### 56015 results for qubit oscillator frequency

Contributors: Ku, Li-Chung, Yu, Clare C.

Date: 2004-08-31

The **qubit**-TLS system starts in its ground state at t = 0 . A microwave π or 3 π pulse (from 0 to 5 ns) puts the **qubit**-TLS system in the **qubit** excited state | 1 that is a superposition of the two entangled states | ψ ' 1 ≡ | 0 e + | 1 g / 2 and | ψ ' 2 ≡ | 0 e - | 1 g / 2 . After the microwaves are turned off, the occupation probability starts **oscillating** coherently. Values of g T L S indicated in the figure are normalized by ℏ ω 10 . The rest of the parameters are the same as in Fig. 1. (a) No energy decay of the excited TLS, i.e., τ p h = ∞ . Coherent **oscillations** with various values of g T L S . (b) **Oscillations** following a π -pulse with τ p h = 40 ns and various values of g T L S . (c) **Oscillations** following a π -pulse and a 3 π -pulse with τ p h = 40 ns and g T L S = 0.004 . The dip in the dot-dash line is one and a half Rabi cycles....(a)-(b): Rabi **oscillations** in the presence of 1/f noise in the **qubit** energy level splitting. Dotted curves show the Rabi **oscillations** without the influence of noise. Panel (c) shows the two noise power spectra S f ≡ | δ ω 10 f / ω 10 | 2 of the fluctuations in ω 10 that were used to produce the solid curves in panels (a) and (b). Rabi **frequency** f R = 0.1 GHz....where the **qubit** energy levels ( | 0 and | 1 ) are the basis states and the noise is produced by a single TLS. Our calculations are oriented to the experimental conditions and the results are shown in Fig. fig:RTN. In Fig. fig:RTNa-c the characteristic fluctuation rate t T L S -1 = 0.6 GHz. Panel fig:RTNa shows that the **qubit** essentially stays coherent when the level fluctuations are small ( δ ω 10 / ω 10 = 0.001 ). Panel fig:RTNa shows that when the level fluctuations increase to 0.006, the Rabi **oscillations** decay within 100 ns. The Rabi relaxation time also depends on the Rabi **frequency** as panel fig:RTNc shows. The faster the Rabi **oscillations**, the longer they last. This is because the low-**frequency** noise is essentially constant over several rapid Rabi **oscillations** . Alternatively, one can explain it by the noise power spectrum S I f . Since the noise from a single TLS is a random process characterized by a single characteristic time scale t T L S , it has a Lorentzian power spectrum...Experimentally, the two TLS decoherence mechanisms (resonant interaction and low-**frequency** level fluctuations) can both be active at the same time. We have calculated the Rabi **oscillations** in the presence of both of these decoherence sources by using the **qubit**-TLS Hamiltonian in eq. ( eq:ham) with a fluctuating ω 10 t that is generated in the same way and with the same amplitude as in Figure fig:RTNb. We show the result in Fig. fig:RTN_decay. By comparing Fig. fig:RTN_decay with Fig. 2b, we note that adding level fluctuations reduces the Rabi amplitude and renormalizes the Rabi **frequency**. The result in Fig. fig:RTN_decay is closer to what is seen experimentally ....Solid line represents Rabi **oscillations** in the presence of both TLS decoherence mechanisms: resonant interaction between the TLS and the **qubit**, and low **frequency** **qubit** energy level fluctuations caused by a single fluctuating TLS. The TLS couples to microwaves ( g T L S / ℏ ω 10 = 0.008 ) and the energy decay time for the TLS is τ p h = 10 ns, the same as in Fig. 2b. The size of the **qubit** level fluctuations is the same as in Fig. fig:RTNb. The dotted line shows the unperturbed Rabi **oscillations**....We do not expect Rabi **oscillations** to be sensitive to noise at **frequencies** much greater than the **frequency** of the Rabi **oscillations** because the higher the **frequency** f , the smaller the noise power and because the Rabi **oscillations** will tend to average over the noise. Rabi dynamics are sensitive to the noise at **frequencies** comparable to the Rabi **frequency**. In addition, the characteristic fluctuation rate plays an important role in the rate of relaxation of the Rabi **oscillations**. It has been shown that t T L S -1 can be thermally activated for TLS in a metal-insulator-metal tunnel junction. If the thermally activated behavior applies here, the decoherence time τ R a b i should decrease as temperature increases. In Fig. fig:RTNd, the characteristic fluctuation rate has been lowered to 0.06 GHz (which is much lower than ω 10 / 2 π ≈ 10 GHz). The noise still causes **qubit** decoherence but affects the **qubit** less than in Fig. fig:RTNc. Fig. fig:RTN shows that the noise primarily affects the Rabi amplitude rather than the phase....Noise and decoherence are major obstacles to the implementation of Josephson junction **qubits** in quantum computing. Recent experiments suggest that two level systems (TLS) in the oxide tunnel barrier are a source of decoherence. We explore two decoherence mechanisms in which these two level systems lead to the decay of Rabi **oscillations** that result when Josephson junction **qubits** are subjected to strong microwave driving. (A) We consider a Josephson **qubit** coupled resonantly to a two level system, i.e., the **qubit** and TLS have equal energy splittings. As a result of this resonant interaction, the occupation probability of the excited state of the **qubit** exhibits beating. Decoherence of the **qubit** results when the two level system decays from its excited state by emitting a phonon. (B) Fluctuations of the two level systems in the oxide barrier produce fluctuations and 1/f noise in the Josephson junction critical current I_o. This in turn leads to fluctuations in the **qubit** energy splitting that degrades the **qubit** coherence. We compare our results with experiments on Josephson junction phase **qubits**....Rabi **oscillations** of a resonantly coupled **qubit**-TLS system with ε T L S = ℏ ω 10 . There is no mechanism for energy decay. Occupation probabilities of various states are plotted as functions of time. (a) P 1 is the occupation probability in the **qubit** state | 1 ; (b) P 0 g is the occupation probability in the state | 0 , g ; (c) P 0 e is the occupation probability of the state | 0 , e ; (d) P 1 g is the occupation probability of the state | 1 , g ; and (e) P 1 e is the occupation probability of the state | 1 , e . Notice the beating with **frequency** 2 η . Throughout the paper, ω 10 / 2 π = 10 GHz. Parameters are chosen mainly according to the experiment in Ref. : η / ℏ ω 10 = 0.0005 , g q b / ℏ ω 10 = 0.01 , and g T L S = 0 . The dotted line in panel (a) shows the usual Rabi **oscillations** without resonant interaction, i.e. η = 0 ....Solid lines show the Rabi **oscillation** decay due to **qubit** level fluctuations caused by a single fluctuating two level system trapped inside the insulating tunnel barrier. The TLS produces random telegraph noise in I o that modulates the **qubit** energy level splitting ω 10 . (a) The level fluctuation δ ω 10 / ω 10 = 0.001 . The characteristic fluctuation rate t T L S -1 = 0.6 GHz. The Rabi **frequency** f R = 0.1 GHz. The dotted lines show the usual Rabi **oscillations** without any noise source. (b) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.6 GHz, and f R = 0.1 GHz. The dotted lines show the usual Rabi **oscillations** without any noise source. (c) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.6 GHz, and f R = 0.5 GHz. (d) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.06 GHz, and f R = 0.5 GHz. Note that the scales of the horizontal axes in (a)-(c) are the same. They are different from that in (d)....We first consider the case of strong driving with g T L S = 0 and with the TLS in resonance with the **qubit**, i.e. ε T L S = ℏ ω 10 . If there is no coupling between the **qubit** and the TLS, then the four states of the system are the ground state | 0 , g , the highest energy state | 1 , e , and two degenerate states in the middle | 1 , g and | 0 , e . If the **qubit** and the TLS are coupled with coupling strength η , the degeneracy is split by an energy 2 η . Figure fig:QB4L shows the coherent **oscillations** of the resonant **qubit**-TLS system. We define a projection operator P ̂ 1 ≡ | 1 , g 1 , g | + | 1 , e 1 , e | so that P ̂ 1 corresponds to the occupation probability of the **qubit** to be in state | 1 as in the phase-**qubit** experiment. Instead of being sinusoidal like typical Rabi **oscillations** (the dotted curve), the occupation probability P 1 exhibits beating (Fig. 1a) because the two entangled states that are linear combinations of | 1 , g and | 0 , e have a small energy splitting 2 η , and this small splitting is the beat **frequency**. Without any source of decoherence, the resonant beating will not decay. Thus far the beating phenomenon has not yet been experimentally verified. The lack of experiment ... Noise and decoherence are major obstacles to the implementation of Josephson junction **qubits** in quantum computing. Recent experiments suggest that two level systems (TLS) in the oxide tunnel barrier are a source of decoherence. We explore two decoherence mechanisms in which these two level systems lead to the decay of Rabi **oscillations** that result when Josephson junction **qubits** are subjected to strong microwave driving. (A) We consider a Josephson **qubit** coupled resonantly to a two level system, i.e., the **qubit** and TLS have equal energy splittings. As a result of this resonant interaction, the occupation probability of the excited state of the **qubit** exhibits beating. Decoherence of the **qubit** results when the two level system decays from its excited state by emitting a phonon. (B) Fluctuations of the two level systems in the oxide barrier produce fluctuations and 1/f noise in the Josephson junction critical current I_o. This in turn leads to fluctuations in the **qubit** energy splitting that degrades the **qubit** coherence. We compare our results with experiments on Josephson junction phase **qubits**.

