Abstract
Open physical systems with balanced loss and gain, described by nonHermitian paritytime \(\left( {{\cal P}{\cal T}} \right)\) reflection symmetric Hamiltonians, exhibit a transition which could engender modes that exponentially decay or grow with time, and thus spontaneously breaks the \({\cal P}{\cal T}\)symmetry. Such \({\cal P}{\cal T}\)symmetrybreaking transitions have attracted many interests because of their extraordinary behaviors and functionalities absent in closed systems. Here we report on the observation of \({\cal P}{\cal T}\)symmetrybreaking transitions by engineering timeperiodic dissipation and coupling, which are realized through statedependent atom loss in an optical dipole trap of ultracold ^{6}Li atoms. Comparing with a single transition appearing for static dissipation, the timeperiodic counterpart undergoes \({\cal P}{\cal T}\)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 \({\cal P}{\cal T}\)symmetric quantum systems.
Introduction
A nonHermitian paritytime reflection symmetric (\({\cal P}{\cal T}\)symmetric) Hamiltonian, that is invariant under combined parity \(\left( {\cal P} \right)\) and timereversal \(\left( {\cal T} \right)\) operations, has been considered as a natural extension of the conventional Hermitian quantum theory to describe an open quantum system with balanced loss and gain^{1,2,3}. \({\cal P}{\cal T}\)symmetric Hamiltonians exhibit many interesting behaviors^{4,5,6,7,8,9}, in which a key property is \({\cal P}{\cal T}\)symmetrybreaking transitions that occur at an exceptional point^{10,11} – a point in the parameter space where two resonant modes of the Hamiltonian become degenerate. A number of seminal studies^{1,12} have shown that the eigenvalues of the Hamiltonian are real in one side of the transition, allowing the \({\cal P}{\cal T}\)symmetric (PTS) phase, while complex eigenvalues appear in the other side with the \({\cal P}{\cal T}\)symmetric broken (PTSB) phase. In recent years, \({\cal P}{\cal T}\)symmetric Hamiltonians have been realized in balanced gain and loss systems with various setups, such as mechanical oscillators^{13}, optical waveguides^{14,15}, optical resonators^{16}, microwave cavities^{17}, lasers^{18}, and optomechanical systems^{19}, or in a statedependent pure lossy system in which the lossy Hamiltonian H′ could be mapped to a \({\cal P}{\cal T}\)symmetric Hamiltonian H_{PT} for passive \({\cal P}{\cal T}\)symmetry breaking^{20,21,22}.
\({\cal P}{\cal T}\)transitions can be induced either by increasing the strength of dissipation or by tuning the periodicity of the dissipation, known as static or Floquet method respectively. By driving the system passing the exceptional point, it is predicted that \({\cal P}{\cal T}\)transitions can reduce the overall dissipation of the system^{20,23,24} and allow topological structures around the exceptional point^{19,25}. Floquet method is particular interesting because timeperiodic modulation can break the continuous time translation symmetry, providing an enriched phase diagram with many fascinating features^{20}.
Here we present an experimental study of \({\cal P}{\cal T}\)symmetrybreaking transitions induced by timeperiodic dissipation or coupling in a twospin system of ultracold atoms. Our experimental results verify that \({\cal P}{\cal T}\)symmetry breaking and restoring transitions can occur by tuning either dissipative or coupling frequency even at vanishingly small dissipation strength. We further map the Floquet \({\cal P}{\cal T}\)phase diagrams by tracing the atom loss of each spin state, and observe the multiphoton resonances and the power broadening associated to the PTSB phase.
Results
\({\cal P}{\cal T}\) transition with static dissipation
We prepare a noninteracting Fermi gas of ^{6}Li atoms at the two lowest ^{2}S_{1/2} hyperfine levels^{26,27}, labeled as ↑〉 and ↓〉. These twospin states are coupled by a radiofrequency (RF) field with a coupling strength of J. A resonant optical beam is used to excite the atoms from ↓〉 to the 2P excited state \(\left e \right\rangle\) and generates the atom loss in ↓〉 with a rate of Γ (Fig. 1a). The Hamiltonian for this dissipative twospin system is given by
where H_{PT}(t) = Jσ_{x} + iΓ(t)σ_{z}/2 is a \({\cal P}{\cal T}\)symmetric Hamiltonian, and I is the unit matrix. The system is prepared with all atoms in ↑〉, and evolves for a time of t. Then the intrap atom numbers, \(n_ \uparrow ^\prime (t)\) and \(n_ \downarrow ^\prime (t)\), are measured by the doubleshot absorption imaging of the twospin states, giving the total atom number \(n{\prime}(t) = n_ \uparrow ^\prime + n_ \downarrow ^\prime\). We map n′(t) to a scaled, normalized atom number n(t) associated to H_{PT}, and then use n(t) to characterize the \({\cal P}{\cal T}\)transitions [See Methods].
