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Congresso del Dipartimento di Fisica
Highlights in Physics 2005
11–14 October 2005, Dipartimento di Fisica, Università di Milano
Generation and decoherence of mesoscopic superposition states in a strongly driven micromaser
F. Casagrande and A. Lulli
INFM
Dipartimento di Fisica, Università di Milano
CNR-INFM – Sezione di Milano
ABSTRACT
A SIMPLE INTRODUCTION
Decoherence, that is the loss of coherence between the states of a quantum system induced by the
environment, is an intriguing subject both for basic investigations on the elusive boundary between the
quantum and the classical worlds, and for applications in quantum information and computation. We
show [1] that the decoherence of mesoscopic superposition states of a cavity field can be observed in a
micromaser where a classical field strongly drives the atoms, that is, in a solvable and experimentally
feasible system [2]. We provide an analytical description in phase space of all stages of atom-pair
correlation measurements at steady-state, where the detection of the first probe atom prepares a
superposition field state, that entangles with the second probe atom. The conditional probabilities for the
detection of the latter atom provide a description of the decoherence of the superposition state, which
occurs in an open system in the presence of pumping, driving, dissipative, and thermal effects. The
decoherence rate is proportional to the squared interaction time, that rules the separation in phase
space between the superposition components, whereas the quantum coherence is unaffected by the
atomic pumping. Hence we further investigate the system when the cavity is not pumped. Starting the
correlation measurements from the vacuum state, the superposition states are maximally separated
Schrödinger cat states, whose decoherence can be thus monitored.
One of the most intriguing problems in modern physics is to explain the emergence of the classical appearance of the macroscopic world from the quantum mechanical laws that rule the behavior
of matter at the microscopic level.
The fundamental features of quantum mechanics, such as linearity and entanglement, imply the occurrence of interference effects and nonlocal correlations, giving rise to seemingly paradoxical
behaviors that we do not see in the world around us. As early as in 1933 Schrödinger pointed out these problems by a caricatural metaphor, in which he described a cat in a closed box containing
a diabolic apparatus correlating his life or death to the excited or unexcited state of an atom. It is then impossible to know if the cat is alive or dead, until the box is opened and only one of the two
possibilities is observed.
The absence of superposition states and hence interference effects in the macroscopic domain originates from the fact that no macroscopic system is isolated; the unavoidable interaction with the
environment induces a decay of quantum coherence, or decoherence, that is so fast to be practically instantaneous. However, the generation and control of systems of mesoscopic size opened
the way to the observation of decoherence in progress. This was done by Haroche and coworkers [3], working in cavity quantum electrodynamics, where systems of atoms and photons can be
controlled with exquisite accuracy. In the experiment they generated superpositions of two coherent cavity field states with the same amplitude but different phases, and were able to monitor their
decoherence for the first time. States of this kind, though not macroscopic, are usually called Schrödinger cats after the famous paradox.
Besides its deep fundamental importance, the study of decoherence plays a central role in quantum information and communication. Here just the most peculiar features of quantum mechanics
allow implementing classically impossible processes, and just decoherence is the main obstacle to the implementation of logic gates and other basic operations in quantum computation.
STRONGLY DRIVEN MICROMASER AND ATOMIC CORRELATIONS
We investigated the Strongly Driven Micromaser [2], in which a beam of two-level atoms crosses
a superconductive microwave cavity where the atomic transition is resonant with one cavity mode.
Like in the micromaser, atoms cross the cavity one at a time. Furthermore, during the passage the
atoms are strongly driven by classical field.
SCHEMATIC SYSTEM SET-UP
GENERATION AND DECOHERENCE OF MESOSCOPIC
SUPERPOSITION STATES
We studied [1] the dynamics of the cavity field states prepared by the detection of outgoing
atoms in the upper or lower state. We described in phase space all stages of atom pair
correlation measurements (see below), in which the detection of the first atom generates a
mesoscopic superposition state of the cavity field, that later entangles with a second atom. The
decoherence of the superposition is described by conditional atomic probabilities. It depends on
the squared interaction time, that is the parameter ruling the separation in phase space between
the components of the states, whereas it is independent of the atomic pumping rate.
DECOHERENCE OF SCHRÖDINGER CAT STATES
We considered the case that the atomic correlation measurements occur in the absence of atomic
pumping, starting from the cavity in a thermal state. This allowed us to isolate and describe the
effects of temperature on decoherence.
We further investigated the system in the limit of negligible temperature, in which the correlation
measurements start from the vacuum state of the cavity field. In this case the system dynamics
depends only on the dimensionless interaction time and can be solved even without phase space
techniques. Furthermore, the mesoscopic superpositions are exactly Schrödinger cats (see
below). Hence the decoherence of cat states can be monitored by an alternative method that can
offer same advantages over the one in the Haroche experiment [3].
Atomic beam oven
QUANTUM CHARACTERISTIC FUNCTION OF THE STEADY-STATE CAVITY FIELD
QUANTUM CHARACTERISTIC FUNCTION OF A CAT STATE
Microwave Cavity
Laser excitation
Field
ionization
e
 
g
External driving field
4
ATOM PAIR CORRELATION FUNCTIONS
The dynamics of this quantum open system was analytically solved in spite of the presence of
driving, pumping, and dissipative effects. One of the main results is the prediction of remarkable
atomic correlations when the system operates at steady state, showing bunching and antibunching effects depending on the atomic states (see below). Another relevant feature is that these
correlations decays much faster than the dissipative decay of the cavity field.
1
  0.05
2
  0.05
e
g
4
Parameters:
1
2
Atomic pumping rate
Dimensionless
interaction time
Mean thermal
photon number
N ex  10
 
n  0.03
e
g
DECOHERENCE RATE
DECOHERENCE SIGNAL
Dissipative rate
    Pe / e     Pe / g    

1
 
1  2 2 
cosh 8 n    e   1
2



  0
 
1  2
cosh 8 n      1
2 
 
2
  
 0 
e
g
 D  2 
2


2
gtint 
2
Coupling frequency
Interaction time
NEXT INVESTIGATIONS
We shall investigate the entanglement of both the driven atom-cavity
field system, at the basis of the decoherence studies, and the atomatom system, to elucidate the nature of atomic correlations.
Dissipative decay
REFERENCES
Parameters:
| |
Dimensionless interaction time

Dimensionless correlation time
[1] F. Casagrande and A. Lulli, Eur. Phys. J. D 36, 123 (2005);
J. Opt. B: Quantum Semiclass. Opt. (2005), in press.
[2] F. Casagrande et al, Phys. Rev. A 69, 023812 (2004);
Opt. Spectrosc. 99, 301 (2005).
[3] M. Brune et al., Phys. Rev. Lett. 77, 4887 (1996).