Abstract
Accurate control of a quantum system is a fundamental requirement in many areas of modern science ranging from quantum information processing to highprecision measurements. A significantly important goal in quantum control is preparing a desired state as fast as possible, with sufficiently high fidelity allowed by available resources and experimental constraints. Stimulated Raman adiabatic passage (STIRAP) is a robust way to realize highfidelity state transfer but it requires a sufficiently long operation time to satisfy the adiabatic criteria. Here we theoretically propose and then experimentally demonstrate a shortcuttoadiabatic protocol to speedup the STIRAP. By modifying the shapes of the Raman pulses, we experimentally realize a fast and highfidelity stimulated Raman shortcuttoadiabatic passage that is robust against control parameter variations. The alloptical, robust and fast protocol demonstrated here provides an efficient and practical way to control quantum systems.
Introduction
Coherent control of the quantum state is an essential task in various areas of physics, such as highprecision measurement^{1,2}, coherent manipulation of atom and molecular systems^{3,4}, and quantum information^{5,6}. In most applications, the basic requirement of coherent control is to reach a given target state with high fidelity as fast as possible. Many schemes have been developed for this purpose, including the adiabatic passage technique, which drives the system along its eigenstate^{7,8,9,10}. One of attractive property of this technique is that the resulting evolution is robust against control parameter variations when the adiabatic condition is fully satisfied. However, the adiabatic passage techniques such as the twolevel adiabatic passage^{10}, threelevel stimulated Raman adiabatic passage (STIRAP)^{11} and their variants are time consuming to realize, which limits their applications in some fast dephasing quantum systems. To overcome this shortcoming, several protocols within the framework of the socalled ‘shortcuttoadiabaticity’^{12} have been proposed to speedup the ‘slow’ adiabatic passage: for instance, counterdiabatic driving (equivalently, the transitionless quantum algorithm)^{13,14,15,16}. Very recently, the acceleration of the adiabatic passage has been demonstrated experimentally in twolevel systems: an energylevel anticrossing for a Bose–Einstein condensate loaded into an accelerated optical lattice^{17} and the electron spin of a single nitrogenvacancy centre in diamond^{18}.
The STIRAP based on the twophoton stimulated Raman transition has several advantages. First, lasers can be focused on a single site in an optical lattice or on a single ion in a linear ion trap, which guarantees individual addressability^{19,20,21}. Second, the STIRAP can couple two states that cannot be directly coupled, such as transferring population between two atomic states with the same parity (which cannot be directly coupled via electric dipole transition)^{22}, or transferring the atomic state to the molecular state^{3}. Furthermore, with large singlephoton detuning, double coherent adiabatic passages exist^{23,24,25}, which guarantees the capacity for state transfer between arbitrary states^{25,26,27}. Interestingly, several theoretical protocols have been proposed to speedup the STIRAP by adding an additional microwave field in various atom and molecular systems^{28,29,30,31}. However, the transfer fidelity will depend on the phase differences among the microwave field, the Stokes and pumping laser pulses for the STIRAP, which are difficult to lock. Furthermore, the combination of the microwave field and Raman lasers makes it difficult to feature the individual addressability of the operation. Therefore, speeding up the STIRAP has not yet been experimentally demonstrated.
Motivated by the goal of a robust, fast, addressable, arbitrary state transfer protocol, we propose a feasible scheme to speedup STIRAP by modifying the shapes of two Raman pulses. We utilize the counterdiabatic driving along with unitary transformation, one of the shortcut techniques to realize adiabatic passages. We then experimentally demonstrate the proposed stimulated Raman shortcuttoadiabatic passage (STIRSAP) protocol in a large singlephoton detuning threelevel Λ system with a cold atomic ensemble. The passage’s robustness against parameter variation is confirmed in our experiments. Fast, robust, individually addressable and arbitrarily transferable between states, the quantum state control protocol demonstrated here is useful for practical applications.
