### 63612 results for qubit oscillator frequency

Contributors: Ralph, J. F., Clark, T. D., Everitt, M. J., Prance, H., Stiffell, P., Prance, R. J.

Date: 2003-06-18

Schematic diagram of persistent current **qubit** inductively coupled to a (low **frequency**) classical **oscillator**. The insert graph shows the time-averaged (Floquet) energies as a function of the external bias field Φ x 1 for the parameters given in the text....We propose a method for characterising the energy level structure of a solid-state **qubit** by monitoring the noise level in its environment. We consider a model persistent-current **qubit** in a lossy resevoir and demonstrate that the noise in a classical bias field is a sensitive function of the applied field....Power spectral density for the low **frequency** **oscillator** at the resonance point ( Φ d c = 0.00015 Φ 0 ) for the three spontaneous decay rates shown in Figure 2: γ = 0.005 , 0.05 , 0.5 per cycle. The other parameters are given in the text...(a) Close-up of the time-averaged (Floquet) energies of the single photon resonance (500 MHz) - solid lines - with the time-independent energies given dotted lines. (b) The output power of the low **frequency** **oscillator** at 300 MHz, as a function of the static magnetic flux bias: γ = 0.005 per cycle (solid line), γ = 0.05 per cycle (crosses), γ = 0.5 per cycle (circles). The other parameters are given in the text...Characterising a solid state **qubit** via environmental noise ... We propose a method for characterising the energy level structure of a solid-state **qubit** by monitoring the noise level in its environment. We consider a model persistent-current **qubit** in a lossy resevoir and demonstrate that the noise in a classical bias field is a sensitive function of the applied field.

Files:

Contributors: Gambetta, J. M., Houck, A. A., Blais, Alexandre

Date: 2010-09-22

A superconducting **qubit** with Purcell protection and tunable coupling...In this letter we will present a three island device that has the properties of a **qubit** (two levels, arbitrary control and measurement) and has the ability to independently tune both the resonance **frequency** and coupling strength g , whilst still exhibiting exponential suppression of the charge noise and maintaining an anharmonicity equivalent to that of the transmon and phase **qubit**. This tunable coupling **qubit** (TCQ) can be tuned from a configuration which is totally Purcell protected from the resonator g = 0 (in a DFS) to a position which couples strongly to the resonator with values comparable to those realized for the transmon. Furthermore, we show that in the DFS position a strong measurement can be performed. The TCQ only needs to be moved from the DFS position when single and two **qubit** gates are required and as such in the off position all multi-**qubit** coupling rates are zero. That is, the TCQ in the circuit QED architecture (see Fig. Fig:TCQ A) is its own tunable coupler to any other TCQ, going beyond the nearest neighbour tunable couplers presented in Refs. and ....Fig:Tune(color online) Matrix element of the collective Cooper pair number operator (A) and transition energy (B) of the dark state as a function of the energy ratio E J + / E C for n ' g + = n ' g - = 0 , E I = - E C and E J - is numerically solved to ensure that only the coupling strength is tuned (blue) and **frequency** (red). Solid lines are from a numerical diagonalization and dashed lines are from the coupled anharmonic **oscillator** model....We present a superconducting **qubit** for the circuit quantum electrodynamics architecture that has a tunable coupling strength g. We show that this coupling strength can be tuned from zero to values that are comparable with other superconducting **qubits**. At g = 0 the **qubit** is in a decoherence free subspace with respect to spontaneous emission induced by the Purcell effect. Furthermore we show that in the decoherence free subspace the state of the **qubit** can still be measured by either a dispersive shift on the resonance **frequency** of the resonator or by a cycling-type measurement....Since charge fluctuations are one of the leading sources of noise in superconducting circuits we want to ensure that quantum information in the TCQ is not destroyed by charge noise. Following Ref. , the dephasing time T φ for the **qubit** and m level will scale as 1 / | ε q m | where ε q m is the the peak to peak value for the charge dispersion of the 0 - 1 and 0 - m transition respectively. The dispersion in the energy levels arises from the gate charges n ' g α and the fact the the potential is periodic. This can not be predicted with the coupled anharmonic **oscillator** model and as such is investigated numerically. We expect, that like the transmon, this will exponentially decrease with the ratio of E J / E C as in this limit the effects of tunneling from one minima to the next becomes exponentially suppressed. This is confirmed in Fig. Fig:Levels B where the have plotted | ε q m | / E q m (the numerical maximum and minimum of the energy level over n ' g α ) as a function of E J / E C for E I = E C and E I = 0 (transmon limit). That is, the TCQ has the same charge noise immunity as the transmon....Currently the most successful superconducting **qubits** are the flux , phase , and transmon as these **qubits** are essentially immune to offset charge (charge noise) by design. The transmon receives its charge noise immunity by operating at a point in parameter space where the energy level variations with offset charge are exponentially suppressed. This suppression has experimentally been observed and resulted in these **qubits** being approximately T 1 limited ( T 2 ≈ 2 T 1 ) in the circuit quantum electrodynamics (QED) architecture . In this architecture the **qubits** are coupled to a coplanar waveguide resonator through a Jaynes-Cummings Hamiltonian operated in the dispersive regime . This resonator acts as the channel to control, couple, and readout the state of the **qubit** (see Fig. Fig:TCQ A)....Since charge fluctuations are one of** the **leading sources of noise in superconducting circuits we want to ensure that quantum information in** the **TCQ is not destroyed by charge noise. Following Ref. , the dephasing time T φ **for** the qubit and m level will scale as 1 / | ε q m | where ε q m is** the **the peak to peak value **for** the charge dispersion of** the **0 - 1 and 0 - m transition respectively. The dispersion in** the **energy levels arises from** the **gate charges n ' g α and** the **fact** the **the potential is periodic. This can not be predicted with** the **coupled anharmonic oscillator model and as such is investigated numerically. We expect, that like** the **transmon, this will exponentially decrease with** the **ratio of E J / E C as in this limit** the **effects of tunneling from one minima to** the **next becomes exponentially suppressed. This is confirmed in Fig. Fig:Levels B where** the **have plotted | ε q m | / E q m (the numerical maximum and minimum of** the **energy level over n ' g α ) as a function of E J / E C **for** E I = E C and E I = 0 (transmon limit). That is, the TCQ has** the **same charge noise immunity as** the **transmon....To achieve tuning of g ~ + we modify the original circuit and replace the Josephson junctions by SQUIDs with Josephson energy E J ± 1 and E J ± 2 (this is hinted at in Fig. Fig:TCQ B). In making this replacement the only change in the above theory is the replacement E J ± → E J ± m a x cos π Φ x ± / Φ 0 1 + d 2 tan 2 π Φ x / Φ 0 with E J ± m a x = E J ± 1 + E J ± 2 , d = E J ± 1 - E J ± 2 / E J ± m a x and Φ x ± is the external flux applied to each SQUID which we assume to be independent (this is not required but simplifies our argument). This independent control allows us to change E J ± independently which in-turn allows independent control on g ~ + and ω q . To illustrate this we consider the symmetric case and plot in Fig. Fig:Tune the normalized coupling strength g ~ + ℏ / 2 e 2 V r m s β (A) and ω ~ q (B) as a function of the ratio E J + / E C when E I = - E C and E J - is numerically solved to ensure that only the coupling rate (blue) and **frequency** (red) vary respectively for both the full numerical (solid) and effective model (dashed). In the full numerical model g ~ + = 2 e 2 V r m s 1 | ( β + n + + β - n - | 0 / ℏ . Here the independent control is clearly observed. Note that while our numerical investigation was only for the symmetric case independent tunable g ~ + (from zero to large values) and ω q will still occur when the device is not symmetric. There is just a different condition on E J ± for the required tuning....In this letter we will present a three island device that has** the **properties of a qubit (two levels, arbitrary control and measurement) and has** the **ability to independently tune both** the **resonance frequency and coupling strength g , whilst still exhibiting exponential suppression of** the **charge noise and maintaining an anharmonicity equivalent to that of** the **transmon and phase qubit. This tunable coupling qubit (TCQ) can be tuned from a configuration which is totally Purcell protected from** the **resonator g = 0 (in a DFS) to a position which couples strongly to** the **resonator with values comparable to those realized **for** the transmon. Furthermore, we show that in** the **DFS position a strong measurement can be performed. The TCQ only needs to be moved from** the **DFS position when single and two qubit gates are required and as such in** the **off position all multi-qubit coupling rates are zero. That is, the TCQ in** the **circuit QED architecture (see Fig. Fig:TCQ A) is its own tunable coupler to any other TCQ, going beyond** the **nearest neighbour tunable couplers presented in Refs. and ....Currently** the **most successful superconducting qubits are** the **flux , phase , and transmon as these qubits are essentially immune to offset charge (charge noise) by design. The transmon receives its charge noise immunity by operating at a point in parameter space where** the **energy level variations with offset charge are exponentially suppressed. This suppression has experimentally been observed and resulted in these qubits being approximately T 1 limited ( T 2 ≈ 2 T 1 ) in** the **circuit quantum electrodynamics (QED) architecture . In this architecture** the **qubits are coupled to a coplanar waveguide resonator through a Jaynes-Cummings Hamiltonian operated in** the **dispersive regime . This resonator acts as** the **channel to control, **couple**, and readout** the **state of** the **qubit (see Fig. Fig:TCQ A)....To achieve tuning of g ~ + we modify** the **original circuit and replace** the **Josephson junctions by SQUIDs with Josephson energy E J ± 1 and E J ± 2 (this is hinted at in Fig. Fig:TCQ B). In making this replacement** the **only change in** the **above theory is** the **replacement E J ± → E J ± m a x cos π Φ x ± / Φ 0 1 + d 2 tan 2 π Φ x / Φ 0 with E J ± m a x = E J ± 1 + E J ± 2 , d = E J ± 1 - E J ± 2 / E J ± m a x and Φ x ± is** the **external flux applied to each SQUID which we assume to be independent (this is not required but simplifies our argument). This independent control allows us to change E J ± independently which in-turn allows independent control on g ~ + and ω q . To illustrate this we consider** the **symmetric case and plot in Fig. Fig:Tune** the **normalized coupling strength g ~ + ℏ / 2 e 2 V r m s β (A) and ω ~ q (B) as a function of** the **ratio E J + / E C when E I = - E C and E J - is numerically solved to ensure that only** the **coupling rate (**blue) and** frequency (red) vary respectively **for** both** the **full numerical (solid) and effective model (dashed). In** the **full numerical model g ~ + = 2 e 2 V r m s 1 | ( β + n + + β - n - | 0 / ℏ . Here** the **independent control is clearly observed. Note that while our numeri...Fig:Levels(color online) A) Eigenenergies of the TCQ Hamiltonian as a function of E I / E C for E J ± = 50 E C . Solid lines are from a numerical diagonalization and dashed lines are from the coupled anharmonic **oscillator** model. B) Charge dispersion | ε q m | as a function of the ratio E J / E C for E I = - E C (solid lines) and E I = 0 (dashed lines)....Fig:Tune(color online) Matrix element of the collective Cooper pair number operator (A) and transition energy (B) of the dark state as a function of the energy ratio E J + / E C for n ' g + = n ' g - = 0 , E I = - E C and E J - is numerically solved to ensure that only the coupling strength is tuned (blue) and frequency (red). Solid lines are from a numerical diagonalization and dashed lines are from the coupled anharmonic **oscillator** model....where ω ~ ± = ω ± + δ ~ ± - δ ± / 2 + δ + + δ - J 2 / 2 μ 2 ± μ / 2 ∓ η / 2 , δ ~ ± = δ + + δ - 1 + η 2 / μ 2 / 4 ± η δ + - δ - / 2 μ and δ ~ c = 2 J 2 δ + + δ - / μ 2 with μ = 4 J 2 + η 2 and the tilde indicating the diagonalized frame. The coupling has induced a conditional anharmonicity δ ~ c , it is this anharmonicity that makes this system different to two coupled **qubits**, it ensures that E 11 is not equal to E 01 + E 10 . Here we have introduced the notation that superscript i j refers to i excitations in the dark mode ( ` ` + " ) and j excitations in the bright mode ( ` ` - " ). The choice of these names will become clearer latter. The dotted lines in Fig. Fig:Levels A are the predictions from this effective model, which agree well with the full numerics. Thus from the effective model the anharmonicities are all around E C provided | J | > | η | . That is the TCQ has not lost any anharmonicity in comparison to the transmon or phase **qubit** and with simple pulse shaping techniques arbitrary control of the lowest three levels will be possible . We will now introduce the notation that the **qubit** is formed by the space | 0 = | 00 ~ , | 1 = | 10 ~ and | m = | 01 ~ is the measurement state....where ω ~ ± = ω ± + δ ~ ± - δ ± / 2 + δ + + δ - J 2 / 2 μ 2 ± μ / 2 ∓ η / 2 , δ ~ ± = δ + + δ - 1 + η 2 / μ 2 / 4 ± η δ + - δ - / 2 μ and δ ~ c = 2 J 2 δ + + δ - / μ 2 with μ = 4 J 2 + η 2 and** the **tilde indicating** the **diagonalized frame. The coupling has induced a conditional anharmonicity δ ~ c , it is this anharmonicity that makes this system different to two coupled qubits, it ensures that E 11 is not equal to E 01 + E 10 . Here we have introduced** the **notation that superscript i j refers to i excitations in** the **dark mode ( ` ` + " ) and j excitations in** the **bright mode ( ` ` - " ). The choice of these names will become clearer latter. The dotted lines in Fig. Fig:Levels A are** the **predictions from this effective model, which agree well with** the **full numerics. Thus from** the **effective model** the **anharmonicities are all around E C provided | J | > | η | . That is** the **TCQ has not lost any anharmonicity in comparison to** the **transmon or phase qubit and with simple pulse shaping techniques arbitrary control of** the **lowest three levels will be possible . We will now introduce** the **notation that** the **qubit is formed by** the **space | 0 = | 00 ~ , | 1 = | 10 ~ and | m = | 01 ~ is** the **measurement state. ... We present a superconducting **qubit** for the circuit quantum electrodynamics architecture that has a tunable coupling strength g. We show that this coupling strength can be tuned from zero to values that are comparable with other superconducting **qubits**. At g = 0 the **qubit** is in a decoherence free subspace with respect to spontaneous emission induced by the Purcell effect. Furthermore we show that in the decoherence free subspace the state of the **qubit** can still be measured by either a dispersive shift on the resonance **frequency** of the resonator or by a cycling-type measurement.

