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Quantum Nondemolition Photon Detection in Circuit QED and the Quantum Zeno Effect

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Quantum Nondemolition Photon Detection in Circuit QED and the Quantum Zeno Effect PHYSICAL REVIEW A 79, 052115 ͑2009͒ Quantum nondemolition photon detection in circuit QED and the quantum Zeno effect Ferdinand Helmer,1 Matteo Mariantoni,2,3 Enrique Solano,1,4 and Florian Marquardt1 1Department of Physics, Center for NanoScience, and Arnold Sommerfeld Center, Ludwig-Maximilians-Universität, Theresienstrasse 37, D-80333 Munich, Germany 2Walther-Meissner-Institut, Bayerische Akademie der Wissenschaften, Walther-Meissner-Str. 8, D-85748 Garching, Germany 3Physics Department, Technische Universität München, James-Franck-Str., D-85748 Garching, Germany 4Departamento de Quimica Fisica, Universidad del Pais Vasco-Euskal Herriko Unibertsitatea, 48080 Bilbao, Spain ͑Received 17 December 2007; revised manuscript received 15 January 2009; published 20 May 2009͒ We analyze the detection of itinerant photons using a quantum nondemolition measurement. An important example is the dispersive detection of microwave photons in circuit quantum electrodynamics, which can be realized via the nonlinear interaction between photons inside a superconducting transmission line resonator. We show that the back action due to the continuous measurement imposes a limit on the detector efficiency in such a scheme. We illustrate this using a setup where signal photons have to enter a cavity in order to be detected dispersively. In this approach, the measurement signal is the phase shift imparted to an intense beam passing through a second cavity mode. The restrictions on the fidelity are a consequence of the quantum Zeno effect, and we discuss both analytical results and quantum trajectory simulations of the measurement process. DOI: 10.1103/PhysRevA.79.052115 PACS number͑s͒: 03.65.Ta, 03.65.Xp, 42.50.Lc I. INTRODUCTION to be already prepared in a certain state. Although the quan- tum Zeno effect also plays a role in the CPB dynamics in Quantum nondemolition ͑QND͒ measurements are ideal ͓19͔, the conclusions and reasoning for the optimal measure- projective measurements that reproduce their outcome when ment rate in our case are different. repeated ͓1,2͔. Using them, it is possible to measure the state Such dispersive photon detection schemes could have a of a system with the minimal disruption required by quantum particularly important application in the field of quantum mechanics. Recent successful experimental demonstrations electrodynamics ͑QED͒ in superconducting circuits. Circuit of QND detection for superconducting qubits and microwave QED setups ͓12,20–22͔ offer the possibility to implement photons ͓3–6͔ are both of fundamental interest and crucial ideas of quantum optics using microwave photons propagat- for the development of quantum communication and infor- ing in transmission lines and they also could form the basis mation processing. When QND detection is applied continu- for novel architectures capable of scalable, fault-tolerant ously to a system that would otherwise undergo some intrin- quantum computing ͓23͔. Proposals for generating nonclas- sic dynamics, quantum jumps are observed, tracing the sical photon states exist or have been implemented quantum evolution in real time ͓7–11͔. As a consequence, the ͓4,21,24,25͔. However, the on-chip single-shot detection of dynamics tends to be frozen, a result now known as the itinerant photons is still missing. quantum Zeno effect ͓12–17͔. Recent experiments have demonstrated dispersive qubit In the present paper, we show that the interplay of these detection ͓3͔ and measurements of photon statistics ͓4͔. phenomena may put interesting constraints on the detection Based on the dispersive coupling strengths demonstrated of itinerant quanta. The specific minimal example we will there, one could employ a superconducting qubit inside a discuss concerns the continuous dispersive QND detection of microwave transmission line resonator to induce a nonlinear single photons passing through a cavity. The crucial distinc- coupling between two modes of the resonator ͑or alterna- tion to be recognized is the following. For localized quanta tively couple two cavities͒, thus creating a dispersive photon ͑e.g., a photon already created inside a cavity ͓8,18͔͒, the detector of the type discussed here. Even though we find a quantum Zeno effect could presumably only enhance the de- limit for the photon detection fidelity of about 30% for the tection by suppressing the decay. However, this no longer simplest scheme considered here, this would still be a con- holds for the detection of itinerant quanta if we require that siderable improvement over the present state of the art. our detector is always working and can detect the quantum without knowing the arrival time in advance. As we will II. MODEL show, in this case the unavoidable back action of the mea- surement device produces a quantum Zeno effect, suppress- We investigate a QND scheme utilizing the nonlinear ing the fidelity of measurements. Kerr-type coupling ͓26–28͔ of two discrete localized modes Similar measurement physics is involved when trying to of a bosonic field. The