Non-adiabatic dynamics of molecules in optical cavities

0.37 MB·

by  Markus Kowalewski

Non-adiabatic dynamics of molecules in optical cavities
Markus Kowalewski,1,a) Kochise Bennett,1 and Shaul Mukamel1,b)
Department of Chemistry, University of California, Irvine, California 92697-2025,
USA
(Dated: 15 January 2016)
Strong coupling of molecules to the vacuum field of micro cavities can modify the potential energy surfaces
openingnewphotophysicalandphotochemicalreactionpathways. Whiletheinfluenceoflaserfieldsisusually
described in terms of classical field, coupling to the vacuum state of a cavity has to be described in terms
of dressed photon-matter states (polaritons) which require quantized fields. We present a derivation of the
non-adiabatic couplings for single molecules in the strong coupling regime suitable for the calculation of the
dressed state dynamics. The formalism allows to use quantities readily accessible from quantum chemistry
6
codes like the adiabatic potential energy surfaces and dipole moments to carry out wave packet simulations
1
in the dressed basis. The implications for photochemistry are demonstrated for a set of model systems
0
2 representing typical situations found in molecules.

n
a
I. INTRODUCTION photondegreesof freedomare heavilyentangledand the
J
molecular bare states do not provide a good basis. The
4
1 Thefateofamoleculeafterexcitationwithlightisde- quantizationoftheradiationfieldhastobetakenintoac-
terminedbyitsexcited(bare)statepotentialenergysur- count. This has been first described theoretically in the
] face. While most molecules make their way back to the Jaynes-Cummings model15 which assumes an electronic
h
ground state by spontaneous emission or non-radiative two level system coupled to a single field mode and has
p
- relaxation, some dissociate, isomerize, or are funneled beenexperimentally appliedto atoms16 Inmolecules the
nt throughconicalintersections(CoIn)1. Thereactivitycan nuclear degrees of freedom must be taken into account
a be manipulated either by chemical modification, chang- as well14. The product basis of electronic and photonic
u ing the environment, or by using photons to interact states is notadequate in the strongcoupling regime. Di-
q with the molecule while it evolves in an excited states. agonalizing the system to the dressed basis recovers po-
[ It has been shown theoretically2 and experimentally3,4 tential energy surfaces but also leads to light induced
1 that light can actively influence the molecular reactiv- avoided crossing and actual curve crossings between the
v ity. The modified photonic vacuum in nano scale fabri- dressedstates,analogoustoavoidedcrossingsandCoIns.
4 cated cavities allows for influencing molecular potential The dynamics of the nuclei, electrons, and photons are
9 energy surfaces in a nondestructive manner and without strongly coupled in the vicinity of these crossings and
6 the useofexternallaserfields. Substantialcouplingscan pose formidable computational challenge.
3
0 be induced between electronic states with just a single Strong coupling to one or more radiation field modes
. photon5. The radiation matter coupling is enhanced in can alter the molecular levels, profoundly affecting the
1 smallcavitymodes6 andthe strongcouplingregimemay basic photophysical and photochemical processes. The
0
6 be realizedevenwhen the fieldis in the vacuumstate7,8. obvious way to achieve this regime is by subjecting the
1 Thestronglycoupledmolecule+fieldstatesareknownas molecule to strong laser fields4,17–19. Alternatively the
: dressed atomic states9 or polaritons10. Recent experi- coupling can be enhanced by placing the molecule in
v
mental developments show promising results, paving the a cavity and letting it interact with the localized cav-
i
X way for strong coupling in the single molecule regime. ity modes. The coupling increases with 1/√V, where V
r Strongcouplingcanbe achievedinnanocavities11,nano is the mode volume. Strong fields are not necessary in
a plasmon antennas12, and nano guides13. Chemical re- this case and the field can be even in the vacuum state.
activity can be influenced in a very distinct way in this The former scenario can be realized with classical fields.
regime. This provides great potential for the manipula- Thispaperfocusesonthislatter,whichinvolvesquantum
tion and control of e.g. the photo-stability of molecules, fields15. A major difference between the two scenarios is
noveltypes oflightinduced CoIns,ormodifying existing the number of photons available in the dressing field. A
CoIns. Specially tailored nano structured materials may stronglaserfieldcangiveriseto multiphoton absorption
then serve as a photonic catalysts that can be used in- and multiphoton ionization pathways that can interfere
steadofchemicalcatalysts. Inarecenttheoreticalpaper with the intended manipulation of the quantum system.
