Coherent Laser Manipulation of Ultracold Molecules 31width to the width of the thermal energy spread, gives the ratio of contributions from thebound state and the unperturbed continuum,
Trang 1Coherent Laser Manipulation of Ultracold Molecules 27atomic gas can be greatly facilitated by a Feshbach resonance The presence of a boundstate imbedded in and resonant with scattering continuum states strongly enhances thecontinuum-bound transition dipole matrix element to an excited electronic state, thusrequiring less laser intensity for efficient transfer In the limit of a wide resonance, whencompared to the thermal spread of collision energies, the dipole matrix element is enhanced
by the Fano parameter q By choosing a tightly bound excited vibrational state, q can be
made much larger than unity, resulting in the intensity of the pump pulse required forefficient adiabatic passage to be∼ 1/q2 times smaller than in the absence of the resonance.Numeical modeling of the adiabatic passage using typical parameters of alkali dimers showsthat intensities of the pump pulse, coupling the continuum to an excited state, of kW/cm2aresufficient for optimal transfer, which is∼100 times smaller than without resonance Optimalpulse durations are severalμs, resulting in energies per pulse ∼ 10μJ for a focus area of 1
mm2
If the Feshbach resonance is narrow compared to the thermal energy spread of collidingatoms, adiabatic passage is hindered by destructive quantum interference The reason is thatelectromagnetically induced transparency significantly reduces the transition dipole matrixelement from the scattering continuum to an excited state in the presence of the boundFeshbach state In the narrow resonance limit, photoassociative adiabatic passage is thereforemore efficient if the resonance is far-detuned
Due to low atomic collision rates at ultracold temperatures, only a small fraction of atomscan be converted into molecules by a pair of photoassociative pulses To convert anentire atomic ensemble, a train of pulse pairs can be applied We estimate that 104−106pulse pairs will associate an atomic gas of alkali dimers with a density 1012 cm−3 in anilluminated volume of 10−2 −10−3mm3in 0.1−10 s, resulting in extremely high productionrates of 105−108 molecules/s High transfer efficiencies combined with low intensities ofadiabatic photoassociative pulses also make the broad resonance limit attractive for quantumcomputation For example, a scheme proposed in (63) can be realized, where qubit statesare encoded into a scattering and a bound molecular states of polar molecules To performone and two-qubit operations, this scheme requires a high degree of control over the system,which our model readily offers
Finally, marrying FOPA and STIRAP is a very promising avenue to produce large amounts ofmolecules, for a variety of molecular species In fact, although we described here examplesbased on magnetically induced Feshbach resonances, such resonances are extremely common,and can be induced by several interactions, such as external electric fields or optical fields.Even in the absence of hyperfine interactions, other interactions can provide the necessarycoupling, such as in the case of the magnetic dipole-dipole interaction in52Cr (64; 65)
4 Conclusions
Precise control over internal and external degrees of freedom of molecules will open theway for new fundamental studies and applications in physics and chemistry As has beenclearly seen with atoms in the recent decades, well-controlled laser fields offer an exquisitecontrol tool over atomic internal and external states, including laser cooling and trapping,coherent manipulation of atomic quantum states and in particular various techniques usedfor quantum information applications, atomic spectroscopy Recent years have witnessedmastering of single atom manipulation in individual traps, including optical dipole traps and
79Coherent Laser Manipulation of Ultracold Molecules
Trang 228 Will-be-set-by-IN-TECHatom chips, and optical lattices, with most manipulation techniques relying on laser fields.There is a great incentive in the atomic and molecular optics community to extend the precisecontrol techniques developed for atoms to molecules We have outlined in this chapter someexperimentally relatively simple laser pulse techniques that can accomplish this task.
