Letter pubs.acs.org/JPCL
Ultrafast Population Inversion without the Strong Field Catch: The Parallel Transfer Bo Y. Chang,† Seokmin Shin,† and Ignacio R. Sola*,‡ †
School of Chemistry (BK21+), Seoul National University, Seoul 151-747, Republic of Korea Departamento de Quı ́mica Fı ́sica, Universidad Complutense, 28040 Madrid, Spain
‡
S Supporting Information *
ABSTRACT: Quantum systems with sublevel structures, like molecules, prevent full population inversion from one manifold of sublevels to the other using ultrafast resonant pulses. We explain the mechanism by which this population transfer is blocked. We then develop a novel concept of geometric control, assuming full or partial coherent manipulation within the manifolds, and show that by preparing specific coherent superpositions in the initial manifold, full population inversion or full population blockade, that is, laser transparency, can be achieved. By properly choosing the relative phases of the initial state, one can interfere in the stimulated emission process, changing the pattern of Rabi oscillations so that full population inversion to the excited electronic state can be achieved almost regardless of the pulse intensity after a minimal threshold value. This is the basis of a novel control mechanism, termed parallel transfer.
M
ultrashort pulse, with a bandwidth larger than the energy spacing of the sublevels. As a typical case, consider a molecule, where the goal is to invert the electronic population from the ground to the excited state with an ultrashort pulse.11−14 Although one may think that the multilevel structure, especially in the case of very different associated time scales (e.g., heavy atoms), does not affect the overall transition, surprisingly the opposite occurs in the strong field case. The unpopulated levels induce Stark shifts7,8 and create large effective detunings from the resonance that limit population absorption. That is, assuming that all of the different sublevels are dipole-allowed, regardless of the strength of the pulse, the maximum population in the excited state that can be reached is typically much smaller than one. We develop here a simple theoretical model with analytical solution15 that allows explanation of this apparently paradoxical observation. Consider that our system consists of two manifolds of sublevels (e.g., the ground and excited electronic states) with Ni and Nf (vibrational) sublevels, respectively. In the ultrafast limit, we may assume that the different sublevels within each manifold are degenerate, ΔE/ℏ ≪ Δω ≈ 0, where ΔE is the energy splitting between adjacent sublevels and Δω is the pulse bandwidth. In the absence of particular selection rules, any sublevel in the ground manifold is coupled to every sublevel of the excited manifold. To simplify the analytical study, we will also assume that all transition dipoles are equal. Then, the equations of motion for every sublevel of the excited manifold |e,k⟩, given by the amplitudes bj(t), and for every
ost quantum systems, as molecules, show a hierarchical spectrum with manifolds of sublevels that can be characterized, at least under certain approximations, by a set of quantum numbers. From the point of view of controlling the dynamics, these systems pose several interesting problems. Quantum control1−4 typically implies the ability to manipulate interfering pathways, which increases with the number of levels that participate in the dynamics as long as the system is controllable.5,6 A multilevel system would therefore offer more control opportunities at the expense of the ability to manipulate within the sublevels. However, in order to address the multiple resonances, ultrashort pulses have to be used. Ultrafast population inversion demands strong fields. However, this comes with a catch: The back reaction of the molecule to the field, even in simple systems, can lead to undesired Stark effects7,8 that prevent the molecular excitation. Our general goal is to investigate whether the accessible structure hinders, or conversely helps, in controlling the system dynamics under strong fields. We will be concerned with coarse-grained goals, where the aim is to transfer the population between the manifolds of levels. In finding the best possible controls, we will assume that the ground manifold is partially controllable, that is, that given some constraints, any possible wave function within a subset of the sublevels can be prepared.9,10 Building on this assumption, we will develop a geometric control approach that allows finding the optimal initial wave functions that maximize the yield of the desired process. This procedure does not prescribe an optimal field but implicitly assumes that a field can be found and makes full use of the multilevel structure. Let us consider a simple and very general process in systems with a congested spectrum, absorption from the ground, initial manifold to the excited, target manifold by means of a strong © XXXX American Chemical Society
