Appendix. Basic properties of linear systems

In this Appendix, to clarify the formalism used in Sec. 3.2, we summarize some well-known basic properties of linear systems linearly coupled to each other or to external classical forces. Much of this material can be found in Sakurai [37], and for its application to quantum-measurement processes in Braginsky and Khalili [19] and Caves et al. [23].

Definition 1 (Linear systems) Any system whose Hamiltonian is at most quadratic in its canon- ical coordinates and momenta is a linear system.

Definition 2 (Linear observables) Any linear combination (either time dependent or time in- dependent) of the canonical coordinates and momenta of a linear system, plus a possible complex number (C-number), is a linear observable of the system.

Denoting all the canonical coordinates and momenta by ˆCⁱ withi= 1,2,· · ·, the Hamiltonian of a linear system can be written as

H(t) =ˆ X

i,j

L^ij₂(t) ˆCⁱCˆ^j+X

i

Lⁱ₁(t) ˆCⁱ+L0(t), (3.122)

where L^ij₂(t) is symmetric ini and j. The equations of motion of the canonical observables in the Heisenberg picture read [we use the fact that ˆC^jH does not depend explicitly on time]:

i~d

dtCˆ^jH(t) = h

Cˆ^jH(t),HˆH(t)i ,

= Uˆ^†(−∞, t)h

Cˆ^jS,HˆS(t)i

Uˆ(−∞, t),

= Uˆ^†(−∞, t)



X

l,m

2L^lm₂ (t)CjlCˆ^mS+X

l

L^l₁(t)Cjl



Uˆ(−∞, t),

= X

l,m

2L^lm₂ (t)CjlCˆ^mH(t) +X

l

L^l₁(t)djl. (3.123)

Here the subscripts S and H stand for Schr¨odinger and Heisenberg pictures respectively, Cjl ≡ [ ˆC^jS,Cˆ^lS] is the commutator between the canonical operators, which is a C-number, and ˆU(−∞, t) is the time-evolution operator which satisfies the Schr¨odinger equation

i~d

dtUˆ(−∞, t) = ˆHSUˆ(−∞, t) (3.124) with initial condition ˆU(−∞,−∞) = 1.The solution to Eq. (3.123) is of the form

Cˆ^jH(t) =X

k

αjk(t) ˆC^kH(−∞) +βj(t) =X

k

αjk(t) ˆC^kS+βj(t), (3.125)

whereαjk(t) andβj(t) are time dependent C-numbers.

For any linear observableAit follows from linearity that ˆAH(t) =P

jaj(t) ˆC^jH(t) +b(t), which, along with Eq. (3.125), leads to:

AˆH(t) =X

j

aj(t) ˆC^jH(t) +b(t) =X

j,k

aj(t)αjk(t) ˆC^kS+X

j

aj(t)βj(t) +b(t). (3.126)

This provides the following theorem:

Theorem 1 At any time the operator of a linear observable in the Heisenberg picture can always be written as a linear combination of operators of the (time-independent) canonical variables in the Schr¨odinger picture plus a possible C-number.

Applying the above theorem to any two linear observablesAandB, recalling thatCjk≡[ ˆC^jS,Cˆ^kS] is a C-number and the commutator between a C-number and any operator is zero, we find

hAˆH(t),BˆH(t⁰)i

=X

j,k

γ^A_j(t)γ_k^B(t⁰)Cjk, (3.127)

which is a C-number. Therefore, the following theorem holds:

Theorem 2 In the Heisenberg picture, the commutator of the operators of any two linear observables at two times is a C-number.

