ls3_unit_7

ls3_unit_7 - THE INTERACTION OF RADIATION AND MATTER:...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 73 R. Victor Jones, May 2, 2000 73 VII. QUANTUM MECHANICAL LANGEVIN EQUATIONS 38 A R UDIMENTARY R ESEVOIR P ROBLEM: 39 40 Consider a complete system which consists of a single simple harmonic oscillator (the system component of "interest") coupled to a reservoir of many simple harmonic oscillators (the system component treated statistically) In particular, the complete interacting system has a simple Hamiltonian of the form H = H laser + H res + H int = h Ω a a + h ω j b j b j j + h g j a b j + g j b j a { } j [ VII-1 ] where a is the system variable of interest (the "state of the system") and the b j 's are the other system variables (the "reservoir states"). System operators commute with reservoir operators at given time, we may easily generate the following set of equations of motion: 38 Much of what follows draws heavily on material in the Arizona Books -- i.e. 1. M. Sargent III, M. O. Scully and W. E. Lamb, Jr., Laser Physics , Addison-Wesley (1974) 2. P. Meystre and M. Sargent III, Elements of Quantum Optics , Springer-Verlag (1992) 3. Weng W. Chow, Stephan W. Koch and Murray Sargent III, Semiconductor-Laser Physics , Springer-Verlag (1994) 39 This model is particular valuable in treating electromagnetic intermode coupling problems, but it also provides guidence in treating more general problems which incorporate atomic states. 40
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 74 R. Victor Jones, May 2, 2000 74 ˙ a t ( ) = i H , a t ( ) [ ] h = i Ω a t ( ) i g j b j t ( ) j [ VII-2a ] ˙ b j t ( ) = i H , b j t ( ) [ ] h = i ω j b j t ( ) ig j a t ( ) [ VII-2b ] Formally integrating Equation [ VII-2b ], we obtain 41 b j t ( ) = b j t 0 ( ) exp i ω j t t 0 ( ) [ ] i g j d ʹ t a ʹ t ( ) exp i ω j t ʹ t ( ) [ ] t 0 t [ VII-3 ] The first term in this equation describes the free evolution of the reservoir states in the absence of any interaction with the system and the second gives the modification of this free evolution as a consequence of the coupling to the system. Inserting Equation [ VII-3 ] into Equation [ VII-2a ] we obtain ˙ a t ( ) = i Ω a t ( ) g j 2 d ʹ t a ʹ t ( ) exp i ω j t − ʹ t ( ) [ ] t 0 t j i g j b j t 0 ( ) exp i ω j t t 0 ( ) [ ] j [ VII-4 ] Here the second summation describes a source of fluctuations arising from the free evolution of the reservoir states and the first is a feedback through the reservoir which might be described as a radiation reaction. To remove the rapid variation in the system variable we transform to the Heisenberg interaction picture -- i.e. we take 42 a t ( ) = A t ( ) exp i Ω t [ ] [ VII-5 ] 41 As is pointed out in Section 14-3 of P. Meystre and M. Sargent III, Elements of Quantum Optics , Springer- Verlag (1992), the statement that "System operators commute with reservoir operators at given time…" is a
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 16

ls3_unit_7 - THE INTERACTION OF RADIATION AND MATTER:...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online