Files:

Contributors: Korotkov, Alexander N.

Date: 2000-03-13

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....Schematic of a **qubit** continuously measured by a detector with output signal I t ....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....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....Using** **Eqs. ( Bayes1)–( Bayes3) and** the **Monte-Carlo

**method (similar**

**to**

**Ref. ) we**

**can**

**calculate**

**in**

**a**

**straightforward**

**way**

**spectral**

**the****density**

**S**I

**ω**

**of**

**detector**

**the****current**

**I**

**t . Solid**

**lines**

**in**

**Fig. M-C**

**show**

**results**

**the****of**

**such**

**calculations**

**for**

**ideal**

**the****detector, η = 1**

**,**and

**weak**

**coupling**between

**the**

**qubit**and

**detector, α = 0.1**

**the****,**where

**α ≡ ℏ**

**Δ**

**I**

**2 / 8**

**S**0

**H ( α**

**is**

**8**

**times**

**less**

**than**

**parameter**

**the****C**

**introduced**

**in**

**Ref. ). One**

**can**

**see**

**that**

**in**

**symmetric**

**the****case, ε = 0**

**,**

**the**

**peak**

**at**

**the**

**frequency**of

**quantum**

**oscillations**

**is**

**4**

**times**

**higher**

**than**

**noise**

**the****pedestal,**

**S**I

**Ω = 5**

**S**0

**while**

**peak**

**the****width**

**is**

**determined**

**by**

**the**

**coupling**strength

**α (see**

**Fig. transition**

**below). In**

**asymmetric**

**the****case, ε ≠ 0**

**,**

**the**

**peak**

**height**

**decreases (Fig. M-C), while**

**additional**

**the****Lorentzian-shape**

**increase**

**of**

**S**I

**ω**

**appears**

**at**

**low**

**frequencies**. The

**origin**

**of**

**this**

**low-**

**frequency**feature

**is**

**slow**

**the****fluctuation**

**of**

**asymmetry**

**the****of**

**ρ**

**11**

**oscillations (Fig. asym). In**

**case**

**ε = 0**

**amplitude**

**the****of**

**ρ**

**11**

**oscillations**

**is**

**maximal (see**

**thick**

**line**

**in**

**Fig. I(t)a), hence**

**there**

**is**

**no**

**such**

**asymmetry**

**and**

**low-**

**the****frequency**feature

**is**

**absent, while**

**spectral**

**the****peak**

**at**

**the**

**frequency**of

**quantum**

**oscillation**

**is**

**maximally**

**high....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....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**....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....Schematic

**of a**qubit continuously measured by a detector with output signal I t ....The

**situation**

**changes**

**as**

**the**

**coupling**between

**detector**

**the****and**

**qubit**increases, α

**1 . The**

**strong**

**influence**

**of**

**measurement**

**destroys**

**quantum**

**oscillations, and**

**Quantum**

**the****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**

**properly**

**the****averaged**

**detector**

**current**

**follows**

**pretty**

**well**

**evolution**

**the****of**

**the**

**qubit**(however,

**the**

**unsuccessful**

**tunneling “attempts” still**

**cannot**

**be**

**directly**

**resolved), and**

**spectral**

**the****density**

**of**

**I**

**t**

**can**

**be**

**calculated**

**using**

**classical**

**the****theory**

**of**

**telegraph**

**noise**

**leading**

**to**

**Lorentzian**

**the****shape**

**of**

**S**I

**ω . Figure**

**transitiona**

**shows**

**gradual**

**the****transformation**

**of**

**spectral**

**the****density**

**with**

**increase**

**the****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....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**

**numerically**

**the****calculated**

**spectral**

**density**

**S**I

**ω**

**of**

**detector**

**the****current**

**for**

**a**

**nonideal**

**detector, η = 0.5 (dashed**

**lines) and**

**for**

**an**

**ideal**

**detector**

**but**

**extra**

**coupling**of

**the**

**qubit**to

**environment**

**the****at**

**temperature**

**T = H (solid**

**lines). The**

**rates**

**γ**

**1**

**and**

**γ**

**2**

**are**

**chosen**

**according**

**to**

**Eqs. ( enveqv1) and ( enveqv2). For**

**symmetric**

**the****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

**spectral**

**the****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. ... 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.

Files:

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....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....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....Power out of the resonator as a function of frequency for different input powers at zero flux bias....**Frequency**-Tunable Josephson Junction Resonator for Quantum Computing ... 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: Kleff, S., Kehrein, S., von Delft, J.

