Two-Color three-pulse Photon Echoes

Two-Color three-pulse Photon Echoes Intensity (normalized) 1 0.8 0.6 0.4 0.2 IR144 in Methanol 0 600 650 700 750 800 850 900 Wavelength (nm) 1 Intensity (normalized) 0.8 0.6 0.4 0.2 DTTCI in Methanol 0 600 650 700 750 800 850 900 Wavelength (nm)

Condensed Phase Dynamics In an ensemble of chromophores, the dynamics for an individual chromophore (i) can be expressed as Full width half maximum of f( ε ) i ω ( t) = ω + δω ( t) + ε eg eg eg i = inhomogeneous width Optical lineshapes and echo measurements relate to: δω(0) δω( t) M ( t) = 2 δω

Stokes Shift and Solvation Dynamics S( t) = υ( t) υ( ) υ(0) υ( ) Optical Lineshapes ωi = ω + δω ( t) + ε eg eg i i For the i th chromophore ω eg δω(0) δω( t) Optical lineshapes and echo M ( t) = measurements relate to 2 δω

Solvation Dynamics in C343 in Water Jimenez, Fleming, Kumar and Maroncelli (1994) Nature 369, 471

Difference Peak Shift For a fixed phase matching direction, i.e., k 3 + k 2 k 1 k s eg k 3 eg k 3 ee T I T II ee k 2 ge τ I τ II eg k 1 gg k 1 k 2 gg Type I scan Echo (Rephasing) (pulse sequence, 1-2-3) Type II scan FID (Non-Rephasing) (pulse sequence, 2-1-3) Peak Shift, fs 20 15 10 5 0 Type I Type II Two Colour -5 0 200 400 600 800 1000 Population Period, fs Difference Peak Shift, fs Difference peak Shift = Type I - Type II 7 6 5 4 3 2 1 0 τ*(t) = τ I *(T I ) -τ II *(T II ) 0 200 400 600 800 1000 Population period, fs

IR144 Methanol 750, 750, 800 15 P e a k s h i f t (f s) 10 5 0-5 0 100 200 300 400 500 P o p u l a t i o n T i m e (fs) Type I Type II Difference

Experimental Difference Peak Shift Data (downhill) 5 Pulse Sequence, 750-750-800 nm τ (Difference Peak Shift, fs) IR144 in Methanol 4 3 2 1 DTTCI in Methanol 0 10 100 1000 10000 Population Period, fs The Difference Peak Shift starts at a near zero value, then rises to a maximum value in ~ 200 fs and then decays to zero for both IR144 and DTTCI in methanol Based on the turnover time, it is suggested that the ultrafast component in methanol is ~ 200 fs

Two-Color 3PEPS as a Probe of Memory Transfer in Spectral Shift Time-Dependent Spectral Shift Memory-Conserved Shift Randomized Shift Transition Density ω probe ω pump = + Energy Two-Color 3PEPS measures correlation dynamics (between transition energies in pumped and probed regions). : de pr ( t) de pu

Homogeneous and Inhomogeneous Distributions of Transition Energies Homogeneous distribution å ( ( E E )) s hom ( w ; j ) = d w - j - e j j j g g e g j e A particular nuclear state in the ground electronic state 2 Inhomogeneous distribution s ( w) = å s hom( w; j ) P j j g ( ) g g g Statistical probability for a molecule to occupy the nuclear state j g Two Mechanisms for Existence of Non-Linear Signals of Two-Color Experiments Interactions of pump and probe lasers have to be made with the same molecule A) Spectral Overlap due to Homogeneous Distribution B) Spectral evolution due to Fluctuation of Inhomogeneous Distribution These two mechanisms are included in the response function formalism in a complicated way

