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A Molecular Stopwatch

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A Molecular Stopwatch PHYSICAL REVIEW X 1, 011002 (2011) Ensemble of Linear Molecules in Nondispersing Rotational Quantum States: A Molecular Stopwatch James P. Cryan,1,2,* James M. Glownia,1,3 Douglas W. Broege,1,3 Yue Ma,1,3 and Philip H. Bucksbaum1,2,3 1PULSE Institute, SLAC National Accelerator Laboratory 2575 Sand Hill Road, Menlo Park, California 94025, USA 2Department of Physics, Stanford University Stanford, California 94305, USA 3Department of Applied Physics, Stanford University Stanford, California 94305, USA (Received 16 May 2011; published 8 August 2011) We present a method to create nondispersing rotational quantum states in an ensemble of linear molecules with a well-defined rotational speed in the laboratory frame. Using a sequence of transform- limited laser pulses, we show that these states can be established through a process of rapid adiabatic passage. Coupling between the rotational and pendular motion of the molecules in the laser field can be used to control the detailed angular shape of the rotating ensemble. We describe applications of these rotational states in molecular dissociation and ultrafast metrology. DOI: 10.1103/PhysRevX.1.011002 Subject Areas: Atomic and Molecular Physics, Chemical Physics, Optics I. INTRODUCTION of the molecule, where circularly polarized photons of one rotational handedness are absorbed and the other handed- We present a method to create a rotating ensemble of ness are stimulated to emit. nondispersing quantum rotors based on ideas of strong- The use of strong-field laser pulses to create rotating J Þ 0 field rapid adiabatic passage. The ensemble has h zi molecular distributions was first considered in a series of and a definite alignment phase angle , which increases papers on the optical centrifuge [2–4]. The optical centri- linearly in time like the hands of a clock or a stopwatch. fuge creates a rotating ensemble by slowly increasing the A linear molecule with an anisotropic polarizability rotational speed of an angular trap. This process was experiences a torque when exposed to a strong laser field described semiclassically by Karczmarek [2]. The optical [1]. The magnitude and direction of the torque depends on the direction and strength of the incident electric field, and the corresponding pendulum potential can be viewed as an (a) (b) angular trap. We consider the response of an ensemble of such molecules to a pair of copropagating counterrotating circularly polarized laser pulses with a fixed frequency difference. The first pulse in the sequence creates an an- gular trap with a minimum in the polarization plane, cos which distorts the ensemble into an oblate distribution [see Fig. 1(c)]. The second pulse turns on before the first pulse turns off, so that the total laser field acquires a slowly rotating linear polarization. The molecules experience a rotating angular trap along the polarization direction, which rotates at half the beat frequency between the two Time (ps) pulses. The rotational speed of the angular trap is fixed, and (c) (d) the trap depth increases with the laser intensity. If the second pulse turns on sufficiently slowly, the ensemble adiabatically follows the eigenstates of the full Hamiltonian. This process adiabatically transforms the planar distribution of molecules into a rotating prolate distribution aligned along the rotating polarization. The quantum mechanical origin of this nondispersing rotation cos2 cos2 cos2 is stimulated Raman couplings among the rotational levels FIG. 1. h xi (blue), h yi (red), and h zi (green) (see text) for (a) short times and (b) long time in the laboratory (solid lines) and the rotating frame (dashed lines). (c) shows the *[email protected] oblate distribution formed by the first pulse at t ¼ 40 ps. (d) shows the prolate distribution in the rotating frame at time Published by the American Physical Society under the terms of t 840 ps. In (c) and (d), the compositeqffiffiffiffiffiffiffi laser field polarization the Creative Commons Attribution 3.0 License. Further distri- x U0 5:73 1:5 B bution of this work must maintain attribution to the author(s) and is along the axis. For this calculation, I 2 ¼ , ¼ @ , the published article’s title, journal citation, and DOI. 1 ¼ 40 ps, 2 ¼ 800 ps. 2160-3308=11=1(1)=011002(6) 011002-1 Published by the American Physical Society CRYAN et al. PHYS. REV. X 1, 011002 (2011) centrifuge technique has recently been adapted to use ÀÁ sin2 V ¼ ½ðjE j2 jE j2Þcos2 adiabatic passage to create a field-free rotating ensemble 2 x y which has been called a ‘‘cogwheel’’ state [5]. Rotating 2 þjEyj þ 2jExjjEyj cos sin; (6a) molecules