⇦Chinese Physics Letters, 2022, Vol. 39, No. 2, Article code 020502 Probing a Dissipative Phase Transition with a Trapped Ion through Reservoir Engineering M.-L. Cai (蔡明磊)1†, Z.-D. Liu (刘子都)1†, Y. Jiang (姜越)1†, Y.-K. Wu (吴宇恺)1, Q.-X. Mei (梅全鑫)1, W.-D. Zhao (赵文定)1, L. He (何丽)1, X. Zhang (张翔)2,1, Z.-C. Zhou (周子超)1,3, and L.-M. Duan (段路明)1* Affiliations 1Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China 2Department of Physics, Renmin University, Beijing 100084, China 3Beijing Academy of Quantum Information Sciences, Beijing 100193, China Received 7 December 2021; accepted 14 January 2022; published online 29 January 2022 †These authors contributed equally to this work.
*Corresponding author. Email: lmduan@tsinghua.edu.cn Citation Text: Cai M L, Liu Z D, Jiang Y et al. 2022 Chin. Phys. Lett. 39 020502    Abstract Dissipation is often considered as a detrimental effect in quantum systems for unitary quantum operations. However, it has been shown that suitable dissipation can be useful resources in both quantum information and quantum simulation. Here, we propose and experimentally simulate a dissipative phase transition (DPT) model using a single trapped ion with an engineered reservoir. We show that the ion's spatial oscillation mode reaches a steady state after the alternating application of unitary evolution under a quantum Rabi model Hamiltonian and sideband cooling of the oscillator. The average phonon number of the oscillation mode is used as the order parameter to provide evidence for the DPT. Our work highlights the suitability of trapped ions for simulating open quantum systems and shall facilitate further investigations of DPT with various dissipation terms.
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 DOI:10.1088/0256-307X/39/2/020502 © 2022 Chinese Physics Society Article Text Dissipation is ubiquitous in physical systems, and is often regarded as an undesired error source in quantum information science. However, well-controlled dissipation can also be helpful resources and has found applications in preparing many-body entangled states,[1,2] quantum information processing[3,4] and the study of nonequilibrium phase transitions.[1,5] In particular, dissipative phase transitions have been observed in various experimental systems such as Bose–Einstein condensate in optical cavities,[6,7] semiconductor microcavities[8,9] and superconducting circuits.[10–12] However, due to the experimental difficulty in harnessing the dissipation, all these experiments utilize the intrinsic dissipation in the system which can not be tuned. A goal that remains outstanding is to demonstrate a dissipative phase transition (DPT) through reservoir engineering to generate a controlled and suitable dissipation. The trapped ion system makes a desirable platform for studying engineered DPT. As one of the earliest physical systems and a leading one for quantum computing, trapped ions support accurate and coherent manipulation of the quantum states,[13,14] and can be well isolated from the environment to provide low intrinsic decoherence.[15] Through optical pumping, dissipation in the spin and the motional modes has also been demonstrated to initialize the system,[15] to prepare desired entangled states[16] and to simulate open system quantum dynamics.