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Quantum Rabi Oscillation
A Direct Test of Field Quantization in a Cavity
Torben Müller
University of Mainz
From a talk for the seminar on quantum optics by Prof. Immanuel Bloch
(Dated: June 23rd, 2004)
Since Planck’s hypothesis, the quantization of radiation is a universally accepted fact of nature.
However, another simple fact granted in all quantum field descriptions, i.e., the discreteness of the
energy of the radiation stored in a cavity mode, has escaped direct observation up to some years
ago. In 1995, M. Brune et al. (Département de Physique de l’Ecole Normale Supérieure, Paris)
reached to observe the Rabi Oscillation of circular Rydberg atoms in vacuum and in small coherent
fields stored in a high Q cavity. Their measured signal exhibited discrete Fourier components at
frequencies proportional to the square root of successive integers. This provided a direct evidence of
field quantization in the cavity. The weights of the Fourier components yielded the photon number
distribution in the field.
I.
INTRODUCTION
The study of the Jaynes-Cummings Hamiltonian,
which describes the the ideal coupling of a two-level atom
to a single quantized mode of the e.m. field, indicates
that a signature of the discrete nature of the field quanta
could be provided by the observation of a single atoms’s
Rabi nutation in a weak radiation field. So, at first, we
are going to treat the atom-field interaction fully quantum mechanically, then discuss the dynamics of the atomfield model for various states of the field and finally obtain an elementary picture of the Cummings collapse and
revivals due to the quantum granularity of the field. In
the last part of this article we will give a brief survey on
the experiment with circular Rydberg atoms in small coherent fields stored in a high Q cavity done by M. Brune
et al. at ENS in 1995.
II.
atom-field Hamiltonian consisting of three parts: The
two-level unperturbed Hamiltonian Hatom , the freefield Hamiltonian Hf ield and the interaction Hamiltonian
Hint :
Htot = Hatom + Hf ield + Hint
The two-level atom Hamiltonian is just given by
Hatom =
Hint
~
= −e~r · E
(1)
~ is the electric field and e~r is the atomic dipolewhere E
moment operator.
The form of the interaction energy remains the same for
~ becomes the electric field operator
quantized fields, but E
Ê:
1
Hf ield = ~ωL (a† a + )
2
(2)
In order to describe the ideal coupling of a two-level
atom to a single quantized field mode we need the total
(4)
(5)
For the single mode field, the interaction Hamiltonian
becomes
Hint = ~(a + a† )(gσ+ + g ∗ σ− )
(6)
where σ+ and σ− are the Pauli spin-flip matrices and
the electric-dipole matrix element [1]
g = −
℘εΩ
sinKz
2~
(7)
and
µ
εΩ =
Hint = −e~r · Ê
1
~ω0 |eihe|
2
where ω0 is the two-level frequency and |eihe| counts
the number of atoms in the upper state |ei.
The free-field Hamiltonian for a single mode can be described by the creation and annihilation operators a† and
a and the light’s frequency ωL [1]:
INTERACTION BETWEEN ATOMS AND
QUANTIZED FIELDS
Considering the interactions of atoms with optical
fields semiclassically the interaction Hamiltonian between an atom and a classical field is given in the dipole
approximation by
(3)
~Ω
²0 V
¶ 21
(8)
Without loss of generality for two-level systems at rest,
we can choose the atomic quantization axis such that the
matrix element g is real.
2
One of the basic approximations in the theory of twolevel atoms is the rotating wave approximation. We can
understand how this approximation works with quantized fields by considering the various terms in the interaction energy (6):
aσ+ corresponds to the absorption of a photon and the
excitation of the atom from the lower state |gi to
the upper state |ei.