Results
STIRAP and STIRSAP protocols
We consider a cold ^{87}Rb atom ensemble (see the Methods section) whose internal energy states 1〉 (2〉) and 3〉 are coupled by pumping pulse Ω_{P}(t) (Stokes pulse Ω_{S}(t)), as shown in Fig. 1a. Two ground states F=1, m_{F}=0〉=1〉; F=2, m_{F}=0〉=2〉 and one excited state 5^{2}P_{3/2} (=3〉) are selected as a typical threelevel Λ system. Under the conditions of rotatingwave approximation and twophoton detuning δ=0, the interaction Hamiltonian of the system in the basis of {1〉, 2〉, 3〉} is given as
where Δ is the singlephoton detuning and is the phase difference between Stokes and pumping lasers, and has been locked to a fixed value in our experiment. In the large detuning condition , the three dressed states of the Hamiltonian (1) can be described as , , and , where mixing angle θ=arctan[Ω_{P}(t)/Ω_{S}(t)] (refs 25, 32). In the usual STIRAP protocol, the Stokes and pumping laser pulses are partially overlapping Gaussian shapes^{11}. If the adiabatic condition is fulfilled, where T is the operation time and T_{π}=2πΔ/(Ω_{P}Ω_{S}), with Ω_{P} and Ω_{S} being the respective peaks of the pulses Ω_{P}(t) and Ω_{S}(t), a highfidelity coherent population transfer from one specific superposition state of 1〉 and 2〉 to another can be realized through adiabatic evolution of the dressed states D〉 and B_{1}〉. This protocol is the double coherent STIRAP^{25} we used in our experiments.
To release the critical requirement but still maintain the highfidelity, one can adopt the shortcut approach to adiabatic passage^{14,15,16}. Under the large detuning condition, the population in excited state 3〉 can be adiabatically eliminated. The Hamiltonian (1) can then be reduced into an effective twolevel system on the basis {1〉, 2〉}, and the Hamiltonian is given by
where the effective detuning and the effective Rabi frequency Ω_{eff}=Ω_{P}(t)Ω_{S}(t)/(2Δ). According to the standard shortcut approach to adiabatic passage, the diabatic transition can be eliminated by adding an appropriate auxiliary counterdiabatic term H_{cd}(t) defined in the Methods section^{12,16}. In our system, this auxiliary term H_{cd}(t) can be realized by adding a microwave field to couple the levels {1〉 and 2〉} (refs 29, 30); however, the aforementioned drawbacks of this method still need to be overcome.
In the Methods section, we describe a feasible approach to realize the shortcut method to adiabatic passage. We find that highfidelity STIRSAP can be achieved if the shapes of the Raman pulses are replaced by
where and are, respectively, the modified effective detuning and Rabi frequency as defined in the Methods section. The modified Raman pulses still satisfy the large detuning condition. With appropriate choices of the parameters and , the system is effectively equivalent to that of adding a supplementary counterdiabatic term H_{cd}(t) (refs 17, 33). The system will thus evolve along its eigenstate of the Hamiltonian H_{0}(t) up to the phase factor for any choice of the protocol parameters, even with very small values of Stokes and pumping fields, and within an arbitrarily short operation time T. According to equation (3), given the original Stokes and pumping pulses with the Gaussianbeam shape shown in Fig. 1b, the modified Stokes and pumping pulses required for STIRSAP can be obtained as shown in Fig. 1c.
Dynamics and characteristics
We now compare the performance of the above STIRAP and STIRSAP protocols. In our experiment, the Stokes pulse and pumping pulse , where 2σ=T/3 is the fullwidth at halfmaximum of the pulse, and is the separation time between the two pulses. We first compare the population transfer dynamics with Raman pulses as shown in Fig. 1b,c. The original parameters of STIRAP are set to be Δ∼2π × 2.5 GHz, Ω_{P}=Ω_{S}=2π × 5 MHz, and hereafter we denote Ω_{0} ≡ 2π × 5 MHz and the corresponding π pulse time . Experimental data (blue and red squares) and theoretical results (dashed and solid lines) are shown together in Fig. 2a. Here the operation time T=0.4 ms, which fails to fulfil the adiabatic criteria. As shown in Fig. 2a, the final transfer efficiency of STIRAP only reaches 36% (blue dashed line). As for the STIRSAP Raman pulses implemented by replacing Ω_{P,S}(t) with in equation (3), the transfer efficiency (the red solid line) can reach 100% since the diabatic transition has been eliminated by effectively adding the Hamiltonian H_{cd}(t). The peak transfer efficiencies of STIRSAP are observed with a twophoton detuning δ=−7 kHz due to acStark shift. The acStark shift can be viewed as a perturbation in our case since it is small compared with Ω_{S,P} and the twophoton bandwidth (∼20 kHz)^{25}. The experimental and theoretical results fit very well with each other. This result clearly shows the remarkable feature of the STIRSAP protocol.
