Communication: Atomic force detection of single-molecule nonlinear opticalvibrational spectroscopyPrasoon Saurabh and Shaul Mukamel

Citation: The Journal of Chemical Physics 140, 161107 (2014); doi: 10.1063/1.4873578 View online: http://dx.doi.org/10.1063/1.4873578 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Communication: Quantitative estimate of the water surface pH using heterodyne-detected electronic sumfrequency generation J. Chem. Phys. 137, 151101 (2012); 10.1063/1.4758805 Multiplexing single-beam coherent anti-stokes Raman spectroscopy with heterodyne detection Appl. Phys. Lett. 100, 071102 (2012); 10.1063/1.3680209 Frequency-modulation atomic force microscopy at high cantilever resonance frequencies using the heterodyneoptical beam deflection method Rev. Sci. Instrum. 76, 126110 (2005); 10.1063/1.2149004 High-frequency mechanical spectroscopy with an atomic force microscope Rev. Sci. Instrum. 72, 3891 (2001); 10.1063/1.1403009 Optical heterodyne detection at a silver scanning tunneling microscope junction J. Appl. Phys. 85, 1311 (1999); 10.1063/1.369332

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THE JOURNAL OF CHEMICAL PHYSICS 140, 161107 (2014)

Communication: Atomic force detection of single-molecule nonlinearoptical vibrational spectroscopy

Prasoon Saurabha) and Shaul Mukamelb)

Department of Chemistry, University of California, Irvine, California 92697, USA

(Received 6 March 2014; accepted 14 April 2014; published online 30 April 2014)

Atomic Force Microscopy (AFM) allows for a highly sensitive detection of spectroscopic signals.This has been first demonstrated for NMR of a single molecule and recently extended to stimulatedRaman in the optical regime. We theoretically investigate the use of optical forces to detect timeand frequency domain nonlinear optical signals. We show that, with proper phase matching, theAFM-detected signals closely resemble coherent heterodyne-detected signals. Applications are madeto AFM-detected and heterodyne-detected vibrational resonances in Coherent Anti-Stokes RamanSpectroscopy (χ (3)) and sum or difference frequency generation (χ (2)). © 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4873578]

Atomic force microscopy of single molecules has seenmany advancements over the past two decades since its the-oretical and experimental advents in early 1990s.1, 2 Opticalforces have been used to create optical lattices for trappingcold atoms and ions.3 Magnetic and optical tweezers are ver-satile tools for measurements of forces and corresponding sin-gle molecule displacements in biological molecules such asDNA and proteins. AFM measurements provide high resolu-tion and conformational control of bio-macromolecules. Non-contact AFM has been used to visualize intermolecular bondssuch as hydrogen bonds.4 Magnetic tweezers are often usedfor analysing DNA topology, and like optical tweezers, canperform 3-dimensional manipulations. These techniques gen-erally measure the displacement of molecular position causedby the forces applied by the probe.5–11 The optical potentialacting on the single molecule by the fields can trap cold atomsin optical lattices.3 Measuring its gradient (the force) consti-tutes a new type of signal.

Here we theoretically investigate how optical forces maybe used to detect nonlinear optical signals. Instead of measur-ing the mechanical displacement of single molecule inducedby the applied force, one can look at the gradient force appliedby the AFM tip to the molecule.12, 13 In such measurements,the spectroscopic information is contained in the pulse param-eters used to create the force. Stimulated Raman resonancesrelated to χ (3) have been recently reported experimentallyby Rajapksa et al.22 The mechanical force associated withdifferent NonLinear Optical (NLO) techniques like FRET-force in force imaging has been investigated.14 A plethoraof methods have been employed for detecting nonlinear op-tical signals. These include homo and hetrodyne detectionof fields,15, 16 incoherent fluorescence detection,17, 18 photoac-coustic detection,19, 20 photoelectron, and current detection.21

We derive general expressions for ultrafast nonlinearoptical spectroscopy with gradient force detection. A sin-gle molecule on a glass cover slide is subjected to sev-

a)Electronic mail: psaurabh@uci.edub)Electronic mail: smukamel@uci.edu

eral collinear laser beams and the variation of the gradientforce with selected parameters such as delays between pulses,phases, or frequencies is measured by AFM tip using vibra-tional tapping mode,22 thus generating multidimensional sig-nals. We show that by manipulating the phases between fields,it is possible in some cases to make the optical potential (andgradient force) to coincide with the heterodyne signal.23

