Observation of a quantum phase transition in the quantum Rabi model with a single trapped ion
M.-L. Cai, Z.-D. Liu, W.-D. Zhao, Y.-K. Wu, Q.-X. Mei, Y. Jiang, L. He, X. Zhang, Z.-C. Zhou, L.-M. Duan
OObservation of a quantum phase transition in the quantum Rabi model with a singletrapped ion
M.-L. Cai ∗ , Z.-D. Liu ∗ , W.-D. Zhao ∗ , Y.-K. Wu , Q.-X. Mei ,Y. Jiang , L. He , X. Zhang , , Z.-C. Zhou , , L.-M. Duan † Center for Quantum Information, Institute for InterdisciplinaryInformation Sciences, Tsinghua University, Beijing 100084, PR China Department of Physics, Renmin University, Beijing 100084, PR China and Beijing Academy of Quantum Information Sciences, Beijing 100193, PR China
ABSTRACT
Quantum phase transitions (QPTs) are usually associated with many-body systems in the ther-modynamic limit when their ground states show abrupt changes at zero temperature with variationof a parameter in the Hamiltonian. Recently it has been realized that a QPT can also occur in asystem composed of only a two-level atom and a single-mode bosonic field, described by the quantumRabi model (QRM). Here we report an experimental demonstration of a QPT in the QRM using atrapped ion. We measure the spin-up state population and the average phonon number of the ion astwo order parameters and observe clear evidence of the phase transition via adiabatic tuning of thecoupling between the ion and its spatial motion. An experimental probe of the phase transition in afundamental quantum optics model without imposing the thermodynamic limit opens up a windowfor controlled study of QPTs and quantum critical phenomena.
INTRODUCTION
Quantum phase transitions (QPTs) have become oneof the focuses of condensed matter physics. Unlike clas-sical phase transitions that occur at finite temperature, aQPT can occur at zero temperature under quantum fluc-tuations [1–3]. When a control parameter, such as theexternal magnetic field or the doping of a component,is scanned across a quantum critical point, the groundstate of the system changes abruptly, characterized by aspontaneous symmetry breaking or a change in the topo-logical order [2, 4].Studies of QPTs usually consider many-body systemsin the thermodynamic limit, with the particle number N approaching infinity [3]. However, it was recently realizedthat a QPT can also occur in a small system with onlytwo constituents, a two-level atom and a bosonic mode,described by the quantum Rabi model (QRM) [5–12]which is one of the simplest models of light-matter inter-actions. Its Hamiltonian can be expressed as (throughoutthis paper we set (cid:126) = 1 for simplicity)ˆ H QRM = ω a σ z + ω f ˆ a † ˆ a + λ (ˆ σ + + ˆ σ − ) (cid:0) ˆ a + ˆ a † (cid:1) , (1)where ˆ a † (ˆ a ) is the bosonic mode creation (annihila-tion) operator and ˆ σ + (ˆ σ − ) is the two-level system rais-ing (lowering) operator; ω a , ω f and λ are the atomictranstion frequency, the field mode frequency and thecoupling strength between the two subsystems, respec- ∗ These authors contribute equally to this work † Corresponding author: [email protected] tively. This model has been widely studied in multi-ple paramter regions with many experimental platforms.When | ω a − ω f | (cid:28) | ω a + ω f | and λ/ω f (cid:28) λ becomes comparable to ω a + ω f , theRWA breaks down leading to the ultra-strong couplingregime ( λ/ω f (cid:38) .
1) and deep-strong coupling regime( λ/ω f (cid:38)
1) [14]. Many exotic dynamical properties inthese regimes have been observed recently in a plenty ofquantum systems such as circuit QED [22–27], photonicsystem [28], semiconductor system [29, 30] and trappedions [31].In the trapped-ion systems, previous works on thesimulation of the QRM have been performed in vari-ous regimes. For ω a = 0 , ω f (cid:54) = 0, the QRM reducesto the spin-dependent force Hamiltonian which is cru-cial in trapped-ion quantum computation [32–35]. For ω a (cid:54) = 0 , ω f = 0, the Dirac equation has been simulatedwith trapped ions [36, 37]. For ω a = 0 , ω f = 0, thecoupling-only regime can be realized and it has been ex-ploited to engineer the Schr¨odinger cat state [38, 39] andthe grid state [40, 41]. By controlling the experimentalparameters, Ref. [31] has access to the ultra-strong andthe deep-strong coupling regimes. However, most of theprevious works focus on the evolution dynamics governedby the QRM Hamiltonian in multiple regimes.Our work realizes the model Hamiltonian in a specialparameter region ω a (cid:29) ω f , which allows the study of aQPT with the phases controlled by the coupling strength λ in the QRM. In Ref. [11], it has been shown that a r X i v : . [ qu a n t - ph ] F e b an order parameter, the rescaled photon number in thebosonic mode, is shown to stay zero in the normal phasewhile acquiring positive values in the superradiant phasewith a spontaneous breaking of the Z parity symme-try. The ground state of the system exhibits nonanalyt-ical behavior at the critical point, supporting a second-order phase transition at zero temperature [11]. We ex-perimentally demonstrate this type of QPT without theconventional thermodynamic limit of a large number ofparticles. Through laser driving near the blue and thered motional sidebands, we use a single trapped Yb + ion to simulate the QRM Hamiltonian with adjustableparameters [14, 31]. We perform a slow quench on thecontrol parameter and measure the average atomic-levelpopulation and the average phonon number as the orderparameters on both sides of the transition point. Theexperiments are repeated for the increasing ratios of ω a and ω f , with the limit ω a /ω f → ∞ analogous to the ther-modynamic limit [11]. From the qualitative behavior ofthe order parameters under the increasing ratios, we ob-tain strong evidence of the QPT in the QRM, althoughthe ratio parameter is still not large enough for a precisescaling analysis of the critical phenomenon. Our worksimulates the QRM in a special parameter region anddevelops a tool for adiabatic passages that allows thecontrolled study of a QPT, and showcases the possibil-ity of exploring the universal QPT properties using thetrapped-ion system, which has a number of tunable ex-perimental knobs that can be used for a controlled studyof the QPT and the critical phenomena under influenceof various effects. RESULTS
The quantum critical point in the quantumRabi model.
