A controlled ac Stark echo for quantum memories

A quantum memory protocol of controlled ac Stark echoes (CASE) based on a double rephasing photon echo scheme via controlled Rabi flopping is proposed. The double rephasing scheme of photon echoes inherently satisfies the no-population inversion requirement for quantum memories, but the resultant absorptive echo remains a fundamental problem. Herein, it is reported that the first echo in the double rephasing scheme can be dynamically controlled so that it does not affect the second echo, which is accomplished by using unbalanced ac Stark shifts. Then, the second echo is coherently controlled to be emissive via controlled coherence conversion. Finally a near perfect ultralong CASE is presented using a backward echo scheme. Compared with other methods such as dc Stark echoes, the present protocol is all-optical with advantages of wavelength-selective dynamic control of quantum processing for erasing, buffering, and channel multiplexing.


Results and Discussion
Before proceeding toward the discussion of CASE, we first analyze on-demand atom phase control via ac Stark effect to erase the first echo (see Figs 1,2 and 3). The basic physics of CCC is introduced in the Supplementary Information (Fig. S2) 17 , and the present quantum memory protocol CASE is discussed in Figs 4, 5 and 6, where the control Rabi pulse for CCC whose pulse area is 2π is resonant between the excited state 2 and an auxiliary ground state 3 (see Fig. 1a). For the discussion, we numerically solve nine time-dependent density matrix equations obtained via Liouville-von Neumann equations under rotating wave approximations for an inhomogeneously broadened Λ-type three-level system interacting with multiple optical pulses (see Methods) 27 . The CASE is an extended version of the ac Stark modulation 23 to a double rephasing scheme combined with CCC, where double rephasing itself has a critical defect of absorptive coherence for the final echo resulting in its impracticality for quantum memories. The generalized Rabi frequency Ω′ by the ac Stark field with a detuning Δ AC in a two-level system is given by Ω ∆ ′ = Ω + AC AC 2 2 , where Ω AC is the Rabi frequency of the ac Stark field. The question then becomes what Φ AC (ac Stark phase shift) is required to silence echo e1.
To resolve this question, we start with a dressed state picture induced by the ac Stark field AC, as seen in Fig. 1a. In a two-level system composed of a ground state ω 1 ( ) 1 and an excited state ω 2 ( ) 2 , the interaction Hamiltonian with an ac Stark field results in dressed states: 28 . This optical system must be inhomogeneously broadened by Δ inh to satisfy the photon echo condition. Thus, for the ac Stark field, the individual atom detuning is denoted by δ ∆′ = ∆ + AC AC j , where δ j is the detuning of the j th atom in Δ inh from the line center. Because Δ inh is symmetric for the resonance frequency, we treat the system as a collection of symmetrically detuned atom pairs ±j across the line center. The generalized Rabi frequency of AC is expressed by Ω′  , and the related frequency of − 2 is ω ω = + ∆ − b S . Here our task is to minimize absorption, while maximizing dispersion (phase shift) by the ac Stark field for the effective phase shift. Any ac-Stark-caused absorption results in coherence dephasing, and thus degrades the echo efficiency. The control pulse C and state 3 are for CASE (discussed in Section D). (b) Pulse sequence for (a). (c) Numerical results of echo erasing (silence) for a π/2 pulse area of ac Stark field AC. (d) Individual atom coherence of (c). For this the optical inhomogeneous broadening of the two-level system is 1.7 MHz (FWHM) and is divided into 201 groups of atoms at 10 kHz spacing for individual calculations. All decay rates are set to zero. Initially, all atoms are in the ground state: ρ 11 (0) = 1. Each pulse duration is set to 0.1 μs. The time of arrival of D, AC, R1 and R2 are 1, 3, 7, and 17 μs, respectively. The AC Rabi frequency is Ω AC = 1.25 MHz, and its detuning is Δ AC = Ω 15 AC . The ac Stark shift Δ S is replaced by the generalized Rabi frequency of the ac Stark field Ω′ = ∆ + Ω AC AC 2 2 . This replacement does not alter the ac Stark physics but accelerates the phase shift time. The e1 is the target echo for erasing. All frequency parameters are multiplied by 2π for all figures. AC field in (a) dresses both ground and excited states in the same way, where the ground state splitting is omitted for simplicity 28 . Hence, a weak Rabi frequency Ω AC compared with the detuning Δ AC is an essential condition (discussed in Section E). ac Stark modulation: echo erasing. Figure 1a,b are the energy level diagram and pulse sequence of the ac Stark modulations, respectively. Figure 1c,d are the corresponding results. Because the second echo e2 is the result of macroscopic rephaing of the first echo e1 by the second rephasing pulse R2, e2 must be affected by any macroscopic coherence burst of e1. This is why the first echo e1 must be erased (silent). To achieve the doubly rephased photon