### 56015 results for qubit oscillator frequency

Contributors: Mao, Wenjin, Averin, Dmitri V., Plastina, Francesco, Fazio, Rosario

Date: 2004-08-21

We see from Eq. ( e62) that for non-interacting **qubits**, the non-vanishing **qubit** bias just shifts the **frequency** position of the liner peaks ( e57) without qualitatively changing their shape. If both the bias and the **qubit**-**qubit** interaction are finite, the bias splits each of the linear peaks in two simple Lorentzians bringing the total number of the finite-**frequency** peaks in the spectrum of the detector output to six as it should be in the generic situation (see, e.g., Fig. fig3)....Output spectra of the non-linear detector measuring two different unbiased **qubits**. Solid line is the spectrum in the case of non-interacting **qubits**. The two larger peaks are the “linear” peaks that correspond to the **oscillations** in the individual **qubits**, while smaller peaks are non-linear peaks at the combination **frequencies**. Dashed line is the spectrum for interacting **qubits**. Interaction shifts the lower-**frequency** liner peak down and all other peaks up in **frequency**. Parameters of the detector-**qubit** coupling are: δ 1 = 0.12 t 0 , δ 2 = 0.09 t 0 , λ = 0.08 t 0 ....Finite **qubit** bias should lead to averaging of the two spectra S I ± ( e27) similar to that discussed in the case of non-interacting **qubits** and illustrated in Fig. fig4....The two spectral densities ( e20) correspond to two possible outcomes of measurement: the **qubits** found in one or the other subspace D ± , the probability of the outcomes being determined by the initial state of the **qubits**. Each of the spectral densities coincides with the spectral density of the linear detector measuring coherent **oscillations** in one **qubit** . Similarly to that case, the maximum of the ratio of the **oscillation** peak versus noise S 0 for each spectrum S I ± ω is 4. As one can see from Eq. ( e20), this maximum is reached when the measurement is weak: | λ | ≪ | t 0 | , and the detector is “ideal”: arg t 0 λ * =0, and only Γ + or Γ - is non-vanishing. If, however, there is small but finite transition rate between the two subspaces that mixes the two outcomes of measurement, the peak height is reduced by averaging over the two spectral densities ( e20). This situation is illustrated in Fig. fig4 which shows the output spectra of the purely quadratic detector, when the subspaces D ± are mixed by small **qubit** bias ε . Since the stationary density matrix ( e14) is equally distributed over all **qubit** states, the two peaks of the spectral densities ( e20) are mixed with equal probabilities, and the maximum of the ratio of the **oscillation** peak heights versus noise S 0 for the combined spectrum S I ω is 2. Spectrum shown in Fig. fig4 for ε = 0.1 Δ 1 (solid line) is close to this limit....An example of the output spectrum of the non-linear detector measuring unbiased **qubits** with different tunneling amplitudes is shown in Fig. fig6. One can see that when the linear and non-linear coefficient of the detector-**qubit** coupling are roughly similar, the linear peaks are more pronounced than the peaks at combination **frequencies**. **Qubit**-**qubit** interaction shifts all but the lower-**frequency** linear peak up in **frequency** and reduces both the amplitudes of the higher-**frequency** peaks and the distance between them....Evolution of the output spectrum of the non-linear detector measuring two identical unbiased **qubits** with the strength ν of the **qubit**-**qubit** interaction. The **qubit**-detector coupling constants δ 1 , 2 are taken to be slightly different to average the spectrum over all **qubit** states. The three solid curves correspond to ν / Δ = 0.0 , 0.1 , 0.2 . In agreement with Eqs. ( e42) – ( e44), the peak at ω ≃ Δ is at first suppressed and then split in two by increasing ν , while the peak at ω ≃ 2 Δ is not changed noticeably by such a weak interaction. Dashed and dotted lines show the regime of relatively strong interaction: ν / Δ = 0.5 and ν / Δ = 1.0 , respectively, that is described by Eqs. ( e46) and ( e47)....We develop a theory of coherent quantum **oscillations** in two, in general interacting, **qubits** measured continuously by a mesoscopic detector with arbitrary non-linearity and discuss an example of SQUID magnetometer that can operate as such a detector. Calculated spectra of the detector output show that the detector non-linearity should lead to mixing of the **oscillations** of the two **qubits**. For non-interacting **qubits** **oscillating** with **frequencies** $\Omega_1$ and $\Omega_2$, the mixing manifests itself as spectral peaks at the combination **frequencies** $\Omega_1\pm \Omega_2$. Additional nonlinearity introduced by the **qubit**-**qubit** interaction shifts all the **frequencies**. In particular, for identical **qubits**, the interaction splits coherent superposition of the single-**qubit** peaks at $\Omega_1=\Omega_2$. Quantum mechanics of the measurement imposes limitations on the height of the spectral peaks....Figure fig5 illustrates evolution of the output spectrum of the non-linear detector measuring identical **qubits** due to changing interaction strength. We see that this evolution agrees with the analytical description developed above. Weak **qubit**-**qubit** interaction ν ≃ κ ≪ Δ suppresses and subsequently splits the spectral peak at ω ≃ Ω while not changing the peak ω ≃ 2 Ω . Stronger **qubit**-**qubit** interaction ν ≃ Δ ≫ κ shifts the ω ≃ 2 Ω peak to higher **frequencies** while moving the two peaks around ω ≃ Ω further apart....Output spectrum of a nonlinear detector measuring two **qubits** with “the most general” set of parameters. Six peaks in the spectrum at finite **frequencies** correspond to six different energy intervals in the energy spectrum of the two-**qubit** system. The zero-**frequency** peak reflects dynamics of transitions between energy levels. Detector parameters are: δ 1 = 0.1 , δ 2 = 0.07 , λ = 0.09 (all normalized to t 0 ). In this Figure, and in all numerical plots below we take Γ + | t 0 | 2 = Δ 1 , Γ - = 0 , and assume that the detector tunneling amplitudes are real....Diagram of a mesoscopic detector measuring two **qubits**. The **qubits** modulate amplitude t of tunneling of detector particles between the two reservoirs....Output spectra of a purely quadratic detector measuring two non-interacting **qubits**. Small **qubit** bias ε 1 = ε 2 ≡ ε (solid line) creates transitions that lead to averaging of the two main peaks at combination **frequencies** Δ 1 ± Δ 2 [see Eq. ( e20)]. Further increase of ε (dashed line) makes additional spectral peaks associated with these transitions more pronounced. The strength of quadratic **qubit**-detector coupling is taken to be λ = 0.15 t 0 . ... We develop a theory of coherent quantum **oscillations** in two, in general interacting, **qubits** measured continuously by a mesoscopic detector with arbitrary non-linearity and discuss an example of SQUID magnetometer that can operate as such a detector. Calculated spectra of the detector output show that the detector non-linearity should lead to mixing of the **oscillations** of the two **qubits**. For non-interacting **qubits** **oscillating** with **frequencies** $\Omega_1$ and $\Omega_2$, the mixing manifests itself as spectral peaks at the combination **frequencies** $\Omega_1\pm \Omega_2$. Additional nonlinearity introduced by the **qubit**-**qubit** interaction shifts all the **frequencies**. In particular, for identical **qubits**, the interaction splits coherent superposition of the single-**qubit** peaks at $\Omega_1=\Omega_2$. Quantum mechanics of the measurement imposes limitations on the height of the spectral peaks.

Data types:

Contributors: Ashhab, S.

Date: 2014-10-02

(color online) **Qubit’s** final excited state probability P obtained from the semiclassical calculation as a function of temperature k B T and coupling strength g , both measured relative to the minimum **qubit** gap Δ . The different panels correspond to different values of the harmonic **oscillator** **frequency**: ℏ ω / Δ = 0.2 (top), 1 (middle) and 5 (bottom)....(color online) Energy level diagram of a coupled **qubit**-**oscillator** system with the **qubit** bias conditions varied according to the LZ protocol....We analyze the dynamics and final populations in a Landau-Zener problem for a two level system (or **qubit**) when this system interacts with one harmonic **oscillator** mode that is initially set to a finite-temperature thermal equilibrium state. The harmonic **oscillator** could represent an external mode that is strongly coupled to the **qubit**, e.g. an ionic **oscillation** mode in a molecule, or it could represent a prototypical uncontrolled environment. We analyze the **qubit**'s occupation probabilities at the final time in a number of different regimes, varying the **qubit** and **oscillator** **frequencies**, their coupling strength and the temperature. In particular we find some surprising non-monotonic dependence on the coupling strength and temperature....We can also see in Fig. Fig:ExcitationProbability02 that for g / Δ 1 the temperature dependence is non-monotonic. In particular, for low temperatures we obtain the intuitively expected increase in excitation probability with increasing temperature, but this trend reverses for higher temperatures. In order to investigate this feature further, we calculate the ** qubit’s** final excited-state probability as a function of the number n of excitation quanta present in the initial state of the

