Tracking the dynamics of an ideal quantum measurement

Tracking the Dynamics of an Ideal Quantum Measurement Fabian Pokorny ,1,* Chi Zhang ,1,†Gerard Higgins ,1,‡Adán Cabello ,2,3,§ Matthias Kleinmann ,4,5,∥and Markus Hennrich 1,¶ 1Department of Physics, Stockholm University, 10691 Stockholm, Sweden 2Departamento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain 3Instituto Carlos I de Física Teórica y Computacional, Universidad de Sevilla, E-41012 Sevilla, Spain 4Department of Theoretical Physics, University of the Basque Country UPV/EHU, E-48080 Bilbao, Spain 5Naturwissenschaftlich–Technische Fakultät, Universität Siegen, 57068 Siegen, Germany (Received 28 March 2019; revised manuscript received 4 September 2019; accepted 24 January 2020; published 25 February 2020; corrected 4 March 2020) The existence of ideal quantum measurements is one of the fundamental predictions of quantum mechanics. In theory, an ideal measurement projects a quantum state onto the eigenbasis of the measurement observable, while preserving coherences between eigenstates that have the same eigenvalue. The question arises whether there are processes in nature that correspond to such ideal quantum measurements and how such processes are dynamically implemented in nature. Here we address this question and present experimental results monitoring the dynamics of a naturally occurring measurement process: the coupling of a trapped ion qutrit to the photon environment. By taking tomographic snapshots during the detection process, we show that the process develops in agreement with the model of an ideal quantum measurement with an average fidelity of 94%. DOI: 10.1103/PhysRevLett.124.080401 Introduction.—What is an ideal measurement in quantum mechanics? What are its inner workings? How does the quantum state change because of it? These have been central questions in the development of quantum mechanics [1]. Notably, today’s accepted answer to the latter question is conceptually different from the one given in the first formalization of quantum mechanics by von Neumann [2]. Then, it was thought that an ideal measurement on a quantum system would inevitably destroy all quantum superpositions. Later, Lüders pointed out [3] that certain superpositions should survive, so that a sequence of ideal measurements would preserve quantum coherence. Lüders’s version is the one accepted today. Even though there is agreement on the theoretical description of an ideal measurement, it is a fundamental question whether, and, if so, how ideal measurements occur as natural processes. This is an especially sensitive question, as measurements do not fall into the domain of unitary time evolution. In this Letter, we demonstrate a natural process that is considered to be an ideal measurement and monitor its dynamics by taking a sequence of snapshots while the process is occurring. These snapshots are tomographically complete and allow us to compare the experimental results with the theoretical prediction of an ideal measurement. As the natural process, we choose a fluorescence measurement of a trapped atomic ion in contrast to a fine-tuned process in an artificially created system [4]. The time evolution of the process then provides insight into how the measurement process occurs naturally. Ideal measurements model an ideal implementation of a quantum observable. In the case of a discrete spectrum, a measurement of the observable Ayields values akaccording to the spectral decomposition A¼PkakΠk, where Πkare mutually orthogonal projections summingto identity, and ak are distinct real numbers. Any measurement requires an interaction of the measurement apparatus with the system and this interaction affects the state of the system. This is true, independent of whether or not the measurement result is recorded by an observer. However, the effect on the state depends on the experimental realization. In practice, a measurement is often a rather violent process that “destroys” the quantum system, for example, the detection of a photon. A Lüders process is the ideal implementation of a measurement in which the state of the system is transformed according to [5] ΞA∶ρ↦X k ΠkρΠk:ð1Þ (For simplicity, we assume that the observer ignores the measurement outcome.) From a theoretical perspective, this process is truly special. On the