Conclusion

To conclude, we have studied two important scenarios that give rise to an effective non-Markovian quantum measurement of the system: (i) a direct continuous measurement of the bath which couples to the system with non-Markovian dynamics; (ii) a direct measurement of the system with correlated probe field. For treating the first scenario, we used Strunz’s method to eliminate the bath degrees of freedom to derive an effective non-Markovian stochastic master equation for the system. We explicitly worked out two interesting examples in cavity QED and optomechanical devices. In addition, we showed the perturbation approach for treating more general nonlinear system-bath

interaction. For the second scenario, we consider both a Gaussian and a non-Gaussian correlated input field. By taking advantage of linear dynamics and converting the influence of measurement in terms of a path integral, we can study non-local correlation of the input field, and derive the conditional Wigner function for the system. It seems to us that there is no transparent way to rewrite it in terms of a solution to a differential equation or Wigner transformation of some master equation, in contrast to the Markovian case. By studying these two scenarios, we can gain a better insight into non-Markovian quantum measurement for more general cases.

Acknowledgement

We thank B.L. Hu and T. Yu for introducing us to this research direction and further discussions on technical details. We thank S.L. Danilishin and F.Ya, Kahlili for fruitful discussions. This work is supported by NSF grants PHY-0555406, PHY-0653653, PHY-0601459, PHY-0956189, PHY- 1068881, as well as the David and Barbara Groce startup fund at Caltech.

4.A Linear continuous quantum measurement

Here we will show how the condition in Eq. (4.1) imposes the requirement that the probe should be a field in a linear continuous quantum measurement. This is adapted from Ref. [9] with small modifications to fit into the context of quantum measurement.

In a linear measurement, we couple the system and the probe linearly. The total Hamiltonian of the system and the probe can be then written as

Hˆ_tot = ˆH_s+ ˆH_p+~gLˆ⊗F .ˆ (4.85)

Here ˆHs and ˆHp are the free Hamiltonians for the system and probe; ˆLand ˆF are some arbitrary operators for the system and probe, respectively;g is the coupling constant.

In order to obtain information about L, the output needs be another operator of the probe, e.g., we denote it by ˆZ and it does not commute with ˆF; otherwise, ˆZ will simply undergo free evolution and contain no information about ˆL. We can then study the dynamics of the output from the Heisenberg equation of motion, and in the interaction picture, it is given by

Z˙ˆI(t) = i

~[ ˆZI(t),HˆI(t)]. (4.86)

The solution reads

ZˆI(t) = ˆZ(0) +i g Z t

0

dt⁰[ ˆZI(t),FˆI(t⁰)] ˆLI(t⁰). (4.87)

Similarly for ˆF_I(t) and ˆL_I(t), we have

FˆI(t) = ˆF(0) +i g Z t

0

dt⁰[ ˆFI(t),FˆI(t⁰)] ˆLI(t⁰), (4.88) Lˆ_I(t) = ˆL(0) +i g

Z t 0

dt⁰[ ˆL_I(t),Lˆ_I(t⁰)] ˆF_I(t⁰). (4.89)

Due to the interaction term being bilinear— ˆL⊗Fˆ, we have

[ ˆZ_I(t), Fˆ_I(t⁰)]≡C_ZF(t, t⁰), [ ˆF_I(t),Fˆ_I(t⁰)]≡C_{F F}(t, t⁰), [ ˆL_I(t),Lˆ_I(t⁰)]≡C_LL(t, t⁰), (4.90)

which are all c-numbers. Eqs. (4.87), (4.88), (4.89) are therefore a set of linear equations, and can be easily solved.

