^{1}, Thomas R. Dyke

^{2}and Andrew H. Marcus

^{3,a)}

### Abstract

Studies of wave packet dynamics often involve phase-selective measurements of coherent optical signals generated from sequences of ultrashort laser pulses. In wave packet interferometry (WPI), the separation between the temporal envelopes of the pulses must be precisely monitored or maintained. Here we introduce a new (and easy to implement) experimental scheme for phase-selective measurements that combines acousto-optic phase modulation with ultrashort laser excitation to produce an intensity-modulated fluorescence signal. Synchronous detection, with respect to an appropriately constructed reference, allows the signal to be simultaneously measured at two phases differing by 90°. Our method effectively decouples the relative temporal phase from the pulse envelopes of a collinear train of optical pulse pairs. We thus achieve a robust and high signal-to-noise scheme for WPI applications, such as quantum state reconstruction and electronic spectroscopy. The validity of the method is demonstrated, and state reconstruction is performed, on a model quantum system—atomic Rb vapor. Moreover, we show that our measurements recover the correct separation between the absorptive and dispersive contributions to the system susceptibility.

We thank Professor Jeff Cina and Professor Michael Kellman for useful discussions, and Cliff Dax for his assistance with the design and implementation of custom electronics. This research is supported by the National Science Foundation CHE-0303715, the American Chemical Society Petroleum Research Fund 40238-AC6, and the Research Corporation RA0314.

I. INTRODUCTION

II. THEORETICAL BACKGROUND

A. Wave packet interferometry (WPI)

III. EXPERIMENTAL METHODS

A. WPI with acousto-optic (AO) phase modulation

B. Phase-sensitive signal detection

C. Quantum state reconstruction

D. Determination of the linear susceptibility

E. PM-WPI instrumentation

IV. DISCUSSION OF RESULTS

A. PM-WPI using spectrally identical pulses

B. Experimental determination of the susceptibility

C. Quantum state reconstruction

V. CONCLUSIONS

### Key Topics

- Monochromators
- 27.0
- Quantum state reconstruction
- 20.0
- Fluorescence
- 17.0
- Excited states
- 16.0
- Phase modulation
- 16.0

## Figures

Energy level diagram for the line transitions (described in text and Table I). For the purposes of our experiments, the system behaves as a three-level atom with ground state , first excited state , and second excited state .

Energy level diagram for the line transitions (described in text and Table I). For the purposes of our experiments, the system behaves as a three-level atom with ground state , first excited state , and second excited state .

Illustration of a train of phase-modulated pulse pairs. Each pulse pair is labeled by the superscript ; the individual pulses are labeled by the subscripts 1 (target pulse) and 2 (reference pulse). A pulse pair is characterized by the interpulse delay , and the relative temporal phase . (A) Both target and reference pulses are spectrally identical and Fourier transform limited. (B) The target pulse is spectrally chirped, while the reference pulse is transform limited.

Illustration of a train of phase-modulated pulse pairs. Each pulse pair is labeled by the superscript ; the individual pulses are labeled by the subscripts 1 (target pulse) and 2 (reference pulse). A pulse pair is characterized by the interpulse delay , and the relative temporal phase . (A) Both target and reference pulses are spectrally identical and Fourier transform limited. (B) The target pulse is spectrally chirped, while the reference pulse is transform limited.

Schematic diagram of the experimental setup for phase modulation (PM-) WPI (described in text). Abbreviations have the following meanings. APD: amplified photodiode; PD: pin photodiode; AO: acousto-optic Bragg cell; BS: beam splitter.

Schematic diagram of the experimental setup for phase modulation (PM-) WPI (described in text). Abbreviations have the following meanings. APD: amplified photodiode; PD: pin photodiode; AO: acousto-optic Bragg cell; BS: beam splitter.

(A) Typical noncollinear pulse autocorrelation measurements (open circles) used to minimize the optical pulse length by predispersion compensation (described in text). The solid curve is the best-fit Gaussian function with . Assuming a Gaussian temporal pulse envelope, the corresponding FWHM pulse width is . (B) Power spectrum measurement of the same pulses measured in (A). The dashed gray curve is the best-fit Gaussian with center wavelength (indicated by the vertical dashed line) and FWHM spectral width . Also shown are the narrow line transitions of the Rb system, with magnitude ratio proportional to the square of the transition dipole moments (see Table I).

