3. Control de una máquina síncrona funcionando como motor paso a paso
3.6. Resultados experimentales
With these capabilities in hand, we now report measurements of gyz(2)(τ) for the light
transmitted by a cavity containing a single trapped atom. We tune the probe Ey p to
(ωp −ω0)/2π = −34 MHz, near −g0, and acquire photoelectric counting statistics
of the field Ez
t by way of two avalanche photodiodes (D1, D2), as illustrated in Fig.
4.1(c). From the record of these counts, we are able to determine gyz(2)(τ) by way
of the procedures discussed in Ref. [52]. The efficiency for photon escape from the cavity isαe2 = 0.6±0.1. The propagation efficiency fromM2 to detectors (D1, D2) is
αP = 0.41±.03, with then each detector receiving half of the photons. The avalanche
photodiodes (D1, D2) have quantum efficiencies αP = 0.49±0.05.
Data are acquired for each trapped atom by cycling through probing, testing, and
cooling intervals (of durations ∆tprobe = 500µs, ∆ttest= 100 µs and ∆tcool = 1.4 ms,
respectively) using a procedure similar to that of Ref. [11]. All probing/cooling cycles
end after an interval ∆ttot = 0.3 s, at which point a new loading cycle is initiated. The
test beam is polarized along ˆzand resonant with the cavity. We select for the presence of an atom by requiring that Tzz(ωp ' ωC1) . 0.35 for the test beam. We employ
only those data records associated with probing intervals after which the presence of
an atom was detected. The intracavity photon number in mode ly due to the probe
field Ey
p on resonance, in the absence of an atom, is 0.21, and the polarizing beam
splitter at the output of the cavity (P BS in Fig. 4.1(c)) suppresses detection of this resonant light by a factor of ∼ 94. A repumping beam transverse to the cavity axis and resonant with 6S1/2, F = 3 → 6P3/2, F0 = 40 also illuminates the atom during
the probe and test intervals. This beam prevents accumulation of population in the
F = 3 ground state caused by the probe off-resonantly exciting theF = 4→F0 = 40 transition.
Fig. 4.3 presents an example ofgyz(2)(τ) determined from the recorded time-resolved
coincidences at (D1, D2). In Fig. 4.3(a), the manifestly nonclassical character of the
transmitted field is clearly observed with a large reduction in g(2)yz(0) below unity, gyz(2)(0) = (0.13±0.11) < 1, corresponding to the sub-Poissonian character of the
2.0 1.5 1.0 0.5 0.0 gyz (2) ( W ) -1.0 -0.5 0.0 0.5 1.0 W (Ps) 2.0 1.5 1.0 0.5 0.0 gyz (2) ( W ) -10 -5 0 5 10 W (Ps) g ( f ) 800 600 400 200 0 f (kHz) (a) (b) (c) ~ "0 "min
Figure 4.3: Intensity correlation functiongyz(2)(τ) for incident excitation with polariza-
tion along ˆyand detection with orthogonal polarization ˆz. (a)g(2)yz(τ) over the interval |τ| ≤1.0µs demonstrates that the transmitted field exhibits both sub-Poissonian pho- ton statistics g(2)yz(0) = (0.13±0.11) < 1 and photon antibunching gyz(2)(0) < gyz(2)(τ)
[49]. (b) gyz(2)(τ) over longer intervals |τ| ≤ 10 µs displays a pronounced modulation
due to axial motion of the trapped atom. (c) The Fourier transform ˜g(f) of gyz(2)(τ)
with the independently determined minimum and maximum frequencies νmin and ν0
for axial motion in a FORT well indicated by the dotted lines.
transmitted field, and with gyz(2)(0)< g(2)yz(τ) as a manifestation of photon antibunch-
ing. gyz(2)(τ) displayed in Fig. 4.3, shown with a 6 ns resolution, has been corrected for
background counts due to detector dark counts and scattered light from the repump-
recorded counts. We find that g(2)(τ) rises to unity at a time τ =τ
B '45ns, which
is consistent with a simple estimate τ− = 2/(γ+κ) = 48 ns based upon the lifetime
for the state |1,−i.
Although for small |τ| our observations of gyz(2)(τ) are in reasonable accord with
the predictions from our theoretical model, there are significant deviations on longer
time scales. Evident in Fig. 4.3(b) is modulation that is not present in the model which arises from the center-of-mass motion of the trapped atom. In support of this
assertion, Fig. 4.3(c) displays the Fourier transform ˜g(f) of gyz(2)(τ) which exhibits
a narrow peak at frequency f0 ' 535 kHz just below the independently determined
frequency ν0 ' 544 kHz for harmonic motion of a trapped atom about an antinode
of the FORT in the axial directionx. This modulation is analogous to that observed in Ref. [53] for g(2)(τ) for the light from a single ion, which arose from micro-motion of the ion in the RF trap.