By applying a linear causal filter K(t) to the output ˆZ_I(t), we can introduce a ˆZ(t) that is directly equal to ˆL(t) plus additional noise terms, and

Z(t) =ˆ Z t

0

dt⁰K(t−t⁰) ˆZI(t) =g⁻¹L(t) + ˆˆ Z0(t) +i g Z t

0

dt⁰CLL(t, t⁰) ˆF0(t⁰) (4.91) where

CLL(t, t⁰)≡[ ˆL0(t),Lˆ0(t⁰)] = [e^iHˆ0t/~Leˆ ^−iHˆ0t/~, e^iHˆ0t0/~Leˆ ^−iHˆ0t0/~] (4.92) is the response function. In generalC_LL(t, t⁰) is non zero, one example being the harmonic oscillator andCxx(t, t⁰) = [ˆx(t),x(tˆ ⁰)] = (mωm)⁻¹sinωm(t−t⁰) withωmbeing the eigenfrequency.

Since ˆZ(t) commutes at different times, ˆZ(t) also commutes at different times:

0 = [ ˆZ(t),Z(tˆ ⁰)] =CLL(t, t⁰) + [ ˆZ(t),Z(tˆ ⁰)] +i g (Z t⁰

0

dτ CLL(t⁰, τ)[ ˆZ(t),F(τ)] +ˆ Z t

0

dτ CLL(t, τ)[ ˆF(τ),Z(tˆ ⁰)]

)

−g² Z t

0

Z t⁰ 0

dτ dτ⁰CLL(t, τ)CLL(t⁰, τ⁰)[ ˆF(τ), F(τˆ ⁰)]. (4.93) As those terms on the left-hand side are different orders of some arbitrary coupling constantg, in order to for this to be satisfied, we therefore require

[ ˆZ(t), Z(tˆ ⁰)] = 0, [ ˆF(t),Fˆ(t⁰)] = 0, [ ˆZ(t),Fˆ(t⁰)] =i δ(t−t⁰), (4.94)

which has been proved in a more rigorous way in Ref [9]. This basically means that we can treat ˆZ(t) at different times as different degrees of freedom (a continuous field); ˆZand ˆFare the corresponding canonical conjugate variables.

4.B SME for measurement of cavity mode

Here we add the details for deriving SME as shown in Eq. (4.8). We start from Eq. (4.9):

ˆ

ρ_a(t+ dt) = 1 2πP[y(t)]

Z

dξ e^−iξy(t)Tr_o

e^iξˆo1(t)

ˆ ρ(t)− i

~

[ ˆH,ρ(t)]dtˆ − 1 2~²

[ ˆH_int, [ ˆH_int,ρ(t)]]dtˆ ²

+O[dt²]. (4.95)

By plugging in the expression for ˆH and ˆHint, we have Tro

e^iξˆo1ρˆ =he^iξˆo1iˆρa, (4.96)

Tr_on

e^iξˆo1[ ˆH_int,ρ]ˆo

=~

√γ(he^iξˆo1oˆ^†iˆaˆρ_a+he^iξˆo1oiˆˆa^†ρˆ_o− hˆo^†e^iξˆo1iˆρ_aaˆ− hˆo e^iξˆo1(t)iρˆˆa^†), (4.97)

and Tro

n

e^iξˆo1[ ˆHint,[ ˆHint,ρ]]ˆ o

=~²γ(he^iξˆo1oˆ^†2iˆa²+he^iξˆo1oˆ^†oiˆˆaˆa^†+he^iξˆo1oˆ^†oiˆˆaˆa^†+he^iξˆo1ˆoˆo^†iˆa^†ˆa+he^iξˆo1ˆo²iˆa^†2) ˆρ_a (4.98) +~²γρˆa(he^iξˆo1oˆ^†2iˆa²+he^iξˆo1ˆo^†oiˆˆaˆa^†+he^iξˆo1ˆo^†oiˆˆaˆa^†+he^iξˆo1ˆoˆo^†iˆa^†aˆ+he^iξˆo1oˆ²iˆa^†2)