To study the QPT, the low-energy ef-fective Hamiltonian in the limit ω a /ω f → ∞ has beenderived in Ref. [11]. When the control parameter g ≡ λ/ √ ω a ω f <
1, the effective Hamiltonian in the normalphase is given by ˆ H np = ω f ˆ a † ˆ a − g ω f (ˆ a + ˆ a † ) / − ω a / g > H sp = ω f ˆ a † ˆ a − ω f (ˆ a + ˆ a † ) / (4 g ) − ω a ( g + g − ) / n f ≡ ( ω f /ω a ) (cid:104) ˆ a † ˆ a (cid:105) ) andthe spin population ( n a = 1 + (cid:104) ˆ σ z (cid:105) ) at ground state asthe order parameters: in the limit ω a /ω f → ∞ , we have n f = 0( n a = 0) when g < n f = ( g − g ) / (4 g )( n a =1 − g − ) for g > Experimental setup.
We use a single Yb + ionconfined in a linear Paul trap to simulate the QRM, as shown in Fig. 1a. By performing the Doppler coolingfollowed by a resolved sideband cooling [18], the spatialmotion of the ion along one of its principal axes x , withthe frequency ω x = 2 π × .
35 MHz, is cooled close tothe ground state. Its motional degree of freedom canbe well described as a quantum harmonic oscillator, andthus serves as the bosonic mode in the QRM. The twohyperfine states in the ground-state manifold S / arechosen as the qubit states, i.e. |↑(cid:105) = | F = 1 , m F = 0 (cid:105) and |↓(cid:105) = | F = 0 , m F = 0 (cid:105) , with a frequency difference ω q ≈ π × . ω q between the two levels [43]; the undesiredteeth of the frequency combs can effectively produce afourth-order AC Stark shift [44], which we carefully mea-sure and compensate in the experiment (see Methods formore details). Two acousto-optic modulators (AOMs)are used to fine-tune the frequencies and the amplitudesof the laser beams for driving the Raman transition.The orientation of the laser beams are chosen suchthat there is a nonzero differential wave vector com-ponent ∆ k x along the x axis. Let us first considera single pair of Raman beams with the frequency andthe phase difference ∆ ω and ∆ φ generating a Rabi fre-quency Ω. The laser-ion coupling Hamiltonian is givenby ˆ H couple = Ω cos(∆ k x · ˆ x − ∆ ω · t + ∆ φ )ˆ σ x [45], whereˆ x = x (ˆ a + ˆ a † ) is the ion-position operator with x be-ing the ground state wave-packet width. Consideringthe Lamb-Dicke approximation η √ n + 1 (cid:28) η ≡ ∆ k x x is the Lamb-Dicke parameter and ¯ n is theaverage phonon number of the motional state (see Sup-plementary Information for more details about the cor-rection of the Lamb-Dicke approximation), we transferˆ H couple into the interaction picture of the uncoupledHamiltonian ˆ H = ω q ˆ σ z / ω x ˆ a † ˆ a , and get the interac-tion Hamiltonian ˆ H r = ( η Ω r / a ˆ σ + e iδ r t + ˆ a † ˆ σ − e − iδ r t )if the frequency difference ∆ ω is tuned close to thered motional sideband with δ r = ω q − ω x − ∆ ω , andˆ H b = ( η Ω b / a † ˆ σ + e iδ b t + ˆ a ˆ σ − e − iδ b t ) when ∆ ω is tunedclose to the blue sideband with δ b = ω q + ω x − ∆ ω .In order to construct the QRM Hamiltonian, we em-ploy the bichromatic Raman beams as shown in Fig. 1driving the red and the blue sidebands simultaneously[32–34] using the specific implementation proposed andrealized recently in Ref. [14, 31], as shown in Fig. 1b. Ifwe set the two Rabi frequencies to be the same Ω r = Ω b =Ω (in the experiment we can calibrate them such thatthe imbalance | Ω r − Ω b | / | Ω r + Ω b | ≤ H rb = ( η Ω / σ + (ˆ ae iδ r t + ˆ a † e iδ b t ) + h.c. which corresponds to the interaction picture Hamilto-nian with respect to the uncoupled Hamiltonian ˆ H (cid:48) = FIG. 1:
Schematic for experimental observation of QPT in the quantum Rabi model. a . Schematic experimentalsetup. The Yb + ion is confined in the middle of a four-blade Paul trap, with the principal axes of the secular motionalong the x , y and z directions. Two counter-propagating 355 nm pulsed-laser beams are focused on the ion, with a nonzerodifferential wave vector component along the x direction. The two laser beams are controlled by two acousto-optic modulators(AOMs). AOM1 is driven by a radio-frequency (RF) signal from a phase-locked loop (PLL) [42] and AOM2 is controlled by anarbitrary waveform generator (AWG). b . Schematic level structure of Yb + . The two qubit states are two S / hyperfineground states |↑(cid:105) = | F = 1 , m F = 0 (cid:105) and |↓(cid:105) = | F = 0 , m F = 0 (cid:105) , at the separation ω q ≈ π × . π ×
33 THz above the P / levels. The differential frequencies of the laser beams aretuned close to the blue and the red motional sidebands, i.e. ω x − δ b and − ( ω x + δ r ) from the carrier transition. The legend atlower right shows clearly that the purple beam and the blue (red) beam form a near-blue-sideband (near-red-sideband) Ramantransition. c . The 355 nm pulsed laser has a frequency-comb structure [43] with the repetition rate ω rep ≈ π × .