Galegoetal.14 pointedouttheimpactofstrongcoupling
We develop a formalism, which allows to express the
on the absorption spectrum of molecules.
dressed states and the non-adiabatic couplings in terms
In the strong coupling regime the molecular and the
of readily accessible molecular properties like the bare
state potential energy surfaces and the transition dipole
moments that can be extracted from standard quantum
a)Electronicmail: [email protected] chemistry calculations. We demonstrate how chemical
b)Electronicmail: [email protected] reactions can be modified by applying this theoretical
2
frameworktotypicalmodelsystems. Wefocusonamod- The eigenstates of H are the dressed (polariton)
JC
erate coupling strength where the dressed state energies states ,n :
c
|± i
arenotwellseparatedbutexperiencecurvecrossingsgiv-
+,n = cosθ e,n +sinθ g,n +1 (6)
ing rise to non-adiabatic dynamics. c c c
| i | i | i
The paper is structured as follows. In section II we ,n = sinθ e,n +cosθ g,n +1 , (7)
c c c
|− i − | i | i
present the formalism by including the nuclear degrees
where the mixing angle θ is
of freedom into the Jaynes-Cummings model. In sec-
tion III we present three models of molecular systems
Ω δ Ω +δ
strongly coupled to the cavity. The photonic catalyst cosθ = n− c, sinθ = n c (8)
model couples a bound state to a dissociative state, ef- r 2Ωn r 2Ωn
fectivelyopeningadecaychannel,decreasingitslifetime.
with the corresponding eigenvalues
Thephotonicboundstatemodeldemonstrateshowstim-
ulated emission from the vacuum state can increase the ~ 1 ~
lifetime of a otherwise unbound state. Finally, forming E±,n = ω0+~ωc nc+ Ωn, (9)
2 (cid:18) 2(cid:19)± 2
light induced conical intersections in a cavity mode is
demonstrated on the formaldehyde molecule. and Ω is the Rabi-frequency
n
Ω = 4g2(n +1)+δ2. (10)
n c c
II. THEORETICAL FRAMEWORK p
The molecule-cavity detuning
WeusetheJaynes-Cummings(JC)model15todescribe
the coupling of the resonator to the molecular dipole δc =ω0 ωc =(Ve Vg)/~ ωc (11)
− − −
transition.
represents the frequency miss-match between the molec-
ulartransitionandthe cavitymode. HereE andE are
H =H +H +H , (1) e g
JC M C I
the eigenvalues of the bares states. We assume that the
whereHM isthemolecularHamiltonianrepresentingtwo cavity is initially in the vacuum state (i.e. nc = 0) and
electronic states omit the photon number nc in the following.
~
H = ω σ†σ σσ† , (2)
M 2 0 − A. The molecular Hamiltonian in the strong coupling
(cid:0) (cid:1)
regime
H is cavity Hamiltonian of a single quantized photon
C
mode
The originalJC model was developed for atomic tran-
1 sitions and does not include nuclear degrees of freedom.
HC =~ωc a†a+ , (3) The molecular potential energy surfacesbecome coupled
(cid:18) 2(cid:19)
whentheelectronicgroundandexcitedstategetintores-
and H describes the interaction between the photon onancewiththecavitymode. Thenuclearandelectronic
I
mode and the molecule motionswillthenbecoupledandtheBorn-Oppenheimer
approximationbreaks down.
