Observing a Topological Transition in Weak-Measurement-Induced Geometric Phases
Yunzhao Wang, Kyrylo Snizhko, Alessandro Romito, Yuval Gefen, Kater Murch
OObserving a Topological Transition in Weak-Measurement-Induced GeometricPhases
Yunzhao Wang, Kyrylo Snizhko,
2, 3
Alessandro Romito, Yuval Gefen, and Kater Murch ∗ Department of Physics, Washington University, St. Louis, Missouri 63130 Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, 76100 Israel Institute for Quantum Materials and Technologies,Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom (Dated: February 12, 2021)Measurement plays a quintessential role in the control of quantum systems. Beyond initializationand readout which pertain to projective measurements, weak measurements in particular, throughtheir back-action on the system, may enable various levels of coherent control. The latter ranges fromobserving quantum trajectories to state dragging and steering. Furthermore, just like the adiabaticevolution of quantum states that is known to induce the Berry phase, sequential weak measurementsmay lead to path-dependent geometric phases. Here we measure the geometric phases induced bysequences of weak measurements and demonstrate a topological transition in the geometric phasecontrolled by measurement strength. This connection between weak measurement induced quan-tum dynamics and topological transitions reveals subtle topological features in measurement-basedmanipulation of quantum systems. Our protocol could be implemented for classes of operations(e.g. braiding) which are topological in nature. Furthermore, our results open new horizons formeasurement-enabled quantum control of many-body topological states.
The geometric phase is a part of the global phasegained by cyclic path of a quantum state, whichonly depends on the trajectory enclosed by the mo-tion in parameter space and not on the traversaltime [1, 2]. A frequently mentioned example is thePancharatnam-Berry phase that emerges from adi-abatic evolution of the system Hamiltonian [3, 4].It was suggested that the Pancharatnam phase canbe viewed in the framework of strong quantum mea-surement backaction [5]. The latter [6–11] is theinevitable disturbance brought by measurement ona certain quantum system. One example thereof isthe projection of a quantum state onto an eigen-state of a strongly-measured observable. More gen-erally, weak measurements only partially modify thequantum state. In either case, strong or weak, theaccumulated disturbance due to a sequence of mea-surements can result in closed-path motion of thequantum state. For a spin 1/2 system the resultinggeometric phase is half the solid angle subtendedby the path in parameter space [4, 12]. While suchmeasurement-induced geometric phases have beenobserved in optical systems [5, 13], a recent theo-retical study [14] has pushed this insight to a newqualitative perspective: the emergence of geometri-cal phases is accompanied by a topological transition[15, 16]. In this letter, we utilize a superconduct-ing transmon circuit to demonstrate and character-ize this measurement-induced topological transition. We consider a series of variable strength measure-ments on a pseudo-spin half system. For the spininitialized in a state | θ, φ = 0 (cid:105) , given by polar andazimuthal angles θ and φ of the Bloch sphere, a seriesof measurements along axes with fixed θ and with φ ranging from 0 to − π (Fig. 1a), has the potential todrag the state along the geodesic lines between theaxes of consecutive measurements. This trajectoryresults in a geometric phase χ related to the en-closed solid angle [17–19]. In the limit of continuousstrong measurements, χ = × π (1 − cos θ ) . How-ever, for weak measurements, the state lags behindthe advancing measurement axis and only partiallymoves along the geodesic line (Fig. 1a). With an ad-ditional, final projective measurement used to closethe path, the surface formed by the set of trajecto-ries along different latitudes either “wraps” (Fig. 1a)or does not “wrap” the Bloch sphere (Fig. 1b). Thistopological transition is controlled by the measure-ment strength and can be represented by a jump inthe Chern number [14], which is the equivalent ofthe winding number for 2 dimensional surfaces. InFigure 1c, we display the predicted geometric phase χ versus the polar angle θ for the extremal casesof strong measurement, zero-strength measurement,and near the topological transition.In order to probe the predicted topological transi-tion, we choose the first three energy levels of a su-perconducting Transmon circuit [20] embedded in a r X i v : . [ qu a n t - ph ] F e b a bc θ (radians) π /2 π /2 −π /2 π3π /2
