^{1}

### Abstract

Significant molecular orientation can be achieved by time-symmetric single-cycle pulses of zero area, in the THz region. We show that in spite of the existence of a combined time-space symmetry operation, not only large peak instantaneous orientations, but also nonzero time-average orientations, over a rotational period, can be obtained. We show that this unexpected phenomenon is due to interferences among eigenstates of the time-evolution operator, as was described previously for transport phenomena in quantum ratchets. This mechanism also works for appropriate sequences of identical pulses, spanning a rotational period. This fact can be used to obtain a net average molecular orientation regardless of the magnitude of the rotational constant.

Financial support from the Spanish Government through the MICINN (Project No. FIS2010-18799) is acknowledged.

I. INTRODUCTION

II. SYMMETRY ANALYSIS FOR POLAR MOLECULES INTERACTING WITH SINGLE-CYCLE PULSES

A. Symmetry properties of quasienergy eigenfunctions

B. Symmetry properties of time-dependent wave packets

III. NUMERICAL RESULTS

A. Molecular orientation by a single-cycle pulse

B. Molecular orientation by a sequence of single-cycle pulses

IV. DISCUSSION

### Key Topics

- Parity
- 10.0
- Eigenvalues
- 5.0
- Hilbert space
- 3.0
- Rotation constants
- 3.0
- Atomic and molecular interactions
- 2.0

## Figures

Instantaneous orientation, ⟨cos θ⟩ for initial state, at *t* _{0} = −*T*/2, |*J* = 0⟩, and eight different single-cycle pulses of zero area. The electric field for each one of these pulses is given by . Parameters of the pulses, in reduced units of time (ℏ/*B*), are: (*a*) σ = 14, *T* = 100, ω = 0.1, μ*E* _{0}/*B* = 2 (solid line), 4 (dashed line), (*b*) σ = 0.45, *T* = 3, ω = 3, μ*E* _{0}/*B* = 5 (solid line), 10 (dashed line), (*c*) σ = 0.045, *T* = 0.3, ω = 30, μ*E* _{0}/*B* = 50 (solid line), 100 (dashed line), and (*d*) σ = 0.0045, *T* = 0.03, ω = 300, μ*E* _{0}/*B* = 500 (solid line), 1000 (dashed line).

Instantaneous orientation, ⟨cos θ⟩ for initial state, at *t* _{0} = −*T*/2, |*J* = 0⟩, and eight different single-cycle pulses of zero area. The electric field for each one of these pulses is given by . Parameters of the pulses, in reduced units of time (ℏ/*B*), are: (*a*) σ = 14, *T* = 100, ω = 0.1, μ*E* _{0}/*B* = 2 (solid line), 4 (dashed line), (*b*) σ = 0.45, *T* = 3, ω = 3, μ*E* _{0}/*B* = 5 (solid line), 10 (dashed line), (*c*) σ = 0.045, *T* = 0.3, ω = 30, μ*E* _{0}/*B* = 50 (solid line), 100 (dashed line), and (*d*) σ = 0.0045, *T* = 0.03, ω = 300, μ*E* _{0}/*B* = 500 (solid line), 1000 (dashed line).

Instantaneous orientation, ⟨cos θ⟩, for initial states corresponding to four different Floquet eigenstates (see Appendix), during a single-cycle pulse. Pulse parameters, in reduced units of time (ℏ/*B*), are: σ = 0.45, *T* = 3, ω = 3, μ*E* _{0}/*B* = 10. The curves correspond to the Floquet eigenstates whose eigenvalues, in *B*/ℏ units, are: −0.17 (black), −0.58 (red), −0.53 (green), and −0.89 (blue). These wave packets are formed essentially, in the four cases, by combining the five lowest rotational eigenstates. Although large peak orientationes are obtained, the integrated orientation over the pulse duration is strictly zero by symmetry.

Instantaneous orientation, ⟨cos θ⟩, for initial states corresponding to four different Floquet eigenstates (see Appendix), during a single-cycle pulse. Pulse parameters, in reduced units of time (ℏ/*B*), are: σ = 0.45, *T* = 3, ω = 3, μ*E* _{0}/*B* = 10. The curves correspond to the Floquet eigenstates whose eigenvalues, in *B*/ℏ units, are: −0.17 (black), −0.58 (red), −0.53 (green), and −0.89 (blue). These wave packets are formed essentially, in the four cases, by combining the five lowest rotational eigenstates. Although large peak orientationes are obtained, the integrated orientation over the pulse duration is strictly zero by symmetry.

Instantaneous orientation, ⟨cos θ⟩, for three sequences of single-cycle pulses and initial state Ψ(−*T*/2) = |*J* = 0⟩. Parameters of the pulses, in reduced units of time (ℏ/*B*), are σ = 0.0045, *T* = 0.01π, ω = 300, μ*E* _{0}/*B* = 1000 (black line), σ = 0.045, *T* = 0.1π, ω = 30, μ*E* _{0}/*B* = 100 (red line), and σ = 0.45, *T* = π, ω = 3, μ*E* _{0}/*B*=10 (green line).

Instantaneous orientation, ⟨cos θ⟩, for three sequences of single-cycle pulses and initial state Ψ(−*T*/2) = |*J* = 0⟩. Parameters of the pulses, in reduced units of time (ℏ/*B*), are σ = 0.0045, *T* = 0.01π, ω = 300, μ*E* _{0}/*B* = 1000 (black line), σ = 0.045, *T* = 0.1π, ω = 30, μ*E* _{0}/*B* = 100 (red line), and σ = 0.45, *T* = π, ω = 3, μ*E* _{0}/*B*=10 (green line).

Time-average orientation, ⟨⟨cos θ⟩⟩, over a rotational period, for two different regimes, as a function of field strength. Parameters, in reduced units of time (ℏ/*B*), are: (*a*) σ = 0.045, *T* = 0.1π, and (*b*) σ = 0.45, *T* = π. Thus, the time average in panel *a* corresponds, at each point, to 10 pulses and in panel *b* to one pulse.

Time-average orientation, ⟨⟨cos θ⟩⟩, over a rotational period, for two different regimes, as a function of field strength. Parameters, in reduced units of time (ℏ/*B*), are: (*a*) σ = 0.045, *T* = 0.1π, and (*b*) σ = 0.45, *T* = π. Thus, the time average in panel *a* corresponds, at each point, to 10 pulses and in panel *b* to one pulse.

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