A prerequisite for many of the new studies is a high phase space density molecular sample in astable internal state, specifically in the ground rovibrational state and preferably in the lowesthyperfine sublevel We have in particular discussed two examples of coherent laser control
of molecular states, multistate chainwise STIRAP and photoassociatice adiabatic passage nearFeshbach resonance, which provide efficient transfer of molecules to the ground rovibrationalstate In chainwise STIRAP the transfer is based on a generalized dark state, which is asuperposition of all ground vibrational levels involved in the process Selecting a special timeorder of the laser pulses coupling vibrational states and optimizing durations and intensitiestransfer efficiencies > 90% are predicted even in the presence of fast collisional decay ofintermediate vibrational states This technique has recently been applied to transfer Cs2Feshbach molecules to the ground rovibrational state with 55% efficiency, limited by technicalissues Additionally, we outlined how the step from the atomic scattering continuum to theground rovibrational molecular state can be done in one coordinated step In the presence
of a Feshbach resonance delocalized scattering states acquire some bound-state character due
to admixture of a bound level associated with a closed channel It strongly enhances theFranck-Condon factor between the initial scattering state and a bound intermediate excitedmolecular state, a technique named Feshbach Optimized Photoassociation We analyzed thetransfer efficiency and intensities of the laser pulses required for optimal transfer both withand without the resonance and found that>70% efficiencies are possible with relatively lowintensity pulses of several W/cm2in the presence of the resonance
5 Acknowledgments
We gratefully acknowledge finantial support from NSF and AFOSR under the MURI awardFA9550-09-1-0588
6 Appendix
A Rotation and dephasing matrices
The Hamiltonian (2) in the case of the two-pulse STIRAP scheme, discussed in Section 2.1has a zero eigenvalue ε0 = 0, describing the dark state, and four eigenvalues, ε1,2 =
Trang 3Coherent Laser Manipulation of Ultracold Molecules 29
where terms of the order of O(Ω2/Ω2)and higher are neglected
The Liouville operator in the bare state basis has a form
B Adiabatic passage in the limits of broad and narrow Feshbach resonances
In this appendix, we discuss Eqs.(26) and (27) for various relative widths of the Feshbachresonance Γ with respect to the thermal energy spread δ of the colliding atoms We first
describe the case of a broad resonance, i.e when the width of the Feshbach resonance greatly
exceeds the thermal energy spread (Γ δ ), and second consider the opposite situation of anarrow resonance (Γ δ ) Finally, we briefly present the case where there is no resonance
B.1 Limit of a broad Feshbach resonance
The typical thermal energy spread for colliding atoms in photoassociation experiments withnon-degenerate gases isδ ∼ 10−100μK The broad resonance case occurs for resonances
having a width of several Gauss (∼ 1 mK), for which we haveΓ/δ ∼ 10−100 A widevariety of systems exhibit broad resonances For instance, they can be found in collision of
6Li atoms at 834 G for the| f =1/2, m f =1/2 ⊗ | f =1/2, m f = −1/2entrance channel(Γ=302 G= 40 mK) and in7Li at 736 G for the| f =1, m f =1 ⊗ | f =1, m f =1entrancechannel (Γ =145 G = 19 mK) We note here that these values ofΓ are slightly different thanthe “magnetic" widthΔB usually given and based on the modelling of the scattering length.
The source function can be readily calculated from Eq.(31) by noticing that the Rabi frequencyterm can be set at = 0 corresponding to the maximum of the Gaussian function in the
integrand Using the function g(q, )defined in Eq.(32), the result takes the form
S w=S0 √
2πδ g(q, 0)sgn( 0− F)e −(t−t0)2δ2/2¯h2 0/¯h−(ω S −ω p ))t
81Coherent Laser Manipulation of Ultracold Molecules
Trang 430 Will-be-set-by-IN-TECH
where Sno−res is the source function without a resonance given below in Eq.(47) Strictlyspeaking, this expression is valid for | F − 0| ≥ δ , but since Γ δ Eq.(38) is a goodapproximation for a wide range of detunings F − 0
The back-stimulation term (34) can be further simplified in the limit of a broad resonance In
this case, both c2(t) andE p(t)change on a time scale∼ 1/δ , i.e slowly compared to the
decay time∼ ¯h/Γ of the exponent Therefore, we can rewrite (34) as:
where Ωno−res = μ2 ˆe p E p /¯h is the continuum-bound Rabi frequency in the absence of
resonance We also added a spontaneous decay termγc2, assuming that the excited molecules
dissociate into high energy continuum states and the resulting atoms leave a trap FromEq.(38), one can see that in a broad resonance case, the source amplitude is enhanced
by the factor g(q, 0) = (q+2( 0− F)/Γ)/
1+4( 0− F)2/Γ2 when compared to theunperturbed continuum case This factor has a maximum at 2( 0− F)/Γ =1/q, with the corresponding maximum value gmax∼ q for q 1
B.2 Limit of a narrow Feshbach resonance
This situation occurs when the width of the resonance is of the order of a few micro-Gauss
or less Examples of narrow resonances include6Li23Na at 746 G for the| f1 =1/2, m f 1 =
1/2| f2 =1, m f 2 = 1channel (Γ = 7.8 mG = 1μK) (66), or6Li87Rb at 882 G for the| f1 =
by the integral The coefficientξ = Γ/√
2δ, which is the ratio of the Feshbach resonance
Trang 5Coherent Laser Manipulation of Ultracold Molecules 31width to the width of the thermal energy spread, gives the ratio of contributions from thebound state and the unperturbed continuum, respectively.