For static dissipation, Γ(t) is a constant value of Γ_{0}. When Γ_{0}/J < 2, the eigenvalues of H_{PT} are real values of \(\pm \sqrt {J^2  {\mathrm{\Gamma }}_0^2{\mathrm{/}}4}\), and n(t) oscillates at frequency \(\pi {\mathrm{/}}\sqrt {J^2  \Gamma _0^2{\mathrm{/}}4}\). The \({\cal P}{\cal T}\) transition occurs at Γ_{0}/J = 2 where the oscillation period diverges. When Γ_{0}/J > 2, the eigenvalues of H_{PT} become complex numbers, and one of the eigenmode exponentially grows [See Supplementary Note 1]. These predictions are verified in our experiments [See Supplementary Fig. 1], and the measured exceptional point agrees with the theoretical model very well [See Supplementary Fig. 2].
The static dissipation experiment is related to the previous quantum zeno effect (QZE) experiments of ultracold atoms with strongloss induced measurement^{28,29,30,31}. In those experiments, 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. Due to a strongloss induced irreversible measurement, the reversetransfer probability from the second state to the first one is treated as zero as well as the occupation of the second state. However, in our dissipation experiment, the transfer probability from the second to the first level is nonzero, and the PTSB phase refers to the slowdown of the decay of the total atom number. Thus, the results cannot be explained purely in terms of QZE, except for the limit of an extremely strong dissipation case, in which the strong atom loss can be treated as an irreversible projection measurement of the second level.
Observation of the \({\cal P}{\cal T}\)transitions with timeperiodic driving
Floquet method enriches the phase diagram of a \({\cal P}{\cal T}\)symmetric system by periodically modulating Hamiltonian H(t) = H(t + T). Previously, the extraordinary structure of the phase diagram has been theoretically predicted^{20,23}, but has never been verified experimentally due to the difficulty of precisely controlling the timedependent dissipation. In our experiment, the optical and RF field provide versatile tools to manipulate the atom loss and coupling of spin levels, so that two types of Floquet Hamiltonians could be implemented: spindependent timeperiodic dissipation and timeperiodic coupling between two spins.
We first study timeperiodic dissipation, in which a squarewave resonant beam is applied to generate timedependent dissipation of the atoms in an optical trap. The coupling strength J is fixed and the dissipation strength is modulated between Γ and 0 with a frequency of Ω_{d} (Fig. 1b). In contrast with the static dissipation, \({\cal P}{\cal T}\)transitions under timeperiodic dissipation depends on the modulation frequency and can occur at vanishingly small dissipation strength with infinite numbers of the resonance peaks [See the Supplementary Note 2]. The primary resonance peak of the \({\cal P}{\cal T}\)transition appears Ω_{d}/J = 2, where the transition behavior of n(t) in the weak dissipation limit \({\mathrm{\Gamma /}}J = 0.2 \ll 2\) is shown in Fig. 1d–f. When Ω_{d}/J is tuned to the PTSB phase, n(t) increases exponentially, in contrast with the PTS phase where n(t) exhibits bounded oscillation n(t) ∝ sin[(Ω_{d} − 2J)t].
In the above cases, the PTSB phases have been observed even when the eigenvalues of H(t) are real all the time. Such \({\cal P}{\cal T}\)symmetry breaking can be determined by the nonunitary timeevolution operator G_{PT} [See Methods], which has two eigenvalues \(\mu _ \pm \propto e^{  i\epsilon _ \pm t}\). \(\epsilon _ \pm\) is the quasienergies of the effective Floquet Hamiltonian [See Supplementary Note 3]. If the magnitude of μ_{±} are equal, \(e^{i\epsilon _ \pm T}\) is a pure phase factor and \(\epsilon _ \pm\) must be the real numbers, which indicate a PTS phase. On the contrary, the unequal magnitude of μ_{±} denote the complex values of \(\epsilon _ \pm\) representing a PTSB phase.
For timeperiodic coupling in the weak dissipation limit, Γ/J is constant and J(t) is modulated at the frequency Ω_{c} (Fig. 1c) and the the primary resonance peak of the PTSB phase is at Ω_{c}/J = 1 (Fig. 1g). When Ω_{c}/J is tuned to the primary resonance region, n(t) shows the similar behavior as timeperiodic dissipation, where the exponential increase of n(t) appears (Fig. 1i), while the PTS phase exhibits the bound oscillation which could be parameterized by n(t) ∝ sin[(Ω_{c} − J)t] in the weak dissipation limit (Fig. 1h). The measurements of that primary resonance of the PTSB phase verify that \({\cal P}{\cal T}\)symmetry transitions can happen with an arbitrary small dissipation under timeperiodic driving. Furthermore, there exist infinite numbers of transitions induced by multiphoton resonances which are investigated as follows.
