Direct measurement of the quantum state of photons in a cavity

Lupei Qin; Zhong Wang; Cheng Zhang; Xin-Qi Li

doi:10.1364/OE.26.007034

1. The state of a system

In quantum theory is described by a quantum wavefunction, which differs drastically from the state description in classical mechanics. Actually the wavefunction represents a knowledge and works perfectly well as a practical tool, however, the underlying physics remains still unclear. The most surprising point is that the quantum state is governed by the simple Schrödinger equation as a universal law. Actually, controllable manipulation of the quantum state has stimulated the advent of the quantum information science and technology.

In addition to manipulating the quantum state based on the law of Schrödinger equation, another important problem is how to determine a unknown state. In general, this is a challenging task, since the quantum state can be determined only by multiple measurements on an ensemble of identically prepared quantum systems, rather than a single shot measurement of the single system. More specifically, to reconstruct the quantum state uniquely, a complete set of probability distributions has to be measured over a range of different representations, by employing the technique of quantum state tomography (QST) [1–4].

For low dimensional states such as the one of a qubit, the task is relatively simple. But for high dimensional states, the job is nontrivial and quite difficult in general. Particular examples include the determination of the optical fields in a cavity [5–14] and of traveling light [15–20], the vibrational states of trapped ions/atoms [21–28] and molecules [29]. In these QST schemes for measuring either the optical fields or the vibrational states, the strategy is to ‘measure’ the Wigner function (but not the wavefunction or density operator of state), by converting the information of the Wigner function into electronic states of atoms and performing fluorescence measurement of the atoms. Viewing that the Wigner function is a class of distributions in phase space, the uncertainty principle forbids to interpret it as real probability distribution [30]. In order to convert it to real physical density matrix, one needs in principle its full information over the phase space, when performing the transformation from the Wigner function to quantum density matrix. This is a demanding task, which requires measuring the Wigner function over a large grid of points in the phase space.

In this work, we propose a scheme to measure directly the quantum wavefunction (but not the Wigner function) of the optical field (photons) in a cavity, first in the solid-state circuit QED then in an atomic cavity QED systems. Importantly, the proposed scheme allows us to access the individual superposition component in Fock state basis, and does not need global reconstruction as usual in the conventional QST scheme. The new scheme is essentially based on the concept of quantum weak values (WVs) [31–33].

Actually, the concept of quantum WVs has been exploited for applications such as ‘direct’ measurement of quantum wavefunctions [34–39]. The basic idea is sequentially measuring two complementary variables of the system. The first measurement is weak, and the second one is strong. The weak measurement gets minor information, which has gentle disturbance and does not collapse the state. The second projective measurement plays a role of post-selection. One of the most desirable features is that, in this new scheme, it is the superposed complex amplitudes in the wavefunction (but not the probabilities) to be extracted from the single round average of the post-selected data of the first weak measurements. Another advantage of the WV-based scheme is the possibility that it does not necessarily need a global reconstruction of the quantum state. Applying this method, experiments have been performed for measuring photon’s transverse wavefunction (a task not previously realized by any method) [34], photon’s polarization state [36, 37], and the high-dimensional orbital angular momentum state of photons [38, 39].

2. Set-up description and basic idea

In Fig. 1 we show schematically the proposed set-up which can be realized with superconducting circuit QED architectures [40–44]. The high-Q cavity in the middle part is prepared in a quantum state of microwave field to be measured. Taking the most natural choice of representation basis, the cavity field state can be expressed as |Ψ〉 = Σ_n c_n|n〉 where |n〉 is the Fock state with n photons. The left and right artificial atoms correspond to the transmon qubits in the circuit QED realization, each of them being stored in its own cavity. The two qubits are designed to couple to the middle cavity to jointly probe the cavity field. More specifically, the left-side (meter) qubit performs weak measurement selectively for Π_n = |n〉〈n| (with “n” a running number), and the right-side (post-selection) qubit generates post-selection to the cavity field. In order to realize the selective monitoring of Π_n, the left-side qubit is dispersively coupled to the middle cavity and the weak interaction with Π_n is implemented by performing, e.g., a σ₁_x rotation to the left qubit by a small angle, by applying a rotating field with frequency in resonance with the n -photon-shifted qubit energy. Then, perform projective measurements of σ₁_x and σ₁_y, respectively, for the left qubit (in ensemble of realizations), via the well established technique of microwave transmission and homodyne detection. Meanwhile, to perform post-selection, the right-side qubit is time-controllably coupled to the middle cavity. Rather than dispersive, here a resonant coupling is proposed. Together with proper rotation to the qubit and homodyne detection of microwave transmission (to projectively measure the qubit state), desired post-selection for the middle cavity state can be realized. Conditioned on the post-selection, the conditional averages of σ₁_x and σ₁_y of the left-side qubit will reveal essential information of the n_th component c_n for the quantum state of the middle cavity.