HI =~g a†σ+aσ† . (4) To obtain the couplings in the dressed state basis we
(cid:0) (cid:1) include the dependence of the nuclear coordinates q =
Here σ† = e g and σ = g e are the creation and (q ,...,q ) into the JC model. The quantities δ , Ω , g
| ih | | ih | 1 N c n
annihilation of a molecular excitation of the molecular dependparametricallyonq,andthe mixing angleθ (Eq.
eigenstates in the electronic subspace g and e . The 8) also becomes a function of the nuclear coordinates.
| i | i
excitationenergybetweenthebareeigenvaluesωg andωe The new dressed potential energy surfaces can then be
is ω0 = ωe ωg. The cavity mode with frequency ωc is expressedintermsofthedressedstateeigenvaluesofEq.
describedb−ythe eigenstates nc 0 , 1 ,... . a† anda 9:
| i≡| i | i
arethebosoniccreationandannihilationoperatorsofthe
cavity mode. The interactionH is given in the rotating 1 ~
I V = (V +V ) Ω (12)
wave approximation (RWA), where g = ε µ /2~ is the ±,0 2 e g ± 2 0
c eg
coupling strength. The RWA holds when δ ω and V =V (13)
c 0 g,0 g
≪
g ω . µ is the molecular transition dipole moment
0 eg
≪
and εc is the cavity vacuum field, where Vg ≡ Vg(q) and Ve ≡ Ve(q) are the bare state
potential energy surfaces of the free molecule.
~ω We follow the standard procedure to derive the non-
c
εc =rVǫ , (5) adiabatic coupling terms in the adiabatic basis1,20,21.
0 Atomicunitsareusedinthefollowing(~=m =4πǫ =
e 0
where V is the resonator mode volume. 1). Insteadofthebareadiabaticelectronicstates,weuse
3
the dressed states from Eqs. 6 denoted φ φ (r,q) , Thesetermsmaybesafelyneglectedwhenthebarestate
k k
| i≡| i
where r = (r ,...,r ) are the electronic coordinates. energiesarewellseparated. Notethatalldiagonalmatrix
1 M
The totalwavefunctionis expandedinthe adiabaticba- elements of f vanish (f =0).
kk
sis: To evaluate the scalar coupling terms h of the sec-
ond derivatives we introduce the following decomposi-
Ψ= ψ (q)φ (r,q) (14)
k k tion,whichbreaksdownthe equationsandsimplifies the
Xk results.
where k runs over the set of dressed states
h(i) =∂ f(i) F(i) (23)
(g,0 , ,0 , e,1 ). The full molecular Hamiltonian kl qi kl − k,l
| i |± i | i
Hˆ =Tˆ+Hˆ (15) The second term F(i) = ∂ φ ∂ φ now contains also
el k,l h qi k| qi li
diagonal contributions:
consists of the nuclear kinetic energy term
Λ2 δ2Λ2
1 ∂2 F(i) =F(i)sin2θ+F(i)cos2θ+ i + c i (24)
Tˆ = (16) +,+ g,g e,e 4 16g2
− 2m ∂q2
Xi i i Λ2 δ2Λ2
F(i) =F(i)cos2θ+F(i)sin2θ+ i + c i (25)
and the electronic part Hˆ with the parametric eigen- −,− g,g e,e 4 16g2
el
values Vg(q) and Ve(q). Taking the matrix elements F(i) =sinθcosθ F(i) F(i) (26)
ΨHˆ Ψ and integrating over r yields: −,+ (cid:16) g,g − e,e(cid:17)
h | | i Λ f(i)
1 ∂ 1 F(i) =F(i)cosθ+ i ge (27)
Hˆ =Tˆ+δ Vˆ + f(i) + h(i) (17) g,+ g,e 4cosθ
kl kl kl m (cid:18) kl ∂q 2 kl(cid:19)
Xi i i Λ f(i)
F(i) = F(i)sinθ+ i ge (28)
where f and h recover the derivative coupling term and g,− − g,e 4sinθ
thescalarcouplingastheyappearinthetheoryofCoIns:
with
fk(il)(q)=hφk(q)|∂qi|φl(q)ir (18) Λ = δc 4g∂g +δ ∆G ∆Gi (29)
hk(il)(q)=hφk(q)|∂q2i2|φl(q)ir (19) i Ω3 (cid:18) ∂qi c i(cid:19)− Ω
No assumptions have been made on the bare electronic f(i) andF(i) containallpossiblecouplings: intrinsicnon-
kl kl
states. This result holds even if V and V undergo a adiabaticcouplingsofthe barestatesandcavityinduced
g e
crossing. In the following we discuss the relevant matrix non-adiabatic couplings. The Hamiltonian eq. 17 thus
elements of f and h in the dressedstates basis and show describes the dynamics in the most general case. The
how the cavity affects the non-adiabatic couplings. only approximation made is the RWA and the condition
Inserting the definitions of the dressed states (Eqs. 6) thatthe systemcannotaccesshigherphotonstates dur-
into Eq. 18 yields the derivative coupling term between ingthetimeevolution. Forverylargedetuningsδc higher
,0 and +,0 : photon states (nc >1) must be taken into account.