echo in Fig. 1b, there are two important tasks with regard to atom phase control. First, echo e1 must be silent so that it does not affect echo e2. Second, echo e2 must be emissive if it is to be radiated out of the medium. To begin with we analytically derive how D-excited atoms' evolutions are described using time-dependent density matrix equations under rotating wave approximations. The key evolution parameter in coherent transients like photon echoes is the detuning δ j in Δ inh : j , where ρ ±j 12 is the density matrix element for the ±j th atoms denoting the coherence between states 1 and 2 . In a Gaussian distributed ensemble, the coherence evolution of symmetrically detuned atom pairs at ±δ j can be expressed by only the evolution term δ ± e i t j for simplicity. Remembering that a π-rephasing pulse R reverses the phase evolution ( , the silent echo can be achieved by inserting either random phase turbulence or a controlled phase shift. In this Report, we use the controlled phase shift. Via the inserted ac Stark field (AC) in Fig. 1b, a phase shift τ Φ ( ) AC is added to each D-excited atom coherence: → . The sign of τ Φ ( ) AC is predetermined by AC detuning, where '−' ('+') represents a blue (red) detuning. This means that each interacting atom is simply frequency shifted by Δ S , and the phase evolution of each atom is accelerated (or decelerated) by the ac Stark-induced phase shift. For direct numerical calculations without any assumption, however, density matrix solutions do not work for the dressed state picture or the ac Stark shift, unless an extra coherent probe field is added for an extended Raman  system. The extended scheme is not for two-level system interactions anymore. Because the ac Stark effect simply adds a phase shift of Φ AC to each atom's phase evolution as described above, we can still keep the original scheme by adapting the generalized frequency Ω′. This Ω′ treatment does not affect the result or violate the physics of ac Stark interactions, but does shorten phase shift time due to Ω′ ∆  S . Another benefit of Ω′ treatment is that the ac Stark field-induced population change can be traced in real time. Although the ac Stark interaction assumes a negligibly small population change, it is good to check whether the ac Stark induced population change affects the echo efficiency. Figure 1c shows the ac Stark-induced photon echo erasing for π Φ = /2 AC . Figure 1d shows coherence evolutions of all individual atoms in Fig. 1c. The phase addition of π/2 by AC to the system coherence ρ 12 (t) results in cancellation of coherence for the first echo e1. This actually effects coherence swapping between the real and imaginary parts of ρ 12 (t), where Reρ 12 is zero for the first echo signal without AC (see the Supplementary Information Fig. S1 12 . Thus, Imρ 12 (absorption) becomes zero at the e1 timing as shown in Fig. 1c,d. Here it should be noted that individual atom phase evolutions are not affected by the macroscopic coherence. Thus, the first task of silent echo e1 is achieved, with the condition of π Φ = /2 AC . In other words, with the unbalanced ac Stark field applied to the double rephasing scheme, the echoes can be dynamically erased in real time.
To understand the photon echo erasing mechanism by AC in Fig. 1, detailed analyses are performed in Fig. 2 for the same parameters as in Fig. 1 except for Ω AC . Figure 2a shows that the first echo (e1) amplitude degrades as Ω AC increases for a fixed AC pulse duration τ up to π Φ = /2 AC , which is denoted by different colored curves: π Φ = π π π π π π cyan red green blue black magenta . To analyze the results in Fig. 2a a simple model is intuitively introduced, in which the e1 amplitude efficiency η is a cosine function of Φ AC : Figure 2b shows the results of equation (1) applied to the Gaussian distributed ensemble, where the colored dots are for the e1 amplitude ratio to the maximum coherence of 0.5 in Fig. 2a. The solid curve is for overall η for all Gaussian distributed atoms, where the optical bandwidth is intentionally reduced by a factor of 2 (∆ . → MHz k Hz : 1 7 850 inh ) to test the bandwidth-dependent damping rate of η. As shown, all colored dots fit well with equation (1), except for the π pulse area indicated by the arrow. The damping of η is due to random phases among coherently excited atoms in Δ inh , such as in the free induction decay. Because the damping in η should be accelerated with bigger Δ inh , this discrepancy for π Φ = the data ( π Φ = AC ) fits well with equation (1) (not shown). From Fig. 2b, we induce the following conclusion: The general condition of the echo erasing (silencing) by the ac Stark field is AC where the damping rate is accelerated by the ensemble broadening Δ inh . This intuitive model of equation (1) has also been confirmed experimentally 29 : . Thus, the echo efficiency can be accurately controlled simply by adjusting the pulse area AC. Moreover, η is insensitive to Δ inh for π Φ~/2 AC . The dashed curve is for the resonant atoms (δ j = 0; ∆′ = ∆ AC AC ) and is provided as a reference.