Fig. 1 Schematic plot for measuring the unknown state of photons in a cavity, say, in the central one which can be expressed in general as |Ψ〉 = Σ_n c_n|n〉 with |n〉 the Fock state of n photons. In connection with the superconducting circuit-QED realization, the two artificial atoms (qubits) in the side cavities are employed to probe the photons state in the central cavity: the left qubit performs weak measurement selectively for Π_n = |n〉〈n|; and the right qubit performs post-selection which will result in a post-selected cavity state |Ψ_f〉 = Σ_n c_n(α_n|n〉 + β_n |n − 1〉). The coupling between the cavities, the required rotations of qubits and their measurements are also schematically indicated, while keeping more detailed explanations referred to the main text.

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3. Weak-value and state determination

Now we present more detailed description for the method how to measure first the weak value of Π_n, then determine the unknown state of the cavity field. As briefly mentioned above, the left-side qubit in Fig. 1 is dispersively coupled to the middle cavity, described by H_int = χa^†aσ₁_z, where a^† and a are the creation and annihilation operators of the single mode cavity photons, σ₁_z is the quasi-spin operator of the left qubit with logic states |e₁〉 and |g₁〉 (another two operators of this qubit are σ₁_x and σ₁_y). The bare energy spacing between |e₁〉 and |g₁〉 is 2Δ₁. As a consequence of ac-Stark effect (or, directly, based on the above dispersive Hamiltonian), the qubit energy will be shifted from Δ₁ to ${\tilde{Δ}}_{1}^{(n)} = Δ_{1} + n χ$ by the Fock state |n〉 of the cavity field.

In order to realize the measurement of Π_n = |n〉〈n|, let us consider a ‘selective’ σ₁_x rotation on the qubit, by applying an external microwave field with frequency in resonance with $2 {\tilde{Δ}}_{1}^{(n)}$ . This induces a measurement coupling between the cavity field and the qubit given by

H_{m e a s} = γ \prod_{n} σ_{1 x} .

In this measurement interaction Hamiltonian, γ is the rotating strength to the qubit, and the projection operator Π_n = |n〉〈n| is from the fact that we selectively rotate the qubit with frequency in resonance with

2 {\tilde{Δ}}_{1}^{(n)}

. More quantitative derivation for Eq. (1) is referred to a latter part in this work.

Under the action of the Hamiltonian Eq. (1), the cavity field and the meter qubit (i.e. the left one in Fig. 1) are subject to a joint evolution. Let us denote the initial state as |Ψ〉 ⊗ |Φ₀〉, where |Φ₀〉 is the state of the meter qubit before switching on the measurement interaction, which is assumed as |Φ₀〉 = |g₁〉. The joint evolution is given by U(τ)(|Ψ〉 ⊗ |Φ₀〉), where $U (τ) = \exp (- i H_{m e a s} τ / ℏ) ≃ 1 - i (\frac{γ τ}{ℏ}) \prod_{n} σ_{1 x}$ in the regime of weak measurement which is characterized by a small parameter of γτ. Conditioned on a post-selection of the cavity field state |Ψ_f〉, which is to be specified soon in the following, the state of the meter qubit is given by

| Φ (τ) 〉 = [| g_{1} 〉 - i (\frac{γ τ}{ℏ}) {〈 Π_{n} 〉}_{w} | e_{1} 〉] / N,

where

N

denotes a normalization factor and the weak value 〈Π_n〉_w reads

{〈 Π_{n} 〉}_{w} = \frac{〈 Ψ_{f} | Π_{n} | Ψ 〉}{〈 Ψ_{f} | Ψ 〉} .