Received: March 30, 2015 Accepted: April 21, 2015
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The Journal of Physical Chemistry Letters sublevel in the initial manifold |g,j⟩, given by the amplitudes aj(t), are the same. In the rotating wave approximation (RWA), for a resonant excitation, ȧj = iΩ(t) ∑k bk(t)/2 and bṅ = iΩ(t) ∑j aj(t)/2, where Ω(t) = μ, (t)/ℏ is called the Rabi frequency.16 However, the initial conditions in the ground manifold, where we assume that a single state, |g,1⟩, is initially populated, break the symmetry so that three different probability amplitudes describe the dynamics, one for the initial state a(t), another for the manifold of excited sublevels b̅(t), and yet another for the remaining (excited vibrational) sublevels of the ground manifold that act as Raman modes, c(t). ̅ We can write Nf
PE(t ) =
(7)
that decays with the number of levels, and the same occurs for the Raman excitation. Larger pulse intensities only increase the frequency of the oscillations, not the maximum amplitudes. This behavior is similar to that found in off-resonant excitation, with an effective detuning created by the Stark shift induced between the unpopulated levels. We confront now the analytical results with numerical results obtained from solving the full Ni + Nf time-dependent Schrödinger equation in the energy representation by a Runge−Kutta method for more general conditions, where the sublevels are not degenerate. Figure 1 shows the population
Ni
|Ψ(t )⟩ = a(t )|g, 1⟩ + b ̅ (t ) ∑ |e, k⟩ + c ̅(t ) ∑ |g, j⟩ k
⎛1 ⎞ 1 sin 2⎜ Nf Ni θ(t )⎟ ⎝ ⎠ Ni 2
j>2
(1)
where b̅(t) and c(t) ̅ are mean probability amplitudes, which behave exactly as every sublevel amplitude. We define now the collective excited |E⟩ and Raman |R⟩ states 1 Nf
|E⟩ =
Nf
∑ |e, k⟩
1 Ni − 1
|R⟩ =
(2)
k Ni
∑ |g, j′⟩ (3)
j>2
which together with the initial state |i⟩ ≡ |g,1⟩ form an orthonormal basis such that |Ψ(t)⟩ = a(t)|i⟩ + B(t)|E⟩ + C(t)| 2 R⟩, with B(t) = (Nf)1/2b̅(t), C(t) = (Ni − 1)1/2c(t), ̅ and |a(t)| + 2 2 |B(t)| + |C(t)| = 1. The effective Hamiltonian in this basis is H=−
1 Nf Ω(t )(|i⟩⟨E| + 2
Ni − 1 |E⟩⟨R| + c.c.)
Figure 1. Population dynamics for a system with (a) Ni = 2 (Nf = 1) and (b) Ni = 7 (Nf = 7). In both cases, Ω0 = 2, corresponding to a pulse area of ( = 5. The energy difference between the levels is ΔE = 0.4.
(4)
dynamics for different systems, using Gaussian pulses. We choose equal energy splittings in both manifolds. The pulse carrier frequency is in resonance for all of the |g,k⟩ → |e,k⟩ transitions. Qualitatively similar results are obtained for other choices, as long as we use broad-band pulses. In this work, the time and energy units are scaled with respect to the pulse duration τ and ℏτ, respectively. Case 1 refers to Ni = 2, ΔE = 0.4 < Δω = 4 ln 2 (where Δω is the bandwidth of the Gaussian pulse), and Nf = 1, and the peak Rabi frequency is Ω0 = 2, for which the pulse area is ( = 5. For comparison, in a molecule with a fundamental vibrational wavenumber of 150 cm−1 ΔE = 0.4 corresponds to the use of a 40 fs FWHM laser. Assuming a reasonable transition dipole of 1 Debye, then the Rabi frequency used corresponds to a peak intensity of 0.43 TW/ cm2, typical of moderately strong femtosecond pulses. In case 2, we use Ni = Nf = 7 with the same energy splitting and pulse parameters as before. The results show (i) efficient Raman transfer for the first case and (ii) population locking in the second case, qualitatively in agreement with eqs 6 and 7. The main effect provoked by the energy splittings is to allow more population flow to the most excited sublevels of the initial and final manifolds because the energy difference partially offsets the effective detuning. However, the effect is too small to qualitatively change the dynamics. Is it possible to optimize the pulse parameters to increase the efficiency of the population transfer? Clearly, as long as the initial state is a single sublevel, eq 7 limits the maximum population that can be transferred using transform-limited
with simple analytic eigenvalues and eigenvectors (also known as dressed states) and no nonadiabatic terms.16 When Ni = 2, the wave function dynamics is particularly interesting, with a population in the manifold of excited states given by Nf
PE(t ) =
∑ |⟨e, k|Ψ(t )⟩|2 = |⟨E|Ψ(t )⟩|2 k
=
⎛ Nf ⎞ 1 sin 2⎜ θ (t )⎟ 2 ⎝ 2 ⎠
(5)
where θ(t) = ∫ t−∞ Ω(t′) dt′, such that θ(∞) is the so-called pulse area, ( .16 The formal derivation of this equation is detailed in the Supporting Information (SI). A maximum of 50% population can reach the excited state, whereas there is Rabi flopping (at twice the period of oscillation) between the initial state and the excited vibrational levels of the ground state or Raman modes ⎛1 PR (t ) = |⟨R|Ψ(t )⟩|2 = sin 4⎜ ⎝2