**oscillator**(Note that this calculation differs from the ones described above in that here we do not use the Boltzmann distribution for the

**initial state). The results are plotted in Fig. Fig:ExcitationProbabilityAsFunctionOfInitialOscillatorExcitationNumber. These results explain the non-monotonic dependence on temperature. For intermediate values of g / Δ (e.g. for g / Δ = 1 ), there is a peak at a small but finite excitation number followed by a steady decrease. As the temperature is increased from zero, the**

**oscillator**’s**final excited-state probability samples the probabilities for increasingly high excitation numbers, and a peak at intermediate values of temperature is obtained. Note that for large excitation numbers, the increase in P as a function of n resumes, and this increase will also be reflected in the temperature dependence....where ω is the characteristic**

**qubit**’s**frequency**of the harmonic

**oscillator**, â and â † are, respectively, the

**annihilation and creation operators, and g is the**

**oscillator**’s**qubit**-

**oscillator**coupling strength. The energy level diagram of this problem is illustrated in Fig. Fig:EnergyLevelDiagram....Another feature worth noting is the temperature dependence of P close to zero temperature. As can be seen clearly in Figs. Fig:ExcitationProbability10 and Fig:ExcitationProbability50, the initial increase in P with temperature is very slow, indicating that it probably follows an exponential function that corresponds to the probability of populating the excited states in the harmonic

**oscillator**(and the same dependence is probably present but difficult to see because of the scale of the x axis in Fig. Fig:ExcitationProbability02). After this initial slow rise, and in particular when k B T ℏ ω , we see a steady rise that in the case of Fig. Fig:ExcitationProbability02 can be approximated as a linear increase in P with increasing T . Importantly, the slope of this increase can be quite large for intermediate g values. From the results shown in Figs. Fig:ExcitationProbability02- Fig:ExcitationProbability50, we find that the maximum slope d P / d k B T / Δ m a x = 0.18 × ℏ ω / Δ -0.57 , and results for other parameter values extending up to ℏ ω / Δ = 20 follow this dependence. The implication of this result can be seen clearly in the middle panel of Fig. Fig:ExcitationProbability02: even when the temperature is substantially smaller than the

**minimum gap Δ , the initial excitation of the low-**

**qubit**’s**frequency**

**oscillator**(stemming from the finite temperature) can cause a large increase in the

**final excited-state probability. This result is in contrast with the exact result of Ref. stating that at zero temperature the**

**qubit**’s**final excited-state probability is given by P L Z regardless of the value of g . The typical temperature scale at which deviations from the LZ formula occur can therefore be much lower than Δ / k B . This result is relevant for adiabatic quantum computing, because it contradicts the expectation that having a minimum gap that is large compared to the temperature might provide automatic protection for the ground state population against thermal excitation. Another point worth noting here is that when ℏ ω**

**qubit**’s**qubit**and

**oscillator**are resonant with each other, yet the initial thermal excitation of the

**oscillator**can result in exciting the

**qubit**at the final time. The excitations in the

**oscillator**are in some sense up-converted into excitations in the

**qubit**as a result of the sweep through the avoided crossing....In addition to solving the Schrödinger equation, we have performed semiclassical calculations where we assume that there is no quantum coherence between the different LZ processes. (Note here that when we replace the isolated

**qubit**with the coupled

**qubit**-

**oscillator**system the single avoided crossing is replaced by a complex network of avoided crossings.) Under this approximation, we only need to calculate the occupation probabilities of the different states, and these probabilities change (according to the LZ formula) only at the points of avoided crossing. This approach greatly simplifies the numerical calculations because the locations and gaps for the different avoided crossings can be determined easily (see e.g.~Fig.~ Fig:EnergyLevelDiagram). The results are shown in Fig. Fig:ExcitationProbabilityFromIncoherentCalculation. The results of this calculation agree generally well with those obtained by solving the Schrödinger equation when ℏ ω / Δ = 1 . For ℏ ω / Δ = 5 , the semiclassical calculation consistently underestimates the excited-state probability, but the overall dependence on temperature and coupling strength is remarkably similar to that shown in Fig. Fig:ExcitationProbability50. We should note that higher values of ℏ ω (not shown) exhibit more pronounced deviations, with side peaks appearing in the dependence of P on g / Δ . The most striking deviation from the results of the fully quantum calculation is seen in the case ℏ ω / Δ = 0.2 (i.e. the case of a low-

**frequency**

**oscillator**). In the semiclassical calculation, there is a rather high peak at a small value of the coupling strength (and sufficiently high temperatures), and the excited-state probability starts decreasing when the coupling strength g becomes larger than ℏ ω . In the fully quantum calculation, however, the peak is located at a much higher value, somewhere between 0.5 and 1 depending on the temperature....(color online) Top:

**Qubit’s**final excited-state probability P as a function of temperature k B T and coupling strength g , both measured relative to the

**minimum gap Δ . Middle: P as a function of k B T / Δ for four different values of g / Δ : 0.1 (red solid line), 0.3 (green dashed line), 1 (blue dotted line) and 2 (magenta dash-dotted line). Bottom: P as a function of g / Δ for three different values of k B T / Δ : 1 (red solid line), 3 (green dashed line), and 5 (blue dotted line). In all the panels, the harmonic**

**qubit**’s**oscillator**

**frequency**is ℏ ω / Δ = 0.2 . The sweep rate is chosen such that P L Z = 0.1 , and this value is the baseline for all of the results plotted in this figure....(color online) The final excited state probability P as a function of the number of excitation quanta n present in the initial state of the

**oscillator**. Here we take ℏ ω / Δ = 0.2 . The different lines correspond to different values of the coupling strength: g / Δ = 0.1 (red solid line), 0.5 (green dashed line), 1 (blue dotted line) and 2 (magenta dash-dotted line)....The probability for the

**qubit**to end up in the excited state at the final time as a function of temperature and coupling strength is plotted in Figs. Fig:ExcitationProbability02- Fig:ExcitationProbability50. As expected from known results , the final excited-state occupation probability P remains equal to 0.1 whenever the temperature or the coupling strength is equal to zero. Otherwise, the coupling to the

**oscillator**causes this probability to increase. A common, and somewhat surprising, trend for all values of ℏ ω / Δ is the non-monotonic dependence on the coupling strength g . As the coupling strength is increased from zero to finite but small values, P increases. But when the coupling strength is increased further, P starts decreasing. Based on the results that are plotted in Figs. Fig:ExcitationProbability02- Fig:ExcitationProbability50, one can expect that in the limit of large g / Δ (and assuming not-very-large values of k B T / Δ ) the excited-state occupation probability will go back to its value in the uncoupled case, i.e. P = 0.1 . This phenomenon is probably a manifestation of the superradiance-like behaviour in a strongly coupled

**qubit**-

**oscillator**system . In the superradiant regime (i.e. the strong-coupling regime), the ground state is highly entangled exactly at the symmetry point (which corresponds to the bias conditions at t = 0 in the LZ problem), but even small deviations from the symmetry point can lead to an effective decoupling between the

**qubit**and resonator with the exception of some state-dependent mean-field shifts. Indeed the maximum values of P reached in Figs. Fig:ExcitationProbability10 and Fig:ExcitationProbability50 occur at coupling strength values that are comparable to the expression for the uncorrelated-to-correlated crossover value, namely g ∼ ℏ ω (and we have verified that the near-linear increase in peak location as a function of

**oscillator**

**frequency**continues up to ℏ ω / Δ = 20 ). This relation does not apply in the case ℏ ω / Δ = 0.2 , shown in Fig. Fig:ExcitationProbability02. In this case, the peak occurs when the coupling strength g is comparable to the minimum gap Δ . It is in fact quite surprising that the excitation peak in the case ℏ ω / Δ = 0.2 occurs at a higher coupling strength than that obtained in the case ℏ ω / Δ = 1 . In order to investigate this point further, we tried values close to ℏ ω / Δ = 1 and found that this value gives a minimum in the peak location (i.e. the peak in P when plotted as a function of g / Δ ). ... We analyze the dynamics and final populations in a Landau-Zener problem for a two level system (or

**qubit**) when this system interacts with one harmonic

**oscillator**mode that is initially set to a finite-temperature thermal equilibrium state. The harmonic

**oscillator**could represent an external mode that is strongly coupled to the

**qubit**, e.g. an ionic

**oscillation**mode in a molecule, or it could represent a prototypical uncontrolled environment. We analyze the

**qubit**'s occupation probabilities at the final time in a number of different regimes, varying the

**qubit**and

**oscillator**

**frequencies**, their coupling strength and the temperature. In particular we find some surprising non-monotonic dependence on the coupling strength and temperature.

