Defining a pulse of particular area

[TheCanadian]

I am reading the book "Super-radiance Multiatomic Coherent Emission" by Benedict et al. and on pg. 32, they discuss the initial conditions for a particular case covered by Burnham and Chiao (1969). It mentions that the system was "excited to a state with angle $\theta_0$ by a short coherent pulse of area $\pi - \theta_0$."

I have looked elsewhere in the text, but have not come across any particular definition/expression for a coherent pulse of a specified area. Would you happen to know what is meant by a "coherent pulse" of specified area and what the analytic form for such a pulse would be? Also, when discussing an incident pulse of a particular area, is this considered to be a function of time or a function of space (i.e. area defined as spatial width or integration over time domain)?

 

[f95toli]

They are referring to the Rabi angle. This terminology is commonly used for quantum two-level systems and refers the fact that the state of such a system can be represented using a so-called Bloch sphere, the 'angle' in this case then refers to the angle of the arrow; a pi-pulse will drive the arrow from pointing to the "north" on the sphere to pointing towards the south pole (which corresponds to the system going from being in the ground state to the excited state)

The area under the pulse is given by Amplitude x time and basically tells you how far the arrow moves.

You should be able to find more information by e.g. looking a the wiki for Rabi oscillations.

 

[TheCanadian]

They are referring to the Rabi angle. This terminology is commonly used for quantum two-level systems and refers the fact that the state of such a system can be represented using a so-called Bloch sphere, the 'angle' in this case then refers to the angle of the arrow; a pi-pulse will drive the arrow from pointing to the "north" on the sphere to pointing towards the south pole (which corresponds to the system going from being in the ground state to the excited state)

The area under the pulse is given by Amplitude x time and basically tells you how far the arrow moves.

You should be able to find more information by e.g. looking a the wiki for Rabi oscillations.

Thank you for the clarification. I've checked this page and it mentions the definition of a $\pi$ pulse as one that acts over a time interval of $t = \frac {\pi}{\omega_1}$, and if the light is resonant with the transition, then $\omega_1 = \frac {\vec{d}\cdot \vec{E_0}}{\hbar}$. But for the pulse itself, is their a particular functional form it takes (e.g. hyperbolic secant, gaussian)? And for the electric field and pulse, as discussed here, is the amplitude defined and/or is it time-varying?

 

[f95toli]

You normally think of the pulses as being rectangular. This means that your waveform would be a sine-wave with a frequency resonant with the transition that is first suddenly turned on and then after some time t turned off. This is indeed how many experiments in e.g. ESR and NMR are implemented (i.e. you have an RF source and then a fast on/off switch).

However, there is a problem with rectangular pulses. A pulse that is rectangular in time will have a complicated frequency spectra (with a fundamental 1/rise time and then harmonics) meaning an quick on/off pulse will contain other frequency components besides that of the sine. Whether or not this is a problem really depend on the systems you are working with (in the case of systems with optical transitions it would probably not matter).

Since it is the area of the pulse that matters the "best" pulse is -in theory- a pulse with a Gaussian shape (because the Fourier transform of a Gaussian is another Gaussian), i.e. no other frequency components are introduced and these are also used. However, this is not always practical (modulating a wave to give it a Gaussian shape is much harder than a simple on/off).

In e.g. NMR people use some very complicated sequences of pulses to manipulate their systems. There are whole libraries of such sequences developed for different purposes.

 

[TheCanadian]

You normally think of the pulses as being rectangular. This means that your waveform would be a sine-wave with a frequency resonant with the transition that is first suddenly turned on and then after some time t turned off. This is indeed how many experiments in e.g. ESR and NMR are implemented (i.e. you have an RF source and then a fast on/off switch).

However, there is a problem with rectangular pulses. A pulse that is rectangular in time will have a complicated frequency spectra (with a fundamental 1/rise time and then harmonics) meaning an quick on/off pulse will contain other frequency components besides that of the sine. Whether or not this is a problem really depend on the systems you are working with (in the case of systems with optical transitions it would probably not matter).

Since it is the area of the pulse that matters the "best" pulse is -in theory- a pulse with a Gaussian shape (because the Fourier transform of a Gaussian is another Gaussian), i.e. no other frequency components are introduced and these are also used. However, this is not always practical (modulating a wave to give it a Gaussian shape is much harder than a simple on/off).

In e.g. NMR people use some very complicated sequences of pulses to manipulate their systems. There are whole libraries of such sequences developed for different purposes.

Thank you, that was very helpful. With regards to the amplitudes of the pulses and electric field, is there any particular value used or normalization applied? You mentioned above the area is (amplitude $\times$ time), but for a sinusoidal pulse, how exactly would that be implemented correctly?