⎞ Nf θ (t )⎟ 2 ⎠
(6)
that is, there is a very efficient Raman Stokes transition. Increasing the number of sublevels in the initial manifold only blocks the population transfer more efficiently. For large Ni, we obtain a population in the excited manifold of 1725
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sublevels follow the pattern of Rabi oscillations coinciding with the population transfer from the initial state but with complete population inversion at multiples of π of the pulse area. For nondegenerate sublevels, the minima of these oscillations do not drop to zero but increase with the number of levels, despite the N−1 factor in eq 7. It is instructive to analyze generic features of the optimized initial states. When the extended pulse area (e = (Ni Nf)1/2( is an odd multiple of π, the optimal initial states have a simple structure. All of their coefficients are equal. This result can be explained analytically when ΔE = 0, following exactly the same steps as those in eq 1. We now write
pulses. The control requires engineering the initial wave function. We will assume that the initial manifold (e.g., the vibrational populations) can be manipulated before the optical pulse Ω(t) is switched on such that we have full controllability within a given subset of states. Typically, this requires the use of laser pulses of very different frequencies. In molecular physics, several control schemes have been proposed that imply creating coherences in the initial electronic state by means of infrared pulses before the optical field is used17−19 and is also reminiscent of some coherent control schemes.20 However, instead of explicitly finding these pulses, we will develop a geometrical approach. We want to maximize the population in the final manifold at time T, given by the functional -
Np
Nf
-=
j
∑ ⟨Ψ(t i)|U(t i , T ; Ω)|e, m⟩⟨e, m|U(T , t i ; Ω)|Ψ(t i)⟩
Nu
k
j
(10)
m
where a(t) ̅ is the mean amplitude of all Np initially populated levels in the initial manifold, whereas Nu = Ni − Np is the set of unoccupied states. The collective initial state defined as
(8)
for fixed Ω(t), with respect to changes in the initial wave function |Ψ(ti)⟩, where ti is the initial time. This amounts to finding U(ti, 0; Ωi)|g,1⟩ = |Ψ(ti)⟩, where U is the time evolution operator. Instead of explicitly finding the new field, we assume controllability and use a variational approach to simply obtain N the rotation matrix Ri|g,1⟩ = |ψi⟩ = ∑j c aij |g,j⟩ (where the sum can be constrained to a subset of the levels of the initial manifold, i.e., Nc ≤ Ni) such that - is maximal. We therefore substitute |Ψ(ti)⟩ by |ψi⟩ in eq 8. The optimization is purely geometrical, and one can use the Rayleigh−Ritz approach. Restricting |ψi⟩ to be normalized (equivalently, Ri to be unitary), we obtain the secular equation, derived in the SI, section II, F|ψi⟩ = χi|ψi⟩, where F has matrix elements Fjk =
Nf
|Ψ(t )⟩ = a ̅ (t ) ∑ |g, 1⟩ + b ̅ (t ) ∑ |e, k⟩ + c ̅(t ) ∑ |g, j⟩
|I⟩ =
1 Np
Np
∑ |g, j⟩ j
(11)
together with the previously defined |R⟩ and |E⟩ collective states, form a orthonormal set such that the Hamiltonian can be written as 1 H=− Nf Ω(t )( Np |I⟩⟨E| + Nu |E⟩⟨R| + c.c.) 2 (12)
with the same eigenvalues as before. Writing the wave function as |Ψ(t)⟩ = A(t)|I⟩ + B(t)|E⟩ + C(t)|R⟩, with A(t) = (Np)1/2a(t), the probability of reaching the excited manifold ̅ at final time is
∑ ⟨g, j|U(0, T ; Ω)|e, m⟩⟨e, m|U(T , 0; Ω)|g, k⟩ m
(9)
PE(∞) =
The solutions are the eigenvectors of F, which give the yields of population transfer χi. In Figure 2, we compare the results of the optimization with the yields obtained from the initial state |i⟩ for degenerate sublevels (ΔE = 0) and with constant energy spacing between adjacent sublevels ΔE = 0.4 for two different systems, Nc = Ni = Nf = 2 and Nc = Ni = Nf = 5. The optimal yields for degenerate
⎛1 ⎞ sin 2⎜ (e⎟ ⎝2 ⎠ Ni
Np
(13)
Whenever (e is an odd multiple of π, full population inversion can be achieved if all of the sublevels of the initial manifold are equally populated and in phase. For different energy spacings, other choices of phases give better results, but the populations remain almost equal. On the other hand, it is simple to prove that when the initial probability amplitudes are all out of phase(such that ∑Nj p aj(0) = 0), then - is minimized, and perfect transparency can be achieved, that is, the population in the excited manifold is zero at all times.21,22 We now return to the original question: Does the presence of accessible sublevels help to control the dynamics? The sublevel structure creates an effective detuning that reduces the amplitude of the Rabi oscillations. Therefore, the larger the energy spacing ΔE, the smaller the back-effect of the unpopulated states and the more the system resembles a simple two-level system. However, by engineering the initial state in the ground manifold, one can obtain full population inversion whenever the number of sublevels is large and regardless of the pulse intensity after some small threshold, as shown in Figure 3a. Whenever ΔE ≠ 0, we can overcome the famous pulse area theorem16 by parallel transfer. Figure 3b shows how the first (and second) minima of the optimized yield of absorption increase with Ni. Only in the degenerate case, Ni does not play any role, and the dynamics is less controllable. Moreover, because the extended area increases