To further characterize the performance of STIRAP and STIRSAP, we plot the transfer efficiencies of them as a function of operation time T in Fig. 2b for a fixed Ω_{P,S}=Ω_{0}. With STIRAP, the transfer efficiency approaches 100% when the operation time is longer than 25T_{0}, where the adiabatic condition is fully satisfied^{11}; however, the efficiency (bluedashed line) will decrease along with the decreasing of T. In particular, it decreases quickly when T<10T_{0}. Remarkably, it is shown in theoretical calculation that the transfer efficiency of STIRSAP (red solid line) can keep constant for any operation time T since the diabatic transition has been eliminated by effectively adding the H_{cd} term through modifying the shape of the pulses accordingly. We confirm the theoretical result with the experimental data for T⩾4T_{0}, where the peak of is around 1.14Ω_{0} for T=4T_{0}.
In principle, both STIRAP and STIRSAP can be sped up to a fixed operation time with fidelity higher than certain value if the peaks of Raman pulses are sufficiently large; however, the resources required are different. With STIRAP, we denote the peak of Ω_{P,S}(t) as Ω_{AP}. Because the characterized time for adiabatic evolution decreases with increasing Ω_{AP}, the operation time can decrease even for a fixed fidelity. By contrast, as shown in Fig. 2b, the operation time for STIRSAP can be arbitrarily small by suitably choosing the peak of the modified Raman pulses . To address the resources required for the speedups, we plot in Fig. 2c the peaks Ω_{AP} (blue dashed line) and (red solid line) required for operation time T, with fidelity no less than 99.4%. It is clear that peak is much smaller than Ω_{AP} for the same operation time with the same high fidelity. This reveals that for the same time T and same fidelity, the resources required for STIRSAP is less than that for STIRAP.
To further compare the performance of STIRAP and STIRSAP, we test the maximum capability of speedup that we could obtain for equal maximum Rabi frequencies, that is, . We theoretically calculate the time T_{AP} of STIRAP to achieve the same high fidelity (99.4%) transfer by sweeping Ω_{AP} and then compare T_{AP} with the operation time T_{SA} for STIRSAP by sweeping . As shown by the green dashed line in Fig. 2d, for the initial Rabi frequency of , which corresponds to a long operation time T_{SA}, the auxiliary Rabi frequency Ω_{a} is small, resulting in only a slight improvement in T_{SA} (see the timederivation term in equation (5) in Methods). However, if we slightly increase , Ω_{a} increases, while the ratio T_{AP}/T_{SA} quickly increases. The ratio is finally stabilized at 5.6, which means that STIRSAP can achieve a speedup 5.6 times that of STIRAP for a fixed Ω_{0}. Although the maximum speedup is achieved when is larger than 2Ω_{0}, an optimal speedup can be achieved by increasing a moderate factor in Ω_{0}. We also plot the difference T_{AP}−T_{SA} (in unit of T_{0}) as shown in Fig. 2d (solid blue line), which reaches its maximum when .
Robustness against imperfection
We now test the stability of the STIRSAP protocol with respect to control parameter variations. To this end, we experimentally measure and theoretically calculate the transfer efficiency by varying one of the protocol parameters in Hamiltonian (1) (that is, the amplitudes and relative time delay of the Stokes and pumping pulses, and singlephoton detuning Δ) while keeping all other parameters unchanged.