(A) Typical noncollinear pulse autocorrelation measurements (open circles) used to minimize the optical pulse length by predispersion compensation (described in text). The solid curve is the best-fit Gaussian function with . Assuming a Gaussian temporal pulse envelope, the corresponding FWHM pulse width is . (B) Power spectrum measurement of the same pulses measured in (A). The dashed gray curve is the best-fit Gaussian with center wavelength (indicated by the vertical dashed line) and FWHM spectral width . Also shown are the narrow line transitions of the Rb system, with magnitude ratio proportional to the square of the transition dipole moments (see Table I).

(A) Spectral density measurement (black) and spectral phase (gray) of a chirped laser pulse resulting from BK7 glass [calculated using Eq. (2.9) and the Sellmeier equation] (Refs. 56 and 57). The best-fit Gaussian (not shown) has a center wavelength and . Also indicated are the monochromator setting , and the Rb line transitions. (B and C) Expected PM-WPI signal [see Eqs. (3.15) and (3.16)] corresponding to the spectral conditions summarized in (A) (see also Table II). The signal is plotted in the complex plane with and axes defined as and , respectively. The signal (, shown in black) is a vector superposition of two counterprecessing components ( and , shown in gray). (B) The initial condition is determined by the spectral overlap of the target pulse with the Rb line transitions. (C) During the evolution period , the resultant traces a quasielliptical trajectory (shown in black), with magnitude and phase function .

(A) Spectral density measurement (black) and spectral phase (gray) of a chirped laser pulse resulting from BK7 glass [calculated using Eq. (2.9) and the Sellmeier equation] (Refs. 56 and 57). The best-fit Gaussian (not shown) has a center wavelength and . Also indicated are the monochromator setting , and the Rb line transitions. (B and C) Expected PM-WPI signal [see Eqs. (3.15) and (3.16)] corresponding to the spectral conditions summarized in (A) (see also Table II). The signal is plotted in the complex plane with and axes defined as and , respectively. The signal (, shown in black) is a vector superposition of two counterprecessing components ( and , shown in gray). (B) The initial condition is determined by the spectral overlap of the target pulse with the Rb line transitions. (C) During the evolution period , the resultant traces a quasielliptical trajectory (shown in black), with magnitude and phase function .

Monochromator setting dependence of the PM-WPI undersampled interferograms for spectrally identical pulses, with and . In-phase data [, gray filled circles], and in-quadrature data [, solid black circles], are superimposed with theoretical curves (gray and black, respectively) given by Eqs. (4.1a) and (4.1b). (A) . The fully sampled interferogram is also shown (light gray). (B) . (C) .

Monochromator setting dependence of the PM-WPI undersampled interferograms for spectrally identical pulses, with and . In-phase data [, gray filled circles], and in-quadrature data [, solid black circles], are superimposed with theoretical curves (gray and black, respectively) given by Eqs. (4.1a) and (4.1b). (A) . The fully sampled interferogram is also shown (light gray). (B) . (C) .

Complex linear susceptibility determined [according to Eqs. (3.28) and (3.29)] from the same undersampled interferograms shown in Fig. 6. The real and imaginary parts of are shown as black and white curves, respectively. Superimposed onto each plot is the Gaussian fit to the laser spectral density with and . (A) . (B) . (C) .

Complex linear susceptibility determined [according to Eqs. (3.28) and (3.29)] from the same undersampled interferograms shown in Fig. 6. The real and imaginary parts of are shown as black and white curves, respectively. Superimposed onto each plot is the Gaussian fit to the laser spectral density with and . (A) . (B) . (C) .

[(A)–(C)] Laser center wavelength dependence of PM-WPI signal for spectrally identical pulses, with and (see Table II). The signal magnitude (gray filled circles) and the signal phase function (solid black circles) are superimposed with the “expected” signal [shown as solid black curves, Eq. (4.6)] and the “reconstructed” signal [shown as dashed gray curves, Eq. (4.5)]. Also shown are the pulse-pulse autocorrelation envelopes (open black circles) and Gaussian fits (solid black curves) used to determine time origins for each data set. (A) . (B) . (C) . [(D)–(F)] Phasor diagram representation of the expected signals [Eq. (4.6)] corresponding to (A)–(C)). The coordinate system axes and component phasor labels are the same as in Figs. 5(b) and 5(c).