Data types:

Contributors: Jordan, Andrew N., Buttiker, Markus

Date: 2005-05-02

The first term has a peak at zero **frequency**, while the second term has a peak at ω = Ω , with width 3 Γ / 2 , and signal -1 / 3 Γ . Bounding this signal in relation to the noise in the individual twin detectors gives | S 1 , 2 Ω | ≤ 2 / 3 S I . The interesting feature of this correlator is that it changes sign as a function of **frequency**. The low **frequency** part describes the incoherent relaxation to the stationary state, while the high **frequency** part describes the out of phase, coherent **oscillations** of the z and x degrees of freedom. The measured correlator S z x , as well as S x x , S z z are plotted as a function of **frequency** in Fig. combo(b,c,d) for different values of ϵ . These correlators all describe different aspects of the time domain destruction of the quantum state by the weak measurement, visualized in Fig. comboa. We note that the cross-correlator changes sign for ϵ = - Δ ....(color online). (a) Time domain destruction of the quantum state by the weak measurement process for ϵ = Δ . The elapsed time is parameterized by color, and (x,y,z) denote coordinates on the Bloch sphere. (b) The measured cross-correlator S z x ω changes sign from positive at low **frequency** (describing incoherent relaxation) to negative at the **qubit** **oscillation** **frequency** (describing out of phase, coherent **oscillations**). (c,d) The correlators S x x , S z z have both a peak at zero **frequency** and at **qubit** **oscillation** **frequency**. We take Γ = Γ x = Γ z = .07 Δ / ℏ . S i j are plotted in units of Γ -1 ....Cross-correlated quantum measurement set-up: Two quantum point contacts are measuring the same double quantum dot **qubit**. As the quantum measurement is taking place, the current outputs of both detectors can be averaged or cross-correlated with each other. ... We investigate the advantages of using two independent, linear detectors for continuous quantum measurement. For single-shot quantum measurement, the measurement is maximally efficient if the detectors are twins. For weak continuous measurement, cross-correlations allow a violation of the Korotkov-Averin bound for the detector's signal-to-noise ratio. A vanishing noise background provides a nontrivial test of ideal independent quantum detectors. We further investigate the correlations of non-commuting operators, and consider possible deviations from the independent detector model for mesoscopic conductors coupled by the screened Coulomb interaction.

Data types:

Contributors: Shen, Li-Tuo, Chen, Rong-Xin, Wu, Huai-Zhi, Yang, Zhen-Biao

Date: 2013-11-05

We study the system involving mutual interaction between three **qubits** and an **oscillator** within the ultrastrong coupling regime. We apply adiabatic approximation approach to explore two extreme regimes: (i) the **oscillator**'s **frequency** is far larger than each **qubit**'s **frequency** and (ii) the **qubit**'s **frequency** is far larger than the **oscillator**'s **frequency**, and analyze the energy-level spectrum and the ground-state property of the **qubit**-**oscillator** system under the conditions of various system parameters. For the energy-level spectrum, we concentrate on studying the degeneracy in low energy levels. For the ground state, we focus on its nonclassical properties that are necessary for preparing the nonclassical states. We show that the minimum **qubit**-**oscillator** coupling strength needed for generating the nonclassical states of the Schr\"{o}dinger-cat type in the **oscillator** is just one half of that in the Rabi model. We find that the **qubit**-**qubit** entanglement in the ground state vanishes if the **qubit**-**oscillator** coupling strength is strong enough, for which the entropy of three **qubits** keeps larger than one. We also observe the phase-transition-like behavior in the regime where the **qubit**'s **frequency** is far larger than the **oscillator**'s **frequency**....(Color online) Energy spectrums for lowest eight levels under the situation with three high-**frequency** **qubits**: ℏ w 0 / E q = 0.01 . The rescaled energy E k / ℏ w 0 with k = 1 , 2 , 3 , . . . , 8 versus the rescaled coupling strength λ / ℏ w 0 is plotted: (a) θ = 0 ; (b) θ = π / 6 ; (c) θ = π / 3 ....(Color online) Schematic of four displaced **oscillators**. The horizontal and vertical axises represent the position and displaced ** oscillator’s** eigenenergy E d o , respectively. Four displaced

**oscillators**are shifted to the left or right from the equilibrium position with a specific constant, where the shift direction is determined by the state of three

**qubits**. The eigenstates (plotted with n no more than 2 ) that have the same value of n are degenerate for the states | A ± 1 (or | A ± 3 ), and have the symmetry divided by the origin point in horizontal axis....adiabatic approximation, three

**qubits**, ultrastrongly coupled, harmonic

**oscillator**...(Color online) Energy spectrums for lowest eight levels under the situation with a high-

**frequency**

**oscillator**: ℏ w 0 / E q = 10 . The rescaled energy E k / ℏ w 0 with k = 1 , 2 , 3 , . . . , 8 versus the rescaled coupling strength λ / ℏ w 0 is plotted: (a) θ = 0 ; (b) θ = π / 6 ; (c) θ = π / 3 ....(Color online) Schematic of the system with three identical

**qubits**coupled to a harmonic

**oscillator**. The j th ( j = 1 , 2 , 3 )

**qubit**with one ground ( | g j ) and one excited states ( | e j ) is coupled to the

**oscillator**with

**frequency**w 0 , where the

**qubit**-

**oscillator**coupling strength is denoted by g or λ ....(Color online) The Q function (upside) and the Wigner function (underside) of the

**state with three high-**

**oscillator**’s**frequency**

**qubits**(i.e., ℏ w 0 / Δ = 0.1 and ϵ = 0 ): (a,d) λ / ℏ w 0 = 0.5 , (b,e) λ / ℏ w 0 = 1 , (c,f) λ / ℏ w 0 = 1.25 . ... We study the system involving mutual interaction between three

**qubits**and an

**oscillator**within the ultrastrong coupling regime. We apply adiabatic approximation approach to explore two extreme regimes: (i) the

**oscillator**'s

**frequency**is far larger than each

**qubit**'s

**frequency**and (ii) the

**qubit**'s

**frequency**is far larger than the

**oscillator**'s

**frequency**, and analyze the energy-level spectrum and the ground-state property of the

**qubit**-

**oscillator**system under the conditions of various system parameters. For the energy-level spectrum, we concentrate on studying the degeneracy in low energy levels. For the ground state, we focus on its nonclassical properties that are necessary for preparing the nonclassical states. We show that the minimum

**qubit**-

**oscillator**coupling strength needed for generating the nonclassical states of the Schr\"{o}dinger-cat type in the

**oscillator**is just one half of that in the Rabi model. We find that the

**qubit**-

**qubit**entanglement in the ground state vanishes if the

**qubit**-

**oscillator**coupling strength is strong enough, for which the entropy of three

**qubits**keeps larger than one. We also observe the phase-transition-like behavior in the regime where the

**qubit**'s

**frequency**is far larger than the

**oscillator**'s

**frequency**.

Data types:

Contributors: Hauss, Julian, Fedorov, Arkady, Hutter, Carsten, Shnirman, Alexander, Schön, Gerd