¡
1
1
~ω0 |eihe| + ~ωL (a† a + )
2
2
†
+~g(aσ+ + a σ− )
(9)
The unperturbed Hamiltonian H0 = Hatom + Hf ield
satisfies the following eigenvalue equations:
µ
¶¸
1
1
H0 |e, ni = ~ ω0 + n + ωL |e, ni
2
2
·
µ
¶¸
1
1
H0 |g, ni = ~ − ω0 + n + ωL |g, ni
2
2
Ω =
Htot =
X
Hn
(12)
n
where Hn acts only on the manifold εn and is given in
the {|e, ni, |g, n + 1i} basis by the matrix

√
g n+1
¡
¢
n − 12 −

ω0
2
(13)
(14)
(15)
p
(ω0 + ωL )2 + 4g 2 (n + 1)
(16)
The energy eigenvectors are
|2ni = cosθn |e, ni − sinθn |g, n + 1i
(17)
|1, ni = sinθn |e, ni + cosθn |g, n + 1i
(18)
where the mixing angle θn is given by (14) and (15).
We have seen in the preceding section that the various
manifolds are
√
g n+1
tan(2θn ) =
ωL − ω0
(10)
In this case |g, ni and |e, ni, the unperturbed Hamiltonian’s eigenstates, are so called atom-field states, i.e.
a tensor product of the atom state and the field state.
|g, ni means, that the atom is in the lower state and n
photon are in the single mode. Accordingly, |e, ni: atom
excited and n photons within the mode.
The interaction energy Hint couples the atom-field
states |e, ni and |g, n + 1i for each value of n, but does
not couple other states such as |e, ni and |g, n − 1i
(aσ− would, but is dropped in the RWA). Hence, we
can consider the atom-field interaction for each manifold
εn = {|e, ni, |g, n + 1i} independently and write Htot as
a sum
ω0
2
where we have introduced the quantized generalized
Rabi flopping frequency
·
(11)
−
1
E1n = ~nωL + Ω
2
1
E2n = ~nωL − Ω
2
a† σ− represents the absorption of a photon and the excitation of the atom.
Htot =
¢
In order to get the eigenvalues of the total Hamiltonian, we can diagonalize this matrix and find the new
eigenvalues:
These combinations correspond to those kept in the rotating wave approximation.
But those last two processes are not energy conserving
and so dropped in this approximation.
Consequently dropping the last two combinations, we
obtain the total atom-filed Hamiltonian:
1
2
Hn = ~  √
g n+1
a† σ− conversely describes the emission of a photon and
the de-excitation of the atom.
aσ− correlates to the emission of a photon and the excitation of the atom from the upper state |ei to the
lower state |gi.
n+
(19)
The states given by (17) and (18) are called dressed
states, namely the eigenstates of the Hamiltonian describing the two-level atom interacting with a singlemode field. We refer to the eigenstates of the unperturbed Hamiltonian, i.e., not including the atom-field interaction as bare states.
III.
JAYNES-CUMMINGS MODEL - QUANTUM
RABI OSCILLATION
The dressed-atom picture developed in section II provides us with very useful physical insight into the dynamics of a two-level atom in a single field mode. Since the
dressed states are the eigenstates of the two-level atom
interacting with a single mode of the radiation field, we
can use them to obtain the state vector of the combined
system as a function of time.
Writing the Schrödinger equation in the integral form
|Ψ(t)i = e−iHtot t/~ |Ψ(0)i
(20)
3
we insert the identity operator expressed in terms of
the dressed-atom states |jni to find
|Ψ(t)i =
∞ X
2
X
e−iEjn t/~ |jnihjn|Ψ(0)i
Rabi flopping in a fully quantized way and comparing
the results to the corresponding semiclassical treatment
[2], we find out that the equations of motion for the
atom-field probability amplitudes have the same form.
(21)
n=0 j=1
where Ejn is given by (14) and (15). We have seen in
the preceding section that the various manifolds εn are
uncoupled. In matrix form and in an interaction picture
rotating at the frequency (n + 21 )ωL , the dressed state
amplitude coefficients inside one such manifold εn read


C2n (t)
C1n (t)


 = 
1
e 2 iΩt
0
0


1
e− 2 iΩt
C2n (0)

 (22)
C1n (0)
Example: In particular in resonance, what we have
assumed in the last section and will assume for the following consideration, for a resonant atom initially in the
upper level and n-1 photons within the mode
1
|Ψ(0)i = √ (|2ni + |1ni)
2
(23)
IV.