2π π χ (r ad i an s ) ZX XYY ZX Y χ XY FIG. 1:
Measurement-induced topological tran-sition. (a) A sequence of measurements along a fixedlatitude drags the state on a trajectory displayed on theBloch sphere (arrows indicate the backaction of the mea-surements for the first two, and last (of 6) measurementsfor one latitude). When an additional, final projectivemeasurement closes the path (green arrow), the state ac-quires a geometric phase ( χ ). Considering all latitudes,these trajectories form a closed surface winding aroundthe Bloch sphere. (b) Weaker measurements result insmaller backaction on the state; as a result the trajecto-ries form a closed surface that does not wrap around theBloch sphere. (c) Dependence of the geometric phase onthe polar angle θ for the measurement sequences withmeasurement strengths slightly below (blue dashed line)and above (black dashed line) the critical value. Theblack solid line shows the case of infinitely strong mea-surements, the blue solid line represents zero measure-ment strength, and faint lines indicate intermediate mea-surement strengths. The insets illustrate the origin ofthe transition. For sufficiently strong measurements theequatorial trajectory circumnavigates the Bloch spherewhile for weak measurements it does not.
3D aluminium cavity [21] as our experiment plat-form (Fig. 2a). In the dispersive limit [22, 23],where the cavity frequency ω r is far detuned fromthe qutrit transition frequencies ω j , the Jaynes-Cummings Hamiltonian becomes: H m = (cid:126) ω r a † a + (cid:88) j (cid:126) ω j | j (cid:105)(cid:104) j | + (cid:88) j (cid:126) ξ j | j (cid:105)(cid:104) j | a † a, (1)where a † a is the cavity photon number operator, | j (cid:105) are the energy eigenstates of the Transmon with en-ergies (cid:126) ω j , and (cid:126) ξ j are the interaction energies be-tween the cavity eigenstates and Transmon energylevels | j (cid:105) , and we consider the lowest three energy γ ge γ gf γ ef γ ( μ s ) - Probe frequency (GHz) |g|f |e
Frequency (GHz) T r an s m i ss i on ( a r b . ) I Q
Qutrit Cavity
I Q |g |f|e
Time ( μ s) P ( e ) |g |f|e a bc ... FIG. 2:
Experiment setup. (a) A superconductingTransmon qutrit is dispersively coupled to a high qualityfactor microwave cavity. A coherent state probe acquiresa qutrit-state ( | g (cid:105) , | e (cid:105) , or | f (cid:105) ) dependent displacementon the I/Q plane. (b) The dispersive interaction re-sults in a state-dependent cavity transmission, allowingfor measurements that resolve one state while leaving theother two unresolved. (c) The measurement is quantifiedvia the frequency dependent dephasing rates of each two-level subspace of Transmon states. The green arrow indi-cates the operating probe frequency, which correspondsto a selective measurement of the | f (cid:105) state, which pre-serves the coherence in the {| g (cid:105) , | e (cid:105)} subspace. A typicalRamsey measurement is shown in the inset. levels, j ∈ { g, e, f } . The effect brought by suchinteraction energies (cid:126) ξ j can be viewed as a qutrit-state-dependent shift on the cavity frequency, en-abling quantum non-demolition weak measurementsof the circuit energy states.As is shown in Figure 2a, when the cavity is probedwith a coherent state, the output signal distributeson the quadrature space of the electromagnetic field( I, Q ) depending on the cavity transmission at themeasurement frequency. The setup is operatedin the strong dispersive regime, where the cavitylinewidth κ (cid:28) ξ j . Figure 2b illustrates that a weakprobe of the cavity near the | f (cid:105) resonance will betransmitted only if the circuit is in the | f (cid:105) state.Therefore, the measurement distinguishes the state | f (cid:105) from both | e (cid:105) and | g (cid:105) , but does not distinguish | e (cid:105) from | g (cid:105) . The selective nature of this measure-ment architecture allows us to reserve one energylevel (e.g. | g (cid:105) ) as a quantum phase reference in or-der to determine the global phase accumulated by astate in the {| e (cid:105) , | f (cid:105)} manifold.In the limit χ f (cid:29) κ the Kraus operators [9] associ-ated with a probe near the | f (cid:105) resonance are givenby M ( r ) z = (cid:114) π σ e − ( r − r ) / σ e − r / σ