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Contributors: Johansson, G., Tornberg, L., Shumeiko, V. S., Wendin, G.

Date: 2006-02-24

Resonant circuits for read-out: a) A lumped element LC-**oscillator** coupled to a driving source and a radio-**frequency** detector through a transmission line. b) The radio-**frequency** single-electron transistor measuring the charge of a charge **qubit** (SCB). The current through the SET determines the dissipation in the resonant circuit. The dissipation is determined by measuring the amplitude of the reflected signal. c) Setup for measuring the quantum capacitance of the charge **qubit**. The **qubit** capacitance influences the resonance **frequency** of the **oscillator**. The capacitance is measured by determining the phase-shift of the reflected signal....up to a constant phase depending on the length of the transmission line. Here Q is the resonator’s quality factor, which for the circuitry in Fig. ResonantCircuitsFig a) is determined by the characteristic impedance on the transmission line Z 0 through Q = ω 0 L C 2 / C c 2 Z 0 . Since there is no dissipation in the **oscillator** we have | Γ ω | = 1 . Driving the **oscillator** at the bare resonance **frequency** ω d = ω 0 the phase-difference between the ground and excited state of the **qubit** will be...The quantum capacitance of the Cooper-pair box is related to the parametric capacitance of small Josephson junctions which is a dual to the Josephson inductance. The origin of the quantum capacitance of a single-Cooper-pair box (SCB) can be understood as follows. Assume that we put a constant voltage V m on the measurement capacitance of the SCB, i.e. we put a voltage source between the open circles in Fig. ResonantCircuitsFigc. The amount of charge on the measurement capacitance q m g / e V m V g will be a nonlinear function of the voltage V m as well as the gate voltage V g and whether the **qubit** is in the ground or excited state. We may define an effective (differential) capacitance...Single-contact flux **qubit** inductively coupled to a linear **oscillator**....We discuss the current situation concerning measurement and readout of Josephson-junction based **qubits**. In particular we focus attention of dispersive low-dissipation techniques involving reflection of radiation from an **oscillator** circuit coupled to a **qubit**, allowing single-shot determination of the state of the **qubit**. In particular we develop a formalism describing a charge **qubit** read out by measuring its effective (quantum) capacitance. To exemplify, we also give explicit formulas for the readout time....At the charge degeneracy point the effective capacitance of the SCB in the ground and excited state differs by 2 C Q m a x . Imbedding the SCB in a resonant circuit as shown in Fig. ResonantCircuitsFig a) and c) we can detect the corresponding change in the **oscillators** resonance **frequency** ω 0 g / e = 1 / L C ± C Q m a x = ω 0 1 ∓ C Q m a x / 2 C , where ω 0 = 1 / L C is the bare resonance **frequency**. The voltage reflection amplitude Γ ω = V o u t ω / V d ω seen from the driving side of the transmission line can for a high quality **oscillator** be written...Circuit diagrams and 2-level energy spectrum of two basic JJ-**qubit** designs: the SCB charge **qubit** with LC-**oscillator** readout (left), and persistent-current flux **qubit** with SQUID **oscillator** readout (right). For the charge **qubit**, the control variable ϵ on the horizontal axis of the energy spectrum (middle) represents the external gate voltage (induced charge), and the splitting is given by the Josephson tunneling energy mixing the charge states. For the flux **qubit**, the variable ϵ represents the external magnetic flux. In both cases, the energy of the **qubit** can be "tuned" and the working point controlled. Away from the origin (asymptotically) the levels represent pure charge states (zero or one Cooper pair on the SCB island) or pure flux states (left or right rotating currents in the SQUID ring)....LagrangianSubsection The circuit for performing read-out through the quantum capacitance is presented in figure fig:circuit. A Josephson charge **qubit** is capacitatively coupled to a harmonic **oscillator**, which is coupled to a transmission line. Through this line, all measurement on the **qubit** is performed. We model the line as a semi-infinite line of LC-circuits in series. The working point of the Josephson junction can be chosen using the bias V g . In writing down the Lagrangian we are free to chose any quantities as our coordinates as long as they give a full description of our circuit. Since we are treating a system including a Josephson junction, the phases Φ i t = ∫ t d t ' V i t ' across the circuit elements are natural coordinates, as discussed by Devoret in ref. ...Double-well potential and energy levels of the flux **qubit** ( f q = π ). ... We discuss the current situation concerning measurement and readout of Josephson-junction based **qubits**. In particular we focus attention of dispersive low-dissipation techniques involving reflection of radiation from an **oscillator** circuit coupled to a **qubit**, allowing single-shot determination of the state of the **qubit**. In particular we develop a formalism describing a charge **qubit** read out by measuring its effective (quantum) capacitance. To exemplify, we also give explicit formulas for the readout time.

