Download A Potassium Atom Four-Level Active Optical Clock Scheme

CHIN. PHYS. LETT. Vol. 30, No. 4 (2013) 040601
A Potassium Atom Four-Level Active Optical Clock Scheme
*
ZHANG Sheng-Nan(张盛楠), WANG Yan-Fei(王彦飞), ZHANG Tong-Gang(张同刚),
ZHUANG Wei(庄伟)** , CHEN Jing-Biao(陈景标)
Institute of Quantum Electronics and State Key Laboratory of Advanced Optical Communication and System
Network, School of Electronics Engineering and Computer Science, Peking University, Beijing 100871
(Received 18 December 2012)
We present an active optical clock scheme with a four-level quantum potassium system. We calculate the population probabilities of each state using the density matrix. At the steady state, 𝜌33 and 𝜌55 are equal to 8.3% and
3.5%, respectively, and the population inversion between the 5𝑆1/2 and 4𝑃3/2 states is built up in the thermal
potassium cell with a 404.7 nm pumping laser. According to the mechanism of the active optical clock, under
the action of the 404.7 nm pumping laser, the scheme can output a 1252.2 nm quantum-limited-linewidth laser,
which can be directly used as an active optical frequency standard.
PACS: 06.30.Ft, 32.30.−r, 42.62.−b
DOI: 10.1088/0256-307X/30/4/040601
Optical clocks have much better stability and
accuracy than microwave clocks. In recent years,
great progress has been made in the study of optical
clocks.[1−13] However, current optical clocks are passive clocks,[1−12] and their stabilities are limited by the
linewidth of the probe laser.[1−15] The thermal noise of
the cavity mode hinders the realization of the available
narrow-linewidth laser.[13−17] In order to solve this
problem, an active optical clock is proposed.[16−21]
Active optical clocks, which are novel optical clocks,
have attracted more attention recently for their special advantages compared with conventional atomic
clocks.[15,17−35] The most interesting point is that the
frequency of an active optical clock is insensitive to
the thermal noise of the cavity mode.[15,17−21] The
quantum-limited-linewidth of active optical clocks can
reach the mHz-level, and it is possible to achieve an
unprecedented-linewidth laser.
For the characteristic advantages of active clocks
compared to passive clocks, many active clock schemes
have been proposed,[16−35] and many neutral atoms
with two, three or four levels at the thermal beam,
laser cooling and trapping configurations have been
investigated recently.[16−35] An active optical clock
based on an atomic beam quantum system has been
studied frequently in the past few years. The stability
and accuracy of a two-level quantum system are limited by the second-order Doppler shift of the thermal
atomic beam. As for a three-level quantum system,
the light shift induced by the pumping laser affects
the stability and accuracy of the optical clock. Fortunately, the four-level quantum system can overcome
these problems. Thus we choose a potassium atom
four-level quantum system for the realization of an
active optical clock, as discussed in the following.
We choose 4𝑆1/2 , 5𝑃1/2 , 5𝑆1/2 and 4𝑃3/2 states of
the K atom as a four-level quantum system. The K
atom four-level quantum system scheme is shown in
Fig. 1. With a 404.7 nm pumping laser operating at
a 4𝑆1/2 to 5𝑃1/2 transition, the atoms are pumped to
5𝑃1/2 and decayed to 4𝑆1/2 , 5𝑆1/2 and 3𝐷3/2 . The
atoms at 5𝑆1/2 and 3𝐷3/2 are decayed to 4𝑃1/2 and
4𝑃3/2 , and finally decayed to the ground state 4𝑆1/2 .
The lifetime of the 5𝑆1/2 state is longer than that of
the 4𝑃3/2 state, so the population inversion is built up
between the 5𝑆1/2 and 4𝑃3/2 states.
In addition, the population inversions are built
up between 5𝑃1/2 –5𝑆1/2 , 5𝑃1/2 –3𝐷3/2 , 5𝑆1/2 –4𝑃1/2 ,
3𝐷3/2 –4𝑃1/2 and 3𝐷3/2 –4𝑃3/2 , respectively. The population inversions between 5𝑃1/2 –5𝑆1/2 and 5𝑃1/2 –
3𝐷3/2 can be used as a gas laser.[36] However, they
cannot be used as the optical frequency standard because of the strong light shift caused by the pumping
laser, which directly interacts with the 5𝑃1/2 state.
We choose population inversion between the 5𝑆1/2 and
4𝑃3/2 states as the object of our study because the
split of the 4𝑃1/2 state is wider than that of the 4𝑃3/2
state in view of hyperfine levels. In order to avoid
mode competition, the population inversion between
the 5𝑆1/2 and 4𝑃1/2 states is a good choice.