Date: 2003-04-08

Structured bath/weak coupling: We now turn to the structured spectral density given by Eq.( eq:density). The main features of the corresponding system [Eq.( eq:system)] can already be understood by analyzing only the coupled two-level-harmonic **oscillator** system (without damping, i.e. Γ = 0 ). For ε = 0 this system exhibits two characteristic **frequencies**, close to Ω and Δ , associated with the transitions 1 and 2 in Fig. fig:correlation_weak(c). These should also show up in the correlation function C ω ; and indeed Fig. fig:correlation_weak(a) displays a double-peak structure with the peak separation somewhat larger than Δ - Ω , due to level repulsion. The coupling to the bath will in general lead to a broadening of the resonances and an enhancement of the repulsion of the two energies. Due to the very small coupling ( α = 0.0006 ) peak positions of C ω in Fig. fig:correlation_weak can with very good accuracy be derived from a second order perturbation calculation for the coupled two-level-harmonic **oscillator** system, yielding the following transition **frequencies** [depicted in inset (c) of Fig. fig:correlation_weak]: ω 1 , + - ω 0 , + = Ω - g 22 Δ 0 / Δ 2 0 - Ω 2 ≈ 0.987 Ω and ω 0 , - - ω 0 , + = Δ 0 + g 22 Δ 0 / Δ 2 0 - Ω 2 ≈ 1.346 Ω . With the two peaks we associate two different dephasing times, τ Ω and τ Δ , as shown in inset (a) and (b) of Fig. fig:correlation_weak....We discuss dephasing times for a two-level system (including bias) coupled to a damped harmonic **oscillator**. This system is realized in measurements on solid-state Josephson **qubits**. It can be mapped to a spin-boson model with a spectral function with an approximately Lorentzian resonance. We diagonalize the model by means of infinitesimal unitary transformations (flow equations), and calculate correlation functions, dephasing rates, and **qubit** quality factors. We find that these depend strongly on the environmental resonance **frequency** $\Omega$; in particular, quality factors can be enhanced significantly by tuning $\Omega$ to lie below the **qubit** **frequency** $\Delta$....Spin-spin correlation function as a function of **frequency** for experimentally relevant parameters discussed in Ref. : α = 0.0006 , Δ 0 = 4 GHz, ε = 0 (this is the so-called “idle state”), Ω = 3 GHz, Γ = 0.02 , and ω c = 8 GHz. The sum rule is fulfilled with an error of less than 1 %. (a) Blow up of the peak region reveals a double peak; (b) blow up of the larger peak, (c) term scheme of a two level system coupled to an **harmonic** **oscillator**, drawn for Δ 0 ≫ Ω ; ( α = 0.0006 corresponds to g / Ω ≈ 0.06 .) -0.9cm...Stronger** **coupling** **to** **bath: Figure fig:times_nobias(b) shows** **τ** **Δ , τ** **Ω , and** **τ** **w** **for** **a** **larger** **coupling** **strength** **of** **α = 0**.**01 . Figure fig:correlation_strong(a) shows** **one** **of** **the** **calculated** **correlation** **functions. Note** **that** **the** **stronger** **coupling** **α** **leads** **to** **a** **larger** **separation, or “level** **repulsion”, between** **the** **Δ - and** **Ω -peaks** **than** **in** **Fig. fig:correlation_weak. The** **inset** **of** **Fig. fig:times_nobias(b) shows** **the** **renormalized** **tunneling** **matrix** **element** **Δ ∞ as** **a** **function** **the** **initial** **matrix** **element** **Δ** **0 . Very** **importantly, for** **Δ** **0** **Ω , Δ** **increases** **during** **the** **flow, whereas** **for** **Δ** **0** **Ω , it** **decreases . This** **behavior** **can** **be** **understood** **from** **the** **fact** **that** **f** **ω** **l** **in** **Eq.( eq:flow1) changes** **sign** **at** **ω = Δ : If** **the** **weight** **of** **J** **ω** **under** **the** **integral** **in ( eq:flow1) is** **larger** **for** **ω > Δ** **0 , which** **is** **the** **case** **if** **Δ Δ** **0 ]. Note** **also, that** **the** **upward** **renormalization** **towards** **larger** **Δ ∞ in** **the** **inset** **of** **Fig. fig:times_nobias(b) is** **stronger** **than** **the** **downward** **one** **towards** **smaller** **values, i**.**e., the** **renormalization** **is** **not** **symmetric** **with** **respect** **to** **Δ** **0 = Ω . The** **reason** **for** **this** **asymmetry** **lies** **in** **the** **fact** **that** **f** **ω** **l** **has** **a** **larger** **weight** **for** **ω Δ . Also** **τ** **Δ** **and** **even** **τ** **w = 1 / J** **Δ ∞ in** **Fig. fig:times_nobias(b) show** **an** **asymmetric** **behavior** **with** **a** **steep** **increase** **at** **Δ** **0 ≈ Ω : dephasing** **times** **for** **Δ** **0 > Ω** **are** **larger** **than** **for** **Δ** **0 Ω** **than** **for** **Δ** **0 Ω , dephasing** **times** **can** **be** **significantly** **enhanced (as** **compared** **to** **Δ** **0 0**.**3** **Ω ) of** **the** **two-level** **system** **to** **the** **harmonic** **oscillator** **in (b)....Exploiting environmental resonances to enhance **qubit** quality factors...Spin-spin correlation function as a function of **frequency** for experimentally relevant parameters discussed in Ref. : α = 0.0006 , Δ 0 = 4 GHz, ε = 0 (this is the so-called “idle state”), Ω = 3 GHz, Γ = 0.02 , and ω c = 8 GHz. The sum rule is fulfilled with an error of less than 1 %. (a) Blow up of the peak region reveals a double peak; (b) blow up of the larger peak, (c) term scheme of a two level system coupled to an harmonic **oscillator**, drawn for Δ 0 ≫ Ω ; ( α = 0.0006 corresponds to g / Ω ≈ 0.06 .) -0.9cm...Stronger** **coupling** **to** **bath: Figure fig:times_nobias(b) shows** **τ** **Δ , τ** **Ω , and** **τ** **w** **for** **a** **larger** **coupling** **strength** **of** **α = 0**.**01 . Figure fig:correlation_strong(a) shows** **one** **of** **the** **calculated** **correlation** **functions. Note** **that** **the** **stronger** **coupling** **α** **leads** **to** **a** **larger** **separation, or “level** **repulsion”, between** **the** **Δ - and** **Ω -peaks** **than** **in** **Fig. fig:correlation_weak. The** **inset** **of** **Fig. fig:times_nobias(b) shows** **the** **renormalized** **tunneling** **matrix** **element** **Δ ∞ as** **a** **function** **the** **initial** **matrix** **element** **Δ** **0 . Very** **importantly, for** **Δ** **0** **Ω , Δ** **increases** **during** **the** **flow, whereas** **for** **Δ** **0** **Ω , it** **decreases . This** **behavior** **can** **be** **understood** **from** **the** **fact** **that** **f** **ω** **l** **in** **Eq.( eq:flow1) changes** **sign** **at** **ω = Δ : If** **the** **weight** **of** **J** **ω** **under** **the** **integral** **in ( eq:flow1) is** **larger** **for** **ω > Δ** **0 , which** **is** **the** **case** **if** **Δ Δ** **0 ]. Note** **also, that** **the** **upward** **renormalization** **towards** **larger** **Δ ∞ in** **the** **inset** **of** **Fig. fig:times_nobias(b) is** **stronger** **than** **the** **downward** **one** **towards** **smaller** **values, i**.**e., the** **renormalization** **is** **not** **symmetric** **with** **respect** **to** **Δ** **0 = Ω . The** **reason** **for** **this** **asymmetry** **lies** **in** **the** **fact** **that** **f** **ω** **l** **has** **a** **larger** **weight** **for** **ω Δ . Also** **τ** **Δ** **and** **even** **τ** **w = 1 / J** **Δ ∞ in** **Fig. fig:times_nobias(b) show** **an** **asymmetric** **behavior** **with** **a** **steep** **increase** **at** **Δ** **0 ≈ Ω : dephasing** **times** **for** **Δ** **0 > Ω** **are** **larger** **than** **for** **Δ** **0 Ω** **than** **for** **Δ** **0 Ω , dephasing** **times** **can** **be** **significantly** **enhanced (as** **compared** **to** **Δ** **0 .3** **Ω ) of** **the** **two-level** **system** **to** **the** **harmonic** **oscillator** **in (b)....Stronger coupling to bath: Figure fig:times_nobias(b) shows τ Δ , τ Ω , and τ w for a larger coupling strength of α = 0.01 . Figure fig:correlation_strong(a) shows one of the calculated correlation functions. Note that the stronger coupling α leads to a larger separation, or “level repulsion”, between the Δ - and Ω -peaks than in Fig. fig:correlation_weak. The inset of Fig. fig:times_nobias(b) shows the renormalized tunneling matrix element Δ ∞ as a function the initial matrix element Δ 0 . Very importantly, for Δ 0 Ω , Δ increases during the flow, whereas for Δ 0 Ω , it decreases . This behavior can be understood from the fact that f ω l in Eq.( eq:flow1) changes sign at ω = Δ : If the weight of J ω under the integral in ( eq:flow1) is larger for ω > Δ 0 , which is the case if Δ Δ 0 ]. Note also, that the upward renormalization towards larger Δ ∞ in the inset of Fig. fig:times_nobias(b) is stronger than the downward one towards smaller values, i.e., the renormalization is not symmetric with respect to Δ 0 = Ω . The reason for this asymmetry lies in the fact that f ω l has a larger weight for ω Δ . Also τ Δ and even τ w = 1 / J Δ ∞ in Fig. fig:times_nobias(b) show an asymmetric behavior with a steep increase at Δ 0 ≈ Ω : dephasing times for Δ 0 > Ω are larger than for Δ 0 Ω than for Δ 0 Ω , dephasing times can be significantly enhanced (as compared to Δ 0 **oscillator** in (b)....Spin-spin correlation function for the structured bath [Eq.( eq:density)] as a function of **frequency** . The maximum height of the middle peak in (b) is ≈ 7.2 . -0.9cm ... We discuss dephasing times for a two-level system (including bias) coupled to a damped harmonic **oscillator**. This system is realized in measurements on solid-state Josephson **qubits**. It can be mapped to a spin-boson model with a spectral function with an approximately Lorentzian resonance. We diagonalize the model by means of infinitesimal unitary transformations (flow equations), and calculate correlation functions, dephasing rates, and **qubit** quality factors. We find that these depend strongly on the environmental resonance **frequency** $\Omega$; in particular, quality factors can be enhanced significantly by tuning $\Omega$ to lie below the **qubit** **frequency** $\Delta$.

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Contributors: Bertet, P., Harmans, C. J. P. M., Mooij, J. E.