The conservative force is given by the gradient of the op-tical potential V(r).3, 12, 14

F(r)gr = −�rV(r). (1)

We first calculate the potential felt by the molecule dueto its nonlinear coupling to electromagnetic fields in contin-uous wave (cw) frequency-domain experiments.24 n classi-cal monochromatic fields induce a polarization P(ω) in themolecule, which then creates an optical potential (V),23

V(r) = −�[ ∫dωP(ω).E∗(r, ω)

] = −�[ ∫dtP(t).E∗(r, t)

].

(2)

The classical electric field can be expanded in modes:

Ei(r, t) =∑

ξi=± εi

∫dωi

2πEξi

i (r, ωi)eiξi (ωi t+φi ),

(3)

Ei(r, ω) =∑

ξi=± 2π.εiEξi

i (r, ωi)eiξiφi δ(ω′ − ξiω),

where Eξi

i (r, ω) is the position dependent envelope of the ithfield and ξ i = ± is the hermiticity (negative for positive fre-quency and positive for negative frequency component) of thefield. φi is the phase angle between the incident classical fieldmodes and εi is the respective polarization vector. The opticalpotential is given by

V(r) = −�∫ ∞

0dωP(ω).E∗(r, ω), (4)

where ω > 0. For comparison, the heterodyne detected signalmeasured at ω′ is given by25

S(r; ω′) = −�∫ ∞

0dωP(ω).E∗(r, ω). (5)

0021-9606/2014/140(16)/161107/5/$30.00 © 2014 AIP Publishing LLC140, 161107-1

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161107-2 P. Saurabh and S. Mukamel J. Chem. Phys. 140, 161107 (2014)

The total induced polarization is the sum of positive and nega-tive frequency components, Ptotal(ω) = P+(ω) + P−(ω). Onecan identify P+(ω) with P(ω) and P−(ω) with P∗(ω). Simi-larly, we can identify E+(ω) with E(ω) and E−(ω) with E∗(ω).The superscripts + and − denote the positive or negative fre-quency components. For a cw measurement involving discreteset of modes, we have

Vd (r; ωj ) = −�∑

jP(ωj ).E∗(r, ωj ), (6)

Sd (r; ω′j ) = −�

∑j

P(ωj ).E∗(r, ωj ). (7)

For the linear response, we get the optical potential andheterodyne signal,

V(1)(r; ωj ) = −∑

jχ (1)′(ωj ).|E(r, ωj )|2, (8)

S(1)het (r; ωj ) = −2

∑jχ (1)′′ (ωj ).|E(r, ωj )|2, (9)

where, linear susceptibility, χ (1)(ω) = χ (1)′(ω) + iχ (1)′′ (ω).Superscripts ′ and ′′ denote the real and imaginary part. For amultilevel system, we have

χ (1)(ω) = 1

¯

∑a,c

P (a)|μac|2ωac

(ωac)2 − (ω + iac)2, (10)

where, P (a) = e−βεa /∑

a e−βεa with εa as the eigen en-ergy of level a, and β = 1/kbT with temperature T. Thelinear optical potential and heterodyne signal have disper-sive and absorptive profiles at ω = ωac as depicted inFig. 1.

We now turn to a second order process (Fig. 2) whereω1 in the IR regime, ω2 and ω3 are in visible region , and ω3

= ω1 + ω2, the optical potential is then given by

V(2)(r; ω1, ω2) = −�[ ∑j(2π )3E1(r, ωj )E2(r, ωj )

E∗3 (r, ωj )

∑p{χ (2)(−ω3; ω1, ω2)}],

(11)

where,∑

p represents the sum over all permutations of fre-quencies {ω1, ω2, −ω3}.χ (2)( − ω3; ω1, ω2) represents SumFrequency Generation (SFG) whereas the other two permuta-tions, when ω2 or ω1 are detected, are Difference FrequencyGeneration (DFG). All possible processes contribute to the

FIG. 1. Optical potential (solid red) and heterodyne signal (dashed blue) in-duced on single chromophore at ¯ωac = 2.5 eV using real and imaginary partof Eq. (10)

