短期安定度

2.5 短期安定度

短期安定度のアラン標準偏差σy(τ)は以下の式に従うことが知られている[13]． σ_y(τ)∝ 1

Q·(S/N)τ⁻¹² (2.6)

ここで，QはCPT共鳴のQ値，S/N はS/N比を表している．したがって，CPT共鳴のS/N比とQ値を測定することで，短期安定度を見積もることができる．

2.5.1 S/N 比

S/N比はシグナル/ノイズで定義される．CPT共鳴のS/N 比には“コントラスト”と呼ばれる指標が使用される(図2.7)．コントラストは百分率で表され，以下のように定義される．

Contrast(%) = Signal

DC level ×100 (2.7)

コントラストをS/N比の指標として使用する理由は，CPT共鳴の支配的なノイズがレーザのAMノイズとレーザの波長変動により生じるFM-AM変換ノイズであり，それらノイズはDCレベルに比例するためである(図2.8)．したがって，CPT共鳴のコントラストを向上させるためには，共鳴振幅を大きくするかDCレベルを下げる必要がある．

CPT共鳴の共鳴振幅は次式に従うことが知られている[14]． Signal∝nΩ⁴

Γ³

2γs+ Ω²/Γ (2.8)

ここで，nはアルカリ原子の密度，Ωはラビ周波数，Γは励起準位の緩和率，γ_sは基底準位の緩和率である．光強度はラビ周波数 Ωの二乗に比例することから，光強度の増加に伴って共鳴振幅は増加する．これは，光強度の増加に伴い遷移レートが上昇し，暗状態になる原子の数が増加するためである．

DCレベルは，CPT共鳴に寄与しないレーザ光度に比例する．CPT共鳴を観測するためには，半導体レーザをRFで変調し，変調により生じた２つのサイドバンドを利用するが，同時にCPT共鳴に寄与しないサイドバンドも同時に生成される．この共鳴に寄与し

26 第2章 CPT原子発振器ないサイドバンドは，原子に吸収されることなくガスセルを通過し，光検出器に入射する．入射した光は背景光となり，DCレベルを上昇させる要因となる．

2.5.2 Q 値

Q値は，共鳴周波数frf と共鳴線幅γ(半値全幅)の比で定義される．共鳴線幅が細いほどQ値は高くなる．

Q = f_rf

γ (2.9)

原子発振器の場合，共鳴周波数f_rf はアルカリ原子の基底準位間の周波数f_hfs と等しい．

自由空間に静止した理想的な条件下では，基底準位の寿命が非常に長いため，共鳴線幅γ は非常に小さい．しかしながら，実際には，アルカリ原子同士のスピン交換衝突γse，バッファガスとの衝突広がりγbuﬀ，壁面との衝突広がりγwallにより広がる．また，励起レーザとの相互作用により，基底準位間が強く結合することで生じるパワーブロードニング γpowerも線幅を広げる要因である．これら要因のうち，γse，γbuﬀ，γwall はガスセルの温度や圧力，寸法に依存する．γ_wallは，ガスセルにバッファガスを封じることや，壁面に緩和防止コーティングを施すことで抑えることができる[49]．パワーブロードニングγ_power はレーザ光強度に比例する(付録Aを参照のこと)[14]．

得られる共鳴線幅γ はこれら広がりの線形結合で

γ =γ_se+γ_buﬀ +γ_wall+γ_power (2.10)

で表される．

図2.9に光強度に対する共鳴線幅特性の測定例を示す．パワーブロードニングによる線幅広がりによりγpower は光強度に対して線形に増加する．光強度がゼロの時に得られる線幅は，光強度に依存しない線幅広がりであり，γse，γbuﬀ，γwallの和である．これら光強度に依存しない線幅の和は0.36 kHzであり，これはパワーブロードニングγpower による広がりに比べ小さい．したがって，CPT共鳴の共鳴線幅は，パワーブロードニングによる線幅広がりが支配的である[14]．

2.5 短期安定度 27

2.5.3 短期安定度の性能指数

短期安定度は共鳴のコントラストとQ値の積で理論的性能が決まる．したがって，性能指数(FoM: Figure of Merit)は以下のように定義される．

F oM =Q·Contrast (2.11)