The amplitude of the Raman pulses for each atom in our system is slightly different since there is a space distribution of laser power around ±5% on the atomic cloud. Here we artificially modify the amplitudes of the Raman pulses as and , (where RR represents resonant Rabi pulses) to simulate the amplitude variation. Figure 3a shows the experimental data (squares) and theoretical results (lines) of the transfer efficiencies as a function of the deviation ɛ for the resonant Raman π pulse (green squares and dotted–dashed line), STIRSAP with T=0.4 ms (blue squares and dashed line) and STIRSAP with T=1 ms (red squares and solid line). As shown in Fig. 3a, the resonant Raman π pulse is very sensitive to the amplitude variation of Rabi frequencies, and the maximum transfer efficiency is <90% due to the intensity space distribution of laser fields. Remarkably, the STIRSAP is less sensitive to the change of , since the system adiabatically evolves along the eigenstate of Hamiltonian H_{0}, which depends only on the ratio of the Stokes and pumping fields. The robustness will be improved if we extend T=0.4–1 ms, because it will be easier for the system to follow the changes of the ratio of the Stokes and pumping fields.
The transfer efficiencies as a function of the separation time are plotted in Fig. 3b. We first measure the transfer efficiency with fixed pulses shapes versus different separation times . The pulses of STIRSAP are generated with parameters =T/10 and T=0.4 ms. The real separation time in our system is achieved by triggering the radio resource with a delay time at a range about ±20% in . We observe the largest 10% reduction in efficiency as shown by the blue squares in Fig. 3b, which accords with the theoretical simulation (blue dashed line). We then measure the transfer efficiency with variable pulse shapes versus different separation times. Here the Raman pulses we use for every separation time are calculated for the STIRSAP according to each specific separation time. Under this condition, the transfer efficiency can be kept to almost 1 as shown by the red curves and squares in Fig. 3b.
We further test the sensitivity of the STIRSAP protocol to the variation of the singlephoton detuning Δ in Hamiltonian. The deviation of the detuning is denoted as Δ′ and can be modified in the range of ±40 MHz in our experiment. The frequency adjustment is implemented by changing the radio frequencies of acoustooptic modulators (AOMs) and the locking points of the pump laser. There are three locking points (F=2 F′=2, F=2 F′=3 and the crossover peak between them) in our setup, and the radio frequencies of AOMs can be continuously varied ±10 MHz around each locking point. Although a specific singlephoton detuning Δ is needed in the calculation of the STIRSAP protocol (equation (3)), as shown in Fig. 3c, the transfer efficiency keeps constant as frequency changes, which indicates that STIRSAP will not suffer from the deviation of the detuning Δ, since the variation of Δ is <1 MHz in the experiments.
As discussed above, in the region where the relative imperfection is <5%, STIRSAP with T=0.4 ms can maintain a fidelity higher than 98%, which shows a good robust feature for potential applications in quantum manipulation.
Double coherent passages and multiple cyclic operation
So far, we have demonstrated that the STIRSAP protocol is fast, robust and has a high fidelity. As a further proof of its fast and highfidelity features, we apply STIRSAP pulses at the maximum speedup point (T=0.4 ms for Ω_{0}) five times to realize backandforth operations in our system. It is noted that the total operation time is limited to 3 ms in our system, mainly due to the expansion of the atomic cloud. For the large singlephoton detuning Λ system, two coherent passages exit. Thus, the state can be cycled back and forth with the same order of Raman pulses. As shown in Fig. 4a, we first pump all the atoms to one of the ground states (1〉) and then repeat the STIRSAP pulse five times. The system will evolve along one eigenstate and then another one. The final population transfer efficiency to the other ground state (2〉) is (95±4)% averaged over five measured data sets, which indicates an average efficiency of 99(6)%.
More interestingly, the STIRSAP protocol with double coherent passages demonstrated here can also be used to drive the superposition state, which is impossible in ordinary STIRAP with zero detuning. As for an example, we experimentally realize a σ_{x} gate between the initial superposition state and the final state with φ_{0} an irrelevant phase. The data driven back and forth for five times are shown in Fig. 4b. Comparing with the ideal population 0.7 in state 1〉, the final population measured after five σ_{x} operations is (68±4)%, which indicates a total transfer efficiency of 96(8)% and an average efficiency of 99(5)%. Note that those multiple cycle operations in Fig. 4a,b cannot be implemented by STIRAP in our system due to the time limit from the expansion of the atomic cloud. The results thus show remarkable advantages of STIRSAP in some quantum systems with short coherent time.