FIG. 2. (a) The three level model scheme used to calculate the χ (2) opticalpotential and heterodyne signal. (b) Loop diagram for SFG and DFG associ-ated to the level diagram in Fig. 2(a)

optical potential. The total heterodyne-detected signals is:

S(2)total(r, ω3; ω1, ω2)

= −�[ ∑j(2π )3E1(r, ωj )

E2(r, ωj )E∗3 (r, ωj )

∑p

χ (2)(−ω3; ω1, ω2)]. (12)

Using the level scheme of Fig. 2 and invoking the Rotat-ing Wave Approximation (RWA), we obtain,

V(2)(r; ω1, ω2)

= −�[

−i

2¯2

∑j(2π )3E1(r, ωj )

E2(r, ωj )E∗3 (r, ωj )μacμcbμba

∑aP (a){

1

(ωIR − ωca + ica)(ωIR + ω

(2)vis − ωab + iab

)+ 1(

ω(3)vis − ωab + iab

)(

−1(ω

(3)vis − ω

(2)vis − ωac + iac

)+ 1(

ω(3)vis − ω

(1)IR − ωca + ica

))}]

. (13)

The sum of the three heterodyne signals (Eq. (12)) isgiven by replacing R with � in Eq. (13) as shown in Fig. 3.

We next turn to the third order optical potential and het-erodyne signal (measured at ω4) for a four wave mixing of

FIG. 3. (a) Optical potential Eq. (13) for cw χ (2) with μac = 6D, μbc

= 1D, ωac=2.3 eV, ωba=10 eV, ωbc=10 eV, ac=5 cm−1, bc=5 cm−1,

and ab = 5 cm−1. (b) Total heterodyne signal (SFG+DFG) for the sameparameters.

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161107-3 P. Saurabh and S. Mukamel J. Chem. Phys. 140, 161107 (2014)

FIG. 4. The three level model system used to calculate the optical potentialand heterodyne χ (3) signal and the contributing Unrestricted loop diagram.

configuration ω4 = ω1 − ω2 + ω3 as shown in Fig. 4, atresonance frequency ω1 − ω2 = ωac and coherent fields(−sgn(ωj)

∑i φi = 0) are given by

V(3)(r; ω1,−ω2, ω3)

= −�[ ∑j(2π )4E1(r, ωj )E∗

2 (r, ωj )

E3(r, ωj )E∗4 (r, ωj )

∑p{χ (3)(−ω4; ω1,−ω2, ω3)}],

(14)

S(3)het (r, ω4; ω1,−ω2, ω3)

= −�[ ∑j(2π )4E1(r, ωj )E∗

2 (r, ωj )

E3(r, ωj )E∗4 (r, ωj )χ (3)(−ω4; ω1,−ω2, ω3)

]. (15)

For the three level system shown in Fig. 4, with ground state(a), vibrational excited state (c) and electronic excited state(b), the CARS resonances of χ (3) for (ω4 = ω1 − ω2 + ω3)is25

χ(3)CARS(−ω4; ω1,−ω2, ω3)

= P (a)|μba|2|μac|2(ω1 − ωba + iηe)

1

(ω1 − ω2 + ω3 − ωbc + iηe)(ω1 − ω2 − ωac + iηR).

(16)

Unlike the heterodyne signal which is measured at a se-lected frequency, the optical potential Eq. (14) is given by asum over all possible permutations of signal frequency forχ (3). The two signals coincide if we change the overall phaseof E1(r, ωj )E∗

2 (r, ωj )E3(r, ωj )E∗4 (r, ωj ) by π

2 and assume theRotating Wave Approximation (RWA) as was done in deriv-ing Eq. (13) for χ (2).

Consider a special case involving only two fields withfrequencies ω1 and ω2, E3(r, ωj ) = E1(r, ωj ) and E2(r, ωj )= E4(r, ωj ), we can rewrite Eqs. (14) and (15) as

V(3)(r; ω1,−ω2)

= −∑

j(2π )4|E1(r, ωj )|2

|E2(r, ωj )|2∑

p{χ (3)′(−ω2; ω1,−ω2, ω1)}, (17)

S(3)total(r; ω1,−ω2)

= −∑

j(2π )4|E1(r, ωj )|2

|E2(r, ωj )|2∑

p{χ (3)′′(−ω2; ω1,−ω2, ω1)}. (18)

AFM provides the same information as heterodyne de-tection but with a much higher sensitivity (∼pN force) thatallows to detect single molecules [SI from Ref. 22]. If the rel-evant field factors are real, for example, in linear regime or inEqs. (17) and (18), then the heterodyne signal is given byimaginary part of susceptibility (χ ′′) and the optical potential(and gradient force) is given by the real part of susceptibil-ity (χ ′). In some cases, it is possible to vary the phases andmake the optical potential (or gradient force) coincide withthe heterodyne signal.