[17,18] Recently, it has been theoretically proposed that a DPT can be observed using two trapped ions,[19] with one ion and a collective oscillation mode forming a quantum Rabi model (QRM),[20,21] and the second ion being laser cooled to provide a controllable dissipation to the bosonic oscillation mode. Despite being a small system, suitable thermodynamic limit of large number of excitations can be defined as the ratio between the spin and the bosonic mode frequencies increases,[19,22] thus allows nonanalytical change across the phase transition point. In this Letter, we propose and experimentally demonstrate a simplified model using only one trapped ion and one oscillator mode, with interleaved pulse sequences of coherent drive and dissipation on the system as shown in Fig. 1. The system can approach a steady state under these two competing effects and, depending on their relative strength, the steady state can have vanishing phonon number or be strongly driven to high phonon populations to break the $Z_2$ symmetry, thus allows a second-order DPT in the intermediate parameter regime.[19,23] Experimental Scheme. In this experiment, the coherent drive is governed by a QRM Hamiltonian[21] $$ \hat{H}_{\mathrm{QRM}}=\frac{\omega_{\rm a}}{2} \hat{\sigma}_{z}+\omega_f \hat{a}^† \hat{a}+\lambda\left(\hat{\sigma}_{+}+\hat{\sigma}_{-}\right)\left(\hat{a} +\hat{a}^†\right),~~ \tag {1} $$ where $\hat{a}^†$ ($\hat{a}$) is the bosonic mode creation (annihilation) operator, $\hat{\sigma}_+$ ($\hat{\sigma}_-$) is the spin raising (lowering) operator; $\omega_{\rm a}$, $\omega_f$ and $\lambda$ are the spin transition frequency, the bosonic mode frequency and the coupling strength between the two subsystems, respectively. This model has been widely studied through quantum simulation in many experimental platforms[21,24–27] including trapped ions.[21,24] In this work, we consider a single ${^{171}\mathrm{Yb}^+}$ ion in a linear Paul trap [for further details about the setup, see the Supplementary Materials (SM)]. The spin is encoded in the $\left|\downarrow\right\rangle =\left|{}^{2}S_{1/2},F=0,m_F=0\right\rangle $ and the $\left|\uparrow\right\rangle =\left|{}^{2}S_{1/2},F=1,m_F=0\right\rangle $ levels of the ion with atomic transition frequency $\omega_0=2\pi\times 12.6\,{\rm GHz}$, and the bosonic mode is represented by a radial oscillation mode with trap frequency $\omega_{\mathrm{m}}=2\pi\times 2.35\,{\rm MHz}$. We first apply Doppler cooling (DC) to bring the ion into the Lamb–Dicke regime.[15] Then we shine bichromatic Raman laser beams onto the ion to form two pairs of Raman transitions with detuning $\delta_{\rm b}$ ($\delta_{\rm r}$) from the blue (red) motional sideband of the oscillation mode. When the two pairs of Raman transitions have the same sideband Rabi frequency $\varOmega_{\mathrm{SB}}$, we get an effective QRM Hamiltonian by identifying $\omega_{\rm a}=(\delta_{\rm b}+\delta_{\rm r})/2$, $\omega_{\rm f}=(\delta_{\rm b}-\delta_{\rm r})/2$ and $\lambda=\varOmega_{\mathrm{SB}}/2$ in an interaction picture with $\hat{H}_0=(\omega_0-\omega_{\rm a})\hat{\sigma}_z/2+(\omega_{\mathrm{m}} -\omega_f)\hat{a}^†\hat{a}$.[24] For the calibration of the model parameters $\delta_{b\,(r)}$ and $\varOmega_{\mathrm{SB}}$, please see the SM.
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Fig. 1. Experimental scheme. (a) An illustration of the experimental sequence in the quantum circuit model. The initial state can be arbitrarily chosen because the final steady state is independent of this choice. Here we initialize the qubit state to $\left|\downarrow\right\rangle $ by optical pumping. Then we apply $N$ rounds of alternating coherent drive (unitary evolution $U_{\rm R}$ under the QRM Hamiltonian) and dissipation (unitary evolution $U_{\rm C}$ under the red sideband driving sandwiched by two optical pumping stages) to bring the system into a steady state. Finally, we measure the average phonon number $\langle a^† a \rangle$ of the phonon steady state. (b) The complete pulse sequence. Two types of initial phonon states are used: a thermal state after Doppler cooling (DC), or a phonon ground state after additional sideband cooling (SBC). To measure the average phonon number in the final steady state, we apply a probe laser beam on the blue sideband and detect the spin state to fit the phonon population[15] (see the SM for details).