COLLAPSE AND REVIVAL
√
The quantum Rabi flopping frequency g n + 1 explicitly shows that different photon number states |ni have
different quantum Rabi flopping frequencies.
Up to now, we have only considered field states with
discrete photon numbers, so called Fock states. But usually, the field presents a dispersion of photon numbers. If
it is thermal, the probability P(n) of finding n photons
in a mode is exponential, while for a coherent field
2
|αi = e−|α|
/2
X αn
√ |ni
n!
n
(28)
it is Poissonian
P (n) = |hn|αi|2
the time derivation is given by:
2
´
1 ³
|Ψ(t)i = √ |2nie−iE2n t/~ + |1nie−iE1n t/~ (24)
2
= e−|α|
we obtain the probabilities
¡ √
¢
|Ce,n−1 (t)|2 = cos2 g n + 1t
(26)
¡ √
¢
|Cg,n (t)|2 = sin2 g n + 1t
(27)
This show how the atom Rabi flops between the upper
and the lower levels within a manifold εn , i.e., the atom
interchanges an energy-quantum with the radiation field.
Note: Rabi oscillation even should be observed, if the
photon number n is equal to zero, i.e., Rabi oscillation in
a vacuum field.
We see that both the dressed-atom and the bare-atom
approaches lead to Rabi flopping as they must, and
in general anything you can study with one basis set
you can study with the other. Furthermore treating
X α2n
n
n!
(29)
where the average photon number is given by
Transforming this state given in the dressed-atom basis
into the bare-atom basis
³
´
1
|Ψ(t)i = |g, ni e−iE2n t/~ + e−iE1n t/~
2
³
´
1
+ |e, n − 1i e−iE2n t/~ + e−iE1n t/~ (25)
2
/2
n̄ = |α|2
(30)
The coherent state is characterized by its minimal uncertainty 4φ 4 n ≥ 1, i.e., minimal fluctuations of phase
and of amplitude. Classical fields are usually described
by coherent fields.
In particular, we consider an initially excited atom interacting with a field initially in a coherent state. Combining the coherent state photon number probability (29)
with the single-photon state result (26), we have the
probability for an excited atom regardless of the field
state:
P2 =
X
P (n)|c2n |2
n
= e−|α|
2
/2
X α2n
n
n!
√
cos2 [g nt]
(31)
For sufficiently intense field and short enough times
t ¿ |α|/g, this sum can be shown to reduce to
P2 '
2 2
1 1
+ cos(2|α|gt)e−g t
2 2
(32)
4
Intuitively this result can be understood by noting that
the range of dominant Rabi frequencies in (31) is from
Ω = g[n̄−4n]1/2 to Ω = g[n̄+4n]1/2 and the probability
in (31) dephases in a time tc such that
t−1
= g[n̄ + 4n]1/2 − g[n̄ − 4n]1/2 ' g
c
(33)
which is independent of n̄. Here we have used the
property of a Poisson distribution 4n = n̄1/2 .
Hence, regarding (32), the Rabi oscillations are
2 2
damped with a Gaussian envelope, e−g t , independent
2
of the photon number n̄ = |α| , a result sometimes called
the Cummings collapse. This collapse is due to the interference of Rabi floppings at different frequencies.
For still longer times, the system exhibits a series of
revivals. Because the photon numbers n are discrete in
the quantum sum (31), the oscillations rephase in the
revival time
nature of the field, so that the atomic evolution is determined by the individual field quanta. Eventually, the
revivals, which are never complete and get broader and
broader, overlap and give a quasi-random time evolution.
It is rather surprising that while the coherent state is
the most classical state allowed by the uncertainty principle, it leads to result qualitatively different from the
classical Rabi flopping formula. In contrast, the very
quantum mechanical number state |ni has the nice semiclassical correspondence. The number state and the classical field share the property of a definite intensity which
is needed to avoid the interferences leading to a collapse.