It is then easier to notice that in the limit of a narrow resonance, the Gaussian function in theintegrand of Eq.(42) is much narrower than the Bessel and Struve functions, which change onthe time scale∼1/ξ Therefore the source term can be aproximated as:
is seen from Eq.(43) that the contribution to the source function from the bound state decays
on the time scale| τ − τ0| ∼ 1/ξ, while the contribution from the unperturbed continuumdecays on the time scale| τ − τ0 | ∼11/ξ
In the limit of a narrow resonance the system (26)-(27) becomes:
B.3 Continuum without resonance
Finally, let us consider the case of a continuum without resonance In this case thecontinuum-bound Rabi frequency Eq.(29) is:
Ω =Ωno−res = μ2 · ˆe p E p /¯h, (46)and the source function is
Sno −res=S0 √
The back-stimulation term (34) reduces to
μ2 · ˆe p /¯h2π¯h E2c2=π¯h |Ωno−res(t )|2
83Coherent Laser Manipulation of Ultracold Molecules
Trang 632 Will-be-set-by-IN-TECHand the system (26)-(27) takes the simple form:
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Trang 94
Fast Charged Particles and Super- Strong Magnetic Fields Generated
by Intense Laser Target Interaction
Vadim Belyaev and Anatoly Matafonov
Central Research Institute of Machine Building
Russian Federation
1 Introduction
The development of a new generation of solid-state lasers has resulted in unique conditions
for irradiating laser targets by light pulses, with radiation intensity ranging from 1017 to 1021
W/cm2 and a duration of 20 - 1000 fs
At such intensities, the laser pulse produces superstrong electric fields which could not be
obtained earlier and considerably exceed the atomic electric field of strength E a = 5.14109
V/cm In these conditions, there arises a new physical picture of laser pulse interaction with
plasma produced when the pulse leading edge or a pre-pulse affects solid targets Laser
radiation is rather efficiently transformed into fluxes of fast charged particles such as
electrons and atomic ions The latter interact with the ambient material of the target, which
leads to the generation of hard X-rays, when inner atomic shells are ionized, and to various
nuclear and photonuclear reactions
One important area in investigating the interaction of sub-picosecond laser pulses with solid
targets is related to the important role which arising superstrong quasistatic magnetic fields
and electronic structures play in laser plasma dynamics This area of research became most
attractive after carrying out the direct measurements of quasistatic magnetic fields on the
Vulcan laser system (Great Britain) (Tatarakis et al., 2002), in particular, after the pinch effect
has been found experimentally in laser plasma (Beg et al., 2004)
The relativistic character of laser radiation with intensity I is realized at the magnitude of a
dimensionless parameter a > 1 This parameter represents the dimensionless momentum of
the electron oscillating in the electric field of linearly polarized laser radiation and can be
expressed as
1 2
0.85
μm 10 W/cm
eE I a
E I
Trang 10Femtosecond–Scale Optics 88
where e and m are the charge and mass of the electron, respectively, E is the amplitude of
electric field strength (in units of V/cm) of laser radiation, is the radiation wavelength (in
m), is the frequency of laser radiation, c is the speed of light, and I is the radiation
intensity (in W/cm2)
Terawatt-power laser systems of moderate size can fulfill the condition a > 1, which corresponds to the electric field strength above 1010 V/cm In such intense fields, the overbarrier ionization of atoms occurs in atomic time on the order of 10-17 s, and the electrons produced are accelerated and reach MeV-range relativistic energies during the laser pulse
The acceleration of atomic ions in femto- and picoseconds laser plasmas constitutes a secondary process It is caused by the strong quasistatic electric fields arising due to spatial charge separation Such separation is related to the motion of a bunch of fast electrons For
laser radiation intensities exceeding I 1018 W/cm2, it is possible to obtain directed beams
of high-energy ions with the energies i > 1 MeV
The generation of high-energy proton and ion beams in laser plasma under the action of ultrashort pulses is a quickly developing field of investigations This is explained, in particular, by their important applications in such fields as proton accelerators, the study
of material structure, proton radiography, the production of short-living radioisotopes for medical purposes, and laser controlled fusion (Umstadter, 2003; Mourou et al., 2006) For
a laser radiation intensity of I 1018 W/cm2, a number of nuclear reactions can be initiated that have only been realized in elementary particle accelerators (Andreev et al., 2001)
Later on, we will consider the principal mechanisms for generating fast charged particles and quasistatic magnetic fields in laser plasmas, as well as experimental results obtained both abroad and on the native laser setup NEODIM in the Central Research Institute of Machine Building (Russ abbr TsNIIMash) (Korolev, Moscow reg.) (Belyaev et al., 2004; Belyaev et al., 2005)
2 Generation of fast electrons in laser plasma