Analytic expression for ac Stark echo: Part I. Figure 3a,b, respectively, show two different cases of unbalanced (asymmetric) and balanced (symmetric) ac Stark fields applied to CASE. If AC2 is turned on before (after) e1 and Ω AC2 = Ω AC1 , then it is called 'balanced' ('unbalanced'). Figure 3c,d are the corresponding results. Figure 3e,f show the details of Fig. 3c,d, respectively, for all individual atoms. As shown in Fig. 3c,e, the unbalanced CASE results in the erasing (silence) of e1 (see the blue center curve). However, the resultant echo e2 is absorptive (notice that e2 has the same sign as D). To understand the physics of atom phase evolution, we can treat the interaction system analytically as follows. For the pulse sequence of D → ACI → RI in Fig. 3a the corresponding phase evolutions of the D-excited atoms are expressed by: . The t k stands for the arrival time of pulse k, and 't′' is the time after t R1 .
Here, in this analytic expression, we use a priori knowledge of a π phase shift by R1 (π pulse area) in a two-level system: . As discussed in Figs 1 and 2 , all Imρ 12 values become zero at ′ = = t T (t 2T) for the e1 timing: . It should be noted that the imaginary value (−i) in this phase evolution notation is actually for Reρ, while the real value is for Imρ, according to the definition j . Remembering the π/2 phase shift between Imρ 12 and Reρ 12 , the π/2 ac Stark field functions are to swap these two quantities, resulting in Imρ 12 = 0 while Reρ 12 is a maximum, as discussed in Fig. 1 (see the swapping of blue and red curves across AC2 in Fig. 3c).
The phase added by AC1, however, can be completely compensated for by AC2, making the final echo e2 unchanged by AC1 via erasing e1. For Φ = Φ AC AC 2 1 , the atom phase evolution by the pulse sequence of AC2 → R2 after R1 is expressed by: and ″ = − ′ + t t T T ( 2 ). The 't″' is the time after t R2 . At ″ = ′ = ′ + t T or t T T ( 2 ( ) ) for the e2 timing, the echo e2 is generated as shown in Fig. 3c: 12 ). However, the echo e2 is absorptive, and cannot be radiated out of the medium. On the other hand, turning on AC2 (Φ = Φ AC AC 2 1 ) before e1 in Fig. 3b has no effect, as shown in Fig. 3d: The balanced ac Stark effect in Fig. 3d is the same as the bare double rephasing as shown in the Supplementary Information Fig. S1. This is due to the complete phase cancellation between AC1 and AC2 before e1 formation. Thus, the on-demand phase manipulations for echo erasing are achieved by using the ac Stark fields.
Analytic expression for ac Stark echo: Part II. The absorptive echo e2 in Fig. 3c can also be driven analytically without using phase evolution terms. Using the phase shift relationship of ρ ρ π ⟶ ⁎  Here the t k stands for the time right after the pulse k. Obviously the final echo e2 (last term) in equations (5) and (6) has the same form as the initial D-excited coherence (first term), resulting in an absorptive echo. After R1, there is no iρ component in Imρ 12 due to the π/2−AC1 in equation (5), which represents the silent (erased) echo e1 as discussed in Figs 1, 2 and 3.

Controlled ac Stark echoes: CASE.