Date: 2005-09-30

(Color online) a) Two flux-**qubits** (shown coupled to their read-out SQUIDs and to their flux control line C i ( i = 1 , 2 )) coupled by a fixed mutual inductance M . b) Parametric coupling scheme : the two flux-**qubits** are now coupled through a circuit that allows to modulate the coupling constant g through the control parameter λ . fig0...a) Circuit proposed in to implement a tunable coupling between two flux-**qubits**. The two **qubits** are directly coupled by a mutual inductance M q q , and also via the dynamical inductance of a DC-SQUID which depends on the bias current I b at fixed flux bias. The total coupling constant g is shown in b) for the same parameters as were considered in as a function of s = I b / 2 I 0 . The dashed line indicates g = 0 . Inset : (dg/ds) as a function of s . fig1...Calculated evolution of the density matrix under the application of an entangling microwave pulse at the **frequency** | Δ 1 - Δ 2 | in the SQUID bias current, with Δ 1 = 5 G H z , Δ 2 = 7 G H z , g t = g 0 + δ g cos 2 π Δ 2 - Δ 1 t and δ g = 100 M H z . For the black curve, g 0 = 0 ; for the grey curve, g 0 = 200 M H z . fig2...We first discuss why the simplest fixed linear coupling scheme as was implemented in the two-**qubit** experiments fails in that respect. Consider two flux **qubits** biased at their flux-noise insensitive point γ Q = π ( γ Q being the total phase drop across the three junctions), and inductively coupled as shown in figure fig0a . The uncoupled energy states of each **qubit** are denoted | 0 i , | 1 i ( i = 1 , 2 ) and their minimum energy separation h Δ i ≡ ℏ ω i . Throughout this article, we will suppose that Δ 1 ≥ Δ 2 . As shown before , the system hamiltonian can be written as H = H q 1 + H q 2 + H I , with H q i = - h / 2 Δ i σ z i ( i = 1 , 2 ) and H I = h g 0 σ x 1 σ x 2 = h g 0 σ 1 + σ 2 + + σ 1 - σ 2 - + σ 1 + σ 2 - + σ 1 - σ 2 + . Here we introduced the Pauli matrices σ x . . z ; i referring to each **qubit** subspace, the raising (lowering) operators σ i + ( σ i - ) and we wrote the hamiltonian in the energy basis of each **qubit**. It is more convenient to rewrite the previous hamiltonian in the interaction representation, resulting in H ' I = exp i H q 1 + H q 2 t / ℏ H I exp - i H q 1 + H q 2 t / ℏ . We obtain...We first discuss why the simplest fixed linear coupling scheme as was implemented in the two-**qubit** experiments fails in that respect. Consider two flux **qubits** biased at their flux-noise insensitive point γ Q = π ( γ Q being the total phase drop across the three junctions), and inductively coupled as shown in figure fig0a . The uncoupled energy states of each** qubit** are denoted | 0 i , | 1 i ( i = 1 , 2 ) and their minimum energy separation h Δ i ≡ ℏ ω i . Throughout this article, we will suppose that Δ 1 ≥ Δ 2 . As shown before , the system hamiltonian can be written as H = H q 1 + H q 2 + H I , with H q i = - h / 2 Δ i σ z i ( i = 1 , 2 ) and H I = h g 0 σ x 1 σ x 2 = h g 0 σ 1 + σ 2 + + σ 1 - σ 2 - + σ 1 + σ 2 - + σ 1 - σ 2 + . Here we introduced the Pauli matrices σ x . . z ; i referring to each

**subspace, the raising (lowering) operators σ i + ( σ i - ) and we wrote the hamiltonian in the energy basis of each**

**qubit****. It is more convenient to rewrite the previous hamiltonian in the interaction representation, resulting in H ' I = exp i H q 1 + H q 2 t / ℏ H I exp - i H q 1 + H q 2 t / ℏ . We obtain...Parametric coupling for superconducting**

**qubit****qubits**...As an example, we now describe how we would generate a maximally entangled state with two flux

**qubits**biased at their flux-noise insensitive points, assuming Δ 1 = 5 G H z and Δ 2 = 7 G H z . We fix the bias current in the SQUID to I b = 2 s 0 I 0 and start with the ground state | 0 1 , 0 2 . We first apply a π pulse to

**qubit**1 thus preparing state | 1 1 , 0 2 . Then we apply a pulse at a

**frequency**Δ 2 - Δ 1 = 2 G H z in the SQUID bias current of amplitude δ s = 0.015 . This results in an effective coupling of strength δ g / 2 = d g / d s δ s / 2 = 50 M H z . A pulse of duration δ t = 5 n s suffices then to generate the state | 0 1 0 2 + | 1 1 1 2 / 2 . We stress that thanks to the large value of the derivative d g / d s , even a small modulation of the bias current of δ I b = 2 I 0 δ s = 15 n A is enough to ensure such rapid gate operation. We performed a calculation of the evolution of the whole density matrix under the complete interaction hamiltonian g t σ x 1 σ x 2 with the parameters just mentioned. We initialized the two

**qubits**in the | 1 1 , 0 2 state at t = 0 ; at t = 10 n s an entangling pulse g t = δ g cos 2 π Δ 1 - Δ 2 t and lasting 20 n s was simulated. The result is shown as a black curve in figure fig2. We plot the diagonal elements of the total density matrix. As expected, ρ 00 , 00 t = ρ 11 , 11 = 0 , and ρ 10 , 10 = 1 - ρ 01 , 01 = cos 2 π δ g / 2 t 2 . We did another calculation for the same

**qubit**parameters but assuming a fixed coupling g 0 = 200 M H z . Following the analysis presented above, we initialized the system in the dressed state | 10 ' and simulated the application of a microwave pulse g t = δ g cos ω M W t at a

**frequency**ω M W = 2.04 G H z taking into account the energy shift of the dressed states. The evolution of the density matrix elements (grey curve in figure fig2) shows that despite the finite value of g 0 , the two

**qubits**become maximally entangled as previously. The evolution is not simply sinusoidal because we plot the density matrix coefficients in the uncoupled state basis. Note also the slightly slower evolution compared to the g 0 = 0 case, consistent with our analysis. This shows that the scheme should actually work for a wide range of experimental parameters....It is straightforward to extend the scheme discussed above to the case of a

**qubit**coupled to a harmonic

**oscillator**of widely different

**frequency**. As an example we consider the circuit studied in which is shown in figure fig3a. A flux

**qubit**is coupled to the plasma mode of its DC SQUID shunted by an on-chip capacitor C s h (resonance

**frequency**ν p ) via the SQUID circulating current J . As discussed in , the coupling between the two systems can be written H I = g 1 I b a + a + g 2 I b a + a 2 σ x . We evaluated g 1 I b for the following parameters : Φ S = 0.45 Φ 0 , I 0 = 1 μ A ,

**qubit**-SQUID mutual inductance M = 10 p H ,

**qubit**persistent current I p = 240 n A , Δ = 5.5 G H z , ν p = 9 G H z as shown in figure fig3b. At I b = I b * = 0 , the coupling constant g 1 vanishes. It has been shown in that when biased at I b = I b * and at its flux-insensitive point, the flux-

**qubit**could reach remarkably long spin-echo times (up to 4 μ s ). On the other hand, the derivative of g 1 is shown in figure fig3b to be nearly constant with a value d g 1 / d I b ≃ - 4 G H z / μ A . Therefore, inducing a modulation of the SQUID bias current δ i cos 2 π ν p - Δ t with amplitude δ i = 50 n A would be enough to reach an effective coupling constant of 100 M H z . The state of the

**qubit**and of the

**oscillator**are thus swapped in 5 n s for reasonable circuit parameters. This process is very similar to the sideband resonances which have been predicted and observed . However, in order to use these sideband resonances for quantum information processing, the quality factor of the harmonic

**oscillator**must be as large as possible, contrary to the experiments described in where Q ≃ 100 . This can be achieved by superconducting distributed resonators for which quality factors in the 10 6 range have been observed . Employing this harmonic

**oscillator**as a bus allows the extension of the scheme to an arbitrary number of

**qubits**, each of them coupled to the bus via a SQUID-based parametric coupling scheme....(Color online) Flux

**qubit**parametrically coupled to an LC

**oscillator**via a DC SQUID. a) Electrical scheme : the

**qubit**(blue loop) is inductively coupled to a DC SQUID shunted by a capacitor and thus forming a LC

**oscillator**. b) Dependence of the coupling constant g 1 as a function of the bias current I b . At the current I b * the coupling constant vanishes. c) Derivative d g 1 / d I b as a function of I b . It stays nearly at a constant value on the current range considered. fig3...We propose a scheme to couple two superconducting charge or flux

**qubits**biased at their symmetry points with unequal energy splittings. Modulating the coupling constant between two

**qubits**at the sum or difference of their two

**frequencies**allows to bring them into resonance in the rotating frame. Switching on and off the modulation amounts to switching on and off the coupling which can be realized at nanosecond speed. We discuss various physical implementations of this idea, and find that our scheme can lead to rapid operation of a two-

**qubit**gate....As an example, we now describe how we would generate a maximally entangled state with two flux

**qubits**biased at their flux-noise insensitive points, assuming Δ 1 = 5 G H z and Δ 2 = 7 G H z . We fix the bias current in the SQUID to I b = 2 s 0 I 0 and start with the ground state | 0 1 , 0 2 . We first apply a π pulse to

**1 thus preparing state | 1 1 , 0 2 . Then we apply a pulse at a**

**qubit****frequency**Δ 2 - Δ 1 = 2 G H z in the SQUID bias current of amplitude δ s = 0.015 . This results in an effective coupling of strength δ g / 2 = d g / d s δ s / 2 = 50 M H z . A pulse of duration δ t = 5 n s suffices then to generate the state | 0 1 0 2 + | 1 1 1 2 / 2 . We stress that thanks to the large value of the derivative d g / d s , even a small modulation of the bias current of δ I b = 2 I 0 δ s = 15 n A is enough to ensure such rapid gate operation. We performed a calculation of the evolution of the whole density matrix under the complete interaction hamiltonian g t σ x 1 σ x 2 with the parameters just mentioned. We initialized the two