one hand, it is the only process implementing a measurement of Athat does not disturb any subsequent refined measurement B[6,7]; that Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by Bibsam. PHYSICAL REVIEW LETTERS 124, 080401 (2020) 0031-9007=20=124(8)=080401(5) 080401-1 Published by the American Physical Society is, a measurement Bwhere each outcome lis at most as likely as a certain outcome klof A, for all states ρ. This implies that, whenever two observables Aand Bare compatible, AB ¼BA, then the respective Lüders processes do not disturb each other, ΞAΞB¼ΞBΞA[3,5]. On the other hand, ΞAis universal: any process ΛAdescribing a measurement of Acan be written as ΛA∶ρ↦ PkΦkðΠkρΠkÞ, where Φkare some probability-preserving processes [5]. While these properties constitute strong theoretical arguments for the special role of the Lüders process, this does not imply that Lüders processes occur in nature, and it does not tell us how nature implements them. An important aspect of the Lüders process is that it leaves any quantum superposition of degenerate eigenstates unaffected. In this sense, the existence of Lüders processes is a nontrivial prediction of quantum mechanics that is usually taken for granted rather than analyzed thoroughly. The structure of a dynamical realization of a Lüders process is best understood by considering the canonical model for measurements [5,8]: A system in state ρis brought into contact with a pointer system, which is in state jΦi. When in contact, the two systems interact via the Hamiltonian HAfor a time τand, after separating the two systems, the pointer system is measured in some fixed basis jωji. This yields the process MA∶ρ↦X jhωjje−iHAτ=ℏðρ⊗jΦihΦjÞeiHAτ=ℏjωji:ð2Þ However, the class of processes covered by Eq. (2) is so general that it can be shown [5] that any quantum process, including the Lüders processes ΞA, can be modeled by Eq. (2). Indeed, if we are able to engineer interactions and pointer systems at will, as in, for example, quantum processors [9,10], the fact that a process cannot be approximated by Eq. (2) would be in contradiction of our present understanding of those systems. Usually, experimental measurement procedures are not Lüders processes, because this would require a careful fine-tuning of the Hamiltonian HAor else the degeneracy of the observable would be lifted by experimental imperfections, which then leads to a process as the one envisioned by von Neumann, in which all coherences are destroyed. For example, in a recent experiment [4], a Lüders-type process was implemented on an artificial three-level quantum system with an overall fidelity of 94%. The interaction to the pointer state (stored in a microwave field transmitted through a cavity) was tuned in such a way that the coherence in a two-dimensional subspace was preserved. In this Letter, we use a fluorescence measurement on an ion to implement a Lüders process. Since the Hamiltonian is given by the natural interaction between the ion and the photon field, we are then particularly interested in the dynamics that leads to the implementation of the Lüders process. Fluorescence measurements.—The best candidate for a natural Lüders process is an interaction-free measurement, that is, a measurement of an observable of the form A¼jϕihϕj, where the event “detection”corresponds to the eigenvalue 1 and the event “no detection”corresponds to the eigenvalue 0. There, it can be expected that the nodetection event does not require any interaction with the eigenspace with eigenvalue 0, and hence superpositions in this subspace are preserved. Our choice for an experiment within which to identify a Lüders process is the coupling of a single trapped ion to the photon environment. To explain why this is a good candidate, it is useful to present a simplified description of the specific physical process in our experiment. For a more accurate theoretical model of this process, see Ref. [11] and the Supplemental Material [12]. We prepare an ion in a superposition of three electronic states: j0i,j1i, and j2i. By driving the transition j0ito a short-lived excited level jei, we facilitate the emission of photons into the environment, via the natural decay jeijn¼0i→j0ijn¼1i. Here, jn¼0iis the initial state of the photon environment and jn¼1iis the state of the photon environment after