00 0 e − r / σ (2)in the {| f (cid:105) , | e (cid:105) , | g (cid:105)} energy eigenstate basis. Here r represents the output signal’s location on the I/Q plane, r is the mean output signal whenthe transmon is in the energy eigenstate | f (cid:105) , σ isthe variance of the output signal. A Kraus op-erator both gives the probability distribution of ameasurement outcome ( P ( r ) = | M ( r ) z | ψ (cid:105) | ) andcharacterizes the measurement backaction on thestate ( | ψ (cid:105) → M ( r ) z | ψ (cid:105) ). Applying such a measure-ment pulse of duration τ reduces coherences be-tween pairs of states, characterized by dephasingfactors exp( − γ ef τ ) = exp( − γ gf τ ) = exp( − r σ ) and exp( − γ ge τ ) = 1 [24].We characterize the strength and selectivity of themeasurement by examining the dephasing rates ofeach pair of states. We drive the cavity with aweak probe, and perform Ramsey measurements oneach pair of levels to determine the dephasing rates γ ge , γ ef , γ gf , in each of the two-level subspaces. InFigure 2c we display these measured dephasing ratesversus probe frequency. The data show enhanceddephasing at each qutrit-state-dressed cavity reso-nance, as expected. We further observe larger back-ground dephasing related to the | f (cid:105) state whichis due to the reduced charge noise insensitivity ofthe higher transmon levels [20]. A cavity probeat frequency ω p / π = 5 . GHz therefore allowsus to realize measurements on the {| e (cid:105) , | f (cid:105)} mani-fold, while preserving coherence within the {| g (cid:105) , | e (cid:105)} manifold. The measurement strength ( γ ef τ ) can betuned via the duration of a single measurement.We now focus on the quantum dynamics of the qubitformed by the {| e (cid:105) , | f (cid:105)} manifold, reserving the state | g (cid:105) as a phase reference. Since the dispersive mea-surements merely provide measurement in the en-ergy basis of the qubit, corresponding to the Z axisof the Bloch sphere, we utilize additional rotationsto perform measurement along any arbitrary axis ofthe qubit [25]. One example of these rotations isshown in Figure 3a for cases of projective and par- tial measurements.Figure 3b displays the experimental sequence. Toform closed path measurement-induced trajectories,we first initialize the qutrit in the state ∝ | g (cid:105) + | θ, φ = 0 (cid:105) , where | θ, φ (cid:105) specifies the qubit state in the {| e (cid:105) , | f (cid:105)} manifold in terms of polar and azimuthalangles. We then apply a sequence of six measure-ments at fixed θ and with decreasing φ chosen towrap the Bloch sphere. Finally, we use rotations anda projective measurement to ensure closed path evo-lution and determine the geometric phase. This final stage of the protocol involves the following steps:i) The first rotation is applied such that the state | θ, φ = − π (cid:105) is rotated into | e (cid:105) . ii) We then apply a π/ rotation to interfere | g (cid:105) (the reference state) and | e (cid:105) (which acquired a measurement-induced phase)followed by iii) projective measurement of | g (cid:105) [26].The resulting geometric phase, χ , and interferencecontrast, c , are determined by the phase and am-plitude of the interference in the {| g (cid:105) , | e (cid:105)} manifold(Fig. 3c).The closed paths that acquire a specific geometricphase are associated with specific trajectories, re-sulting from specific sequences of measurement read-outs, implying the need for postselection. This post-selction is implicitly enforced via the measurementarchitecture that