In irradiating a target by a high-intensity ultrashort laser pulse, the radiation energy is rather efficiently converted into the energy of fast electrons which later partially transfer their energy to the atomic ions of the target Presently, several mechanisms are being discussed concerning the generation of fast electrons when a laser pulse affects plasma with
a density well above the critical value If the laser pulse is not accompanied by a pre-pulse (the case of high contrast), then the laser radiation interacts with plasma of a solid-state density, possessing a sharp boundary In this case, the mechanism of `vacuum heating' is
realized (Brunel, 1987), as is the so-called vB mechanism (Wilks et al., 1992) (here, B is the
magnetic field induction of the laser field) caused by a longitudinal ponderomotive force
acting along the propagation direction of the laser pulse) This vB mechanism becomes
substantial at relativistic intensities where the energy of electron oscillations is comparable
with or exceeds the electron rest energy mc2 = 511 keV - that is, for the parameter a > 1 [see
formula (1)] In addition, fast electrons can be generated on the critical surface of plasma at a plasma resonance ( Gus’kov et al., 2001; Demchenko et al., 2001) if the vector of the laser radiation electric field has a projection along the density gradient (usually at an inclined incidence of laser radiation to target) and the laser frequency coincides with the plasma
Trang 11Fast Charged Particles and Super-Strong Magnetic
frequency In contrast to the ponderomotive vB mechanism, vacuum heating and
resonance absorption arise at nonrelativistic (substantially lower, with a < 1) intensities as
well In the case of the ponderomotive mechanism, the average energy of fast electrons can
be estimated as the maximum energy of transverse electron oscillations in an
electromagnetic field, which in the general case takes a relativistic value In a underdense
part of the laser plasma, we have
1 2 2
.2
mc Q
In the overdense part of the plasma, the ponderomotive heating of electrons is noticeably
weaker due to a difficult penetration of the laser field into this region
In the case of vacuum heating, the maximum energy of an electron flying into the depths of
a dense target is given by the formula similar to equation (3), however, with a different
numerical factor
There is one more mechanism for generating fast electrons in the underdense part of plasma
in front of a target due to the betatron resonance in the arising magnetic field (Pukhov,
2003) In this regime, electrons are accelerated by the transverse ultrarelativistic electric field
of the laser wave in the direction of wave polarization, and the azimuthal magnetic field
produced by the current of fast electrons is responsible for the magnetic part of the Lorentz
force This force turns electrons in such a way that they gradually change to the opposite
direction of motion In the case of an exact betatron resonance, the reflection occurs at the
instant when the transverse electric field changes its direction, so the electrons are
accelerated at all times This mechanism yields an energy of fast electrons three times
greater than formula (3) does:
1 2 2 0
mc Q
There are also further mechanisms of electron acceleration that require special experimental
conditions, for example, the wake field acceleration (Esarey, 1996; Amiranoff, 1998) In the
case of resonance absorption, the electric field near the plasma critical surface is much
stronger than that of incident laser radiation The result is that the heating of electrons upon
their impact with atomic ions is greater than follows from formulae (3) and (4)
Trang 12Femtosecond–Scale Optics 90
Electrons are also accelerated by a transverse ponderomotive force (acting in the radial
direction) due to a focal distribution of laser intensity This acceleration leads to the
maximum electron energy also expressed by formula (3) (in the underdense part of plasma)
if electrons succeed in acquiring this energy moving from the focus to the periphery during
the laser pulse Thus, the duration of a laser pulse should meet the inequality mR/eE
(in the nonrelativistic case) Here, R is the radius of the focal spot of a laser beam This
inequality holds for picosecond- and longer-duration laser light pulses with an intensity on
the order of 1016 W/cm2 In fields with an intensity of 1018 W/cm2, the right-hand side of
this inequality reaches dozens of femtoseconds, whereas in the overdense part of plasma
this ponderomotive force is noticeably weaker
We have discussed the above mentioned mechanisms in more detail in our article (Belyaev
et al., 2008)
We suggested and investigated the new mechanism of high-energy electrons formation in
ultra-high intensity laser pulse interaction with solid targets (Belyaev, 2004) This
investigation is an attempt to reveal and describe, based on the model suggested, the
high-energy electron formation mechanism in laser plasmas so as to derive theoretical
dependences which would represent specific relations between the parameters of fast
electrons, laser radiation and target substance
Any theory can be accepted only after reliable experimental verification The degree of
reliability is determined not only by the sufficient diversity of independent experimental
data, but also by the ability to choose out of these data those best representative of the
overall pattern Analysis of numerous experiments to measure energy of fast electrons
formed in laser plasmas shows that with a particular laser facility, given its available
