Home
Search
Collections
Journals
About
Contact us
My IOPscience
Corrigendum: Quantum Bayesian rule for weak measurements of qubits in superconducting circuit QED (2014 New J. Phys. 16 123047)
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 New J. Phys. 17 059501 (http://iopscience.iop.org/1367-2630/17/5/059501) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 192.161.162.242 This content was downloaded on 26/03/2017 at 08:37 Please note that terms and conditions apply.
You may also be interested in: Quantum Bayesian rule for weak measurements of qubits in superconducting circuit QED Peiyue Wang, Lupei Qin and Xin-Qi Li
New J. Phys. 17 (2015) 059501
doi:10.1088/1367-2630/17/5/059501
CORRIGENDUM
OPEN ACCESS RECEIVED
22 April 2015
Corrigendum: Quantum Bayesian rule for weak measurements of qubits in superconducting circuit QED (2014 New J. Phys. 16 123047)
ACCEPTED FOR PUBLICATION
22 April 2015 PUBLISHED
15 May 2015
Peiyue Wang, Lupei Qin and Xin-Qi Li Department of Physics, Beijing Normal University, Beijing 100875, People’s Republic of China E-mail:
[email protected]
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
We noted an error in the integrand in equation (18). That is, the phase factor Φ2 (tm ) in the Bayesian rule for the off-diagonal element should be constructed as
( )
∫0
Φ 2 tm = −
tm
∼ Γba (t ) Iφ (t )dt .
∼ In this result, instead of ξ (t ) as proposed previously, we amend it here by Iφ (t ) which is determined as follows. First, within the ‘polaron’ transformation scheme, the output current reads Iφ (t ) = − Γci 〈σz 〉 + κ ∣ μ ∣ cos(θμ − φ) + ξ (t ), where μ = αe (t ) + α g (t ) ≡ ∣ μ ∣ e iθμ . Second, the ensemble-average of ρge (tm ) over the stochastic ‘field’ ξ (t ), or equivalently over the stochastic output current Iφ (t ), should be consistent with the result by averaging equation (14). Therefore, instead of extracting a formal solution from equation (14) as tm ∼ Φ2 (tm ) = −∫ Γba (t ) ξ (t ) dt , we need to replace ‘ ξ (t )’ with Iφ (t ) = Iφ (t ) − I¯φ (t ), where 0 ∼ I¯φ (t ) = κ ∣ μ ∣ cos(θμ − φ) thus Iφ (t ) = − Γci 〈σz 〉 + ξ (t ). Accordingly, in the suggested approximation of 1 tm ¯ equation (19), the average current I¯ (tm ) is replaced by I¯ (tm ) = ∫ dtIφ (t ). tm 0 We may elaborate further the above correction idea by taking the bad-cavity limit. In this case, all the rates in equation (14) are time-independent. Ensemble-averaging equation (14) over the stochastic Wiener variable ξ (t ) would vanish the last two stochastic terms and lead thus to ρge (tm ) ∼ e−Γd tm , i.e., the overall dephasing factor. Now let us show how this result can be recovered by ensemble-averaging the conditional state based on the Bayesian rule. First, consider the bare Bayesian rule of equation (10). Ensemble-averaging the stochastic integrated current Im, we obtain ρ∼ge (tm ) ∼ e−Γci tm 2. Then, inserting e−iΦ2 (tm ) and making the ‘joint’ ensembleaverage, we have ρge (tm ) ∼ e−(Γci + Γba ) tm 2. Finally, multiplying this ensemble-average dephasing factor by D (tm ), i.e., the purity degradation factor given by equation (13), we recover the overall dephasing factor e−Γd tm by noting that Γci + Γba = Γm. For the (I , Q) two quadrature measurement, the phase factor Φ2 (tm ) of equation (26) is correct. However, in order to be in similar form of the single quadrature result, we propose to reexpress equation (26) as
( )
Φ 2 tm = −
∫0
tm
∼ Γm (t ) 2 Qm (t ) dt ,
∼ where Qm (t ) = Qm (t ) − κ ∣ μ (t ) ∣ sin θμ, while Qm (t ) = Iφ (t ) ∣φ = π 2 = κ ∣ μ (t ) ∣ sin θμ + ξ2 (t ). At last, we mention that all the numerical results presented in the article are correct, because of the assumed initial state (∣ e〉 + ∣ g 〉 ) 2 which holds 〈σz 〉 = 0. For initial states with 〈σz 〉 ≠ 0, the above correction for the tm
single quadrature measurement will affect the results. That is, if using the previous Φ2 (tm ) = −∫ Γba (t ) ξ (t )dt 0 tm ∼ (not replacing ξ (t ) by Iφ (t )), there would be a more phase factor, exp[i ∫ dt Γba (t ) Γci (t ) 〈σz 〉 ], which, in the 0 case 〈σz 〉 ≠ 0, will affect both the quantum trajectories and the ensemble-averaged result. However, as carefully explained above, the self-consistence-required correct result should not have this extra phase factor.
© 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