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Contributors: Neto, O. P. de Sa, de Oliveira, M. C., Caldeira, A. O.

Date: 2011-04-27

with Ω t = ω - E J ℏ c o s π Φ x t Φ 0 , as given by the second order terms from Eq. ( 10). From Eq. ( 11) it is possible to understand that as the **qubit** is brought closer to resonance with the resonator field, it will imprint an accumulated phase on it, given by Im θ ± t conditioned on the **qubit** state | ± . In Fig. fig4 we depict the numerical results for those two conditioned accumulated phases. We see that practically only when the **qubit** is in the state | - the phase in α is changed. With an appropriate accumulation of -3 π , as shown in Fig. fig4a, it is possible to create a state if the **qubit** is initially in the | - state. We have consistently checked that this approximation is indeed very good, not only for small α , if we respect a balance between the field intensity and the operation time. Moreover, we observed that around the time of optimal phase accumulation, t o p = 7.5 ns , the real part of θ ± t , related to damping or amplification, is negligible, ( ≈ 10 -3 - 10 -4 ), so that in Eq. ( 11), f ± t = 0 at the instant of measurement as shown in the Appendix, and shall not be considered from now on. To generate the superposition of coherent states for the field, we note that due to the low temperature of the system it is easy to prepare the **qubit** state initially in . Obviously the decoupled coherent state of the resonator and the **qubit** state may be written as . If the aforementioned pulse is applied to the **qubit**, coupling it to the resonator field through the evolution given by Eq. ( 10), we shall have U(t 0+ t,t 0)—0 — Pulse —- —- + —+ —2 14, or [ which is an entangled state between the **qubit** and the resonator field. Consequently, the resonator field can be left in an odd or even superposition of coherent states depending on the detection of the **qubit** state with a single electron transistor . Field state decoherence and **qubit** relaxation probing Certainly, the exact preparation of such a state is compromised by external noise. In contrast to experiments with microwave fields and Rydberg atoms, dissipative effects are most noticeable for the **qubit** states, which can flip from the ground state to the excited one and vice versa due to thermal effects and inductive coupling of the **qubit** to the external circuit. While the relaxation time of the **qubits** are of the order of 10 -6 s , the relaxation of the resonator field is of the order of 10 -3 s and can, in principle, be neglected. If compared with the time of the pulses for the accumulated phase of the field, Δ t ≈ 7.5 × 10 -9 s , the relaxation of the atom is negligible as well, but certainly will be important if any further manipulation is to be executed. So in fact the effects of dissipation are more relevant after the pulse is applied, i.e., when Φ x t = 0 , and will affect the probability to detect a given **qubit** state, and consequently the generation of an appropriate field. To understand the effects of noise in the system, we couple the **qubit** two-state to a bath of harmonic **oscillators** in an adaptation of the standard spin-boson model with Ohmic dissipation , through the Hamiltonian H = H S + H A R + H R . Here H S is given by Eq. ( s), presented in reference , with the convention at the degeneracy point , and . Eq. ( s) can be conveniently rewriten as...Probabilities of charge **qubit** detections and consequently of postselected | 0 L , or | 1 L field states. The decoherence of the preselected field state is given by 2 P 0 t - 1 ....We demonstrate how a superposition of coherent states can be generated for a microwave field inside a coplanar transmission line coupled to a single superconducting charge **qubit**, with the addition of a single classical magnetic pulse for chirping of the **qubit** transition **frequency**. We show how the **qubit** dephasing induces decoherence on the field superposition state, and how it can be probed by the **qubit** charge detection. The character of the charge **qubit** relaxation process itself is imprinted in the field state decoherence profile....Phase of the resonator coherent field, due to the time dependent interaction with the **qubit** prepared in a ) the ground state, and b ) the excited state. The pulse Φ x t **oscillates** for a half period ( Δ t ≈ 7.5 ns ) with **frequency** 8 π × 10 6 Hz ....When the atomic Hamiltonian is diagonalized, the first two energy levels as a function of the gate charge n g ≡ C g V g / 2 e are described in fig. fig2, where the vertical axis represents the energy and the horizontal represents the gate charge which is limited by the gate voltage. Changing the basis through a rotation, σ z → σ x and σ x → - σ z , and going to the rotating frame with the field **frequency**, ω , through R f = exp i ω t σ z + a a , gives...Cavity quantum electrodynamics in superconducting circuits offer a exquisite playground for quantum information processing, and has provided the first coherent coupling between an “artificial atom”, the charge **qubit**, and a field mode of a resonator . Mappings of **qubit** states , and also tests for fundamental problems, such as the Purcell effect and photon number state resolving have also been achieved. The setup employed in all those remarkable experiments is shown in Fig. fig1, where a niobium transmission line resonator is capacitively coupled to a source (on the left) and to a drain (on the right). The resonator is also capacitively coupled to the charge states of a SQUID . The advantage of employing a SQUID is that the charge states can be addressed and manipulated in such a way to be set close or far from resonance with a given resonator field mode by an externally applied classical magnetic flux. By considering only the ground and the first excited states near the charge degeneracy point, the superconducting device can be well approximated by a two level system (Fig. 1b), here addressed as a **qubit**. In this regime the Hamiltonian describing a quantized electromagnetic field mode coupled to the charge **qubit** is given by...(a) Schematic setup, with the central transmission line (resonator) capacitively coupled to the source and drain, and capacitively coupled to a SQUID-type **qubit**. (b) Variable energy levels of the **qubit** ( 0 < n g < 1 ), with an external classical magnetic flux Φ x t . The resonator field is always blue detuned from that transition. ... We demonstrate how a superposition of coherent states can be generated for a microwave field inside a coplanar transmission line coupled to a single superconducting charge **qubit**, with the addition of a single classical magnetic pulse for chirping of the **qubit** transition **frequency**. We show how the **qubit** dephasing induces decoherence on the field superposition state, and how it can be probed by the **qubit** charge detection. The character of the charge **qubit** relaxation process itself is imprinted in the field state decoherence profile.