Date: 2007-01-02

The systems considered are shown in Fig. fig:system. To be specific we first analyze the Rabi driven flux **qubit** coupled to an LC-**oscillator** (Fig. fig:systema) with Hamiltonian...Average number of photons in the resonator as function of the driving detuning δ ω and amplitude Ω R 0 . Peaks at δ ω > 0 correspond to lasing, dips at δ ω **qubit** are Δ / 2 π = 1 GHz, ϵ = 0.01 Δ , and Γ 0 / 2 π = 125 kHz, the **frequency** and line-width of the resonator are ω T / 2 π = 6 MHz and κ / 2 π = 1.7 kHz, the coupling constant is g / 2 π = 3.3 MHz and the temperature of the resonator T = 10 mK. The inset shows the bistability of the photon number for Ω R 0 / 2 π = 7 MHz. The dashed line represents the unstable solution....So far we described a flux **qubit** coupled to an LC **oscillator**, but our analysis applies equally to a nano-mechanical resonator capacitively coupled to a Josephson charge **qubit** (see Fig. fig:systemb). In this case σ z stands for the charge of the **qubit**, and both the coupling to the **oscillator** and the driving are capacitive, i.e., involve σ z . To produce capacitive coupling between the **qubit** and the **oscillator**, the latter is metal coated and charged by a voltage source . The dc component of the gate voltage V g puts the system near the charge degeneracy point where the dephasing due to the 1 / f charge noise is minimal. Rabi driving is induced by an ac component of V g . Realistic experimental parameters are expected to be very similar to the ones used in the examples discussed above, except that a much higher quality factor of the resonator ( ∼ 10 5 ) and a much higher number of quanta in the **oscillator** can be reached. This number will easily exceed the thermal one, thus a proper lasing state with Poisson statistics, appropriately named SASER , is produced. One should then observe the usual line narrowing with line width given by κ N t h / 4 n ̄ ∼ κ 2 N t h / Γ 1 . Experimental observation of this line-width narrowing would constitute a confirmation of the lasing/sasing....For a superconducting **qubit** driven to perform Rabi **oscillations** and coupled to a slow electromagnetic or nano-mechanical **oscillator** we describe previously unexplored quantum optics effects. When the Rabi **frequency** is tuned to resonance with the **oscillator** the latter can be driven far from equilibrium. Blue detuned driving leads to a population inversion in the **qubit** and a bi-stability with lasing behavior of the **oscillator**; for red detuning the **qubit** cools the **oscillator**. This behavior persists at the symmetry point where the **qubit**-**oscillator** coupling is quadratic and decoherence effects are minimized. There the system realizes a "single-atom-two-photon laser"....In Fig. 3dphoton we summarize our main results obtained by solving the Langevin (Fokker-Plank) equations . The number of photons n ̄ is plotted as a function of the detuning δ ω of the driving **frequency** and driving amplitude Ω R 0 . It exhibits sharp extrema along two curves corresponding to the one- and two-photon resonances, Ω R = ω T - 4 g 3 n ̄ and Ω R = 2 ω T - 4 g 3 n ̄ . Blue detuning, δ ω > 0 , induces a strong population inversion of the **qubit** levels, which in resonance leads to one-**qubit** lasing. In experiments the effect can be measured as a strong increase of the photon number in the resonator above the thermal values. On the other hand, red detuning produces a one-**qubit** cooler with photon numbers substantially below the thermal value. Near the resonances we find regions of bi-stability illustrated in the inset of Fig. 3dphoton. In these regions we expect a telegraph-like noise due to random switching between the two solutions....Several recent experiments on quantum state engineering with superconducting circuits realized concepts originally introduced in the field of quantum optics and stimulated substantial theoretical activities . Josephson **qubits** play the role of two-level atoms, while **oscillators** of various kinds replace the quantized light field. Motivated by one such experiment , we investigate a Josephson **qubit** coupled to a slow LC **oscillator** (Fig. fig:system a) with eigenfrequency (in the MHz range) much lower than the ** qubit’s** energy splitting (in the GHz range), ω T ≪ Δ E . The

**qubit**is ac-driven to perform Rabi

**oscillations**, and the Rabi

**frequency**Ω R is tuned close to resonance with the

**oscillator**. For this previously unexplored regime of

**frequencies**we study both one-photon (for Ω R ≈ ω T ) and two-photon (for Ω R ≈ 2 ω T )

**qubit**-

**oscillator**couplings. The latter is dominant at the “sweet" point of the

**qubit**, where due to symmetry the linear coupling to the noise sources is tuned to zero and dephasing effects are minimized . When the

**qubit**driving

**frequency**is blue detuned, δ ω = ω d - Δ E > 0 , we find that the system exhibits lasing behavior; for red detuning the

**qubit**cools the

**oscillator**. Similar behavior is expected in an accessible range of parameters for a Josephson

**qubit**coupled to a nano-mechanical

**oscillator**(Fig. fig:systemb), thus providing a realization of a SASER (Sound Amplifier by Stimulated Emission of Radiation)....The systems. a) In the circuit QED setup of Ref. an externally driven three-junction flux

**qubit**is coupled inductively to an LC

**oscillator**. b) In an equivalent setup a charge

**qubit**is coupled to a mechanical resonator. ... For a superconducting

**qubit**driven to perform Rabi

**oscillations**and coupled to a slow electromagnetic or nano-mechanical

**oscillator**we describe previously unexplored quantum optics effects. When the Rabi

**frequency**is tuned to resonance with the

**oscillator**the latter can be driven far from equilibrium. Blue detuned driving leads to a population inversion in the

**qubit**and a bi-stability with lasing behavior of the

**oscillator**; for red detuning the

**qubit**cools the

**oscillator**. This behavior persists at the symmetry point where the

**qubit**-

**oscillator**coupling is quadratic and decoherence effects are minimized. There the system realizes a "single-atom-two-photon laser".

Data types:

Contributors: Hauss, Julian, Fedorov, Arkady, André, Stephan, Brosco, Valentina, Hutter, Carsten, Kothari, Robin, Yeshwant, Sunil, Shnirman, Alexander, Schön, Gerd

Date: 2008-06-07

Average number of photons in the resonator as function of the driving detuning δ ω and amplitude Ω R 0 . Peaks at δ ω > 0 correspond to lasing, while dips at δ ω **qubit**: Δ / 2 π = 1 GHz, ϵ = 0.01 Δ , Γ 0 / 2 π = 125 kHz, the resonator: ω T / 2 π = 6 MHz, κ / 2 π = 0.34 kHz, and the coupling: g / 2 π = 3.3 MHz. The bath temperature is T = 10 mK....Dressed states of a driven **qubit** near resonance. Here m is the number of photons of the driving field, which is assumed to be quantized....In experiments with the same setup as shown in Fig. fig:systema) but in a different parameter regime the mechanisms of Sisyphus cooling and amplification has recently been demonstrated . Due to the resonant high-**frequency** driving of the **qubit**, depending on the detuning, the **oscillator** is either cooled or amplified with a tendency towards lasing. The Sisyphus mechanism is most efficient when the relaxation rate of the **qubit** is close to the **oscillator**’s**frequency**. In contrast, in the present paper we concentrate on the “resolved sub-band" regime where the dissipative transition rates of the **qubits** are much lower than the **oscillator**’s**frequency**....Average number of photons n ̄ versus the detuning. The blue curves are obtained from the Langevin equations ( dot alpha) and ( dot alpha2). They show the bistability with the solid curve denoting stable solutions, while the dashed curve denotes the unstable solution. The red curve is obtained from a numerical solution of the master equation ( eq:Master_Equation). The driving amplitude is taken as Ω R 0 / 2 π = 5 MHz. The parameters of the **qubit**: Δ / 2 π = 1 GHz, ϵ = 0.01 Δ , Γ 0 / 2 π = 125 kHz, the resonator: ω T / 2 π = 6 MHz, κ / 2 π = 1.7 kHz, N t h = 5 , and the coupling: g / 2 π = 3.3 MHz....So far we described an LC **oscillator** coupled to a flux **qubit**. But our analysis equally applies for a nano-mechanical resonator coupled capacitively to a Josephson charge **qubit** (see Fig. fig:systemb). In this case σ z stands for the charge of the **qubit** and both the coupling to the **oscillator** as well as the driving are capacitive, i.e., involve σ z . To produce the capacitive coupling between the **qubit** and the **oscillator**, the latter could be metal-coated and charged by the voltage source V x . The dc component of the gate voltage V g puts the system near the charge degeneracy point where the dephasing due to the 1 / f charge noise is minimal. Rabi driving is induced by an a c component of V g . Realistic experimental parameters are expected to be very similar to the ones used in the examples discussed above, except that a much higher quality factor of the resonator ( ∼ 10 5 ) and a much higher number of quanta in the **oscillator** can be reached. This number will easily exceed the thermal one, thus a proper lasing state with Poisson statistics, appropriately named SASER , is produced. One should then observe the usual line narrowing with line width given by κ N t h / 4 n ̄ ∼ κ 2 N t h / Γ ~ 1 . Experimental observation of this line-width narrowing would constitute a confirmation of the lasing/sasing....Average number of photons in the resonator as function of the ** qubit’s** relaxation rate, Γ 0 at the one-photon resonance, Ω R = ω T for g 3 = 0 and N t h = 5 . The dark blue line shows the numerical solution of the master equation, the light blue solid line represents the solution of the Langevin equation, Eq. ( dot alpha ). The green and red dashed curves represent respectively the saturation number n 0 and the thermal photon number N t h . The parameters are as in Fig. fig:compar (except for Γ 0 )....Superconducting

**qubits**coupled to electric or nanomechanical resonators display effects previously studied in quantum electrodynamics (QED) and extensions thereof. Here we study a driven

**qubit**coupled to a low-

**frequency**tank circuit with particular emphasis on the role of dissipation. When the

**qubit**is driven to perform Rabi

**oscillations**, with Rabi

**frequency**in resonance with the

**oscillator**, the latter can be driven far from equilibrium. Blue detuned driving leads to a population inversion in the

**qubit**and lasing behavior of the

**oscillator**("single-atom laser"). For red detuning the

**qubit**cools the

**oscillator**. This behavior persists at the symmetry point where the

**qubit**-

**oscillator**coupling is quadratic and decoherence effects are minimized. Here the system realizes a "single-atom-two-photon laser"....Also in situations where the