radiation intensity, fast electron maximum energy can be determined most closely
Generally, it is electron maximum energy values that are most widely presented in
experimental investigations This is motivated not only by experimenters’ striving to get
extreme record-breaking output parameters, but also by the possibility to most closely
determine the electron maximum energy around their spectrum extrapolation at specified
intensity of laser radiation incident on a target On this basis we will establish our
theoretical model of the maximum-energy electron formation process for a given laser
radiation intensity
Without going into details of magnetic field generation mechanisms, it can be noted that a
vortical electron structure develops eventually in plasma Given the applied electric field
(constituent of the incident laser radiation) and the dominance of tunnel ionization, a great
number of electrons (practically determined by solid density) are accelerated This current of
electrons generates a magnetic field which bends their trajectory Under certain conditions
these trajectories can close at skin-layer depth within larmor-radius circle The high electron
density and, correspondingly, the circular current strength cause super-strong magnetic
fields generation
Condition for such fields generation can be written as a condition for electron movement
around such a circle in the form of a balance between the centrifugal force and the Lorentz
force:
2,
mV eVB
Trang 13Fast Charged Particles and Super-Strong Magnetic
where r = /2, - skin-layer thickness, e, m, V – charge, mass, electron velocity, c – velocity
of light, B – magnetic induction in the electron orbit
Taking electromagnetic field penetration depth to be equal to incident radiation wave
length , we have r/ 2
Given the relationship between mass and velocity, the kinetic energy change due to the
action of the forces applied is always equal to
where V – electron velocity;
- use of generalized momentum
e c
, (8)
where p – ordinary momentum (8), A – vector-potential;
- magnetic field B cylindrical symmetry: B X = 0; B Y = 0; X
To find the electron maximum kinetic energy at specified intensity of laser radiation
incident on the target we need the maximum value of B – magnetic field induced in laser
plasma This value can be estimated using the energy conservation law
Omitting calculations we can use the following formula easy to keep in mind:
Trang 14Femtosecond–Scale Optics 92
where intensity J expressed in W/cm2, – in micrometer, kinetic energy – MeV
Graph of this dependence show Fig 1 by curve 1
Fig 1 Dependence of electron kinetic energy on laser radiation intensity
Consider limiting cases
1,510
KIN J
and graph of this dependence show on Fig 1 by curve 3
Equations obtained for small (< m0c2) and large (> m0c2) values of kinetic energy agree with
those in use for calculations of particle energy in a cyclotron and in a betatron,
Trang 15Fast Charged Particles and Super-Strong Magnetic
correspondingly In both cases electrons are accelerated under the action of an electric field
In a cyclotron, this is a periodically changing electric field applied externally In a betatron,
this is a vortex electric field occurring with axisymmetric magnetic field rise in time In laser
plasmas a magnetic field is generated giving rise to a vortex electric field accelerating
electrons Thus the laser-plasma electron acceleration mechanism resembles the betatron
case
Equation (10) for electron kinetic energy was derived on the assumption that the electron
acceleration is governed only by the laser radiation incident on the target without
considering the processes going within the target substance, specifically, ionization process
Formally, it is reflected in the fact that the skin-layer size is determined by the laser
true only at the first stage of interaction with the substance when a vortical electron
structure develops on skin-layer scales, its characteristic size being in accordance with (16)
This structure is unstable and there is a possibility of its transformation to smaller-scale
structures This process is known as a dynamic pinch
It is demonstrated in (Belyaev & Mikhailov, 2001) that in case of laser plasmas produced by
the action of high-intensity (J > 1016 W/cm2) laser radiation of ultrashort duration ( < 10-12
sec) on a solid target this process is of quantum nature and can be described by the diffusion
equation Without going into the process nature, note that under tunnel ionization the
vortical electron structure generated on skin-layer scales (16) transforms to another one, its
characteristic size now being determined by the ionization frequency as an effective
frequency at the next stage of laser radiation interaction with the substance, i.e at the stage
of tunnel ionization development:
2
Assuming that the vortical electron structure transformation process goes with the magnetic
flow kept unchanged, we have
where B i – magnetic field within the vortical structure, its characteristic size l i, being
determined by (17) Such a vortical structure provides the following kinetic energy to the
electrons:
2 0
Equation (19) determines the maximum energy of the small group (tail) of high-energy
electrons This dependence can be represented via the energy or ionization potential of the
target substance atoms: