Direct measurement of the quantum state of the electromagnetic field in a superconducting transmission line

F. de Melo; F. Toscano; L. Aolita; L. Davidovich

REVIEW

Direct measurement of the quantum state of the electromagnetic field in a superconducting transmission line

F. de Melo, L. Aolita, F. Toscano, L. Davidovich

Quantum Physics · Submitted 23 Feb 2006 · 21 references

Pith's verdict: A clean, honest proposal for direct Wigner tomography in circuit QED with always-on coupling; the main limitation is stated and mitigated.

T0 · AI review

How this review was made: An AI referee (deepseek-v4-flash) read the paper on August 28, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 7 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDquant-ph/0602195 (version 1)

Pith IDQLWJN53D, Pith's permanent record of this paper and its review

CitationF. de Melo, F. Toscano, L. Aolita, L. Davidovich. "Direct measurement of the quantum state of the electromagnetic field in a superconducting transmission line." arXiv:quant-ph/0602195 (2006). https://pith.science/paper/QLWJN53D

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Dear colleague,

This paper does one genuinely useful thing: it adapts the Lutterbach-Davidovich Wigner-function protocol to the situation where the qubit cannot be switched off from the resonator, which is exactly the circuit QED case. The trick is inserting a free dispersive evolution between the two π/2 pulses, so the parity information transfers to the qubit before the second pulse rotates it. The derivation of Eq. (5) is a straight calculation from the effective Hamiltonian, with no fitted parameters, and the numerical simulation using the exact Jaynes–Cummings model is the right independent check. For the three states tested—vacuum, Fock n=1, and a cat state—the reconstruction works, and with the proposed second parameter set the tails are no longer degraded.

The soft spot is the photon-number bound on the π/2 pulse: the rotation axis gets a z-component proportional to n, and the condition n << |ε_{π/2}|/g − 1/2 must hold for every Fock component after the displacement. The authors state this clearly, and they are honest that the first parameter set fails at larger displacements, which shows up as bad tails in Fig. 2. The stress-test worry is that the bound is not quantified beyond the statement, and the SET back-action is left unmodeled. That is fair but minor: the second parameter set puts the demonstrated states comfortably inside the validity region, and the SET issue is a standard experimental concern, not a flaw in the protocol. A fuller treatment could give an error bound vs. |α|, but that is an extension, not a necessary fix.

The citation pattern is sound; the self-citation to Lutterbach–Davidovich is prior work and relevant, not gratuitous. The paper reads as a serious theoretical proposal from people who know the experimental constraints. It is not a demonstration, but it does not pretend to be.

This paper deserves a serious referee. It is a useful contribution to the then-emerging circuit QED toolbox, and a careful referee could ask for a supplemental error analysis. I would bring it to our reading group, and I would cite it as the standard reference for always-on Wigner tomography. My verdict: accept after minor revision, with the caveat that the authors should either quantify the tail error or state explicitly the regime of validity for the second parameter set.

Best, [Your name]

Referee report

Significance

This paper addresses a real gap in early circuit-QED experiments: direct tomographic access to the intracavity field state. The main result is attractive because Eq. (5) is derived from the effective Hamiltonian without free parameters, and the numerical check uses the exact Jaynes-Cummings dynamics rather than the approximate dispersive model. The specification of two concrete parameter sets, one from existing experiments and one proposed, makes the proposal actionable. The paper is also honest about its limitations, including the photon-number bound on the π/2-pulse approximation and the omission of decoherence in the simulations. If the protocol is realized, it would enable direct Wigner-function tomography and decoherence monitoring in superconducting resonators.