are also analogous to Trojan wave packets in E 2 E 2 E t E t cos Á!t ; Rydberg atoms, first considered by Kalinski [6,7]. The rich j xj j yj ¼j 1ð Þjj 2ð Þj ð Þ (6b) 2 2 dynamics of driven rotor systems has also been recently 1 jE1ðtÞj jE2ðtÞj jE j2 ¼ þ studied by Topcu and Robicheaux [8]. y 2 2 2 The analysis here presents many interesting properties of these rotating ensembles that have not been reported for jE1ðtÞjjE2ðtÞj cosðÁ!tÞ ; (6c) the optical centrifuge. The orientation of the rotating po- E E 1 E t E t sin Á!t ; tential maps to the phase difference between the counter- j xjj yj¼2j 1ð Þjj 2ð Þj ð Þ (6d) rotating laser fields and the rotational speed is directly proportional to the laser frequency difference, so that this where Á! ¼j!1 À !2j, is the polar angle, and is the system can be used in timing and metrology applications. azimuthal angle, and again, the z axis is along the propa- We present the theory behind the creation of a rotating gation direction of the laser field. In Eq. (6) we have ensemble and a full density matrix calculation. integrated over the laser period, leaving only the slow frequency Á!. It is illustrative to consider the Hamiltonian in a frame II. METHOD Á! rotating with speed ¼ 2 . This is done through a gauge The effective Hamiltonian for a polarizable linear rigid transformation, UR ¼ expðil0=@Þ, where l0 ¼ I and rotor subject to an intense laser pulse of arbitrary polariza- I is the moment of inertia of the rotor. In this frame the tion is given by Hamiltonian in Eq. (1) is still appropriate, but there is an additional centrifugal term due to fictitious forces, H ¼ H0 þ V; (1) y 2 ðHÞR ¼ URHUR ¼ BJ þVR À Jz ¼ðH0ÞR þVR; (7) where 2 and the induced potential is greatly simplified to resemble H0 ¼ BJ ; (2a) a simple pendulum: Á V E~ 2; 2 ¼ j kj (2b) ÀÁ sin 2 V ¼ ½ðjE j2 jE j2Þcos2 þjE j2; R 2 x y y and we only include the terms relevant to the dynamics of (8a) the rotational state [9]. The angular momentum is quan- E 2 E 2 E t E t ; tized along the direction of propagation of the laser pulse, j xj j yj ¼j 1ð Þjj 2ð Þj (8b) 2 2 taken as the z axis. H0 is the field-free rotational 1 E1ðtÞ E2ðtÞ jE j2 ¼ þ jE ðtÞjjE ðtÞj : Hamiltonian, B is the rotational constant, and V is the y 2 2 2 1 2 (8c) potential induced by the laser electric field E~. The differ- ential polarizability of the rotor is Á ¼ð k À ?Þ, and Again in Eq. (8) we integrate over the laser period. ~ Ek is the projection of the electric field along the rotor axis. The Hamiltonian for this system is strikingly similar to Centrifugal distortion terms are absent since the rotors are that of a Trojan (nonspreading) state first proposed by rigid. Kalinski and Eberly [6,7]. The original works on Trojan If we consider two counterrotating circularly polarized wave packets in Rydberg atoms considered circularly po- fields, the total field is given by larized microwave fields which create a nonspreading state. ~ 1 We now proceed to calculate the dynamical evolution of E ¼ pffiffiffi f½E1ðtÞ cosð!1tÞþE2ðtÞ cosð!2tÞx^ 2 the first few eigenvalues and low-lying eigenstates of this system. For ease of calculation we consider only the turn- þ½E1ðtÞ sinð!1tÞE2ðtÞ sinð!2tÞy^g; (3) on of the laser fields. The field envelope functions EiðtÞ are sin2 where EiðtÞ is the electric field envelope of each pulse. This pulses described by equation can be rewritten as: 2 t E1ðtÞ¼E0 ½1ÀÂðtÀ1Þsin þÂðtÀ1Þ ; (9a) ~ ~ ~ 21 E ¼ Ek þ E? ¼ Exx^ þ Eyy;^ (4) 2 ðtÀonÞ E2ðtÞ¼E0 ½ÂðtÀonÞÂðtÀon À2Þsin 22 Ek ¼ sinðEx cos þ Ey sinÞ: (5) þÂðtÀon À2Þ ; (9b) Then Eq. (2b) reduces to 011002-2 ENSEMBLE OF LINEAR MOLECULES IN NON- ... PHYS. REV. X 1, 011002 (2011) where ÂðtÞ is the Heaviside step function. We perform (a) (b) calculations in both the laboratory and rotating frames. In either case the full density matrix is propagated using a split operator technique with a 100 fs step size. III. RESULTS cos h The adiabaticity condition for the first pulse is 1 2B . For the second pulse the adiabaticity condition is more complicated. We find that the turn-on, 2, should take place over multiple rotations of the linear field. In addition, 2 must also be slow compared to the energy spacing of the pendular states in the rotating frame. [] Within a single manifold of states with the same Jz M cos2 cos2 cos2 quantum number , the splitting between states increases FIG. 2. h xi (blue), hqffiffiffiffiffiffiffi yiq(red),ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and h zi (green) linearly with the field strength [10]. However, the splitting U Á jE1ðtÞjjE2ðtÞj 1:5 B (see text) as a function of I 2 ¼ 2I 2 . (a) ¼ @ between the ground states in different M manifolds de- B and (b) ¼ 3:5 .
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