|− i | i
The non-adiabaticcouplingsmay be further simplified
∆G δ2 δ ∂g
f(i) = i 1 c c (20) in specific parameter regimes. Assuming that the bare
−,+ 4g (cid:18) − 4g2+δc2(cid:19)− 4g2+δc2∂qi states are well separated in energy and do not undergo
any curve crossings, all terms f(i), F(i), F(i), and F(i)
where ∆G = ∂ (V V ) is the gradient difference. g,e g,g e,e g,e
The dresseid stateqicoeup−lingg has two contributions: The may be neglected. The Fg(,ie) terms are usually neglected
first term is governed by the gradient difference of the inmoleculardynamicssimulationsandquantumdynam-
two bare states PESs, whereas the second term depends ics of the bare states. Note that F−,+ does not con-
on the gradientof the transition dipole moment through tain any contribution from the cavity. F(i) vanishes for
±,±
∂qig. The latter vanishes in the Condon approximation small gradient differences and in the Condon approxi-
but may be substantial in regions where the transition mation and may also be neglected in most cases, since
dipole varies rapidly with q. Note that Eq. 20 does they only make a minor contribution to the shape of the
not contain any coupling terms involving the bare state PESs. Dropping all F terms leads to the approximate
crossings (f(i)) since these couplings vanish due to the Hamiltonian:
g,e
orthogonalityofthe photonstates. This is incontrastto
1 ∂ ∂
the couplings between the groundand the dressedstates Hˆ =Tˆ+δ Vˆ + 2f(i) + f(i)
kl kl kl 2m (cid:18) kl ∂q ∂q kl (cid:19)
which solely contain the bare state derivative couplings Xi i i i
but no contribution from the cavity: (30)
f(i) =f(i)cosθ (21) The hermitian Hamilton operator (Eq. 30) will be used
g,+ g,e
in the following to calculate the wave packet dynam-
f(i) = f(i)sinθ (22)
g,− − g,e ics. Hamiltonians with this structure are commonly
4
a) a)
8
S 6 |+i
6 2
5
V [eV] 4 S1 V [eV] 4 |−i
2 3
S S
0 1
0 2
1 2 3 4 5 6 2 3 4 5 6
q [Å] q [Å]
b) b)
4
0
3
i
−
µ 2 ∂|q -0.5
+|
h
1
-1
0
2 3 4 5 6
1 2 3 4 5 6
q [Å]
q [Å]
FIG. 2. (a) Dressed state PESs for the photonic catalyst
FIG. 1. (a) Bare state PESs used for the photonic catalyst
model. TheS state isnow coupled tothedissociative state.
model as well as thephotonic bound state model. The mini- 2
(b) Non-adiabatic coupling matrix element for the polariton
mumof theS stateisdisplaced by0.3˚Awith respect tothe
2 states. Dominated by the gradient difference term of eq. 20.
groundstate. (b)Transitiondipolecurveusedinthedifferent
Parameters: g=54meV
models. TheparametersforthemodelaregiveninAppendix
A.
cavitycoupling andthe effects onthe non-adiabaticcou-
used to simulate the dynamics in the vicinity of Coni- plings. The level structure of the dressed states creates
cal intersections20 by numerical propagationof the wave new pathways for the nuclear dynamics and new tran-
function. This is done by using a grid in the nuclear co- sitions for spectroscopic measurements. Our goal is to
ordinates, rather than expanding in nuclear eigenstates usetheinfluence ofthe cavitytomodify thereactivityof
which scales unfavorably with the number of nuclear a molecule. Photodissociation in the dressed state basis
modes. Hereafter we use this approach. can then be enhanced or suppressed.