695 MHz.With small frequency adjustments in the AOMs, the desired Raman transitions can be achieved between distant teeth of thefrequency combs. d . Relative positions of the carrier transition (black) and two motional sidebands (red and blue) in solidlines and the bichromatic Raman-transition frequencies (red and blue) in dashed lines. − ( δ b + δ r )ˆ σ z / − ( δ b − δ r )ˆ a † ˆ a/ H Irb = δ b + δ r σ z + δ b − δ r a † ˆ a + η Ω2 (ˆ σ + + ˆ σ − ) (cid:0) ˆ a + ˆ a † (cid:1) . (2)We clearly see the transformed Hamiltonian is exactlythe QRM Hamiltonian if we identify ω a = ( δ b + δ r ) / ω f = ( δ b − δ r ) / λ = η Ω /
2. From our definition, thecontrol parameter is g ≡ λ/ √ ω a ω f = 2 η Ω / (cid:112) δ b − δ r .Since the uncoupled Hamiltonian ˆ H (cid:48) commutes with ourdesired observables, the spin (ˆ σ z ) and the phonon (ˆ a † ˆ a )population, their measurements will not be affected bythis transformation [14]. By controlling the experimentalparameters δ b , δ r and Ω, we can achieve the simulationin the regime ω a (cid:29) ω f where an observation of a QPT ispossible. Observation of quantum phase transition from the spin population.
To observe the QPT from thenormal phase to the superradiant phase in the QRMHamiltonian, we consider two measurable order parame-ters, the spin-up state population (1 + (cid:104) ˆ σ z (cid:105) ) / (cid:104) ˆ a † ˆ a (cid:105) [11]. As the controlparameter g rises from zero to above the quantum crit-ical point, the Z parity symmetry is broken and thesetwo values at the ground state will accordingly increasefrom zero to a non-zero value. However, it is hard to pre-pare the ground state of a general Hamiltonian [47], andsince the energy gap closes at the quantum critical point,we are not able to adiabatically scan the control parame-ter across this point without generating the quasi-particleexcitations into the system [11]. Therefore, in this exper-iment we perform slow quench on the control parameteras suggested by Ref. [11], and compare the measured val-ues with the theoretical predictions.First we set δ b = 2 π × . δ r = 2 π × . FIG. 2:
Spin-up state population versus sidebandRabi frequency.
By setting δ b = 2 π × . δ r =2 π × . R = ω a /ω f = 25fixed. As we increase the sideband Rabi frequency Ω SB (bot-tom axis) linearly with time (top axis), i.e Ω SB = Ω max t/τ q where Ω max = 2 π × . τ q = 2 msare two pre-determined parameters, the control parameter g ( t ) = 2Ω SB ( t ) / (cid:112) δ b − δ r goes up accordingly. With a dura-tion time t , we prepare a target state under g ( t ) and measurethe spin-up state population by florescence detection. Ev-ery orange dot is the average of 20 rounds of measurementsof the spin-up state population, corrected by subtracting the1 .
0% dark-state detection error as the background; the errorbar is estimated as one standard deviation of the 20-roundoutcomes (see Supplementary Information for more detailsabout the error bar estimation). For each round of measure-ment, we repeat the experiment sequence for 500 shots andtake the average. The blue curve is the theoretical value bydirectly solving the time-dependent Schr¨odinger equation un-der the QRM Hamiltonian. The vertical dashed line is anindication of the quantum critical point g c = 1 (correspond-ing to Ω cSB = 2 π ×
10 kHz). The inset shows the florescencedetection scheme of Yb + ions [46]. which corresponds to a ratio R ≡ ω a /ω f = 25 between theatomic transition frequency and the field mode frequencyin the QRM. Under this finite ratio, the energy gap atthe quantum critical point becomes finite which is around0 . ω f = 2 π × . .
25 ms such that the preparedstate does not deviate too much from the true groundstate. After sideband cooling, we initialize the ion in theground state |↓ , n = 0 (cid:105) . Then we linearly increase thesideband Rabi frequency such that Ω SB ( t ) ≡ η Ω( t ) =Ω max t/τ q where Ω max = 2 π × . τ q = 2 ms are two pre-determined parameters. Inother words, the time to reach the critical point Ω cSB = (cid:112) δ b − δ r / π ×
10 kHz is about 1 . g ( t ) = Ω SB ( t ) / Ω cSB . Hence with a duration time t , we generate the target state under aspecific coupling strength of the QRM and measure theorder parameters.The spin-up state population can be measured by aresonant driving on the | S / , F = 1 (cid:105) → | P / , F = 0 (cid:105) cyclic transition of the Yb + ion and a detection ofthe scattered photon counts [46]. The result is shownin Fig. 2. Every orange data point is the average of 20rounds of measurements of the spin-up state populationand has been corrected by subtracting the 1 .
0% dark-state detection error which arises from the small resid-ual off-resonant coupling of the detection laser to thebright state [46] as the background. For each round ofmeasurement, the outcome is acquired by averaging over500 shots of the experiment sequence. The error bar isestimated by one standard deviation of the 20 rounds.We clearly observe the increase of the order parameter(1 + (cid:104) ˆ σ z (cid:105) ) / R , which agrees wellwith the numerical simulation (the blue curve in Fig. 2from numerically solving the time-dependent Schr¨odingerequation of the QRM Hamiltonian). Observation of quantum phase transition fromthe phonon number.