The intrinsic phonon decoherence rate is estimated to be around $0.2\,{\rm kHz}$ in our system. In order to engineer a strong and controllable dissipation in the system, we employ the sideband cooling process[15] where a laser pulse resonant to the red-sideband transition is sandwiched by two optical pumping stages of the ions to reset the spin state to $\left|\downarrow\right\rangle $. In an interaction picture with $\hat{H}_0^\prime=\omega_0\hat{\sigma}_z/2+\omega_{\mathrm{m}}\hat{a}^†\hat{a}$, the resonant driving on the red sideband can be represented by $\hat{H}_{c}=\varOmega_{c}\left(\hat{a} \hat{\sigma}_{+} +\hat{a}^† \hat{\sigma}_{-}\right)/2$, hence after time $\tau_{\rm c}$, an initial state with $n$ phonons will evolve into $\left|\downarrow\right\rangle \left|n\right\rangle \to \cos (\sqrt{n}\varOmega_{\rm c} \tau_{\rm c}/2)\left|\downarrow\right\rangle \left|n\right\rangle - i\sin (\sqrt{n}\varOmega_{\rm c} \tau_{\rm c}/2) \left|\uparrow\right\rangle \left|n-1\right\rangle $. Now after resetting the spin state again through optical pumping, the probability to reduce a phonon is $\sin^2 (\sqrt{n}\varOmega_{\rm c} \tau_{\rm c}/2)\approx n \varOmega_{\rm c}^2\tau_{\rm c}^2/4$ assuming $n\lesssim 1/(\varOmega_{\rm c}\tau_{\rm c})^2$ (see the SM for further discussion when this assumption is broken down), which resembles a phonon damping term with the Lindblad operator $\hat{L}=\varOmega_{\rm c}\sqrt{\tau_{\rm c}}\hat{a}/2$. Therefore this sideband cooling mechanism can be modeled as a master equation $\dot{\rho}_{\rm m}=\hat{L}\rho\hat{L}^†-\{\hat{L}^†\hat{L},\rho_{\rm m}\} / 2$, where $\rho_{\rm m}$ is the reduced density matrix in the phonon subspace, together with a reset of the spin state to $\left|\downarrow\right\rangle $ after each cycle. This process offers a much stronger dissipation than the intrinsic one for both the spin and the bosonic modes with extraordinary controllability, thus allows us to explore the rich DPT phenomena. Note that small violation of the above approximation condition will slightly decrease the cooling rate for high-phonon-number states, but it shall not change the qualitative behavior of the phase transition. Through the alternating application of the coherent drive and the artificial dissipation, the system is expected to reach a steady state such that the observables no longer change as we increase the number of cycles. Throughout this work, we consider the average phonon number in the bosonic mode as the order parameter to indicate the phase transition. It can be measured by probing the blue motional sideband (note that at the end of the preceding sideband cooling stage, we have already reset the spin state to $\left|\downarrow\right\rangle $) and detecting the spin state to fit the phonon population,[15] as sketched in Fig. 1(b). Also note that the steady state is expected to be independent of the choice of the initial state. We consider two possible initial states, where the phonon state can be either a thermal state generated from Doppler cooling (DC), or the ground state from Doppler cooling followed by sideband cooling (DC+SBC). Experimental Results. As shown in many previous works (see, e.g., Refs. [19,24]), in such a finite-component system, a thermodynamic limit can be defined as $R\equiv\omega_{\rm a}/\omega_{\rm f}=(\delta_{\rm b}+\delta_{\rm r})/(\delta_{\rm b}-\delta_{\rm r})$ approaches infinity to allow a large number of excitations. We start from $R=25$ in Fig. 2 by setting $\delta_{\rm b}=2\pi\times26\,{\rm kHz}$ and $\delta_{\rm r}=2\pi\times24\,{\rm kHz}$. We fix the duration of the coherent driving stage in each cycle to be $\tau=20\,{\rm µ s}$ and vary the sideband Rabi frequency $\varOmega_{\mathrm{SB}}$ to study the phase transition. As for the dissipation stage, we drive the red sideband at the sideband Rabi frequency $\varOmega_{\rm c}=2\pi\times 20\,{\rm kHz}$ for $\tau_{\rm c}=5\,{\rm µ s}$, which, together with the two $3\,{\rm µ s}$ optical pumping pulses and the idle time in between, makes up the total $\tau_{\rm d}=13\,{\rm µ s}$ duration.
