Perturbative solution to the optical fields: Effective operator for atoms Typically, in an interferometric process, the light-atom interaction time is very

ATOM INTERFEROMETRY: LINEAR QUANTUM MEASUREMENT THEORY AND THE PRECISION LIMIT

5.5 Dynamics of the effective operators: A more exact treatment

5.5.1 Perturbative solution to the optical fields: Effective operator for atoms Typically, in an interferometric process, the light-atom interaction time is very

short compared to the free evolution time of the atom cloud, and the center-of-mass velocity of the atom cloud is very low, typically∼ 2 cm/s. Therefore to the leading order, we can treat the atom center-of-mass motion to be static during the interaction process, that is,𝑣_𝐴 ≈ 𝑣_𝐵 ≈ 0. We also note that the spatial size of optical fields is much larger than the size of the atom cloud, therefore we can approximate the mean value of the optical fields to be almost constants during the interaction process. For this calculation, we only care about control field because it transfers noise to the next atom-light interaction kernel, while different kernels interact with different passive fields, as shown in Fig. 5.3.

These equations can be solved in a perturbative way. For the equation of motion of the control field, the formal solution of the first-order perturbation is given by:

𝛿 𝜙⁺

𝑐(𝑡+𝑧, 𝑧)−𝛿 𝜙⁺

𝑐(𝑡−𝜖 ,−𝜖) = 𝑔_𝑐

∫ 𝑧

−𝜖

𝑑𝑦 𝛿[𝜙⁻

𝐴(𝑡+𝑦, 𝑦)𝜙⁺

𝐵(𝑡+𝑦, 𝑦)𝜙⁺

𝑝(𝑡+𝑦, 𝑦)],

(5.49)

where the atom cloud mostly distributed in [−𝜖 , 𝜖], as shown in Fig. 5.5 (we move the coordinate origin to the atom center-of-mass position). For brevity, we remove the tilde on the operators. In the following, all operators are the slowly varying

✏ ✏

z

t atom cloud

(y, t+y) passive field

In Out

control field

Figure 5.5: Atom-light interaction kernel. The atoms are undergoing the state swapping interaction, and during the very short interaction time (typically∼ 𝜇s), the A and B atoms do not have enough time to fly apart. Our effective atom operators Φ are defined through integration over the atomic cloud profile on the spatial direction. The discussion in the Section II will reduce the problem to be an effective model describing the interaction of the effective atom operators with the incoming light fields.

amplitude operators. Expanding the r.h.s. to the first order, we obtain:

𝑔_𝑐

∫ ^𝑧

−𝜖

𝑑𝑦𝜙¯⁺

𝑝𝜙¯⁺

𝐵(𝑡+𝑦, 𝑦)𝛿 𝜙⁻

𝐴(𝑡+𝑦, 𝑦) +𝑔_𝑐

∫ ^𝑧

−𝜖

𝑑𝑦𝜙¯⁺

𝑝𝜙¯⁻

𝐴(𝑡+𝑦, 𝑦)𝛿 𝜙⁺

𝐵(𝑡+𝑦, 𝑦) +𝑔_𝑐

∫ ^𝑧

−𝜖

𝑑𝑦𝜙¯⁺

𝐵(𝑡+𝑦, 𝑦)𝜙¯⁻

𝐴(𝑡+𝑦, 𝑦)𝛿 𝜙⁺

𝑝(𝑡+𝑦, 𝑦).

