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 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 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 !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 cosð !tÞ ; (6c) the optical centrifuge. The orientation of the rotating po- E E 1 E t E t sin !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 ! ¼j!1 !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 !. It is illustrative to consider the Hamiltonian in a frame II. METHOD ! rotating with speed ¼ 2 . This is done through a gauge The effective Hamiltonian for a polarizable linear rigid transformation, UR ¼ expð il0 =@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 Jz ¼ðH0ÞR þVR; (7) where
2 and the induced potential is greatly simplified to resemble H0 ¼ BJ ; (2a) a simple pendulum: V E~ 2; 2 ¼ j kj (2b) 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 ¼ð k ?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!2tÞ x^ 2 the first few eigenvalues and low-lying eigenstates of this system. For ease of calculation we consider only the turn- þ½E1ðtÞ sinð!1tÞ E2ðtÞ sinð!2tÞ y^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 ½1 ðt 1Þ sin þ ðt 1Þ ; (9a) ~ ~ ~ 2 1 E ¼ Ek þ E? ¼ Exx^ þ Eyy;^ (4) 2 ðt onÞ E2ðtÞ¼E0 ½ ðt onÞ ðt on 2Þ sin 2 2 Ek ¼ sin ðEx cos þ Ey sin Þ: (5) þ ðt on 2Þ ; (9b) Then Eq. (2b) reduces to
011002-2 ENSEMBLE OF LINEAR MOLECULES IN NON- ... PHYS. REV. X 1, 011002 (2011) where ðtÞ 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 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 . In both plots, 1 ¼ 40 ps, 2 ¼ 800 ps. creases with increasing field strength. As the second pulse @ turns on, the different M manifolds are coupled by the sin2 cos2 operator. This means that the spacing between ground states of different M manifolds will determine the (a) (b) adiabatic time scale. In addition to the energy shift caused by the external field, in the rotating frame the energy of each M manifold is shifted by the centrifugal term of the Hamiltonian, Eq. (7). For a given rotational speed, the @ manifold with M closest to B 1 has a ground state energy near the M ¼ 0 manifold ground state energy. Therefore, we find the adiabaticity condition for the second pulse is h 2 ; (10) ð ;E0Þ @Kð Þ where ð ;E0Þ is the splitting between the M ¼ 0 and @ (c) (d) M ¼ Kð Þ¼Int½ B 1 manifolds of the pendular states formed by a circularly polarized field in a nonrotating frame. In this work, 1 ¼ on ¼ 40 ps and 2 ¼ 800 ps, to satisfy the adiabaticity conditions. The results of the calculations are shown in Figs. 1–3. These figures show the projection of the ensemble x cos2 Tr sin2 cos2 y 2 2 2 onto the axis ½h xi¼ ð Þ , axis FIG. 3. hcos xi (blue), hcos yi (red), and hcos zi (green) 2 2 2 2 ½hcos yi¼Trð sin sin Þ , and z axis ½hcos zi¼ (see text) for (a) short times and (b) long times in the laboratory Trð cos2 Þ in both the rotating and laboratory frames (solid lines) and the rotating frame (dashed lines). (c) shows the [9]. Figures 1 and 3 also show the calculated probability distribution at t ¼ 165 ps. (d) shows the prolate distribution in density of molecules, jc ð ; Þj2 ¼h ; j j ; i at differ- the rotating frame at time t 840 ps. In (c) and (d), the com- x ent times during the pulse sequence. qpositeffiffiffiffiffiffiffi laser polarization is along the axis. For this calculation, U0 2:45 3:5 B 40 ps 800 ps We find that the degree of alignment along the rotating I 2 ¼ , ¼ @ , 1 ¼ , 2 ¼ . polarization axis increases linearly with the potential well 2 E0 depth, U0 ¼ 2 , as in the nonrotating, linearly polarized case. The rotational frequency has a more complicated IV. DISCUSSION 2:5B effect on the system: above a critical value of @ ¼ The quantum behavior for > crit can be understood crit, there is a marked change in the evolution of the initial by viewing the system in the rotating frame given in state through the adiabatic pulse sequence, which is shown Eqs. (7) and (8). To simplify this discussion we will con- in Fig. 2. The wave function splits and acquires and addi- sider the transformation of a pure state, which we can tional node. The exact value of crit does have a slight obtain by assuming a rotational temperature of 0 K. dependence on E0, and the value quoted above is for Using this simplification, we can now restrict ourselves 2 E0 74B. to only considering states with even values of the quantum
