is a time-independent coefficient, and the eigenvalues of the rest matrix are given by (13)

Supplementary Note 4: Mapping the PT -symmetry phase diagram
It is relatively simple to map the phase diagram of the static dissipation, in which the PTS and PTSB phases are separated by a single exceptional point. However, the phase diagram of Floquet Hamiltonians are extraordinary rich, in which the (Γ 0 , Ω) parameter-plane has an infinite number of the PTS and PTSB regions separated by lines of exceptional points [2].
We map the phase diagram by tracing the time evolution of the the normalized atom number n (t). Supplementary Figure 5a shows the time evolution of n (t) in the vicinity of the primary resonance Ω ≈ 2J for a square-wave modulated dissipation. We record n (t) for five modulation frequencies. For all frequencies, the state-dependent dissipation Γ 0 = 0.22J is an order of magnitude smaller than the transition point 2.0J. It is found that the total atom loss-rate decreases dramatically as Ω → 2J and reaches a minimum at the transition point. The loss rate increases again when the modulation frequency is increased further more in the PTSB phase. Supplementary Figure 5b shows n (Γ 0 , Ω) at a fixed time-point t f . As Γ 0 increases, the center position of the peaks of n (t) remain pegged at the transition point and the widths of n (t) increases. The peaks of n (t) indicate the appearance a long-lived mode in the PT Hamiltonians, and thus signal the PTSB phase. A Gaussian fit is used to extract the half-width at half-maximum (HWHM) of each peak. The inset in Supplementary Figure 5b shows that the extracted HWHM is linearly proportional to the loss strength as predicted by the theoretical model. It is confirmed that the fixed-time, frequency-dependent atom number n (Ω) can be a good indicator to characterize the phase diagram of the Floquet Hamiltonian. However, for finite probe time, the width of the residual atom number does not equal the width of the PTSB phase predicted by theoretical calculations. The width of the residual atom number gets narrower for the longer probing times, only approaching the width of the PTSB phase for ideal infinite probe time. In our experiments, we usually extend the probe time to the point when about ten percents of the atoms left which gives us a fairly well approximation of the width of the PTSB phase.
The decrease of the decay rate of the total atom number in the PTSB phase reminds of quantum Zeno effect (QZE) previously observed in cold atom experiments [3][4][5][6]. However, the difference between our experiments and those QZE experiments exists: QZE refers to the reduction of the rate of transferring from one state to a second state by the projection measurement of the second state. Since the perturbation is the projection measurement of the second state, the reverse-transfer probability from the second state to the first state and the occupation of the second state are both treated as zero. In comparison, PT -symmetric Hamiltonian experiments covers the crossover from the weak dissipation to the strong dissipation, and the transfer probability from the second to the first level is usually nonzero. Instead of the observation of the slow-down of the state transferring, the PTSB phase refers to the slow-down of the decay of the total atom number. In this sense, QZE can be treated as the extremely strong dissipation limit of our studies in which the strong atom loss can be treated as an irreversible projection measurement of the second level. It is also expected that QZE phenomena can expanded for more general case by using the dissipation based point of view instead of measurement based point of view, in which the analysis of PT phase transition can be used to study QZE-like phenomena in a pure dissipative system [7].