+ 2~²γ(hˆo^†e^iξˆo1oˆ^†iˆaρˆ_aaˆ+hˆo e^iξˆo1oˆ^†iˆaˆρ_aˆa^†+hˆo^†e^iξˆo1ˆoiˆa^†ρˆ_aˆa+hˆo e^iξˆo1oiˆˆa^†ρˆ_aˆa^†) (4.99)

where the averagehAi ≡ˆ Tro[ ˆAρo(t)] is over the continuous optical field. Since we assume that the optical field is in a vacuum state⁶, we havehˆo²₁i=hˆo²₂i= _2dt^{1 7} and those averages over the optical field can be easily worked out explicitly. Specifically,

he^iξˆo1i=e^−ξ

2

4dt, he^iξˆo1oiˆ = 0, he^iξˆo1oˆ^†i= iξ

√ 2 dte^−ξ

2 4dt,

he^iξˆo1oˆ²i= 0, he^iξˆo1oˆ^†2i=− ξ² 2dt²e^−ξ

2

4dt2, he^iξˆo1oˆoˆ^†i= 1 2dte^−ξ

2 4dt2,

he^iξˆo1oˆ^†ˆoi= 0, hˆo e^iξˆo1ˆo^†i= 1

dt − ξ² 2dt

e^−ξ

2

4dt2, hˆo^†e^iξˆo1oiˆ = ξ² 4dt²e^−ξ

2 4dt.

All the other terms, e.g. hˆoe^iξˆo1i, can be obtained by taking the complex conjugate ofhe^iξˆo1oˆ^†iand settingξ→ −ξ, so we will not list all of them. We then need to integrate overξ, and we have

Z

dξe^−iξy−ξ

2

4dtξ=−4iydt√

πdt e^−y2dt, (4.100)

Z

dξe^−iξy−ξ

2

4dtξ²= 4(dt−2y²dt²)√

πdt e^−y2dt. (4.101)

6In reality, it is a coherent state, but we are only interested in fluctuation around the steady-state amplitude.

7We have used the fact thatδ(0)dt= 1

By taking the trace of Eq. (4.95) and using the fact that Tr_a[ ˆρ_a(t)] = 1, we obtain, up to the order of dt,

P[y(t)] = rdt

π e^−y2dt(1−2i√

γhˆa+ ˆa^†iydt)≈ rdt

π e^−(y+i

√γhˆa−ˆa^†i)²dt. (4.102)

Therefore, the measurement result can be viewed as a classical random process, as written in Eq.

(4.9):

y(t)dt=−i√

γhˆa−ˆa^†idt+ dW /√

2 (4.103)

where dW is the Wiener increment and dW² = dt. After replacing y(t) inthe above formula by Eq. (4.95), it turns out all the counter rotating terms vanishe because the right-hand-side of Eq.

(4.101) vanishes to the first order of dt, and finally we get the stochastic master equation for the cavity mode, shown in Eq. (4.8):

d ˆρ_a(t) =−i[∆ˆa^†ˆa,ρˆ_a(t)]dt−γ[ˆa^†ˆaˆρ_a(t) + ˆρ_a(t)ˆa^†aˆ−2ˆaˆρ_a(t)ˆa^†]dt

−ip

2γ[ˆaˆρa(t)−ρˆa(t)ˆa^†− hˆa−ˆa^†iˆρa(t)]dW. (4.104)

4.C Operator O ˆ in the linear coupling case

To derive the operator ˆO, it is most convenient to use the stochastic Schr¨odinger equation counterpart of Eq. (4.12) and derive directly∂α^∗|ψ(α^∗)i. The stochastic Schr¨odinger equation reads

d|ψi=−i

~

( ˆH_x+ ˆH_a+ ˆH_int)|ψidt−γ(ˆa^†aˆ−2hˆa^†iˆa+hˆa^†ihˆai)|ψidt−ip

2γ(ˆa− hˆai)|ψidW. (4.105)

We move into the interaction picture of ˆHx+ ˆHa+i~γˆa^†a, and define the evolution operatorˆ U(t) = expˆ

−i

~

( ˆHx+ ˆHa−i~γˆa^†ˆa)t

. (4.106)