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Contributors: Greenberg, Ya. S., Il'ichev, E., Izmalkov, A.

Date: 2005-07-21

Fast Fourier transform of at different amplitudes G / h of low-**frequency** field....As an example we show below the time evolution of the quantity σ Z t = Z t , obtained from the numerical solution of the equations ( sigmaZ), ( sigmaY), and ( sigmaX) where we take a low **frequency** excitations as G t = G c o s ω L t . The calculations have been performed with initial conditions σ Z 0 = 1 , σ X 0 = σ Y 0 = 0 for the following set of the parameters: F / h = 36 MHz, Δ / h = 1 GHz, Γ / 2 π = 4 MHz, Γ z / 2 π = 1 MHz, ϵ / Δ = 1 , Z 0 = - 1 , δ / 2 π = 6.366 MHz, ω L / Ω R = 1 . As is seen from Fig. fig1 in the absence of low **frequency** signal ( G = 0 ) the **oscillations** are damped out, while if G ≠ 0 the **oscillations** persist....The Fourier spectra of these signals are shown on Fig. fig2 for different amplitudes of low **frequency** excitation. For G = 0 the Rabi **frequency** is positioned at approximately 26.2 MHz, which is close to Ω R = 26.24 MHz. With the increase of G the peak becomes higher. It is worth noting the appearance of the peak at the second harmonic of Rabi **frequency**. This peak is due to the contribution of the terms on the order of G 2 which we omitted in our theoretical analysis....Time evolution of . (thick) G=0, (thin) G / h = 1 MHz. The insert shows the undamped **oscillations** of at G / h = 1 MHz....We have analyzed the interaction of a dissipative two level quantum system with high and low **frequency** excitation. The system is continuously and simultaneously irradiated by these two waves. If the **frequency** of the first signal is close to the level separation the response of the system exhibits undamped low **frequency** **oscillations** whose amplitude has a clear resonance at the Rabi **frequency** with the width being dependent on the damping rates of the system. The method can be useful for low **frequency** Rabi spectroscopy in various physical systems which are described by a two level Hamiltonian, such as nuclei spins in NMR, double well quantum dots, superconducting flux and charge **qubits**, etc. As the examples, the application of the method to a nuclear spin and to the readout of a flux **qubit** are briefly discussed....The comparison of analytical and numerical resonance curves calculated for low **frequency** amplitude, G / h = 1 MHz and different dephasing rates, Γ are shown on Fig. fig3. The curves at the figure are the peak-to-peak amplitudes of **oscillations** of Z t calculated from Eq. ( ZOmega) with g ˜ ω = g δ ω + ω L + δ ω - ω L / 2 , where δ ω is Dirac delta function. The point symbols are found from numerical solution of Eqs. ( sigmaZ),( sigmaY),( sigmaX). The widths of the curves depend on Γ (see the insert) and the positions of the resonances coincide with the Rabi **frequency**. A good agreement between numerics and Eq. ZOmega, as shown at Fig. fig3, is observed only for relative small low **frequency** amplitude G / h , for which our linear response theory is valid. ... We have analyzed the interaction of a dissipative two level quantum system with high and low **frequency** excitation. The system is continuously and simultaneously irradiated by these two waves. If the **frequency** of the first signal is close to the level separation the response of the system exhibits undamped low **frequency** **oscillations** whose amplitude has a clear resonance at the Rabi **frequency** with the width being dependent on the damping rates of the system. The method can be useful for low **frequency** Rabi spectroscopy in various physical systems which are described by a two level Hamiltonian, such as nuclei spins in NMR, double well quantum dots, superconducting flux and charge **qubits**, etc. As the examples, the application of the method to a nuclear spin and to the readout of a flux **qubit** are briefly discussed.

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Contributors: Osborn, K. D., Strong, J. A., Sirois, A. J., Simmonds, R. W.

Date: 2007-03-04

Power out of the resonator as a function of **frequency** for different input powers at zero flux bias....V in, rms / V out, rms versus V out, rms 2 at Φ a = 0 and the low power resonance **frequency** for zero flux bias (f=8.2423Ghz). The circles show the experimental data, and the line shows the expected theoretical result for parameters determined by fits to figure 3 and figure 4....Measured resonance **frequency** as a function of flux bias....We have fabricated and measured a high-Q Josephson junction resonator with a tunable resonance **frequency**. A dc magnetic flux allows the resonance **frequency** to be changed by over 10 %. Weak coupling to the environment allows a quality factor of $\thicksim$7000 when on average less than one photon is stored in the resonator. At large photon numbers, the nonlinearity of the Josephson junction creates two stable **oscillation** states. This resonator can be used as a tool for investigating the quality of Josephson junctions in **qubits** below the single photon limit, and can be used as a microwave **qubit** readout at high photon numbers. ... We have fabricated and measured a high-Q Josephson junction resonator with a tunable resonance **frequency**. A dc magnetic flux allows the resonance **frequency** to be changed by over 10 %. Weak coupling to the environment allows a quality factor of $\thicksim$7000 when on average less than one photon is stored in the resonator. At large photon numbers, the nonlinearity of the Josephson junction creates two stable **oscillation** states. This resonator can be used as a tool for investigating the quality of Josephson junctions in **qubits** below the single photon limit, and can be used as a microwave **qubit** readout at high photon numbers.