**qubits**in the | 1 1 , 0 2 state at t = 0 ; at t = 10 n s an entangling pulse g t = δ g cos 2 π Δ 1 - Δ 2 t and lasting 20 n s was simulated. The result is shown as a black curve in figure fig2. We plot the diagonal elements of the total density matrix. As expected, ρ 00 , 00 t = ρ 11 , 11 = 0 , and ρ 10 , 10 = 1 - ρ 01 , 01 = cos 2 π δ g / 2 t 2 . We did another calculation for the same

**parameters but assuming a fixed coupling g 0 = 200 M H z . Following the analysis presented above, we initialized the system in the dressed state | 10 ' and simulated the application of a microwave pulse g t = δ g cos ω M W t at a**

**qubit****frequency**ω M W = 2.04 G H z taking into account the energy shift of the dressed states. The evolution of the density matrix elements (grey curve in figure fig2) shows that despite the finite value of g 0 , the two

**qubits**become maximally entangled as previously. The evolution is not simply sinusoidal because we plot the density matrix coefficients in the uncoupled state basis. Note also the slightly slower evolution compared to the g 0 = 0 case, consistent with our analysis. This shows that the scheme should actually work for a wide range of experimental parameters....It is straightforward to extend the scheme discussed above to the case of a

**coupled to a harmonic oscillator of widely different**

**qubit****frequency**. As an example we consider the circuit studied in which is shown in figure fig3a. A flux

**is coupled to the plasma mode of its DC SQUID shunted by an on-chip capacitor C s h (resonance**

**qubit****frequency**ν p ) via the SQUID circulating current J . As discussed in , the coupling between the two systems can be written H I = g 1 I b a + a + g 2 I b a + a 2 σ x . We evaluated g 1 I b for the following parameters : Φ S = 0.45 Φ 0 , I 0 = 1 μ A ,

**qubit**-SQUID mutual inductance M = 10 p H ,

**qubit**persistent current I p = 240 n A , Δ = 5.5 G H z , ν p = 9 G H z as shown in figure fig3b. At I b = I b * = 0 , the coupling constant g 1 vanishes. It has been shown in that when biased at I b = I b * and at its flux-insensitive point, the flux-

**qubit**could reach remarkably long spin-echo times (up to 4 μ s ). On the other hand, the derivative of g 1 is shown in figure fig3b to be nearly constant with a value d g 1 / d I b ≃ - 4 G H z / μ A . Therefore, inducing a modulation of the SQUID bias current δ i cos 2 π ν p - Δ t with amplitude δ i = 50 n A would be enough to reach an effective coupling constant of 100 M H z . The state of the

**and of the oscillator are thus swapped in 5 n s for reasonable circuit parameters. This process is very similar to the sideband resonances which have been predicted and observed . However, in order to use these sideband resonances for quantum information processing, the quality factor of the harmonic oscillator must be as large as possible, contrary to the experiments described in where Q ≃ 100 . This can be achieved by superconducting distributed resonators for which quality factors in the 10 6 range have been observed . Employing this harmonic oscillator as a bus allows the extension of the scheme to an arbitrary number of**

**qubit****qubits**, each of them coupled to the bus via a SQUID-based parametric coupling scheme....We will now discuss the physical implementation of the above ideas. Simple circuits based on Josephson junctions, and thus on the same technology as the

**qubits**themselves, allow to modulate the coupling constant at G H z

**frequency**. To be more specific in our discussion, we will focus in particular on the scheme discussed in , and show that the very circuit analyzed by the authors (shown in figure fig1a) can be used to implement our parametric coupling scheme. Two flux-

**qubits**of persistent currents I q , i and energy gaps Δ i ( i = 1 , 2 ) are inductively coupled by a mutual inductance M q q . They are also inductively coupled to a DC-SQUID with a mutual inductance M q s . The SQUID loop (of inductance L ) is threaded by a flux Φ S , and bears a circulating current J . The critical current of its junctions is denoted I 0 . Writing the hamiltonian in the

**qubit**energy eigenstates at the flux-insensitive point, equation (2) in now writes H = - h / 2 Δ 1 σ z 1 + Δ 2 σ z 2 + h g σ x 1 σ x 2 , where g = M q q | I q 1 I q 2 | + M q s 2 | I q 1 I q 2 | R e ∂ J / ∂ Φ s I b / h . In figure fig1b we plot the coupling constant g as a function of the dimensionless parameter s = I b / 2 I 0 for the same parameters as in : I 0 = 0.48 μ A , L = 200 p H , I q 1 = I q 2 = 0.46 μ A , M q q = 0.25 p H , M q s = 33 p H , Φ s = 0.45 Φ 0 . We see that g strongly depends on s . In particular g s 0 = 0 for a specific value s 0 . On the other hand the derivative d g / d s is finite (for instance, d g / d s s 0 = 7 G H z ) as can be shown in the inset of figure fig1b. Biasing the system at s 0 protects it against 1 / f flux-noise in the SQUID loop and noise in the bias current. At GHz

**frequencies**, the noise power spectrum of s is ohmic due to the bias current line dissipative impedance, and has a resonance due to the plasma

**frequency**of the SQUID junctions. This resonance is in the 40 G H z range for typical parameters and should not affect the coupled system dynamics....While in the scheme proposed by Rigeti et al. quantum gates are realized with a fixed coupling constant g , our scheme relies on the possibility to modulate g by varying a control parameter λ . This gives us the possibility of realizing two-

**qubit**operations with arbitrary fixed

**frequencies, which is particularly attractive for flux-**

**qubit****qubits**. We first assume that we dispose of a “black box" circuit realizing this task, as shown in figure fig0b, actual implementation will be discussed later. Our parametric coupling scheme consists in modulating λ at a

**frequency**ω / 2 π close to Δ 1 - Δ 2 or Δ 1 + Δ 2 . Supposing λ t = λ 0 + δ λ cos ω t leads to g t = g 0 + δ g c o s ω t , with g 0 = g λ 0 and δ g = d g / d λ δ λ . Then, if ω is close to the difference in

**frequencies ω = ω 1 - ω 2 + δ 12 while | δ 12 | < < | ω 1 - ω 2 | , a few terms in the hamiltonian eq1:ham will rotate slowly. Keeping only these terms, we obtain...We will now discuss the physical implementation of the above ideas. Simple circuits based on Josephson junctions, and thus on the same technology as the**

**qubit****qubits**themselves, allow to modulate the coupling constant at G H z

**frequency**. To be more specific in our discussion, we will focus in particular on the scheme discussed in , and show that the very circuit analyzed by the authors (shown in figure fig1a) can be used to implement our parametric coupling scheme. Two flux-

**qubits**of persistent currents I q , i and energy gaps Δ i ( i = 1 , 2 ) are inductively coupled by a mutual inductance M q q . They are also inductively coupled to a DC-SQUID with a mutual inductance M q s . The SQUID loop (of inductance L ) is threaded by a flux Φ S , and bears a circulating current J . The critical current of its junctions is denoted I 0 . Writing the hamiltonian in the

**energy eigenstates at the flux-insensitive point, equation (2) in now writes H = - h / 2 Δ 1 σ z 1 + Δ 2 σ z 2 + h g σ x 1 σ x 2 , where g = M q q | I q 1 I q 2 | + M q s 2 | I q 1 I q 2 | R e ∂ J / ∂ Φ s I b / h . In figure fig1b we plot the coupling constant g as a function of the dimensionless parameter s = I b / 2 I 0 for the same parameters as in : I 0 = 0.48 μ A , L = 200 p H , I q 1 = I q 2 = 0.46 μ A , M q q = 0.25 p H , M q s = 33 p H , Φ s = 0.45 Φ 0 . We see that g strongly depends on s . In particular g s 0 = 0 for a specific value s 0 . On the other hand the derivative d g / d s is finite (for instance, d g / d s s 0 = 7 G H z ) as can be shown in the inset of figure fig1b. Biasing the system at s 0 protects it against 1 / f flux-noise in the SQUID loop and noise in the bias current. At GHz frequencies, the noise power spectrum of s is ohmic due to the bias current line dissipative impedance, and has a resonance due to the plasma**

**qubit****frequency**of the SQUID junctions. This resonance is in the 40 G H z range for typical parameters and should not affect the coupled system dynamics....While in the scheme proposed by Rigeti et al. quantum gates are realized with a fixed coupling constant g , our scheme relies on the possibility to modulate g by varying a control parameter λ . This gives us the possibility of realizing two-

**qubit**operations with arbitrary fixed

**qubit**

**frequencies**, which is particularly attractive for flux-

**qubits**. We first assume that we dispose of a “black box" circuit realizing this task, as shown in figure fig0b, actual implementation will be discussed later. Our parametric coupling scheme consists in modulating λ at a

**frequency**ω / 2 π close to Δ 1 - Δ 2 or Δ 1 + Δ 2 . Supposing λ t = λ 0 + δ λ cos ω t leads to g t = g 0 + δ g c o s ω t , with g 0 = g λ 0 and δ g = d g / d λ δ λ . Then, if ω is close to the difference in

**qubit**

**frequencies**ω = ω 1 - ω 2 + δ 12 while | δ 12 | < < | ω 1 - ω 2 | , a few terms in the hamiltonian eq1:ham will rotate slowly. Keeping only these terms, we obtain ... We propose a scheme to couple two superconducting charge or flux

**qubits**biased at their symmetry points with unequal energy splittings. Modulating the coupling constant between two

**qubits**at the sum or difference of their two

**frequencies**allows to bring them into resonance in the rotating frame. Switching on and off the modulation amounts to switching on and off the coupling which can be realized at nanosecond speed. We discuss various physical implementations of this idea, and find that our scheme can lead to rapid operation of a two-

**qubit**gate.

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Contributors: Reuther, Georg M., Zueco, David, Hänggi, Peter, Kohler, Sigmund