According to the mechanism of the active optical
clock,[16−35] the cavity loss rate Γ 𝑐 is much larger than
the frequency gain bandwidth of the laser medium Γ𝑔 ,
namely, the active optical clock is a laser working in
the bad-cavity regime where 𝑎 = Γ𝑐 /Γ𝑔 ≫ 1. We use
a 404.7 nm laser as the pumping laser, and the Rabi
frequency is Ω = 𝑑𝐸/~, where 𝑑 is the electric dipole
matrix between the 4𝑆1/2 and 5𝑃1/2 states. 𝐸 is the
electric field strength of the pumping laser. The light
intensity of the pumping laser is
𝐼=
2𝜋ℎ𝑐Ω 2
.
3𝜆321 Γ21
(1)
The saturation light intensity is
* Supported
𝐼s =
𝜋ℎ𝑐Γ21
2
= 0.6 mW/cm .
3𝜆321
by the National Natural Science Foundation of China under Grant Nos 10874009 and 11074011.
author. Email: [email protected]
© 2013 Chinese Physical Society and IOP Publishing Ltd
** Corresponding
040601-1
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CHIN. PHYS. LETT. Vol. 30, No. 4 (2013) 040601
In this study, we mainly use the density matrix equations for the theoretical calculation of the K atom fourlevel quantum system. The equations are written as
𝑑𝜌11
𝑑𝑡
𝑑𝜌22
𝑑𝑡
𝑑𝜌33
𝑑𝑡
𝑑𝜌44
𝑑𝑡
𝑑𝜌55
𝑑𝑡
𝑑𝜌66
𝑑𝑡
𝑑𝜌′12
𝑑𝑡
𝑑𝜌′′12
𝑑𝑡
= − Ω 𝜌′12 + Γ21 𝜌22 + Γ51 𝜌55 + Γ61 𝜌66 ,
= Ω 𝜌′12 − (Γ21 + Γ23 + Γ24 )𝜌22 ,
= Γ23 𝜌22 − (Γ35 + Γ36 )𝜌33 ,
the classical laser theory, the system will finally output a 1252.2 nm laser after the stimulated radiation exceeds threshold and meets the condition of
self-oscillation. The atom-cavity
coupling constant
√︀
𝑔 [37,38] is given as 𝑔 = (𝜇/~) ~𝜔35 /2𝜖0 𝑉mode , where
3
𝜔35 = 2𝜋𝑐/𝜆35 and 𝜇2 = 3𝜋~𝑐3 𝜖0 Γ35 /𝜔35
. Thus
2
2
𝑔 = 3𝑐Γ35 𝜆35√
/(8𝜋𝑉mode ). The laser emission coefficient is sin2 ( 𝑛 + 1𝑔𝜏𝑖 ),[37,38] where 𝜏𝑖 = 1/(Γ3 +
Γ5 ) =17.1 ns. The cycle time 𝜏𝑐 of the four-level quantum system is given as
= Γ24 𝜌22 − (Γ45 + Γ46 )𝜌44 ,
𝜏𝑐 =
= Γ35 𝜌33 + Γ45 𝜌44 − Γ51 𝜌55 ,
𝑑𝜌′11
= − Ω 𝜌′12 + Γ21 𝜌′22 + Γ51 𝜌′55 + Γ61 𝜌′66 ,
𝑑𝑡
𝑑𝜌′22
= Ω 𝜌′12 − (Γ21 + Γ23 + Γ24 )𝜌′22 ,
𝑑𝑡
𝑑𝜌′33
= Γ23 𝜌′22 − (Γ35 + Γ36 )𝜌′33
𝑑𝑡
√
𝜌′ − 𝜌′55
− 33
sin2 ( 𝑛 + 1𝑔𝜏𝑖 )
𝜏𝑐
′
𝑑𝜌44
= Γ24 𝜌′22 − (Γ45 + Γ46 )𝜌′44 ,
𝑑𝑡
𝑑𝜌′55
= Γ35 𝜌′33 + Γ45 𝜌′44 − Γ51 𝜌′55
𝑑𝑡
√
𝜌′ − 𝜌′55
sin2 ( 𝑛 + 1𝑔𝜏𝑖 )
+ 33
𝜏𝑐
𝑑𝜌′66
= Γ36 𝜌′33 + Γ46 𝜌′44 − Γ61 𝜌′66 ,
𝑑𝑡
1
1
𝑑𝜌′12
= Ω (𝜌′11 − 𝜌′22 ) + 𝜌′′12 Δ − Γ21 𝜌′12 ,
𝑑𝑡
2
2
1
𝑑𝜌′′12
= − 𝜌′12 Δ − Γ21 𝜌′′12 ,
𝑑𝑡
2
√
𝑑𝑛 𝜌′33 − 𝜌′55
=
(5)
sin2 ( 𝑛 + 1𝑔𝜏𝑖 ) − Γ𝑐 𝑛,
𝑑𝑡
𝜏𝑐
(3)
where the diagonal element 𝜌𝑖𝑖 represents the population probability at |𝑖⟩, 𝜌′12 and 𝜌′′12 are the imaginary
and real parts of the nondiagonal element 𝜌12 , respectively, and Δ = 𝜔21 − 𝜔 is the frequency detuning of
the pumping laser on a transition frequency. It is set
to be 0 for simplicity.