Operatorswhichrepresentmolecularpropertiescanbe
expressed in the bare state basis by transforming them
into the dressed state basis using Eqs. 6 to 8. The tran- A. Photonic Catalyst
sition dipole moments then read:
In the first model (Fig. 1(a)) we assume that a bound
+,0µˆg,0 =cosθµ (31)
h | | i eg state S is accessible by a dipole transition from the
2
,0µˆg,0 = sinθµ (32)
eg groundstate S . The dissociative state S does not have
h− | | i − 0 1
+,0µˆ ,0 =cosθsinθ(µee µgg) , (33) a transitiondipole moment withthe groundstate, butis
h | |− i −
accessible from S . The cavity couples the states S and
2 2
where µ is the bare state electronic transition dipole
eg S1 through the transition dipole moment shown in Fig.
moment and µgg and µee are dipole moments of the 1(b). Thecavitymodefrequencyωcissetbeinresonance
ground and excited state respectively.
at the minimum of S (1.45eV) and with a maximum
2
cavity coupling of g = 54meV. The states g S and
1
| i ≡
e S are used along with Eq. 13 to form the dressed
2
III. PHOTOCHEMISTRY IN THE STRONG COUPLING s|tiat≡es, shown in Fig. 2(a). The resulting dressed states
REGIME S , , and + undergo an avoided crossing close to
1
| i |−i | i
resonance, while their shape remains similar to the bare
Inthefollowingwepresentcalculationsonthreesimple states. Thecorrespondingnon-adiabaticcouplingmatrix
model systems to illustrate the basic possibilities of the elementf (Eq. 18), whichis responsible for the tran-
+,−
5
4
a)
1
u]
a 2
[
µ aa0.5
ρ
0
2 3 4 5 6
0
q [Å]
0 0.5 1 1.5 2
t [ps]
FIG. 3. Transition dipole momentsin thedressed statebasis
b)
for the photonic catalyst model: µg+ (red), µg− (blue),µ−+ 0
(black).
u.]
a. -0.5
sition between the dressed states is shown in Fig. 2(b). [
N
The initially dark state S now becomes radiatively ac- S
1
cessible from S throughthe non-adiabatic couplings via
0
-1
the S state. It is evident that the dressed states are
2 0 100 200 300 400 500 600 700
coupledtoeachotherinthe regionwherethebarestates
T [fs]
are close to resonance with the cavity mode. The upper
dressed state – whose shape still resembles the shape of
FIG.4. (a)Populationsofthedressedstates|+i(red)and|−i
theS state–isthusnotstableanymoreandthemolecule
2 (blue) in thephotoniccatalyst model vs. time. Thedecay of
candissociate throughthe non-adiabaticcoupling to the
the|+istatecan befittedtoabi-exponentialmodelyielding
unbound lower dressed state.
thetime constants228fs and 42ps. (b)Transient absorption
Figure 3 displays the relevant transformed transition signalindepenenceoftheprobedelayT and. Thelaserisset
dipolemomentscalculatedfromEqs. 31to33. Allcurves toberesonant betweenS andthe|±istate(1.5eV)andhas
1
show a dip around 2.2˚A where the non-adiabatic cou- apulselengthof10fs(FWHM).Thedashedlineisthesignal
pling and thus the mixing between the molecular states for a wave packet in theS2 bare state potential.
andthe photonstates is strongest. The transitiondipole
betweenthedressedstatesvanishesifthecavitycoupling
vanishes. The coupling to the cavity creates a new tran- excitationfromS toS causesimmediatedissociationof
0 1
sition and modified dynamics which can be probed with the molecule in the bare state model. Setting the cavity
time resolved spectroscopy. modeonresonancewithS andS at 2eVcreatesaset
0 1
≈
The excited state evolution was simulated by wave of dressed states, which experience an avoided crossing
packet dynamics on a spatial grid (for details see Ap- (Fig. 5(a)) with a non-adiabatic coupling matrix ele-
pendixB).Figure4(a)depictsthedynamicsafterimpul- ment (Fig. 5(b)) peaking at the crossing at 2.9˚A. The
sive excitation from the ground state (S0) to the dressed lower dressed state resembles the ground state around
states . The initial population pattern is caused by the Franck-Condon point and forms a bound state po-
|±i
the mixing ofthe transitiondipole moments,followedby tential. The upper dressed state now also appears as a
arapiddecayoftheupperdressedstatecausedbythedis- partially bound state potential, which is coupled to the
sociative/unbound character of the lower dressed state. dissociative curve by the avoided crossing.