The wave function in the Schr¨odinger and the interaction picture are related by

|ψi= ˆU(t)|ψiI. (4.107)

In contrast to the usual interaction picture, ˆU is not unitary and the operator transforms as ˆaI(t) = Uˆ⁻¹(t)ˆaUˆ(t),instead of ˆaI = ˆU^†ˆaUˆ. In the interaction picture, we have

d|ψiI =−ig( ˆLIˆa^†_I+ ˆL^†_IˆaI) +γ(2hˆa^†iˆaI− hˆa^†ihˆai)|ψidt−ip

2γ(ˆaI − hˆai)|ψidW. (4.108)

In the coherence basis of the cavity mode, it can be rewritten as d|ψ(α^∗)iI =−ig( ˆLIe^{i(∆−iγ)t}α^∗+ ˆL^†_Ie^{−i(∆−iγ)t}∂α^∗)|ψ(α^∗)iIdt

+γ[2e^{−i(∆−iγ)t}hˆa^†i∂α^∗− hˆa^†ihˆai]|ψ(α^∗)iIdt (4.109)

−ip

2γ[e^{−i(∆−iγ)t}∂_α^∗− hˆai]|ψ(α^∗)iIdW ≡Hˆeff(α^∗)|ψ(α^∗)iIdt. (4.110)

We use the following Ansatz:

∂α^∗|ψ(α^∗)iI ≡ −iOˆI(t, α^∗)|ψ(α^∗)iI. (4.111) By comparing the resulting equation, we find out that the Schr¨odinger picture operator ˆO(t, α^∗), which is defined in Eq. (4.17), and the interaction picture operator ˆO_I(t, α^∗) are related by

O(t, αˆ ^∗) =e^{−i(∆−iγ)t}e^−~iHˆxtOˆI(t, α^∗)e^~iHˆxt. (4.112)

To derive ˆO, we use the consistent condition [2]:

d

dt[∂_α^∗|ψ(α^∗)i_I] =∂_α^∗ d

dt|ψ(α^∗)i_I

, (4.113)

which gives d dt

OˆI(t, α^∗) =i ∂α^∗Hˆeff(α^∗) + [ ˆHeff(α^∗),OˆI(t, α^∗)]

=gLˆIe^{i(∆−iγ)t}+p

2γe^{−i(∆−iγ)t}(hˆa^†i −ip

2γdW)∂α^∗O(t, αˆ ^∗)

−ig[ ˆL_Ie^{i(∆−iγ)t}α^∗−iLˆ^†_Ie^{−i(∆−iγ)t}Oˆ_I(t, α^∗),Oˆ_I(t, α^∗)]. (4.114)

For the case considered in the main text, ˆL= ˆσ− and ˆσ^I₋= ˆσ e^−iωqt. We assume that

OˆI(t, α^∗) =f(t)ˆσ₋≡Oˆ0(t), (4.115)

which is independent of α^∗. A more systematic approach is to expand ˆO_I(t, α^∗) in terms ofα^∗ as in Ref. [33]. From the consistent condition, we obtain

f˙(t) =g e^{−i(ωq−∆+iγ)t}+g e^{i(ωm−∆+iγ)t}f²(t). (4.116)

with initial conditionf(0) = 0. Finally, by using Eq. (4.112), one can easily find out that

Oˆ0(t) =e^{i(ωq−∆+iγ)t}f(t)ˆσ₋. (4.117)

4.D Operator O ˆ for optomechanical interaction

Here we show the details for deriving the ˆO operator for the optomechanical device considered in Section 11.3. The general procedure is similar the atom-cavity case, and the only complexity arises due to the dependence of ˆO onα^∗, but only to the linear order.