The indeterminacy in the field phase associated with the
number state, but not with the classical field, is not important for Rabi flopping since the atom and the field
maintain a precise relative phase in the absence of decay
processes. In contrast, the coherent state field features
a minimum uncertainty phase, but its minimum uncertainty intensity causes the atom-field relative phase to
diffuse away.
The remarkable fact is that these collapse and revivals
have recently started to be observed experimentally in
cavity QED experiments.
V.
FIG. 1: Cummings collapse and revival for a coherent field
with n̄ = 10 photons [3]
t−1
' 4π/g
r
= 4πn̄[n̄ + 4n]1/2 − g[n̄1/2 tc
(34)
as illustrated in FIG. 1.
This revival property is a much more unambiguous signature of quantum electrodynamics than the collapse:
Any spread in field strengths will dephase Rabi oscillations, but the revivals are entirely due to the grainy
EXPERIMENTS
As we have seen, the study of the Jaynes-Cummings
Hamiltonian indicates that a signature of the discrete nature of field quanta could be provided by the observation
of a single atom’s Rabi nutation in a weak radiation field.
But however, the discreteness of the energy of the radiation stored in a cavity mode has up to now escaped
direct observation. Obviously, a detector more subtle
than an ordinary linear photodetector counting clinks is
required.
But in 1995, M. Brune et al. (Département de
Physique de l’Ecole Normale Supérieure, Paris) reached
to observe the Rabi Oscillation of circular Rydberg atoms
in vacuum and in small coherent fields stored in a high Q
cavity. Their measured signal exhibited discrete Fourier
components at frequencies proportional to the square
root of successive integers. This provided a direct evidence of field quantization in the cavity. The weights
of the Fourier components yielded the photon number
distribution in the field [3].
The setup of this experiment, given in FIG. 2 , is cooled
to 0.8 K. Rubidium atoms , effusing from oven O, are prepared by a time resolved process into the circular Rydberg state e (principle quantum number 51) in the box B.
At a repetition rate of 660 Hz, 2 µs long pulses of Rydberg
atoms start from B with a Maxwellian velocity spread
(mean velocity 350 m/s). The atoms cross the cavity C
made of two niobium superconducting mirrors (diameter
5 cm, radius of curvature 4 cm, mirror separation 2.75
cm). This cavity, whose axis is vertical, sustains the two
TEM900 modes with orthogonal linear polarization and
transverse Gaussian profiles. the lower frequency is tuned
5
FIG. 2: Sketch of the experimental setup [3]
into resonance with the e to g transition between adjacent Rydberg states with principle quantum numbers 51
and 50 (51.099 GHz), see FIG.3. The other mode serves
for stabilizing and fine tuning. The mode Q factor is
7 · 107 , corresponding to a photon lifetime Tcav = 220µs,
which is longer than the atom-cavity interaction time. A
very stable source S is used to inject continuously into
the cavity a small coherent field with a controlled energy
varying from zero to a few photons. The atoms are detected one by one after the cavity by state selective field
ionization (detector D) and the transfer rate from e to
g is measured (Ionization-energy for n=51 is lower than
for n=50) [3].
In circular Rydberg atoms, the valence electron is confined near the classical Bohr orbit. These atoms have a
long radiative lifetime (32 and 30 ms for e and g), which
makes atomic relaxation negligible during the atom transit time across the apparatus, and are strongly coupled
to radiation (very large dipole moment: 1250 u.a.). The
m=49 to m=50 transition represents a two-level system
(in a weak electric field). However, the preparation is
somewhat complex, 53 photons: see FIG. 3
In order to get a time resolved signal the control of
the atom-cavity interaction time t is essential by velocity
selection based on optical pumping: see FIG. 4. The
signal is recorded corresponding to three sequences of
interaction time between 0 and 90 µs. Finally, the three
parts are then combined and checked that they merge
smoothly.
The signals are presented in FIG. 5 [3]. FIGS. 5 (A) to
5(D) show the Rabi nutations for increasing field amplitudes. Figure 5(A) presents the nutation in cavity vacuum (with very small corrections due to thermal field effects). Four oscillations are observed. The signal exhibits
the reversible spontaneous emission and reabsorption of
a single photon in an initially empty cavity mode, an
effect predicted by the Jaynes Cummings model, as we
have seen before.