The oscillation pattern is caused by the wave packet os-
The transition dipole moments are shown in Fig. 6.
cillation in the + state, passing through the coupling
Due to the large detuning δ in the Franck-Condon re-
| i c
region. Figure4(b)showsthetransientabsorptionsignal
gion,thelowerdressedstatehasaweaktransitiondipole
(see Appendix C) probing the system via the g state.
moment with respect to the S state. The state char-
| i 0
The signal shows a clear decay of stimulated emission acter change at the crossing at 2.9˚A manifests itself in
modulated by the wave packet motion in the + state.
the rapid change of the transition dipole moments (the
| i
crossing of the red and blue curve in Fig. 6.).
The population dynamics after excitation is shown in
B. Photonic Bound States
Fig. 7(a)). The clear distribution of the dipole moments
betweengroundstate andthe dressedstates leads to the
Inthesecondmodel,wereversetherolesofboundand upper dressed state population upon impulsive excita-
unbound states to create a situation where a purely dis- tion. The quasi-bound character of the upper dressed
sociativestatecanbestabilized(i.e. increaseitslifetime) state becomes clear from Fig. 7(a)): Instead of imme-
via cavity coupling to a bound state. We use the model diate dissociation the upper dressed state acquires a sig-
from Fig. 1, but only considering states S and S . An nificant lifetime. The population of + leaks into
0 1
| i |−i
6
a) a)
5 1
4 |+i
V] 3 ρaa0.5
e
V [ 2 |−i
1 S 0
0
0 0.5 1 1.5 2
0 t [ps]
2 3 4 5 6
b)
q [Å] 0
b)
1
u.]
a. -0.5
[
−i 0 SN
|
∂q
+| -1
h -1 0 100 200 300 400 500 600 700
T [fs]
-2
2 3 4 5 6 FIG.7. (a)Populationofthedressedstates|+i(red)and|−i
(blue) in thephotoniccatalyst model vs. time. Thedecay of
q [Å]
the|+istatecan befittedtoabi-exponentialmodelyielding
thetimeconstants234fsand5.2ps. (b)Transientabsorption
FIG. 5. (a) Dressed state PESsfor thephotonicbound state
signal. Laser is set to be resonant between S and the |±i
1
model. (b) Non-adiabatic coupling matrix element causing
state (1.5eV) and has a pulse length of 10fs (FWHM). The
transition between thedressed states.
dashedlineisthesignalforawavepacketintheS barestate
1
potential
4
u] principlealsocreateacrossing,whichexhibitsadegener-
a 2 acybetweenthedressedstates. Thisisthebasicrequire-
[
µ menttoobtainaCoIn,i.e. thecouplingbetweentheadi-
abatic electronic states has to vanish at the intersection
0 point1. In our third example of light-induced CoIns17,23
2 3 4 5 6 this condition is fulfilled by rotating the molecule with
q [Å] respect to the polarization vector of the driving field.
By inspecting Eq. 13 we identify another type of light
induced CoIn: Setting the cavity on resonance at a nu-
FIG.6. Tranistiondipolemomentsinthedressedstatebasis:
µg+ (red),µg− (blue), µ−+ (black). clear configuration where the transition dipole moments
vanishes yields a degenerate point in the dressed state
basis. This can be achieved by choosing an electronic
transition which is dipole forbidden at a certain config-
on a picosecond time scale. The corresponding transient
uration of high symmetry and becomes allowed as the
absorption signal is shown in Fig. 7(b)) along with the
symmetry is lowered. We now demonstrate this case for
signal for the bare state system (dashed curve).