A Simple ad hoc Model for the Dynamics of Correlation Function Total Signal = P hom ( t) R hom ( t) + P inhom ( t) R inhom ( t) Total Correlation Function P ( t) P ( t) hom inhom de pr ( t) de = de pr ( t) de + de ( ) hom pr t de P ( t) + P ( t) P ( t) + P ( t) pu pu pu hom inhom hom inhom inhom At short times, P ( t) > > P ( t) hom inhom At longer times, Phom ( t) < < Pinhom ( t) Inhomogeneous distribution fluctuates with time due to random fluctuation of the statistical distribution of the nuclear states, which is described by a stochastic approach. 1 de pr ( t) de pu = dw dw 1 2 ( w - w( t) 2 )( w - w 1 pu) W ( w E 2 ) ( ; 2 1) ( 1 ) ( ) inhom pr pr P w t w W w E pu s w abs 1 N ( t) ò ò 2 1 [( ω2 ω ) ( ω1 ω ) M ( t)] P( ω2; t ω1) = exp 2 2 2 2 2 π (1 M ( t) ) 2 (1 M ( t) ) P( ω ;0 ω ) = δ ( ω ω ) 2 1 2 1 Skinner et al, J. Chem. Phys. 106, 2129 (1997)

Dynamics of Conditional Probability for the Inhomogeneous Distribution inhom Full Response Function 0 100 200 300 400 Time (fs) 500 Rephasing Capability Normalized Difference Peak Shift depr (t )depu 600 Homogeneous broadening domain : No common transitions between the pump and the probe (no rephasing capability) Rise in Two-Color Difference Peak Shift ~ Inertial Solvation Dynamics Uphill and Downhill difference peak shifts should have distinct behavior for systems with a systematic red shift

Model Calculations for Difference Peak Shift (downhill) τ (T), Difference Peak Shift, fs 10 8 6 4 2 0-2 Difference Peak Shift = TypeI - Type II 50 100 200 300 500 M ( T) = λexp[( t / τ )] 2 0 200 400 600 800 1000 Population Period, fs Empirical formula: T ~ τ g{ c1 log( c2 / λ) + c3}, turnover τ = Gaussian Time Constant, λ = reorganization energy g = Frequency difference between the two pulses Adding exponentials and vibrations does not alter the turnover time significantly. Therefore, we can extract information of the Gaussian parameters from the turnover time. g

Simulation model for the Difference Peak Shift 5 Pulse Sequence: 750-750-800 nm IR144 in Methanol 4 Difference Peak shift, fs 3 2 1 Difference Peak Shift 5 4 3 2 1 0 0 0 1000 2000 3000 4000 5000 Population Period, fs 0 100 200 300 400 500 600 700 800 900 1000 Population Period, fs Simulation scheme: Type I and II peak shifts were calculated using a Gaussian (220 fs, λ = 150 cm -1 ),exponential 1 (2500 fs, λ = 75 cm -1 ), exponential 2 (9500 fs, 70 cm -1 ), 35 intramolecular modes (λ tot ~ 400 cm -1 )

Two Color Difference Peak Shift for Nile Blue 800 600 Experiment A Simulation B Acetonitrile 400 Detuning (pump-probe), cm -1 200 0-200 800 600 400 0 3 6 9 C -4 0 4 8 12 D Ethylene glycol 200 0-200 0 50 100 150 200 0 50 100 150 200 250 Population Time, fs

υ υ Two-Color Three-Pulse Photon Echoes 1. Electronic Mixing in Molecular Complexes A1 Site Representation J B1 Eigen State Representation µ µ υ Two interactions with 0 µ one with 0 υ 2. Correlation between initial and final states e.g. disordered energy transfer system A0 B0 0 0 Bra State Transition Ket State Transition Detect at 0 frequency υ Energy Transfer system Peak Shift (fs) 40 35 30 25 20 15 10 5 S = f ( G) S( G) vs. S» < f ( G) > S ( k( )) G G< G > φ( Γ) phase factor S( Γ) population kinetics 1-color 3PEPS A B 0 0 200 400 600 800 1000 Population Time (fs) Peak Shift (fs) 25 20 15 10 5 0 2-color 3PEPS A B -5 0 200 400 600 800 1000 Population Time (fs) G <..> Γ Ensemble averages over distribution of static energies Donors and acceptors become correlated via energy transfer (even though initially uncorrelated)

One-color three-pulse photon echo peakshift (1C3PEPS) e Coherence state Coherence state g Interaction with 1st pulse 2nd (Population state) Strong correlation between the two coherence states (long-lasting memory) large peakshift Weak correlation between the two coherence states (quick memory loss) small peakshift 3PEPS profile closely follows the transition frequency correlation function 3rd M ( t) = ω(0) ω( t) 2 ω