preserves the coherence in the sub-space {| g (cid:105) , | e (cid:105)} , while destroying coherences with | f (cid:105) . As an example, consider the regime of a verystrong measurement where | r | (cid:29) σ . Here, thereare effectively two measurement outcomes, r (cid:39) r and r (cid:39) , with backaction projecting onto | f (cid:105) orthe {| g (cid:105) , | e (cid:105)} manifold respectively. When the out-come is “null”, r (cid:39) , the population in the {| g (cid:105) , | e (cid:105)} manifold is preserved, ultimately contributing to theinterference used to infer the geometric phase. Inthis case, there is backaction on the {| e (cid:105) , | f (cid:105)} sub-space corresponding to a segment of the closed tra-jectory. However, when the outcome is r (cid:39) r , thepopulation in the {| g (cid:105) , | e (cid:105)} manifold is eliminated,so that no contribution of the trajectory to the off-diagonal term between | g (cid:105) and {| e (cid:105) , | f (cid:105)} remains.Thus, only with a series of measurement outcomes r (cid:39) (null-outcome path) can the resulting geo-metric phase be observed, while the other paths arenaturally excluded from interference. As discussedin Methods, such selection occurs for any measure-ment strength, effectively giving the null-outcomepath geometric phase, without any postselection onspecific sequences of measurement readouts.In order to probe the topological transition we recordthe geometric phase χ and interference contrast c for different trajectory latitudes and measurement R ge π /2 R ef, φ =0 θ R ef, φ =-2 π i /(N+1) θ M z R ef, φ =-2 π i /(N+1) −θ R ge, =[0,2 π ] π /2 Π g i=[1,N], N=6 φ ge ππ /2 3 π /2 2 π c χ P ( g ) |g|f|e R ef, φ =0 - θ a b c R ef, φ =-2 π i /(N+1) θ R ef, φ =-2 π i /(N+1) −θ M z Z X Y i=6 φ ge M θ , φ M θ , φ FIG. 3:
Experimental sequence (a) Measurements along an arbitrary axis (solid arrows) are composed of rotationsbefore and after measurement along the Z axis (dashed arrows). Partial (projective) measurements are indicated inblue (red). (b) The full experiment consists of a sequence of 6 measurements in the qubit manifold at decreasingazimuthal angles φ . (c) The geometric phase, χ , and interference contrast, c , are determined from the final inter-ference pattern between | g (cid:105) and | e (cid:105) , where P ( g ) is the probability of obtaining outcome g from the final projectivemeasurement, Π g . χ (radians) c M ea s u r e m en t s t r eng t h : γ e f τ θ (radians) θ (radians) a b π /2 π π /2 π FIG. 4: (a) The geometric phase under varying mea-surement strength ( γ ef τ ) gained with sequential mea-surements along different latitudes of the Bloch spherefor polar angle → π . The approximate location of thetransition is marked by a dashed circle. (b) The cor-responding contrast of the interference vanishes at thetransition. strengths. The results are displayed in Figure 4. Inthe limit of strong measurement, exp( − γ ef τ ) → ,the measurement backaction is sufficent to allow thequantum state to follow the measurement axis lead-ing to monotonous increasing geometric phase withpolar angle (as sketched in Fig. 1a and c). In theinfinitely weak measurement limit, γ ef τ ∼ thereis no measurement backaction, hence there is no ob-served dependence of the geometric phase on the po-lar angle. Between these two limits we encounter thetopological transition, which appears as a π phase winding about a singularity point. Exactly at thispoint the phase is ill-defined, which can only happenif the contrast vanishes, which we indeed observe (cf.Fig. 4b). The singularity point belongs to θ = π/ .We observe that, as a function of the measurementstrength, the phase at θ = π/ exhibits an abruptjump of size π , in agreement with the theory pre-dictions (cf. Fig. 1c). This jump corresponds to thecritical measurement strength that drags the statehalf way around the Bloch sphere (cf. the insets ofFig. 