Date: 2008-06-17

We propose a scheme for monitoring coherent quantum dynamics with good time-resolution and low backaction, which relies on the response of the considered quantum system to high-**frequency** ac driving. An approximate analytical solution of the corresponding quantum master equation reveals that the phase of an outgoing signal, which can directly be measured in an experiment with lock-in technique, is proportional to the expectation value of a particular system observable. This result is corroborated by the numerical solution of the master equation for a charge **qubit** realized with a Cooper-pair box, where we focus on monitoring coherent **oscillations**....(color online) (a) Fidelity defect δ F = 1 - F and (b) time-averaged trace distance between **the** driven and **the** undriven density operator of **the** CPB for various driving amplitudes as a function of **the** driving **frequency**. All other parameters are **the** same as in Fig. fig:oscillation....eq:7, to reflect **the** unperturbed time evolution of σ x 0 with respect to **the** **qubit**. Although **the** condition of high-**frequency** probing, Ω ≫ ω q b , is not strictly fulfilled and despite **the** presence of higher charge states, **the** lock-in amplified phase φ o u t t and **the** predicted phase φ 0 h f t are barely distinguishable for an appropriate choice of parameters as is shown in Fig. fig:oscillation(a)....eq:7 allows one to retrieve information about the coherent **qubit** dynamics in an experiment. Figure fig:oscillation(a) shows the time evolution of the expectation value Q ̇ t for the initial state | ↑ ≡ | 1 , obtained via numerical integration of the master equation ...Time-Resolved Measurement of a Charge **Qubit**...CPB in the presence of the ac driving which in principle may excite higher states. The driving, due to its rather small amplitude, is barely noticeable on the scale chosen for the main figure, but only on a refined scale for long times; see inset of Fig. fig:oscillation(a). This already insinuates that the backaction on the dynamics is weak. In the corresponding power spectrum of Q ̇ depicted in Fig. fig:oscillation(b), the driving is nevertheless reflected in sideband peaks at the **frequencies** Ω and Ω ± ω q b . In the time domain these peaks correspond to a signal cos Ω t - φ o u t t . Moreover, non-**qubit** CPB states leads to additional peaks at higher **frequencies**, while their influence at **frequencies** ω Ω is minor. Experimentally, the phase φ o u t t can be retrieved by lock-in amplification of the output signal, which we mimic numerically in the following way : We only consider the spectrum of ξ o u t in a window Ω ± Δ Ω around the driving **frequency** and shift it by - Ω . The inverse Fourier transformation to the time domain provides φ o u t t which is expected to agree with φ h f 0 t and, according to Eq. ...In order to quantify this agreement, we introduce the measurement fidelity F = φ o u t σ x 0 , where f g = ∫ d t f g / ∫ d t f 2 ∫ d t g 2 1 / 2 with time integration over the decay duration. Thus, the ideal value F = 1 is assumed if φ o u t t and σ x t 0 are proportional to each other, i.e. if the agreement between the measured phase and the unperturbed expectation value σ x 0 is perfect. Figure fig:fidelity(a) depicts the fidelity as a function of the driving **frequency**. As expected, whenever non-**qubit** CPB states are excited resonantly, we find F ≪ 1 , indicating a significant population of these states. Far-off such resonances, the fidelity increases with the driving **frequency** Ω . A proper **frequency** lies in the middle between the **qubit** doublet and the next higher state. In the present case, Ω ≈ 15 E J / ℏ appears as a good choice. Concerning the driving amplitude, one has to find a compromise, because as A increases, so does the phase contrast of the outgoing signal...CPB in **the** presence of **the** ac driving which in principle may excite higher states. The driving, due to its rather small amplitude, is barely noticeable on **the** scale chosen for **the** main figure, but only on a refined scale for long times; see inset of Fig. fig:oscillation(a). This already insinuates that **the** backaction on **the** dynamics is weak. In **the** corresponding power spectrum of Q ̇ depicted in Fig. fig:oscillation(b), **the** driving is nevertheless reflected in sideband peaks **at** **the** frequencies Ω and Ω ± ω q b . In **the** time domain these peaks correspond to a signal cos Ω t - φ o u t t . **Moreover**, non-**qubit** CPB states leads to additional peaks **at** higher frequencies, while their **influence** **at** frequencies ω Ω is minor. Experimentally, **the** phase φ o u t t can be retrieved by lock-in amplification of **the** output signal, which we mimic numerically in **the** following way : We only consider **the** spectrum of ξ o u t in a window Ω ± Δ Ω around **the** driving **frequency** and shift it by - Ω . The inverse Fourier transformation to **the** time domain provides φ o u t t which is expected to agree with φ h f 0 t and, according to Eq. ...Although later on we focus on **the** dynamics of a superconducting charge **qubit** as sketched in Fig. fig:setup, our measurement scheme is rather generic and can be applied to any open quantum system. We employ **the** system-bath Hamiltonian...(color online) Decaying **qubit** **oscillations** with initial state | ↑ in a weakly probed CPB with 6 states for α = Z 0 e 2 / ℏ = 0.08 , A = 0.1 E J / e , E C = 5.25 E J and N g = 0.45 , so that E e l = 2.1 E J and ω q b = 2.3 E J / ℏ . (a) Time evolution of the measured difference signal Q ̇ ∝ ξ o u t - ξ i n (in units of 2 e E J / ℏ ) of the full CPB and its lock-in amplified phase φ o u t (**frequency** window Δ Ω = 5 E J / ℏ ), compared to the estimated phase φ h f 0 ∝ σ x 0 in the **qubit** approximation. The inset resolves the underlying small rapid **oscillations** with **frequency** Ω = 15 E J / ℏ in the long-time limit. (b) Power spectrum of Q ̇ for the full CPB Hamiltonian (solid) and for the two-level approximation (dashed)....(color online) Decaying qubit oscillations with initial state | ↑ in a weakly probed CPB with 6 states for α = Z 0 e 2 / ℏ = 0.08 , A = 0.1 E J / e , E C = 5.25 E J and N g = 0.45 , so that E e l = 2.1 E J and ω q b = 2.3 E J / ℏ . (a) Time evolution of the measured difference signal Q ̇ ∝ ξ o u t - ξ i n (in units of 2 e E J / ℏ ) of the full CPB and its lock-in amplified phase φ o u t (**frequency** window Δ Ω = 5 E J / ℏ ), compared to the estimated phase φ h f 0 ∝ σ x 0 in the qubit approximation. The inset resolves the underlying small rapid oscillations with **frequency** Ω = 15 E J / ℏ in the long-time limit. (b) Power spectrum of Q ̇ for the full CPB Hamiltonian (solid) and for the two-level approximation (dashed)....Although later on we focus on the dynamics of a superconducting charge **qubit** as sketched in Fig. fig:setup, our measurement scheme is rather generic and can be applied to any open quantum system. We employ the system-bath Hamiltonian...In order to quantify this agreement, we introduce **the** measurement fidelity F = φ o u t σ x 0 , where f g = ∫ d t f g / ∫ d t f 2 ∫ d t g 2 1 / 2 with time integration over **the** decay duration. Thus, **the** ideal value F = 1 is assumed if φ o u t t and σ x t 0 are proportional to each other, i.e. if **the** agreement between **the** measured phase and **the** unperturbed expectation value σ x 0 is perfect. Figure fig:fidelity(a) depicts **the** fidelity as a function of **the** driving **frequency**. As expected, whenever non-**qubit** CPB states are excited resonantly, we find F ≪ 1 , indicating a significant population of these states. Far-off such resonances, **the** fidelity increases with **the** driving **frequency** Ω . A proper **frequency** lies in **the** middle between **the** **qubit** doublet and **the** next higher state. In **the** present case, Ω ≈ 15 E J / ℏ appears as a good choice. Concerning **the** driving amplitude, one has to find a compromise, because as A increases, so does **the** phase contrast of **the** outgoing signal...(color online) (a) Fidelity defect δ F = 1 - F and (b) time-averaged trace distance between the driven and the undriven density operator of the CPB for various driving amplitudes as a function of the driving **frequency**. All other parameters are the same as in Fig. fig:oscillation....We propose a scheme for monitoring coherent quantum dynamics with good time-resolution and low backaction, which relies on the response of the considered quantum system to high-**frequency** ac driving. An approximate analytical solution of the corresponding quantum master equation reveals that the phase of an outgoing signal, which can directly be measured in an experiment with lock-in technique, is proportional to the expectation value of a particular system observable. This result is corroborated by the numerical solution of the master equation for a charge **qubit** realized with a Cooper-pair box, where we focus on monitoring coherent oscillations....eq:7, to reflect the unperturbed time evolution of σ x 0 with respect to the **qubit**. Although the condition of high-**frequency** probing, Ω ≫ ω q b , is not strictly fulfilled and despite the presence of higher charge states, the lock-in amplified phase φ o u t t and the predicted phase φ 0 h f t are barely distinguishable for an appropriate choice of parameters as is shown in Fig. fig:oscillation(a). ... We propose a scheme for monitoring coherent quantum dynamics with good time-resolution and low backaction, which relies on the response of the considered quantum system to high-**frequency** ac driving. An approximate analytical solution of the corresponding quantum master equation reveals that the phase of an outgoing signal, which can directly be measured in an experiment with lock-in technique, is proportional to the expectation value of a particular system observable. This result is corroborated by the numerical solution of the master equation for a charge **qubit** realized with a Cooper-pair box, where we focus on monitoring coherent **oscillations**.