Two-color three-pulse photon echo peakshift (2C3PEPS) H Energy b B J h g Pulses of two different colors generate photo echo signal associated with correlation between ω H and ω B Energy fluctuation of two excitonically mixed states are correlated due to electronic coupling Electronic coupling constant (J) can be determined by 2C3PEPS without knowing the site energies (E h and E b )of the two chromophores

One- and two-color three-pulse photon echo peakshift experiment Photon echo peakshift: coherence time (τ) where a photon echo signal is maximum One-color three- pulse photon echo peakshift (1C3PEPS) Two-color Three-pulse Photon Echo Peakshift (2C3PEPS) λ 1 =λ 2 =λ 3 Sensitive to environment around a chromophore and energy transfer process λ 1 =λ 2 λ 3 Sensitive to coupling between two chromophores

Strongly Coupled: Exciton Dimer 1e2e> c 1 1e2g> + c 2 1g2e> 2e> 1e> c 1 1e2g> c 2 1g2e> 1g> 2g> 1g2g> Molecule 1 Molecule 2 Dimer 1.4 1.2 1 0.8 0.6 0.4 0.2 0 550 600 650 700 750 Electronic Coupling, J, can be extracted from shifts of absorption peaks. This requires knowledge of original site energies.

Pump Probe Spectroscopy of Bis-Phthalocyanine: Excited State and Wavepacket Dynamics N. Ishikawa, Chuo University

LuPc 2 Dimer Absorption spectra shows monomer s single peak is split in the dimer E / 10 5 M -1 cm -1 2.5 2.0 1.5 1.0 Monomer Dimer Inset: Sandwich structure of Lutetium bisphthalocyanine 0.5 0.0 400 500 600 700 Wavelength (nm)

Two-Color Echoes on Excitonic Systems: Theory Mixing between states with site energies ε 1 and ε 2 and coupling J can be quantified by θ. 2J tan(2 θ ) = eq eq ε ε 1 2 µ (cos( θ ) + sin( θ ) ) 0 0 + + + B1 B2 B µ ν ( sin( θ) + cos( θ) ) 0 0 + + + B1 B2 B ν Energy fluctuations at one monomer shift both exciton states in the same direction; the magnitude depends on the coupling. Stronger mixing yields stronger correlation. M. Yang, G. R. Fleming, J. Chem. Phys. 110, 2983-2990 (1999).

Coupling Strength Coupling strength relates to the renormalization coefficient, C mn, for the line broadening function, g(t). C 2 2 µν = Cνµ = 2 sin θ cos θ Dependence of g(t) on coupling causes the one-color and two-color peak shift to increase with increased coupling. C C ( T ) µν * τ τ two τ two( T ) ( T ) + τ ( T ) one Coupling Strength, Cµν C µυ 0.3 0.2 0.1 2Color, = 1Color + 2Color 0.0 0 100 200 300 400 Population Time, T (fs) M. Yang, G. R. Fleming, J. Chem. Phys. 110, 2983-2990 (1999).

1-and 2-Color (620, 620, 700nm) Photon Echo Peak Shift 1-color and 2-color peakshifts of LuPc 2 are very similar 30 20 LuPc 1-color 2 Oscillation, of similar period in both measurements, but approximately π out of phase Peakshift (fs) 10 2-color 0 0 100 200 300 400 500 Population Time (fs)

0.6 The Mixing Coefficient C τ* (experiment) Electronic structure of Lu(P c) 2 Estimating C µυ C τ (T) 0.4 0.5 10 100 J EX-CR C R + > EX + > 0.2 0.3 0.0 0.1 0 20 40 60 80 100 Population Time (fs) Experimental C τ* is approximately 0.45 Dip at longer time due to out-ofphase oscillation between 1- and 2-color g > If all dipole strength comes from EX + >, the ratio of absorption intensities gives 2 Tan ( θ ) C µυ 0.43 From wavefunction coefficients of Ishikawa et. al C µυ 0.40

Modulation of the Electronic Coupling τ τ τ * one * tw o ( T ) = ( T ) = (1-C ) a ( T ) + 2 C N ( T ) µν µν {(1 C µν ) C 0} {(1 C µν ) C 0} 3 2 3 2 (C a ( T ) 2 C N ( T ) µν µν 1 N ( T ) = < q 2 12 ( T ) q12 (0) > h 2 q12 Γ T = e cosω 2 h * τ two( T ) 2 N( T ) = Cµν 1 ( T ) + τ two( T ) a( T ) * * one π π vib T Minhaeng Cho π phase shift ( a( T ) = real part of C( t) ) Time(fs) 20 15 10 5 0 One-color PEPS Two-color PEPS Ratio 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2-5 0 500 1000 1500 Time Time(fs) (fs) 0.1 0 0.5 1 1.5 2 2.5 3 Log(time) in fs

Wavelength resolved pump-probe signals for - LuPc 2 1400 615 700 Population Time (fs) 1200 1000 800 600 400-0.010-0.005 0.000 0.005 0.010 0.030 0.035 0.028 0.032 200 0 580 600 620 640 660 680 700 720 740 Wavelength (nm) 580 600 620 640 660 680 700 720 740 The beats on the red side of the high energy band are exactly out of phase with the beats on the red side of the low energy band, but are in phase with the beats on the blue side of the low energy band

Modulated Electronic Structure Simple model with a single coordinate, x,coupled linearly to the states E EX +, E CR+ and the coupling between these two states V EX CR x( t) = x cos( ωt)exp( t τ ) o o EX + = EX + + EX + E E E x E = E + E x o CR+ CR+ CR+ V = V + V x 0 EX CR EX CR EX CR New, time-dependent potential curves for 1E + and 2E + F E CR+ = -60 E EX+ = 60 V EX-CR = 30 Non-Condon effects (modulation of transition moments). If the signs of E EX + and E CR + are opposite the motions of 1E + and 2E + are anti-correlated. 0 600 620 640 660 680 700 720 740

ω 1 ω 2 Physical Origin Ground State Wave-Packet Symmetric nitrogen-metal mode; Fluctuating coupling

1C and 2C3PEPS on P-oxidized reaction center of Rb. Sphaeroides P + B b B a X X ~3ps ~150fs ~100fs Energy H h b B H b Q b <1ps H a Qa g b g h Excitonic Coupling Fe 2+ 1C3PEPS H band: λ 1 =λ 2 =λ 3 =750nm B band: λ 1 =λ 2 =λ 3 =800nm Uphill: 2C3PEPS λ 1 =λ 2 =800nm, λ 3 =750nm Downhill: λ 1 =λ 2 =750nm, λ 3 =800nm

Simulation of 3PEPS based on density matrix & ρ( t) = i[ H + H ( t), ρ( t)] R[ ρ( t)] S Laser int Bath Directly simulate P (3) 3PPE (t) from laser field-driven dynamics: S( τ, T ) ~ dt P ( t) 0 (3) 3PPE 2 Peakshift Pros compared to conventional response function formalism: Pulse-overlap effects included by considering laser fields explicitly. Complete dynamics of the system treated by non-markovian equations of motion.

Simulation for 1C3PEPS and obtained bath parameters 35 30 25 H band (Experiment) H band (Simulation) B band (Experiment) B band (Simulation) Most of the dynamics happen in less than 300fs due to the rapid energy transfer (H B P + ) Peakshfit(fs) 20 15 10 5 0 0 50 100 150 200 250 300 T(fs) Proper inclusion of pulse-overlap effects improved reproduction of the peakshifts measured at small T Determination of bath parameters for H and B, respectively B H λ 75 1500 250 Ω 100 220 50 Γ 150 10 150 σ 60 60 λ: electron-phonon coupling constant Ω: characteristic frequency Γ: damping constant σ: static disorder

Determination of coupling constant J between H & B by 2C3PEPS Peakshift(fs) 15 10 5 0 Downhill Experiment Simulation (J=200cm -1 ) Simulation (J=250cm -1 ) Simulation (J=300cm -1 ) -5 0 50 100 150 200 250 300 T(fs) 20 J=250cm -1 gives best fit to both measurements Peakshift(fs) 15 10 5 0 Uphill Negative peakshift due to the rapid dynamics B P + -5-10 0 50 100 150 200 250 300 T(fs) Y.-C. Cheng et al., J. Phys. Chem. A 111, 9499 (2007)

Nonlinear spectroscopy incident fields QM system signal field ~ i ω P(ω) detection polarization P different aspects probed by TA, TG, 3PEPS etc. r r r P ( ω) ε dω dω χ ( ω,, ω ) E( ω ) E( ω ) δ ( ω ω ) ( n) ( n) = 0 1L n 1 K n 1 L n s obtain complete information about system by measuring signal intensity and phase