**qubit**, e.g., a Josephson charge

**qubit**, is coupled to a nano-mechanical

**oscillator**(Fig. fig:systemb) it either cools or amplifies the

**oscillator**. On one hand, this may constitute an important tool on the way to ground state cooling. On the other hand, this setup provides a realization of what is called a SASER ....Recent experiments on quantum state engineering with superconducting circuits realized concepts originally introduced in the field of quantum optics, as well as extensions thereof, e.g., to the regime of strong coupling , and prompted substantial theoretical activities . Josephson

**qubits**play the role of two-level atoms while electric or nanomechanical

**oscillators**play the role of the quantized radiation field. In most QED or circuit QED experiments the atom or

**qubit**transition

**frequency**is near resonance with the

**oscillator**. In contrast, in the experiments of Refs. , with setup shown in Fig. fig:systema), the

**qubit**is coupled to a slow LC

**oscillator**with

**frequency**( ω T / 2 π ∼ MHz) much lower than the

**level splitting ( Δ E / 2 π ℏ ∼ 10 GHz). The idea of this experiment is to drive the**

**qubit**’s**qubit**to perform Rabi

**oscillations**with Rabi

**frequency**in resonance with the

**oscillator**, Ω R ≈ ω T . In this situation the

**qubit**should drive the

**oscillator**and increase its

**oscillation**amplitude. When the

**qubit**driving

**frequency**is blue detuned, the driving creates a population inversion of the

**qubit**, and the system exhibits lasing behavior (“single-atom laser"); for red detuning the

**qubit**cools the

**oscillator**. A similar strategy for cooling of a nanomechanical resonator via a Cooper pair box

**qubit**has been recently suggested in Ref. . The analysis of the driven circuit QED system shows that these properties depend strongly on relaxation and decoherence effects in the

**qubit**....a) In the setup of Ref. an externally driven three-junction flux

**qubit**is coupled inductively to an LC

**oscillator**. b) A charge

**qubit**is coupled to a mechanical resonator....The systems to be considered are shown in Fig. fig:system. A

**qubit**is coupled to an

**oscillator**and driven to perform Rabi

**oscillations**. To be specific we first analyze the flux

**qubit**coupled to an electric

**oscillator**(Fig. fig:systema) with Hamiltonian ... Superconducting

**qubits**coupled to electric or nanomechanical resonators display effects previously studied in quantum electrodynamics (QED) and extensions thereof. Here we study a driven

**qubit**coupled to a low-

**frequency**tank circuit with particular emphasis on the role of dissipation. When the

**qubit**is driven to perform Rabi

**oscillations**, with Rabi

**frequency**in resonance with the

**oscillator**, the latter can be driven far from equilibrium. Blue detuned driving leads to a population inversion in the

**qubit**and lasing behavior of the

**oscillator**("single-atom laser"). For red detuning the

**qubit**cools the

**oscillator**. This behavior persists at the symmetry point where the

**qubit**-

**oscillator**coupling is quadratic and decoherence effects are minimized. Here the system realizes a "single-atom-two-photon laser".

Data types:

Contributors: Griffith, E. J., Ralph, J. F., Greentree, Andrew D., Clark, T. D.

Date: 2005-10-04

In the example figure (Fig. fig:qubosc1d2d), the control bias is varied from left to right for a low **frequency** **oscillator** circuit (1.36GHz). For each bias point the simulation is reinitialised, the stochastic time evolution of the system density matrix is simulated over 1500 **oscillator** cycles. Then the **oscillator** and **qubit** charge expectation values are extracted to obtain the power spectrum for each component, with a **frequency** resolution of 4.01MHz. The power spectra for each time series are collated as an image such that the power axis is now represented as a colour, and the individual power spectra are vertical ‘slices’ through the image. The dominant **frequency** peaks become line traces, therefore illustrating the various avoided crossings, mergeings and intersections. The example figure shows the PSD ‘slice’ at Bias = 0.5187 , the broadband noise is readily apparent and is due to the discontinuous quantum jumps in the **qubit**. The bias **oscillator** peak (1.36GHz) is most prominent in the **oscillator** PSD, as would be expected, but it is also present in the **qubit** PSD. It should also be noted that most features are present in both the **qubit** and **oscillator**, including the noise which is generated by the quantum jumps and the quantum state diffusion processes. Interestingly, the **qubit** PSD is significantly stronger than the **oscillator** PSD, however, a larger voltage is generated by the smaller charge due to the extremely small island capacitance, V q = q / C q ....fig:mwRamp (Color online) **Oscillator** PSD as a function of the applied microwave drive **frequency** f m w , for microwave amplitudes A m w = 0.0050 (A) and A m w = 0.0100 (B). It is important to notice that there are now two **frequency** axes per plot, a drive (H) and a response (V). Of particular interest is the magnified section which shows clearly the distinct secondary splitting in the sub-GHz regime. This occurs due to a high **frequency** interaction seen in the upper plots, where the lower Rabi sideband of the microwave drive passes through the high **frequency** **oscillator** signal. The maximum splitting occurs when the Rabi amplitude is a maximum, hence this is observed for a very particular combination of bias and drive, which is beneficial for charactering the **qubit**. Most importantly, this would not be observed with a conventional low **frequency** **oscillator** configuration as the f m w - f o s c separation would be too large for the Rabi **frequency**. ( κ = 5 × 10 -5 )....Fig. fig:mwRamp is presented in a similar manner as Fig. fig:BiasRamp. However there are now two **frequency** axes: the horizontal axis represents the **frequency** of the applied microwave drive field, and the vertical axis is the **frequency** response. It should be remembered that the microwave **frequency** axis is focused near the **qubit** transition **frequency** ( f q u b i t ≈ 3.49GHz) and the diagonally increasing line is now the microwave **frequency**....Autler Townes effect, charge **qubit**, characterisation, **frequency** spectrum...fig:QubitOscEnergy A two level **qubit** is coupled to a many level harmonic **oscillator**, investigated for two different **oscillator** energies. Firstly, the **oscillator** resonant **frequency** is set to 1.36GHz, this more resembles the conventional configuration such that the fundamental component of the **oscillator** does not drive the **qubit**. However, we also investigate the use of a high **frequency** **oscillator** of 3.06GHz which can excite this **qubit**. In addition, **qubit** is constantly driven by a microwave field at 3.49GHz to generate Rabi **oscillations** and in this paper we examine the relation between these three fields....A theoretical spectroscopic analysis of a microwave driven superconducting charge **qubit** (Cooper-pair box coupled) to an RLC **oscillator** model is performed. By treating the **oscillator** as a probe through the backreaction effect of the **qubit** on the **oscillator** circuit, we extract **frequency** splitting features analogous to the Autler-Townes effect from quantum optics, thereby extending the analogies between superconducting and quantum optical phenomenology. These features are found in a **frequency** band that avoids the need for high **frequency** measurement systems and therefore may be of use in **qubit** characterization and coupling schemes. In addition we find this **frequency** band can be adjusted to suit an experimental **frequency** regime by changing the **oscillator** **frequency**....fig:qubosc1d2d (Color online) **Oscillator** and **Qubit** power spectra slices for Bias = 0.5187, using the low **frequency** **oscillator** circuit f o s c = 1.36 GHz. The solid lines overlay the energy level separations found in Fig. fig:EnergyLevel. ( κ = 5 × 10 -5 ). As one would expect, the bias **oscillator** peak at 1.36GHz is clearly observed in the **oscillator** PSD, but only weakly in the **qubit** PSD. Likewise the **qubit** Rabi **frequency** is found to be stronger in the **qubit** PSD. However it is important to note that the **qubit** dynamics such as the Rabi **oscillations** are indeed coupled to the bias **oscillator** circuit and so can be extracted. In addition, it is recommended to compare the layout of the most prominent features with Fig. fig:BiasRamp....fig:BiasRamp (Color online) **Oscillator** PSD as a function of bias, for microwave amplitudes A m w = 0.0025 (A) and A m w = 0.0050 (B). The red lines track the positions (in **frequency**) of significant power spectrum peaks (+10dB to +15dB above background), the overlaid black and blue lines are the **qubit** energy and microwave transition (Fig. fig:EnergyLevel). Unlike Fig. fig:qubosc1d2d, in these figures the 3.06GHz **oscillator** circuit can now drive the **qubit** (Fig. fig:EnergyLevel) and so creates excitations which mix with the microwave driven excitations creating a secondary splitting centred on f m w - f o s c (430MHz). This feature contains the Rabi **frequency** information in the sidebands of the splitting, but now in a different and controllable **frequency** regime. In addition, the intersection of the two differently driven excitations (illustrated in the magnified sections), opens the possibility of calibrating the biased **qubit** against a fixed engineered **oscillator** circuit, using a single point feature. ( κ = 5 × 10 -5 )....In a previous paper , a method was proposed by which the energy level structure of a charge **qubit** can be obtained from measurements of the peak noise in the bias/control **oscillator**, without the need of extra readout devices. This was based on a technique originally proposed for superconducting flux **qubits** but there are many similarities between the two technologies. The **oscillator** noise peak is the result of broadband noise caused by quantum jumps in the **qubit** being coupled back to the **oscillator** circuit. This increase in the jump rate becomes a maximum when the Rabi **oscillations** are at peak amplitude, this should only occur when the **qubit** is correctly biased and the microwave drive is driving at the transition **frequency**. Therefore by monitoring this peak as a function of bias, we can associate a bias position with a microwave **frequency** equal to that of the energy gap, hence constructing the energy diagram (Fig. fig:EnergyLevel)....fig:Jumps (Color online) (A) **Oscillator** power spectra when the coupled **qubit** is driven at f m w = 5.00 GHz. An increase in bias noise power ( f o s c = 1.36 GHz) can be observed when Rabi **oscillations** occur, the more frequent quantum jump noise couples back to the **oscillator**. (B) Bias noise power peak position changes as a function of f m w , the microwave drive **frequency**. Therefore, it is possible to probe the **qubit** energy level structure by using the power increase in the **oscillator** which is already in place, eliminating the need for additional measurement devices. However, it should be noted that the surrounding **oscillator** harmonics may mask the microwave driven peak. ( κ = 1 × 10 -3 ). ... A theoretical spectroscopic analysis of a microwave driven superconducting charge **qubit** (Cooper-pair box coupled) to an RLC **oscillator** model is performed. By treating the **oscillator** as a probe through the backreaction effect of the **qubit** on the **oscillator** circuit, we extract **frequency** splitting features analogous to the Autler-Townes effect from quantum optics, thereby extending the analogies between superconducting and quantum optical phenomenology. These features are found in a **frequency** band that avoids the need for high **frequency** measurement systems and therefore may be of use in **qubit** characterization and coupling schemes. In addition we find this **frequency** band can be adjusted to suit an experimental **frequency** regime by changing the **oscillator** **frequency**.