presence of a quantum inside the read out the phonon state of a nanoresonator using a single signal mode gives rise to a frequency shift of the detection electron transistor ͑SET͒ and a Cooper pair box ͑CPB͒ as a mode, which can be observed dispersively via the phase shift coupler ͓19͔. However, there are some distinct differences to of a beam transmitted through that mode ͑see Fig. 1͒. In turn, the scheme considered here. The most prominent difference the signal mode frequency fluctuates due to the detection is that we analyze a scheme to detect itinerant quanta ͑in our beam’s shot noise. As a consequence, the incoming signal case photons͒ and therefore do not assume the signal mode photon will be reflected with a probability that rises with 1050-2947/2009/79͑5͒/052115͑10͒ 052115-1 ©2009 The American Physical Society HELMER et al. PHYSICAL REVIEW A 79, 052115 ͑2009͒ III. STOCHASTIC MASTER EQUATION In this section we derive a stochastic master equation for this measurement situation. Starting from a standard Lind- blad master equation, involving both driving and decay for the two modes, we can derive a stochastic master equation that keeps track of the individual measurement signal ob- tained in each individual run of the experiment. Before pre- FIG. 1. ͑Color online͒ Schematic sketch of the model. Two cav- senting that equation ͓Eq. ͑2͒ below͔, let us briefly describe ity modes are coupled anharmonically. The detector mode is irradi- in qualitative terms the purpose of using a stochastic master ated with a strong coherent field that suffers a phase shift whenever equation. Imagine a general measurement situation where the a photon is present in the signal mode. system in question cannot be directly projectively measured, but instead one has to couple a measurement apparatus ͑most ͒ coupling strength and detection beam intensity. often a meso- or macroscopic apparatus to the system. Only This incarnation of the quantum Zeno effect generates a this apparatus can then be projectively measured. Assuming trade-off that yields the highest detection efficiency at inter- that apparatus and coupling to the system are indeed suitable mediate coupling strengths. In that way, such dispersive to extract information about a certain system observable via schemes for itinerant quanta turn out to be similar to weak a projective measurement on an apparatus observable, during measurements using general linear detectors and amplifiers such a measurement the following happens: We let system ͓2͔. and apparatus, initially assumed to be in a product state, We proceed as follows. ͑i͒ We numerically evaluate quan- interact according to the coupling Hamiltonian for some time ⌬ tum jump trajectories for the phase-shift signal in a minimal interval t. Afterwards, the state will have changed, gener- model of a QND photon detector and analyze the fraction of ally into an entangled state between system and measure- detected photons, observing the trade-off described above. ment apparatus. It is by this entanglement that information ͑ii͒ We interpret these findings using an analytical approxi- about the system state is transferred into the measurement mation. ͑iii͒ Finally, we briefly comment on possible experi- apparatus. Then we imagine to projectively measure the ͑ ͒ mental realizations. measurement apparatus. This has several consequences: i We consider a system of two cavity modes with a Kerr- the measurement apparatus experiences a projection onto one ͑ ͒ type coupling of strength g, of the usually smoothly distributed eigenstates of the appa- ratus observable; ͑ii͒ the system and the measurement appa- ratus are now again in a product state; ͑iii͒ this evidently also ͑ 1 1 changed the system state this is called measurement back ˆ ប␻ͩ ͪ ប␻ ͩ ͪ ប ˆ ͒ H = nˆ + + det nˆ det + + gnˆnˆ det + Hdrive+decay. action , however not necessarily into an eigenstate of the 2 2 system. It is important to realize that this change is also ran- ͑1͒ dom, as it is conditional on the measurement result we ob- tained when projectively measuring the apparatus. Very often and especially in the situation we consider, the These modes might represent two different electromagnetic coupling between system and measurement apparatus is field modes inside an optical or microwave cavity, the modes weak, meaning that during one measurement as we just de- of two adjacent cavities ͓29͔, or even two anharmonically scribed, the system state only changes very little. This is then coupled modes of a nanomechanical resonator. Photons in referred to as a “weak” measurement. When we now imagine the signal mode ͑frequency ␻, number operator nˆ͒ and the that we take the continuum limit ⌬t→0, it is intuitively clear ͑␻ ͒ detector mode det, nˆ det decay by leaking out of the cavity. that the infinitesimally small kicks the system experiences The anharmonic Kerr-type coupling arises generically when should lead to stochastic dynamics. We can therefore expect introducing any nonlinear medium, such as an atom, a qubit, ͑i͒ having a stochastic term enter the master equation which or a quantum dot, into a cavity and has been studied for the leads to ͑ii͒ diffusionlike behavior of the system state and purpose of QND measurements in quantum optics ͓26–28͔.
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