Figure 2. Yield of absorption as a function of the pulse area starting in the ground sublevel |g,1⟩ (black line) or using an optimized initial state that allows parallel transfer (orange line). In (a), Ni = 2, and in (b), Ni = 5. In both cases, Nf = Ni. Dotted lines represent the case ΔE = 0, while solid lines are for ΔE = 0.4. 1726
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In Figure 4a, we show the yield of population transfer as a function of the pulse area for different “control subspaces”. Now, Ni = 20, but the controller has only access to the first Nc sublevels (or to the odd-numbered sublevels due to, e.g., a unspecified selection rule or symmetry). The case Nc = 20 gives maximum yield, while Nc = 1 implies no control over the initial state. The yields are deteriorated as the ability to control the system decreases, and this effect cannot be overcome by increasing the pulse area. On the contrary, often best results are obtained with (e = π. Moreover, adding external constraints, such as access to only an odd number of levels, returns lower values of the yields. Figure 4b shows the results of simulations on a realistic molecular model of Na2 (details of the model are given in the SI) using a 40 fs pulse of 650 nm. The maximum yield of absorption is relatively large for such an ultrashort pulse but saturates at higher intensities due to the Stark shifts and barely exceeds 50%. However, a large increase in the yield and rate can be achieved by preparing an optimal initial superposition involving just the ground and first excited vibrational states, and practically full population inversion can be obtained when playing with the first 10 vibrational states. In this case, as the Franck−Condon factors are different and have different signs, the optimal initial superposition state involves different amplitudes and phases of the vibrational levels. In summary, we have shown that the quantum structure of multilevel systems such as molecules may hamper the success of population transfer. By engineering the initial state, one can avoid the detrimental Stark effects and modify the pattern of Rabi oscillations. This is not achieved by brute force (increasing the pulse intensity) but by preparing specific quantum superposition states that enhance the absorption rate via parallel transfer. Full population blockade and, in fact, laser transparency can also be achieved in similar manners. The control over the dynamics increases with the ability to manipulate every sublevel of the ground manifold and is reduced when there is limited control over the sublevels. However, even a small initial coherence within the initial manifold of levels can be used to improve the rate and yield of population absorption to the excited state. This explains the success in control schemes that capitalize on the use of infrared pulses before the optical pulse drives the electronic transition and also suggests that by using a very strong impulsive IR pulse ahead of the electronic transition, the rate of the transfer can be significantly increased.17−19 However, the parallel transfer mechanism can be applied in many other molecular scenarios. On the other hand, by optimizing the shape of the pulse, it is often possible to create the necessary coherences to overcome the Stark shift effects. Typically, these coherences will involve vibrational states of both the ground and excited manifolds and not only of the ground electronic state. In many situations, limiting the complexity of the control to the form of the initial wave function will not render the simplest control strategies, and we do not claim that, in general, parallel transfer will be advantageous over pulse shaping. However, the geometrical optimization methodology proposed is universal and can be easily extended to the treatment of different substructures or degrees of freedom. It can also be used together with quantum optimal control theory. By taking into account the structure of the molecular Hamiltonian, the geometrical optimization can be used as an effective tool to gain information concerning the key features that limit the way in which we can manipulate complex quantum systems.