Importantly, the weak value of Π_n in Eq. (2) plays a role of rotation parameter to the meter qubit. Using standard method, this complex parameter can be extracted from the averages of the meter qubit, 〈σ₁_x〉_Φ = 〈Φ(τ)|σ₁_x|Φ(τ)〉 and 〈σ₁_y〉_Φ = 〈Φ(τ)|σ₁_y|Φ(τ)〉. After simple algebra, we obtain

{〈 Π_{n} 〉}_{w} = (\frac{ℏ}{2 γ τ}) (i {〈 σ_{1 x} 〉}_{Φ} - {〈 σ_{1 y} 〉}_{Φ}) .

The averages 〈σ₁_x〉_Φ and 〈σ₁_y〉_Φ can be obtained via an ensemble of projective measurements within the ‘natural’ basis |e₁〉 and |g₁〉 of the qubit. However, before the projective measurements, a respective σ₁_x or σ₁_y rotation (basis rotation) should be exerted on the qubit. Another point associated with the weak value 〈Π_n〉_w is that the measurement records are collected only if the post-selection of the cavity state |Ψ_f〉 is successful. In our proposal, the average success probability of post-selection is about 50%, which is high among the various weak-value-related applications.

Now we address the post-selection for the cavity field state, via a couple of procedures in order as follows. (i) Switch on for a time period of resonant coupling between the cavity field and the ‘post-selection’ qubit (the right one in Fig. 1). We assume this qubit prepared initially in the ground state |g₂〉. The coupling interaction leads to a Rabi rotation: |g₂〉|n〉 → α_n|g₂〉|n〉 + β_n|e₂〉|n − 1〉. (ii) Perform, for instance, π/2-pulse σ_2y rotation to the qubit, which is described by the unitary transformation $U (θ) = e^{- i \frac{θ}{2} σ_{2 y}}$ . After (i) and (ii), the joint state of the cavity and qubit reads:

| Ψ_{2} 〉 = \frac{1}{\sqrt{2}} \sum_{n = 0}^{\infty} c_{n} [(α_{n} | n 〉 - β_{n} | n - 1 〉) | g_{2} 〉 + (α_{n} | n 〉 + β_{n} | n - 1 〉) | e_{2} 〉] .

(iii) Perform a projective measurement on the qubit and select the result of |g₂〉 we obtain the cavity state as

| Ψ_{f} 〉 = [c_{0} α_{0} | 0 〉 + \sum_{n = 1}^{\infty} c_{n} (α_{n} | n 〉 - β_{n} | n - 1 〉)] / N .

This is the post-selected state of the cavity photons.

Inserting it into Eq. (3), up to a common normalization factor, we obtain

\begin{matrix} {〈 \prod_{0} 〉}_{w} & = & {| c_{0} |}^{2} {[α_{0} - (c_{1} / c_{0}) β_{1}]}^{*}, \\ {〈 \prod_{1} 〉}_{w} & = & {| c_{1} |}^{2} {[α_{1} - (c_{2} / c_{1}) β_{2}]}^{*}, \\ ⋮ \\ {〈 \prod_{n} 〉}_{w} & = & {| c_{n} |}^{2} {[α_{n} - (c_{n + 1} / c_{n}) β_{n + 1}]}^{*} . \end{matrix}

Up to a common normalization factor, like other WV-based state tomographic schemes [34–39], this set of iterative expressions allows us to determine sequentially c₁, c₂, ⋯ c_n, based on the measured weak values 〈Π_n〉_w (note that all the α_n and β_n are known coefficients). Compared to the conventional tomographic method, which cannot access the individual components of the superposed state, the present iterative expressions hold the advantage of permitting us to access the single components without global reconstruction, viewing the fact that it is the relative ratios of the amplitudes in the quantum superposition that represent the real information relevant to observable effects. Actually, in a quantum superposed state, the ratio of neighboring components is equivalent to the relative amplitude with respect to a common normalization factor.

