Final Exam Problem Set

OPTI 544
Final Exam, 29 April, 2015
Jessen
I
Consider a 4-level laser with "32 >> "10 >> "21 . The lasing mode is resonant with the lasing
!
transition at " = 1.0 #10 $6 m , the lasing medium is unpolarized ( p21 oriented randomly with
respect to the laser field), and the spontaneous decay rate for the lasing transition is
A21 = 1.0 ! 10 6 s "1 . The atom density is N = 1.0 "1016 m #3 , the cavity length is L = 0.1m , the
!
cavity
cross
sectional
area is A = 0.5 "10 #6 m 2 , and the cavity mirrors have transmissions and
!
!
reflectivities T1 = 0.00 , R1 = 1.00 (high reflector) and T2 = 0.01, R2 = 0.99 (output coupler).
There are no cavity losses other than
! through the output coupler. You may
! use results from class
or homework without!re-deriving them.
! averaged damping rate " , the
! threshold
! gain gt , and the threshold inversion
(a) !Find the time
"N t . Give expressions in terms of the laser parameters as well as numerical values. (10%)
(b)
Assume the laser is operated high above threshold, such that !10 >> P, ! " >> ! 21 .
!
Keeping only leading order!terms, find an expression
for the intracavity photon flux " .
Calculate the output power for a pumping rate P = 1.0 "10 8 . For the latter give both an
expression in terms of the laser parameters and a numerical value. (10%)
(c)
Calculate the laser line width in Hz using the Schawlow-Townes formula. (5%)!
!
!
II
Consider a homodyne detection setup as shown on the right,
consisting of a 50/50 beam splitter and a coherent state input
! (2) in port 2. The detector measures the photon number
difference between ports 3 and 4, and its output signal is thus
proportional to the observable Mˆ = aˆ3+ aˆ3 ! aˆ4+ aˆ4 .
(a)
Assume the field in port 1 is a coherent state ! (1) .
Write down the density operators !ˆ1 , !ˆ 2 and !ˆ in that
describe the state of mode 1, mode 2, and the joint 2mode state, respectively. Then use the density operator
formalism to derive an expression for the expectation
value Mˆ (10%)
(b)
Next, generalize the result in (a) to the case where !ˆ1 is a statistical mixture of coherent
states ! j (1) , each occurring with probability P(! j ) . (10%)
(c)
Finally, using the results from (a) and (b), find the expectation value Mˆ for an input state
that is a superposition of two coherent states,
!ˆ1 = ( " 1 (1) + " 2 (1) )( " 1 (1) + " 2 (1) ) # ,
(
and for an input state that is an incoherent mixture of those same states,
!ˆ1 = 12 " 1 (1) " 1 (1) + 12 " 2 (1) " 2 (1) .
Compare and discuss the two results! (10%)
1
)
! = 2 1 + Re !" "1 (1) " 2 (1) #$ ,
III
In this problem we explore a quantum version of the electron oscillator model, i. e. an atom
where the internal motion of the electron relative to the nucleus is described by a onedimensional quantum mechanical harmonic oscillator, instead of the classical harmonic oscillator
used in the Lorentz model. The quantum oscillator has frequency !0 , creation and annihilation
ˆ .
operators bˆ † , bˆ , and position operator xˆ = x0 (bˆ† + b)
(a)
Consider the dipole matrix elements pmn = m pˆ n between the harmonic oscillator states.
What are the selection rules for electric-dipole transitions, i. e. which pmn 's are zero and
which are non-zero? Write out the corresponding matrix for the dipole operator in the
basis of oscillator states n , for n ! 5 . (10%)
We put the quantum electron oscillator inside an optical cavity, where it interacts with a
quantized mode of the electromagnetic field with frequency ! , creation and annihilation
ˆ . For simplicity, we assume the
operators aˆ † , aˆ , and electric field operator Eˆ = E0 (aˆ † + a)
electric field is parallel to the electron oscillator motion.
(b)
Write down the Hamiltonian for the system consisting of the electron oscillator and the
quantized field mode, including the electric dipole interaction between the oscillator and
the field. Compare to the Jaynes-Cummings Hamiltonian for a two-level atom coupled to a
quantized mode of the electromagnetic field. (10%)
(c)
Now assume ! = !0 . Based on what you found in (b), what do you expect will happen if
the system at t = 0 has one quantum of excitation in the electron oscillator, and zero quanta
of excitation in the field? Sketch the populations of the states nosc = 1 n field = 0 and
nosc = 0 n field = 1 as function of time. (10%)
IV
A Cs atom with transition wavelength ! = 852nm and excited state decay rate A21 = 3.28 !10 7 s
is located at an antinode of the field in a resonant optical cavity with length l = 100 µm and
effective cross sectional area A = ! (2 µm)2 . We assume the field polarization is parallel to the
electric dipole matrix element, and that the cavity mirrors are perfect, lossless reflectors.
(a) Calculate the vacuum Rabi frequency and compare to A21 . Is the system in a regime that
permits the observation of coherent vacuum Rabi oscillations? (10%)
(b)
A weak probe beam is passed through the cavity. Sketch the transmission as the probe
frequency ! is tuned around the atom/cavity resonance frequency ! 0 . What phenomenon
are you seeing? (5%)
2