Download Quantum Theory of the Coherently Pumped Micromaser

University of West Hungary
Department of Physics
Quantum Theory of the
Coherently Pumped
Micromaser
István Németh and János Bergou
CEWQO 2008
Belgrade, 30 May – 03 June, 2008
University of West Hungary
Department of Physics
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 today’s 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 noncoherent 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)
Model
University of West Hungary
Department of Physics
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 τ  1/r.
The n-th atom is injected into the maser cavity at time tn with the initial density matrix:
 aa
(n)
atom (tn )  
i tn

e
 ba
ab ei t
bb
Here ρaa and ρbb are the populations and ρab=( ρba)* are the maximally allowed
coherences for a given population. Furthermore ν 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 ωab). The parameter λ (0 λ1) determines
the degree of the injected coherence. If λ=0 no atomic coherence and if λ=1 the
maximal atomic coherence is injected into the micromaser.
*Phys. Rev. A 40, 237 (1989)
n

.

The Master Equation
University of West Hungary
Department of Physics
)
d k ,l (tmethods
Various
were developed to obtain the master equation for the density
 Ak ,l ,1,1 k 1,l 1 (t )  Ak ,l ,1,0 k 1,l (t )  Ak ,l ,0,1 k ,l 1 (t )  Ak(1),l ,0,0 k ,l (t )
operator
dt  of the cavity field (Phys. Rev. A 40, 5073 (1989); Phys. Rev. A 46, 5913 (1992); Phys.
Rev. A 52, 602 (1995)(2)
). For non-resonant pumping and Poissonian arrivals they all
 Ak ,l ,0,0 k ,l (t )  Ak ,l ,0,1 k ,l 1 (t )  Ak ,l ,1,0 k 1,l (t )  Ak ,l ,1,1 k 1,l 1 (t ) .
lead to the same master equation. Which in the interaction picture, after
transforming
the explicitly time dependent terms away ( eia a( t    /2)) reads as
N
(
)
(
)
†
Ak ,l ,1,1 
Ak ,l ,1,0 
Ak ,l ,0,1 
Ak(1),l ,0,0 
Ak(2),l ,0,0 
Ak ,l ,0,1 
Ak ,l ,1,0 
Ak ,l ,1,1 
ex
(1  u ) Sk Sl  nth kl ,
g
Ck  cos( n02  k )  i
n0
2
n02  k
N ex
 1  u 2 Sk Cl ,
k
2
Sk 
sin( n02  k ).
N ex
n02  k
2 
 1  u Ck S l ,
2
N ex
k l
(1  u )(1  CkCl )  i(k  l )  (nth  1)
,
2
2
N ex
(k  1)  (l  1)
(1  u )(1  Ck 1Cl1 )  nth
,
2
2
N ex
 1  u 2 Ck 1Sl 1 ,
2
N ex
 1  u 2 Sk 1Cl1 ,
2
N ex
(1  u ) Sk 1Sl 1  (nth  1) (k  1)(l  1),
2
ab
sin( n02  k ),
University of West Hungary
Department of Physics
Parameters of the model
t   t
(g e
 i g
i
a†  σ -  g e g a  σ + )
r
N ex 

u   aa  bb

 g 
n0 

nth
0  ab
2g
0  

Time
isthe
scaled
to
cavity
decay
time.
Gives
The
It
is the
magnitude
atomic
parameter
mean
number
number
inversion
detuning
which
describes
ofthe
the
of
ofatoms
parameter.
thermal
determines
complex
which
the
passing
effective
photons.
gives
atom
the
through
field
theΓ
the
degree
cavity
ofconstant,
the
during
injected
the
cavity
coherence.
time
Ifand
λ=01/
no
coupling
is
interaction
photon
phase
the
cavity-damping
shift
number
phase
accumulated
shift
ofappearing
adue
constant
single
during
to decay
the
atom
inwhich
detuning
the
Jaynescavity
the
ofΓ.
atomic
coherence
and
λ=1
thethe
maximal
Cummings
arises
cavity
the
decay
empty
field.
due
time
cavity
to
Hamiltonian.
between
the frequency
coupling
theif oscillation
ofand
cavity
of
atomic
the
field
β