4. Alternative set-up of atomic cavity QED system

The direct scheme of state tomography proposed above can be similarly applied to the state-of-the-art atomic cavity-QED set-up [45]. The basic idea is schematically illustrated in Fig. 2. The high-Q cavity (‘C ’) is prepared in a state described in general by |Ψ〉 = Σ_n c_n|n〉. This cavity field is probed first by crossing an atom (meter atom) through it (as shown by the upper panel of Fig. 2), then by a subsequent post-selection atom (the lower panel of Fig. 2). The second low-Q Ramsey cavity (‘R’) is employed to rotate the crossing atoms (sequentially, first the meter and then the post-selection atoms) between |g_j〉 and |e_j〉 (j = 1, 2), by introducing π/2 classical Rabi pulses.

Fig. 2 Schematic illustration for implementing the proposed scheme in atomic cavity-QED set-up. The high-Q cavity (‘C’) is prepared in an initial state described in general by |Ψ〉 = Σ_n c_n|n〉, while the second low-Q Ramsey cavity (‘R’) is employed to rotate the atomic states of the crossing atoms by introducing external classical fields. The upper and lower panels show, respectively, a meter and post-selection atom crossing sequentially the two cavities.

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We may detail the weak probe and post-selection of the cavity state, respectively, as follows. (i) For the meter atom (prepared in ground state |g₁〉 before entering the cavity C), the dispersive coupling with the cavity field generates an ac Stark shift nχ between |g₁〉 and |e₁〉, where n is the photon numbers and χ the dispersive coupling strength. When the meter atom crosses the cavity C, shine a classical laser field into the cavity to rotate selectively, i.e., n-dependently, the meter atom weakly by an amount of small angle (small γτ in Eq. (2)). Then, let the meter atom cross the Ramsey cavity R, experience a π/2 pulse of σ₁_x and σ₁_y rotations (in the sense of ensemble realizations), and suffer a final ionization measurement of |g₁〉 or |e₁〉. The ensemble averages, conditioned also on the result (e.g., |g₂〉) of the subsequent post-selection atom, give us the key results |σ₁_x〉_Φ and |σ₁_y〉_Φ required in Eq. (4).

(ii) In order to generate the post-selection state |Ψ_f〉 for the cavity field, a post-selection atom (following the meter atom) is sending to cross the both cavities C and R. In C, this atom experiences a resonant interaction with the cavity photon; while in R, it suffers a π/2 Rabi pulse for σ₂_y rotation. After these, the atom is subject to a final ionization measurement. Selecting the result of |g₂〉, we obtain the post-selection state |Ψ_f〉 for the cavity field, given by Eq. (6).

5. On the ‘selective’ rotation

We now present a derivation for the Hamiltonian shown by Eq. (1). Let us return to the starting Hamiltonian of the meter qubit (the first one) coupling to the cavity mode and in the presence of driving by external field, $H = Δ_{1} σ_{1 z} + χ σ_{1 z} a^{†} a + [(γ e^{i ω t}) σ_{1}^{-} + h . c .]$ , where the second term describes the dispersive coupling of the meter qubit to the cavity mode and the third term is the external driving (with frequency ω). We may regard the first two terms as free Hamiltonian, H₀ = Δ₁ σ₁_z + χσ₁_za^†a and express it as a sum from subspaces expanded by {(|e₁〉, n, |g₁, n〉} (with n = 0, 1, 2, ⋯: $H_{0} = \otimes_{n} H_{0}^{(n)} = \otimes_{n} {\tilde{Δ}}_{1}^{(n)} σ_{1 z}^{(n)}$ , where ${\tilde{Δ}}_{1}^{(n)} = Δ_{1} + n_{χ}$ and $H_{0}^{(n)}$ reads

H_{0}^{(n)} = (\begin{matrix} {\tilde{Δ}}_{1}^{(n)} & 0 \\ 0 & - {\tilde{Δ}}_{1}^{(n)} \end{matrix}),

Now, including the driving term and in the rotating frame with respect to

{\tilde{H}}_{0} = \frac{ω}{2} σ_{1 z}

, we can express the Hamiltonian in the n_th subspace as

H^{(n)} = (\begin{matrix} {\tilde{Δ}}_{1}^{(n)} - \frac{ω}{2} & γ \\ γ & (- {\tilde{Δ}}_{1}^{(n)} - \frac{ω}{2}) \end{matrix}) .