Here, U(~r) =U0sin2(2πx/λC2) exp(−2ρ
2/w2
C2) is the FORT potential, which gives
rise to an anharmonic ladder of vibrational states with energies {Em}. There are ' 100 bound states in the axial dimension for radial coordinate ρ ≡ py2+z2 = 0.
The anharmonicity leads to the observed offset f0 < ν0 due to the distribution of
energies for axial motion in the FORT well. Indeed, the frequency νmin = (Emmax −
Emmax−1)/hat the top of the well is approximately half that at the bottom of the well,
ν0 = (E1−E0)/h. By comparing the measured distribution of frequencies exhibited
by ˜g(f) with the calculated axial frequencies {νm}, we estimate that those atoms
from which data was obtained are trapped in the lowest lying axial states m . 10, which corresponds to a maximum energy E/kB ∼ 250 µK. This energy estimate
is consistent with other measurements of g(2)yz(τ) that we have made, as well as the
Fourier transform of the record of the transmitted intensity and the transmission
spectra of Ref. [11].
We have demonstrated photon blockade for the transmission of an optical cavity
strongly coupled to a single trapped atom [38, 39, 40, 41, 45, 46, 8]. The observed nonclassical photon statistics for the transmitted field result from strong nonlinear
blockade for electron transport [36, 37, 54]. Extensions of our work include operation
in a pulsed mode as was analyzed in Ref. [38], thereby realizing a source for single photons “on demand.” As we improve the effectiveness of our cooling procedure, we
should be able to explore the dependence of gyz(2)(τ) on probe detuning, ωp −ω0, as
Chapter 5
Detailed Theory of Photon
Blockade
In Chapter 4, we employ a model of a two-mode cavity coupled to an atom with mul- tiple internal states. In this chapter, we make explicit the coupling used in the model
Hamiltonian to determine the eigenvalues displayed in Fig. 4.1(b). This Hamiltonian was also incorporated into the master equation for the damped, driven system used to
compute the theoretical results in Fig. 4.2(b). We compare the effects of a two-mode cavity to the effect of driving the atom (rather than the cavity) in a Jaynes-Cummings
system. We also present an extension to our atom-cavity model which includes the effect of cavity birefringence and FORT induced ac-Stark shifts in the atomic states.
The modified cavity transmission and intensity correlation function are presented for comparison. Finally, we present a theoretical result for the time dependence of the
intensity correlation function.
5.1
Eigenvalues
This section is adapted from Ref. [55].
We approximate the atom-cavity coupling to be a dipole interaction. We define the atomic dipole transition operators for the 6S1/2, F = 4→6P3/2, F0 = 50 transition
in atomic Cesium as Dq = 4 X mF=−4 |F = 4, mFihF = 4, mF|µq|F0 = 50, mF +qihF0 = 50, mF +q|, (5.1)
where q = {−1,0,1} and µq is the dipole operator for {σ−, π, σ+}-polarization, re-
spectively, normalized such that for the cycling transition hF = 4, mF = 4|µ1|F0 =
50, mF = 5i = 1. The matrix element of the dipole operator hF = 4, mF|µq|F0 =
50, m0Fi is equivalent to the Clebsch-Gordan coefficient for adding spin 1 to spin 4 to reach total spin 5, namely hj1 = 4, j2 = 1;m1 =mF, m2 =q|jtotal= 5;mtotal =m0Fi.
The Hamiltonian of a single atom coupled to a cavity with two degenerate orthog-
onal linear modes is
H4→50 = ~ωA 5 X m0 F=−5 |F0 = 50, m0FihF0 = 50, m0F|+~ωC1(a † a+b†b) (5.2) +~g0(a†D0+D † 0a+b † Dy+Dy†b)
where Dy = √i2(D−1+D+1) is the dipole operator for linear polarization along the
y-axis. We are using coordinates where the cavity supports ˆyand ˆz polarizations and ˆ
x is along the cavity axis. The annihilation operator for the ˆz (ˆy) polarized cavity mode is a (b).
Assuming ωA = ωC1 ≡ ω0, we find that the lowest eigenvalues of H4→50 have a
relatively simple structure. In the manifold of zero excitations, all nine eigenvalues are zero. In manifolds withn excitations, the eigenvalues are of the form En,k =n~ω0+
~g0ε (n)
k , where ε
(n)
k is a numerical factor and k is an index for distinct eigenvalues.
