(Dated: July 15, 2014)

Transcription

1 Experimental test of the quantum Jarzynski equality with a trapped-ion system Shuoming An 1, Jing -Ning Zhang 1, Mark Um 1, Dingshun Lv 1, Yao Lu 1, Junhua Zhang 1, Zhangqi Yin 1, H. T. Quan,3, and Kihwan Kim 1 1 Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, eijing , P. R. China School of Physics, Peking University, eijing , P. R. China 3 Collaborative Innovation Center of Quantum Matter, eijing, , P. R. China Dated: July 15, 014 I. THE ADIAATIC LUE-SIDEAND TRANSITION IN PROJECTIVE MEASUREMENT We realize the adiabatic blue-sideband transition by changing intensity and detuning of the laser beams as shown in the Fig. 1a. The time-varying detuning shown in Fig. 1a is realized by adding a time dependent phase t φt = δtdt. The duration T of the adiabatic transition is 7 times of that of the π pulse π/ωbsb 0 max of bluesideband transition, which is 87.5 µs and the δ 0 is 1.5 Ω bsb max, which is π55.6 khz. We experimentally study the fidelity of the adiabatic operation by observing the transition probability from,n to,n+1 with phonon number n =0, 1,, 5. The table I summarizes the experimental results and the comparison with the numerical simulations. The discrepancies between experimental results and simulations of the adiabatic blue-sideband transitions are originated mainly by the heating process and imperfection of Fock state preparation. The Fock state fidelities are also degraded mainly by heating effect during the preparation, which is summarized in the table of section III. FIG. 1: a The time dependence of the laser intensity Ωt =Ω bsb max sin π t and the detuning δt T =δ0 cos π t for the adiabatic T blue-sideband transition. b The example of the transition between, upper red point and, 3 on the loch sphere. TALE I: The comparison between the simulation results and the experimental population transfers from,n to,n+1 by the adiabatic blue-sideband transition. In the simulation, we include the effect of AC stark shift and the influence from the other motional mode. In the experiment, we measure the probability of state after the adiabatic transfer. The main discrepancies come from the imperfection of initial phonon number states and the heating process during the operation., 0, 1, 1,,, 3, 3, 4, 4, 5 simulations % experiments % In experimental implementation, we consider the limited duration of the whole process and the AC stark shift from the off-resonant coupling to the carrier transition. In order to make the state vector always follow the instantaneous NATURE PHYSICS 1

2 Hamiltonian, we add another weak auxiliary laser with a π/ phase difference from the strong main laser 1]. Its power is β =0.075 times of the strong one, which is decided by experiments. esides the rapid adiabatic passage, we consider the AC Stark shift correction to δt, which comes from the off-resonant coupling to the carrier transition. When we measure the frequency of the blue sideband ω bsb with the Rabi rate Ω bsb max, it already contains the AC Stark shift. So we should firstly get the net blue sideband frequency ω bsb + Ωbsb max ω X η, then add the calculated time dependent AC Stark shift and also the detuning showed in Fig. 1a. The final result δt from the blue sideband transition is: δt = Ωbsb max ω X η Ωbsb max ω X η β + sin π T t + δ 0 cos π T t. 1 II. ERROR CORRECTIONS IN THE PROJECTIVE MEASUREMENT The main errors in the phonon projective measurement come from two sources: imperfection in qubit-state detection and heating of phonon state during the measurement. When there is no error in detection, the population in n state is determined by P n = P n+1 P n 1 j D P D, where P D is the probability of detecting bright dark state in j-th measurement and P j j D =1 P. Due to the detection errors, the measured probability of detecting bright state P j,m is modified to p P j + 1 p D1 P j 6], where p p D of detecting state as bright dark state in the experiment is 97 99%. With the relation P j j,m =P 1 p D /p 1 p D, the P n without detection errors is attained. Here we do not include the alternation of phonon distribution due to the detection error, which introduce high phonon states and limit the precision of the measurement below 10 3 level. The phonon state changes due to the heating process during the fluorescence detection period, 375 µs can be tracked and reversed by calculation. In our experimental condition n 0.1, we did not correct the heating errors since the increase of average phonon number during the total duration of measurement is estimated to be less than 10, which is on the similar order of uncertainty in average phonon number. III. THE EXPERIMENT SEQUENCE TO MEASURE THE TRANSITION PROAILITY The detailed experiment sequence contains the following steps after the projective measurements on phonons: 1. Prepare the dark phonon ground state, 0 by Doppler cooling and resolved sideband cooling.. Generate the number state,n from n = 0 to n = Apply a π/ carrier pulse to make the spin along the eigenstate of σ x x,n. 4. Shift the trap center by qubit-state dependent force that linearly increases from zero to a fixed maximum at various speeds. 5. Adiabatically reduce the laser intensity to zero to bring the phonon distribution back to the lab frame. 6. Apply a π/ carrier pulse with a π phase difference with respect to the previous π/ carrier pulse. The spin comes baczk to. 7. Drive the blue or red sideband transition and observe the time evolutions to obtain the final phonon distribution by maximal likelihood fitting. IV. FOCK STATE PREPARATION To generate the Fock state n, we apply n times of π pulses of sequential blue-sideband and carrier transitions. The Rabi frequency of blue-sideband transition is Ω n,n+1 = e η / η 1 n+1 L 1 nη Ω, where η = kx 0 is the Lamb- Dicke parameter for X-vibrational mode, L 1 nη is the generalized Laguerre polynomial, and Ω is the carrier Rabi frequency ]. In the Lamb-Dicke regime, Ω n,n+1 n +1ηΩ. We generate number states up to n = 0 and find the Lamb-Dicke parameter η = by fitting the results with the exact formula as shown in Fig.. In our experiment for the Jarzynski equality test, we generate the Fock states from n = 0 up to n = 5. We measure NATURE PHYSICS