(
N
,
u
,

,

,
n
,

,
n
th )
to the environment,
atomic
transition
empty
cavity
coherence
frequency.
fieldexand
ismodeled
injected
the injected
into
by0 athe
reservoir
signal.
in thermal equilibrium.
micromaser.
University of West Hungary
Department of Physics
Trapping states of the coherently and
non-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 Θ 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

q
n  (nq  1)
2
0
Snq 1  0,
,
q  1, 2,3,

q  1, 2,3,
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.
downward and upward
trapping states
Therefore, in the presence of the thermal
reservoir, setting β unambiguously determines
the steady state.
*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)
Steady state, analytic solution*
University of West Hungary
Department of Physics
We assume that Θ 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 (ΘT) we do not
restrict the generality of the discussion since for every Θ and ε we can find a ΘT such that  Θ - ΘT < ε.
Let us assume, that we know all k ,0  0,k (k  0,1,2, , nq ) entries
(the boundary values). If so, by solving simple linear equations,
dt k ,0  0 (k  0,1,2, , nq  1) starting with k=0 we can express the
k 1,1 (k  0,1,2, , nq  1) entries in terms of the boundary values. In
addition, we also obtain a condition from dt  nq ,0  0 which must be
satisfied by the boundary values.

d k ,l (β, t )
0
dt 
k , l  0,1,2, , nq
After successive repetition of this tedious but simple procedure for the
first nq columns, we determined all entries of the density matrix in terms
of the boundary values, and obtained nq conditions for the nq+1
boundary values. The last condition needed to determine the boundary
values uniquely, and thus the
whole density matrix, is given by the
nq
normalization condition  k 0 k ,k  1 .
By solving the nq+1 conditions we determine all nq+1 boundary values
which, in turn, we use to compute all  k ,l (β) entries of the steady state
field density matrix of the coherently pumped micromaser.
Notes:
This method is the extension of the one used in the case of the noncoherently pumped micromaser: here we chose all boundary values,
not only ρ0,0, and generate the conditions to determine them all.
In the case of resonant pumping, when ab    0 , in steady
state all entries of the field density matrix are real.
*Phys. Rev. A 72, 023823 (2005)
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Department of Physics
The photon distribution and the purity of the
steady state of the coherently pumped micromaser
Coherent pumping
Non-coherent pumping
Coherent pumping
S / Smax 
k B Tr   ln  
k B ln  nq  1
,
Non-coherent pumping
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Department of Physics
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:

   k ,k  N (t )
1 

P ( , t ) 
 N (t )e iN  , where  N (t )   k  N

2 N 
 
k , k  N (t )

k 0
for N  0,1, 2,
for N  1, 2,
.
We consider a relative phase measurement performed on the micromaser field. We assume that an
injected signal with known phase, Φ0(t), provides a reference phase. Using this we write
†
†
 (t )  eia a0 (t )  (t )eia a0 (t ) , and calculate the phase distribution function corresponding to  (t )
P(0  , t )  P(, t ).
*J. Math. Soc. Phys. 40, 4688 (1999); J. Math. Soc. Phys. 41, 7352 (2000)
Moments of periodic operators
A solution to the problem
 circular
arg(C )
if C  0

Indeterminate if C  0,
where
2
C   ei P(, t )d ,
0
and for the spread of the phase
D 
2
2
0
 circular
i
2
e  C P ( , t ) d   1  C ,
D

arctan

C


 2
2
if C  0
if C  0.
Calculating    P(, t )d  but this fails to
provide a consistent result; Φ depends on the
choice of the 2π interval used to evaluate the
integral. Φ
University of West Hungary
Department of Physics
University of West Hungary
Department of Physics
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)
 t  ab   R 
(t )  0 (t ) 
0R 



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Department of Physics
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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Department of Physics
Wigner functions
Non-coherent pumping
Coherent pumping
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Department of Physics
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.