We again use the stochastic Schr¨odiner equation in the interaction picture of~(ω0−iγ)ˆa^†ˆawhich is

d|ψ(α^∗)iI =−i

~ pˆ²

2m+1 2mω_m²xˆ²

|ψ(α^∗)iIdt−ig[e^{i(ω0−iγ)t}α^∗+e^{−i(ω0−iγt)}∂_α^∗]ˆx|ψ(α^∗)iIdt +γ[2e^{−i(∆−iγ)t}hˆa^†i∂α^∗− hˆa^†ihˆai]|ψ(α^∗)iIdt−ip

2γ[e^{−i(∆−iγ)t}∂_α^∗− hˆai]|ψ(α^∗)iIdW.

(4.118) The ˆOoperator is similar to that which was obtained in Ref. [2] and it is a linear function of ˆx,pˆ andα^∗. We use the following ansatz and derive those functions by using the consistent condition:

∂α^∗|ψ(α^∗)iI =−i[f0(t) +fx(t)ˆx+fp(t)ˆp+f1(t)α^∗]|ψ(α^∗)iI ≡ −iOˆI(t, α)|ψ(α^∗)iI. (4.119) The Schrodinger picture operator ˆO(t, α^∗) and the interaction picture operator ˆOI(t, α^∗) here are related by

O(t, αˆ ^∗) =e^{−i(∆−iγ)t}[f0(t) +fx(t)ˆx+fp(t)ˆp+e^{−i(∆−iγ)t}f1(t)α^∗]≡Oˆ0(t) +O1(t)α^∗, (4.120)

in which we define operator ˆO₀and functionO₁by using the notation similar to Ref. [33]—subscripts 0 and 1 indicate the power dependence ofα. From the consistent condition, those functionsf satisfy the following equations

f˙0(t)dt=e^{−i(∆−iγ)t}[2γhˆa^†if1(t)dt−ip

2γ f1(t)dW−i g f0(t)fp(t)dt], (4.121) f˙x(t) =ωmfp(t) +g e^{i(∆−iγ)t}−i g e^{−i(∆−iγ)t}[f1(t) +fx(t)fp(t)], (4.122) f˙_p(t) =−ω_mf_x(t)−i g f_p²(t)e^{−i(∆−iγ)t}, (4.123) f˙1(t) =g f2(t)e^{i(∆−iγ)t}−i g f2(t)f3(t)e^{−i(∆−iγ)t} (4.124)

with null initial condition.

To derive the SME, we need to average over α. By using the fact that Mα[α^∗ρˆx(α^∗, α)] = Mα[∂αρˆx(α^∗, α)] = ˆρO†, we obtainˆ

Oˆρˆ= ˆO0ρˆ+O1ρˆOˆ^†, (4.125) ˆ

ρOˆ^† = ˆρOˆ^†₀+O^∗₁Oˆ^†₀ρ.ˆ (4.126)

We can then obtain Eqs. (4.27) and (4.28), namely

Oˆρˆx= [O1ρˆxOˆ₀^†+ ˆO0ρˆx]/[1− |O1|²], (4.127) ˆ

ρxOˆ^†= [O^∗₁Oˆ0ρˆx+ ˆρxOˆ₀^†]/[1− |O1|²]. (4.128)

4.E Operator O ˆ

_k

in the linear coupling case

Here we derive the ˆOkin the linear coupling case. The corresponding stochastic Schr¨odinger equation reads

d|ψi=−i

~( ˆHx+ ˆHa)|ψidt−iX

k

gk( ˆLˆa^†_k+ ˆL^†ˆak)|ψidt

−X

kk⁰

√γkγk⁰(ˆa^†_kˆak⁰−2hˆa^†_kiˆak⁰+hˆa^†_kihˆak⁰i)|ψidt+X p

2γk(ˆak− hˆaki)|ψidW. (4.129)

Similar to the single-mode case, we move into the interaction picture of ˆHx+ ˆHa+i~P

kk⁰

√γkγk⁰ˆa^†_kˆak⁰, and define the evolution operator

Uˆ(t) = exp

"

−i

~

Hˆx+ ˆHa+i~ X

kk⁰

√γkγk⁰ˆa^†_kˆak⁰

! t

#

. (4.130)