To convert the absorptive echo e2 into an emissive one, the present quantum memory protocol CASE is introduced in Fig. 4. As shown in Fig. 4a, a control 2π pulse C is added after the second rephasing pulse R2 (but before e2), where the function of the control pulse is to induce coherence inversion via an optical Rabi flopping between the excited state 2 and an auxiliary state 3 (see Fig. 1a). The mechanism of the control pulse has been intensively studied under the name of CDR [14][15][16][17][18][19] , where the control pulse inverts the system coherence:

12
. Thus, the absorptive echo (e2) in Fig. 3c becomes emissive, as shown in Fig. 4b. Echo e2 has exactly the same form as the original two-pulse photon echo, but without the population inversion as a direct result of double rephasing (see also Supplementary Information Fig. S3). The first echo e1 is completely erased by AC1 π Φ = ( / 2) AC1 due to ρ ρ ↔ Im Re 12 12 as discussed in Figs 1, 2 and 3. The details of coherence inversion are shown in Fig. 4c for a detuned individual atom at δ j = 150 kHz in Fig. 4b. By the control pulse C, the signs of both Reρ 12 and Imρ 12 are exactly reversed, which is known as CCC (see also the Supplementary Information Fig. S3(d)).
For the final e2 in CASE of Fig. 4, the system population is nearly the same as that for data absorption owing to double rephasing (will be discussed in Section E), i.e., there is no population inversion. Due to the large detuning of ac Stark fields the population redistribution via AC is negligible. Any absorption (population) change by the ac Stark fields, however, should affect the echo efficiency due to the phase turbulence. The comparison between the ac Stark induced population change and phase shift is discussed in Section E. By splitting 2π-C into two π-Cs, a near perfect ultralong quantum memory protocol can be achieved, which is discussed in Section F. CASE: Weak-field limit. Figure 5 shows multiple bit quantum memory of CASE for a weak field limit, whose binary code of the data Ds is 1101. Each data pulse is extremely weak, where the pulse area is Φ = π D 20 for each pulse, corresponding to a 250 kHz Rabi frequency with a fixed pulse duration of 0.1 μs. Other parameters are noted in the figure caption. Figure 5a is the pulse sequence, and Fig. 5b is the corresponding numerical results. As shown in Fig. 5b, the consecutive data pulses are retrieved in the same order as the data pulse sequence owing to the double rephasing mechanism. Compared to the reversed order in two-pulse photon echoes such as CRIB [1][2][3][4][5] and AFC [6][7][8][9] , this same order sequence in multiple-bit storage offers an important benefit for direct information processing without additional conversion processes. The dotted line in Fig. 5b represents the population (ρ 22 ) evolution in the excited state 2 , where the population difference between the regions right after Ds and after R2 is negligibly small at ~2% (discussed in Fig. 5e). Figure 5c,d show the coherence evolutions of all individual atoms from Fig. 5b. As shown in Fig. 5c, the ac Stark pulse AC1 adds a particular phase of π/2 to each atom, resulting in a silent echo e1 as discussed in Figs 1, 2, 3 and 4. As discussed in Figs 3 and 4, the phase shift due to AC1 is completely cancelled by AC2 owing to the rephasing process by R1, resulting in e2 remaining intact. Figure 5e is the expansion of the dotted circle in Fig. 5b for the population (ρ 22 ) change of 1% by AC1. Due to the weak Rabi frequency Ω AC , the population change (absorption) is minimized, while the π/2 phase shift (dispersion) can be obtained. Balancing the absorption and dispersion is the key role of the ac Stark field in CASE, where any population change by the ac Stark fields must affect e2. Thus, the ac Stark field-induced coherence dephasing due to absorption must be carefully controlled for it to be minimized, especially for single photon-based quantum memory applications. Figure 5f shows coherence inversion by the 2π-C control pulse via Rabi flopping between states 2 and an auxiliary ground (spin) state 3 . Owing to the (single) Rabi flopping there is no population change in ρ 22 before and after C. However, the system coherence ρ 12 is inverted as mentioned in Fig. 4c. In the middle of C, the population ρ 22 in the excited state 2 is completely transferred into the auxiliary spin state 3 ( ρ ρ . Thus, not only optical dephasing, but also optical evolution stops at this mid-point (discussed in Section F with regard to storage time extension). These are the novel characteristics of CCC, distinguishing a three-level system from the two-level system. The physics of coherence inversion in Fig. 5f originates from two-photon coherence in a resonant Raman system 30,31 . In such a two-photon coherence system, the harmonic oscillation period for the coherence is based on a 4π modulus (see also the Supplementary Information Fig. S3(e) and (f)) 17,30,31 . Here in CCC, only half of the complete process is performed by one leg (C) of the Raman pulse pair for ρ 22 , resulting in a half phase shift, π.