Data types:

Contributors: Mitra, Kaushik, Lobb, C. J., de Melo, C. A. R. Sá

Date: 2007-12-06

In the preceeding analysis we neglected the effect of the local environment by setting Y i n t ω = 0 . As a result, the low-**frequency** value of T 1 is substantially larger than obtained in experiment . By modeling the local environment with R 0 = 5000 ohms and L 0 = 0 we obtain the T 1 versus ω 01 plot shown in Fig. fig:three. Notice that this value of R 0 brings T 1 to values close to 20 ns at T = 0 . The message to extract from Figs. fig:two and fig:three is that increasing R 0 as much as possible and increasing the **qubit** **frequency** ω 01 from 0.1 Ω to 2 Ω at fixed low temperature can produce a large increase in T 1 ....Schematic drawing of the phase **qubit** with an RLC isolation circuit....The circuit used to describe intrinsic decoherence and self-induced Rabi **oscillations** in phase **qubits** is shown in Fig. fig:one, which correponds to an asymmetric dc SQUID . The circuit elements inside the dashed box form an isolation network which serves two purposes: a) it prevents current noise from reaching the **qubit** junction; b) it is used as a measurement tool....In the limit of T = 0 , we can solve for c 1 t exactly and obtain the closed form c 1 t = L -1 s + Γ - i ω 01 2 + Ω 2 - Γ 2 s s + Γ - i ω 01 2 + Ω 2 - Γ 2 - κ Ω 4 π i / Γ where L -1 F s is the inverse Laplace transform of F s , and κ = α / M ω 01 × Φ 0 / 2 π 2 ≈ 1 / ω 01 T 1 , 0 . The element ρ 11 = | c 1 t | 2 of the density matrix is plotted in Fig. fig:four for three different values of resistance, assuming that the **qubit** is in its excited state such that ρ 11 0 = 1 . We consider the experimentally relevant limit of Γ ≪ ω 01 ≈ Ω , which corresponds to the weak dissipation limit. Since Γ = 1 / 2 C R the width of the resonance in the spectral density shown in Eq. ( eqn:sd-poles) is smaller for larger values of R . Thus, for large R , the RLC environment transfers energy resonantly back and forth to the **qubit** and induces Rabi-**oscillations** with an effective time dependent decay rate γ t = - 2 ℜ c ̇ 1 t / c 1 t ....We study decoherence effects in a dc SQUID phase **qubit** caused by an isolation circuit with a resonant **frequency**. The coupling between the SQUID phase **qubit** and its environment is modeled via the Caldeira-Leggett formulation of quantum dissipation/coherence, where the spectral density of the environment is related to the admittance of the isolation circuit. When the **frequency** of the **qubit** is at least two times larger than the resonance **frequency** of the isolation circuit, we find that the decoherence time of the **qubit** is two orders of magnitude larger than the typical ohmic regime, where the **frequency** of the **qubit** is much smaller than the resonance **frequency** of the isolation circuit. Lastly, we show that when the **qubit** **frequency** is on resonance with the isolation circuit, an oscillatory non-Markovian decay emerges, as the dc SQUID phase **qubit** and its environment self-generate Rabi **oscillations** of characteristic time scales shorter than the decoherence time....fig:three T 1 (in nanoseconds) as a function of **qubit** **frequency** ω 01 . The solid (red) curves describes an RLC isolation network with parameters R = 50 ohms, L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, and **qubit** parameters C 0 = 4.44 pF, R 0 = 5000 ohms and L 0 = 0 . The dashed curves correspond to an RL isolation network with the same parameters, except that C = 0 . Main figure ( T = 0 ), inset ( T = 50 mK) with Ω = 141 GHz....fig:four Population of the excited state of the **qubit** as a function of time ρ 11 t , with ρ 11 t = 0 = 1 for R = 50 ohms (solid curve), 350 ohms (dotted curve), and R = 550 ohms (dashed curve), and L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, C 0 = 4.44 pF, R 0 = ∞ and L 0 = 0 ....fig:two T 1 (in seconds) as a function of **qubit** **frequency** ω 01 . The solid (red) curves describes an RLC isolation network with parameters R = 50 ohms, L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, and **qubit** parameters C 0 = 4.44 pF, R 0 = ∞ and L 0 = 0 . The dashed curves correspond to an RL isolation network with the same parameters, except that C = 0 . Main figure ( T = 0 ), inset ( T = 50 mK) with Ω = 141 GHz....In Fig. fig:two, T 1 is plotted versus **qubit** **frequency** ω 01 for spectral densities describing an RLC (Eq. eqn:spectral-density-isolation) or Drude (Eq. eqn:sd-drude) isolation network at fixed temperatures T = 0 (main figure) and T = 50 mK (inset), for J i n t ω = 0 corresponding to R 0 ∞ . In the limit of low temperatures k B T / ℏ ω 01 ≪ 1 , the relaxation time becomes T 1 ω 01 = M ω 01 / J ω 01 . From Fig. fig:two (main plot) several important points can be extracted. First, in the low **frequency** regime ( ω 01 ≪ Ω ) the RL (Drude) and RLC environments produce essentially the same relaxation time T 1 , R L C 0 = T 1 , R L 0 = T 1 , 0 ≈ L 1 / L 2 R C 0 , because both systems are ohmic. Second, near resonance ( ω 01 ≈ Ω ), T 1 , R L C is substantially reduced because the **qubit** is resonantly coupled to its environment producing a distinct non-ohmic behavior. Third, for ( ω 01 > Ω ), T 1 grows very rapidly in the RLC case. Notice that for ω 01 > 2 Ω , the RLC relaxation time T 1 , R L C is always larger than T 1 , R L . Furthermore, in the limit of ω 01 ≫ m a x Ω , 2 Γ , T 1 , R L C grows with the fourth power of ω 01 behaving as T 1 , R L C ≈ T 1 , 0 ω 01 4 / Ω 4 , while for ω 01 ≫ Ω 2 / 2 Γ , T 1 , R L grows only with second power of ω 01 behaving as T 1 , R L ≈ 4 T 1 , 0 Γ 2 ω 01 2 / Ω 4 . Thus, T 1 , R L C is always much larger than T 1 , R L for sufficiently large ω 01 . Notice, however, that for parameters in the experimental range such as those used in Fig fig:two, T 1 , R L C is two orders of magnitude larger than T 1 , R L , indicating a clear advantage of the RLC environment shown in Fig fig:one over the standard ohmic RL environment. Thermal effects are illustrated in the inset of Fig. fig:two where T = 50 mK is a characteristic temperature where experiments are performed . The typical values of T 1 at low **frequencies** vary from 10 -5 s at T = 0 to 10 -6 s at T = 50 mK, while the high **frequency** values remain essentially unchanged as the thermal effects are not important for ℏ ω 01 ≫ k B T ....These environmentally-induced Rabi **oscillations** are a clear signature of the non-Markovian behavior produced by the RLC environment, and are completely absent in the RL environment because the energy from the **qubits** is quickly dissipated without being temporarily stored. These environmentally-induced Rabi **oscillations** are generic features of circuits with resonances in the real part of the admittance. The **frequency** of the Rabi **oscillations** Ω R a = π κ Ω 3 / 2 Γ is independent of the resistance since Ω R a ≈ Ω π L 2 C / L 1 2 C 0 , and has the value of Ω R a = 2 π f R a ≈ 360 × 10 6 rad/sec for Fig. fig:four. ... We study decoherence effects in a dc SQUID phase **qubit** caused by an isolation circuit with a resonant **frequency**. The coupling between the SQUID phase **qubit** and its environment is modeled via the Caldeira-Leggett formulation of quantum dissipation/coherence, where the spectral density of the environment is related to the admittance of the isolation circuit. When the **frequency** of the **qubit** is at least two times larger than the resonance **frequency** of the isolation circuit, we find that the decoherence time of the **qubit** is two orders of magnitude larger than the typical ohmic regime, where the **frequency** of the **qubit** is much smaller than the resonance **frequency** of the isolation circuit. Lastly, we show that when the **qubit** **frequency** is on resonance with the isolation circuit, an oscillatory non-Markovian decay emerges, as the dc SQUID phase **qubit** and its environment self-generate Rabi **oscillations** of characteristic time scales shorter than the decoherence time.