Files:

Contributors: Rao, D. D. Bhaktavatsala

Date: 2007-06-19

(Color online) Concurrence with time. The time-dependent concurrence, for the Bell-states | B 1 = 1 2 | ↑ ↓ + | ↓ ↑ | B 2 = 1 2 | ↑ ↑ + | ↓ ↓ , | B 3 = 1 2 | ↑ ↑ - | ↓ ↓ , is plotted for two values of Δ ω = ω - ω 0 . The loss of entanglement is slowest at the resonant **frequency** δ ω = 0 , for all the states, and it increases as the r.f **frequency** shifts away from ω 0 . The bath is unpolarized with uniform coupling strengths between the **qubits** and bath-spins. In plotting the above we have taken N = 20 , ω = 100 , ω 1 = 10 and g / ℏ = 1 . The **frequencies** are in the units of MHz....(Color online) Concurrence with time. The time-dependent concurrence for an initial singlet shared between two non-interacting **qubits** is plotted for three different cases, (i) when the r.f **frequencies** are tuned to their resonance values i.e., δ ω 1 2 = 0 , (ii) the r.f **frequencies** are slightly away from resonance and (iii) much away from their resonant **frequencies**. In the above δ ω 1 2 = ω 0 1 2 - ω 1 2 , where ω 0 1 2 , are the fields at the local sites of the **qubits**. The initial states of the baths are unpolarized each consiting of N = 20 spins and the local fields experienced by the **qubits** are ω 0 1 = 100 , ω 0 1 = 110 respectively....(Color online) Qubit polarization with time. The initial bath state is unpolarized consisting of N = 20 spins. Two different distributions for the** qubit**-bath couplings are considered, where ∑ k g k = g for both the distributions. In plotting the above we have taken ω 0 = 10 3 , ω 1 = 10 and g / ℏ = 40 , which are in the units of mHz. The decoherence of one-qubit states ( 1 - P S t 2 ), can be inferred directly from the above figure....(Color online) Probability distribution of the resonant **frequencies** given in Eq. resf. The bath consists of N = 20 spins. The distributions for the **qubit**-bath couplings are normalized such that ∑ k g k = g , where g / ℏ = 20 MHz. In plotting the above we have set ω = ω 0 ....(Color online) The z -component of **qubit** polarization with time. The plot shows the decay of Rabi **oscillations** with time. The bath is unpolarized consisting of N = 20 spins. Two different distributions for the **qubit**-bath couplings are considered, where ∑ k g k = g for both the distributions. In plotting the above we have taken ω 0 = 10 3 , ω 1 = 10 and g / ℏ = 40 , which are in the units of MHz...(Color online) Probability distribution of the resonant frequencies given in Eq. resf. The bath consists of N = 20 spins. The distributions for the** qubit**-bath couplings are normalized such that ∑ k g k = g , where g / ℏ = 20 MHz. In plotting the above we have set ω = ω 0 ....(Color online) Concurrence with time. The time-dependent concurrence, for the Bell-states | B 1 = 1 2 | ↑ ↓ + | ↓ ↑ | B 2 = 1 2 | ↑ ↑ + | ↓ ↓ , | B 3 = 1 2 | ↑ ↑ - | ↓ ↓ , is plotted for two values of Δ ω = ω - ω 0 . The loss of entanglement is slowest at the resonant frequency δ ω = 0 , for all the states, and it increases as the r.f frequency shifts away from ω 0 . The bath is unpolarized with uniform coupling strengths between the qubits and bath-spins. In plotting the above we have taken N = 20 , ω = 100 , ω 1 = 10 and g / ℏ = 1 . The frequencies are in the units of MHz....(Color online) The transition probability as a function of r.f **frequency** ω . Initial bath states with different polarizations are considered. The resonant **frequency** varies with the bath polarization as ω = ω 0 + g N P B / 2 . In plotting the above we have taken N = 20 , ω 0 = 100 , ω 1 = 10 and g / ℏ = 1 . The **frequencies** are in the units of MHz....Controlled dynamics of **qubits** in the presence of decoherence...(Color online) The transition probability as a function of r.f frequency ω . Initial bath states with different polarizations are considered. The resonant frequency varies with the bath polarization as ω = ω 0 + g N P B / 2 . In plotting the above we have taken N = 20 , ω 0 = 100 , ω 1 = 10 and g / ℏ = 1 . The frequencies are in the units of MHz....An exactly solvable model for the decoherence of one and two-**qubit** states interacting with a spin-bath, in the presence of a time-dependent magnetic field is studied. The magnetic field is static along $\hat{z}$ direction and oscillatory in the transverse plane. The transition probability and Rabi oscillations between the spin-states of a single **qubit** is shown to depend on the size of bath, the distribution of **qubit**-bath couplings and the initial bath polarization. In contrast to the fast Gaussian decay for short times, the polarization of the **qubit** shows an oscillatory power-law decay for long times. The loss of entanglement for the maximally entangled two-**qubit** states, can be controlled by tuning the **frequency** of the rotating field. The decay rates of entanglement and purity for all the Bell-states are same when the **qubits** are non-interacting, and different when they are interacting....(Color online) Concurrence with time. The time-dependent concurrence for an initial singlet shared between two non-interacting qubits is plotted for three different cases, (i) when the r.f frequencies are tuned to their resonance values i.e., δ ω 1 2 = 0 , (ii) the r.f frequencies are slightly away from resonance and (iii) much away from their resonant frequencies. In the above δ ω 1 2 = ω 0 1 2 - ω 1 2 , where ω 0 1 2 , are the fields at the local sites of the qubits. The initial states of the baths are unpolarized each consiting of N = 20 spins and the local fields experienced by the qubits are ω 0 1 = 100 , ω 0 1 = 110 respectively....An exactly solvable model for the decoherence of one and two-**qubit** states interacting with a spin-bath, in the presence of a time-dependent magnetic field is studied. The magnetic field is static along $\hat{z}$ direction and oscillatory in the transverse plane. The transition probability and Rabi **oscillations** between the spin-states of a single **qubit** is shown to depend on the size of bath, the distribution of **qubit**-bath couplings and the initial bath polarization. In contrast to the fast Gaussian decay for short times, the polarization of the **qubit** shows an oscillatory power-law decay for long times. The loss of entanglement for the maximally entangled two-**qubit** states, can be controlled by tuning the **frequency** of the rotating field. The decay rates of entanglement and purity for all the Bell-states are same when the **qubits** are non-interacting, and different when they are interacting. ... An exactly solvable model for the decoherence of one and two-**qubit** states interacting with a spin-bath, in the presence of a time-dependent magnetic field is studied. The magnetic field is static along $\hat{z}$ direction and oscillatory in the transverse plane. The transition probability and Rabi **oscillations** between the spin-states of a single **qubit** is shown to depend on the size of bath, the distribution of **qubit**-bath couplings and the initial bath polarization. In contrast to the fast Gaussian decay for short times, the polarization of the **qubit** shows an oscillatory power-law decay for long times. The loss of entanglement for the maximally entangled two-**qubit** states, can be controlled by tuning the **frequency** of the rotating field. The decay rates of entanglement and purity for all the Bell-states are same when the **qubits** are non-interacting, and different when they are interacting.

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Contributors: Alexander N. Korotkov

Date: 2005-03-01

Quantum feedback of a double-dot **qubit**...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. ... 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: Saito, Keiji, Wubs, Martijn, Kohler, Sigmund, Kayanuma, Yosuke, Hanggi, Peter