There are 29 states in the n = 1 manifold, but due to degeneracy k has only 13 distinct values, k ∈ {−6, . . .6}; in the n = 2 manifold there are 49 states but k ∈ {−11, . . .11}. The number of states in any manifold can be understood by considering how the excitations can be distributed among the atom and the two cavity modes. For example, in the n = 1 manifold, the atom can be in one of its 9 ground states (mF ∈ {−4, . . .4}) and either cavity modely or mode lz can have one photon (giving
k ε(1)k ηk(1) ε(2)k ηk(2) 0 0 7 0 5 1 0.667 1 0.516 1 2 0.683 2 0.556 2 3 0.730 2 0.662 2 4 0.803 2 0.805 2 5 0.894 2 0.966 3 6 1 2 0.978 2 7 1.014 2 8 1.073 2 9 1.155 2 10 1.265 2 11 1.414 2
Table 5.1: Numerical factorsε(kn) for the eigenvalues of the Hamiltonian H4→50 in Eq.
5.2, together with their degeneracies η(kn).
while both cavity modes are in the vacuum state, yielding a total of 29 states. Table 5.1 lists numerical values for ε(1k,2) as well as their respective degeneracies ηk(1,2). The numerical factors and degeneracies have the symmetries ε(−nk) =−ε(kn) and η−(nk) =ηk(n). The resulting eigenvalues En,k for n={0,1,2} are displayed in Fig. 4.1(b).
Although these eigenvalues are certainly not sufficient for understanding the com- plex dynamics associated with the full master equation, they do provide some insight
into some structural aspects of the atom-cavity system. For example, the eigenvalues ε(1)±6 =±1 correspond to the vacuum-Rabi splitting for the states|1,±ifor a two-state atom coupled to a single cavity mode [cf., Fig. 4.1(a)]. The one-photon detunings for transitions from the n = 1 → n0 = 2 manifold are largest for the eigenstates associated with ε(1)±6. Indeed, just as for the two-state atom with one cavity mode, transitions from the eigenstates at±g0have frequency detunings±(2−
√
2)g0relative
to the nearest states in then0 = 2 manifold (atε(2)±11 =±√2, respectively). Hence, as a function of probe frequency ωp, the eigenvalue structure in Table 5.1 suggests that
the ratio of two-photon to one-photon excitation would exhibit a minimum around ωp =ω0±g0, resulting in reduced valuesg(2)(0)<1, which the full calculation verifies
10
-310
010
310
6g
zz (2)(0),
g
yz (2)(0)
60x10
-340
20
0
T
zz,
T
yz-1.0
0.0
1.0
(
p-
0)/g
0zz
yz
Figure 5.1: Tzz and g (2)zz(0) (dashed), and Tyz and g
(2)
yz(0) (red) versus normalized
probe detuning (see Chapter 4 for definition of variables). We consider an F = 4 → F0 = 50 transition driven by linearly polarized light in a cavity containing two modes of orthogonal polarization that are frequency degenerate. Parameters are (g0, κ, γ)/2π = (50,1,1) MHz. The probe strength is such that the intracavity
photon number on resonance without an atom is 0.05. The blue dotted line indicates g(2)(0) = 1 for Poissonian statistics.
For excitation to the other eigenstates in the n = 1 manifold, such blockade is not evidenced in Fig. 4.2(b). A contributing factor suggested by the structure of eigenvalues in Table 5.1 is interference of one and two-photon excitation processes.
For example, excitation at ωp ' ω0±g0/4 results in two-photon resonance for the
eigenstates associated with ε(2)±1 ' ±0.5, and to photon bunching with g(2)(0) 1 as
confirmed by our full calculation of photon statistics.
Fig. 5.1 provides a global perspective of these various effects. Here, we calculate
in Fig. 4.2(b), but now with coherent coupling g0 much larger than the dissipative
rates (κ, γ) and well beyond what we have achieved in our experiments, g0/κ =
g0/γ = 50. At ωp =ω0 ±g0, g (2)
yz(0) ' 0.002 in evidence of the previously discussed
photon blockade suggested by the eigenvalue structure in Table 5.1. As anticipated, large photon bunching results near ωp ' ω0 ±g0/4 associated with the two-photon
resonance to reach the eigenstates with ε(2)±1 ' ±0.5. Between these two extremes for the eigenvalues with the largest and smallest nonzero magnitudes (ω0 ±g0 ≤ ωp ≤ ω0±g0/4),g
(2)
yz(0) displays a complex structure involving multiple excitation pathways
through states in the n = 1 manifold to reach states in the n0 = 2 manifold. The extremely large peak at ωp =ω0 is discussed in Refs. [50, 46].