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Fig. 2. Dynamics and steady state properties at the ratio $R=25$. We set the experimental parameters $\delta_{\rm b}=2\pi\times26\,{\rm kHz}$ and $\delta_{\rm r}=2\pi\times24\,{\rm kHz}$, which correspond to a ratio $R\equiv\omega_{\rm a}/\omega_{\rm f}=(\delta_{\rm b}+\delta_{\rm r})/(\delta_{\rm b}-\delta_{\rm r})=25$. (a) The dynamics of the system approaching the steady state indicated by the average phonon number. The initial phonon state is a thermal state prepared by Doppler cooling. The horizontal axis is the number of rounds of coherent driving and dissipation, as indicated by the inset. In each round, the QRM Hamiltonian is applied for $\tau=20\,{\rm µ s}$ with a sideband Rabi frequency $\varOmega_{\mathrm{SB}}=2\pi\times9.0\,{\rm kHz}$, and the $\tau_{\rm d}=13\,{\rm µ s}$ sideband cooling process consists of a $\tau_{\rm c}=5\,{\rm µ s}$ driving on the red sideband at $\varOmega_{\rm c}=2\pi\times 20\,{\rm kHz}$ together with the optical pumping and the idle stages. (b) The same plot as (a) for an initial phonon ground state prepared by additional sideband cooling. Note the two SBCs in the inset have different meanings, the former SBC means a multi-pulse sequence for ground state cooling and the latter SBC means a single pulse operation for dissipation. Each dot represents one measured data with the error bar indicating one standard deviation (1 S.D.). The fitting line follows $\bar{n}=Ae^{-N/N_0}+B$, where $A$, $B$ and $N_0$ are fitting parameters, with the shaded region showing a 0.9 confidence level band of the fitting. In both cases, the average phonon number approaches roughly the same value of about $\bar{n}=3.2$ after about $N=50$ cycles, suggesting a steady state independent of the initial state. (c) and (d) The steady state average phonon number for the two initial states as in (a) and (b) respectively versus the sideband Rabi frequency $\varOmega_{\mathrm{SB}}$ in the QRM Hamiltonian. The other parameters are unchanged. We fix the number of rounds to 200 to ensure that the final average phonon number has saturated. The colored dots with error bars representing 1 S.D. are the experimental results and the colored lines are from the numerical simulation. The vertical dashed line indicates the numerically computed phase transition point (see the SM for details). A crossover between the two phases can be clearly observed.
In Figs. 2(a) and 2(b), we present two examples with $\varOmega_{\mathrm{SB}} = 2\pi\times9.0\,{\rm kHz}$ for how the average phonon number approaches the steady state value, starting from a thermal state and the ground state of the phonon mode, respectively. Regardless of the initial state, the steady state average phonon number saturates at around $\bar{n}=3.2$, thus verifies that our engineered dissipative term can lead to a unique steady state. We further fit the data by an exponential decay $\bar{n}=Ae^{-N/N_0}+B$, where $N$ is the number of rounds while $A$, $B$ and $N_0$ are fitting parameters. The fitting results are shown as the central curves with the shaded areas representing a 0.9 confidence level band. It is evident that in these two examples, the system is already reasonably close to the steady state (or at least the value of the average phonon number, which is the relevant observable for the DPT) after about 50 rounds. The saturation rate also depends on the driving and the cooling parameters, hence in the next step when we scan $\varOmega_{\mathrm{SB}}$ to study the change in the average phonon number of the steady state, we increase the number of rounds to 200 to ensure saturation. In Figs. 2(c) and 2(d) we plot the average phonon number in the steady state versus the sideband Rabi frequency $\varOmega_{\mathrm{SB}}$ in the QRM, again for the initial thermal state and the phonon ground state, respectively. The measured steady state phonon numbers match well with the numerical simulations which have already included the decoherence effect of motion (see the SM for detailed discussion) and indicate a smooth crossover between the two phases. The vertical dashed line indicates the phase transition point from the numerical calculation (see the SM for