Date: 2007-03-22

The** **diabatic** **energies** **cross, but** **the** **adiabatic** **energies ± 1** **2** **v** **2** **t** **2 + Δ** **2** **for** **Δ ≠ 0** **form** **an** **avoided** **crossing, as** **sketched** **in** **Fig. fig:energies. The** **adiabatic** **theorem** **states** **that** **the** **splitting** **Δ** **prevents** **transfer** **of** **population** **between** **the** **adiabatic** **eigenstates** **in** **the** **adiabatic** **limit** **ℏ** **v ≪ Δ** **2 , in** **other** **words** **if** **the** **sweep** **occurs** **slowly** **enough. A** ****qubit**** **prepared** **at** **t = - ∞ in** **the** **initial** **ground** **state | ↑ will** **then** **end** **up** **in** **the** **final** **ground** **state | ↓ . Beyond** **the** **adiabatic** **regime, the** **dynamics** **can** **be** **rather** **complex. Nevertheless, the** **population** **of** **the** **diabatic** **states** **at** **t = ∞ can** **be** **calculated** **exactly** **and** **is** **determined** **by** **the** **Landau-Zener** **transition** **probability...(Color online) Sketch of the diabatic energy levels as a function of time for a **qubit** coupled to a single harmonic **oscillator**. Energies of the states in the “up cluster” increase. These states correspond to the **qubit** state | ↑ . Energies decrease in the “down cluster”, where the **qubit** state is | ↓ . According to the “no-go-up theorem” app:nogoup, the initial state | ↑ , 0 + evolves to a superposition in which | ↑ , 0 + is the only “up” state. Energies within a band are separated by the **oscillator** energy ℏ Ω . For a **qubit** coupled to an **oscillator** bath, the corresponding crossing clusters would be continuous bands of states....(Color online) Landau-Zener dynamics for a **qubit** with Δ = 0 , in all three cases shown off-diagonally coupled via σ x to three **oscillators**. The various **oscillator** **frequencies** Ω j are given in units of v / ℏ . All coupling strengths have the same value γ j = ℏ v / 3 . The dotted line marks the analytical final transition probability corresponding to Eq. transverse result....By making a controlled point defect or line defect defect in the vicinity of the atom that breaks the periodicity of the photonic crystal, a narrow defect mode may be created within the spectral gap, as sketched in Fig. fig:Jphotoniccrystal. Ideally, this would allow cavity QED experiments to be performed within a photonic crystal, and progress is made in this direction. We propose to do LZ sweeps of the atomic **frequency** ω A t around the defect **frequency** but within the band gap. This will allow the creation of atom-defect entanglement and of single photons in the defect mode, in quite the same way as in circuit QED....LZ** **sweeps** **pas** **resonances** **of** **nonlinear **oscillators** are** **of** **practical** **interest, since** **nonlinear **oscillators** are** **currently** **used** **for** **the** **readout** **of** **flux** **qubits. For** **a** **numerical** **test** **of** **the** **predicted** **final** **transition** **probability, we** **take** **the** **situation** **in** **which** **the** ****qubit**** **is** **diagonally** **coupled** **to** **two** **of** **these** **nonlinear **oscillators**. Figure fig3** **shows** **the** **corresponding** **time-evolution** **of** **the** **probability** **P ↑ → ↓ t** **for** **the** ****qubit**** **to** **be** **in** **the “down” state. It** **also** **shows** **the** **dynamics** **in** **case** **the** ****qubit**** **couples** **to** **two** **linear **oscillators** with** **the** **same** **parameters, except** **that** **now** **β** **1 , 2 = 0 . Furthermore, the** **effect** **of** **a** **diagonally** **coupled** **spin** **bath, a** **special** **case** **of** **Eq. ...The diabatic energies cross, but the adiabatic energies ± 1 2 v 2 t 2 + Δ 2 for Δ ≠ 0 form an avoided crossing, as sketched in Fig. fig:energies. The adiabatic theorem states that the splitting Δ prevents transfer of population between the adiabatic eigenstates in the adiabatic limit ℏ v ≪ Δ 2 , in other words if the sweep occurs slowly enough. A **qubit** prepared at t = - ∞ in the initial ground state | ↑ will then end up in the final ground state | ↓ . Beyond the adiabatic regime, the dynamics can be rather complex. Nevertheless, the population of the diabatic states at t = ∞ can be calculated exactly and is determined by the Landau-Zener transition probability...General** **coupling. When** **the **oscillators** neither** **couple** **purely** **off-diagonally ( ϑ = π / 2 ) nor** **purely** **diagonally ( ϑ = 0 ), the** **Landau-Zener** **probability** **generally** **exhibits** **a** **non-monotonic** **dependence** **on** **the** **tunnel** **coupling** **Δ . This** **is** **shown** **in** **Figure fig2** **for** **various** **angles** **ϕ** **and** **ϑ . Most** **interesting** **is** **the** **comparison** **to** **the** **non-dissipative** **case, which** **as** **we** **saw** **coincides** **with** **the** **result** **for** **diagonal** **coupling ( ϑ = 0 ): Any** **dissipative** **Landau-Zener** **probability** **lower** **than** **the** **curve** **for** **ϑ = 0** **is** **counterintuitive. Such** **situations** **occur, however: for** **several** **values** **of** **ϑ** **and** **for** **a** **sufficiently** **large** **tunnel** **splitting** **Δ , the** **bath** **coupling** **reduces** **P ↑ → ↓ ∞ , i.e. dissipation** **enhances** **the** **population** **of** **the** **final** **excited** ****qubit**** **state....Interestingly enough, the Landau-Zener tunneling probability is then fully determined by the integrated spectral density S . In particular, there is no dependence on the **oscillator** **frequencies** Ω j . This result is nicely illustrated in the simple example of Figure fig:three_osc, showing Landau-Zener dynamics of a **qubit** that is coupled to only three **oscillators**. The **oscillator** **frequencies** are varied, while the **qubit**-**oscillator** couplings are kept constant. The dynamics at intermediate times depends on the **oscillator** **frequencies**, but the final transition probability does not....Dissipative Landau-Zener transitions of a **qubit**: bath-specific and universal behavior...(Color online) Landau-Zener dynamics for a **qubit** with Δ = 0 , in all three cases shown off-diagonally coupled via σ x to three **oscillators**. The various oscillator frequencies Ω j are given in units of v / ℏ . All coupling strengths have the same value γ j = ℏ v / 3 . The dotted line marks the analytical final transition probability corresponding to Eq. transverse result....LZ sweeps pas resonances of nonlinear **oscillators** are of practical interest, since nonlinear **oscillators** are currently used for the readout of flux **qubits**. For a numerical test of the predicted final transition probability, we take the situation in which the **qubit** is diagonally coupled to two of these nonlinear **oscillators**. Figure fig3 shows the corresponding time-evolution of the probability P ↑ → ↓ t for the **qubit** to be in the “down” state. It also shows the dynamics in case the **qubit** couples to two linear **oscillators** with the same parameters, except that now β 1 , 2 = 0 . Furthermore, the effect of a diagonally coupled spin bath, a special case of Eq. ...Interestingly** **enough, the** **Landau-Zener** **tunneling** **probability** **is** **then** **fully** **determined** **by** **the** **integrated** **spectral** **density** **S . In** **particular, there** **is** **no** **dependence** **on** **the** ****oscillator**** ****frequencies**** **Ω** **j . This** **result** **is** **nicely** **illustrated** **in** **the** **simple** **example** **of** **Figure fig:three_osc, showing** **Landau-Zener** **dynamics** **of** **a** ****qubit**** **that** **is** **coupled** **to** **only** **three **oscillators**. The** ****oscillator**** ****frequencies**** **are** **varied, while** **the** ****qubit**-**oscillator**** **couplings** **are** **kept** **constant. The** **dynamics** **at** **intermediate** **times** **depends** **on** **the** ****oscillator**** ****frequencies**, but** **the** **final** **transition** **probability** **does** **not....(Color online) Sketch of spectral density for an atom near a local defect in a photonic crystal with a band gap. The quadratic free-space spectral density is modified by the crystal that creates a spectral gap around ω 0 . A narrow defect mode inside a broader band gap allows a controlled atom-defect interaction via LZ sweeps of the atomic transition **frequency** ω A t ....(Color online) Sketch of spectral density for an atom near a local defect in a photonic crystal with a band gap. The quadratic free-space spectral density is modified by the crystal that creates a spectral gap around ω 0 . A narrow defect mode inside a broader band gap allows a controlled atom-defect interaction via LZ sweeps of the atomic transition frequency ω A t ....By** **making** **a** **controlled** **point** **defect** **or** **line** **defect** **defect** **in** **the** **vicinity** **of** **the** **atom** **that** **breaks** **the** **periodicity** **of** **the** **photonic** **crystal, a** **narrow** **defect** **mode** **may** **be** **created** **within** **the** **spectral** **gap, as** **sketched** **in** **Fig. fig:Jphotoniccrystal. Ideally, this** **would** **allow** **cavity** **QED** **experiments** **to** **be** **performed** **within** **a** **photonic** **crystal, and** **progress** **is** **made** **in** **this** **direction. We** **propose** **to** **do** **LZ** **sweeps** **of** **the** **atomic** **frequency** **ω** **A** **t** **around** **the** **defect** **frequency** **but** **within** **the** **band** **gap. This** **will** **allow** **the** **creation** **of** **atom-defect** **entanglement** **and** **of** **single** **photons** **in** **the** **defect** **mode, in** **quite** **the** **same** **way** **as** **in** **circuit** **QED....(Color online) Time evolution of spin-flip probability for a **qubit** with Δ = 0.5 ℏ v diagonally coupled to two harmonic **oscillators**, two nonlinear **oscillators**, and seven spins, respectively. The harmonic **oscillators** are specified by Ω 1 = 0.1 v / ℏ , Ω 2 = 0.5 v / ℏ , γ 1 = 2 ℏ v , and γ 2 = 6 ℏ v , while the nonlinear **oscillators**, in addition, have β 1 = β 2 = 3 ℏ v . The values of the B j ν and the γ j z are randomly chosen from the range - ℏ v / 10 ℏ v / 10 . In all three cases, the transition probability converges to the universal value PLZuniversal....We study Landau-Zener transitions in a **qubit** coupled to a bath at zero temperature. A general formula is derived that is applicable to models with a non-degenerate ground state. We calculate exact transition probabilities for a **qubit** coupled to either a bosonic or a spin bath. The nature of the baths and the **qubit**-bath coupling is reflected in the transition probabilities. For diagonal coupling, when the bath causes energy fluctuations of the diabatic **qubit** states but no transitions between them, the transition probability coincides with the standard LZ probability of an isolated **qubit**. This result is universal as it does not depend on the specific type of bath. For pure off-diagonal coupling, by contrast, the tunneling probability is sensitive to the coupling strength. We discuss the relevance of our results for experiments on molecular nanomagnets, in circuit QED, and for the fast-pulse readout of superconducting phase **qubits**....General coupling. When the **oscillators** neither couple purely off-diagonally ( ϑ = π / 2 ) nor purely diagonally ( ϑ = 0 ), the Landau-Zener probability generally exhibits a non-monotonic dependence on the tunnel coupling Δ . This is shown in Figure fig2 for various angles ϕ and ϑ . Most interesting is the comparison to the non-dissipative case, which as we saw coincides with the result for diagonal coupling ( ϑ = 0 ): Any dissipative Landau-Zener probability lower than the curve for ϑ = 0 is counterintuitive. Such situations occur, however: for several values of ϑ and for a sufficiently large tunnel splitting Δ , the bath coupling reduces P ↑ → ↓ ∞ , i.e. dissipation enhances the population of the final excited **qubit** state....(Color online) Sketch of the diabatic energy levels as a function of time for a **qubit** coupled to a single harmonic oscillator. Energies of the states in the “up cluster” increase. These states correspond to the **qubit** state | ↑ . Energies decrease in the “down cluster”, where the **qubit** state is | ↓ . According to the “no-go-up theorem” app:nogoup, the initial state | ↑ , 0 + evolves to a superposition in which | ↑ , 0 + is the only “up” state. Energies within a band are separated by the oscillator energy ℏ Ω . For a **qubit** coupled to an oscillator bath, the corresponding crossing clusters would be continuous bands of states. ... We study Landau-Zener transitions in a **qubit** coupled to a bath at zero temperature. A general formula is derived that is applicable to models with a non-degenerate ground state. We calculate exact transition probabilities for a **qubit** coupled to either a bosonic or a spin bath. The nature of the baths and the **qubit**-bath coupling is reflected in the transition probabilities. For diagonal coupling, when the bath causes energy fluctuations of the diabatic **qubit** states but no transitions between them, the transition probability coincides with the standard LZ probability of an isolated **qubit**. This result is universal as it does not depend on the specific type of bath. For pure off-diagonal coupling, by contrast, the tunneling probability is sensitive to the coupling strength. We discuss the relevance of our results for experiments on molecular nanomagnets, in circuit QED, and for the fast-pulse readout of superconducting phase **qubits**.

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