Minor objections (7)

[Title and abstract] The word 'electromagnetic' is split as 'electromagne tic' in the title and abstract; this should be corrected.
[Eq. (5) derivation] The text says that the population difference is 'straightforwardly calculated' but does not show the intermediate algebra. Given the sequence of rotating frames and the nontrivial phase φ, an appendix or a few displayed equations would let readers verify Eq. (5) independently.
[Fig. 2 and numerical simulation] The simulation details are underspecified; the authors should state the Hilbert-space truncation, the integration method, and whether damping is included (the text later says decoherence is not taken into account). This is needed for reproducibility.
[Paragraph after Eq. (4)] The condition n ≪ |ε_π/2|/g − 1/2 is identified as necessary for each Fock component, but the error introduced when it is violated is not estimated. Since the initial displacement raises the mean photon number by |α|², this condition limits the phase-space region where Eq. (5) is accurate. The second parameter set keeps the demonstrated states safe, and the first set’s failure is visible in Fig. 2, but a quantitative error estimate would make the 'full reconstruction' claim precise.
[Readout paragraph] The paper proposes a single-electron transistor (SET) for measuring Pe and Pg, relying on Ref. [19] for the claim that it can be turned on only at the measurement time and that its back-action is minimized. Because the readout is an essential part of the protocol, a brief quantitative estimate of the SET-induced decoherence or a more direct citation to a modeling study would strengthen the feasibility argument.
[Table I caption] The quantity γ^{-1} is called the 'atom lifetime'; for a CPB qubit the standard term is 'qubit relaxation time'.
[Final paragraph] The suggestion that the protocol could be used to continuously monitor decoherence should be qualified, since the measurement itself projects the field state; the authors should clarify whether repeated preparation and measurement are intended.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

A superconducting qubit can directly read the full quantum state of a microwave resonator field.

The reading

This paper proposes an experimental procedure to directly measure the Wigner function of the electromagnetic field in a superconducting transmission-line resonator by encoding it into the population difference of a Cooper-pair box qubit. A sequence of two classical-field pulses and a free dispersive evolution makes the qubit population difference equal to the displaced-parity (Wigner) function at a point in phase space, so that scanning the displacement reconstructs the full quantum state. Because the qubit always remains coupled to the resonator, this bypasses the standard cavity-QED method's need to send atoms through the cavity. The paper supports the proposal with numerical simulations using experimentally realistic parameters, showing good reconstruction for vacuum, one-photon, and Schrödinger-cat states.

What carries the argument

The load-bearing object is the dispersive Jaynes-Cummings Hamiltonian with driving, Eqs. (3)–(4), in which the qubit rotation axis has a photon-number-dependent $z$-component $(g^2/\Delta)(2\hat{a}^\dagger\hat{a}+1)$. The protocol steers the system through four steps: (1) a coherent displacement $\hat{D}(\alpha)$ with the qubit rotation axis along $z$; (2) a $\pi/2$ pulse about an equatorial axis, valid when the photon-number-dependent $z$-component is negligible; (3) a free dispersive evolution of duration $t_P = \pi\Delta/2g^2$, which entangles field and qubit; and (4) a second $\pi/2$ pulse. The identity in Eq. (5) then converts the qubit population difference into the displaced-parity expectation value that defines the Wigner function.

Extended reading notes

Core claim

The paper's central claim is that a single observable—the population difference of a Cooper-pair box qubit—can directly yield the Wigner function of the electromagnetic field in a superconducting transmission-line resonator. After a displacement of the field, two $\pi/2$ pulses on the qubit separated by a free dispersive evolution produce the identity $P_e - P_g = \mathrm{Tr}[\sin(\varphi_1-\varphi_2)\,\hat{\rho}\,\hat{D}(\beta)^\dagger e^{-i\pi \hat{a}^\dagger \hat{a}} \hat{D}(\beta)]$ (Eq. 5). For a phase difference $\varphi_1-\varphi_2 = \pi/2$, this quantity is exactly the Wigner function of the initial field at the phase-space point $-\beta$. Repeating the measurement for different drive amplitudes and phases $\beta$ therefore reconstructs the complete quantum state of the field.

Load-bearing premise

During the two $\pi/2$ pulses, the qubit rotation axis must be effectively independent of photon number, requiring $n \ll |\varepsilon_{\pi/2}|/g - 1/2$ for every Fock state in the displaced field; the paper's own simulation shows this condition is violated for larger displacement amplitudes, degrading the reconstructed Wigner function's tails.