details). For a real transition from the normal phase to the superradiance phase[19] across this line, we expect that the average phonon number would have a nonanalytical increment when the ratio $R$ approaches infinity. This is similar to the quantum phase transition in the closed QRM,[24] where the nonanalytical behavior happens on the ground state rather than the steady state. However, due to the finite ratio $R=25$ we adopted here, we can only see a smooth crossover behavior. To acquire further evidence of the DPT, we study the finite frequency scaling behavior under the increasing ratio $R$, which corresponds to the system size in the conventional thermodynamic limit. Here we fix $\delta_{\rm b}-\delta_{\rm r}=2\pi\times 2\,{\rm kHz}$ and increase $\delta_{\rm b}+\delta_{\rm r}$ up to $2\pi\times 200\,{\rm kHz}$ and study the scaling behavior of the average phonon number, as shown in Fig. 3. Again we consider two different initial states and find that the steady state properties are unaffected. The experimental results agree well with the numerical prediction within about 1 S.D. The main error source of the deviation can be referred to the SM. Again, the numerical simulation results have already considered the decoherence effect of motion. As we can see, the change in the average phonon number becomes sharper with increasing $R$. However, the phonon number shows a nonmonotonic behavior with increasing $R$ near the critical point, this is further investigated in the SM. Moreover, the numerical simulation shows that for $g$ below the critical point $g_{\rm c}$, the average phonon number converges to a finite values as we increase $R$, while for $g>g_{\rm c}$, the steady state phonon number diverges in the limit $R \to \infty$. These behaviors indicate a DPT in the thermodynamic limit. More numerical and experimental results are presented in the SM to prove the existence of phase transition in this model.
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Fig. 3. Average phonon number under the increasing ratio $R$. Here we repeat the measurement for the steady state average phonon number in Fig. 2 for different ratio parameters $R=50,\,75,\,100$ by keeping $(\delta_{\rm b}-\delta_{\rm r})/2\pi=2\,{\rm kHz}$ fixed while increasing $(\delta_{\rm b}+\delta_{\rm r})/2\pi$ from $50\,{\rm kHz}$ to $100\,{\rm kHz}$, $150\,{\rm kHz}$ and $200\,{\rm kHz}$. Under each $R$, we measure the average phonon number in the steady state versus the sideband Rabi frequency $\varOmega_{\mathrm{SB}}$ starting from (a) a thermal state or (b) the phonon ground state similar to Fig. 2. The horizontal axis is the dimensionless coupling $g\equiv 2\varOmega_{\mathrm{SB}}/\sqrt{\delta_{\rm b}^2-\delta_{\rm r}^2}$, and the vertical dashed line indicates the numerically computed phase transition point at $g_{\rm c} \approx 1.35$. The colored dots are experimental data with error bars representing 1 S.D. The colored curves are the results from numerical simulations under the same parameters. The measured data agree with the numerical results within about 1 S.D., and we can see that the sharpness of the curve increases with the ratio parameter. The similarity between (a) and (b) again verifies that the steady states are independent of the initial states.
Finally we look into the influence of the cooling rate, which is enabled by our engineered reservoir, at the fixed frequency ratio $R=50$. With other parameters unchanged, the cooling rate can be easily tuned by varying the red sideband Rabi frequency $\varOmega_{\rm c}$. Here we start from the phonon ground state for simplicity since the steady state properties have been verified above to be independent of the initial state. For various $\varOmega_{\rm c}$ from $2\pi \times 10\,{\rm kHz}$ to $2\pi \times 20\,{\rm kHz}$, we measure the average steady-state phonon number versus the dimensionless coupling strength $g\equiv 2\varOmega_{\mathrm{SB}}/\sqrt{\delta_{\rm b}^2-\delta_{\rm r}^2}$, with the experimental results shown in Fig. 4. Clearly, with larger cooling rates, the transition point (crossover region for finite $R$) moves toward higher $g$ due to the competition between the coherent driving and the dissipation terms.