formaldehyde. In its planar equilibrium structure the
lowest energy transition from the 1A state to the 1A
1 2
state transition is dipole forbidden. Every vibrational
C. Photoninduced Conical Intersections
mode which is not of the A irreducible representation
1
breaks the C symmetry (B , B ) and can be expected
2v 1 2
So far we have demonstrated the non-adiabatic cou- to make the transition dipole allowed. In Fig. 8 the
plingsinducedbythecavityintermsavoidedcurvecross- potentials and transition dipole moments are shown vs.
ings, which stem from the fact that a non-vanishing theout-of-planemotion(B )ofthehydrogenatoms. Set-
1
dipole in the coupling region creates a splitting between tingthecavityinresonancewiththeforbiddentransition
the dressed states (see Eq. 9). However, by choosing a thus creates a vacuum field, light induced CoIn, which
point of vanishing transition dipole moments one can in we call photoninduced CoIn. The corresponding dressed
7
a) 4.3
V] 4.25
e
V [
4.2
-60 -40 -20 0 20 40 60
b) 1
V]
e 0.5
V [
0
-60 -40 -20 0 20 40 60
c) 0.2
µ 0
-0.2
-60 -40 -20 0 20 40 60
FIG. 10. Photoninduced CoIn between the dressed states in
q [°]
formaldehyde in dependence of two vibrational modes: CH
2
out-of-planemotion(φ≡q )andtheCH asymmetricstretch
1 2
FIG. 8. Bare state PESs for the photoninduced CoIn model motion (∆≡q ).
2
calculated at the CAS(4/4)/MRCI/6-311G* level of theory
with theprogram packageMOLPRO22: (a) Thedoublemin-
imum of the S state. (b) The ground S ground state of
1 0 statesandthenon-adiabaticcouplingmatrixelementare
formaldehyde. q istheangleof theout-of-planemotion. The
shown in Fig. 9. The degenerate point appears at the
cavity is set in resonance at q = 0. (c) Transition dipole
planar configuration (q = 0) along with a peaking non-
moment S →S . 1
0 1 (1) (1)
adiabatictransitionmatrixelementf . Notethatf
−,+ −,+
diverges when the detuning is exactly zero. Choosing a
second vibrational mode which also breaks the symme-
a) 4.3 trywillcreateatransitiondipole momentandwillresult
in a typical cone-shaped PES. We demonstrate this fea-
turefortheasymmetricstretchmotionoftheCH group
V [eV]4.25 (inB2F)i.g.T1h0e. resulting PES of the dressed states is2shown
4.2
-60 -40 -20 0 20 40 60 IV. CONCLUSIONS AND OUTLOOK
q [°]
b) 0.03
We have developed a theory, which can be applied to
compute the non-adiabatic dynamics of single molecules
−i0.02 strongly coupled to a single cavity mode. The for-
∂|q malism expressed in terms well known derivative cou-
+|
h0.01 plings is suitable for various simulations protocols like
full quantum propagation and semi-classical methods
like for example surface hopping24 and ab initio multi-
0
-60 -40 -20 0 20 40 60 ple spawning25. The quantities required to express the
q [°]
molecular system in terms of dressed states, i.e. the po-
tentialenergysurfacesandtransitiondipolemomentscan
FIG. 9. (a) Dressed state PESs (red: |+i, blue |−i) for the
bedirectlyobtainedfromstateoftheartquantumchem-
photoninducedCoInmodelindependenceoftheout-of-plane
istrymethods. ThederivationsaredonewithintheRWA
angleφ. ThedashedlineindicatestheS barestate. (b)Non-
1
adiabatic coupling matrix element for the polariton states. andtheassumptionthattheultrafastdynamicstimescale
The splitting vanishes at q = 0, hence the degeneracy. The isshorterthanthelifetimeofthephotonmode. However,
cavity coupling is chosen to be gmax =434meV. wecouldidentify situationswheretheJCHamiltonianis
notadequatelydescribedbytwophotonstatesandhigher
8
manifolds must be taken into account for the procedure
TABLE I.Parameters for thepotentials shown in Fig. 1.
toconverge. AbreakdownoftheRWAcanbecausedby
various factors: Large detunings give rise to off-resonant i Di[eV] ai[˚A−1] q0,i[˚A] V0[eV]
terms (a†σ† and aσ) in the ultra strong coupling regime
S 3.0 1 2.0 0
0
(g ω ) give rise to the Bloch-Siegert shift and ground
sta≈te mcodifications. The off-resonant regime can be eas- S1 0.01 2.43 2.5 3
S 3.0 1 2.3 4.5
ilyaccessedbycouplingclosetotheCoIn,whiletheultra 2
strongcoupling regime might be difficult to reachdue to
technical limitations of the nano-structures, like for ex-
The respective parameters are given in tab. I.
ampletheachievablesizeofthemodevolume. Moreover,
The transition dipole shown in Fig. 1(b) is defined by
the applicable field strength is limited by the ionization
the sigmoid function:
potential of a molecule.