[(A)–(C)] Laser center wavelength dependence of PM-WPI signal for spectrally identical pulses, with and (see Table II). The signal magnitude (gray filled circles) and the signal phase function (solid black circles) are superimposed with the “expected” signal [shown as solid black curves, Eq. (4.6)] and the “reconstructed” signal [shown as dashed gray curves, Eq. (4.5)]. Also shown are the pulse-pulse autocorrelation envelopes (open black circles) and Gaussian fits (solid black curves) used to determine time origins for each data set. (A) . (B) . (C) . [(D)–(F)] Phasor diagram representation of the expected signals [Eq. (4.6)] corresponding to (A)–(C)). The coordinate system axes and component phasor labels are the same as in Figs. 5(b) and 5(c).

[(A)–(C)] Laser center wavelength dependence of PM-WPI signal for spectrally distinct pulses, with (see Table II). The signal magnitude (gray filled circles) and the signal phase function (solid black circles) are superimposed with the “expected” signal [shown as solid black curves, Eq. (4.6)] and the “reconstructed” signal [shown as dashed gray curves, Eq. (4.5)]. Also shown are the pulse-pulse autocorrelation envelopes (open black circles) and Gaussian fits (solid black curves) used to determine time origins for each data set. (A) and . (B) and . (C) and . [(D)–(F)] Phasor diagram representation of the expected signals [Eq. (4.6)] corresponding to (A)–(C). The coordinate system axes and component phasor labels are the same as in Figs. 5(b) and 5(c).

[(A)–(C)] Laser center wavelength dependence of PM-WPI signal for spectrally distinct pulses, with (see Table II). The signal magnitude (gray filled circles) and the signal phase function (solid black circles) are superimposed with the “expected” signal [shown as solid black curves, Eq. (4.6)] and the “reconstructed” signal [shown as dashed gray curves, Eq. (4.5)]. Also shown are the pulse-pulse autocorrelation envelopes (open black circles) and Gaussian fits (solid black curves) used to determine time origins for each data set. (A) and . (B) and . (C) and . [(D)–(F)] Phasor diagram representation of the expected signals [Eq. (4.6)] corresponding to (A)–(C). The coordinate system axes and component phasor labels are the same as in Figs. 5(b) and 5(c).

(Color) Comparison between PM-WPI signals obtained from excitation using a chirped target pulse and transform-limited pulse . [(A) and (B)] Phase function and magnitude data for the chirped and transform-limited experiments (shown as open circles and squares) are superimposed with the “expected” signals (solid curves) and “reconstructed” signals (dashed gray curves) described by Eqs. (4.6) and (4.5), respectively.

(Color) Comparison between PM-WPI signals obtained from excitation using a chirped target pulse and transform-limited pulse . [(A) and (B)] Phase function and magnitude data for the chirped and transform-limited experiments (shown as open circles and squares) are superimposed with the “expected” signals (solid curves) and “reconstructed” signals (dashed gray curves) described by Eqs. (4.6) and (4.5), respectively.

## Tables

Physical constants associated with optical line transitions. is the transition index, is the transition frequency, is the transition wavelength, is the transition dipole moment (in units of D), is the lifetime, and is the natural linewidth (Ref. 45).

Physical constants associated with optical line transitions. is the transition index, is the transition frequency, is the transition wavelength, is the transition dipole moment (in units of D), is the lifetime, and is the natural linewidth (Ref. 45).

State reconstruction parameters and results. For these data, . is the laser center wavelength, is the magnitude ratio of the expected target state amplitudes, is the relative phase of the expected target state amplitudes, is the magnitude ratio of the reconstructed target state amplitudes, is the relative phase of the reconstructed target state amplitudes, and is the fidelity of the reconstructed state [Eq. (4.4)].

State reconstruction parameters and results. For these data, . is the laser center wavelength, is the magnitude ratio of the expected target state amplitudes, is the relative phase of the expected target state amplitudes, is the magnitude ratio of the reconstructed target state amplitudes, is the relative phase of the reconstructed target state amplitudes, and is the fidelity of the reconstructed state [Eq. (4.4)].

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