Editorial extensions

If this is right

Scanning the drive amplitude and phase $\beta$ yields the Wigner function over the entire phase space, giving complete knowledge of the resonator field state from a single observable.
The same four-step pulse sequence also produces Schrödinger-cat-like field states entangled with the qubit; detecting the qubit projects the field onto a coherent superposition of two coherent states.
Because the protocol is much faster than the cavity lifetime for the proposed parameters, it can monitor decoherence continuously and in real time.
The method extends to any qubit-resonator system where the dispersive approximation holds, since no access to the inside of the resonator is required.
Numerical simulations with realistic parameters show the protocol can reconstruct vacuum, one-photon, and Schrödinger-cat states, indicating readiness for experimental implementation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The protocol's accuracy is fundamentally limited by the photon-number condition for the $\pi/2$ pulses; pushing this limit would require stronger drives or larger detunings, which may be experimentally reachable with current technology.
The same encoding sequence could be adapted to other solid-state qubits, such as flux or phase qubits, as long as the dispersive-interaction and rotation-axis conditions can be met.
Because each phase-space point is obtained from a single population measurement, the protocol offers a resource-efficient route to continuous decoherence monitoring compared with full tomography.

Original abstract

We propose an experimental procedure to directly measure the state of an electromagnetic field inside a resonator, corresponding to a superconducting transmission line, coupled to a Cooper-pair box (CPB). The measurement protocol is based on the use of a dispersive interaction between the field and the CPB, and the coupling to an external classical field that is tuned to resonance with either the field or the CPB. We present a numerical simulation that demonstrates the feasibility of this protocol, which is within reach of present technology.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,980 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.05 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We propose an experimental procedure to directly measure the state of an electromagnetic field inside a resonator, corresponding to a superconducting transmission line, coupled to a Cooper-pair box (CPB). The measurement protocol is based on the use of a dispersive interaction between the field and the CPB, and the coupling to an external classical field that is tuned to resonance with either the field or the CPB. We present a numerical simulation that demonstrates the feasibility of this protocol, which is within reach of present technology.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose an experimental procedure to directly measure the state of an electromagnetic field inside a resonator, corresponding to a superconducting transmission line, coupled to a Cooper-pair box (CPB).	contribution 0.99	0.19	0.16	asserted 0.57
s1	The measurement protocol is based on the use of a dispersive interaction between the field and the CPB, and the coupling to an external classical field that is tuned to resonance with either the field or the CPB.	method 1.00	0.25	0.05	asserted 1.00
s2	We present a numerical simulation that demonstrates the feasibility of this protocol, which is within reach of present technology.	result 0.47	0.87	0.52	asserted 0.96

What kind of paper the classifier reports this is: method or tool 0.98 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (21)

[1]

Y. Makhlin, G. Schon, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001)

View on Pith 2001
[2]

A. Wallraﬀ et al. , Nature 431, 162 (2004)

View on Pith 2004
[3]

Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Phys. Rev. Lett. 87, 246601 (2001); Y. Yu et al. , Science 296, 889 (2002); D. Vion et al. , Science 296, 886 (2002); A. Wall- raﬀ et al. , Phys. Rev. Lett. 95, 060501 (2005)

View on Pith 2001
[4]

Y. A. Pashkin et al. , Nature 421, 823 (2003)

View on Pith 2003
[5]

T. Yamamoto et al. , Nature 425, 941 (2003)

View on Pith 2003
[6]

J. Q. You, J. S. Tsai, and F. Nori, Phys. Rev. Lett. 89, 197902 (2002)

View on Pith 2002
[7]

A. Blais et al. , Phys. Rev. A 69, 062320 (2004)

View on Pith 2004
[8]

M. F. Santos et al. , Phys. Rev. A 63, 033813 (2001)

View on Pith 2001
[9]

M. Mariantoni et al. , cond-mat/0509737 (2005)

View on Pith Pith review arXiv 2005
[10]

M. Khanbekyan et al. , Phys. Rev. A 69, 043807 (2004)

View on Pith 2004
[11]

E. Wigner, Phys. Rev. 40, 749 (1932)

View on Pith 1932
[12]