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Contributors: de Groot, P. C., Lisenfeld, J., Schouten, R. N., Ashhab, S., Lupascu, A., Harmans, C. J. P. M., Mooij, J. E.

Date: 2010-08-06

Controlled manipulation of quantum states is central to studying natural and artificial quantum systems. If a quantum system consists of interacting sub-units, the nature of the coupling may lead to quantum levels with degenerate energy differences. This degeneracy makes **frequency**-selective quantum operations impossible. For the prominent group of transversely coupled two-level systems, i.e. **qubits**, we introduce a method to selectively suppress one transition of a degenerate pair while coherently exciting the other, effectively creating artificial selection rules. It requires driving two **qubits** simultaneously with the same **frequency** and specified relative amplitude and phase. We demonstrate our method on a pair of superconducting flux **qubits**. It can directly be applied to the other superconducting **qubits**, and to any other **qubit** type that allows for individual driving. Our results provide a single-pulse controlled-NOT gate for the class of transversely coupled **qubits**....Coupled **qubit** system and transitions. a, Optical micrograph of the sample, showing two flux **qubits** colored in blue and red. The inset shows part of each **qubit** loop, both containing four Josephson tunnel junctions. Overlapping the **qubit** loops, in light-grey, are the SQUID-based **qubit**-state detectors. In the top right and bottom left are the two antennas from which the **qubits** are driven. b, Energy level diagram of the coupled **qubit** system. Arrows of the same color indicate transitions of the same **qubit** and are degenerate in **frequency**. c, Pulse sequence used for the coherent excitation of the **qubits**. The first pulse is resonant with **qubit** 1. The second pulse, applied from both antennas simultaneously with independent amplitudes and phases, is resonant with **qubit** 2. After the second pulse the state of both **qubits** is read out. d, The normalized transition strengths of the four transitions in b as a function of the net driving amplitudes a 1 / a 1 + a 2 for ϕ 2 - ϕ 1 = 0 . For ϕ 2 - ϕ 1 = π the dashed and solid lines are interchanged. The black dotted lines indicate the locations of the darkened transitions....Transition strength tuning and darkened transitions. a-c, Rabi **frequency** dependence on φ 2 - φ 1 for three different amplitude-ratios. The color scale represents the Fourier component of P s w , 2 τ 2 . **Qubit** 1 is prepared with a π / 2 -rotation. Markers X 0 and X 1 indicate the conditions for a darkened transition on 00 ↔ 01 and 10 ↔ 11 respectively. d-f P s w , 2 versus the durations τ 1 and τ 2 . The white solid and dashed lines indicate a π - and 2 π -rotation of **qubit** 1, respectively. The driving conditions are as marked by Y left arrow (d), X 0 (e) and X 1 (f)....Driving from a single antenna. Measurement of the state of the **qubits**, represented by switching probabilities P s w , 1 and P s w , 2 , after applying a pulse of duration τ 1 resonant with **qubit** 1, followed by a pulse of duration τ 2 resonant with **qubit** 2. a, P s w , 1 , showing coherent **oscillations** of **qubit** 1 induced by pulse 1. The white solid and dashed lines indicate a π - and 2 π -rotation respectively. For pulse 2, **qubit** 1 only shows relaxation. b, P s w , 2 , showing coherent **oscillations** induced by pulse 2. After an odd number of π -rotations on **qubit** 1, the **oscillation** **frequency** is higher than after an even number of π -rotations. For superposition states of **qubit** 1, a beating pattern of the two **oscillations** is observed. c-f, Level occupations Q of the four different levels. Note that a value of 0.2 has been added to Q 11 to improve visibility. ... Controlled manipulation of quantum states is central to studying natural and artificial quantum systems. If a quantum system consists of interacting sub-units, the nature of the coupling may lead to quantum levels with degenerate energy differences. This degeneracy makes **frequency**-selective quantum operations impossible. For the prominent group of transversely coupled two-level systems, i.e. **qubits**, we introduce a method to selectively suppress one transition of a degenerate pair while coherently exciting the other, effectively creating artificial selection rules. It requires driving two **qubits** simultaneously with the same **frequency** and specified relative amplitude and phase. We demonstrate our method on a pair of superconducting flux **qubits**. It can directly be applied to the other superconducting **qubits**, and to any other **qubit** type that allows for individual driving. Our results provide a single-pulse controlled-NOT gate for the class of transversely coupled **qubits**.

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Contributors: Hoffman, Anthony J., Srinivasan, Srikanth J., Gambetta, Jay M., Houck, Andrew A.

