From Molecular Symmetry Breaking to Symmetry Restoration by Attosecond Quantum Control

Abstract

It is well known that laser pulses can break electronic structure symmetry of atoms and molecules, specifically by superposing two (or more) states with different irreducible representations (IRREPs). Recently, our theory group together with the experimental group of Professor Kenji Ohmori, our partner at the Institute for Molecular Science in Okazaki, have shown that laser pulses can also achieve the reverse process, symmetry restoration. For this purpose, the pulse for symmetry restoration was designed as a copy of the pulse for symmetry breaking, with attosecond precision of the proper time delay. Here we develop the theory for new scenarios coherent molecular symmetry breaking and restoration. The extensions are from previous applications of weak circularly polarized ultrafast (fs) laser pulses with Gaussian shapes to intense linearly polarized ultrafast laser pulses that do not need to be transform-limited, e.g. they may have down- and up-chirps. As a proof-of-principle, quantum dynamics simulations demonstrate restoration of $D_{6h}$ symmetry of the aligned model benzene molecule, after laser induced $D_{6h} \rightarrow C_{2v}$ symmetry breaking. The success depends on two criteria: (i) the laser pulse that restores symmetry is designed as a time-reversed copy of the pulse that breaks symmetry; and (ii) their time delay must be chosen with attosecond precision such that the wavefunction at the central time between the pulses is a superposition of two states with different IRREPs but with the same or with opposite phases (modulo $2\pi$).

Appendix: From the Conditions of Time Reversibility (7.36) and Phase Matching (7.20) to Symmetry Restoration (7.25)

The purpose of this Appendix is to show that the two conditions (7.20) and (7.36) guarantee that the symmetry of the system’s electronic structure is restored by the second linearly polarized laser pulse after it had been broken by the first laser pulse. For the proof, we shall assume that the system is initially in the electronic ground state; the inital symmetry is, therefore, the symmetry of the ground state. The first laser pulse transfers part of the population from the ground state (using Dirac notation: $\vert \varPsi _g \rangle$) to an excited state ($\vert \varPsi _e\rangle$). The system is thus prepared in a superposition of $\vert \varPsi _g \rangle$ and $\vert \varPsi _e\rangle$. We consider the case where $\vert \varPsi _g \rangle$ and $\vert \varPsi _e\rangle$ have different irreducible representations IRREPS - that means the superposition state can no longer have the symmetry $\text {IRREP}_g$ of the ground state, nor does it have the symmetry $IRREP_e$ of the excited state—symmetry is broken. Finally, we use (7.20) and (7.36) in order to derive expression (7.25) for the population of $\vert \varPsi _e\rangle$ after the two pulses, depending on the time delay between the pulses. We shall show that for special time shifts of the second pulse ($t' = 0, T, 2T, \ldots$ etc in (7.25)) the population of $\vert \varPsi _e\rangle$ is equal to zero - that means the system is in the ground state; hence it is back to the symmetry of $\vert \varPsi _g\rangle$: the symmetry that was broken by the first pulse is restored by the second pulse.

For a self-contained presentation of this Appendix, let us recall that condition (7.36) means that the matrix representation $\mathbf {H}(t)$ of the Hamiltonian $\hat{H}(t)$ of the laser driven system is time reversed,

$$\begin{aligned} \mathbf {H}(t) = \mathbf {H}(-t). \end{aligned}$$

(7.49)

It has been shown in the main text that the time reversibility of the Hamilton matrix (7.49) is a consequence of our design of the laser pulse for symmetry restoration as time reversed, linearly polarized copy of the pulse that breaks symmetry, (7.32). Note that this construction of the second laser pulse yields the time reversibility (7.49) as a general phenomenon that holds irrespectively of the form of the first laser pulse. The special versions that have been employed in the main text i.e. non-chirped as well as chirped laser pulses with Gaussian shapes ((7.28), (7.29), (7.33) and (7.34)) just serve as examples. Their linear polarizations are important, however, because they imply that $\mathbf {H}(t)$ is real-valued, cf. (7.37),

$$\begin{aligned} \mathbf {H}(t) = \mathbf {H}^*(t). \end{aligned}$$

(7.50)

This is different from circularly polarized laser pulses that yield complex-valued Hamilton matrices, cf. [1]. As a consequence, the present proof differs slightly from the proof given in [1], even though the concepts and the steps are parallel to each other.