Discussion
In summary, we have theoretically proposed and experimentally demonstrated an useful protocol to speedup conventional ‘slow’ STIRAP in a large singlephoton detuning threelevel system through transitionless passage. The STIRSAP demonstrated here is faster than STIRAP and more robust as compared with resonant Raman π pulses. Furthermore, the existence of double coherent passages provides a feasible way to control arbitrary quantum states. Fast, high in fidelity and robust against control parameter variations, the STIRSAP protocol is promising for practical applications in quantum control, in quantum information processing and even in chemical interaction control.
Methods
Cold atomic ensemble controlled by Raman lasers
Our experimental system shown in Fig. 1a is similar to the one described in our previous work^{25}. The ^{87}Rb atoms are trapped by a magnetooptical trap. Two Raman lasers (Stokes and pumping lasers), respectively, couple two ground states (1〉, 2〉) with the excited state (3〉). The Raman lasers are set to be twophoton resonance (δ=0) and large singlephoton detuning (Δ∼2π × 2.5 GHz) from the excited state. The frequency of the Stokes laser is further locked to the pumping laser with a stable beating frequency (bandwidth is <0.1 kHz) through optical phaselocked loop technique. The shapes of Raman pulses are controlled by two AOMs (Fig. 1a), which are driven by a radio source (Rigol, DG4162). The radio source has a frequency stability smaller than 2 p.p.m. and a maximum frequency output of 160 MHz.
With a bias field B_{z} about 0.1 G, twophoton Raman transition between magnetic sublevels of F=1〉 and F=2〉 is split by 140 kHz, which allows us to selectively transfer population between F=1, m_{F}=0〉 and F=2, m_{F′}=0〉. Population is measured with the fluorescence collected by a photodiode. To eliminate the total population fluctuation, the populations of F=1, m_{F}=0〉 and F=2, m_{F′}=0〉 are measured simultaneously in the experiments for normalization.
Detailed STIRSAP method
Under the large detuning condition, the threelevel Λ system reduces to an effective twolevel system described by the Hamiltonian (2). According to the theory of shortcuttoadiabatic passage, the diabatic transition can be eliminated by adding a counterdiabatic term given as ^{16,29}, which will lead the system evolution along the eigenstate λ_{n}〉 (={D〉, B_{1}〉} here ) for any time T. For our system, the counterdiabatic term can be realized by adding a microwave field to couple the levels 1〉 and 2〉 (refs 29, 30). Given this, the counterdiabatic term H_{cd} should be given by
where
represents the Rabi frequency of the auxiliarydriving field and its phase . The phase relation requires one to lock the phase between the microwave field and the Raman lasers, which is quite complicated.
To overcome these drawbacks, we develop a much simpler approach to realize the shortcut method to adiabatic passage. We note that H_{cd} can be absorbed into the variation of the original field to form a total Hamiltonian, H(t)=H_{0}(t)+H_{cd}(t), given by
where with φ(t)=arctan(Ω_{a}(t)/Ω_{eff}(t)). It implies that the additional microwave field to achieve H_{cd} is not necessary. We may simply modify both the phase and the amplitude of the Raman lasers to effectively add the H_{cd} term and thus realize the shortcuttoadiabatic passage protocol. Moreover, we further show that the precise control of the timedependent phase γ(t), which is still complicated, can be released. To this end, we apply the unitary transformation^{13,17,33}
which amounts a rotation around the Z axis by γ and eliminates the σ_{y} term in the Hamiltonian (6). After the transformation, we obtain an equivalent Hamiltonian with equation (6), , that is,
where the modified effective detuning and effective Rabi frequency . In the derivation, is used. The wavefunction related to the Hamiltonian is , where is the wavefunction related to the Hamiltonian H(t) in equation (6). Since the unitary transformation U(t) is diagonal and the elements are just phase factors, population measured in the basis {1〉, 2〉} should be the same for both and .