Data types:

Contributors: Ashhab, S.

Date: 2013-12-25

where we have defined the total spin operators J ̂ α = ∑ σ ̂ α / 2 . In the limit ℏ ω 0 / Δ → 0 , all the results concerning the low-energy spectrum of the resonator remain unchanged; one could say that the reduction of the coupling strength by the factor N is compensated by the strengthening of the spin raising and lowering operators by the same factor because of the collective behaviour of the **qubits**. In particular, the transition occurs at the critical coupling strength given by Eq. ( Eq:CriticalCouplingStrength). Because the **qubits** now have a larger total spin (when compared to the single-**qubit** case), spin states that are separated by small angles can be drastically different (i.e. have a small overlap). In particular, the overlap for N **qubits** is given by cos 2 N θ / 2 . By expanding this function to second order around θ = 0 , one can see that for small values of θ the relevant overlap is lower than unity by an amount that is proportional to N . This dependence translates into the dependence of the **qubit**-**oscillator** entanglement on the coupling strength just above the critical point. The entanglement therefore rises more sharply in the multi-**qubit** case (with the increase being by a factor N ), as demonstrated in Fig. Fig:EntropyLogLog....We consider the phase-transition-like behaviour in the Rabi model containing a single two-level system, or **qubit**, and a single harmonic **oscillator**. The system experiences a sudden transition from an uncorrelated state to an increasingly correlated one as the **qubit**-**oscillator** coupling strength is varied and increased past a critical point. This singular behaviour occurs in the limit where the **oscillator**'s **frequency** is much lower than the **qubit**'s **frequency**; away from this limit one obtains a finite-width transition region. By analyzing the energy-level structure, the value of the **oscillator** field and its squeezing and the **qubit**-**oscillator** correlation, we gain insight into the nature of the transition and the associated critical behaviour....(Color online) The logarithm of the von Neumann entropy S as a function of the logarithm of the quantity λ / λ c - 1 , which measures the distance of the coupling strength from the critical value. The red solid line corresponds to the single-**qubit** case, whereas the other lines correspond to the multi-**qubit** case: N = 2 (green dashed line), 3 (blue short-dashed line), 5 (purple dotted line) and 10 (dash-dotted cyan line). All the lines correspond to ℏ ω 0 / Δ = 10 -7 . The slope of all lines is approximately 0.92 when λ / λ c - 1 = 10 -4 . The ratio of the entropy in the multi-**qubit** case to that in the single-**qubit** case approaches N for all the lines as we approach the critical point....The energy level structure in the single-**qubit** case is simple in principle. In the limit ℏ ω 0 / Δ → 0 , one can say that the energy levels form two sets, one corresponding to each **qubit** state. Each one of these sets has a structure that is similar to that of a harmonic **oscillator** with some modifications that are not central in the present context. In particular the density of states has a weak dependence on energy, a situation that cannot support a thermal phase transition. If the temperature is increased while all other system parameters are kept fixed, **qubit**-**oscillator** correlations (which are finite only above the critical point) gradually decrease and vanish asymptotically in the high-temperature limit. No singular point is encountered along the way. This result implies that the transition point is independent of temperature. In other words, it remains at the value given by Eq. ( Eq:CriticalCouplingStrength) for all temperatures. If, for example, one is investigating the dependence of the correlation function C on the coupling strength (as plotted in Fig. Fig:SpinFieldSignCorrelationFunction), the only change that occurs as we increase the temperature is that the **qubit**-**oscillator** correlations change more slowly when the coupling strength is varied....where p ̂ is the ** oscillator’s** momentum operator, which is proportional to i â † - â in our definition of the operators. The squeezing parameter mirrors the behaviour of the low-lying energy levels. In particular we can see from Fig. Fig:SqueezingParameter that only when ℏ ω 0 / Δ reaches the value 10 -5 does the squeezing become almost singular at the critical point....(Color online) The von Neumann entropy S as a function of the

**oscillator**

**frequency**ℏ ω 0 and the coupling strength λ , both measured in comparison to the

**qubit**

**frequency**Δ . One can see clearly that moving in the vertical direction the rise in entropy is sharp in the regime ℏ ω 0 / Δ ≪ 1 , whereas it is smooth when ℏ ω 0 / Δ is comparable to or larger than 0.1....The tendency towards singular behaviour (in the dependence of various physical quantities on λ ) in the limit ℏ ω 0 / Δ → 0 is illustrated in Figs. Fig:ColorPlot- Fig:SqueezingParameter. In these figures, the entanglement, spin-field correlation function, low-lying energy levels (measured from the ground state) and the

**squeezing parameter are plotted as functions of the coupling strength. It is clear from Figs. Fig:EntropyLinear and Fig:SpinFieldSignCorrelationFunction that when ℏ ω 0 / Δ ≤ 10 -3 both the entanglement (which is quantified through the von Neumann entropy S = T r ρ q log 2 ρ q with ρ q being the**

**oscillator**’s**reduced density matrix) and the correlation function C = σ z s i g n a + a † rise sharply upon crossing the critical point . The low-lying energy levels, shown in Fig. Fig:EnergyLevels, approach each other to form a large group of almost degenerate energy levels at the critical point before they separate again into pairs of asymptotically degenerate energy levels. This approach is not complete, however, even when ℏ ω 0 / Δ = 10 -3 ; for this value the energy level spacing in the closest-approach region is roughly ten times smaller than the energy level spacing at λ = 0 . The squeezing parameter is defined by the width of the momentum distribution relative to that in the case of an isolated**

**qubit**’s**oscillator**. For consistency with Ref. , we define it as ... We consider the phase-transition-like behaviour in the Rabi model containing a single two-level system, or

**qubit**, and a single harmonic

**oscillator**. The system experiences a sudden transition from an uncorrelated state to an increasingly correlated one as the

**qubit**-

**oscillator**coupling strength is varied and increased past a critical point. This singular behaviour occurs in the limit where the

**oscillator**'s

**frequency**is much lower than the

**qubit**'s

**frequency**; away from this limit one obtains a finite-width transition region. By analyzing the energy-level structure, the value of the

**oscillator**field and its squeezing and the

**qubit**-

**oscillator**correlation, we gain insight into the nature of the transition and the associated critical behaviour.

Data types:

Contributors: Saito, S., Meno, T., Ueda, M., Tanaka, H., Semba, K., Takayanagi, H.