Next we consider another or-der parameter, the average phonon number. After theslow quench of the QRM Hamiltonian, a short opticalpumping pulse of 5 µ s is applied to pump the internalstate of the ion (qubit state) into |↓(cid:105) [46] with negligi-ble effect on the motional state (phonon state) popu-lation. Then we drive the blue-sideband transition be-tween |↓ , n (cid:105) and |↑ , n + 1 (cid:105) ( n = 0 , , · · · ) for varioustime interval t . By fitting the resultant spin-up statepopulation, we can reconstruct the population of differ-ent phonon states, thus calculating the average phononnumber [17, 18, 38, 39, 48–52].With the same experimental parameters as above, theresults are shown in Fig. 3. Each black dot in Fig. 3a isthe calculated average phonon number from the phononpopulation distribution with the error bar estimated byone standard deviation. In Fig. 3b we show an examplefor the blue sideband signal of the leftmost data pointin Fig. 3a. The measured spin-up state population isfitted by the blue curve to give the phonon state popula-tion { p k } ( k = 0 , , · · · ) with a suitable truncation. Thefitting result is shown in Fig. 3c with a covariance ma-trix (inset) representing the correlation between different p k ’s, from which we further deduce the standard devi-ation of the average phonon number, assuming a jointGaussian distribution [53]. More details can be foundin Methods. As we can see, for this data point we get avery low average phonon number, consistent with the factthat it is deep in the normal phase. Similarly, Fig. 3d andFig. 3e show the results for the rightmost data point inFig. 3a. Here we get much faster oscillation at the begin-ning of the blue sideband data owing to the much higher FIG. 3:
Average phonon number versus sideband Rabi frequency.
Again we set δ b = 2 π × . δ r =2 π × . R = ω a /ω f = 25. With the same quench process as above, we prepare the targetstates and measure the corresponding average phonon numbers. a . Each black dot is a measured average phonon number for aspecific ground state. Its value and the error bar are determined according to (b-e). The blue curve is the theoretical result bysolving the time-dependent Schr¨odinger equation. The inset shows the blue sideband scheme for analyzing the phonon numberdistribution: before the measurement, we optically pump the spin state into |↓(cid:105) [46] with tiny influence to the phonon statepopulation; then we drive the blue sideband transition for various time interval and fit the obtained spin-up state populationto extract the phonon distribution. For the leftmost data point in the normal phase, b presents the experimental data (blackdots, averaged over 200 shots) and the fitted curve (blue line), and c shows the fitted population p k ( k = 0 , , · · · ) with thecovariance matrix shown in the inset. The error bar in (a) is computed from this covariance matrix as one standard deviationfor the average phonon number. Similarly d and e show the results for the rightmost data point in (a) in the superradiantphase. More details can be found in Methods. phonon number population (the sideband Rabi oscilla-tion frequency ∼ √ n + 1 η Ω) in the superradiant phase,as well as much faster decay since the phonon numberhas a wider distribution. In this case we get larger un-certainty in each fitted p k . However, they are stronglycorrelated as shown by the off-diagonal elements of thecovariance matrix (inset of Fig. 3e), and we still get areasonable error bar for the average phonon number. Fi-nally, in Fig. 3a we further compare the measured averagephonon number with the theoretical values from numer-ically solving the time-dependent Schr¨odinger equation.Again these results agree well within the error bars.It should be pointed out that the fourth order AC Starkshift induced by the laser beams is not zero in our setup[44], and will increase as we gradually turn up the cou-pling strength of the QRM in the above experiments.Therefore they cannot be compensated by a static fre-quency shift in the laser beams, but require a dynamiccompensation by phase modulation of the laser as shownin Methods. Also note that for our slow quench dynamicsto maintain quantum coherence, the total quench time τ q should be shorter than the motional decoherence time τ d of the trapped ion. The motional coherence of our sys-tem is largely affected by the 50 Hz noise from the ACpower line. Therefore we use a line-trigger to lock the experimental sequence to the AC signal from the powerline, which extends the motional decoherence time to over5 ms. Scaling of the order parameter with respect tovarious experimental parameters.
Finally we con-sider the scaling of the order parameter with respect todifferent experimental parameters. For this purpose, theaverage phonon number is the preferred observable be-cause it can vary in a wider range than the spin-up statepopulation. Our results are summarized in Fig. 4 wherewe change the ratio parameter R , the total quench time τ q and the motional decoherence time of the ion τ d , whilekeeping the other parameters the same. Figure 4a con-siders different ratios R = ( δ b + δ r ) / ( δ b − δ r ) by keepingthe critical sideband Rabi frequency Ω cSB = 2 π ×
10 kHzfixed. Hence we can deduce δ b ( r ) = Ω cSB ( √ R ± / √ R )from the ratio parameter R . As expected, the sharp-ness of the curve and the final average phonon numberare positively correlated with the ratio parameter, andapproach nonanalytical behavior in the limit R → ∞ (see Supplementary Information for a further discussionabout the finite-ratio scaling). In Fig. 4b, we vary thequench time τ q to study its effect on the order param-eter. A shorter quench time leads to a larger deviationfrom the adiabatic evolution, thus the prepared state has FIG. 4:
Average phonon number versus sideband Rabi frequency under different experimental parameters.
Each dot is an average phonon number measured in the same way as in Fig. 3, with the error bar representing one standarddeviation. In a , b and c , we vary the ratio parameter R , the total quench time τ q and the motional decoherence time τ d ,respectively, while keeping the other parameters the same as those in Fig. 3. a . We keep Ω cSB = 2 π ×
10 kHz and τ q = 2 ms.Then we need δ b = 2 π × . δ r = 2 π × . R = 15 and δ b = 2 π × . δ r = 2 π × . R = 5. b . We keep Ω cSB = 2 π ×
10 kHz, R = 25, but use different quench time τ q . c . We keep Ω cSB = 2 π ×
10 kHz, R = 25and τ q = 2 ms, but vary the motional decoherence time τ d by turning on ( τ d = 5 . τ d = 0 . a and b are from numerical simulation without considering themotional decoherence, similar to the τ d = ∞ curve in c . The other two curves in c include the motional decoherence effect bynumerically solving a Lindblad master equation (see Methods for more details). The difference between τ d = ∞ and τ d = 5 . τ q = 2 ms, thus justifies our simplification of τ d = ∞ for a , b and the previous numericalsimulations. larger deviation from the true ground state. Only forlong enough quench time can the prepared states havelarge enough overlap with the real ground states, henceshow the clear evidences of the QPT. In Fig. 4c we studythe influence of finite motional decoherence time τ d ofthe trapped ion. To keep the quantum nature of the sys-tem during the slow quench dynamics, the quench timeshould be within the coherence time of the system. Asis mentioned above, the motional coherence of our trapis largely affected by the 50 Hz noise from the AC powerline. By locking the experimental sequence to the 50 Hzreference, the coherence time is above 5 ms; while if weturn off the locking, the coherence time will drop below1 ms. This phenomenon is also reported in Ref. [54]. Weconduct the experiments with the locking turned on andoff, respectively. As expected, the sharpness of the curvereduces for shorter coherence time. The results agreewell with the theoretical prediction for a motional deco-herence time τ d = 5 . τ d = 0 . τ d = ∞ . This curve is very close to that for τ d = 5 . τ q ≤ DISCUSSION
To sum up, we have successfully observed a QPT fromthe normal phase to the phonon superradiance phase as-sociated with the QRM simulated by a single trappedion. Through slow quench dynamics, we measure thespin-up state population and the average phonon num-ber as the order parameters and observe them changingfrom near zero to large values when the control parameteris tuned across the quantum critical point. For the aver-age phonon number, the change becomes sharper whenthe ratio parameter increases, analogous to approachingcloser to a thermodynamic limit. The strong controlla-bility of the trapped-ion system also allows us to vary theexperimental parameters and study their influence on thephase transition. We also note that in Ref [12], a methodto observe the universal scaling with spin-up state popu-lation was proposed. However, considering some techni-cal difficulties, it is not possible for our system to observethe critical phenomena currently (see Supplementary In-formation for more discussions about this). To furtherstudy the finite-ratio scaling, we will either need to re-duce the experimental noise and to upgrade the experi-mental setup to get more accurate results near the criti-cal point for larger frequency ratio R ; or we may need todevelop different scaling methods which use data pointsfarther away from the critical point. Our work is a firststep towards the more detailed studies of the QPT in theQRM, including the critical dynamics and the universalscaling [11, 12]. With reservoir engineering [52, 57], it isalso possible to observe the dissipative phase transitionin the QRM [58]. Besides, our method can be directlyextended to study the QPT in the many-body versionof the QRM, i.e. the Dicke model [5, 59, 60] when weincrease the number of the trapped ions. METHODS
AC Stark shift compensation.
Our 355 nm pulsedlaser has a frequency comb structure with a repetitionrate ω rep ≈ π × .
695 MHz and a bandwidth of about200 GHz. It can be used to bridge the transition be-tween the two qubit levels with a frequency differencearound ω q ≈ π × . ω AOM1 , which is dynamically varied to compensate thefluctuation of the repetition rate ω rep [42], and AOM2leads to a frequency shift ω AOM2 ,r ( b ) for the red (blue)component of the bichromatic laser beams. The closestdifferential frequencies to the sideband transitions will be∆ ω r ( b ) = n × ω rep + ω AOM1 − ω AOM2 ,r ( b ) with n = 107,the span number of the frequency-comb pairs as shownin Fig. 1c.As we have mentioned in the main text, when tun-ning the sideband Rabi frequency from zero to a specificvalue, the AC Stark shift induced by the off-resonantcoupling of the undesired frequency-comb pairs will alsoincrease continuously. This is a common shift to δ r and δ b , which changes δ r + δ b and hence the ratio parame-ter R . For the 355 nm pulsed laser we use, when thesideband Rabi frequency is set to 2 π × . π ×
10 kHz measured bythe standard Ramsey method [61]. Such a large shifthas non-negligible effect on the order parameters andmust be compensated during the slow quench dynamics.Before each round of experiment, we calibrate the ACStark shift ∆ ac under the QRM Hamiltonian with differ-ent sideband Rabi frequencies Ω SB and fit it accordingto ∆ ac = α Ω where α is a proportionality constant.Then when performing the slow quench experiment, wecorrect the frequency of the blue (red) component inthe bichromatic beams as ω b ( r ) ( t ) = ω b ( r ) (0) + ∆ ac ( t ),to make the detuning δ b ( r ) fixed. This can be realizedby phase modulating the driving RF signals on AOM2,which can be conveniently implemented by an AWG asshown in Fig. 1a with a pre-determined waveform loadingto its memory. The waveform for the pulse is given by A ( t ) cos (cid:16) ω AOM2 ,r ( b ) t − (cid:82) t ∆ ac ( t )d t (cid:17) , where ω AOM2 ,r ( b ) isa pre-set driving frequency of AOM2 at the beginning ofthe experiment and the driving amplitude A ( t ) ∝ Ω SB ( t )is also calibrated before the experiment. Phonon number distribution measurement.
Tomeasure the phonon number of a quantum state of thespin-phonon system, we trace out the spin part by opti-cally pumping it to |↓(cid:105) [46] within a duration of 5 µ s sothat its influence to the motional state can be neglected.Then we apply a blue sideband pulse with various dura-tion t and measure the resultant spin-up state population P ↑ ( t ). It can be fitted by [18, 31, 51] P ↑ ( t ) = 12 (cid:34) − k max (cid:88) k =0 p k e − γ k t cos(Ω k,k +1 t ) (cid:35) , (3)where p k is the occupation of the phonon number state | k (cid:105) , γ k is a number-state-dependent empirical decay rateof the Rabi oscillation where we adopt a commonly usedform γ k ∝ ( k + 1) . [18, 31, 51], Ω k,k +1 = √ k + 1Ω SB is the number-state-dependent sideband Rabi frequency,and k max is the cutoff in the phonon number. If the hy-perparameter k max in the fitting model is too small, wewill lose the high-phonon population and thus limited toa small average phonon number; however, if k max is cho-sen too large, the uncertainty in the fitting will increasebecause we need to fit more parameters; and the risk ofmisjudgement of high-phonon population from the noiseof the blue-sideband signals will also increase (see Sup-plementary Information for more details about the choiceof k max ).After fitting the phonon state population P =( p , p , · · · ) T with its covariance matrix Σ, we can com-pute the average phonon number ¯ n = N · P where N = (0 , , · · · ) is a row vector representing the phononnumber basis. Assuming the fitted parameters follow ajoint Gaussian distribution [53] (see Supplementary In-formation for more details about this assumption), wecan estimate the variance of ¯ n as σ n = N Σ N T . Error analysis and numerical simulation.
Toconsider the motional decoherence effect, we numericallysolve the master equation with the Lindblad superopera-tor L [ ˆ O ]ˆ ρ ≡ ˆ O ˆ ρ ˆ O † − ˆ O † ˆ O ˆ ρ/ − ˆ ρ ˆ O † ˆ O/ ρ ( t ) = − i [ ˆ H, ˆ ρ ( t )]+ L [ √ m ˆ a † ˆ a ]ˆ ρ , where Γ m = 1 /τ d is the dephasing rate with the decoherence time τ d . InFig. 4c with the line-trigger on (off), we set τ d = 5 . . L [ √ γn th ˆ a † ] + L [ (cid:112) γ ( n th + 1)ˆ a ] [62] and L [ (cid:112) q ˆ σ + ˆ σ − ] [63], respec-tively, where γn th ≈ γ ( n th + 1) is the motional heatingrate which is below 50 s − and Γ q is the qubit decoher-ence rate which is below 20 s − in our system. As wehave mentioned in the main text, the effects of these twoterms are negligible from numerical simulation. All theLindblad superoperators we used in the master equationjust represent the results in the lab frame (describing theexperimental decay), and does not represent decay in thesimulated system frame (describing the QRM decay).The fluctuation of the trap frequency (motional modefrequency ω x ), which is within 2 π × R , because the trapfrequency fluctuation is asymmetrical for δ r and δ b (seeFig. 1d), causing δ b − δ r to change, thus the ratio pa-rameter. Under 2 π ×
150 Hz trap frequency fluctuation,the uncertainty for R = 25 , , ± . ± .
82 and ± .
16, respectively. Other sources of errors can be fromthe phonon number fitting beacuse some noise in theblue-sideband signals may be incorrectly recognized asa high-phonon population and cause the fitting error;and from the fluctuation of the AC Stark shift due tothe fluctuation of the laser repetition rate and the laserintensity. Consider a 1% sideband Rabi freqeuncy fluctu-ation (i.e. 1% of 2 π × . π ×
30 Hz fluctuation of the repetition rate, the stan-dard deviation of the fluctuated AC Stark shift from atheoretical calculation [44] can reach about 2 π ×
400 Hz.Under this value, the ratio parameter uncertainty for R = 25 , , ± . ± .
15 and ± .
09, respectively.
Data Availability:
The data that support the find-ings of this study are available from the correspondingauthors upon reasonable request.
Code Availability:
The code used for numerical sim-ulations is available from the corresponding authors uponreasonable request.
Acknowledgements:
This work was supported bythe National key Research and Development Program ofChina (2016YFA0301902), the Beijing Academy of Quan-tum Information Sciences, the Frontier Science Centerfor Quantum Information of the Ministry of Educationof China, and the Tsinghua University Initiative Scien-tific Research Program. X.Z. acknowledges in additionsupport from the National Natural Science Foundation ofChina (11704408, 91836106) and the Beijing Natural Sci-ence Foundation (Z180013). Y.K.W. acknowledges sup-port from Shuimu Tsinghua Scholar Program and the In-ternational Postdoctoral Exchange Fellowship Program.
Competing interests:
The authors declare thatthere are no competing interests.
Author Information:
Correspondence and requestsfor materials should be addressed to L.M.D. ([email protected]).
Author Contributions:
L.M.D. proposed and su-pervised the project. M.L.C., W.D.Z., Q.X.M., Y.J.,L.H., X.Z., Z.C.Z. carried out the experiment. Z.D.L.and Y.K.W. carried out the theoretical analysis. M.L.C.,Y.K.W., and L.M.D. wrote the manuscript.
SUPPLEMENTARY INFORMATIONNote on the error bar estimation
Error bar estimation in the spin population ex-periment.
In the spin population experiment, there aremainly two types of experimental noise we are consid-ering: one is the intrinsic quantum fluctuation and theother is the extrinsic fluctuation of control parametersand environmental parameters. During one round of theexperiment, the system is relatively stable and we aremainly concerned with the quantum projection noise [65].It arises because the quantum state is not an eigenstateof the observable, say, the spin-up state population andthus by repeating the experiment we get different out-comes even if we prepare the same quantum state. Thisnoise can be suppressed by increasing the number of mea-surements. By averaging over 500 shots in each exper-imental round, we get the average spin-up state popu-lation with the quantum projection noise suppressed to1 / √ Error bar estimation in the phonon number ex-periment.
When estimating the error bar of the averagephonon number, we need to make an assumption aboutthe distribution of the experimental noise. Under thecommon assumption of independent and identically dis-tributed Gaussian noise of the experimental data, it canbe shown that the fitted parameters also follow a jointGaussian distribution (see e.g. Theorem 2.1 of Ref. [53].),which is what we use in this work. We want to empha-size that this assumption is used in lots of experimentswhen extracting parameters by fitting the experimentaldata, and is implicitly used in many scientific computingsoftwares like MATLAB when fitting parameters.
Note on the choice of the k max in the phononnumber distribution fitting We use the lowest cutoff number that can ensure thetotal occupation of all the Fock states to be above 95%as k max in the phonon number distribution fitting. Wetake the phonon number distribution of the state with thelargest average phonon number in this experiment as anexample to show how we choose a proper k max . In Fig. 5a,the extracted average phonon number is 11 . ± .
71 whilethe total occupation (cid:80) k max k =0 p k is around 95 .
6% with acutoff number 23 (which can be seen from the horizontalaxis). When we continue to increase the cutoff numberto 24 (Fig. 5b) and 25 (Fig. 5c), the results of the phononnumber distribution are nearly the same with the totaloccupation around 95 .
9% and the average phonon num-ber 11 . ± .
73 (0 . | (cid:105) becomes very large, indicat-ing that overfitting occurs. Also, according to the numer-ical simulation, the total occupation number above theFock state | (cid:105) (including | (cid:105) ) is only 0 . | (cid:105) is occupiedand the only non-zero occupation is p . We can easilysee that p is the contrast of the sinusoidal spin-up statepopulation curve used to extract the occupation number.However, due to the SPAM error, the contrast must beless than 1. In our system, the SPAM error is around2% (an average of 1% dark-state detection error and 3%bright-state detection error), which means the contrastof the spin-up state population curve is only 96%. Thisexplains the relatively low total occupation. Note on the correction for the Lamb-Dickeapproximation
All of our discussions in the main text are based on thecondition that the single trapped ion is in the Lamb-Dickeregime. In this regime, the extension of the ion’s wavefunction is much smaller than the laser’s wavelength, orthis limitation can be written as η √ n + 1 (cid:28) η is the Lamb-Dicke parameter and ¯ n is the aver-age phonon number of the motional state. In our system,the Lamb-Dicke parameter is around 0 .
07. However, inour experiment, the maximum average phonon numberexceeds ten, which means η √ n + 1 is around 0 .
3, mak-ing the non-linear terms of η a non-negligible effect to theentire model Hamiltonian. In the following, we considerthe corrections to the numerical results of the two orderparameters due to the non-linear effect.When we consider the non-linear terms, the totalHamiltonian of the QRM simulated by a single trapped ion reads [66]:ˆ H NQRM = ω a σ z + ω f ˆ a † ˆ a + λ (ˆ σ + + ˆ σ − ) (cid:16) ˆ f ˆ a + ˆ a † ˆ f (cid:17) , (4)where the non-linear effect is embodied in the function[67] ˆ f (ˆ a, ˆ a † ) = e − η / ∞ (cid:88) l =0 (cid:0) − η (cid:1) l l !( l + 1)! ˆ a † l ˆ a l . (5)When we only consider the first expansion term, i.e. l = 0and neglect the term e − η / , the Hamiltonian reduces tothe linear QRM. Here, we implement a numerical simu-lation additionally considering an l = 1 term.As shown in Fig. 6, with the same experimental pa-rameters as in the main text, we simulate the effect onthe spin-up state population and the average phononnumber during the quench dynamics. As we can see, inthe normal phase, the phonon number is small enoughthat both the two order parameters in the non-linearmodel (NLM) show good consistency with those in thelinear model (LM). In the superradiant phase, with theincrease of the average phonon number, the non-lineareffect becomes more and more significant. In our simu-lation, we find that the maximum relative deviation ofthe average phonon number between the NLM and theLM ( | ¯ n LM − ¯ n NLM | / ¯ n LM ) is about 17 %. However, thedeviation near the critical point is only about 2 %, whichis small enough compared with other errors discussed inthe Methods.In conclusion, the non-linear terms in the simulatedQRM causes a small but non-negligible deviation whenthe average phonon number is large ( (cid:38) Note on the scaling analysis
Scaling analysis with spin population.
We notein Ref. [12], spin population is used to analyze the scal-ing effect of the QPT in the QRM. However, some of theexperimental parameters and conditions in Ref. [12] arerather stringent for our system. There are mainly threeconditions that are currently not achievable in our sys-tem. First, in Ref. [12] the bosonic mode frequency ˜ ω / π ( ω f / π in our notation) is set to 200 Hz to realize largefrequency ratio R of 50 to 400 under realistic couplingstrength. This is comparable to the trap frequency fluc-tuation (around 150 Hz) and even smaller than the fluc-tuation of the estimated AC Stark shift (around 400 Hz,see Methods) in our system, and therefore will lead tolarge error. Second, under such large frequency ratios,the required adiabatic evolution time of about 250 ms is0 FIG. 5:
Phonon number distribution with different cutoff number k max . The phonon number distribution with cutoffnumber k max = 23 in a , k max = 24 in b , k max = 24 in c and k max = 24 in d . The error bar is one standard deviation fromthe fitting program. The extracted average phonon number is 11 . ± .