(5.50)

The classical atomic fields can be written as (under the slow motion approximation):

¯ 𝜙⁺⁽⁻⁾

𝐴/𝐵(𝑡 , 𝑦) = 𝑓_𝑎(𝑦)𝛼¯^(∗)

𝐴/𝐵(𝑡), (5.51)

where

𝑓_𝑎(𝑦)= Δ^1/2𝑎

(2𝜋)^1/4exp

−1 4Δ²_𝑎𝑦²

. (5.52)

Since𝑦 ∈ [−𝜖 , 𝜖]and𝜖 1, we can expand

¯

𝛼_𝐴/𝐵(𝑡+𝑦) ≈𝛼¯_𝐴/𝐵(𝑡) +𝑦𝛼¤¯_𝐴/𝐵(𝑡). (5.53)

Note that |𝑦𝛼¤¯_𝐴/𝐵(𝑡) | ∼ Ω𝑦𝛼¯_𝐵/𝐴(𝑡) and Ω𝑦 1, we can simplify the terms in Eq. (5.50) to be:

𝑔_𝑐𝜙¯⁺

𝑝

¯ 𝛼_𝐵(𝑡)

∫ 𝑧

−𝜖

𝑑𝑦 𝑓_𝑎(𝑦)𝛿 𝜙⁻

𝐴(𝑡 , 𝑦) +𝛼¯^∗

𝐴(𝑡)

∫ 𝑧

−𝜖

𝑓_𝑎(𝑦)𝛿 𝜙⁺

𝐵(𝑡 , 𝑦)

+𝑔_𝑐

∫ ^𝑧

−𝜖

𝑑𝑦𝛼¯^∗

𝐴(𝑡)𝛼¯_𝐵(𝑡)𝑓²

𝑎(𝑦)𝛿 𝜙⁺

𝑝(𝑡+𝑦, 𝑦).

(5.54)

Now let us define theeffective atom operatorsto be:

𝛿Φ^±_𝐴/𝐵(𝑡) =lim𝑧→𝜖

∫ ^𝑧

−𝜖

𝑑𝑦 𝑓_𝑎(𝑦)𝛿 𝜙^±

𝐴/𝐵(𝑡 , 𝑦). (5.55) As we shall see later, these effective operators have a nice property that the com- mutation relation of the associated creation and annihilation operators normalizes to one. The physical interpretation of the effective operators is that it describes the whole wavepacket of the atom field. Using these effective operators, the input- output relation for a light ray passing through the atomic cloud can be written as (for the passive field, it can be calculated in the same way):

𝛿 𝜙⁺

𝑐out(𝑡) −𝛿 𝜙⁺

𝑐in(𝑡) =𝑔_𝑐𝜙¯⁺

𝑝[𝛼¯_𝐵(𝑡)𝛿Φ⁻_𝐴(𝑡) +𝛼¯^∗

𝐴(𝑡)Φ⁺_𝐵(𝑡)]

+𝑔_𝑐𝛼¯^∗

𝐴(𝑡)𝛼¯_𝐵(𝑡)

∫ ^𝜖

−𝜖

𝑑𝑦 𝑓²

𝑎(𝑦)𝛿 𝜙⁺

𝑝(𝑡+𝑦, 𝑦), 𝛿 𝜙⁺

𝑝out(𝑡) −𝛿 𝜙⁺

𝑝in(𝑡) =𝑔_𝑝𝜙¯⁺

𝑐[𝛼¯^∗

𝐵(𝑡)𝛿Φ⁺_𝐴(𝑡) +𝛼¯_𝐴(𝑡)Φ⁻_𝐵(𝑡)]

+𝑔_𝑝𝛼¯_𝐴(𝑡)𝛼¯^∗

𝐵(𝑡)

∫ ^𝜖

−𝜖

𝑑𝑦 𝑓²

𝑎(𝑦)𝛿 𝜙⁺

𝑝(𝑡−𝑦, 𝑦).

(5.56)

The negative frequency branches simply obey the Hermitian conjugate of the above equations. The ratio between the second term and the first term on the r.h.s. of the equation is∼p

𝑁_𝑎/𝑁_𝐿 1, therefore it can be safely ignored.