Note that in terms of this decomposition, the total Hamiltonian simply reads H = ⊗_nH⁽ⁿ⁾.

Consider now the initial state, |g₁〉 ⊗ |Ψ〉 = |g₁〉 ⊗ (Σ_n c_n|n〉). If we choose the frequency of the driving field in resonance with the shifted energy of the qubit by n photons, i.e., $ω = 2 {\tilde{Δ}}_{1}^{(n)}$ , only the state component in the n_th subspace will be affected by the driving field. That is, |g₁, n〉 is rotated by a small amount as

| g_{1}, n 〉 \to | g_{1}, n 〉 - i (γ τ / ℏ) | e_{1}, n 〉 .

Here, we expanded the unitary evolution operator U(τ) to the first order, which is valid in the weak measurement limit. Other components in |g₁〉 ⊗ |Ψ〉, owing to large detuning from the frequency of the driving field, are not affected by the driving field. Putting these together, we have

\begin{array}{l} U (τ) [| g_{1} 〉 \otimes (\sum_{n} c_{n} | n 〉)] \\ = \sum_{n^{'} \neq n} c_{n^{'}} | g_{1}, n^{'} 〉 + c_{n} [| g_{1}, n 〉 - i (γ τ / ℏ) | e_{1}, n 〉] \\ = | g_{1} 〉 \otimes | Ψ 〉 - i (γ τ / ℏ) c_{n} | e_{1}, n 〉 \\ = | g_{1} 〉 \otimes | Ψ 〉 - i (γ τ / ℏ) | e_{1} 〉 \otimes ({\prod^{^}}_{n} | Ψ 〉) . \end{array}

This allows us to construct the effective rotating Hamiltonian, Eq. (1), which leads to the selective rotation given by Eq. (2).

Finally, let us explain how the state in the subspace with large energy detuning can be free from the influence of the rotating field. In the rotating frame with frequency $ω = 2 {\tilde{Δ}}_{1}^{(n)}$ , the detuning of the n′-photon-shifted qubit energy from ω is characterized by nonzero energies of the qubit states |e₁〉 and |g₁〉, $E_{e_{1}, g_{1}} = \pm ϵ$ , where $ϵ = | {\tilde{Δ}}_{1}^{(n^{'})} - ω / 2 | = | n^{'} - n | χ$ . Then, after a simple algebra, the transition probability from |g₁〉 to |e₁〉 is obtained as

P_{e_{1}} (t) = {(γ / \tilde{γ})}^{2} \sin^{2} (\tilde{γ} t),

where

\tilde{γ} = \sqrt{ϵ^{2} + γ^{2}}

. In the special case of resonant driving (i.e. ϵ = 0) and for weak measurement limit, we have

P_{e_{1}} (t) = \sin^{2} (γ t) ≃ {(γ t)}^{2} .

For nonzero energy detuning, we reexpress the transition probability as

P_{e_{1}} (t) = {(γ t)}^{2} [\frac{\sin^{2} (\tilde{γ} t)}{{(\tilde{γ} t)}^{2}}] .

Let us assume that the weak measurement transition given by the upper result Eq. (13) is realized by weak coupling (with small γ). Then, under the condition of strong dispersive coupling χ/γ ≫ 1, the

\tilde{γ}

in the lower result Eq. (14) can be approximated as

\tilde{γ} ≃ ϵ = | n^{'} - n | χ \equiv m χ

. Now, importantly, if we properly design the coupling strength and time to make γt a small parameter and χt ≃ π, based on Eq. (14) we find that, for the n′(≠, n)-photon-shifted qubit state, the transition from |g₁〉 to |e₁〉 is to be strongly suppressed owing to sin²(mχt)/(mχt)² → 0. Therefore, via this type of design, we can realize the ‘selective’ rotation of the n-photon-shifted state.