a photon has been emitted by the decay. We write the initial state of the system and the photon environment as jΨi¼ðα0j0iþα1j1iþα2j2iÞjn¼0i:ð3Þ Since only the level j0iparticipates in the driving j0i↔jei and the subsequent decay, the state after the driving is jΨmi¼α0j0iðg0jn¼0iþg1jn¼1iÞ þðα1j1iþα2j2iÞjn¼0i;ð4Þ where g1is related to the probability of photon scattering by Pscatt ¼jg1j2, and jg0j2þjg1j2¼1. Ignoring the photon environment, we obtain the reduced state ρm¼0 B @ jα0j2α0α 1g0α0α 2g0 α 0α1g 0jα1j2α1α 2 α 0α2g 0α 1α2jα2j2 1 C A:ð5Þ The coherence between levels j1iand j2iis preserved, while the coherences between j0iand j1iand between j0i and j2idecay with jg0j. Intuitively, if at least one photon is scattered into the environment, then jg0j2¼0and the coupling to the photon environment corresponds to the measurement of the qutrit projection ðj0ih0j;1−j0ih0jÞ. In general, g0depends on the strength of the interaction and, for a weak or short interaction, jg0j2is between zero and one. These intermediate values are then given by the evolution leading to the Lüders process. Because the probability for scattering is PHYSICAL REVIEW LETTERS 124, 080401 (2020) 080401-2 Pscatt ¼1−jg0j2, the implemented measurement is the generalized measurement ðE1;E 0Þwith E1¼Pscattj0ih0j and E0¼1−E1. The numerical value of g0depends on the experimental configuration, and its computation requires a more rigorous model of the measurement process. A formula for g0is given in Eq. (7) below and the interaction strength is varied in the experiment such that Pscatt takes values between 0.33 and 1.0. Since our computation of the state ρmis generic for any initial state, we can read off the measurement process Λmas Λm∶ρ↦ffiffiffiffiffiffi E1 pρffiffiffiffiffiffi E1 pþGffiffiffiffiffiffi E0 pρffiffiffiffiffiffi E0 pG†;ð6Þ with G¼eiφj0ih0jþj1ih1jþj2ih2jand φ¼argðg0Þ.For Pscatt →1, this process implements the Lüders process ΞA for the observable A¼j0ih0jand hence constitutes an ideal measurement. For Pscatt <1, the process is a transitional form between ΞAand the trivial process ρ↦ρ. Then, not only is the coherence between j1iand j2ipreserved, but also partially the coherences between j0iand the other two qutrit levels. The occurrence of the terms ffiffiffiffiffiffi Ek pρffiffiffiffiffiffi Ek pin Λm is in accordance with the canonical form of such transitional processes [5], while the phase eiφintroduced by the unitary Gis specific to this measurement process. Experimental setup.—Our setup consists of a single 88Srþ ion confined in a linear Paul trap, similar to Ref. [13]. Figure 1(a) shows the level scheme of the ion with the qutrit states encoded in electronic states of the ion. The measurement process is implemented using a pulse of 422 nm laser light, which facilitates coupling of the electronic state j0i and the photon environment, while the qutrit levels j1iand j2iare left unaffected. We use the same laser light during the fluorescence detection step. The variable g0is obtained from a quantum optical model of the measurement (see the Supplemental Material [12]). This yields g0≈exp − Ω2 2Γþ4iΔt· exp ðiϕrÞ;ð7Þ where Ωis the Rabi frequency driving the j0i↔jei transition, Γ¼2π×21.65 MHz [14] is the decay rate of state jei,Δ¼2π×ð52ÞMHz is the detuning from resonance, t¼1μs is the length of the laser pulse, and the phase ϕrresults from the ac-Stark shifts induced by repump laser fields at 422 and 1092 nm, which are present during the measurement process. The intensities and detunings of the repump fields are tuned such that Imðg0Þ≈0 (see Supplemental Material [12]). Values of Ωand Δare determined from the 5S1=2↔5P1=2spectral line shape. We track the dynamics of the measurement process by carrying out process tomography as the power of the 422 nm laser used for coupling j0i↔jeiis varied. For the process tomography [Fig. 1(b)], the ion is prepared in j0i, then rotated (unitary Ui) to one of the nine initial states jψiiusing pulses of 674 nm laser light (see the Supplemental Material [12]). Then the measurement process is carried out; a 422 nm laser pulse is applied for 1μs. Finally, the state tomography is carried out by applying rotation U† jfollowed by fluorescence detection for 500 μs. The measurement process uses a shorter pulse length than the fluorescence detection step, since the photons scattered during the measurement process do not need to be detected. During state tomography the rotation U† jfollowed by a measurement of ðj0ih0j;1−j0ih0jÞ acts as measurement operator ðjψjihψjj;1−jψjihψjjÞ. The nine initial states are each measured by the nine different operators; this is sufficient to characterize the experimental process. Results.