|2> 5P1/2
Γ23=4.40T106 s-1
λ23=2720.4 nm
|3> 5S1/2
Γ24=1.74T106 s-1
λ24=3159.2 nm
|4> 3P3/2
Γ45=4.09T106 s-1
λ45=1177.0 nm
Γ36=7.15T106 s-1
Γ46=2.09T107 s-1
λ46=1169.0 nm
λ36=1243.2 nm
Γ35=1.42T107 s-1
λ35=1252.2 nm
Γ21=1.98T106 s-1
λ21=404.7 nm
|1> 4S1/2
(4)
According to the semiclassical approximation
theory,[37,38] the equation for the emitted photons and
density matrix under small gain conditions are listed
as follows:
= Γ36 𝜌33 + Γ46 𝜌44 − Γ61 𝜌66 ,
1
1
= Ω (𝜌11 − 𝜌22 ) + 𝜌′′12 Δ − Γ21 𝜌′12 ,
2
2
1
= − 𝜌′12 Δ − Γ21 𝜌′′12 ,
2
1
1
1
1
+
+
+ .
Γ2
Γ3
Γ5
Ω
|5> 3P3/2
|6> 3P3/2
Γ61=3.69T107 s-1
λ61=769.9 nm
Γ51=3.72T107 s-1
λ51=766.5 nm
Fig. 1. The scheme of a K atom four-level quantum system. The population inversion is built up between the
5𝑆1/2 and 4𝑃3/2 states with a 404.7 nm pumping laser.
Γ𝑖𝑗 represents the spontaneous decay rate from |𝑖⟩ to |𝑗⟩,
and 𝜆𝑖𝑗 represents the transition wavelength between |𝑖⟩
and |𝑗⟩.
By solving these equations, we obtain the numerical solutions shown in Fig. 2. We can see that the
population probabilities at each state become steady
after 2 µs, and they are shown in Table 1. At a steady
state, 𝜌33 and 𝜌55 are equal to 8.3% and 3.5%. Therefore, the population inversion is built up between 𝜌33
and 𝜌55 .
Table 1. The population probabilities at steady states.
𝜌11 𝜌22 𝜌33
𝜌44
𝜌55
𝜌66
42% 40% 8.3% 2.7% 3.5% 3.2%
Once the population inversion is built up, the
stimulated radiation will occur by adding an optical resonant cavity to the system. According to
where the diagonal element 𝜌′𝑖𝑖 represents the number of atoms at the corresponding state, which is different from 𝜌𝑖𝑖 , 𝑛 is the number of emitted photons,
and Γ𝑐 is the cavity loss rate, Γ𝑐 = 𝑎Γ𝑔 . The frequency gain bandwidth of laser medium Γ𝑔 is given as
Γ𝑔 = Γ𝑠𝑝 = 14.2 MHz. In this study, we set 𝑎 = 100.
The total atom number 𝑁 of the K atom in the
cavity mode is different in different temperature and
mode volume conditions. We obtain that the cell
pressure[39] of the K atom is log 𝑃 = 9.967 − 4646𝑇 −1 ,
when 298 < 𝑇 < 337 K, and is log 𝑃 = 9.408 −
4453𝑇 −1 , when 337 < 𝑇 < 600 K. The state equation
of the ideal gas is 𝑃 𝑉 = 𝑁𝑣 𝑘B 𝑇 . Here 𝑁𝑣 is the atom
number in the volume 𝑉 . We find that the atom number density 𝑛𝑎 of the K atom is 2.5×1019 m−3 at 435 K.
Thus, the total atom number 𝑁 of the K atom is
5×1011 with the mode volume 𝑉mode = 0.2×10−7 m3 .
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CHIN. PHYS. LETT. Vol. 30, No. 4 (2013) 040601
By solving Eq. (5), we obtain the relationship between average photon number and time, and the relationship between photon number at the steady state
and Γ𝑐 , 𝑁 , respectively.
we present an active optical clock scheme based on a
K four-level quantum system by adding a Fabry–Perot
resonator to the system, as shown in Fig. 5.