When a small coherent field is injected [FIGS. 5(B) and
5(C)], the signal is no longer sinusoidal, as it would be
for an atom interacting with a classical field. In FIGS.
5(C) and 5(D), after a first oscillation, a clear collapse
and revival feature is observed.
FIGS. 5(a) to 5(d) show the Fourier transform of the
nutation signal, obtained after symmetrization with respect
√ to√t = 0. Discrete peaks at frequencies ν=47 KHz,
ν 2, ν 3, and even 2ν are clearly observable, revealing directly the quantized nature of the field up to three
photons. The frequency ν is in good agreement with the
expected value. The total area of the Fourier transform
curve remains constant, as required by P(n) normalization.
The time dependent signals are fitted
√ by a sum of
damped sinusoids, with frequencies ν n + 1, n varying
from 0 to 5 [solid lines in FIGS. 5(A) to 5(D)]. The agreement is also very good. From the relative weights of
the terms in these fits, the photon number probabilities
are determined, shown in FIGS. 5(α) to 5(δ). When no
field is injected [FIG.5(α)], this distribution fits the thermal radiation law (solid line) with very small average
photon number n̄=0.06(±0.01), corresponding well with
the value deduced from the cavity temperature. With
an injected coherent field [FIGS. 5(β) to 5(δ)], there is
a very good agreement with the experimental data and
the Poisson law (solid lines), providing an acurate value
of the mean photon number in each case: 0.40(±0.02),
0.85(±0.04) and 1.77(±0.15).
This experiment can also be viewed as a measurement
of the atom-cavity spectrum, deduced from the JaynesCummings Hamiltonian. The excited levels of this system are organized in doublets, separated by one field
quantum. The splittings of doublets corresponding
to
√
√
increasing energies are precisely hn, hν 2 , hν 3 ,... .
The Rabi nutation is thus a quantum beat signal, resulting from the coherent excitation and detection of linear
superpositions of all these levels. The spectral component at frequency n, the only one to be excited if the
field is in the vacuum state, reveals the splitting of the
first manifold, already observed in direct spectroscopic
investigations (vacuum Rabi splitting).
This resonant experiment dramatically shows once
more that circular Rydberg atoms are very sensitive
probes of millimeter wave fields, able to measure not only
the mean field intensity with subphoton sensitivity, but
also to determine accurately its statistics [3].
[1 ]Elements of Quantum Optics, P. Meystre, M. sargent
III (Springer, Third Edition, 1999)
[2 ] Licht-Atom Wechselwirkung im Zwei-Niveau System, I. Bloch (Lecture Notes, University of Mainz,
2003)
[3 ] Quantum Rabi Oscillation: A Direct Test of Field
Quantization in a Cavity, M. Brune et al., Phys.
Rev. Lett., 76, 11 (1996)
[4 ] Cavity Quantum Electrodynamics Group, LKB,
ENS, Paris
6
FIG. 3: Preparation of circular states and circular Rydberg states [4]
FIG. 4: Velocity selection based on optical pumping [4]
7
FIG. 5: (A), (B), (C), and (D): Rabi nutation signal representing Pe,g , for fields with increasing amplitudes. (A) No injected field
and 0.06(±0.01) thermal photon on average; (B), (C), and (D) coherent fields with 0.40(±0.02), 0.85(±0.04) and 1.77(±0.15)
photons on average. The points are experimental [errors bars in (A) only for clarity]; the solid
√ lines
√ correspond to theoretical fits
(see text). (a), (b), (c), (d) Corresponding Fourier transforms. Frequencies ν=47 KHz, ν 2, ν 3, and even 2ν are indicated by
vertical dotted lines. Vertical scales are proportional to 4, 3, 1.5, and 1 from (a) to (d). (α), (β), (γ), (δ) Corresponding photon
number distribution inferred from experimental signals (points). Solid lines show the theoretical thermal (α) or coherent [(β),
(γ), (δ)] distributions which best fit the data. [3]