—For analyzing the experiment, a process Λis conveniently represented by its Choi matrix [5], χ¼X 2 i;j¼0 Λ½jiihjjsys⊗jiihjjaux:ð8Þ (a) (b) FIG. 1. Experimental realization of the process tomography of a Lüders measurement. (a) Level scheme of 88Srþ. A magnetic field of 0.36 mT applied along the trap axis splits the mJ¼−5=2and mJ¼−3=2Zeeman sublevels of the metastable states 4D5=2by 2π×6.0MHz. Together with the 5S1=2mJ¼−1=2ground state, these states form the qutrit basis states j0i,j1i, and j2i. The qutrit transitions are driven by a narrow-linewidth laser at 674 nm (linewidth <2π×600 Hz) tuned to either the j0i↔j1ior j0i↔j2itransition. A 422 nm laser driving the j0i↔jei≡ 5P1=2transition is used to induce the Lüders measurement process and the fluorescence detection during state tomography. To prevent loss of population from the qutrit subspace during the measurement process, 1092 nm laser light and circularly polarized 422 nm laser light are used to repump population from 4D3=2 and 5S1=2mJ¼þ1=2to j0i. (b) Experimental sequence for the process tomography. First, the ion is prepared in one of nine initial states (see Supplemental Material [12]) followed by a measurement that implements snapshots of an ideal measurement process, and finally, state tomography is performed using a qutrit rotation and fluorescence detection. PHYSICAL REVIEW LETTERS 124, 080401 (2020) 080401-3 This matrix is positive semidefinite for any physical process and yields a complete characterization of Λ, since Λ½ρ¼traux½χð1⊗ρTÞ with ρT¼Pi;j jiihjjhjjρjii.IfΛ is a probability-preserving process, that is, trðΛ½ρÞ ¼ trðρÞ¼1, then the Choi matrix obeys trsysðχÞ¼1aux. For the measurement process Λmin this Letter, the Choi matrix reads χm¼j00ih00jþjξihξjþg0j00ihξjþg 0jξih00j;ð9Þ where jξi¼j11iþj22i. The system is prepared in jψii, the process Λmis carried out, and subsequently, the system is measured using the observable jψjihψjj. This is repeated N¼1000 times for each i,j, with measurement outcome 1 occurring ni;j times, and thus resulting in the experimental outcome frequency fi;j ¼ni;j=N. We use fi;j to reconstruct the experimental Choi matrix χexp with χexp ¼arg minχPi;jðpi;j −fi;jÞ2, where the probabilities pi;j are given by pi;j ¼hψjjΛ½jψiihψijjψji ¼tr½χjψjihψjj⊗jψiihψijT:ð10Þ In Fig. 2, we show a graphical representation of the different snapshots, namely, at about 33%, 56%, 91%, and 100% of the maximal measurement interaction strength. For the weakest interaction, the process is very similar to the trivial channel (with only hiijχjjji¼1). With increasing strength of the measurement interaction, the terms responsible for preserving the coherence between j0i and j1i, and j0iand j2i(i.e., h11jχj00iand h22jχj00i), decay more and more. This shows that, with increasing interaction strength, the coherence between j0iand j1iis more and more destroyed, while any superposition of j1iand j2iremains mostly unaltered. For further illustration, we present the evolution of the quantum states ð1=ffiffiffi 2 pÞðj1iþij2iÞ and ð1=ffiffiffi 2 pÞðj0iþij2iÞ in Fig. 3 of the Supplemental Material [12]. There, the loss of the coherence involving j0iand the preservation of the coherence between j1iand j2ican be observed directly. In addition, we calculate the Uhlmann fidelity (rescaled by 1 9) of the experimental Choi matrix and the model to be, on average, 94%. For details, see the Supplemental Material [12]. χexp does not perfectly obey the probability-preserving constraint trsys χ¼1aux, due to statistical fluctuations and imperfections in the experimental setup. A likelihood ratio test shows that the deviation from trsys χ¼1aux is statistically significant by between 4 and 8 standard deviations. This can be attributed to imperfect state preparation. While the effect is statistically significant, it is clear from the high number of samples that the systematic deviation is overall very small and can be neglected. A comparison of the χexp reconstructed from experimental data with our model predictions, including the respective uncertainties, can be found in the Supplemental Material [12]. Conclusion.