Let us also recall that (7.20) means that at the central time $t_c (= 0)$ between the two laser pulses, the wave function of the system with broken symmetry is a superposition of $\vert \varPsi _g \rangle$ and $\vert \varPsi _e \rangle$ with the same (real-valued) phase $\eta$ for $\vert \varPsi _g \rangle$ and $\vert \varPsi _e \rangle$, and without populations of any complementary states $\vert \varPsi _d \rangle , d \ne g,e$ (the subscript “d” means “different from g and e”),

$$\begin{aligned} \vert \varPsi (t_c = 0)\rangle&= c_g(t_c = 0) \vert \varPsi _g\rangle + c_e(t_c = 0)\vert \varPsi _e\rangle \nonumber \\&\equiv e^{i\eta } C_g \vert \varPsi _g\rangle + e^{i\eta } C_e \vert \varPsi _e\rangle \end{aligned}$$

(7.51)

with real-valued amplitudes $C_g$ and $C_e$. (The general version of condition (7.20) has another option, i.e. the phase factor for $\vert \varPsi _e \rangle$ could be $e^{i(\eta \pm \pi ) }= - e^{i\eta }$, but this case is equivalent to (7.51) because the minus sign could be used to re-define the amplitude, $C'_e = -C_e$, and this would not change the rest of the derivation.) The amplitudes are normalized, $C_g^2 + C_e^2 = 1$.

Equation (7.51) is a special case of the general expansion of the time dependent electronic wavefunction $\vert \varPsi (t) \rangle$ in terms of electronic eigenstates $\vert \varPsi _k \rangle$,

$$\begin{aligned} \vert \varPsi (t)\rangle = \sum _k c_k(t) \vert \varPsi _k \rangle \end{aligned}$$

(7.52)

where the sum over k includes states g, e, and possibly several different states d. The eigenstates $\vert \varPsi _k \rangle$ are solutions of the time independent Schrödinger equation (TISE)

$$\begin{aligned} H_e \vert \varPsi _k \rangle = E_k \vert \varPsi _k \rangle \end{aligned}$$

(7.53)

where

$$\begin{aligned} H_e = T_e + V_C \end{aligned}$$

(7.54)

is the electronic Hamiltonian of the field-free system, with kinetic energy operator $T_e$ and Coulomb interaction $V_C$ of all particles (electrons and nuclei.) The time dependent wavefunction $\vert \varPsi (t) \rangle$ is evaluated as solution of the time dependent Schrödinger equation (TDSE)

$$\begin{aligned} i \hbar \dfrac{\partial }{\partial t} \vert \varPsi (t)\rangle = H(t) \vert \varPsi (t)\rangle \end{aligned}$$

(7.55)

with model Hamiltonian

$$\begin{aligned} H(t) = H_e - \mathbf {d} \cdot \mathbf {\epsilon }(t) \end{aligned}$$

(7.56)

where $\mathbf {d}$ is the electronic dipole operator, and

$$\begin{aligned} \mathbf {\epsilon }(t)=\mathbf {\epsilon }_b(t)+\mathbf {\epsilon }_r(t) \end{aligned}$$

(7.57)

is the sum of the electric fields of the two laser pulses that break and restore symmetry. The TDSE is solved subject to the initial condition

$$\begin{aligned} \vert \varPsi (t_i)\rangle = \vert \varPsi _g\rangle \end{aligned}$$

(7.58)

i.e. initially the system is in its ground state. The initial time $t = t_i < 0$ is chosen well before the laser pulse that breaks symmetry.

Inserting the expansion (7.52) into the TDSE (7.55) yields the algebraic version of the TDSE,

$$\begin{aligned} i \hbar \dfrac{d}{dt} \mathbf {c}(t) = \mathbf {H}(t)\mathbf {c}(t) \end{aligned}$$

(7.59)

where $\mathbf {c}(t) = (c_g(t), c_e(t), c_d(t), \ldots )^\intercal$ denotes the column vector of coefficients $c_k(t)$. The order of the components $c_k(t)$ in $\mathbf {c}(t)$ is arbitrary; for convenience, we use the order $k = g, e$ followed by the components d that are different from g and e. The Hamilton matrix $\mathbf {H}(t)$ in the TDSE (7.59) has elements

$$\begin{aligned} H_{kl}(t) = \langle \varPsi _k \vert \mathbf {H}(t) \vert \varPsi _l \rangle = E_k \delta _{kl} - \mathbf {d}_{kl} \cdot \mathbf {\epsilon }(t) \end{aligned}$$

(7.60)

where $\mathbf {d}_{kl} = \langle \varPsi _k \vert \mathbf {d} \vert \varPsi _l\rangle$ is the $k\rightarrow l$ transition dipole matrix element.