An interesting result implied in equation (8) to further simplify the experimental protocol, which will be proven in the next section, is that we can realize shortcuttoadiabatic passage by replacing Ω_{S}(t) and Ω_{P}(t) in Hamiltonian (1) with modified Raman pulses , . By solving the following equations
we obtain the results of equation (3). Therefore, we can achieve STIRSAP by replacing the original Raman pulse shapes Ω_{S,P}(t) with as described in equation (3).
We should point out that after modifying Raman pulse shapes the STIRSAP protocol is robust against the control parameter variation but is not necessarily optimal. STIRSAP might be further optimized by using inverse engineering^{34,35}. Finally, similar STIRSAP protocols can also be implemented with ordinary singlephoton resonant STIRAP of the threelevel system, which can be reduced to an effective twolevel system due to its intrinsic SU(2) symmetry^{36}.
Dynamics of the three Hamiltonians
We here prove that the STIRSAP protocol can be directly achieved by the realization of equation (8). To this end, we compare the dynamics of the three Hamiltonians H_{0}(t), H(t) and . For any 2 × 2 Hamiltonian H′, we can relate it with an effective magnetic field B′ by the relation , that is,
The unit vector of the effective magnetic field is defined as . Replaced H′ with the Hamiltonian H_{0}(t) in equation (2) (the Hamiltonian H(t) in equation (6)), we can obtain such effective magnetic field for H_{0}(t) [H(t)], and the results are plotted in Fig. 5, where Ω_{P}=Ω_{S}=2π × 5 MHz, Δ=2π × 2.5 GHz and T=0.4 ms.
Furthermore, we denote as the wavefunction related to the Schrodinger equation , and similar denotations for and , then the spin polarizations can be defined as
We numerically solve the Schrödinger equations for those Hamiltonians with the initial states given by and the initial effective magnetic field is along the z direction. The numerical results of the spin polarizations are plotted in Fig. 5. If the adiabatic condition is fully filled, 〈n_{0}(t)〉 should follow the direction of , but as shown in Fig. 5, 〈n_{0}(t)〉 for T=0.4 ms does not overlap . However, both 〈n(t)〉 and follow along the trajectory of . Therefore, rather than following or , both 〈n(t)〉 and follow the adiabatic dynamics of the Hamiltonian H_{0}(t). We thus demonstrate that both H(t) and can in principle be used to realize STIRSAP protocol, but is easier to be manipulated in the experiments.
Data availability
The data that support the findings of this study are available from the corresponding author on request.
Additional information
How to cite this article: Du, Y.X. et al. Experimental realization of stimulated Raman shortcuttoadiabatic passage with cold atoms. Nat. Commun. 7:12479 doi: 10.1038/ncomms12479 (2016).
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Acknowledgements
We thank Li You for fruitful discussions. This work was supported by the NSF of China (Grants No. 11474107, No. 11474153 and No. 11474193), the NKRDP of China (Grant No. 2016YFA0301800 and No. 2016YFA0302800), the Shuguang Program (Grant No. 14SG35), the SRFDP (Grant No. 2013310811003), the Program for Eastern Scholar, the GDSBPYSTIT (Grant No.2015TQ01X715), the GNSFDYS (Grant No. 2014A030306012), the FOYTHEG (Grant No. Yq2013050), the PRNPGZ (Grant No. 2014010), the PCSIRT (Grant No. IRT1243), NSF of Guangdong province (Grant No. 2016A030310462) and SRFYTSCNU (Grant No. 15KJ15).
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Y.X.D., Z.T.L., X.X.Y., Q.X.L., W.H. and H.Y. designed and carried out the experiments; Y.X.D., Z.T.L., Y.C.L., X.C. and S.L.Z. developed the STIRSAP protocol and performed the numerical simulations; Y.X.D., X.C., H.Y. and S.L.Z. wrote the paper and all authors discussed the contents; X.C., H.Y. and S.L.Z. supervised the whole project.
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Du, YX., Liang, ZT., Li, YC. et al. Experimental realization of stimulated Raman shortcuttoadiabatic passage with cold atoms. Nat Commun 7, 12479 (2016). https://doi.org/10.1038/ncomms12479
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DOI: https://doi.org/10.1038/ncomms12479
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