We now apply the results to the three-level model systemshown in Fig. 4, with ground state (a), vibrational excited state(c), and electronic excited state (b),χ (3) for stimulated CARSfor two beam (ω4 = 2ω1 − ω2) within the RWA is25

χ(3)CARS(−ω2; ω1)

= P (a)|μba|2|μac|2(2ω1 − ω2 − ωab′ + iηab′ )

1

(ω1 − ω2 − ωac + iηac)(ω1 − ωba + iηba), (19)

χ(3)CARS(−ω1; ω2)

= P (c)|μba|2|μac|2(2ω2 − ω1 − ωac + iηac)

1

(ω2 − ω1 − ωbb′ + iηbb′ )(ω2 − ωcb + iηcb). (20)

Since the electronic transition frequencies ωbc and ωab aremuch higher than the vibrational frequency ωac, we canrewrite Eq. (16) for optical potential as, Eq. (21), whereχ

(3) = �[

∑p χ

(3)CARS(−ω2; ω1)]. We get the total heterodyne

signal by simply replacing R in Eq. (21) with �,

V(3)(r; ω1,−ω2)= − [∑j(2π )4|E1(r, ωj )|2|E2(r, ωj )|2χ (3)

].

(21)

Raman resonances are observed when ω1 − ω2 = ±ωac

where ωac is the difference between two ground vibrationalstates. The calculated optical potential and heterodyne sig-nals are shown in Fig. 5. The gradient force due to this Ramanresonance was calculated in Ref. 22 using phenomenologicalnonlinear polarizability. Our expression is recast in terms ofthe general third order polarizability χ (3) and allows to cal-culate the optical potential (or gradient force) for other tech-niques as well.

We next turn to time-domain measurements which usetemporally well separated impulsive pulses. The induced nthorder optical potential V(r; tj ) and heterodyne signal S(r; tj)

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161107-4 P. Saurabh and S. Mukamel J. Chem. Phys. 140, 161107 (2014)

7.5 5.0 2.5 0.0 2.5 5.0 7.57.5

5.0

2.5

0.0

2.5

5.0

7.5

7.5 5.0 2.5 0.0 2.5 5.0 7.57.5

5.0

2.5

0.0

2.5

5.0

7.5

FIG. 5. (a) Optical potential (Eq. (21)) for the model system of Fig. 4,at Raman Resonances for four wave mixing with two frequenciesω1 and ω2 (�1 = ω1, �2 = ω1 − ω2; �3 = 2ω1 − ω2)with, μac = 6D, μbc = 1D, ωac = 0.2014 eV, ωbc = 20 eV, ωba ≈ 20 eV,

ωbb′ ≈ 0eV ; ηab = 0.1 cm−1 and ηac = 0.2 cm−1. (b) Total Heterodyne sig-nal for the same parameters by changing R to � in Eq. (21).

are given by

V(n)(r; tj ) = −�¯(−n)

[∫dtP(n)(t).E∗(r, t)

](22)

S(n)het (r; tj ) = − 2n

¯n�

[∫dtP(n)(t).E∗(r, t)

].

Signals are now parameterized by the time delays betweenpulses t1, t2, t3 rather than their frequencies. We can writethe optical potential and corresponding heterodyne signals, tofirst order,

V(1)(r; t1) = −�[∫

dt

∫ ∞

0dτ1S

(1)(τ1)E∗(t)E(τ1)

](23)

S(1)het (r; t1) = −�

[∫dt

∫ ∞

0dτ1S

(1)(τ1)E∗(t)E(τ1)

],

in second order,

V(2)(r; t1, t2) = −�[ ∫

dt

∫ ∞

0dτ1

∫ ∞

0dτ2S

(2)(τ1, τ2)

E1(τ1)E2(τ2)E∗3(t)

]