The algebraic version of the initial condition (7.58) is

$$\begin{aligned} \mathbf {c}(t_i) = (1,0,0, \ldots )^\intercal \end{aligned}$$

(7.61)

Likewise, the algebraic version of condition (7.20) and (7.51) is

$$\begin{aligned} \mathbf {c}(t_c = 0) =e^{i\eta } (C_g, C_e, 0, 0,\ldots ) \end{aligned}$$

(7.62)

The TDSE (7.59) propagates the initial state (7.61) at $t_i$ to the state (7.62) at $t_c$ that is generated by the laser pulse that breaks symmetry. There are various ways how to express the propagation from (7.61) to (7.62). A rather general expression is

$$\begin{aligned} \mathbf {c}(t_c=0) = \mathbf {U}(t_c-t_i) \mathbf {c}(t_i). \end{aligned}$$

(7.63)

where $\mathbf {U}(t_c - t_i)$ is a unitary transformation that depends on the time interval $t_c - t_i = - t_i$. Condition (7.20) (=(7.51)) implies that $\mathbf {U}(t_c - t_i)$ can be written as block-diagonal unitary matrix,

$$\begin{aligned} \mathbf {U}(t_c-t_i) = \begin{pmatrix} \mathbf {U}_{ge}(t_c-t_i) &{} \mathbf {0} \\ \mathbf {0} &{} \mathbf {U}_d(t_c-t_i) \end{pmatrix} \end{aligned}$$

(7.64)

The notations of the block diagonal sub-matrices indicate that at the time $t_c - t_i$ after the initial time $t_i$, the net effect is population transfer between states g and e, as well as population transfer between the different states labelled d, but without any population transfers from states g or e to d, or vice versa. Note that (7.63) does not tell us anything about what has happened between the times $t_i$ and $t_c$ - for example the population transfer between state g and e may well have been mediated by transient populations of state(s) d—see e.g. the experimental example in [1].

The unitarity of $\mathbf {U}(t_c - t_i)$ implies that its block diagonal matrices are also unitary. As a consequence, $\mathbf {U}_{ge}(t_c - t_i)$ takes the form

$$\begin{aligned} \mathbf {U}_{ge}(t_c-t_i)=e^{i\eta }\begin{pmatrix} C_g &{} -C_e \\ C_e &{} C_g \end{pmatrix} \end{aligned}$$

(7.65)

Another way to express the propagation from (7.61) to (7.62) is

$$\begin{aligned} \mathbf {c}(t_c=0) = \hat{T} e^{-i \int _{t_i} ^0 dt' \mathbf {H}(t')/\hbar } \mathbf {c}(t_i). \end{aligned}$$

(7.66)

where $\hat{T}$ denotes the time-ordering operator. The complex conjugate of (7.66) is

$$\begin{aligned} \begin{aligned} \mathbf {c}^*(0)&= \hat{T} e^{i \int _{t_i} ^0 dt' \mathbf {H}(t')/\hbar } \mathbf {c}(t_i)\\&= \hat{T} e^{i \int _{t_i} ^0 dt' \mathbf {H}(-t')/\hbar } \mathbf {c}(t_i)\\&= e^{-2i \eta }\mathbf {c}(0) \end{aligned} \end{aligned}$$

(7.67)

In the first (7.67), we use the fact that $\mathbf {H}(t)$ and also $\mathbf {c}(t_i)$ are real-valued, cf. (7.50) and (7.61). The second (7.67) exploits the time reversibility (7.36) of the Hamilton matrix, (7.49). The third (7.67) uses the condition (7.20) for the phases, (7.62). Inversion of (7.67) yields

$$\begin{aligned} \begin{aligned} \mathbf {c}(t_i)&= \hat{T} e^{-i \int _0 ^{-t_i} dt' \mathbf {H}(t')/\hbar } \mathbf {c}^*(0) \\&= e^{-2i \eta } \hat{T} e^{-i \int _0 ^{t_f} dt' \mathbf {H}(t')/\hbar } \mathbf {c}(0)\\&= e^{-2i \eta } \mathbf {c}(t_f) \end{aligned} \end{aligned}$$