011002-3 CRYAN et al. PHYS. REV. X 1, 011002 (2011) numbers J and M, since the potential V [Eqs. (6) and (8)] Hamiltonian, where the two eigenvalues exhibit an avoided only couples states with J ¼ 0, 2, M ¼ 0, 2 [9]. crossing. In the rotating frame we can write the Hamiltonian as In this calculation, all of the molecules start in the ground state of the laboratory frame, jJ ¼ 0;M¼ 0i. HR ¼ Hp Jz; (11) But, this is not necessarily the ground state of the rotating frame due to the fictitious forces associated with a non- H where p is the pendular Hamiltonian given by inertial reference frame. In fact, when the rotational speed of the trap is above crit the molecules are in the first 2 2 2 2 Hp ¼ BJ sin ½E1ðtÞE2ðtÞcos þjEyðtÞj : (12) excited state of the system in the rotational frame prior to 2 the turn-on of the second field. Therefore, an adiabatic transformation will leave the system in the first excited We will expand the solution of HR in the basis of state of the rotational frame Hamiltonian, after passing solutions to Hp. Figure 4 shows the diagonal matrix ele- through the avoided crossing described above. This state ments of HR expressed in this basis (the basis of the non- has a node in the angular coordinates, which is shown in rotating pendular potential). For > crit, there is a value of the electric field for which these two matrix elements Fig. 3. Now the structure of the ensembles shown in Figs. 1 become degenerate. This degeneracy is lifted in the full and 3 can be clearly understood in terms of the avoided crossings described above. This result is qualitatively different from the previously reported optical centrifuge, where the system remains in the ground state of the instantaneous rotational frame Hamiltonian due to the adiabatic increase of the angular speed of the trap. However, the method presented here is truly adiabatic, in that if the distribution begins in the ground state of the field-free Hamiltonian in the laboratory frame, then it will be in the ground state of the Hamiltonian (a) which includes the field that is also in the laboratory frame. If > crit, then the rotational and laboratory frame ground states are not the same. Also, the sharp nodal features seen here are suppressed at finite temperature, where an incoherent superposition of initial M states ef- fectively fills in the nodes that appear above crit. In rapid adiabatic passage, the order of operations of adiabatic transformations is important. For the system described in this work, there are two parameters (angular trap depth, U0, and rotational speed, ) which can be (b) varied adiabatically. The difference between the method presented here and the optical centrifuge is in the order of adiabatic transformations. The optical centrifuge first adia- batically increases the angular trap depth and then the rotational speed, which leaves the ensemble in the ground state in the rotating frame. The method presented here starts with a rotating polarization and then slowly changes the rotating angular trap depth. The final state of the system depends on the rotational speed of the trap. In either method, the final parameters of the system are the same, (c) but the final states can be very different. The parameter space is shown in Fig. 5, along with the paths followed by both the optical centrifuge method and the method pre- sented here. FIG. 4. Diagonal matrix elements of the rotating frame This analysis has focused on the dynamical evolution of Hamiltonian, HR Eq. (7), expanded in the basis of eigenstates the initial state through the pulse sequence to a final state of of the stationary pendulum, Hp Eq. (12) (dashed lines). When the system where the magnitude of the two counterrotating the rotating frame Hamiltonian is diagonalizied, the eigenvalues fields are equal. If the two pulses have unequal magnitudes, exhibit avoided crossing when > crit (solid lines). then the total field resembles a slowly rotating linear B ¼ 1:75 (a), 2.7 (b), and 3.75 (c) in units of @ . polarization superposed on a residual circularly polarized
011002-4 ENSEMBLE OF LINEAR MOLECULES IN NON- ... PHYS. REV. X 1, 011002 (2011)
Nondispersing rotational states expand these techniques to a rotating reference frame. This type of rotating ensemble will allow investigation into the breakdown of the axial recoil approximation. By dissociating the rotating en- −2 semble with a femtosecond probe pulse, the validity of −4 the axial recoil approximation can be tested for various −6 rotational speeds. −8 In addition, a rotating ensemble could be used as a tool
−10 to find the timing between two laser pulses from different sources. In this application, the rotating ensemble could be −12 established using a long pulse ( 4ns) infrared laser −14 0 source. Two other femtosecond laser sources then ionize −16 and dissociate the rotating linear molecules. The dissocia- −18 2 ω tion products are collected with a position sensitive 0 p 1 4 detector, and the angular displacement between ion frag- 2 Ω 3 ments maps directly to the relative time delay between 4 6 the two pulses. Relative delays could be measured down to 20 fs resolution, with the appropriate choice of target gas and frequency shift. The nondispersive properties and FIG. 5. Energy eignvalues of the rotatingq frameffiffiffi Hamiltonian, uniform rotational speed of the molecular state are not U HR Eq. (7), in units of B, in the ( , !p ¼ I ) parameter space. dependent on intensity, and so spatial averaging over the B laser focus does not destroy these features. The detailed The x and y axes are in units of @ , and the color of the curve represents the splitting between the eignevalues. Also shown are angular distribution does depend on intensity, and, in par- the paths followed by the optical centrifuge [2] (green) and the ticular, the bifurcation of the wave packet shown in Fig. 2 method described here (magenta). The insets show jc j2 for will only appear in parts of the focused laser beam where different points in the parameter space. the intensity exceeds a critical threshold. However, if the focal spot size of the probe is much smaller than the adiabatic pump laser, the focal volume averaging effects field. This type of field creates two angular traps of differ- are minimal. ent strength, one in the dimension and one in the As a further extrapolation of this technique, we could dimension. The dynamic evolution adiabatically follows envision a pulse sequence which uses the optical centrifuge the ground state of the system throughout the pulse se- method to move the distribution into the ground state of the quence, and we notice an interesting behavior of the rotational frame Hamiltonian, and then turn off the pulse ground state of the system when the two fields are unbal- envelopes, while maintaining the constant frequency dif- anced and > crit. Figure 3 clearly shows that for small E ference, as in the technique described in this article. The values of 2, the ensemble of molecules localizes around ensemble will now be left in the ground state of the rota- the unstable equilibrium point of the classical pendulum. tional frame field-free Hamiltonian, which is the state For such small values of E2, the angular well depth in the jJ ¼ 2;M¼ 2i so long as > crit. A subsequent appli- U E t E t dimension, ¼ 2 1ð Þ 2ð Þ, is much smaller than the cation of the same pulse sequence will return the ensemble centrifugal term Jz of the Hamiltonian. The presence of to the laboratory ground state, jJ ¼ 0;M ¼ 0i. Actually this centrifugal term causes a dynamical stabilization of the the ground state of the rotational frame field-free unstable equilibrium point, but as E2 grows the stabiliza- ðJþ1ÞB Hamiltonian changes every time ¼ @ , so by varying tion is lost and the ensemble migrates back toward local- the frequency difference of the counterrotating pulses, it ization around the stable equilibrium point of the classical should be possible to move the population to any maximal pendulum. M-state, i.e., jJ; M ¼ Ji, by choosing the appropriate rota- tional speed. V. CONCLUSIONS We have presented a method of strong-field adiabatic ACKNOWLEDGMENTS control of linear molecules which results in a nondispers- The authors would like to thank C. Bostedt and R. ing rotational state with hJzi¼I . This state can be Coffee for discussions on timing and metrology applica- thought of as rotating at a constant angular speed in tions that motivated us to initiate this work, and Y. the polarization plane. Silberberg for discussions on adiabatic passage. This re- Adiabatic control of molecular rotational wave packets search is supported through the PULSE Institute at the has already been used to align molecules in the laboratory SLAC National Accelerator Laboratory by the U.S. frame to facilitate molecular frame measurements. [11,12] Department of Energy, Office of Basic Energy Science.
011002-5 CRYAN et al. PHYS. REV. X 1, 011002 (2011)
[1] H. Stapelfeldt and T. Seideman, Colloquium: Aligning [8] T. Topcu and F. Robicheaux, Multiphoton Population Molecules with Strong Laser Pulses, Rev. Mod. Phys. Transfer in Systems Violating the Classical Twist 75, 543 (2003). Condition: A Comparative Study of Separatrix Crossing [2] J. Karczmarek, J. Wright, P. Corkum, and M. Ivanov, in Phase Space, Phys. Rev. E 83, 046607 (2011). Optical Centrifuge for Molecules, Phys. Rev. Lett. 82, [9] D. Daems, S. Gue´rin, E. Hertz, H. R. Jauslin, B. 3420 (1999). Lavorel, and O. Faucher, Field-Free Two-Direction [3] D. M. Villeneuve, S. A. Aseyev, P. Dietrich, M. Spanner, Alignment Alternation of Linear Molecules by M. Y. Ivanov, and P.B. Corkum, Forced Molecular Elliptic Laser Pulses, Phys. Rev. Lett. 95, 063005 Rotation in an Optical Centrifuge, Phys. Rev. Lett. 85, (2005). 542 (2000). [10] B. Friedrich and D. Herschbach, Alignment and Trapping [4] M. Spanner, K. M. Davitt, and M. Y. Ivanov, Stability of of Molecules in Intense Laser Fields, Phys. Rev. Lett. 74, Angular Confinement and Rotational Acceleration of a 4623 (1995). Diatomic Molecule in an Optical Centrifuge, J. Chem. [11] L. Holmegaard, J. L. Hansen, L. Kalhoj, S. L. Kragh, H. Phys. 115, 8403 (2001). Stapelfeldt, F. Filsinger, J. Kupper, G. Meijer, D. [5] M. Lapert, S. Gue´rin, and D. Sugny, Field-Free Quantum Dimitrovski, and M. Abu-samha et al., Photoelectron Cogwheel by Shaping of Rotational Wave Packets, Phys. Angular Distributions from Strong-Field Ionization of Rev. A 83, 013403 (2011). Oriented Molecules, Nature Phys. 6, 428 (2010). [6] M. Kalinski and J. H. Eberly, New States of Hydrogen in a [12] J. L. Hansen, H. Stapelfeldt, D. Dimitrovski, M. Circularly Polarized Electromagnetic Field, Phys. Rev. Abu-samha, C. P.J. Martiny, and L. B. Madsen, Time- Lett. 77, 2420 (1996). Resolved Photoelectron Angular Distributions from [7] M. Kalinski and J. H. Eberly, Trojan Wave Packets: Strong-Field Ionization of Rotating Naphthalene Mathieu Theory and Generation from Circular States, Molecules, Phys. Rev. Lett. 106, 073001 (2011). Phys. Rev. A 53, 1715 (1996).
011002-6