Decoherence measurement of single phonons using spectral interference
F. C. Waldermann, B. J. Sussman, J. Nunn, V. O. Lorenz, K. C. Lee, K. Surmacz, K. H. Lee, D. Jaksch, I. A. Walmsley, P. Spizziri, P. Olivero, S. Prawer
aa r X i v : . [ qu a n t - ph ] J u l Measuring Phonon Dephasing withUltrafast Pulses
F. C. Waldermann, Benjamin J. Sussman, ∗ J. Nunn, V. O. Lorenz,K. C. Lee, K. Surmacz, K. H. Lee, D. Jaksch, and I. A. Walmsley
Clarendon Laboratory, University of Oxford,Parks Road, Oxford, OX1 3PU, UK
P. Spizziri, P. Olivero, and S. Prawer
Center for Quantum Computer Technology, School of Physics,The University of Melbourne, Parkville, Victoria 3010, Australia
Abstract
A technique to measure the decoherence time of optical phonons in a solid is presented. Phononsare excited with a pair of time delayed 80 fs, near infrared pulses via spontaneous, transient Ramanscattering. The fringe visibility of the resulting Stokes pulse pair, as a function of time delay, isused to measure the phonon dephasing time. The method avoids the need to use either narrowband or few femtosecond pulses and is useful for low phonon excitations. The dephasing time ofphonons created in bulk diamond is measured to be τ =6.8 ps (∆ ν = 1 .
56 cm − ). . INTRODUCTION Phonons are a fundamental excitation of solids that are responsible for numerous electric,thermal, and acoustic properties of matter. The lifetime of optical phonons plays an im-portant role in determining these physical properties and has been the subject of extensivestudy. A technique to measure phonon dephasing times is presented here that utilizes ul-trafast infrared laser pulses: Transient Coherent Ultrafast Phonon Spectroscopy (TCUPS)offers a number of conveniences for measuring phonon dephasing. TCUPS utilizes com-mercially available ultrafast pulses (80 fs) and hence does not require a narrow band orextremely short lasers to achieve high spectral or temporal resolution. As well, TCUPS issuitable for measurements in the single phonon excitation regime. The large sampling areaand long sampling distance increase the generated Stokes power and avoid sample heating,which is a concern for low temperature studies. Diamonds are well known for their ex-traordinary physical properties and, as well, offer interesting prospects for use in QuantumInformation applications . As such, diamond has been selected here as the material fordemonstration of TCUPS.Two methods have previously been utilized to measure phonon lifetimes: high-resolutionRaman spectroscopy and differential reflectivity measurements. The first is the traditionaltechnique, where the optical phonon lifetime is obtained from high-resolution linewidthmeasurements of the first-order Raman peak, usually conducted using narrowband excita-tion lasers and high-resolution spectrometers . The alternative technique, working in thetime-domain, can directly show the temporal evolution of the surface vibrations of solids .A femtosecond pump pulse is used to excite a phonon population. The reflectivity (ortransmittivity) of a subsequent probe pulse displays a time dependence that follows thevibrational frequency and phonon population. This method was used to study the phonondecay in various solids , their symmetry modes , and their interaction with charge carriers and with other phonons . In these experiments, impulsive stimulated Raman scatteringhas been established as the coherent phonon generation mechanism .The time-domain experiments utilize the impulsive regime, i.e . laser pulse lengths muchshorter than the phonon oscillation period (inverse phonon frequency). This requirementcan be challenging for the application of the differential reflectivity technique to materialswith high phonon energies, as laser systems with very short pulse lengths are required2 IG. 1: Experimental setup. An oscillator pulse is split into two time delayed pulses and focusedthrough the diamond sample. Not shown, a bandpass filter cleans the oscillator pulse before thediamond and a longpass filter rejects the pump and transmits the Stokes before the spectrometer. ( e.g. , for diamond, sub-10 fs pulses are required to resolve a phonon frequency of 40 THz).TCUPS operates in the transient Raman scattering regime, i.e. , pulse lengths much shorterthan the phonon decoherence time, which is usually orders of magnitude slower than thephonon oscillation period (about 25 fs for diamond) . Stimulated Raman scattering, whichimplies large phonon excitations, is often employed in dephasing measurements in order toachieve good signal-to-noise ratios. High phonon population numbers, often referred to as hot phonons , can be subject to an increased decay rate, as previously observed for GaN.By contrast, TCUPS investigates the properties of a coherent phonon excitation by directanalysis of the Stokes spectra generated in the Raman process. The use of single photondetectors extends the sensitivity of the experiment to low phonon populations, including thesingle phonon level. II. EXPERIMENT
The diamond was classified as a type Ib high pressure high temperature (HPHT) sam-ple with a nitrogen impurity concentration of less than 100 ppm. The Stokes shift ofdiamond is 1332 cm − and the Raman gain coefficient for diamond has been reported as g = 7 . × − cm / MW (corrected for λ = 800 nm). With pump pulse energies ranging over1 . . . .
380 pJ, the collinear Stokes emission is calculated as 0 . . . . . IG. 2: Diamond energy level schematic. Ground state phonons are excited with the incident788 nm pump, via a Raman transition, to the optical phonon mode, emitting an 880 nm Stokespulse
The experimental setup is depicted in fig. 1. Phonons are excited, via a Raman transition,with a pair of time-delayed 80 fs, 788 nm pulses (fig. 2) from a commercial Ti:SapphireOscillator (Coherent Mira). The pulses are focussed into a 2 × × λ is as expected for two time-delayed coherent pulses: ∆ λ = λ /cτ (see also(4), below). For the excitation pair, λ is the centre wavelength of the pump (fig. 3b) and forthe generated output Raman pair, λ is the centre wavelength of the Stokes (fig. 3d). The4 IG. 3: Example spectral interference for a delay τ = 0 .
39 ps. Spectra of the broadband excitationlaser (left) and the Stokes signal of diamond (right). The single pulse data in (a) and (c) show thepump laser spectrum and the corresponding broadband Raman spectrum. Spectral interferencefringes appear for coherent pulse pairs in (b) and (d). fringe spacing of the Raman output corresponds to the Stokes peak wavelength, confirmingthat the process is a measure of the coherence of the Raman process. Figure 4a shows thefringe visibility reduction as a function of time delay. The fringe visibility V = exp( − Γ | τ | )is plotted in fig. 4b. The visibility has been renormalized using the laser visibility for eachdelay to account for beam walk-off and the spectrometer resolution which artificially reducesvisibility, due to a sampling effect from the finite pixel size of the spectrometer CCD. III. THEORY
The observed interference visibility can be considered from two perspectives. In the first,the visibility decay arises due to fluctuations of the phase in the classical fields. Each inputlaser pulse excites optical phonons, via a Raman transition (fig. 2), which in turn causes theemission of two Stokes pulses. That is, the Raman interaction maps the electric field of thetwo input pulses to two output Stokes shifted pulses E Stokes = E ( t ) + E ( t ) . (1)5
73 874 875 876 877
Wavelength (nm) - a r m s p ec t r a ( w i t h o ff se t; a r b . u . ) Delay (ps) V i s i b ili t y (a) (b)(c) S t o kes I n t e n s i t y ( a r b . u . ) Stokes Shift (cm ) –1 FIG. 4: Decoherence measurement. (a) Stokes spectra for two pump pulses with various delays τ ,recorded with a 1800 lines/mm grating. The decrease of spectral interference visibility of the Stokessignal is due to decoherence of the optical phonons created. The respective visibilities are plottedin (b), obtained by curve-fitting the spectra. Asterisks denote data points, the continuous line anexponential decay fit. Part (c) shows a high-resolution Raman spectrum for the same diamond(dots) with a Lorentzian fit (line). The phase of the first Stokes pulse E is determined spontaneously, but the phase (andamplitude at stimulated intensities) of the second pulse E is influenced by the coherencemaintained in the phonon of the system following the first pulse, so that the output fieldmay also be rewritten as E Stokes ( t ) = E ( t ) + e iθ E ( t − τ ) (2)where τ is the time delay between the input pulses and θ is the spontaneously fluctuatingphase difference between the pulses. The spectral intensity of the Stokes pulse pair | E Stokes ( ω ) | = 2 | E ( ω ) | (1 + cos( ωτ + θ )) (3)6ontains interference fringes whose position depends on the relative phase θ . Shot-to-shot,decoherence causes spontaneous fluctuations in θ and the fringe pattern loses visibility. Atlonger delays τ , the fluctuations increase, eventually reducing the visibility of any integratedfringe pattern to zero. Assuming a Lorentzian lineshape with width Γ for the distributionof the phase shift, the shot-to-shot averaged spectral intensity is broadened to D | E Stokes ( ω ) | E shots = 2 | E ( ω ) | (cid:16) e − Γ | τ | cos( ωτ ) (cid:17) . (4)The phase fluctuations cause a reduction of the fringe visibility.Alternatively, the fluctuating phase perspective can be connected with the second, quan-tum field perspective. This formalism can also be made applicable in the stimulated regime.The observed spectral intensity expectation value is proportional to the number of Stokesphotons: D | E Stokes ( ω ) | E ∝ D A † A E (5)where the lowering operator A ( ω ) is a sum of the first A and second A pulse mode loweringoperators: D | E Stokes ( ω ) | E ∝ D A † A E + D A † A E + 2 ℜ D A † A E . (6)The final, correlated term ( cf . the decay term in (4)) measures the phonon coherence thatremains in the system between pulses. During the evolution of the system, the starting timephonon mode B † (0) is ‘mixed’ into the Stokes photon modes A i due to application of thelaser field. The mixed-in term is then subsequently the source for spontaneous emission. Thesource term is the same for both pulses, but during the period between pulses the coherenceis reduced due to crystal anharmonicity and impurities. For the correlation, the relevantterms to lowest perturbative order are (see appendix for the equations of motion) A ≈ A (0) − igτ pump B † (0) (7) A ≈ A (0) e iωτ − igτ pump B † (0) e − Γ τ (8)from which the correlation term can be evaluated as D A † A E ≈ g τ pump D B (0) B (0) † E = g τ pump h N B (0) + 1 i e − Γ τ (9)where N B (0) is the initial number of phonons, which in this case is the nearly zero thermalpopulation. This result links the phonon decoherence rate Γ with the fluctuating phaseperspective linewidth Γ from (4). Therefore, measuring the reduction of the fringe visibilityis a direct measure of the phonon dephasing time.7 V. DISCUSSION
The TCUPS measurement indicates a phonon dephasing time of 1 / Γ = 6 . ± . ν = 1 .
56 cm − . The literature has reported a great deal of varia-tion in linewidth measurements for diamond, varying from at least 1.1 cm − to as high as4 .
75 cm − . Here, the TCUPS (fig. 4b, ∆ ν = 1 .
56 cm − ) and conventional Raman spec-trum (fig. 4c, ∆ ν = 1 .
95 cm − ) show comparatively good agreement. The lifetime measuredhere is slightly shorter than the decay rate calculated theoretically by Debernardi et al. for an ideal crystal (∆ ν = 1 .
01 cm − or 1 / Γ = 10 . e.g. , schemes that transfer optical entanglement to matter .The spectral interference pattern persists for low excitation levels, i.e., a phonon exci-tation probability per mode of p <
1. (The case of p ≫ , althoughusing spatial, not spectral interference.) A constant visibility with excitation power canbe seen at low excitation level (fig. 5). TCUPS can therefore be employed to measurethe decoherence properties of single optical phonons, overcoming the need for large phononpopulations for lifetime measurements of phonon decoherence.The excitation probability per mode was much smaller than 1, ranging over p ≈ − . . . − due to the large number of phonon modes in the Brillouin zone for which Stokesscatter is detected ( ∼ ). This level is in fact smaller than the thermal population levelof the optical phonons at room temperature, given by p thermal = [exp( E vib /k B T ) − − ≈ . verage Excitation Power (mW) single phonon regime FIG. 5: Power dependence. The power dependence of the Stokes interference visibility (at constantdelay τ = 0 .