(7.68)

where $t_f = - t_i$ is the final time after the second pulse. As a consequence, the corresponding populations $P_k(t) = \vert c_k(t)\vert ^2$ of states $k = g,e,d, \ldots$ at $t=t_f$ are the same as the initial probabilities,

$$\begin{aligned} P_k(t_f) = P_k(t_i), k = g,e,d,\ldots \end{aligned}$$

(7.69)

In particular, we have

$$\begin{aligned} P_g(t_f) = 1 \end{aligned}$$

(7.70)

and

$$\begin{aligned} P_e(t_f) = 0 \end{aligned}$$

(7.71)

for the final populations of the ground and excited states. Equation (7.70) implies symmetry restoration by the second laser pulse after symmetry breaking by the first laser pulse—q.e.d.

Next let us consider the case when the second pulse is delayed by time shift $t'$ with respect to the ideal timing—in most cases, this will violate the conditions (7.20) and (7.62). We shall now show that—as one may anticipate—this will prohibit symmetry restoration. The time shift $t'$ has several consequences: (i) The new final time is $t'_f = t_f + t'$. (ii) The system evolves in field-free (ff) environment (because the two laser pulses do not overlap), from time $t_c = 0$ till $t'$. The corresponding unitary matrix $\mathbf {U}_{ff}(t' - t_c)$ for propagation from $t_c$ to $t'$ has elements

$$\begin{aligned} \mathbf {U}_{ff,kl}(t'- t_c) = e^{-iE_k t'/\hbar } \delta _{kl} \end{aligned}$$

(7.72)

The diagonality implies that $\mathbf {U}_{ff}(t' - t_c)$ has the same block-diagonal structure as $\mathbf {U}(t_c - t_i)$, (7.64). (iii) The total time evolution operator for propagation from $t_i$ via $t_c = 0$ and $t'$ to $t'_f$ is

$$\begin{aligned} \begin{aligned} \mathbf {U}(t'_f - t_i)&= \mathbf {U}(t'_f - t') \mathbf {U}_{ff}(t'-t_c) \mathbf {U}(t_c - t_i)\\&= \mathbf {U}(t_f-t_c) \mathbf {U}_{ff}(t'-t_c) \mathbf {U}(t_c-t_i)\\ \end{aligned} \end{aligned}$$

(7.73)

In the second (7.73) we use the fact that time evolution operators depend on time intervals (here $t'_f - t' = t_f - t_c)$, but not on the initial times (i.e. not on $t'$ or on $t_c$). Now the first and second (7.68) imply the relation

$$\begin{aligned} \mathbf {U}^{-1}(t_c - t_i) = \mathbf {U}^\dagger (t_c - t_i) = \mathbf {U}(t_f - t_c ) \end{aligned}$$

(7.74)

In (7.74), we use the unitarity of $\mathbf {U}(t_c - t_i)$. Inserting (7.74) into (7.73) yields

$$\begin{aligned} \begin{aligned} \mathbf {U}(t'_f - t_i)&= \mathbf {U}^{-1}(t_c-t_i) \mathbf {U}_{ff}(t'-t_c) \mathbf {U}(t_c-t_i)\\&= \mathbf {U}^{\dagger }(t_c-t_i) \mathbf {U}_{ff}(t'-t_c) \mathbf {U}(t_c-t_i) \end{aligned} \end{aligned}$$

(7.75)

The block-diagonal structure of the unitary time evolution matrices implies that the relation (7.75) holds as well for the block-diagonal sub-matrices. Inserting (7.65) for the sub-matrix $\mathbf {U}_{ge}(t_c - t_i)$ into (7.75) yields the final result for the population of the excited state

$$\begin{aligned} P_e(t'_f) = 4 P_g(t_c) P_e(t_c) \cdot \left[ \frac{1}{2} - \frac{1}{2}\cos \left( \frac{2\pi t'}{T} \right) \right] \end{aligned}$$

(7.76)

cf. (7.25). Apparently, $P_e(t'_f)$ is periodic with period T and with amplitude $2 P_g(t_c) P_e(t_c)$. The special case (7.71) is realized for zero time shift $t' = 0$, and then periodically for $t' = T, 2T$, etc. q.e.d..