3 n,n 1 khz n,n 1 n 1 Η n,n 1 e Η Η n 1 L n 1 Η n FIG. : Ω n,n+1 for different n. The red line is obtained by fitting the measured blue-sideband Rabi frequencies dots with the exact formula and the blue line comes from the approximate expression n + 1. the fidelity of the Fock state by fitting the blue-sideband transitions with Eq. 4 in the main text. We define the infidelity as 1 P n, where P n is the population in the Fock state n. The infidelities of the prepared Fock states are summarized in Table II. We simulate the Fock state preparation with master equation including the phonon heating effect and the off-resonant coupling to red sideband, carrier transition, blue sideband of the other radial Y motional mode ω Y ω X + π380 khz. According to our simulation, the most significant error comes from the heating. Fortunately, small Fock states, which have relatively negligible errors, contribute to the majority of the equation e βw F. TALE II: Infidelities of Fock state preparation. n Measured infidelity % Simulated infidelity % V. THE ADIAATIC PROCESS TO THE LA FRAME The adiabatic process means the populations for different eigenstates do not change during the evolution. We can realize this protocol by varying the parameters of Hamiltonian slowly compared to the characteristic time of the quantum system. Since main infidelity comes from heating in our experiment, however, it is necessary to shorten the duration of the adiabatic process. In our experiment, instead of using the slow process, we linearly decrease the laser intensity to zero in T a = π ν. We find the result is the same to that of the ideal slow adiabatic process. This can be explained by the following time evolution operation during the returning process. We can express the time evolution in the form of displacement operator Dα = expαâ α â a common phase neglected 3]. When we linearly increase the laser intensity in T w and linearly decrease in T a, the displacement is written by α = i kx 0Ω max e it wν 1 it w ν 1 T w ν + eitwν 1 e itaν + it a ν T a ν where Ω max = π378 khz is the maximal carrier Rabi frequency and the detuning ν = π0.0 khz. The first item is the displacement by the increasing laser intensity and the second one by the decreasing intensity. Note that the α is the same for T a =+ or k π ν k =1,, 3... As a consequence, the shortest time that we can bring the system to the lab frame is T a = π ν = 50 µs. We experimentally confirm this rapid effective adiabatic process with two settings: change T w and T a together with the same amount and change T w with fixed T a = 50µs. We measure the final phonon distribution by fitting the NATURE PHYSICS 3