The wave function and the operator in the Schr¨odinger and the interaction picture are related by

|ψi= ˆU|ψiI, ˆoI = ˆU⁻¹oˆU .ˆ (4.131)

Specifically, the annihilation operator for the bath ˆak transforms as Uˆ⁻¹(t)ˆa_kUˆ(t) =X

k⁰

e^−iMkk0tˆa_k⁰, Uˆ⁻¹(t)ˆa^†_kUˆ(t) =X

k⁰

e^iMkk0taˆ^†_k0 (4.132)

where the dynamical matrix M is defined in Eq. (5.9). By using the coherent state basis for the bath, the stochastic master equation in the interaction picture reads

d|ψ(~α^∗)i_I =−iX

kk⁰

g_k

Lˆ_Ie^iMkk0tα^∗_k0+ ˆL^†_Ie^−iMkk0t∂_α^∗

k0

|ψ(α~^∗)i_Idt+ X

kk⁰k⁰⁰

√γ_kγ_k⁰(2hˆa^†_kie^−iMk0k00t∂_α^∗

k00

− hˆa^†_kihˆa_k0i)|ψ(~α^∗)iIdt+X

kk⁰

p2γ_k(e^−iMkk0t∂_α^∗

k0− hˆa_ki)|ψ(α~^∗)iIdW ≡ Heff(α~^∗)|ψ(~α^∗)iI. (4.133) Similarly, we use the following ansatz

∂α^∗_k|ψ(~α^∗)iI =−iOˆ⁰_Ik(t, ~α^∗)|ψ(~α^∗)iI. (4.134)

It is related to the Schr¨odinger picture operator ˆO_k⁰(t, α^∗) by Oˆ_k⁰(t, ~α^∗) =X

k⁰

e^−iMkk0te^−i~HˆxtOˆ_Ik⁰ 0(t, ~α^∗)e^i~Hˆxt. (4.135)

From the consistent condition:

d

dt[∂α^∗_k|ψ(~α^∗)iI] =∂α^∗_k

d

dt|ψ(~α^∗)iI

, (4.136)

we get d dt

Oˆ⁰_Ik(t, α^∗) =i ∂_α^∗

k

Hˆ_eff(α~^∗) + [ ˆH_eff(~α^∗),Oˆ⁰_Ik(t, ~α^∗)]

=X

k⁰

g_k⁰Lˆ_Ie^iMkk0t−iX

k⁰k⁰⁰

g_k⁰h

Lˆ_Ie^iMk0k00tα^∗_k00−iLˆ^†_Ie^−iMk0k00tOˆ⁰_Ik00(t, ~α^∗),Oˆ⁰_Ik(t, ~α^∗)i .

(4.137) For the simple case considered in the main text, ˆL= ˆb, it is straightforward to get

Oˆ_Ik⁰ (t, ~α) =fk(t)ˆb (4.138)

with

f˙k(t) =X

k⁰

gk⁰e^{−i(ωm−Mkk0)t}−fk(t)X

k⁰k⁰⁰

gk⁰e^{i(ωmt−Mk0k00)t}fk⁰⁰(t). (4.139) By using Eq. (4.135), the Schr¨odinger picture operator is given by

Oˆk(t) =X

k⁰

e^{i(ωm−Mkk0)t}fk⁰(t)ˆb. (4.140)

This gives rise to Eq. (4.117) in the single-mode case.

4.F Perturbative solution of O ˆ

_k

In this section, we will try to derive ∂α^∗_kρˆx and hˆaki in the case of the many-degrees-of-freedom bath. To illustrate the procedure, we first consider the case of the single-degree-of-freedom bath.