図 2.10に光強度に対する性能指数特性の測定例を示す．連続励起の性能指数は，光強度が小さいときに上昇傾向となるが，光強度が大きいときでは低下傾向に転じる．これは，光強度が小さいときではコントラストの上昇が支配的であるのに対し，光強度が大きいときでは，パワーブロードニングによるQ値の低下が支配的になるためである．したがって，連続励起の性能指数が最大となる光強度がある．このように，連続励起ではコントラストとQ値を同時に改善するのは困難である[14]．

28 第2章 CPT原子発振器

35

^th

Annual Precise Time and Time Interval (PTTI) Meeting

469

0.1 1 10

N o is e [n A

RMS

/H z

1/2

]

DC Level [ µ A]

The two principal noise mechanisms, shot noise and laser frequency noise, can thus be differentiated by their power-law dependence on the laser intensity. In order to determine their relative impact in the CSAC device, we’ve made measurements of noise as a function of DC level in our testbed apparatus.

Figure 2 shows the RMS noise, measured in a 1 Hz bandwidth at 500 Hz, as a function of laser power, over two orders of magnitude spanning the designed operating region of the CSAC. Also shown is a fit to a power law model of (S/N) which indicates N ∝ (DC level)

^0.998

. We conclude that, for typical CSAC components and operating conditions, the principal noise mechanism in the CSAC is laser frequency noise and, for practical design purposes, N ∝ (DC level).

Figure 2. Noise vs. DC level.

Signal

Light Transmission

Laser Frequency DC Level

Linewidth,

γ

For the purposes of definition, Figure 3 shows schematically the elements of the CPT resonance signal. The CPT spectrum is dominated by the Gaussian laser absorption feature, whose width may be several GHz due to a combination of Doppler and pressure gas broadening. The Lorentzian CPT signal sits atop the Doppler absorption feature and is characterized by its amplitude, designated “Signal,” and linewidth, γ.

We define the “contrast,” η, of the CPT resonance as Signal divided by DC level.

Figure 3. CPT resonance features.

With these definitions, and the aforementioned linear relationship between DC level and noise, we arrive at the appropriate figure-of-merit, ξ, for predicting short-term stability of the CSAC:

γ η ξ = / ,

where η and γ are measured in ppm and Hz respectively. This leads to the relationship

2 /

1 1

)

( ∝ τ ⁻ τ ξ

σ _y ,

where the constant of proportionality is determined by the spectrum of frequency noise on the VCSEL.

This presents a valuable relationship, from the perspective of CSAC engineering, in that it permits the evaluation of different atomic transitions and species and operating conditions without explicit dependence on the noise properties of the particular laser source employed in the measurement.

図2.7 CPT共鳴のコントラスト(文献[3]より引用)

35

^th

Annual Precise Time and Time Interval (PTTI) Meeting

469

0.1 1 10

Noise [nA RMS/Hz1/2 ]

DC Level [µA]

The two principal noise mechanisms, shot noise and laser frequency noise, can thus be differentiated by their power-law dependence on the laser intensity. In order to determine their relative impact in the CSAC device, we’ve made measurements of noise as a function of DC level in our testbed apparatus.

Figure 2 shows the RMS noise, measured in a 1 Hz bandwidth at 500 Hz, as a function of laser power, over two orders of magnitude spanning the designed operating region of the CSAC. Also shown is a fit to a power law model of (S/N) which indicates N ∝ (DC level)

^0.998

. We conclude that, for typical CSAC components and operating conditions, the principal noise mechanism in the CSAC is laser frequency noise and, for practical design purposes, N ∝ (DC level).

Figure 2. Noise vs. DC level.

Signal

Light Transmission

Laser Frequency DC Level

Linewidth, γ

For the purposes of definition, Figure 3 shows schematically the elements of the CPT resonance signal. The CPT spectrum is dominated by the Gaussian laser absorption feature, whose width may be several GHz due to a combination of Doppler and pressure gas broadening. The Lorentzian CPT signal sits atop the Doppler absorption feature and is characterized by its amplitude, designated “Signal,” and linewidth, γ.

We define the “contrast,” η, of the CPT resonance as Signal divided by DC level.

Figure 3. CPT resonance features.

With these definitions, and the aforementioned linear relationship between DC level and noise, we arrive at the appropriate figure-of-merit, ξ, for predicting short-term stability of the CSAC:

γ η ξ = / ,

where η and γ are measured in ppm and Hz respectively. This leads to the relationship

2 /

1 )

( ∝ τ

⁻

τ ξ

σ

,

where the constant of proportionality is determined by the spectrum of frequency noise on the VCSEL.