Multiphoton resonances with timeperiodic dissipation
For \({\cal P}{\cal T}\)transitions with timeperiodic dissipation, there exist infinite numbers of the PTSB phases induced by multiphoton process in a nonHermitian Rabi model^{20}. Their widths have been predicted to decrease with the index number of the the resonances. For a squarewave modulation, the widths of the PTSB phases in the weak dissipation limit are
where Ω_{n} = 2J/n is the resonance peak under zero dissipation with n as the odd number 1, 3, 5 …. Γ is the magnitude of the squarewave dissipation [See Supplementary Note 2]. Figure 2 show the broadening of the PTSB phases for one (primary), three, and fivephoton resonances. To measure the width of the resonances, the residual atom number n(t_{f}, Ω) is probed at a fixed time point t_{f} for various modulation frequencies Ω [See Supplementary Note 4]. It is noted that, for the purpose of mapping the phase diagram, it is ideal to choose t_{f} as large as possible so that n(t_{f}) can reflect the longterm dynamics. However, because we map a pure lossy system to a \({\cal P}{\cal T}\)symmetric Hamiltonian, t_{f} must be remained in a finite range for the reasonable signaltonoise ratio of the unscaled atom number n′(t) [See Supplementary Note 5]. In our experiment, we choose t_{f} to be larger than several oscillation periods so that n(t_{f}, Ω) can present the trend of increasing in the PTSB phase. As shown in Fig. 2a, the half width at half maximum (HWHM) of the primary resonance is proportional to the strength of timeperiodic dissipation. Such behavior is the nonHermitian analog of the resonance broadening induced by the Bloch–Siegert shifts of a strong driving Hermitian system. The width of the residual atom number also depends on the probe time and gets narrower for the longer probing times, which approaches the width of the PTSB phase predicted by theoretical calculations. For the finite probe time, the width is a qualitatively measure of the dependence of the PTSB regime on the dissipation strength [See Supplementary Note 4].
Comparing with the Hermitian system where the multiphoton resonance is difficult to be observed at the weak driving limit, in a nonHermitian system, the timeperiodic dissipation significantly broadens the width of the PTSB phase so that the multiphoton resonance could be observed clearly with the weak dissipation. The widths of the three(Fig. 2b), five (Fig. 2d) photon resonances, agree with the theoretical phase diagram very well. At the exact resonant frequencies, the exponentially increase of n(t) with a very small dissipation strength are recorded in Fig. 2c (threephoton) and Fig. 2e (fivephoton) manifesting the PTSB phase.
Multiphoton resonances with timeperiodic coupling
The phase diagram of timeperiodic coupling is studied by modulating the coupling between the twospin states. The resonance widths of the PTSB phases are given by
where Ω_{n} = 2J/n is the resonant peak with the even integer n = 2, 4, 6 .... Eq. (3) indicates that the PTSB phases of the timeperiodic coupling have the wider width than that of timeperiodic dissipation, which scales with the multiphoton index number n as 1/n instead of 1/n^{2} for timeperiodic dissipation [See Supplementary Note 2]. The first four multiphoton resonances are shown in Fig. 3a, where the widths of the PTSB phases increase with dissipation. The increasing of n(t) at the resonance frequencies are fitted by the exponential curves in Fig. 3b to verify the PTSB phase.
It is interesting to find that, the resonant peaks Ω_{n} = 2J/n of the timeperiodic dissipation have odd integers n = 1, 3, 5 ..., but the timeperiodic coupling ones have even integers n = 2, 4, 6 .... The odd or even rule can be explained by a simple picture. For example, in the timeperiodic coupling case of the weak loss limit, all atoms are initially in the lossless (up) state and the coupling is turned on for nπ/(2J). If n is even, then the coupling is on exactly for the time of multiple 2π Rabi pulses such that all atoms are back to the up state, and will remain in this state for the next half cycle of couplingoff. During this half cycle, no atoms are lost so that the scaled total atom number increase because the scaling assumes equal loss in both spin states. Overall, the system spends more time in the lossless (up) state when n is even numbers. This is not the case for n to be odd, where on average the system spends the same amount of time in both up and down states because the coupling is turned on for the odd numbers of π pulse.