6. Applications and feasibility

Existing tomographic scheme of cavity field is largely based on measuring the Wigner function in phase space. For example, in the recent works based on the circuit QED architecture [46, 47], applying the method proposed in Ref. [12], the Wigner functions of the microwave cavity states were measured. In particular, in Ref. [47], arbitrary quantum superposition of Fock states was prepared and and demonstrated via measuring first the Wigner function of the cavity field then converting it into the density matrix in Fock state basis. In order to perform this conversion, one needs to digitalize the phase space and gain by measurement the database of a large grid of points. And, for each of these points, one must perform the usual ensemble measurements. Moreover, one is unable to scan all the points in the phase space. This incomplete database may affect the accuracy of the conversion, e.g., which may have been involved in the occurrence of the undesirable non-vanishing imaginary parts of the off-diagonal matrix elements for the specific state demonstrated in Ref. [47].

The essential advantage of the WV-based scheme is the capability of direct access into the individual Fock-state component of the cavity field, not needing a global reconstruction, e.g., based on the incomplete database of phase space. As a specific example, applying our present method to the example studied in Ref. [47] will be of great interest, especially demonstrating that one can locally identify the Fock-state component without global reconstruction. Our method may be applied as well to the atomic cavity QED system studied by Haroche et al [48], by extending their study to the superposition of Fock states. In general speaking, the WV-based scheme has advantage for measuring the highly quantum-mechanical state of cavity field consisting of a few Fock-state components, especially for its local characterization.

In concern with the feasibility in experiments, the basic coupling elements between the qubit/atom and the cavity field have been well established in the present state-of-the-art technology, as employed in Refs. [46–48]. One of the subtle issues in practice is the accurate reset/prepartion of the initial state of the cavity field, after each weak measurement and postselection. This is because the second postselection would destroy the cavity state, despite the negligible influence on it of the first weak measurement. The reset can be fulfilled by properly driving the cavity by external field, and/or coupling it to qubits (e.g. in the solid-state circuit QED architecture), or sending a stream of atoms to cross through the cavity to excite cavity photons (e.g. in the case of atomic cavity QED set-up). The accuracy of the reset will set up the upper limit of tomography quality, as in any other tomographic schemes, owing to the probability nature of the quantum wavefunction. Other issues in experiment needing attention include properly performing both the σ_x and σ_y rotations – this can be realized by modulating the phase of the driving field by π/2, and precisely tuning the selective frequency of the weak measurement in resonance with Δ₁ + nχ. This frequency tuning can be implemented by (i) altering the frequency of the driving field, and/or (ii) modulating the level spacing of the qubit by gate voltage control (in the case of circuit QED set-up).

Finally, we remark that in concern with the extra procedure of postselection which is usually regarded as a shortcoming involved in the WV-based scheme, our present proposal holds a feature of high efficiency, viewing that the postselection of the cavity field is fulfilled by selecting one from the two states of the second qubit/atom. This high efficiency of postselection should benefit a lot to the practical realization of the present scheme, by regarding the possible high dimensions of the state of the cavity photons.

7. Summary

We have proposed a scheme to measure the quantum state of photons in a cavity. The scheme is essentially based on the concept of quantum weak values, which allows direct access to the individual superposition components in Fock state basis, not needing a global reconstruction as the conventional method of quantum state tomography. Compared to existing schemes of measurement of the Wigner function, the present scheme does not need the conversion from phase space to physical representation. It would be of great interest to realize the proposal in the state-of-the-art superconducting circuits.

Funding

This work was supported by the NNSF of China under Nos. 11675016 & 21421003.

References and links

1. K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847 (1989). [CrossRef]

2. G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471 (1997). [CrossRef]

3. A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, “Nonmaximally Entangled States: Production, Characterization, and Utilization,” Phys. Rev. Lett. 83, 3103 (1999). [CrossRef]

4. M. Hofheinz, H Wang, M Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, J. M. Martinis, and AN Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature 459, 546 (2009). [CrossRef] [PubMed]

5. M. Wilkens and P. Meystre, “Nonlinear atomic homodyne detection: A technique to detect macroscopic superpositions in a micromaser,” Phys. Rev. A 43, 3832 (1991). [CrossRef] [PubMed]

6. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244 (1993). [CrossRef] [PubMed]