Third-order spectroscopy t LO freq.-dom. local oscillator = 4 coh. pop. echo time time time τ T t 1 2 3 sig homodyne detection: E sig (ω) 2 or even frequency-integrated t LO time-domain local oscillator heterodyne (time domain): E sig (t) + E LO (t+t LO ) 2 problem: t LO needs to be scanned heterodyne (frequency domain): multichannel frequency detection LO-position t LO fixed E sig (ω) + E LO (ω) exp{-iωt LO } 2

4=LO Two-Dimensional Heterodyne Spectroscopy coh. time pop. time echo time τ T t 1 2 3 sig 1 2 3 4 sample spectrometer spherical mirror 1&2 OD3 4 3 3&4 2 diffractive optic (DO) 2 f delay 2 delay 1 1

Experimental setup sample lens delay 2 Opt. Lett. 29 (8), in press (2004) diffractive optic (DO) delay 1 spherical mirror

Advantages of setup problem small signals delay accuracy phase stability scattering solution heterodyne advantage (record signals < 100 aj) small-angled wedges (position repeatability: 5 as) diffractive-optics-based setup (2D background < 2%) subtraction by automated shutter first implementation of tunable 2D spectroscopy in the visible region

2D data analysis raw data step 2: FFT -1 (along τ) Re Im step 1a: slice spectrum (for each τ) FFT cut & FFT -1 step 1b: get I(ω), Φ(ω)

Time-delay accuracy Nile Blue heterodyned signal recorded within separate scans lab time: 20 min 15 min 10 min 5 min 0 min no shift of fringe pattern: interferometric stability and reproducibility

Scattering subtraction Desired heterodyned signal comes from (E 4 * E sig ) But heterodyned signal involves scatter E 1,,E 3 : E 1 +E 2 +E 3 +E 4 +E sig 2 3.07 without subtraction Remove scattering by E 1 +E 2 +E 3 +E 4 +E sig 2 - E 3 +E 4 2 - E 1 +E 2 +E 4 2 with subtraction ω t / fs -1 3.27-3.27-3.07 ω τ / fs -1-3.27-3.07 ω τ / fs -1

Phasing (PP versus 2D) Nile Blue in acetonitrile (T = 0 fs) positive signal: emission or ground-state bleach negative signal: excited-state absorption overall phase determines Re {S 2 }: absorption / emission Im {S 2 }: refraction

probability distribution S 2 (ω τ,t, ω t ) 2: wait for population time T 3: emit or bleach at frequency ω t 2D spectrum here: inhomogeneous system remembers excitation after time T homogeneous linewidth 1: excite at frequency ω τ inhom. linewidth

Nile Blue (experiment) 3.07 Abs Re ω t / fs -1 3.17 3.27 3.07 Pha Im ω t / fs -1 3.17 3.27-3.27-3.17-3.07 ω τ / fs -1-3.27-3.17-3.07 ω τ / fs -1 T T = = 100 30 40 50 60 70 80 90 15 20 05 fs

Theory of 3-level systems f ω fe d fe e ω eg d eg g g ff (t) f-bath g fe (t) g ee (t) e-bath laser: ω 0, τ p g(t) has contributions from vibrations: (ω v, λ v, τ v ) solvent: Gaussian (λ sg, τ sg ) Exponential (λ se, τ se ) t τ 1 = 2 h 0 0 g( t) dτ dτ C( τ ) regular energy-gap correlations correlations between excited states { } C ( t) = Tr U ( t) U (0) ρ ee eg eg gg v, s { } C ( t) = Tr U ( t) U (0) ρ ff fg fg gg v, s { } C ( t) = Tr U ( t) U (0) ρ fe fg eg gg v, s

Nile Blue (simulation) 2-level simulation Re Im Re Im experiment 0 fs 100 fs 3-level simulation ω fe = ω eg +100 cm -1 g fe (t) = 0 3-level simulation ω fe = ω eg +100 cm -1 g fe (t) = g ee (t)

Dipole strength and g fe T = 0 fs Re Im T = 100 fs Re Im d fe = d eg g fe (t) = 0 ω fe = ω eg d fe = 1.5 d eg g fe (t) = 0 ω fe = ω eg d fe = 1.5 d eg g fe (t) = g ee (t) ω fe = ω eg