Date: 2011-08-12

Along this contour of constant **qubit** **frequency**, the **qubit**-cavity coupling strength, g 10 - 00 , changes due to the quantum interference of the two transmon-like halves of the TCQ. In Fig. figure2b, we measure the **frequency** response of the **qubit** while moving along the parameterized contour and can clearly see that the dressed **qubit** **frequency** remains 7.500 G H z . Moreover, in this constant power measurement, the amplitude of the response is related to the coupling strength between the **qubit** and the superconducting resonator. When the coupling is small, little response is seen because the **qubit** cannot be driven. The disappearance of a signal corresponds to the situation where the **qubit**-cavity coupling is tuned through zero....figure1 Energy level diagram for the TCQ showing the hybridized energy levels. The transitions that have a high probability of occurring are indicated by arrows. Considering that the system is primarily in the | 00 or | 10 ˜ states, single-photon transitions leading out of these two states have the maximum transition probabilities and they are indicated by arrows. The red, solid arrows indicate transitions with low coupling strengths and the blue, dashed arrows indicate transitions with high coupling strengths. The levels shown here are for the bare energy levels of the device; there are no effects of coupling to a cavity. In this work, the | 00 and | 10 ˜ states are used as the logical states of the **qubit**....In this work, we are mainly concerned with changing only the coupling strength of the **qubit** to the cavity while keeping the **qubit** **frequency** fixed. Since the flux controls allow for a wide range of coupling strengths and dressed **qubit** transition **frequencies**, it is necessary to find the control subspace that corresponds to constant dressed **qubit** **frequency**. This subspace accounts for any dispersive shifts due to changes in **qubit**-cavity coupling. To accomplish this, we use standard dispersive readout techniques of cQED: monitoring the amplitude and phase of cavity transmission while applying a second spectroscopy tone. Here, though, we keep the spectroscopy tone at a constant **frequency** of 7.500 G H z while sweeping the two control fluxes. When the dressed **qubit** **frequency**, which is a function of the two control fluxes, is resonant with the 7.500 G H z spectroscopy tone, a change in the cavity transmission is measured . Over a wide range of control voltages, it is then possible to extract a contour that corresponds to where the dressed **qubit** **frequency** is 7.500 G H z ; such a contour is shown in Fig. figure2a....figure2 (a) Observed cavity transmission versus the two control voltages with a fixed spectroscopy tone at 7.5 G H z . Both the dressed **qubit** **frequency** and coupling strength are functions of the control voltages. The contour shows where the **qubit** is resonant with the 7.5 G H z tone and is therefore driven between the ground and excited states. (b) Measured dressed **frequency** response of the **qubit** while moving along the 7.5 G H z contour in Fig. 1. The dressed **qubit** **frequency** remains constant at 7.5 G H z . The amplitude of the response is related to the coupling strength between the **qubit** and the superconducting resonator. The point where the signal disappears corresponds to coupling strengths where the **qubit** cannot be driven by the spectroscopy tone. The dotted and dashed lines indicate g 10 - 00 control values where the measurements were performed for Fig. 3....figure3 Rabi **oscillations** for three different **qubit**-cavity coupling strengths and a fixed dressed **qubit** **frequency** of 7.5 G H z . Panels (a), (b), and (c) correspond to the dashed, dot-dash, and dotted lines in Fig. 2, respectively. In (a), a spectroscopy power of -32 dBm is used. To keep the number of **oscillations** approximately the same for the lower **qubit**-cavity coupling strength in (b), the spectroscopy power is increased to -22 dBm. In panel (c), 27 dBm more power than that in (a) is applied and no **oscillations** are observed. Given the measurement noise, we put a bound of 1 / 10 of a Rabi **oscillation**....figure4 (a) Observed Rabi **oscillations** when the **qubit** starts in the g 10 - 00 = 0 state and is simulataneously moved to a large g 10 - 00 state and driven by a 7.5 G H z spectroscopy pulse of varying amplitude. The fast flux pulse is 60 ns in duration and is followed by an identical pulse of the opposite sign so that the total pulse integral is zero; these zero integral pulses help reduce slow transients. (b) Pulsed measurements showing the probability of the **qubit** being in the excited state as a function of delay following a π -pulse. The **qubit** starts in the g 10 - 00 = 0 state and is excited with a π -pulse in the manner described in (a); a pulsing scheme is included as an inset to the figure. The measured T 1 is 1.6 μ s. (c) Hahn echo measurements with the **qubit** starting in the g 10 - 00 = 0 state. Each of the pulses in the Hahn sequence is synchronized with a pair of fast flux pulses. A pulsing scheme is included as an inset to the figure. The measured T 2 time is 1.9 μ s....Using these fast flux bias pulses, we first measure T 1 by applying a π -pulse that is synchronized with the fast flux pulse, and measure the **qubit** excitation probability after a delay. We measure T 2 using a Hahn echo experiment. The **qubit** is returned to the g 10 - 00 state after each pulse in the Hahn echo sequence. The results of these measurements and the pulse schemes are shown in Fig figure4b, c. The measured T 1 and T 2 times are 1.6 and 1.9 μ s , respectively. The times are only slightly shorter than the 1.9 and 2.8 μ s times recorded at high g 10 - 00 without any fast flux pulses....We demonstrate coherent control and measurement of a superconducting **qubit** coupled to a superconducting coplanar waveguide resonator with a dynamically tunable **qubit**-cavity coupling strength. Rabi **oscillations** are measured for several coupling strengths showing that the **qubit** transition can be turned off by a factor of more than 1500. We show how the **qubit** can still be accessed in the off state via fast flux pulses. We perform pulse delay measurements with synchronized fast flux pulses on the device and observe $T_1$ and $T_2$ times of 1.6 and 1.9 $\mu$s, respectively. This work demonstrates how this **qubit** can be incorporated into quantum computing architectures....Time domain measurements provide a more quantitative assessment of any residual coupling at the g 10 - 00 = 0 point. The rate of Rabi driving is proportional to the coupling strength g 10 - 00 and the applied drive amplitude as per the equation Ω R a b i = g 10 - 00 n , where n is the number of drive photons . In Fig figure3, we demonstrate Rabi **oscillations** at three different points on the constant **frequency** contour; these three points are marked on Fig. figure2b. Figure figure3a, b show Rabi **oscillations** at high and medium coupling respectively. In the two panels, the **oscillation** rate is kept nearly constant by increasing the applied rf spectroscopy power by 10 dB to compensate for the reduction in **qubit**-cavity coupling. Fig. figure3c shows the measurement at the g 10 - 00 = 0 point, with 27 dB more rf power than at the high coupling point. No excitation is visible. Given the measurement noise, we should easily be able to detect a tenth of a Rabi **oscillation**; that we see no excitation puts a lower bound on the change in the Rabi rate of a factor of 80. Together with the much higher excitation power, we estimate that the coupling is at least 1500 times smaller at the g 10 - 00 = 0 point compared with the high coupling point. If several **qubits** were in a single cavity, this tuning provides protection against single **qubit** gate errors in one **qubit** while a second **qubit** is driven. ... We demonstrate coherent control and measurement of a superconducting **qubit** coupled to a superconducting coplanar waveguide resonator with a dynamically tunable **qubit**-cavity coupling strength. Rabi **oscillations** are measured for several coupling strengths showing that the **qubit** transition can be turned off by a factor of more than 1500. We show how the **qubit** can still be accessed in the off state via fast flux pulses. We perform pulse delay measurements with synchronized fast flux pulses on the device and observe $T_1$ and $T_2$ times of 1.6 and 1.9 $\mu$s, respectively. This work demonstrates how this **qubit** can be incorporated into quantum computing architectures.