7. M. Freyberger and A. M. Herkommer, “Probing a Quantum State via Atomic Deflection,” Phys. Rev. Lett. 72, 1952 (1994). [CrossRef] [PubMed]

8. S. M. Dutra and P. L. Knight, “Atomic probe for quantum states of the electromagnetic field,” Phys. Rev. A 49, 1506 (1994). [CrossRef] [PubMed]

9. G. Breitenbach, T. Müller, S. F. Pereira, J. Ph. Poizat, S. Schiller, and J. Mlynek, “Squeezed vacuum from a monolithic optical parametric oscillator,” J. Opt. Soc. B 12, 2304 (1995). [CrossRef]

10. P. J. Bardroff, E. Mayr, and W. P. Schleich, “Quantum state endoscopy: Measurement of the quantum state in a cavity,”, “Simulation of quantum-state endoscopy”, Phys. Rev. A 51, 4963 (1995). [CrossRef] [PubMed]

11. P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J.M. Raimond, and S. Haroche, “Direct Measurement of the Wigner Function of a One-Photon Fock State in a Cavity,” Phys. Rev. Lett. 89, 200402 (2002). [CrossRef] [PubMed]

12. L. G. Lutterbach and L. Davidovich, “Method for Direct Measurement of the Wigner Function in Cavity QED and Ion Traps,” Phys. Rev. Lett. 78, 2547 (1997). [CrossRef]

13. M. Franca Santos, L. G. Lutterbach, S. M. Dutra, N. Zagury, and L. Davidovich, “Reconstruction of the state of the radiation field in a cavity through measurements of the outgoing field,” Phys. Rev. A 63, 033813 (2001). [CrossRef]

14. A. A. Semenov, D. Yu. Vasylyev, W. Vogel, M. Khanbekyan, and D.-G. Welsch, “Leaky cavities with unwanted noise,” Phys. Rev. A 74, 033803 (2006). [CrossRef]

15. K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, “Direct measurement of the Wigner function by photon counting,” Phys. Rev. A 60, 674 (1999). [CrossRef]

16. E Mukamel, K Banaszek, I A. Walmsley, and C Dorrer, “Direct measurement of the spatial Wigner function with area-integrated detection,” Opt. Lett. 28, 1317 (2003); [CrossRef] [PubMed]

17. B J. Smith, B Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365 (2005). [CrossRef]

18. M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett. 34, 1444 (2009). [CrossRef] [PubMed]

19. A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, P. Traina, S. Olivares, Matteo G. A. Paris, and G Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009). [CrossRef]

20. K Laiho, K N Cassemiro, D Gross, and C Silberhorn, “Probing the Negative Wigner Function of a Pulsed Single Photon Point by Point,” Phys. Rev. Lett. 105, 253603 (2010). [CrossRef]

21. C. A. Blockley, D. F. Walls, and H. Risken, “Quantum Collapses and Revivals in a Quantized Trap,” Europhys. Lett. 77, 509 (1992). [CrossRef]

22. J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Quantum collapse and revival in the motion of a single trapped ion,” Phys. Rev. A 49, 1202 (1994). [CrossRef] [PubMed]

23. S. Wallentowitz and W. Vogel, “Reconstruction of the Quantum Mechanical State of a Trapped Ion,” Phys. Rev. Lett. 75, 2932 (1995). [CrossRef] [PubMed]

24. J. F. Poyatos, R. Walser, J. I. Cirac, P. Zoller, and R. Blatt, “Motion tomography of a single trapped ion,” Phys. Rev. A 53, R1966 (1996). [CrossRef] [PubMed]

25. C. D’Helon and G. J. Milburn, “Reconstructing the vibrational state of a trapped ion,” Phys. Rev. A 54, R25 (1996). [CrossRef]

26. P. J. Bardroff, C. Leichtle, G. Schrade, and W. P. Schleich, “Endoscopy in the Paul Trap: Measurement of the Vibratory Quantum State of a Single Ion,” Phys. Rev. Lett. 77, 2198 (1996). [CrossRef] [PubMed]

27. D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental Determination of the Motional Quantum State of a Trapped Atom,” Phys. Rev. Lett. 77, 4281 (1996). [CrossRef] [PubMed]