S(2)het (r; t1, t2) = −�

[ ∫dt

∫ ∞

0dτ1

∫ ∞

0dτ2S

(2)(τ1, τ2)

E1(τ1)E2(τ2)E∗3(t)

], (24)

finally, in third order,

V(3)(r; t1, t2, t3)

= −�[ ∫

dt

∫ ∞

0dτ1

∫ ∞

0dτ2

∫ ∞

0dτ3

S(3)(τ3, τ2, τ1)E∗4(t)E1(τ1)E∗

2(τ2)E3(τ3)

](25)

S(3)het (r; t1, t2, t3)

= −�[ ∫

dt

∫ ∞

0dτ1

∫ ∞

0dτ2

∫ ∞

0dτ3

S(3)(τ3, τ2, τ1)E∗4(t)E1(τ1)E∗

2(τ2)E3(τ3)

]. (26)

We again focus on vibrational resonances and assumeelectronically off-resonant frequencies. For χ (2) (Fig. 2), if wehave two temporally resolved impulsive electric fields withdelay t1 and total phase

∑ni φi = 0, we can then write the

P(2)(r, t1) as26–28

P(2)(r, t1) = E1E2∑

a P (a)μab|μca|2Iac(t1)eiω1t1 , (27)

where we define, P (a) = e−βεa /∑

a e−βεa and Iνν ′ (t)= exp[−iωνν ′ t1 − γνν ′ t1]. For simplicity, the r dependence isengraved in the envelope E and has not been explicitly written.Now we can write the second order optical potential measuredas a function of t at t = t1 with t2 = 0 as,

V(2)(r, t1)=�¯(−2)[iE1E2E∗

3

∑aP (a)μacμcbμbc|Iac(t1)|2eiω1t1

].

(28)

Heterodyne signals are given by replacing R in Eqs. (28) and(29) with �. One could observe the Raman resonance at con-jugate frequency (�1) by Fourier transforming Eq. (28) withrespect to t1 as

V(2)(r,�1) =∫ ∞

0dt1V(2)(r, t1)ei�1t1 . (29)

For three temporally separated electric fields (Fig. 4), weassume that the field envelopes Ei(t) does not allow matter toevolve during the pulse and the fields are long enough so thattheir spectral bandwidth is narrow enough to show vibrationalRaman resonances. Using coherent (

∑ni φi = 0) and impul-

sive (Ei(t) = Eiδ(t)) fields, we can immediately write the Ra-man resonance term for optical potential using loop diagramsin Fig. 4 and Eq. (25), as Eq. (30),

V(3)(r, t2) = �[

−1

¯3|E1|2|E∗

2 |2∑

aP (a)|μab|2|μca|2

|Ica(t2)|2ei(ω1−ω2)t2

]. (30)

The heterodyne-detected signal corresponding toEq. (30) is obtained by simply replacing R in Eq. (30) with�. If we Fourier transform Eq. (30), similar to Eq. (29),but with respect to t2 using conjugate frequency (�2) instead,we can observe the vibrational Raman resonance at �2

= ωac. In our case, this will give a similar optical potentialand heterodyne-detected signal as shown in Fig. 6 for χ (2)

processes but with a different prefactor. Since we consideredelectronically off-resonant processes, second and third orderoptical potentials can simply be parameterized by conjugatefrequencies �1 or �2 respectively, as in Eqs. (29) and(30), giving one dimensional spectra. When extended toelectronically resonant system, one would be able to get

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161107-5 P. Saurabh and S. Mukamel J. Chem. Phys. 140, 161107 (2014)

FIG. 6. Time resolved χ (2) (Eqs. (28) and (29)) with similar parameters asFig. 3.

multidimensional time-domain spectra with variable pulsedelays.

In summary, we showed that AFM detection closely re-sembles coherent heterodyne signals with appropriate phasematching. This is in contrast with other incoherent detectiontechniques like Florescence,15 photo-acoustic,19, 20 and photo-electron detection.21

The authors wish to thank H. K. Wickramasinghe andE. Potma for useful discussions. They gratefully acknowl-edge the support of the Chemical Sciences, Geosciences andBiosciences Division, Office of Basic Energy Sciences, Of-fice of Science, U.S. Department of Energy. In addition, theyalso thank the National Science Foundation (Grant No. CHE-1058791) and the National Institute of Health (Grant No. GM-59230) for their support.

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