This presents a valuable relationship, from the perspective of CSAC engineering, in that it permits the evaluation of different atomic transitions and species and operating conditions without explicit dependence on the noise properties of the particular laser source employed in the measurement.

図2.8 DCレベルに対するノイズ特性(文献[3]より引用)

2.5 短期安定度 29

0.0 0.5 1.0 1.5 2.0 2.5

0.0 0.4 0.8 1.2 1.6 2.0 2.4

Light intensity (mW/cm ² )

Linewidth (kHz)

図2.9 光強度に対する共鳴線幅特性例

0.0 0.5 1.0 1.5 2.0 2.5

0.0 0.5 1.0 1.5 2.0

Light intensity (mW/cm ² )

Figure of merit (a.u.)

図2.10 光強度に対する性能指数特性例

30 第2章 CPT原子発振器

ドキュメント内 Coherent Population Trapping 原子発振器の特性改善に関する研究矢野雄一郎 (ページ 35-40)

2.5 短期安定度

2.5.1 S/N 比

2.5.2 Q 値

2.5.3 短期安定度の性能指数

35

Annual Precise Time and Time Interval (PTTI) Meeting

469

N o is e [n A

/H z

]

DC Level [ µ A]

Figure 2 shows the RMS noise, measured in a 1 Hz bandwidth at 500 Hz, as a function of laser power, over two orders of magnitude spanning the designed operating region of the CSAC. Also shown is a fit to a power law model of (S/N) which indicates N ∝ (DC level)

. We conclude that, for typical CSAC components and operating conditions, the principal noise mechanism in the CSAC is laser frequency noise and, for practical design purposes, N ∝ (DC level).

Figure 2. Noise vs. DC level.

γ

We define the “contrast,” η, of the CPT resonance as Signal divided by DC level.

Figure 3. CPT resonance features.

With these definitions, and the aforementioned linear relationship between DC level and noise, we arrive at the appropriate figure-of-merit, ξ, for predicting short-term stability of the CSAC:

γ η ξ = / ,

where η and γ are measured in ppm and Hz respectively. This leads to the relationship

2 /

1 1

)

( ∝ τ − τ ξ

σ y ,

where the constant of proportionality is determined by the spectrum of frequency noise on the VCSEL.

This presents a valuable relationship, from the perspective of CSAC engineering, in that it permits the evaluation of different atomic transitions and species and operating conditions without explicit dependence on the noise properties of the particular laser source employed in the measurement.

35

Annual Precise Time and Time Interval (PTTI) Meeting

469

Figure 2 shows the RMS noise, measured in a 1 Hz bandwidth at 500 Hz, as a function of laser power, over two orders of magnitude spanning the designed operating region of the CSAC. Also shown is a fit to a power law model of (S/N) which indicates N ∝ (DC level)

. We conclude that, for typical CSAC components and operating conditions, the principal noise mechanism in the CSAC is laser frequency noise and, for practical design purposes, N ∝ (DC level).

Figure 2. Noise vs. DC level.

We define the “contrast,” η, of the CPT resonance as Signal divided by DC level.

Figure 3. CPT resonance features.

With these definitions, and the aforementioned linear relationship between DC level and noise, we arrive at the appropriate figure-of-merit, ξ, for predicting short-term stability of the CSAC:

γ η ξ = / ,

where η and γ are measured in ppm and Hz respectively. This leads to the relationship

1

)

( ∝ τ

τ ξ

σ

,

where the constant of proportionality is determined by the spectrum of frequency noise on the VCSEL.

This presents a valuable relationship, from the perspective of CSAC engineering, in that it permits the evaluation of different atomic transitions and species and operating conditions without explicit dependence on the noise properties of the particular laser source employed in the measurement.

0.0 0.5 1.0 1.5 2.0 2.5

0.0 0.4 0.8 1.2 1.6 2.0 2.4

Light intensity (mW/cm 2 )

Linewidth (kHz)

0.0 0.5 1.0 1.5 2.0 2.5

0.0 0.5 1.0 1.5 2.0

Light intensity (mW/cm 2 )

Figure of merit (a.u.)

( ∝ τ ⁻ τ ξ

σ _y ,

Light intensity (mW/cm ² )

Light intensity (mW/cm ² )