28. J. F. Kanem, S. Maneshi, S. H. Myrskog, and A. M. Steinberg, “Phase Space Tomography of Classical and Nonclassical Vibrational States of Atoms in an Optical Lattice,” J. Opt. B 7, S705 (2005). [CrossRef]

29. T. J. Dunn, I. A. Walmsley, and S. Mukamel, “Experimental Determination of the Quantum-Mechanical State of a Molecular Vibrational Mode Using Fluorescence Tomography,” Phys. Rev. Lett. 74, 884 (1995). [CrossRef] [PubMed]

30. A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299 (2009). [CrossRef]

31. Y. Aharonov, D. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351 (1988). [CrossRef] [PubMed]

32. I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a “weak measurement” of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112 (1989). [CrossRef]

33. Y. Aharonov and L. Vaidman, “Properties of a quantum system during the time interval between two measurements,” Phys. Rev. A 41, 11 (1990). [CrossRef] [PubMed]

34. J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188 (2011). [CrossRef] [PubMed]

35. J. S. Lundeen and C. Bamber, “Procedure for Direct Measurement of General Quantum States Using Weak Measurement,” Phys. Rev. Lett. 108, 070402 (2012). [CrossRef] [PubMed]

36. G. S. Thekkadath, L. Giner, Y. Chalich, M. J. Horton, J. Banker, and J. S. Lundeen, “Direct Measurement of the Density Matrix of a Quantum System,” Phys. Rev. Lett. 117, 120401 (2016). [CrossRef] [PubMed]

37. J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterisation of polarisation states of light via direct measurement,” Nature Photonics 7, 316 (2013). [CrossRef]

38. M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nature Communications 5, 3115 (2014). [CrossRef] [PubMed]

39. M. Malik and R. W. Boyd, Quantum Imaging Technologies, arXiv:1406.1685; Rivista del Nuovo Cimento 37, 5 (2014) p. 273

40. A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Phys. Rev. A 69, 062320 (2004). [CrossRef]

41. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162 (2004). [CrossRef] [PubMed]

42. M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert, K. Geerlings, T. Brecht, K. M. Sliwa, B. Abdo, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, “Quantum Back-Action of an Individual Variable-Strength Measurement,” Science 339, 178 (2013). [CrossRef] [PubMed]

43. K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi, “Observing Single Quantum Trajectories of a Superconducting Quantum Bit,” Nature 502, 211 (2013). [CrossRef] [PubMed]

44. D. Tan, S. J. Weber, I. Siddiqi, K. Molmer, and K.W. Murch, “Prediction and Retrodiction for a Continuously Monitored Superconducting Qubit,” Phys. Rev. Lett. 114, 090403 (2015). [CrossRef] [PubMed]

45. S. Haroche and J. M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University, 2006). [CrossRef]

46. L. Sun, A. Petrenko, Z. Leghtas, B. Vlastakis, G. Kirchmair, K. M. Sliwa, A. Narla, M. Hatridge, S. Shankar, J. Blumoff, L. Frunzio, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “ Tracking photon jumps with repeated quantum non-demolition parity measurements,” Nature 511, 444 (2014). [CrossRef] [PubMed]

47. W. Wang, L. Hu, Y. Xu, K. Liu, Y. Ma Shi-Biao Zheng, R. Vijay, Y. P. Song, L.-M. Duan, and L. Sun, “Converting quasiclassical states into arbitrary Fock state superpositions in a superconducting cavity,” Phys. Rev. Lett. 118, 223604 (2017). [CrossRef]

48. C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune, J. Raimond, and S. Haroche, “Real-time quantum feedback prepares and stabilizes photon number states,” Nature 477, 73 (2011) [CrossRef] [PubMed]

Direct measurement of the quantum state of photons in a cavity

Abstract

1. The state of a system

2. Set-up description and basic idea

3. Weak-value and state determination

4. Alternative set-up of atomic cavity QED system

5. On the ‘selective’ rotation

6. Applications and feasibility

7. Summary

Funding

References and links

Cited By

Figures (2)

Equations (14)

Optics Express