Some basic possibilities for the manipulation of the 4
µ(q)= (A2)
excited state photo chemistry have been demonstrated 1+exp[2.4575(q 4.232)]
for photo dissociationmodel systems. The life time with −
respecttodissociationcanpotentiallybesignificantlyin-
fluenced by the cavity coupling. Non-adiabatic coupling Appendix B: Quantum Propagation
between the dressed states is the cause for the coupling
and the effect on the nuclear dynamics. The population The wave packet propagations are carried out on a
transfer between the dressed states may also be viewed numericalgridusingtheHamiltonianfromEq. 30where
via stimulated emission caused by the vacuum state of the kinetic energy is given by
the photon mode.
Single molecule strong coupling is an experimentally Tˆ = 1 d2 (B1)
challenging regime and has not been demonstrates yet −2mdq2
to the extent necessary to influence chemistry. However,
with m=3650 being the reduced mass. For all dissocia-
coupling of a ensemble of N molecules to the mode of
a micro resonator, which is enhanced by a factor √N5 tive potentials the kinetic energy term Tˆ is replaced by
a perfectly matched layer27 to avoid spurious reflections
shows promising results. The collective chemistry in a
at the edge of the grid. The time evolution is calculated
cavity is a many body effect, which needs further inves-
with an Arnoldi propagationscheme28,29.
tigation. Its theoretical treatment is more challenging
since all particles are coupled and the dimensionality in-
crease with the number of particles. Finally the super
Appendix C: Transient Absorption Spectrum
radiant26 regime might be used to engineer the reactiv-
ity of molecules in a novel way.
The transient absorption signal is linear in the probe
intensity and given as the frequency integrated rate of
ACKNOWLEDGMENTS change in the photon number (for further details see
Ref.30):
The support from the National Science Foundation 2 ∞ t
(grant CHE-1361516) and support of the Chemical Sci- SN(T)=− ~IZ dtZ dτE∗(t−T)E(τ −T)
ences,Geosciences,andBiosciencesdivision,OfficeofBa- −∞ −∞
sic Energy Sciences, Office of Science, U.S. Department Ψ0 U†(t,0)µˆU(t,τ)µˆ†U(τ,0)Ψ0 (C1)
×h | | i
of Energy through award No. DE-FG02-04ER15571 is
where Ψ =(µˆ +µˆ )Ψ is the initial wavefunc-
gratefully acknowledged. The computational resources 0 g+ g− S0,v=0
tion prepared by impulsive excitation from the vibra-
and the support for Kochise Bennett was provided by
tional ground state of the S potential and U(t,t′) =
DOE. M.K gratefully acknowledges support from the 0
AlexandervonHumboldtfoundationthroughtheFeodor exp iHˆ(t t′) propagates the system from t′ to t.
Lynenprogram. Wewouldliketothankthegreenplanet The(cid:16)p−ropaga−torU(cid:17)is implement bya numericalpropaga-
cluster(NSFGrantCHE-0840513)forallocationofcom- tion as described in the previous section. The operator
pute resources. µˆ µ (q) includes transition dipole moments between
ij
≡
the relevant electronic stats and depends on the nuclear
coordinates. The probe field is given by
Appendix A: Model parameters
(t)=e−iωLt−t2/2σ2, (C2)
E
The model potentials shown in Fig. 1(a) are obtained
where ω is the center frequency and σ the temporal
from Morse potentials: L
width of the laser pulse and T is the delay with respect
V (q)=D [1 exp( a (q q ))]2+V (A1) to the initial state preparation.
i i i 0,i 0
− − −
9
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