4 4 blue-sideband transition with coherent state distribution. The experimental results agree with the predictions by Eq.. a b FIG. 3: Test of the rapid adiabatic process. a Average phonon depending on the change of T w = T a. The red solid line is the prediction by Eq.. b Average phonon depending on the change of T w with T a = 50 µs fixed, which is equivalent to the real experiment process. As shown in Fig. 3a, when the time for ramping the laser intensity up and down is too short µs, the ion does not respond for the force. AS the T w and T a increase, the displacement increases 5 and 10 µs. After a certain duration 0 µs, the displacement reduces as the T w and T a increase 30 and 40 µs. At T w = T a = π ν 50 µs, the ion is not displaced. As shown in Fig. 3b, the rapid pushing results in a high phonon excitation. ased on the results in Fig. 3b, we decide the duration τ =5, 5, 45 µs of applying force to the maximum value for the protocols of far-from equilibrium, intermediate, and quasi-static process, respectively, for the test of Jarzynski equality. VI. MAXIMAL LIKELIHOOD FITTING AND ERROR ESTIMATION Reconstruction of phonon distribution is the key step to find the correct transition probability. In order to get a credible result we measure the time-evolutions of blue-sideband transition and red-sideband transition. We utilize the maximal likelihood method to obtain the phonon distribution from the interferences among different Fock states in the time evolution signal. The fitting functions are shown in Eq. 4 in the main text. We find all the parameters by minimizing the rms deviation χ of the measurement results, which is defined by χ = 1 N N b P b exp t i P fitt b i N r + P r exp t i P fitt r i 3 i=1 where N b N r is the number of points on the bluered transition curve and N = N b + N r. We also use the maximal likelihood method to get the error bar for each fitting parameter. As an example, if we want to fit a coherent state and get the error bar for α, we replace p n by e α / α n n!. After we get the least squares fitting results of all parameters, we scan each parameter across a range of ±10% around the least squares fitting and calculate the corresponding χ. Then we plot different χ versus α in Fig. 4. The set is bounded from below by a parabolic curve. Then we do a parabolic fitting and get the best estimation of α. In order to know the error bar for each fitting parameter, we calculate the χ st i of each t i point: χ st i = Then we get the standard deviation of this set of χ s: i=1 P fit t i P exp t i 4 σχ s= χ s χ s N 5 4 NATURE PHYSICS

5 Χ Σ Α Α FIG. 4: χ as a function of α and the maximal likelihood fitting scheme. Α Considering the repetition In this experiment repetitionr=00 of each point, we get the 95% confidence interval of χ : σχ = 1.96σχ s N R 6 It means if χ deviates from the best fitting more than σ, the fitting parameter is considered as erroneous. In this way, we get the error bar of each fitting parameter Fig. 4. In the experiment, we focus on the phonon distribution p n. Here we show an example of the raw data and the maximal likelihood fitting for initial Fock state n i = 4, the pushing time τ = 5 µs in Fig. 5. FIG. 5: a Red and b blue sideband transition curves for n i = 5 and τ = 5 µs. c The final phonon distribution and uncertainty of each Fock-state population obtained by the maximal likelihood fitting discussed in section V. VII. ERROR ANALYSIS IN THE APPLICATION OF THE WORK Here we present the analysis of errors in final populations of Fock states measured by fitting the blue and red sideband transitions through the maximal likelihood method after the application of the work. We mainly compare the experimental results to those by numerical simulation. In the numerical analysis, we include the heating effect, population transfer and AC stark shift by off-resonant coupling of the laser beam for force to carrier, blue and red sideband transitions. Spontaneous emission and AC stark shift from the coupling of the laser beams between the S 1/ and P 1/ of 171 Yb +, the effect by laser intensity fluctuation and trap frequency fluctuations are not included due to the reasons that are discussed in the end of the section. We also discuss the limitation in the maximal likelihood fitting methods. NATURE PHYSICS 5

6 We define the error in the displaced number state after the application of work as 1 n max n P exp n np ideal n n 7 where n max is the cutoff of the fitting, P exp n n and P ideal n n are the measured and theoretical probabilities to be in the final phonon state n, respectively, after the application of the work from the initial phonon state n. The experimental results and simulations are summarized in Table III. According to the simulation, the main infidelities come from the heating of the motional mode, which also reduces the fidelity of Fock states as shown in Table II. TALE III: Infidelities in the transfer probabilities unit % τ Measured Errors Simulation n i=0 n i=1 n i= n i=3 n i=4 n i=5 n i=0 n i=1 n i= n i=3 n i=4 n i=5 5µs µs µs The heating process is considered to be caused by the electric-field noise from the trap electrodes 4]. Its time evolution can be described by the following master equation 5]: ρt = γ â n ρtâ ρtââ ââ ρt + γ n + 1 âρtâ ρtâ â â âρt 8 where γ is the coupling strength between the ion motion and the thermal reservoir, n is the average phonon of the thermal reservoir. We use the master equation to simulate the effect of heating during the application of work. If we consider the room temperature reservoir, which is relevant to our experimental situation, the n is extremely large. Then γ n γ n+1, which is the 13 Hz heating rate from the measurements shown in Fig. a in the main text. The heating rate can be reduced by using larger trap reducing γ, cooling the trap reducing n or cleaning the electrodes reducing γ 7]. In Table III, the infidelities about a few percentage level in the populations of high phonon states are explained mainly by the heating effect. Off-resonant coupling has two effects: off-resonant population transfer and the AC Stark shift. The qubit-state dependent force is realized by the simultaneous application of laser beams with the beat-note frequencies close to blue and red sideband transitions. The population driven to carrier transition by these laser beams is about Ω/ ωx + Ω And the population driven to the other radial mode Y ω Y = ω X + π380 khz is The errors from the population transfer are reducing the contrast on the time evolutions of the blue and red sideband transitions and do not change the distribution of Fock state population to the same order. When we apply the laser beams with the frequency near to blue and red sideband transitions, the AC Stark shift caused by off-resonant coupling to the carrier is perfectly canceled due to symmetry of these transitions with respect to carrier. However, there is another source of AC stark shift between qubit states from the laser beams due to coupling to the transition between S 1/ and P 1/ states of 171 Yb + ion. Though the AC Stark shift compared with the carrier Rabi rate is pretty small since the detuning for P 1/ is π18 THz, it introduces unwanted phase drift during the increase of the intensity of laser beams. We compensate the AC stark shift by adding of another pair of blue and red sideband transitions with the beat-note frequency detuned by π00 khz from the X-motional sideband. Then we keep the total intensity of the laser beams constant during the application of work and the return process to the lab frame. We choose the 00 khz detuning, which is far enough not to drive any obvious transition. The errors from the spontaneous emission is negligible, since the probability to scatter one photon during blue and red sideband transitions π pulse is The fluctuation of laser intensity is % and the fluctuation of the trap frequency is 0.1 khz in an hour. Since we calibrate the trap frequency just before the application work, it is not necessary to include the drift of trap frequency in the numerical simulation. The errors in small Fock state mainly comes from the inaccuracies in the maximal-likelihood method of fitting. The phonon distributions fitted by the blue and red sideband transitions always show random populations on high phonon number states around a couple of % level. We have carefully investigated the the maximal-likelihood method of fitting by varying the number of phonon states included in the fitting. The fitting results become inaccurate when more than 0 phonon number states are included since the Rabi frequency difference is not sufficient to distinguish in the duration of measurements. For the far-from equilibrium work, we have to include up to 15 phonon number 6 NATURE PHYSICS

7 states for the fitting, since the population up to n = 13 is not negligible. We have also studied the dependency of the measured phonon distributions on the model of phenomenological decay rate γ: constant decay or dependency on phonon number states. We did not observe a noticeable difference beyond the error bars from the different models of decay. VIII. WORK DISTRIUTION FOR A DRAGGED HARMONIC OSCILLATOR IN A NONEQUILIRIUM PROCESS Let us consider the following Hamiltonian Ht = P + 1 M e M eν X + ftx = P + 1 M e M e νx + ft ft M e ν M e ν 9 Here the work parameter is switched according to a protocol ft =gt/t f, where gτ is a continuous differentiable function of τ 0, 1]. From t = 0 to t = t f the work parameter is switched from ft =0=0toft f. The quantum version of this Hamiltonian is Eq.17 of Ref. 8]. First of all, we would like to introduce the rapidity parameter of the force protocol Eq. of 8] z = M e ν ˆ 1 tf dsfs ν expiνs From Eq. 33 of Ref. 8] see also Ref. 9], we get the characteristic function of the work distribution G c t f,t 0 β,u =exp iu ft f M e ν +e iu ν 1 z 4 z sin νu/ e β ν In the classical limit 0, the characteristic function can be simplified to G c t f,t 0 β,u =exp iu ft f M e ν + iu ν z 4 z νu/. 1 β ν The work distribution for the dragged classical harmonic oscillator in an arbitrary nonequilibrium process is given by the Fourier transform of Eq. 1. ˆ P W = due iuw exp iu ft f M e ν + iu ν z 4 z νu/ β ν ˆ = du exp iu W + ft f M e ν β W + ft f M eν ν z exp 4 ν z ν z. ] exp ν z u β This is the work distribution for an the dragged classical harmonic oscillator in an arbitrary nonequilibrium process. It can be seen that it is a Gaussian distribution with the center of the distribution at W = ft f M eν + ν z. As a self-consistent check, we first want to check the work distribution in a sudden switch process t f 0. In the sudden switch process, lim z = ft f t f 0 ν M e ν. 14 ] 13 NATURE PHYSICS 7

8 Substituting Eq. 14 into Eq. 1 we obtain G c t f,t 0 β,u =exp ft f u βm e ν. 15 From this characteristic function of work distribution 15, we obtain the work distribution for the dragged classical harmonic oscillator in a sudden switch process via a Fourier transformation ˆ ] P W = due iuw ft f u exp βm e ν ] 16 exp βm eν W ft f. The work distribution in a sudden switch process can also be understood intuitively: For a system initially characterized by a point X 0,P 0 in phase space, if the process is a sudden switch, from Eq. 9 we know the work done on the system is equal to the change of the energy of the system W x 0,p 0 =ft f X That is, the work done during the sudden switch process is proportional to the position coordinate of the particle. For a initial thermal state canonical ensemble, the particles obey oltzmann distribution in position space. That is, a Gaussian distribution. P x 0 exp 1 ] βm eν X0. 18 Substituting Eq. 17 into Eq. 18 we obtain P W exp βm eν W ] ft f. 19 The work distribution from the intuitive analysis agrees with the result obtained from our general result 16. Also, from Eq. 13 we can tell in the quasistatic limit, the work distribution obeys delta distribution, with the center at W = ft M eν. This agrees with out intuition, too see also Refs. 8 10]. Hence, we believe in our model, the work distribution is always Gaussian even when the process is beyond the linear response regime. It is worth mentioning that the above discussion are about an isolated harmonic oscillator. For an open harmonic oscillator, the work distribution is also always Gaussioan 11]. Hence, the Gaussian distribution of the work distribution even beyond the linear response regime is unique to this model, because the potential is harmonic. For other potentials, such as a quartic potential, this is not true. For a generic model, the work distribution obey Gaussian distribution only when the process is in the linear response regime and the system is an open system in a heat reservoir. For a quantum system, however, the work distribution deviates from the Gaussian distribution, especially when the initial temperature is extremely low. The deviation from the Gaussian distribution for the quantum dragged harmonic oscillator can be seen in the Fig. 5 of Ref.8]. The red curve deviates from Gaussian more prominently than the blue and the black curve because the red curve represents a lower temperature. 1] M. V. erry, J. Phys. A: Math. Theor. 4, ] D. Leibfried, et al, Rev. Mod. Phys. 75, ] P. J. Lee, et al, J. Opt. : Quantum Semiclass. Opt. 7 S , ] Q. A. Turchette, et al, Phys. Rev. A 61, ] Q. A. Turchette, et al, Phys. Rev. A 6, ] C Shen et al New J. Phys ] D.A. Hite, et al, Phys. Rev. Lett., 109, ] P. Talkner, P. Sekhar urada, and P. Hanggi, Phys. Rev. E 78, ] P. Talkner, P. Sekhar urada, and P. Hanggi, Phys. Rev. E 79, ] M. Campisi, P. Talkner, and P. Hanggi, Phil. Trans. R. Soc. A 369, ] O. Mazonka, C. Jarzynski, arxiv:cond-mat/ NATURE PHYSICS