Date: 2005-08-19

We performed a spectroscopy measurement of the **qubit** with long (50 ns) single-**frequency** microwave pulses. We observed multi-photon resonant peaks ( Φ q b 1.5 Φ 0 ) in the dependence of P s w on f M W 1 at a fixed magnetic flux Φ q b . We obtained the **qubit** energy diagram by plotting their positions as a function of Φ q b / Φ 0 (Fig. Fig2(a)). We took the data around the degeneracy point Φ q b ≈ 1.5 Φ 0 by applying an additional dc pulse to the microwave line to shift Φ q b away from 1.5 Φ 0 just before the readout, because the dc-SQUID could not distinguish the **qubit** states around the degeneracy point. The top solid curve in Fig. Fig2(a) represents a numerical fit to the resonant **frequencies** of one-photon absorption. From this fit, we obtain the **qubit** parameters E J / h = 213 GHz, Δ / 2 π = 1.73 GHz, and α = 0.8. The other curves in Fig. Fig2(a) are drawn by using these parameters for n 1 = 2, 3, and 4....Next, we used short single-**frequency** microwave pulses with a **frequency** of 10.25 GHz to observe the coherent quantum dynamics of the **qubit**. Figures Fig2(b) and (c) show one- and four-photon Rabi **oscillations** observed at the operating points indicated by arrows in Fig. Fig2(a) with various microwave amplitudes V M W 1 . These data can be fitted by damped **oscillations** ∝ exp - t p / T d cos Ω R a b i t p , except for the upper two curves in Fig. Fig2(b). Here, t p and T d are the microwave pulse length and **qubit** decay time, respectively. To obtain Ω R a b i , we performed a fast Fourier transform (FFT) on the curves that we could not fit by damped **oscillations**. Although we controlled the **qubit** environment, there were some unexpected resonators coupled to the **qubit**, which could be excited by the strong microwave driving or by the Rabi **oscillations** of the **qubit**. We consider that these resonators degraded the Rabi **oscillations** in the higher V M W 1 range of Fig. Fig2(b). Figure Fig2(d) shows the V M W 1 dependences of Ω R a b i / 2 π up to four-photon Rabi **oscillations**, which are well reproduced by Eq. ( eq2). Here, we used only one scaling parameter a (10.25 GHz) = 0.013 defined as a f M W 1 ≡ 4 g 1 α 1 / ω M W 1 V M W 1 , because it is hard to measure the real amplitude of the microwave applied to the **qubit** at the sample position. The scaling parameter a f M W 1 reflects the way in which the applied microwave is attenuated during its transmission to the **qubit** and the efficiency of the coupling between the **qubit** and the on-chip microwave line. In this way, we can estimate the real microwave amplitude and the interaction energy between the **qubit** and the microwave 2 ℏ g 1 α 1 by fitting the dependence of Ω R a b i / 2 π on V M W 1 . These results show that we can reach a driving regime that is so strong that the interaction energy 2 ℏ g 1 α 1 is larger than the **qubit** transition energy ℏ ω q b ....Experimental results with single-**frequency** microwave pulses. (a) Spectroscopic data of the **qubit**. Each set of the dots represents the resonant **frequencies** f r e s caused by the one to four-photon absorption processes. The solid curves are numerical fits. The dashed line shows a microwave **frequency** f M W 1 of 10.25 GHz. (b) One-photon Rabi **oscillations** of P s w with exponentially damped **oscillation** fits. Both the **qubit** Larmor **frequency** f q b and the microwave **frequency** f M W 1 are 10.25 GHz. The external flux is Φ q b / Φ 0 = 1.4944. (c) Four-photon Rabi **oscillations** when f q b = 41.0 GHz, f M W 1 = 10.25 GHz, and Φ q b / Φ 0 = 1.4769. (d) The microwave amplitude dependence of the Rabi **frequencies** Ω R a b i / 2 π up to four-photon Rabi **oscillations**. The solid curves represent theoretical fits. Fig2...The measurements were carried out in a dilution refrigerator. The sample was mounted in a gold plated copper box that was thermalized to the base temperature of 20 mK ( k B T **frequency** microwave pulses, we added two microwaves MW1 and MW2 with **frequencies** of f M W 1 and f M W 2 , respectively by using a splitter SP (Fig. Fig1(b)). Then we shaped them into microwave pulses through two mixers. We measured the amplitude of MW k V M W k at the point between the attenuator and the mixer with an oscilloscope. We confirmed that unwanted higher-order **frequency** components in the pulses, for example | f M W 1 ± f M W 2 | , 2 f M W 1 , and 2 f M W 2 are negligibly small under our experimental conditions. First, we choose the operating point by setting Φ q b around 1.5 Φ 0 , which fixes the **qubit** Larmor **frequency** f q b . The **qubit** is thermally initialized to be in | g by waiting for 300 μ s, which is much longer than the **qubit** energy relaxation time (for example 3.8 μ s at f q b = 11.1 GHz). Then a **qubit** operation is performed by applying a microwave pulse to the **qubit**. The pulse, with an appropriate length t p , amplitudes V M W k , and **frequencies** f M W k , prepares a **qubit** in the superposition state of | g and | e . After the operation, we immediately apply a dc readout pulse to the dc-SQUID. This dc pulse consists of a short (70 ns) initial pulse followed by a long (1.5 μ s) trailing plateau that has 0.6 times the amplitude of the initial part. For Φ q b **qubit** is detected as being in | e , the SQUID switches to a voltage state and an output voltage pulse should be observed; otherwise there should be no output voltage pulse. By repeating the measurement 8000 times, we obtain the SQUID switching probability P s w , which is directly related to P e t p for the dc readout pulse with a proper amplitude. For Φ q b > 1.5 Φ 0 , P s w is directly related to 1 - P e t p ....We next investigated the coherent **oscillations** of the **qubit** through the parametric processes by using short two-**frequency** microwave pulses. Figure Fig3(a) [(b)] shows the Rabi **oscillations** of P s w when the **qubit** Larmor **frequency** f q b = 26.45 [7.4] GHz corresponds to the sum of the two microwave **frequencies** f M W 1 = 16.2 GHz, f M W 2 = 10.25 GHz [the difference between f M W 1 = 11.1 GHz and f M W 2 = 18.5 GHz] and the microwave amplitude of MW2 V M W 2 was fixed at 33.0 [50.1] mV. They are well fitted by exponentially damped **oscillations** ∝ exp - t p / T d cos Ω R a b i t p . The Rabi **frequencies** obtained from the data in Fig. 3(a) [(b)] are well reproduced by Eq. ( eq3) without any fitting parameters (Fig. Fig3(c) [(d)]). Here, we used Δ , which was obtained from the spectroscopy measurement (Fig. Fig2(a)) and used a (10.25 GHz) = 0.013 and a (16.2 GHz) = 0.0074 [ a (11.1 GHz) = 0.013 and a (18.5 GHz) = 0.0082], which had been obtained from Rabi **oscillations** by using single-**frequency** microwave pulses with each **frequency**. Those results provide strong evidence that we can achieve parametric control of the **qubit** with two-**frequency** microwave pulses....(a) Scanning electron micrograph of a flux **qubit** (inner loop) and a dc-SQUID (outer loop). The loop sizes of the **qubit** and SQUID are 10.2 × 10.4 μ m 2 and 12.6 × 13.5 μ m 2 , respectively. They are magnetically coupled by the mutual inductance M ≈ 13 pH. (b) A circuit diagram of the flux **qubit** measurement system. On-chip components are shown in the dashed box. L ≈ 140 pH, C ≈ 9.7 pF, R I 1 = 0.9 k Ω , R V 1 = 5 k Ω . Surface mount resistors R I 2 = 1 k Ω and R V 2 = 3 k Ω are set in the sample holder. We put adequate copper powder filters CP and LC filters F and attenuators A for each line. Fig1...Experimental results with two-**frequency** microwave pulses. (a) [(b)] Two-photon Rabi **oscillations** due to a parametric process when f q b = f M W 2 + - f M W 1 . The solid curves are fits by exponentially damped **oscillations**. (c) [(d)] Rabi **frequencies** as a function of V M W 1 , which are obtained from the data in Fig. Fig3(a) [(b)]. The dots represent experimental data when V M W 2 = 16.9, 23.5, 33.0, and 52.0 [50.1, 62.9, 79.1, and 124.7] mV from the bottom set of dots to the top one. The solid curves represent Eq. ( eq3). The inset is a schematic of the parametric process that causes two-photon Rabi **oscillation** when f q b = f M W 2 + - f M W 1 . Fig3...Parametric control of a superconducting flux **qubit** has been achieved by using two-**frequency** microwave pulses. We have observed Rabi **oscillations** stemming from parametric transitions between the **qubit** states when the sum of the two microwave **frequencies** or the difference between them matches the **qubit** Larmor **frequency**. We have also observed multi-photon Rabi **oscillations** corresponding to one- to four-photon resonances by applying single-**frequency** microwave pulses. The parametric control demonstrated in this work widens the **frequency** range of microwaves for controlling the **qubit** and offers a high quality testing ground for exploring nonlinear quantum phenomena. ... Parametric control of a superconducting flux **qubit** has been achieved by using two-**frequency** microwave pulses. We have observed Rabi **oscillations** stemming from parametric transitions between the **qubit** states when the sum of the two microwave **frequencies** or the difference between them matches the **qubit** Larmor **frequency**. We have also observed multi-photon Rabi **oscillations** corresponding to one- to four-photon resonances by applying single-**frequency** microwave pulses. The parametric control demonstrated in this work widens the **frequency** range of microwaves for controlling the **qubit** and offers a high quality testing ground for exploring nonlinear quantum phenomena.

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