Figure 3. Maximum population inversion achieved by parallel transfer as a function of the extended pulse area for systems with a different number of sublevels Ni = Nf = N and ΔE = 0.4. In (a), N = 2 (dotted line), 5 (green line), and 20 (orange line). In (b), we show how the first and second minima of the absorption yield increase with Ni. As shown in (a), the first minimum for Ni = 20 occurs practically at the first maximum for smaller Ni.
with the number of sublevels, the absorption rate is much faster. In the given example, with Nc = Ni = Nf = 20, full population inversion is already obtained with a pulse area of ( ≈ π/10 (see Figure 4a), corresponding to a pulse 100 times less
Figure 4. Maximum population inversion achieved by parallel transfer with limited control of the ground manifold, that is, using a different number of sublevels Nc. In (a), the results for the general symmetric model with Ni = Nf = 20 sublevels is shown as a function of the pulse area. In (b), we show the results of applying the theory to maximize electronic absorption of the A band of Na2 as a function of the pulse peak amplitude.
intense than what would be needed in the two-level system! Therefore, using parallel transfer, we can essentially move into a new ultrafast weak field regime. On the other hand, one should remember that the ability to optimize the yield with increasing Ni (for fixed ΔE) is at the expense of a finer optimization of the initial state. In a molecular scenario, particularly in ultrafast electronic absorption, it will be difficult to prepare arbitrary quantum superpositions including every vibrational level of the ground electronic state. How are the optimal solutions when we can only manipulate a subset of the ground sublevels to create the initial wave function corresponding to the case Nc < Ni? 1727
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Excitation and a Femtosecond Ultraviolet Laser Pulse. J. Chem. Phys. 1992, 97, 8285−8295. (18) Meyer, S.; Engel, V. Vibrational Revivals and the Control of Photochemical Reactions. J. Phys. Chem. A 1997, 101, 7749−7753. (19) Elghobashi, N.; González, L. Breaking the Strong and Weak Bonds of OHF− Using Few-Cycle IR + UV Laser Pulses. Phys. Chem. Chem. Phys. 2004, 6, 4071−4073. (20) Brumer, P.; Shapiro, M. Control of Unimolecular Reactions Using Coherent Light. Chem. Phys. Lett. 1986, 126, 541−546. (21) Boller, K. J.; Imamoglu, A.; Harris, S. E. Observation of Electromagnetically Induced Transparency. Phys. Rev. Lett. 1991, 66, 2593−2596. (22) Eberly, J. H.; Pons, M. L.; Haq, H. R. Dressed-Field Pulses in an Absorbing Medium. Phys. Rev. Lett. 1994, 72, 56−59.
ASSOCIATED CONTENT
S Supporting Information *
Analytical expressions of the populations in the degenerate multilevel system model, derivation of the optimal initial wave function, and details of the molecular model of Na2. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b00651.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the NRF Grant funded by the Korean government (2007-0056343), the International Cooperation Program (NRF-2013K2A1A2054518), the Basic Science Research Program (NRF-2013R1A1A2061898), the EDISON Project (2012M3C1A6035358), and the MICINN Project CTQ2012-36184. I.R.S. acknowledges support from the Korean Brain Pool Program.
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