51 ps, reduced due to a limited alignment) showing that the experiment can be carriedout at arbitrarily low phonon excitation levels (the green horizontal line is plotted to guide theeyes). The inset shows the dependence of the Stokes pulse energy on the average pump power(single pump pulses). The linear power dependence shows that the scattering is in the spontaneousRaman regime. The fraction of pump power converted into collinear Stokes light was measured tobe less than 10 − . further excitation. The linewidth increase due to phonon-phonon interaction is negligibleat ambient temperatures in diamond due to the low population level . An increase in theRaman linewidth of diamond due to temperature has been reported to begin at around T ≈
300 K. At T ≈
800 K, it is more than twice the zero temperature linewidth. Atroom temperature, the phonon decay is only marginally enhanced by an acoustical phononpopulation. This insensitivity to a thermal background is in contrast to the differentialreflectivity method, where thermal phonons lead to additional noise as both thermal and9oherent phonons lead to a change of the material reflectivity.TCUPS is a convenient approach to determining the quantum coherence properties ofoptical phonons in Raman active solids. The measurement technique relies solely on spon-taneous Raman scattering and is therefore useful down to the single phonon levels. Inparticular, TCUPS enables the measurement of the decoherence time of phonons, which isof paramount importance in many Quantum Information Processing schemes. Spectral inter-ference of the Stokes light from pump pulse pairs is used to measure the Raman linewidthof the material, while maintaining a coherent excitation due to ultrafast excitation. Thephonon lifetime of diamond was measured as 6 . Q -factor of Q = ν/ Γ ∼ Q and the low thermal population atroom temperature make it feasible for proof-of-principle demonstrations of typical quantumoptics schemes, such as collective-excitation entanglement in the solid state.Acknowledgements. This work was supported by the QIPIRC and EPSRC (grant numberGR/S82176/01), EU RTN project EMALI, and Toshiba Research Europe. Appendix: Phonon-photon equations of motion
Consider an incident pump laser that Raman scatters off a phonon field of the diamondto produce an output Stokes field. The equations of motion for Stokes field A ( t ) and thephonon field B ( t ) are linked by the pump coupling g via :˙ A ( t ) = − igB † ( t ) (10)and ˙ B ( t ) = − igA † ( t ) − Γ B ( t ) + F † ( t ) . (11)The dephasing rate Γ is due to crystal anharmonicity and impurities. The Langevin operator F has been added to maintain the normalization of B in the presence of decay, allowingthe phonon to decohere, but keeping the operator norm, via the commutation relation[ B, B † ] = 1, constant. The formal solutions are: B = B (0) e − Γ t + Z t e − Γ( t − t ′ ) h − igA † ( t ′ ) + F † ( t ′ ) i dt ′ (12)and A = A (0) − i Z t gB † ( t ′ ) dt ′ . (13)10or brevity, the time argument has been dropped from the solutions. In the weak ( gτ pump ≪
1) and transient (Γ τ pump ≪
1) pump pulse limit, the incident laser leaves the phonon operatorapproximately in the vacuum state B (0) and the phonon operator solution at lowest orderis B ≈ B (0) e − Γ t + Z t e − Γ( t − t ′ ) F † ( t ′ ) dt ′ . (14)The Stokes field to first order is then A ≈ A (0) − igτ pump B † (0) e − Γ t − i Z t Z t ′ ge − Γ( τ − t ′′ ) F ( t ′′ ) dt ′′ dt ′ (15)where the coupling g in the second term has been taken as a constant step for the durationof the pump. The initial Stokes operator A (0) annihilates the vacuum, but the solution for A mixes in a component of the phonon raising operator B † (0), which acts as a source forthe spontaneous Raman scattering.The decoherence rate Γ represents the dephasing of the phonon raising operator B † .The phonon number N B = B † B therefore decays at a rate 2Γ. The corresponding spectralfrequency linewidth is ∆ ν = Γ /π . ∗ Electronic address: [email protected]; Also at National Research Council of Canada, Ottawa,Ontario K1A 0R6, Canada J. E. Field, editor.
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