Furthermore, we introduce the creation and annihilation operators that correspond to those effective atom operators and also the optical operators. Remember that the operators here are related to the creation and annihilation operators as follows:

𝛿Φ⁺_𝐴/𝐵 = ˆ 𝐴_𝐴/𝐵(𝑡)

√2𝜔_𝑎

, 𝛿 𝜙⁺

𝑐/𝑝in = 𝑎ˆ_𝑐/𝑝in(𝑧, 𝑡)

√2𝜔_𝐿

, (5.57)

where we take assumptions 𝜔_𝑐0 ≈ 𝜔_𝑝0 = 𝜔_𝐿 and note that ¯Φ⁺

𝐴/𝐵 = 𝛼¯_𝐴/𝐵/(2𝜔_𝑎). Now we want to rewrite the equations of motion of the atom fields in a more concise way, using these creation and annihilation operators. First, let us check the dimension of the above defined creation and annihilation operators. We have

𝐴ˆ_𝐴/𝐵(𝑡) =

∫

𝑑 𝑧 𝑓_𝑎(𝑧)𝑎ˆ_𝐴/𝐵(𝑧, 𝑡), (5.58)

where ˆ𝑎_𝐴/𝐵(𝑧, 𝑡)has the standard commutation relation on one time slice𝑡: [𝑎ˆ_𝐴/𝐵(𝑧, 𝑡),𝑎ˆ^†

𝐴/𝐵(𝑧⁰, 𝑡)] =𝛿(𝑧−𝑧⁰). (5.59) Using this commutation relation and the normalization condition for 𝑓_𝑎(𝑧), it is straightforward to show that:

[𝐴ˆ_𝐴/𝐵(𝑡),𝐴ˆ^†

𝐴/𝐵(𝑡)]=1. (5.60)

Note that ˆ𝐴is a dimensionless operator, while ˆ𝑎_𝑐in(𝑧, 𝑡)has the dimension[Length]^−1/2. The gravitational wave community is more familiar with the operator satisfying [𝑎˜(𝑡),𝑎˜^†(𝑡⁰)] = 𝛿(𝑡 − 𝑡⁰), so it is important to note that this is an equal time commutation relation for propagating fields, and the operators here are related by

˜

𝑎_𝑐in/𝑎ˆ_𝑐in = 𝑐^1/2 where 𝑐 is the speed of light. In the following, we will use the ˜𝑎 and replace the tilde with hat, i.e., ˜𝑎→ 𝑎ˆ. Then Eqs. (5.56) can be translated to:

ˆ

𝑎_𝑐out−𝑎ˆ_𝑐i=𝑖 𝜒_𝐿𝑒^{−𝑖 𝜑𝑝}[𝛼¯_𝐵(𝑡)𝐴ˆ^†

𝐴+𝛼¯^∗

𝐴(𝑡)𝐴ˆ_𝐵], ˆ

𝑎_𝑝out−𝑎ˆ_𝑝i =𝑖 𝜒_𝐿𝑒^{−𝑖 𝜑𝑐}[𝛼¯^∗

𝐵(𝑡)𝐴ˆ_𝐴+𝛼¯_𝐴(𝑡)𝐴ˆ^†

𝐵],

(5.61)

where we have 𝜒_𝐿 = |𝑔_𝑐|𝜙¯_𝐿p

𝜔_𝐿/𝜔_𝑎 under the approximation that𝜔_𝑐 = 𝜔_𝑝 = 𝜔_𝐿 and |𝜙¯⁺

𝑝| = |𝜙¯⁺

𝑐| = 𝜙¯_𝐿 = 𝑎¯_𝐿/√

2𝜔_𝐿. Comparing this equation with Eq. (5.10), we know that the 𝜒in the effective Hamiltonian has the form:

𝜒→ −𝑔/(2𝜔_𝐿) =𝑔_𝑐 =𝑔_𝑝. (5.62) 5.5.2 Deriving the evolution of atom fields using field theory

Using Eq. (5.55), we integrate the perturbation equations of atom fields to obtain the perturbation equations ofeffective atomic operators:

𝜕_𝑡𝛿Φ⁺_𝐴+𝑖Ω𝑒^{𝑖 𝜑}𝛿Φ⁺_𝐵 =𝑔_𝑎𝜙¯⁺

𝑝

∫ 𝜖

−𝜖

𝑑 𝑧 𝑓_𝑎(𝑧)Φ¯⁺_𝐵(𝑡 , 𝑧)𝛿 𝜙⁻

𝑐(𝑡 , 𝑧) +𝑔_𝑎𝜙¯⁻

𝑐

∫ 𝜖

−𝜖

𝑓_𝑎(𝑧)Φ¯⁺_𝐵(𝑡 , 𝑧)𝛿 𝜙⁺

𝑝(𝑡 , 𝑧),

𝜕_𝑡𝛿Φ⁺_𝐵+𝑖Ω𝑒^{−𝑖 𝜑}𝛿Φ⁺_𝐴 =𝑔_𝑎𝜙¯⁻

𝑝

∫ 𝜖

−𝜖

𝑑 𝑧 𝑓_𝑎(𝑧)Φ¯⁺_𝐴(𝑡 , 𝑧)𝛿 𝜙⁺

𝑐(𝑡 , 𝑧) +𝑔_𝑎𝜙¯⁺

𝑐

∫ 𝜖

−𝜖

𝑓_𝑎(𝑧)Φ¯⁺_𝐴(𝑡 , 𝑧)𝛿 𝜙⁻

𝑝(𝑡 , 𝑧).

(5.63)

Now, we substitute the formal solution of𝛿 𝜙^±

𝑐/𝑝into the above equations, which will reveal the structure of the optical back-action on the atom fields.

Let us take the first term on the r.h.s. of the𝛿Φ𝐴equation as an example. It has the following form after substituting𝛿 𝜙⁻

𝑐(𝑡 , 𝑧): 𝑔_𝑎𝜙¯⁺

𝑝

∫ ^𝜖

−𝜖

𝑑 𝑧 𝑓_𝑎(𝑧)Φ¯⁺_𝐵(𝑡 , 𝑧)𝛿 𝜙⁻

𝑐(𝑡 , 𝑧)

=𝑔_𝑎𝜙¯⁺

𝑝𝛼_𝐵(𝑡)

∫ 𝜖

−𝜖

𝑑 𝑧 𝑓²

𝑎(𝑧)

𝛿 𝜙⁻

𝑐in(𝑡) +𝑔_𝑐𝛼_𝐴(𝑡)𝛼^∗

𝐵(𝑡)

∫ 𝑧

−𝜖

𝑑𝑦 𝑓²

𝑎(𝑦)𝛿 𝜙⁻

𝑝(𝑦) +𝑔_𝑐𝜙¯⁻

𝑝

∫ ^𝑧

−𝜖

𝑑𝑦 𝑓_𝑎(𝑦) [𝛼_𝐴(𝑡)𝛿 𝜙⁻

𝐵(𝑦, 𝑡) +𝛼^∗

𝐵(𝑡)𝛿 𝜙⁺

𝐴(𝑦, 𝑡)]

.

(5.64)

Note that the 𝑓_𝑎(𝑦) takes a Gaussian form symmetric around 𝑦 = 0, which means that we can write:

∫ ^𝜖

−𝜖

𝑑 𝑧 𝑓²

𝑎(𝑧)

∫ ^𝑧

−𝜖

𝑑𝑦 𝑓_𝑎(𝑦)𝛿 𝜙⁺

𝐴(𝑦, 𝑡)= 1

2𝛿Φ⁺_𝐴(𝑡), (5.65) where we have used the normalization condition for 𝑓_𝑎(𝑧), and we have:

𝑔_𝑎𝜙¯⁺

𝑝

∫ 𝜖

−𝜖

𝑑 𝑧 𝑓_𝑎(𝑧)Φ¯⁺_𝐵(𝑡 , 𝑧)𝛿 𝜙⁻

𝑐(𝑡 , 𝑧) (5.66)