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Contributors: Oh, Sangchul, Kim, Jaewan

Date: 2006-05-03

The harmonic **oscillator** plays two competing roles: (i) making two **qubits** entangled by inducing the two-**qubit** coupling I e f f , (ii) having two **qubits** decoherent through entanglement with them, i.e. Γ R t . The role of the imaginary part Γ I t is different from that of the real part Γ R t . While Γ R t makes the two **qubit** decoherent, i.e. decaying the off-diagonal elements, Γ I t causes the two-**qubit** coupling to fluctuate as shown in Fig. Fig2(b). These could be uncovered by examining the reduced density matrix of the two **qubits**...(color online). (a) Maximum concurrence of the two **qubits** (b) entropy of two **qubits** at equilibrium as a function of α and ω 0 / ω c . (c) exp - Γ R ∞ as a function of temperature T and ω 0 / ω c for α = 0.25 ....As ω / λ becomes large, C goes to C i d e a l and S to 0 as depicted in Fig. Fig2. This is explained by means of the Born-Oppenheimer approximation that is based on the assumption of the weak coupling between the two **qubits** and the environment. The **frequency** ω of the harmonic **oscillator** is larger than that of the two **qubits**, θ . The harmonic **oscillator** **oscillates** very fast in comparison with the two **qubits**. So the two **qubits** feel the harmonic **oscillator** stays in the same state. However, the condition of ω ≫ λ is not a unique way to make the two **qubits** entangled maximally. Surprisingly, we find the maximum entanglement of the two **qubits**, C = 1 , at θ t = π / 4 under the condition that ω / λ = 4 n with n = 1 , 2 , as shown in Figs. Fig2(a) and Fig2(d). This is due to the fact that the **frequency** θ for entangling the two **qubits** is commensurate with the **frequency** ω of the harmonic **oscillator**. If the condition of ω / λ = 4 n , n = 1 , 2 , is not met, then the concurrence C does not reach 1 at θ t = π / 4 and 0 at θ t = π / 2 due to Γ t as shown in Fig. Fig2(b). The **oscillation** period of C (red solid line) does not coincide with that of the ideal case C i d e a l (thick black line). This implies that the two **qubit** coupling fluctuates due to Γ I t . For ω / λ **qubits** could not be entangled. Fig. Fig2(c) shows the case of ω = λ ....Fig. Fig2(d) shows two competing roles of the harmonic **oscillator**. Let us define the average concurrence C a v g for a period τ ≡ π / 2 θ by C a v g ≡ 1 τ ∫ 0 τ C t d t . For the first period 0 ≤ θ t **frequencies** ω and θ , S a v g and C a v g show the behavior of the stair case....The environment is characterized by the spectral density function J ω = ∑ j λ j 2 δ ω - ω j . As shown in the inset of Fig. Fig3-(a), let us consider an Ohmic environment with a gap ω 0 and an exponential cutoff function of the cutoff **frequency** ω c...The harmonic **oscillator** remains isolated always from the system, but it induces the indirect interaction between the two **qubits** and thus entanglement between them . For a pure two **qubits**, concurrence, an entanglement measure, reads C i d e a l = 2 | a d - e i 4 θ t b c | . Here the subscript, ideal, stands for the case of the **qubits** in a pure state. If a = b = c = d = 1 / 2 , we have C i d e a l = | sin 2 θ t | as shown in Fig. Fig2....Two circles refer to two **qubits** and an oval is an environment. The coupling between **qubits** and the common environment (solid arrow) induces the indirect interaction between two **qubits** (dashed arrow)....We study a system of two **qubits** interacting with a common environment, described by a two-spin boson model. We demonstrate two competing roles of the environment: inducing entanglement between the two **qubits** and making them decoherent. For the environment of a single harmonic **oscillator**, if its **frequency** is commensurate with the induced two-**qubit** coupling strength, the two **qubits** could be maximally entangled and the environment could be separable. In the case of the environment of a bosonic bath, the gap of its spectral density function is essential to generate entanglement between two **qubits** at equilibrium and for it to be used as a quantum data bus. ... We study a system of two **qubits** interacting with a common environment, described by a two-spin boson model. We demonstrate two competing roles of the environment: inducing entanglement between the two **qubits** and making them decoherent. For the environment of a single harmonic **oscillator**, if its **frequency** is commensurate with the induced two-**qubit** coupling strength, the two **qubits** could be maximally entangled and the environment could be separable. In the case of the environment of a bosonic bath, the gap of its spectral density function is essential to generate entanglement between two **qubits** at equilibrium and for it to be used as a quantum data bus.

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Contributors: Alexander N. Korotkov

Date: 2005-03-01

We discuss an experimental proposal on quantum feedback control of a double-dot **qubit**, which seems to be within the reach of the present-day technology. Similar to the earlier proposal, the feedback loop is used to maintain the coherent **oscillations** in the **qubit** for an arbitrary long time; however, this is done in a significantly simpler way. The main idea is to use the quadrature components of the noisy detector current to monitor approximately the phase of **qubit** **oscillations**....Solid lines: synchronization degree D (and in-phase current quadrature 〈X〉) as functions of F for several values of the detection efficiency ηeff. Dashed and dotted lines illustrate the effects of the energy mismatch (ε≠0) and the **frequency** mismatch (Ω≠Ω0).
... We discuss an experimental proposal on quantum feedback control of a double-dot **qubit**, which seems to be within the reach of the present-day technology. Similar to the earlier proposal, the feedback loop is used to maintain the coherent **oscillations** in the **qubit** for an arbitrary long time; however, this is done in a significantly simpler way. The main idea is to use the quadrature components of the noisy detector current to monitor approximately the phase of **qubit** **oscillations**.

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Contributors: Korotkov, Alexander N.

Date: 2000-03-13

Schematic of a **qubit** continuously measured by a detector with output signal I t ....A particular realization of the evoltion of ρ 11 t due to continuous measurement for ε / H = 1 , α = 0.1 and η = 1 . Notice the fluctuation of both the phase and the asymmetry of **oscillations**....The situation changes as the coupling between the detector and **qubit** increases, α 1 . The strong influence of measurement destroys quantum **oscillations**, and the Quantum Zeno effect develops, so that for α ≫ 1 the **qubit** performs random jumps between two localized states (see Fig. I(t)b). In this case the properly averaged detector current follows pretty well the evolution of the **qubit** (however, the unsuccessful tunneling “attempts” still cannot be directly resolved), and the spectral density of I t can be calculated using the classical theory of telegraph noise leading to the Lorentzian shape of S I ω . Figure transitiona shows the gradual transformation of the spectral density with the increase of the coupling α for a symmetric **qubit**, ε = 0 , and an ideal detector, η = 1 . The results for an asymmetric **qubit**, ε / H = 1 , are shown in Fig. transitionb....has an obvious relation to the average square of the detector current variation due to **oscillations** in the measured system. Notice, however, that this integral is twice as large as one would expect from the classical harmonic signal, since one half of the peak height comes from nonclassical correlation between the **qubit** evolution and the detector noise. Classically, Eq. ( integral) would be easily understood if the signal was not harmonic but rectangular-like, which is obviously not the case. Actually, the detector current shows neither clear harmonic nor rectangular signal distinguishable from the intrinsic noise contribution. Figure I(t)a shows the simulation of ρ 11 evolution (thick line) together with the detector current I t . Since I t contains white noise, it necessarily requires some averaging. Thin solid, dotted, and dashed lines show the detector current averaged with different time constants τ a : τ a Ω / 2 π = 0.3 , 1, and 3, respectively. For weak averaging the signal is too noisy, while for strong averaging individual **oscillation** periods cannot be resolved either, so quantum **oscillations** can never be observed directly by a continuous measurement (although they can be calculated using Eqs. ( Bayes1)–( Bayes2)). This unobservability is revealed in the relatively low peak height of the spectral density of the detector current....The curves in Fig. transition as well as the dashed curves in Fig. M-C are calculated using the conventional master equation approach which gives the same results for the detector spectral density as the Bayesian formalism (we will prove this later). In the conventional approach we should assume no correlation between the detector noise and the **qubit** evolution (the last term in Eq. ( 3contrib) is absent) while the correlation function K z ̂ τ should be calculated considering z t not as an ordinary function but as an operator. Then the calculation of z ̂ t z ̂ t + τ can be essentially interpreted as follows. The first (in time) operator z ̂ t collapses the **qubit** into one of two eigenstates which correspond to localized states, then during time τ the **qubit** performs the evolution described by conventional Eqs. ( conv1)–( conv2), and finally the second operator z ̂ t + τ gives the probability for the **qubit** to be measured in one of two localized states. (Of course, this procedure can be done purely formally, without any interpretation.) Notice that there is complete symmetry between states “1” and “2” even for ε ≠ 0 (in particular, in the stationary state ρ 11 = ρ 22 = 1 / 2 ), so the evolution after the first collapse can be started from any localized state leading to the same contribution to the correlation function. In this way we obviously get K z ̂ τ = ρ 11 τ - ρ 22 τ where ρ i i is the solution of Eqs. ( conv1)–( conv2) with the initial conditions ρ 11 0 = 1 and ρ 12 0 = 0 ....The detector current spectral density S I ω for η = 1 and different coupling α with (a) symmetric ( ε = 0 ) and (b) asymmetric ( ε / H = 1 ) **qubit**....Figure envir shows the numerically calculated spectral density S I ω of the detector current for a nonideal detector, η = 0.5 (dashed lines) and for an ideal detector but extra coupling of the **qubit** to the environment at temperature T = H (solid lines). The rates γ 1 and γ 2 are chosen according to Eqs. ( enveqv1) and ( enveqv2). For the symmetric **qubit**, ε = 0 , the results of two models practically coincide. In contrast, the solid and dashed lines for ε = 2 H significantly differ from each other at low **frequencies**, while the spectral peak at ω ∼ Ω is fitted quite well....We consider a two-level quantum system (**qubit**) which is continuously measured by a detector and calculate the spectral density of the detector output. In the weakly coupled case the spectrum exhibits a moderate peak at the **frequency** of quantum **oscillations** and a Lorentzian-shape increase of the detector noise at low **frequency**. With increasing coupling the spectrum transforms into a single Lorentzian corresponding to random jumps between two states. We prove that the Bayesian formalism for the selective evolution of the density matrix gives the same spectrum as the conventional master equation approach, despite the significant difference in interpretation. The effects of the detector nonideality and the finite-temperature environment are also discussed....Using Eqs. ( Bayes1)–( Bayes3) and the Monte-Carlo method (similar to Ref. ) we can calculate in a straightforward way the spectral density S I ω of the detector current I t . Solid lines in Fig. M-C show the results of such calculations for the ideal detector, η = 1 , and weak coupling between the **qubit** and the detector, α = 0.1 , where α ≡ ℏ Δ I 2 / 8 S 0 H ( α is 8 times less than the parameter C introduced in Ref. ). One can see that in the symmetric case, ε = 0 , the peak at the **frequency** of quantum **oscillations** is 4 times higher than the noise pedestal, S I Ω = 5 S 0 while the peak width is determined by the coupling strength α (see Fig. transition below). In the asymmetric case, ε ≠ 0 , the peak height decreases (Fig. M-C), while the additional Lorentzian-shape increase of S I ω appears at low **frequencies**. The origin of this low-**frequency** feature is the slow fluctuation of the asymmetry of ρ 11 **oscillations** (Fig. asym). In case ε = 0 the amplitude of ρ 11 **oscillations** is maximal (see thick line in Fig. I(t)a), hence there is no such asymmetry and the low-**frequency** feature is absent, while the spectral peak at the **frequency** of quantum **oscillation** is maximally high. ... We consider a two-level quantum system (**qubit**) which is continuously measured by a detector and calculate the spectral density of the detector output. In the weakly coupled case the spectrum exhibits a moderate peak at the **frequency** of quantum **oscillations** and a Lorentzian-shape increase of the detector noise at low **frequency**. With increasing coupling the spectrum transforms into a single Lorentzian corresponding to random jumps between two states. We prove that the Bayesian formalism for the selective evolution of the density matrix gives the same spectrum as the conventional master equation approach, despite the significant difference in interpretation. The effects of the detector nonideality and the finite-temperature environment are also discussed.

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