Another issue to discuss is the imperfect rephasing by R due to the atom detuning δ j in Δ inh , resulting in imbalanced population swapping. To minimize the Δ inh effect on imbalanced swapping a bigger Rabi frequency Ω R must be used. In Figs 5 and 6, Ω R is increased by one order from 5 MHz to 50 MHz. Thus, the spectral channel bandwidth for signal multiplexing in CASE may be limited by the rephasing Ω R rather than the ac Stark detuning Δ AC . Considering the ~10 GHz Δ inh in rare-earth doped solids, the possible number of spectral channels for quantum memory multiplexing reaches a few hundreds, though. To avoid rephasing-induced unwanted quantum noises, spectral preparation for an anti-hole in a wide spectral pit may be needed for each spectral channel 32 . Considering three hyperfine states in the ground level of rare earth-doped solids, this spectral preparation, however, is not challenging 30-32 . Backward CASE: Near perfect, ultralong quantum memory. In Fig. 6 we discuss near perfect ultralong CASE by replacing the single 2π-C in Fig. 5 by two time-delayed π-C pulses, where the pulse sequence is shown in the top panel. All other parameters are the same as in Fig. 5. The corresponding results are shown in the middle panel, in which the storage time is extended by the delay ′ − T t t ( ) C C 2 1 of C2 assuming no spin dephasing. In an actual solid medium, however, the spin dephasing (due to spin inhomogeneous broadening) is severe, as demonstrated in a Pr 3+ -ion doped Y 2 SiO 5 crystal 8,31 . The spin dephasing can be minimized by simply applying Zeeman fields, so that T′ can be as long as spin phase decay time 33 . As discussed in Fig. 5f, the π-pulse of C1 stops all-optical dephasing process until C2 is turned on. As shown in the bottom panel, no optical coherence change happens between C1 and C2 (see also the dotted box in the middle panel). Even though the dynamic decoupling (DD) technique 26 is limited to quantum information processing with many gate operations, DD can be used for quantum repeaters, where T′ with DD can be extended up to the spin population decay time, which is in the order of a minute in Pr:YSO 26 .
For near unity retrieval efficiency a backward echo scheme using the control pulse pair C1 and C2 has been suggested 1,2,14,16 and experimentally demonstrated 31 for a single rephasing scheme. Unlike rephasing-based backward echo techniques 1, 2, 10, 11 , the present backward CASE relies on Cs via nonlinear quantum optics 31 . The phase matching conditions for the backward CASE are as follows: SCIEntIfIC REPORtS | 7: 7655 | DOI:10.1038/s41598-017-08051-5 is the wave vector (angular frequency) of the pulse j. From equations (7) and (8)  ) is confirmed, but the propagation direction between D and e2 is not exactly opposite: . This propagation deviation, however, is within the interaction volume inside the medium owing to ~10 −8 in the frequency ratio between them, resulting in a ~0.1 mrad deviation 31 . Considering both ~mm-length optical medium with ~0.1 mm-wide interaction cross section, equation (7) is satisfied for potentially any angles. This fact has already been demonstrated in many (rare-earth) nonlinear optics experiments and has drawn much attention recently for its use in high-etendue medical imaging 34 . If this mechanism were rephasing-based, i.e., ) as in refs 1 and 10. the phase mismatching would never allow echo formation since , where L is the medium length. Thus, the present backward CASE has the practical benefit of spatial multiplexing for a wide angle in multi-spectral channels.
Due to the fundamental limitation of the density matrix equations applicable to only the time domain, Maxwell-Bloch (MB) equations must be used for the retrieval efficiency calculations, which is beyond the scope of the present Report. Thus, we have separately provided the retrieval efficiency of backward CDR echoes in a double rephasing scheme in ref. 19. Because the ac Stark field has nothing to do with the retrieval efficiency based on equations (7) and (8) due to the complete phase cancellation before R2, the retrieval efficiency η of the present backward CASE is given by ref. 19: where αL represents the optical depth of the medium. Therefore in an optically dense medium α  L ( 1 ) the retrieval efficiency of backward CASE is almost unity. Here, an optically dense medium is also necessary for all ensemble-based quantum memory techniques for complete data transfer via full absorption. Although the echo efficiency has nothing to do with the fidelity, the near perfect retrieval efficiency provides a critical condition for fault-tolerant quantum computing with thousands of gate operations 20 and loophole-free quantum communications via quantum interface 21 .