=𝑔_𝑎𝜙¯⁺

𝑝𝛼_𝐵(𝑡)𝛿 𝜙⁻

𝑐in(𝑡) + 1

2|𝑔_𝑎𝑔_𝑐||𝜙¯_𝑝|²

𝛼_𝐴(𝑡)𝛼_𝐵(𝑡)𝛿Φ⁻_𝐵(𝑡) + |𝛼_𝐵(𝑡) |²𝛿Φ⁺_𝐴(𝑡) . Here, we ignore the ¯𝜙⁻

𝑝 term since it is much smaller than the other terms. In a similar way, we have:

𝑔_𝑎𝜙¯⁻

𝑐

∫ ^𝜖

−𝜖

𝑑 𝑧 𝑓_𝑎(𝑧)Φ¯⁺_𝐵(𝑡 , 𝑧)𝛿 𝜙⁺

𝑝(𝑡 , 𝑧) (5.67)

=𝑔_𝑎𝜙¯⁻

𝑐𝛼_𝐵(𝑡)𝛿 𝜙⁺

𝑝in(𝑡) − 1

2|𝑔_𝑎𝑔_𝑐||𝜙¯_𝑐|²

𝛼_𝐴(𝑡)𝛼_𝐵(𝑡)𝛿Φ⁻_𝐵(𝑡) + |𝛼_𝐵(𝑡) |²𝛿Φ⁺_𝐴(𝑡) . Finally, we have the equations of motion for atom fields as:

𝜕_𝑡𝛿Φ⁺_𝐴+𝑖Ω𝑒^{𝑖 𝜑}𝛿Φ⁺_𝐵 =𝑔_𝑎𝜙¯_𝐿𝛼_𝐵(𝑡) [𝑒^{−𝑖 𝜑𝑝}𝛿 𝜙⁻

𝑐in(𝑡) +𝑒^{𝑖 𝜑𝑐}𝛿 𝜙⁺

𝑝in(𝑡)],

𝜕_𝑡𝛿Φ⁺_𝐵+𝑖Ω𝑒^{−𝑖 𝜑}𝛿Φ⁺_𝐴 =𝑔_𝑎𝜙¯_𝐿𝛼_𝐴(𝑡) [𝑒^{𝑖 𝜑𝑝}𝛿 𝜙⁺

𝑐in(𝑡) +𝑒^{−𝑖 𝜑𝑐}𝛿 𝜙⁻

𝑝in(𝑡)].

(5.68)

Then we can obtain the exact form of equations of motion for atom cloud:

𝜕_𝑡𝐴ˆ_𝐴+𝑖Ω𝑒^{𝑖 𝜑}𝐴ˆ_𝐵 =𝑖 𝜒_𝐵(𝑡) [𝑒^{−𝑖 𝜑𝑝}𝑎ˆ^†

𝑐in(𝑡) +𝑒^{𝑖 𝜑𝑐}𝑎ˆ_𝑝in(𝑡)],

𝜕_𝑡𝐴ˆ_𝐵+𝑖Ω𝑒^{−𝑖 𝜑}𝐴ˆ_𝐴 =𝑖 𝜒_𝐴(𝑡) [𝑒^{𝑖 𝜑𝑝}𝑎ˆ_𝑐in(𝑡) +𝑒^{−𝑖 𝜑𝑐}𝑎ˆ^†

𝑝in(𝑡)],

(5.69) where𝜑=𝜑_𝑐−𝜑_𝑝is the phase difference between control and passive lights, and

𝜒_𝐴/𝐵(𝑡) = r 𝜔_𝑎

𝑐𝜔_𝐿

|𝑔_𝑎|𝜙¯_𝐿𝛼_𝐴/𝐵(𝑡). (5.70) The 𝛼_±(𝑡) = 𝛼_±(0)exp[∓Ω𝑡] and 𝜒₀ := p

𝜔_𝑎/𝑐𝜔_𝐿|𝑔_𝑎|𝜙¯_𝐿. Using the relation Eq. (5.62), we can also map Eq. (5.69) to Eq. (5.7).

5.6 Conceptual comparison with the laser interferometer GW detector