71 while the total occupation (cid:80) k max k =0 p k is around95 .
6% in a . The results of the phonon number distribution are nearly the same with the total occupation around 95 .
9% andthe average phonon number 11 . ± .
73 (0 .
74) in b and c . However, when the cutoff number is set to 26 ( d ), the phononnumber distribution dramatically changes and the error bar of the occupation of the Fock states after | (cid:105) becomes very large,indicating that overfitting occurs.FIG. 6: The spin-up state population and the average phonon number versus the sideband Rabi frequencywith/without non-linear effect.
Here we set the experimental parameters the same as the main text with δ b = 2 π × . δ r = 2 π × . R = 25. The total quench time τ q = 2 ms with the sideband Rabi frequencyincreases linearly from zero to Ω max = 2 π × . η = 0 . a and b are Fig. 2 and Fig. 3a inthe main text with an additional numerical result of the non-linear QRM, respectively. We can see clearly that in the normalphase, the phonon number is small enough that both the two order parameters in the non-linear model (NLM) show goodconsistency with those in the linear model (LM). In the superradiant phase, with the increase of the average phonon number,the non-linear effect becomes more and more significant and causes a non-negligible deviation of the two order parametersbetween the NLM and the LM. R = 25 to implement the spin popula-tion experiment and show the overall behavior in Fig. 2in the main text. Here we further supplement some ex-perimental data around the critical point together with anumerical simulation according to Ref. [12]. We summa-rize the results in Fig. 7a. The figure is a S s ( G )- G plotwhere S s ( G ) ≡ P ( ↑ ) | g − | − and G ≡ R | g − | / with g the coupling strength and P ( ↑ ) the spin-up state pop-ulation. The blue, yellow and green points are numericalsimulation results and the red curve is an analytical linewith a slope − / S s ( G ) is lim G → S s ( G ) ∝ G − / ,i.e. there is a universal critical exponent -2/3). Thenumerical results agree well with the analytical line ex-cept the numerical result with R = 100. This is becausewhen the ratio R is too large, the carrier term in thetrapped-ion simulation will cause the simulated Hamil-tonian to deviate from the real QRM model and this iswhy Ref. [12] propose a standing-wave laser configura-tion to suppress the influence of the carrier term. Theblack points with error bar are calculated from the ex-perimental results with R = 25. The error bar is es-timated as the error bar of the spin-up state popula-tion P ( ↑ ) (which is the raw data taken from the experi-ment) multiplied by the corresponding | g − | − which issupposed to be accurate. The four experimental points(from left to right) are all very close to the criticalpoint g c = 1, where their raw data values of ( g, P ( ↑ )) are (0 . , . ± . . , . ± . . , . ± . . , . ± . S s ( G ) and G on | g − | . In conclusion, the precision of the current exper-iment prevents us from observing the universal scalinglaw with spin population. Scaling analysis with average phonon number.
We present a numerical simulation of the finite-ratio scal-ing of the average phonon number near the critical point g c = 1 and show the result in Fig. 8. The red points arethe numerical results with the system size (indicated bythe ratio R ≡ ω a /ω f ) ranging from 5 to 1000 and the fit-ting result shows that the slope of the fitting line is 0.48.The blue points with error bar are the experimental re-sults. Under the current achievable ratio R , the differencebetween these points is on the same order of magnitudeas the error bar, indicating they are vulnerable to theexperimental noises. Thus these points cannot be usedto extract the critical exponent. Also, we note that thefitted slope of 0.48 from the numerical simulation dataactually deviates from the true critical exponent 1/3 inthe regime R → ∞ in analytic calculation (see Ref. [11]).In order to see this precise exponent, the ratio R in thenumerical simulation needs to exceed 10 . Due to suchlarge ratio, the adiabatic ground state preparation mayneed a duration orders of magnitude larger than the co-herence time of the system. Hence it is not achievable forour system currently to observe the precise scaling effectand to extract the critical exponent with average phononnumber. We can only observe the overall behavior of thephonon number variation curves with three different ra-tios (5, 15 and 25), and as expected the curve becomessharper with larger ratio (see Fig. 4a in the main text). Note on the Ramsey interferometric measurementfor motional coherence
We use the commonly used Ramsey method [68] tomeasure the motional coherence time with or without theline-trigger on. We apply two pi/ τ in between and then measure thespin population. By varying the time interval τ , we ob-tain the Ramsey fringes shown in Fig. 9. We fit the re-sult by an attenuated sinusoid curve Ae − t/τ d cos( ωt + φ )where A , τ d , ω and φ are the fitting parameters. Thecoherence time τ d is extracted from the fitted curve. InFig. 9a, the estimated coherence time is around 0 . . a b FIG. 7:
Scaling analysis with spin-up state population. a . The S s ( G )- G plot, where S s ( G ) ≡ P ( ↑ ) | g − | − and G ≡ R | g − | / with g the coupling strength and P ( ↑ ) the spin-up state population. The blue, yellow and green points arenumerical simulation results and the red curve is an analytical line with a slope − /
3, which is a critical exponent [12]. Thenumerical results agree well with the analytical line except the numerical result with R = 100. This is because when the ratio R is too large, the carrier term in the trapped-ion simulation will cause the simulated Hamiltonian to deviate from the realQRM model [12]. The black points with error bar are calculated from the experimental results with R = 25. The error baris estimated as the error bar of the spin-up state population P ( ↑ ) multiplied by the corresponding | g − | − . b . The fourexperimental data presented in a near the critical point g c = 1, with their raw values of ( g, P ( ↑ )) being (0 . , . ± . . , . ± . . , . ± . . , . ± . S s ( G ) and G on | g − | .FIG. 8: Finite-ratio scaling of the average phononnumber near the critical point.
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