The corresponding stochastic Schr¨odinger equation in the coherence basis of the cavity mode reads d|ψ(α^∗)i=− i

~

( ˆH_x+ ˆH_a)|ψ(α^∗)idt−ig( ˆLα^∗+ ˆL^†∂_α^∗)|ψ(α^∗)idt

−γ(α^∗∂_α^∗−2hˆa^†i∂_α^∗+hˆa^†ihˆai)|ψ(α^∗)idt +p

2γ(∂α^∗− hˆai)|ψ(α^∗)idW. (4.141)

We do not use the interaction picture, as it is more convenient to study in the Schr¨odinger picture for this case.

In order to seek a perturbative solution to ∂α^∗|ψ(α^∗)i, we need to find a small dimensionless parameter. We notice that the characteristic memory time scale for the oscillator interacting with the cavity mode is γ⁻¹, after which the cavity mode is refreshed by the external field. Since the coupling strength between the oscillator and the cavity mode isg, the small dimensionless parameter is≡g/γ. In addition, we know thatg enters|ψ(α^∗)ias ofgα^∗(from the interaction term), we can make a Taylor expansion of|ψ(α^∗)ias α, which is equivalent to expansion in terms of⁸, i.e.,

|ψ(α^∗)i=|ψi(⁰) +α^∗∂α^∗|ψi(¹) +1

2α^∗2∂_α²∗|ψi(²) +· · ·. (4.142) By taking the partial derivative of Eq. (4.146), up to the first order ofg, we have

d(∂α^∗|ψi) =−i

~( ˆHx+~∆−i~γ)(∂α^∗|ψi)dt−igL|ψidt,ˆ (4.143) and we obtain

(∂_α^∗|ψ(t)i) = (∂_α^∗|ψ(0)i)−ig Z t

0

dt⁰e−i(∆−iγ)(t−t⁰)e^{−iHˆx(t−t0)/~}L|ψ(tˆ ⁰)i

=−ig Z t

0

dt⁰e−i(∆−iγ)(t−t⁰)L(tˆ ⁰−t)|ψ(t)i (4.144)

with ˆL(t⁰−t)≡e^{−iHˆx(t−t0)/~}Leˆ ^{iHˆx(t−t0)/~}. We assume that the oscillator and the cavity mode are separable initially, and the cavity mode is in a vacuum state, in which case|ψ(0)iis independent of α^∗and ∂α^∗|ψ(0)i= 0. Up to the first order of the interaction strength, we get

O(t) =ˆ g Z t

0

dt⁰e−i(∆−iγ)(t−t⁰)L(tˆ ⁰−t) +O[g²]. (4.145) Now, we can move on the the case of the bath with many degrees of freedom. The corresponding stochastic Schr¨odinger equation in the coherent state basis is

d|ψ(α^∗)i=− i

~

( ˆH_x+ ˆH_a)|ψ(α^∗)idt−iX

k

g_k( ˆLα^∗_k+ ˆL^†∂_α^∗

k)|ψ(α^∗)idt−X

kk⁰

√γ_kγ_k⁰(α^∗_k∂_α^∗

k0 −2hˆa^†_ki∂α^∗

k0

+hˆa^†_kihˆak⁰i)|ψ(α^∗)idt+X

k

p2γk(∂α^∗_k− hˆaki)|ψ(α^∗)idW. (4.146)

Up to the first order of interaction strengthg_k, we have

d(∂α^∗_k|ψi) =−i

~( ˆHx+~ωk)(∂α^∗_k|ψi)dt−√ γk

X

k⁰

√γk⁰(∂α^∗

k0|ψi)dt−igkL|ψidtˆ +O[g_k²]. (4.147)

8This is identical to Ting’s expansion of ˆOin terms of the random process [33].