F-P resonator
Laser
1
Probability
0.05
0.8
Probability
ρ11
0.6
Output
ρ55
0.04
0.03
ρ44
ρ66
0.02
Fig. 5. The experimental configuration of an active optical clock based on a K atom four-level quantum system. The cavity length 𝐿 is 5 cm. The left mirror is antireflected for a 404.7 nm laser and high-reflected for 1252.2
nm, and the right mirror is high-reflected for a 404.7 nm
laser and has 25% reflectivity for 1252.2 nm.
0.01
ρ22
0
0
2
4
6
8
Time (10-7s)
0.4
0.2
ρ33
0
0
0.5
1
1.5
2
Time (ms)
Fig. 2. The change rule of population probabilities at
each state with Rabi frequency Ω = 2 × 107 s−1 .
Average photon number (106)
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
Time (ms)
Photon number at steady state (106)
Fig. 3. The average photon number with Ω = 2×107 s−1 ,
𝑔 = 9 × 104 s−1 , Γ𝑐 = 1.42 × 109 s−1 and 𝑁 = 5 × 1011 .
4
0
20
N (1011)
40
60
10
15
3.9
3.8
3.7
3.6
3.5
3.4
0
5
Γc (108s-1)
Fig. 4. The average photon number with Ω = 2×107 s−1 ,
𝑔 = 9 × 104 s−1 , Γ𝑐 = 1.42 × 109 s−1 .
Figure 3 shows that the photon number stabilizes
at 3.48 × 106 after 2 × 10−6 s with Ω = 2 × 107 s−1 ,
𝑔 = 9 × 104 s−1 , Γ𝑐 = 1.42 × 109 s−1 and 𝑁 = 5 × 1011 .
Figure 4 shows the photon number at the steady state
with the increase in Γ𝑐 and 𝑁 . Above all, the photon
number at the steady state is related to 𝑔, Γ𝑐 and 𝑁 .
Thanks to the convenience of a thermal gas cell,
The free spectral range FSR is FSR = 𝑐/(2 𝐿) =
3 GHz with 𝐿 = 5 cm. If we ignore the loss of the cell,
then the reflectivity fineness is
√
𝜋 𝑟1 𝑟2
FSR
𝐹𝑟 =
=
,
(6)
Γ𝑐
1 − 𝑟1 𝑟2
where 𝑟1 and 𝑟2 are the right mirror’s reflectivity of
404.7 nm and 1252.2 nm, respectively. We can obtain 𝑟1 = 100% and 𝑟2 = 25% in consideration of
Γ𝑐 = 1.42 GHz. This can be explained by the fact
that the left mirror is anti-reflected for the 404.7 nm
laser and high-reflected for 1252.2 nm, and the right
mirror is high-reflected for the 404.7 nm laser and 25%
for 1252.2 nm. In this experimental configuration, the
objective 1252.2 nm laser radiates from the right mirror. However, the cell in fact brings loss to the system.
From the experiments, we know that the light intensity decreases 10% to 20% when the laser goes through
the cell once. Taking the loss caused by the cell into
consideration, we need 𝑟2 equal to 31%, 35% and 39%
when the loss rates of the cell are 10%, 15% and 20%,
respectively.
The active optical clock gradually displays its special advantages in many fields.[16−35] It works in the
bad-cavity regime 𝑎 ≫ 1, and according to the definition of quantum-limited-linewidth,[17] we find that the
quantum-limited-linewidth of active optical clocks is
far narrower than that of conventional optical clocks.
In addition, the frequency of an active optical clock,
whose stability is not limited by the linewidth of the
probe laser, is insensitive to the thermal noise of the
cavity mode.
In summary, we presented an active optical clock
scheme based on a K atom four-level quantum system,
which is formed by 4𝑆1/2 , 5𝑃1/2 , 5𝑆1/2 and 4𝑃3/2 .
With a 404.7 nm pumping laser operating at a 4𝑆1/2
to 5𝑃1/2 transition, the population inversion between
the 5𝑆1/2 and 4𝑃3/2 states is built up. At the steady
state, 𝜌33 and 𝜌55 equal 8.3% and 3.5%, respectively.
Additionally, we present a experimental configuration
of the active optical clock by adding an F-P resonator.
In this study, we focus on the K atom quantum system for two main reasons. Firstly, most of the atomic
structure parameters of alkali metals,[25−32] including
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CHIN. PHYS. LETT. Vol. 30, No. 4 (2013) 040601
K atoms, have been studied very exactly. Secondly,
the melting point of the K atom is low, and it can
be stored in a cell. The cell stored with K atoms can
be used in spectroscopy experiments, and in addition
the cell is very cheap. To reduce the influence of the
Doppler effect, the thermal gas cell can be replaced by
a magneto-optical trap.
The authors thank Wang Dong-ying for helpful discussions.
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