Conclusion
In summary a new quantum memory protocol of controlled ac Stark echoes (CASE) was presented in a double rephasing photon echo scheme with an unbalanced ac Stark pulse pair and controlled Rabi pulse(s), where the control Rabi pulse(s) converts the absorptive echo into an emissive one. The function of ac Stark pulse is to erase (silence) the first echo so that the first echo does not affect the final echo. An exact equation of ac Stark pulse area for erasing the first echo was induced. Using time-dependent density matrix equations, multiple-bit CASE was successfully demonstrated in a weak field limit without any approximations. The multiple-bit retrieval order was the same as for the data bit sequence, providing an essential benefit of CASE for direct quantum information processing without additional conversion steps. In addition to full numerical calculations, analytical evaluations were also given for the same results to provide conceptual understanding. For an extended version of CASE, storage-time extension was discussed by splitting the control Rabi pulse into two time-delayed π-control pulses. In this storage-time extended version, a near unity retrieval efficiency was obtained in an optically dense medium via the backward CASE scheme, where wide angle flexibility offers the practical benefits of spatial multiplexing for spectral channels 35 . Thus, combined with its intrinsic property of spectral multiplexing, the present quantum memory protocol of CASE can be applied to various multi-mode quantum memory applications utilizing all spectral, temporal, and spatial domains 36 . The near perfect retrieval efficiency of the present scheme should contribute to fault-tolerant quantum computing and loophole-free quantum communications via a quantum interface.

Methods
For the numerical analyses, the time-dependent density matrix equations, ) 27 , and they were numerically solved without approximations. The following equations are for the coherence terms of ρ ij in a lambda-type, three-level system interacting with several resonant/off-resonant optical fields:  Here Ω 1 is the Rabi frequency of the optical field between the ground state 1 and the excited state 2 , Ω 2 is the Rabi frequency of the control field between the auxiliary ground state 3 and the excited state 2 , and δ 1 (δ 2 ) is the atom detuning from the field Ω 1 (Ω 2 ). For visualization purpose and simplification, all decay terms are neglected.
To satisfy the ensemble-based quantum memory protocols particularly photon echoes, the interaction medium must be inhomogeneously broadened by Δ inh , where Δ inh is Gaussian distributed. For this, we used a practical value of ∆ = . 1 7 MHz inh for the anti-hole in a wide spectral pit. To minimize the population swapping imbalance via rephasing pulse Ω R due to atom inhomogeneity Δ inh , we used a bigger Rabi frequency of Ω = 50 MHz R for ultraweak data pulses in Figs 5 and 6. The ac Stark fields also may deteriorate system coherence via unwanted absorption. To limit the ac Stark-induced absorption change at ~1%, we used the following numbers: Ω = .
∆ = 0 1 MHz; 5 MHz AC AC . In these calculations, all frequency parameters are multiplied by 2π. Ultraweak consecutive data pulses at Ω = π D 20 each were used for practical CASE to reveal any coherence noise due to strong classical pulses of the ac Stark AC, rephasing R, and control C. The time step for the density matrix calculations was set to 0.01 μs. The data, rephasing, and ac Stark pulse durations were each set to 0.1 μs. The optical inhomogeneous width Δ inh was divided into 401 groups at a step of 10 kHz to denote the detuning δ j for each atom group. For the plots in all figures, all atom groups were calculated independently, and then summed according to δ j -dependent weights in the Gaussian distribution. To overcome the limitation of density matrix-based numerical calculations for ac Stark shift, we use the generalized Rabi frequency (Ω′) as the ac Stark phase shift (Δ S ): . In this Ω′ treatment, nothing changes except that the phase-shift time is shortened without violating the physics. Moreover, the ac Stark-affected population change can be seen for the effect of echo efficiency. Here any population change should affect phase shift. For comparison with the Ω′-based shortened time, a test version of the ac Stark effect in a Raman system was shown in Supplementary Information Fig. S4, where the phase shift of τ Φ ( )