The similar picture is also applied to the timeperiodic dissipation case, where the atom loss is turned on for a time duration of nπ/(2J) with n being odd numbers. This amount of time is odd numbers of π Rabi pulse. With the proper choice of phase, the atom loss can only present when the majority of the atoms are in the lossless (loss) state so that the scaled total atom number increase (decrease) exponentially. Either increasing or decreasing depends on the phase of the dissipation, which corresponds the two eigenstates in the PTSB phase. In the experiments, we usually optimize the phase of the timeperiodic modulation to obtain the strongest signal, resulting from ensuring the largest overlap between the initial state and the slowly decaying eigenmode of the corresponding Floquet Hamiltonian. For more general initial states, such as a balanced mixture of up and down state, as long as there is a nonzero overlap between the initial state and the slow mode, the \({\cal P}{\cal T}\)symmetrybreaking signatures of the slowdecaying will be visible in the longtime limit.
Discussion
The lasercooled ultracold atoms provide a clean and well controllable platform for studying \({\cal P}{\cal T}\)symmetric Hamiltonians. Previously, phase transitions observed in cold atom systems are usually driven by tunable interparticle interactions and, in principle, occur only in the thermodynamic limit. However, \({\cal P}{\cal T}\)symmetric breaking transitions, in contrast, can occur in a single twolevel system with localized loss. The fate of the former transitions in the presence of such a loss has not been fully understood, as is the fate of latter transition in the presence of interparticle interactions. Investigating the interplay between these two classes of transitions will require quantum simulators with tunable interparticle interactions and engineered statedependent dissipation, both of which can be realized with certain species of ultracold atoms, such as fermionic ^{6}Li atoms used in this experiment. As a starting point of this route, we apply statedependent dissipation to ultracold ^{6}Li atoms to study \({\cal P}{\cal T}\)transitions in a twolevel system. With the advantages of modulating the resonant optical and RF field as the versatile tools for timeperiodic driving, we could manipulate the atom loss as well as the coupling of spin levels, and experimentally map both timeperiodic dissipation and coupling. The phase diagram of the static dissipation and timeperiodic dissipation (coupling) are explored by tracing the timeevolution of the atoms. While the single exceptional point under static dissipation is determined as usual, our results verify remarkably rich phase diagrams with multiple Floquet \({\cal P}{\cal T}\)transitions associated to timeperiodic driving. It is shown that the PTSB phases can be induced by judiciously selected temporal profiles of statedependent dissipation or coupling with vanishingly small strength of the dissipation. The multiphoton resonant structure of \({\cal P}{\cal T}\)transitions are demonstrated. Such Floquet method thus provide an experimental platform to study timedependent \({\cal P}{\cal T}\)symmetric Hamiltonians.
Our system has potential to be extended to more complex situations: one is to study the topological phenomena associated to nonHermitian Hamiltonian and the other is to explore an interacting system with a vanishingly small, timemodulated dissipation. For the formal one, if we use the unresonant RF pulses to couple the spin levels, a detuning term will appear in the diagonal part of the Hamiltonian, and we can adiabatically encircle the exceptional points by changing the detuning and dissipation simultaneously to observe the topological phenomena associated to the nonHermitian systems^{19,25,32,33,34,35,36,37,38}. For the latter one, the interplay between the \({\cal P}{\cal T}\)transition and the BECBCS (Bose–Einstein condensate to Bardeen–Cooper–Schrieffer pairing) crossover can be investigated by sweeping the ultracold Fermi gas from the noninteracting limit (presented here) to the unitary, stronglyinteracting limit^{39}. This approach, where a singleparticle, statedependent loss is used in conjunction with strong interparticle interactions, provides exciting opportunities to explore physical phenomena in open manybody quantum systems.
Methods
Experimental system
We prepare a dissipative twolevel system with a noninteracting Fermi gas. ^{6}Li atoms are prepared in the two lowest hyperfine states, ↑〉 ≡ F = 1/2, m_{F} = 1/2〉 and ↓〉 ≡ F = 1/2, m_{F} = −1/2〉, in a magnetooptical trap. The precooled atoms are then transferred into a crossedbeam optical dipole trap made by a 100 Watt fiber laser. The bias magnetic field is swept to 330 G to implement an evaporative cooling^{26}. The trap potential is lowered to generate a final trap depth of 2.2 μK. In order to null the interaction between the two hyperfine states, the magnetic field is fast swept to 527.3 G, where the swave scattering length of the ↑〉 and ↓〉 states is zero^{27}. The lifetime of the noninteracting Fermi gas is about 20 s, which is three orders of magnitude longer than our typical experimental time. So when the dissipative optical field is absent, this noninteracting Fermi gas can be treated as a closed, twolevel quantum system. To prepare a single component Fermi gas in the ↑〉 state as the initial state, we apply a 5 ms optical pulse with −2π × 30 MHz detuning from the ↓〉 → ^{2}P_{3/2} transition to blow away atoms in the ↓〉 state. We typically have about N = 2.0 × 10^{5} atoms in a pure ↑〉 state at temperature T ≈ 0.8 μK and T/T_{F} ≈ 0.5 with T_{F} is the Fermi temperature.