It is a linearly coupled equation for∂_α^∗

k|ψi. The solution can be formally written as

∂_α~^∗|ψ(t)i=−i Z t

0

dt⁰exp

−i

~

( ˆH_xI+~M)(t−t⁰)

~

gL|ψ(tˆ ⁰)i=−i Z t

0

dt⁰e^{−iM(t−t0)}~gL(tˆ ⁰−t)|ψ(t)i, (4.148) where matrixMis defined in Eq. (5.9). We therefore obtain

O(t) =ˆ X

k⁰

Z t 0

dt⁰e^{−iMkk0(t−t0)}g_kL(tˆ ⁰−t) ˆρ_x+O[g_k²]. (4.149)

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Chapter 5

Revealing non-Markovianity of open quantum systems via local operations

Non-Markovianity, as an important feature of general open quantum systems, is usually difficult to quantify with limited knowledge of how the plant, which we are interested in, interacts with its environment, the bath. It often happens that the reduced dynamics of the plant attached to a non-Markovian bath becomes indistinguishable from the one with a Markovian bath, if we let the plant-bath system freely evolve. Here we show that non- Markovianity can be revealed via applying local unitary operations on the plant—they will influence the plant dynamics at later times due to memory of the bath. This not only allows us to show non-Markovianity in those systems that are perviously considered as being Markovian, but also sheds light on protecting and recovering quantum coher- ence in non-Markovian systems, which will be useful for quantum-information processing.

Based on preprint by H. Yang, H. Miao, and Y. Chen, arXiv:1111.6079

5.1 Introduction

Recently, there have been many important theoretical and experimental studies of non-Markovian open quantum systems in the literature [1–5]. This is largely motivated by the quest for quantum information processing protocols that are robust under decoherence, as the basic information stor- ing unit—the qubit—often interacts with a non-Markovian environment with which the noises at different times are correlated. If we can arbitrarily control the plant, e.g., the atomic spin, the non-Markovianity of the bath, in principle, allows us to completely decouple the spin from the envi- ronment, which is known as dynamical decoupling [6–8]. A good understanding and quantification of non-Markovian dynamics in open quantum systems can therefore lead to novel designs of quantum

devices that are less susceptible to environmentally-induced decoherence.

To quantify non-Markovianity, Breueret al. proposed a measure based upon the evolution of the trace distance Tr|ˆρ1(t)−ρˆ2(t)|/2 between two different initial quantum states of the plant ˆρ1(0) and ˆ

ρ2(0) [3]. An increase in the trace distance gives a unequivocal signature of non-Markovianity, as it indicates that information flows from the environment back to the plant. There is another measure proposed by Rivaset al.[4]. The authors introduced an ancilla to entangle but not interact with the plant, whereas the plant is still interacting with the bath—an increase in the entanglement between the plant and the ancilla during evolution signifies the existence of non-Markovian dynamics between the plant and bath. Both measures have been compared theoretically [9, 10] and also tested in a recent novel experiment by Liuet al.[5].

These two measures focus on the reduced dynamics of the plant and do not provide details on how the bath and plant interact with each other. As the reduced dynamics contain limited knowledge of the bath and also critically depend on the initial state, it often happens that the plant dynamics is highly degenerate among different non-Markovian systems. In other words, the plant can effectively behave in the same way even when it is attached to vastly different baths—e.g., one a Markovian bath and the other a non-Markovian bath. In this letter, we propose a new criterion for non-Markivianity by exploring the memory effect in non-Markovian dynamics: the dynamics is non-Markovian if a local unitary operation on the plant at a given moment can induce non-local influence on the plant dynamics at later times. This allows us to reveal non-Markovianity in systems in which the reduced dynamics of the plant appear to be Markovian before applying local operations—e.g., it satisfies the time-local master equation with effective damping ratesγ_i(t)>0:

˙ˆ

ρp(t) =−(i/~)[ ˆHp(t),ρˆp(t)] +P

iγi(t) ˆLiρˆp(t), (5.1) where Lindblad terms ˆLiρˆp= 2 ˆAiρˆpAˆ^†_i − {Aˆ^†_iAˆi,ρˆp}with ˆAibeing plant operators [11, 12].

Figure 5.1: (Color online.) A schematic showing how the reduced dynamics of the plant emerges from the full dynamics of the plant-bath system by tracing over the bath state at each step.