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Fig. 4. Average phonon number under different cooling rates. Here we fix the ratio $R=50$ and tune the cooling rate by varying the red sideband Rabi frequency $\varOmega_{\rm c}$. With different $\varOmega_{\rm c}$, we measure the change of the average steady-state phonon number versus the dimensionless coupling strength $g$. The colored dots are experimental data with error bars representing 1 S.D. The colored curves are the results from numerical simulations under the same parameters. The measured data agree with the numerical results within about 1 S.D., and we can see that as the cooling rate increases, the transition point (crossover region) is shifted to higher $g$ due to the competing effect between the driving and the dissipation terms.
Discussion and Conclusions. To sum up, we propose and demonstrate a DPT model with an artificially engineered reservoir using a single trapped ion. We first verify the steady states of the ion motion and clearly observe a crossover between two different phases of the phonon mode. Second, we implement the finite frequency scaling to study the DPT where the crossover becomes sharper with an increasing frequency ratio. Finally, we observe the behavior of the crossover under different dissipative rates via tuning the sideband cooling rate. To our knowledge, our experiment is the first experimental probe of a DPT through reservoir engineering with adjustable dissipation rates. It shows the advantage of strong controllability of the trapped-ion system for the simulation of open quantum systems and shall facilitate further investigations of DPT under various engineered dissipation terms. Also, the demonstrated scheme here is a universal method very similar to those used in dissipative nonclassical-state engineering[28] and the pulsed-CPT scheme in quantum metrology.[29,30] In this sense, the scheme can be well adapted to other research fields where a controlled dissipation is desired. The observation of a critical phenomenon near the transition point and experimentally extracting the critical exponent are basically limited by the frequency ratio we can achieve under the current experimental conditions. With the suppression of the fluctuation of the experimental parameters ($\delta_{\rm b}$, $\delta_{\rm r}$), we can further decrease the denominator of the ratio ($\delta_{\rm b}-\delta_{\rm r}$) without too much deviation, hence an observation of the critical phenomena becomes possible. This work was supported by the Beijing Academy of Quantum Information Sciences, the National Key Research and Development Program of China (Grant No. 2016YFA0301902), Frontier Science Center for Quantum Information of the Ministry of Education of China, and Tsinghua University Initiative Scientific Research Program. Y.-K. W. acknowledges support from Shuimu Tsinghua Scholar Program and International Postdoctoral Exchange Fellowship Program (Talent-Introduction Program). References Quantum states and phases in driven open quantum systems with cold atomsPreparation of entangled states by quantum Markov processesQuantum computation and quantum-state engineering driven by dissipationQuantum memories based on engineered dissipationDissipative phase transition in a central spin systemReal-time observation of fluctuations at the driven-dissipative Dicke phase transitionDynamical phase transition in the open Dicke modelProbing a Dissipative Phase Transition via Dynamical Optical HysteresisSignatures of a dissipative phase transition in photon correlation measurementsObservation of the Photon-Blockade Breakdown Phase TransitionObservation of a Dissipative Phase Transition in a One-Dimensional Circuit QED LatticeObservation of the Crossover from Photon Ordering to Delocalization in Tunably Coupled ResonatorsHigh-Fidelity Quantum Logic Gates Using Trapped-Ion Hyperfine QubitsHigh-Fidelity Universal Gate Set for Be 9 + Ion QubitsQuantum dynamics of single trapped ionsDissipative production of a maximally entangled steady state of two quantum bitsAn open-system quantum simulator with trapped ionsQuantum simulation of dynamical maps with trapped ionsDissipative phase transition in the open quantum Rabi modelQuantum Rabi Model with Trapped IonsQuantum Simulation of the Quantum Rabi Model in a Trapped IonCritical dynamical properties of a first-order dissipative phase transitionSpectral theory of Liouvillians for dissipative phase transitionsObservation of a quantum phase transition in the quantum Rabi model with a single trapped ionExperimentally simulating the dynamics of quantum light and matter at deep-strong couplingPhotonic Realization of the Quantum Rabi ModelStrong Light-Matter Coupling in Subwavelength Metal-Dielectric Microcavities at Terahertz FrequenciesQuantum harmonic oscillator state synthesis by reservoir engineeringCoherent Population Trapping of a Single Nuclear Spin Under Ambient ConditionsCoherent population trapping with a controlled dissipation: applications in optical metrology
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