To generate Rabi oscillation between the twospin states, we couple them via an RF field with frequency ω and coupling strength J. An optical beam resonant with the ↓〉 → ^{2}P_{3/2} transition is used to create the number dissipation (atom loss) in the ↓〉 state. The resonantphoton recoil energy of 3.5 μK is ~50% larger than the trap depth, so the atom that absorb a photon will leave the trap quickly, resulting a statedependent atom loss. The RF coupling strength J is measured in the absence of the dissipative optical field, while the atomnumber loss rate 2Γ is measured in the absence of the RF coupling. Figure 4a shows the Rabi oscillation with Rabi frequency 2J. Figure 4b shows the atom numbers \(n_ \downarrow ^\prime (t) = n_ \downarrow ^\prime (0){\mathrm{exp}}(  2{\mathrm{\Gamma }}t)\) with a constant dissipative optical field that only couples the ↓〉 state to the continuum. These measurements are used to calibrate the values of J and Γ for the dissipative twostate Rabi system.
Theoretical model
The dissipative twostate system is described by a nonHermitian Hamiltonian (ħ = 1)
where ω_{0} = 2π × 75.6 MHz is the hyperfine splitting at 527.3 G. When the RF driving is close to the resonance, that is ω ≈ ω_{0}, with the rotating wave approximation in the interacting picture, H(t) = −iΓ(t)/2 + H_{PT}(t), where the nonHermitian, \({\cal P}{\cal T}\)symmetric Hamiltonian is given by (ħ = 1) \(H_{{\mathrm{PT}}} = J\sigma _{\mathrm{x}} + i{\mathrm{\Gamma }}(t)\sigma _{\mathrm{z}}{\mathrm{/}}2 = {\cal P}{\cal T}H_{{\mathrm{PT}}}{\cal P}{\cal T}\), where \({\cal P} = \sigma _{\mathrm{x}}\) and \({\cal T} = \ast\) denotes complex conjugation operation. Starting with an initial state ψ(0)〉, the decaying atom numbers for the two states are given by \(n_\sigma ^\prime (t) \equiv \left {\left\langle \sigma \rightG\prime (t)\left {\psi (0)} \right\rangle } \right^2\) where
is the nonunitary timeevolution operator obtained via the timeordered product. It is also useful to define scaled atom number n_{σ}(t) = 〈σG_{PT}(t)ψ(0)〉^{2} where G_{PT}(t) is the corresponding timeevolution operator for H_{PT}(t). It follows that \(n_\sigma (t) = n_\sigma ^\prime (t) \times {\mathrm{exp}}\left( {{\int}_0^t {\kern 1pt} {\mathrm{\Gamma }}(t\prime )dt\prime {\mathrm{/}}2} \right)\). In a \({\cal P}{\cal T}\)symmetric system, the \({\cal P}{\cal T}\)symmetric phase is signaled by nondecaying, oscillatory n_{σ}(t) and the \({\cal P}{\cal T}\)broken phase is signaled by an exponentially increasing n_{σ}(t).
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
L.L. is a member of the Indiana University Center for Spacetime Symmetries (IUCSS). L.L. received supports from National Natural Science Foundation of China under Grant No. 11774436, Sun Yatsen University Discipline Construction Fund, Sun Yatsen University Three Major Construction Fund, Indiana University Collaborative Research Grant. Y.N.J. received NSF grant no. DMR1054020. J.Li. received supports from National Natural Science Foundation of China under Grant No. 11804406.
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L.L. and Y.N.J. conceived the idea and supervised the study. J.Li. and L.d.m. set up experiments and performed measurements. L.L. designed and supervised the experiments. A.K.H., J.Li. and Y.N.J. carried out theoretical modeling. J.Li. and J.Liu. analyzed the data. J.Li., L.L. and Y.N.J. contributed to writing the manuscript.
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Li, J., Harter, A.K., Liu, J. et al. Observation of paritytime symmetry breaking transitions in a dissipative Floquet system of ultracold atoms. Nat Commun 10, 855 (2019). https://doi.org/10.1038/s41467019085961
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