We are interested in the evolution of a linear system subject to a classical external linear force or linearly coupled to another independent linear system. A force-susceptibility kind of formulation can be introduced in these cases (as is done by Braginsky and Khalili, see Sec. 6.4 of Ref. [19]). We shall describe the system using a perturbative approach. Thus we write the total Hamiltonian in the Schr¨odinger picture as ˆHS= ˆH0S+ ˆVS(t), where ˆVS(t) is treated as a perturbation with respect to the zeroth order Hamiltonian ˆH0S. It is generally convenient to introduce the so-called Interaction picture (see, e.g., Sections 5.5 and 5.6 of Ref. [37]), in which the evolution operator ˆUI is defined by the relation ˆU(−∞, t)≡Uˆ0(−∞, t) ˆUI(−∞, t), where ˆU0(−∞, t) is the evolution operator associated with ˆH0S and ˆU is defined by Eq. (3.124). Then, ˆUI(−∞, t) satisfies the equations

i~d

dtUˆI(−∞, t) = ˆVI(t) ˆUI(−∞, t), UˆI(−∞,−∞) = 1, (3.128)

with ˆVI(t)≡Uˆ₀^†(−∞, t) ˆVS(t) ˆU0(−∞, t). The solution of Eq. (3.128) can be written as a perturbative expansion,

UˆI(−∞, t) = 1 + 1 i~

Z t

−∞

dt1VˆI(t1) + µ1

i~

¶2Z t

−∞

dt1

Z t1

−∞

dt2VˆI(t1) ˆVI(t2) +· · ·,

= X∞ n=0

1 n!

µ1 i~

¶n

T (·Z t

−∞

dt1VˆI(t1)

¸ⁿ)

, (3.129)

whereTdenotes the time-ordered product [38]. The Heisenberg operator associated with any observ- ableA, evolving under the full Hamiltonian ˆH, is linked to the corresponding Heisenberg operator evolving under the Hamiltonian ˆH0by the relation ˆAH(t) = ˆU_I^†(−∞, t) ˆA⁽⁰⁾_H (t) ˆUI(−∞, t), where the superscript (0) on the observableA denotes that the evolution is due to ˆH0. Inserting Eq. (3.129) into the above equation, we get

AˆH(t) = ˆA⁽⁰⁾_H (t) + i

~ Z t

−∞

dt1

hVˆI(t1),Aˆ⁽⁰⁾_H (t)i +

µi

~

¶2Z t

−∞

dt1

Z t1

−∞

dt2

hVˆI(t2),h

VˆI(t1),Aˆ⁽⁰⁾_H (t)ii +· · ·+

µi

~

¶nZ t

−∞

dt1

Z t1

−∞

dt2· · · Z tn−1

−∞

dtn

hVˆI(tn),h

· · ·,h

VˆI(t2),h

VˆI(t1),Aˆ⁽⁰⁾_H (t)ii

· · ·ii

+· · ·. (3.130)

For a linear system subject to an external classical linear forceG(t), the interaction term is ˆVI(t) =

−xˆ⁽⁰⁾_H G(t). Plugging this expression into Eq. (3.130) and using Theorem 2, it is straightforward to deduce that the second and all higher-order terms in Eq. (3.130) vanish and the first order perturbation gives the exact solution. Hence, we obtain the following theorem:

Theorem 3 Consider a linear system subject to a classical generalized force G(t), whose Hamilto- nian is given by Hˆ = ˆH0−x G(t), whereˆ xˆ is a linear observable. Then, for any linear observable A, the Heisenberg operatorˆ AˆH(t) can be written as the sum of its free-evolution part, Aˆ⁽⁰⁾_H (t), plus a term which is due to the presence of the external force, i.e.,

AˆH(t) = ˆA⁽⁰⁾_H (t) + i

~ Z t

−∞

dt⁰CAx(t, t⁰)G(t⁰), (3.131)

whereCAx(t, t⁰)is a C-number, called the (time-domain) susceptibility, given explicitly by

CAx(t, t⁰)≡[ ˆA⁽⁰⁾_H (t),xˆ⁽⁰⁾_H (t⁰)]. (3.132)

Let us now suppose that we have two independent linear systems P (e.g., the probe) and D (e.g., the detector), which by definition are described by two different Hilbert spaces HP and HD. We introduce the Hilbert space H=HP ⊗ HD and define for any operator ˆxof the system P the corresponding operator acting onHas ˆx⊗ˆ1, while for any operator ˆF of the systemDwe introduce

the operator ˆ1⊗Fˆ which acts onH. Henceforth, we shall limit ourselves to interaction termsV, in the total Hamiltonian ˆH = ˆH_P + ˆH_D + ˆV, of the form: ˆV =−xˆ⊗Fˆ, with ˆx and ˆF acting on P andD, respectively. Using Eq. (3.130) with ˆVI(t) =−xˆ⁽⁰⁾_H (t) ˆF_H⁽⁰⁾(t), noticing that (i) the zeroth order Heisenberg operators of two observables living in different Hilbert spaces commute and (ii) the zeroth order Heisenberg operators of two linear observables living in the same Hilbert space have a C-number commutator, we derive the following theorem:

Theorem 4 Consider two independent linear systemsP andD, and two linear observables,xˆ ofP andFˆ ofD. Suppose that the two systems are coupled by a term −xˆ⊗Fˆ, i.e., the Hamiltonian of the composite systemP + DreadsHˆ = ˆH_P+ ˆH_D−xˆ⊗F .ˆ Then, for any linear observableAˆof the systemP andBˆ of the systemD, their full Heisenberg evolutions are given by

AˆH(t) = ˆA⁽⁰⁾_H (t) + i

~ Z t

−∞

dt⁰CAx(t, t⁰) ˆFH(t⁰), BˆH(t) = ˆB_H⁽⁰⁾(t) + i

~ Z t

−∞

dt⁰CBF(t, t⁰) ˆxH(t⁰), (3.133) whereAˆ⁽⁰⁾_H andBˆ_H⁽⁰⁾ stand for the free Heisenberg evolutions, and the susceptibilities are defined by

CAx(t, t⁰)≡[ ˆA⁽⁰⁾_H (t),xˆ⁽⁰⁾_H (t⁰)], CBF(t, t⁰)≡[ ˆB_H⁽⁰⁾(t),Fˆ_H⁽⁰⁾(t⁰)]. (3.134)

In the case where the zeroth order Hamiltonian is time independent, it is easy and convenient to express the above formalism in the Fourier domain. We first notice that for a time independent ˆH0, Uˆ0(t, t+τ) =e^{−iHˆ0τ /~}and for any two linear observables ˆA1and ˆA2 we haveCA1A2(t+τ, t⁰+τ) = CA1A2(t, t⁰), i.e.,CA1A2(t, t⁰) depends only ont−t⁰. Defining the Fourier transform of any observable A(t) asˆ

A(Ω)ˆ ≡ Z +∞

−∞

dt e^iΩtA(t)ˆ , (3.135)

Eq. (3.131) becomes ˆAH(Ω) = ˆA⁽⁰⁾_H (Ω) +RAx(Ω)G(Ω) while Eq. (3.133) can be recast in the form AˆH(Ω) = ˆA⁽⁰⁾_H (Ω) +RAx(Ω) ˆFH(Ω), BˆH(Ω) = ˆB_H⁽⁰⁾(Ω) +RBF(Ω) ˆxH(Ω), (3.136)

whereRAB(Ω) is the susceptibility in the Fourier-domain, given by

RAB(Ω) = i

~ Z +∞

−∞

dτ e^iΩτΘ(τ)CAB(0,−τ) = i

~ Z +∞

0

dτ e^iΩτCAB(0,−τ), (3.137) with Θ(τ) the step function. For future reference, let us point out two properties whichRAB(Ω) satisfies and that we use repeatedly in Sec. 3.2:

R^∗_AB(Ω) =RAB(−Ω), h

Aˆ⁽⁰⁾_H (Ω1),Bˆ_H⁽⁰⁾(Ω2)i

=−2πi~δ(Ω1+ Ω2) [RAB(Ω1)−RBA(Ω2)]. (3.138)

To deduce the first identity in Eq. (3.138), we consider the complex (Hermitian) conjugate of Eq. (3.137) and use the Hermiticy of ˆA⁽⁰⁾_H (t) and ˆB⁽⁰⁾_H (t). For the second identity in Eq. (3.138), we take the double Fourier transform of [ ˆA⁽⁰⁾_H (t1),Bˆ_H⁽⁰⁾(t2)] with respect to t1 and t2, and then using Eq. (3.137) we find that the region corresponding tot1 > t2 in the double integral yields the RAB

term of Eq. (3.138), while the region corresponding tot1< t2 gives theRBAterm.