2D Spectroscopy of Aggregates MOLECULAR AGGREGATES WEAKLY COUPLED LH2 Complex Absorption spectra of BIC monomer and J-aggregates STRONGLY COUPLED Two-exciton Band 2e Linear chain of 2 level molecules with electrostatic dipole-dipole interaction One-exciton Band 1e Ground state g

H (q) = N n ε n n + n = 1 Off-diagonal Electrostatic J-AGGREGATE HAMILTONIAN N m, m n n = 1 m J mn n N + n H ( q) n + H ph (q) n= 1 SITE BASIS: el ph nn Diagonal Electron-Phonon Diagonal Exciton-Phonon EXCITON BASIS: Off-Diagonal Exciton-Phonon EXCITON WAVEFUNCTIONS Renormalization Factors Cause Exchange Narrowing Overlap Factors Define Relaxation Higher Exciton States are Strongly Delocalized Exchange-Narrowing is Stronger for Higher (More Delocalized) Exciton States Relaxation is Faster for Higher Exciton States

BIC J-Aggregates, Real 2D Spectra + -

Absolute Value 2D Spectra BIC J-Aggregates 70% 50% Magnified central part of 2D spectrum Diagonal Elongation: Static Disorder Width Broadening at Higher Frequencies: Exciton Relaxation Detailed Width Profile: Frequency-Dependent Dynamics

Absolute Value 2D Plots BIC J-aggregates T = 50 fs T = 500 fs Absolute value of the 2D plots; population periods T=50, 500 fs. Evolution of the 2D plots is mostly determined by the exciton relaxation during the population period.

Frequency-Dependent Exciton Relaxation Fast Mode, no Relaxation: Small Changes with T, Minimum Width at Higher Frequencies Simulation with relaxation & Experiment: - Minimum width to the lower Frequency of 2D Peak & Asymmetric Shape: Relaxation induced Life-Time Broadening is stronger at Higher Frequency - Increased Width: Exciton Relaxation in Population Period - Reduced Width: No Exciton Relaxation

H (q) N N = ε n + n= 1 m, n= 1 m n J-Aggregate Hamiltonian n n m J n Off-diagonal Electrostatic mn EXCITON BASIS: k n= 1 N N = ϕ n= 1 kn SITE BASIS: + n H ( q) n + H ph (q) el ph nn Diagonal Electron-Phonon n n> - monomer #n is excited Diagonal Exciton-Phonon Off-Diagonal Exciton-Phonon Exchange Narrowing: Reduction of the Fluctuation Amplitude Relaxation: Hopping Between States FREQUENCY- DEPENDENT DYNAMICS

Experimental 2D Spectrum Anti-Diagonal Width ω 1 2D Spectra From J-Aggregates for T=50fs and T=500fs Broadening of Higher Frequency Part is Due to Exciton Relaxation

Discrimination of Fluctuational and Relaxational Dynamics EXPERIMENT SIMULATION without RELAXATION a) fast mode b) slower mode SIMULATION with RELAXATION Relaxation Broadens High Frequency Part Fluctuation Only Dynamics High Frequency Part Shrinks

Frequency-Dependent Delocalization, Relaxation and Exchange-Narrowing Delocalization Varies from 5 to 30 monomers across the Absorption Band of J-aggregates Relaxation is Increased at Higher Frequencies Dynamic Fluctuations are Reduced at Higher Frequencies by Exchange- Narrowing

Best Fit: Experiment & Simulations Experiment is Reproduced Theoretically over Complete R (3) Model Parameters(only 1 independent): Disorder: σ = 250 cm -1 Gaussian Mode Correlation Function: λ = 250 cm -1, τ = 50 fs (liquid water: 30 70fs) Coupling: J = 1080 cm -1 (from absorption) (σ & λ determine absorption spectrum width) Frequency-Dependent Dynamics is Correctly Reproduced Monomer Static Disorder and the Approximate Correlation Function are Retrieved Solid lines are plotted at +10,20, % and dashed at -10,20, % of the absolute 2D spectrum maximum [at given time T ] Dynamics of One and Two Exciton Contributions are Correctly Reproduced [Positive and Negative Spots in the 2D Real Spectrum ]