Quantum Theory of the Coherently Pumped Micromaser

1
Quantum Theory of the Coherently Pumped
Micromaser
University of West HungaryDepartment of Physics

• István Németh and János Bergou

CEWQO 2008 Belgrade, 30 May 03 June, 2008
2
Introduction and motivation

• The single-atom maser or micromaser consists of
  a stream of two-level Rydberg atoms and a single
  mode of a high-Q cavity. The system has many
  advantages
• The interaction of the maser field and the
  passing atoms is described by the Jaynes-Cummings
  Hamiltonian, the most commonly used interaction
  Hamiltonian of theoretical quantum optics.
• The microwave cavities used in todays
  experiments have long decay times (approx. 0.3
  s), therefore effects occur on a macroscopic time
  scale in which the dynamics can be observed in
  great detail.
• This system's ability to coherently transfer
  quantum states between atoms and photons made it
  relevant in the context of quantum computation as
  well.
• Although considerable work, both theoretical and
  experimental, has been devoted to this system,
  with a few exceptions, most cases involved only
  non-coherent pumping. As a result, the density
  matrix describing the field remained always
  diagonal, preventing the appearance of
  coherences, which are central to quantum
  information processing.

Phys. Rev. A 34, 3077 (1986) Phys. Rev. Lett.
64, 2783 (1990)
3
Model

• The two-level atoms, initially prepared in a
  proper form of the atomic coherence, are
  randomly injected into the micromaser cavity at a
  rate r low enough that at most one atom at a time
  is present inside the cavity and allowed to
  interact with a single mode of
• the maser field for a time period of t ? 1/r.

The n-th atom is injected into the maser cavity
at time tn with the initial density matrix
Here ?aa and ?bb are the populations and ?ab(
?ba) are the maximally allowed coherences for a
given population. Furthermore ? is the frequency
of the classical field used to prepare the atomic
coherence (this frequency is not necessarily the
same as the atomic transition frequency ?ab). The
parameter ? (0? ??1) determines the degree of
the injected coherence. If ?0 no atomic
coherence and if ?1 the maximal atomic coherence
is injected into the micromaser.
Phys. Rev. A 40, 237 (1989)
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The Master Equation

• Various methods were developed to obtain the
  master equation for the density operator of the
  cavity field (Phys. Rev. A 40, 5073 (1989) Phys.
  Rev. A 46, 5913 (1992) Phys. Rev. A 52, 602
  (1995)). For non-resonant pumping and Poissonian
  arrivals they all lead to the same master
  equation. Which in the interaction picture, after
  transforming the explicitly time dependent terms
  away ( ) reads as

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Parameters of the model
The magnitude of the complex atom field coupling
constant, appearing in the Jaynes-Cummings
Hamiltonian.
Time is scaled to the cavity decay time. G is the
cavity-damping constant which arises due to the
coupling of the cavity field to the environment,
modeled by a reservoir in thermal equilibrium.
Gives the number of atoms passing through the
cavity during the cavity decay time 1/ G.
The atomic inversion parameter.
The parameter which determines the degree of the
injected coherence. If ?0 no atomic coherence
and if ?1 the maximal atomic coherence is
injected into the micromaser.
The parameter which determines the interaction
phase of a single atom and the cavity field.
The parameter describes the effective photon
number shift due to the detuning of the empty
cavity frequency and the atomic transition
frequency.
It is the scaled detuning which gives the phase
shift accumulated during the cavity decay time
between the oscillation of the empty cavity field
and the injected signal.
The mean number of thermal photons.
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Trapping states of the coherently
andnon-resonantly pumped lossy micromaser
The steady state formed in a micromaser is the
result of two competing processes, the pumping
and the decay due to the cavity losses. Under
general conditions in the absence of either
process steady state cannot be reached (except of
course the vacuum state). However, if in the
absence of a thermal reservoir (decay process) we
restrict the interaction phase T in such a way
that the coupling between given rows and columns
of the field density matrix cancels (trapping
states), a steady state will be reached
tangent and cotangent states
Interaction with the thermal reservoir introduces
coupling between the entries of the same
diagonal. In particular, when the thermal
reservoir is at zero temperature, the interaction
serves as a decay channel and thus all but the
entries in the partition that includes the
non-decaying vacuum state decay over time.
Therefore, in the presence of the thermal
reservoir, setting ß unambiguously determines the
steady state.
downward and upward trapping states
J. Opt. Soc. Am. B 3, 906 (1986) Opt. Lett. 13,
1078 (1988) Phys. Rev. Lett. 63, 934 (1989)
Phys. Rev. A 41, 3867 (1990)
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Steady state, analytic solution
We assume that T satisfies the trapping state
condition, and that the thermal reservoir is at
zero temperature, thus the steady state of the
field is localized in the first diagonal
partition bounded by ??? and ?nq?. By limiting
our investigation to only trapping state
producing interaction phases (TT) we do not
restrict the generality of the discussion since
for every T and e we can find a TT such that ? T
- TT ?lt e.
Phys. Rev. A 72, 023823 (2005)
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The photon distribution and the purity of the
steady state of the coherently pumped micromaser
Coherent pumping
Coherent pumping
Non-coherent pumping
Non-coherent pumping
9
Phase Properties of the Coherently Pumped
Micromaser
It is quite clear that the strong coherence
present in this system also calls for an
understanding of its features in terms of quantum
phase. Studying the phase properties of quantum
fields is arguably one of the most controversial
subjects in physics. The reason for that is the
lack of a well defined Hermitian phase operator.
The problem is rooted in the fundamental but
unnecessarily restrictive concept that a quantum
observable must be represented by a self-adjoint
operator. Under this condition there is no
spectral measure which is covariant under the
shifts generated by the number observable of a
single-mode field. Relaxing the condition,
considering the quantum phase observable as a
normalized positive operator measure, however,
led to successful studies of various properties
of phase observables by Lahti and
Pellonpää. Therefore the starting point of our
investigation is the probability measure
corresponding to the canonical phase observable
which is defined by the probability density
function
J. Math. Soc. Phys. 40, 4688 (1999) J. Math.
Soc. Phys. 41, 7352 (2000)
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Moments of periodic operators
A solution to the problem
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The phase distribution of the steady state of
the coherently pumped micromaser (classical
features)
Phase locking scheme II. (field leads)
Phase locking scheme I. (atoms lead)
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The phase distribution of the steady state of the
coherently pumped micromaser (non-classical
features)
Bifurcations
Phase transitions of the phase
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Wigner functions
Non-coherent pumping
Coherent pumping
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Comparing the results of the semiclassical model
and the quantum model

• Stable points of the semiclassical model for
  resonant pumping.
• Photon and phase distributions for resonant
  pumping provided by the quantum model.