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Home , Coherent state, Fock state

Lecture 5 Summer Term 2019 Andrey Surzhykov Robert M¨uller

Coherent states

As we have seen and you will still proove number states do not have a well defined electric field. Thus, number states are not suitable to describe the quantized version of a classical field E(r, t). Can we find another that corresponds to E(r, t) in the ? The intuitive way to achieve that would be a state with a huge number of N  1. However, as we have shown, the fact that hn|Eˆ(r, t)|ni = 0 does not depend on the number. Instead, since Eˆ depends on a sum ofa ˆ anda ˆ†, we could construct a state that consists of Fock states with numbers |ni and |n + 1i. Definition: Coherent states |αi are eigenfunctions of the annihilation operatora ˆ, so that: aˆ |αi = α |αi (1) Because Fock states are a complete ONS, we can expand a in terms of number states, so that: ∞ X |αi = cn |ni . (2) n=0 If we act with the annihilation on this expansion, we get:

∞ ∞ X X √ aˆ |αi = cnaˆ |ni = ncn |n − 1i . (3) n=0 n=1 Because |αi is an eigenfunction ofa ˆ we find the condition:

∞ ∞ ∞ X √ X X ncn |n − 1i = α cn |ni = α cn−1 |n − 1i . (4) n=1 n=0 n=1 By comparison of the coefficients of the same , we find the recurrence relation: √ ncn = αcn−1, (5) which we can rewrite: αn cn = c0 √ (6) n!

The only free parameter is the first coefficient c0, which can be fixed by normalization:

∞ ∞ ∗ n n0 ∞ 2 n 2 X X (α ) α 0 2 X (|α| ) 2 |α|2 ! hα|αi = |c0| √ hn|n i = |c0| = |c0| e = 1 (7) 0 n! n=0 n0=0 n!n ! n=0

1 2 − |α| and therefore c0 = e 2 . Thus, a coherent state is constructed by:

∞ ∞ 2 n 2 n † n − |α| X α − |α| X α (ˆa ) |αi = e 2 √ |ni = e 2 √ |0i (8) n=0 n! n=0 n!

No let us show, that any state can be expanded in coherent states. Therefore we write:

Z X Z dα |αi hα| = dα |ni hn|αi hα|mi hm| n,m Z X |ni hm| 2 = √ dα e−|α| αnα∗m n!m! n,m | {z } =πn!δnm X = π |ni hn| n Therefore we have found that the unity operator in terms of coherent states is: 1 Z dα |αi hα| = Iˆ (9) π

The electric field of a coherent state The aim of the whole operation above was to construct a state with the proper classical limit. So let us calculate the expectation value of the electric field operator:

r ω hα|Eˆ(r, t)|αi = i ~ hα|aˆ ei(r·k−ωt) − aˆ† e−i(r·k−ωt)|αi 2ε0V r ω = i ~ α ei(r·k−ωt) − α∗ e−i(r·k−ωt) 2ε V 0 (10) r ω = i ~ |α| ei(r·k−ωt+φ) − e−i(r·k−ωt+φ) 2ε0V r ω = −2 ~ |α| sin(r · k − ωt + φ), 2ε0V which corresponds to the classical electrical field. In the exercises you will show, that the variance of the electric field is ω hα|∆Eˆ2(r, t)|αi = ~ . (11) 2ε0V This corresponds to the variance of the vacuum and thus shows the definition of coherent states as states with minimal uncertainty.

2 Figure 1: A Fock or number state in corresponds to a circular probability q 1 distribution around zero. The variance has a radius of n + 2 .

Photon statistics of coherent states Since we can expand coherent states into number states, it is a straightforward question to ask, what the expectation value of the photon number is in such a state. As you will show it corresponds to: hα|nˆ|αi = |α|2 ≡ n (12) We can use this relation to find an expression for the of the photon number. The probability to find n photons in a coherent state |αi can be calculated as follows: 2 2 m 2 − |α| X α 2 Pn = | hn|αi | = e √ hn|mi m=0 m! 2n 2 |α| = e−|α| n! nn = e−n n! This corresponds to a Poisson-distribution!

Fock and coherent states in phase space As we know from , each configuration at point r with momentum p can be represented by a point (r, p) in phase space. We can perform a similar consider-

3 ation in . Let 1 r → ηˆ = aˆ +a ˆ† 1 2 1 p → ηˆ = aˆ − aˆ† 2 2i

From these conjugated operatorsη ˆ1 andη ˆ2 we can build a 2D-phase space associated with a single-mode field. The expectation values of both operators with respect to a Fock state are

hn|ηˆ1/2|ni = 0. The variance, however, is non-vanishing: 1 hn|∆ˆη2 |ni = (2n + 1). 1/2 4

Sinceη ˆ1 andη ˆ2 do not commute we find the incertainty: 1 ∆ˆη2 ∆ˆη2 ≥ 1 2 16 Therefore we find for a number state in phase space the following relation: 1 ∆ˆη2 + ∆ˆη2 = n + (13) 1 2 2

q 1 which describes a circle with radius n + 2 , as depicted in Fig. 1. The description of a coherent state in phase space requires a bit more work. For the expectation values we find the interesting relation:

1 ∗ hα|ηˆ1|αi = (α + α ) = Re(α) 2 (14) 1 hα|ηˆ |αi = (α − α∗) = Im(α) 2 2i This means that we can map the complex α-space to the phase space of a coherent state. Let us, therefore, use the polar representation of α:

α = |α| eiθ

The uncertainty√ of the radial distance |α| corresponds to the standard deviation of the photon number n. Becauseη ˆ1 andη ˆ2, again, do not commute, there is also a phase uncertainty. The variance of both operators is given by 1 ∆ˆη2 = , (15) 1/2 4 which is the same as the vacuum uncertainty, which again shows that (i) coherent states are states of minimal uncertainty and (ii) as seen in Fig. 2 coherent states can be viewed

4 Figure 2: A coherent state |αi in phase space has a circular probability distribution. The center of this distribution is at (Re(α), Im(α). The 1 − σ-radius is the same as for the vacuum √1 . 2 as vacuum states displaced in phase space. Indeed we can construct an operator Dˆ(α) that, acted on the vacuum state, performs exactly that shift, i.e.: Dˆ(α) |0i = |αi (16) From Eq. (8) one might think that Dˆ(α) = exp(−|α|2/2) exp(αaˆ†). This, however, is not true, because this exponential operator would not be unitary. Instead we define: Dˆ(α) = eαaˆ†−α∗aˆ (17) With the Baker-Campbell-Hausdorff formula

Aˆ+Bˆ 1 [A,ˆ Bˆ] Bˆ Aˆ e = e 2 e e (18) and Aˆ = α∗aˆ and Bˆ = αaˆ† we see that this corresponds to

2 − |α| αaˆ† α∗aˆ Dˆ(α) = e 2 e e , (19) where we note that the exponent of the annihilation operator acted on the vacuum state has no effect and, thus, we retain Eq. (8) and, thus, have obtaines the unitary shifting operator we were looking for.

Squeezed states In the previous section both phase-space variables of a coherent state had the same variance: 1 ∆ˆη2 = . (20) 1/2 4

5 Figure 3: A squeezed state |α, ξi in phase space has an asymmetric probability distribu- tion. The center of this distribution is at (Re(α), Im(α). The 1 − σ-distance from the center is modified by a factor of e−|ξ| or e|ξ|, respectively.

For various applications, however, it may be useful to construct states that have an 1 uncertainty smaller than 4 with respect to one obervable, while it is not important if another is defined less sharply. Therefore we aim to construct states, where: 1 ∆ˆη2 = γ 1 4 1 (21) ∆ˆη2 = , 2 4γ

2 2 1 where γ ≤ 1. So that the product of both still satisfies h∆ˆη1i h∆ˆη2i ≥ 16 . With a complex number ξ we can define the squeezing operator:

h 2i 1 ξ∗aˆ2−ξ(aˆ†) Sˆ(ξ) = e 2 , (22) which is a nonlinear operator ina ˆ anda ˆ†. For consistency this operator has to be unitary:

Sˆ†(ξ) = Sˆ(−ξ) Sˆ†(ξ)Sˆ(ξ) = Sˆ(ξ)Sˆ†(ξ) = I.ˆ

Since we have seen that a coherent state is just a displaced vacuum state, we can for sim- plicity study the action of this operator on the quantum vacuum to generate a squeezed vacuum state: |ξi = Sˆ(ξ) |0i (23)

6 Now let us proof that those states are, indeed, squeezed states. Therefore let us calculate the expectation values:

1 ˆ† † ˆ hξ|ηˆ1|ξi = h0|S (ξ)(ˆa +a ˆ )S(ξ)|0i 2 (24) 1 hξ|ηˆ |ξi = h0|Sˆ†(ξ)(ˆa − aˆ†)Sˆ(ξ)|0i 2 2i

This, however, leaves us with the problem to calculate the operators Sˆ†(ξ)ˆaSˆ(ξ) and Sˆ†(ξ)ˆa†Sˆ(ξ). In order to do this we can use the decomposition of exponential operators: 1 h i 1 h h ii eAˆBˆ e−Aˆ = Bˆ + [A,ˆ Bˆ] + A,ˆ [A,ˆ Bˆ] + Aˆ A,ˆ [A,ˆ Bˆ] + ... (25) 2! 3! h i ˆ ˆ † ˆ 1 ∗ 2 †2 With B =a ˆ or B =a ˆ , respectively and A = 2 ξ aˆ − ξ aˆ we find:

Sˆ†(ξ)ˆaSˆ(ξ) =a ˆ + ξaˆ† Sˆ†(ξ)ˆa†Sˆ(ξ) =a ˆ† + ξ∗a.ˆ

Therefore: hξ|ηˆ1/2|ξi = 0 (26) and, following analogue considerations:

2 1 −2r hξ|∆ˆη1|ξi = e 4 (27) 1 hξ|∆ˆη2|ξi = e2r, 2 4 where we have assumed that arg(ξ) = 0 and r is the so-called squeezing parameter. Finally we need to mention, that, of course, not only the vacuum can be squeezed, but any coherent state by applying the squeezing operator:

|α, ξi = Sˆ(ξ) |αi (28)

Such a state, that is squeezed in η1-direction is shown in Fig. 3.

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