—Fluorescence detection is the standard way to measure a qubit in an ion trap or in similar setups that enable quantum information processing. This process is a prime example of an ideal measurement process: The system that is subject to the measurement is brought into contact with a macroscopic pointer state by facilitating a strong interaction between state j0iof the system and the photon environment. A measurement of the photon environment then reveals the state of the system. The predictions of quantum mechanics for ideal measurements become apparent when a qutrit is employed: Any coherence between states j1iand j2i, which are not coupled to the photon environment, should persist. We verified this property by performing process tomography of the coupling of electronic states of a single trapped ion to the photon environment. The theoretically ideal measurement process only occurs in the case of a strong interaction. If the interaction is weakened, then an intermediate process between the ideal measurement process and the identity process occurs. In the mathematical theory of measurements [5], such ideal measurements have a canonical “square root”form; see (a) (b) (c) (d) FIG. 2. Choi matrices χexp reconstructed from the experimental data. The colors indicate the phase, g0is calculated according to our model, see Eq. (7). (a)–(d) The coupling strength of j0ito the photon environment is increased and coherences involving state j0iare destroyed. (d) Similar to the ideal Lüders process, and from (d) →(a) the process becomes more similar to the trivial process ρ↦ρ. PHYSICAL REVIEW LETTERS 124, 080401 (2020) 080401-4 Eq. (6) with G¼1. The quantum optical model for a weak measurement predicts a similar form. In process tomographies for those intermediate processes, we obtain an overall process fidelity with the predicted model of approximately 94%. We thus quantitatively demonstrated how a measurement can be implemented in a natural process while preserving coherence. By taking snapshots of the measurement process itself, we also demonstrated that this process is not instantaneous. A similar conclusion was reached in [15],in which it was found that quantum jumps evolve coherently and are not instantaneous. A future direction of research is to test the predictions of ideal quantum measurements beyond what we have tested here: In the current experiment, the coherence-preserving measurement works relatively effortlessly, since the measurement process only affects a single state of our qutrit. This corresponds to an interaction-free measurement, where it is only measured whether or not the system is in state j0i. It could also be possible to find an ideal measurement process in nature, where all eigenvalues of the measured observable are twofold degenerate, and therefore, the coherence between the degenerate eigenstates needs to be preserved. Whether such processes exist as natural processes and can be implemented with a fidelity comparable to our experiment is an open question. We thank Mohamed Bourennane and Markus Müller for discussions. This work was supported by the Spanish Ministry of Economy, Industry and Competitiveness MINECO and the European Regional Development Fund FEDER: European Regional Development Fund through Grants No. FIS2017-89609-Pand No. FIS2015-67161-P, by the FQXi Large Grant “The Observer Observed: A Bayesian Route to the Reconstruction of Quantum Theory,”by the EU (ERC Consolidator Grant No. 683107/TempoQ and ERC Starting Grant No. 258647/GEDENTQOPT), by the Basque Government (Grant No. IT986-16), by the Swedish Research Council (Trapped Rydberg Ion Quantum Simulator), by QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union's Horizon 2020 Programme (ERyQSenS), and by the project “Photonic Quantum Information”(Knut and Alice Wallenberg Foundation, Sweden). *[email protected] †[email protected].se ‡[email protected].se §[email protected] ∥[email protected] ¶[email protected] [1] Quantum Theory and Measurement, edited by J. A. Wheeler and W. H. 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Correction: The license statement contained an omission and has been fixed. PHYSICAL REVIEW LETTERS 124, 080401 (2020) 080401-5