4.4 Non-Markovian quantum measurement: the first scenario
4.4.3 Generalization to bath with many degrees of freedom
In the previous two sections, we used two specific examples to illustrate non-Markovian measurement in the first scenario—due to non-Markovian dynamics between the system and bath, a measurement on the bath gives rise to non-Markovian stochastic (conditional) evolution of the system quantum state. In those examples, the cavity mode serves as the bath, and it only has one degree of freedom.
In this section, we will generalize the result to the case with a many-degrees-of-freedom bath which is linearly coupled to the probe field. We consider the following Hamiltonian for such a case:
Hˆ = ˆHs+X
k
~ωkaˆ†kˆak+X
k
~gk( ˆLˆa†k+ ˆL†ˆak) +X
k
~√
γk[ˆakˆo†(t) + ˆa†ko(t)].ˆ (4.32)
Here ˆHs is the free part of the system Hamiltonian; ˆL is an arbitrary operator of the system; gk
is the coupling strength between the system and the bath which consists of different modes with frequency ωk; γk is the coupling strength between the bath and the probe field ˆo. Note that this Hamiltonian is still quite special. It excludes other possibilities: (i) some of the bath modes might not be coupled to the probe field, in which case we cannot access the information of those modes and they will contribute to decoherence of the system; (ii) different modes might be coupled to different probe fields, in which case we have multiple measurement output channels. In principle, we can include those complicated cases, but right now, we will just focus on this specific Hamiltonian, and postpone the discussion of more general cases until later.
The derivation of non-Markovian SME for the system is almost parallel to the single-degree- of-freedom case, and we can formally write down the reduced stochastic master equation for the system:
d ˆρs=−i
~
[ ˆHs,ρˆs] dt−X
k
gk( ˆL†Oˆsρˆs+ ˆρsOˆ†kL−ˆ OˆkρˆsLˆ†−LˆˆρsOˆk†)dt+X
k
p2γk[ ˆOkρˆs+ ˆρsOˆ†k−hOˆk+ ˆOk†iρˆs]dW.
(4.33) The operator ˆOk is defined through the following ansatz:
Mα~[∂α∗kρˆs(t, ~α)] =−iOˆk(t) ˆρs(t). (4.34) Generally, it is quite difficult to find a closed form for ˆOk in terms of the system operator. Only in the following two cases, can we obtain relatively a simple expression for ˆOk: (i) ˆLis a linear function of the canonical variables of the system—linear coupling case; (ii) gk/γk < 1—the weak coupling case, which allows us to find a perturbative solution. We will discuss these two cases in more detail.
Linear coupling.— We use the atom as an example with ˆHs=~ω2qσˆz, and choose ˆL= ˆσ−. As shown in the Appendix 4.E, we can obtain
Oˆk(t) =X
k0
[ei(ωqI−M)t]kk0fk0(t)ˆσ−, (4.35)
with
f˙k(t) =X
k0
gk0[e−i(ωqI−M)t]kk0 −fk(t)X
k0k00
gk0[ei(ωqI−M)t]kk0fk00(t) (4.36)
and initial conditionfk(0) = 0. Here we have introduced identity matrixIand matrixM:
M≡
ω1−iγ1 −i√
γ1γ2 · · ·
−i√
γ1γ2 ω2−iγ2 ... ... · · · . ..
. (4.37)
This recovers the result in subsection 4.4.1 when the bath has only one mode.
Similarly, if the system is a harmonic oscillator and ˆL= ˆbis the annihilation operator, we will end with the same expression, expect thatωq is replaced by the oscillator frequency and ˆσ− is replaced by ˆb.
Weak coupling— When the operator ˆLis a nonlinear function of the system canonical variables, there will be no straightforward route to derive ˆOk. If the coupling is weak, we can use the perturbation method and expand the solution in series of coupling strength. As shown in Appendix 4.F, we obtain
Oˆk(t) =X
k0
Z t 0
dτ[e−iMτ]kk0gk0L(−τ) ˆˆ ρs(t) +O[gk2], (4.38)
where ˆL(−τ) = e−iHˆsτ /~L eˆ iHˆsτ /~ under free evolution and the matrix elementMkk0 = ωkδkk0− i√
γkγk0 with δkk0 being the Kronecker delta. Basically, ˆ%k is equal to ˆLρˆs convoluted with the Green’s function of the bath. In other words, we are effectively coupled to a dynamical quantity of the system that is shaped by the bath—a quantum filter. One can therefore engineer the bath to measure desired observables of the system, e.g., a QND observable, as illustrated in the following two examples.
The first example is measuring mechanical energy quantization considered in Refs. [16–19], aiming at unequivocally demonstrating the quantumness of a macroscopic mechanical oscillator. In the proposed experiment, the position of a mechanical oscillator is quadratically coupled to a cavity mode, namely,
Hˆint=~gxˆ2(ˆa+ ˆa†). (4.39) If the cavity bandwidth γ is less than the mechanical frequency ωm, we expect a direct probe of the slowly-varying part of ˆx2which is proportional to the QND variable (energy or equivalently the phonon number ˆN). Indeed, from ˆx(−τ) = ˆxcosωmτ−pˆsinωmτ,
Oˆ =g Z t
0
dτ e−γτxˆ2(−τ) ˆρ≈ g γ
Nˆρ,ˆ (4.40)
where we have ignored terms proportional toe−γt, as the characteristic measurement time scale is
γ−1. The leading-order SME for the oscillator reads:
d ˆρ=−i[ωmN ,ˆ ρ] dtˆ −geff[ ˆX2, [ ˆN ,ρ]]dtˆ +p
2geff[{N ,ˆ ρ} −ˆ 2hNiˆˆ ρ]dW (4.41) with geff =g2/γ. Note that such a measurement is not an exact QND measurement, because we have [ ˆX2,[ ˆN ,ρ]] instead of the usual Lindblad term [ ˆˆ N ,[ ˆN ,ρ]]. This term describes a two-phononˆ process that induces quantum jumps. However, after numerically solving this SME, we find that it does not have significant effects, and a QND measurement can indeed be effectively realized. This is in accord with the argument by Martin and Zurek [16]—the two-photon process happens at 2ωm, which is strongly suppressed due to a small cavity bandwidthγ.
The second example is measuring the QND observable of a free mass—the momentum ˆp. This is of particular interest in quantum-limited force measurement with mechanical probes, e.g., detecting gravitational waves [20]. By monitoring the momentum change, one can detect the force signal without quantum back action, enabling surpassing of the Standard Quantum Limit (SQL) [21]. To achieve this, we can couple the position ˆx of the free mass with two coupled cavity modes ˆa1 and ˆ
a2, of which the interaction Hamiltonian is given by:
Hˆint=~ωs(ˆa1ˆa†2+ ˆa†1ˆa2) +~gx(ˆˆ a1+ ˆa†1), (4.42)
where ωs is the coupling constant between two cavity modes. The cavity mode ˆa1 is coupled to external probe field. From Eq. (4.8), we derive that:
Oˆ= 2g Z t
0
dτ e−γτcosωsτ 2
x(−τˆ ) ˆρ≈ 4g
ωs2x(0) ˆ˙ˆ ρ, (4.43) where we have used the stationary-phase approximation by assumingωsγ, and also ignored terms proportional toe−γt. The effective observable is therefore equal to the momentum, as ˆp=mx(0).˙ˆ Indeed, such a coupled-cavity scheme has been proposed as the so-called “speed-meter” for advanced gravitational-wave detectors [22].