F. Toscano, R. L. de Matos Filho, and L. Davidovich, Phys. Rev. A 71, 010101(R) (2005)

View on Pith 2005
[13]

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969); 177, 1882 (1969)

View on Pith 1969
[14]

J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73, 565 (2001)

View on Pith 2001
[15]

L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997)

View on Pith 1997
[16]

G. Nogues et al. , Phys. Rev. A 62, 054101 (2000)

View on Pith 2000
[17]

A. S. Sorensen et al. , Phys. Rev. Lett. 92, 063601 (2004); L. Tian et al. , Phys. Rev. Lett. 92, 247902 (2004); A. N. Cleland and M. R. Geller, Phys. Rev. Lett. 93, 070501 (2004)

View on Pith 2004
[18]

Strictly speaking, we should impose this condition to e v- ery photon number n present in the state; or, alterna- tively, ask that the distribution in photon number be very localized around n

View on Pith
[19]

O. Astaﬁev et al. , Phys. Rev. B 69, 180507(R) (2004)

View on Pith 2004
[20]

L. Davidovich et al. , Phys. Rev. A 53, 1295 (1996)

View on Pith 1996
[21]

Yu-xi Liu et al. , cond-mat/0507496

View on Pith Pith review arXiv

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Y. Makhlin, G. Schon, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001)

View on Pith 2001

[2] [2]

A. Wallraﬀ et al. , Nature 431, 162 (2004)

View on Pith 2004

[3] [3]

Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Phys. Rev. Lett. 87, 246601 (2001); Y. Yu et al. , Science 296, 889 (2002); D. Vion et al. , Science 296, 886 (2002); A. Wall- raﬀ et al. , Phys. Rev. Lett. 95, 060501 (2005)

View on Pith 2001

[4] [4]

Y. A. Pashkin et al. , Nature 421, 823 (2003)

View on Pith 2003

[5] [5]

T. Yamamoto et al. , Nature 425, 941 (2003)

View on Pith 2003

[6] [6]

J. Q. You, J. S. Tsai, and F. Nori, Phys. Rev. Lett. 89, 197902 (2002)

View on Pith 2002

[7] [7]

A. Blais et al. , Phys. Rev. A 69, 062320 (2004)

View on Pith 2004

[8] [8]

M. F. Santos et al. , Phys. Rev. A 63, 033813 (2001)

View on Pith 2001

[9] [9]

M. Mariantoni et al. , cond-mat/0509737 (2005)

View on Pith Pith review arXiv 2005

[10] [10]

M. Khanbekyan et al. , Phys. Rev. A 69, 043807 (2004)

View on Pith 2004

[11] [11]

E. Wigner, Phys. Rev. 40, 749 (1932)

View on Pith 1932

[12] [12]

F. Toscano, R. L. de Matos Filho, and L. Davidovich, Phys. Rev. A 71, 010101(R) (2005)

View on Pith 2005

[13] [13]

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969); 177, 1882 (1969)

View on Pith 1969

[14] [14]

J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73, 565 (2001)

View on Pith 2001

[15] [15]

L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997)

View on Pith 1997

[16] [16]

G. Nogues et al. , Phys. Rev. A 62, 054101 (2000)

View on Pith 2000

[17] [17]

A. S. Sorensen et al. , Phys. Rev. Lett. 92, 063601 (2004); L. Tian et al. , Phys. Rev. Lett. 92, 247902 (2004); A. N. Cleland and M. R. Geller, Phys. Rev. Lett. 93, 070501 (2004)

View on Pith 2004

[18] [18]

Strictly speaking, we should impose this condition to e v- ery photon number n present in the state; or, alterna- tively, ask that the distribution in photon number be very localized around n

View on Pith

[19] [19]

O. Astaﬁev et al. , Phys. Rev. B 69, 180507(R) (2004)

View on Pith 2004

[20] [20]

L. Davidovich et al. , Phys. Rev. A 53, 1295 (1996)

View on Pith 1996

[21] [21]

Yu-xi Liu et al. , cond-mat/0507496

View on Pith Pith review arXiv