1c). Near the transition, the state after the finalprojection involves averaging over trajectories thateither encircled the Bloch sphere, acquiring a geo-metric phase of π , and those that did not, acquiringzero geometric phase. This leads to the observedvanishing contrast.We have investigated measurement enabled quan-tum dynamics where the dynamics carry a topo-logical character. Despite the stochastic nature ofthese weak measurement dynamics, the resulting ge-ometric phases feature a robust and sharp transi-tion which is immune to non-universal details of theplatform. A salient feature of our protocol is thatright at the transition point, coherent features of themeasurement-induced phase are washed out, under-scoring the role of fluctuations at the critical tran-sition point. While our single qubit experiment hassimple topology, our work advances the possibilitythat many body systems with non-trivial topologymight also exhibit such robust topological transi-tions. Such investigations would reveal the interplaybetween the topological nature of many-body phasesand the measurement-induced dynamics. This sug-gests the possibility to utilize such topological effectsfor protected quantum information processing. Acknowledgements —This research was supported byNSF Grant No. PHY- 1752844 (CAREER) andused of facilities at the Institute of Materials Scienceand Engineering at Washington University. A.R.acknowledges the support of the U.K. Engineeringand Physical Sciences Research Council (EPSRC)via Grant No. EP/P010180/1. Y.G. acknowledgesfunding by the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation)– 277101999–TRR 183 (project C01) and EG 96/13-1, by the Is-rael Science Foundation (ISF), by the NSF GrantNo. DMR-2037654, and the US-Israel BinationalScience Foundation (BSF), Jerusalem, Israel. K.S.acknowledges funding by Deutsche Forschungsge-meinschaft (DFG, German Research Foundation) –277101999–TRR 183 (project C01) and GO 1405/6-1. ∗ Electronic address: [email protected][1] Y. Aharonov and J. Anandan. Phase change dur-ing a cyclic quantum evolution.
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Stabilizing higher Transmon states against charge noise —Although a superconducting Transmon circuit isdesigned to reduce the charge noise sensitivity of the | e (cid:105) state exponentially in the ratio between E J /h =13 . GHz and E C /h = 285 MHz [20], the third energy level | f (cid:105) may still be affected by charge noise. Weobserve both increased dephasing associated with the | f (cid:105) state (Fig. 2c), as well as abrupt transitions in the | f (cid:105) energy and associated fluctuations. We stabilize the experiment against these fluctuations by trackingthe Ramsey pattern in the {| e (cid:105) , | f (cid:105)} manifold and sorting the acquired data in a post-processing step. Calibrating dynamical phase accumulation —The quantum evolution in our experiment that takes place over ∼ µ s of evolution is associated with the accumulation of large dynamical phases (on the order of radians).This dynamical phase is measured in a rotating frame associated with microwave generator used to performTransmon rotations. This results in an effective cancellation of the dynamical phase. To confirm thiscancellation, we perform a reference experiment at each data point using rotation sequences with fixed φ = 0 instead of the sequential φ = [0 , − π ] . The reference experiment makes use of the same number and strengthof sequential measurements at one point on the latitude instead of winding around the Z axis. Figure 5 showsthat this relative dynamical phase is smooth and of order a fraction of one radian, confirming that dynamicalphase does not appreciably contribute to our observed topological transition. The premise is that, whileno geometric phase is accumulated, the dynamic phase gained is the same as in the actual experimentalprotocol. The phase singularity lying at θ = π/ (cf. Fig. 4) further confirms that the dynamic phasesdo not affect the measurement-induced dynamics in our protocol, cf. Refs. [27, 28]. The observed stripefeatures in the reference phase and contrast are likely due to residual dynamical phase associated with theanharmonicity of the Transmon. Selective averaging theory —The Kraus operator of the selective measurement as a function of the outputsignal’s position on the IQ plane r is given by M ( r ) z = ˜Ψ( r ) 0 00 Ψ( r ) 00 0 Ψ( r ) , (3)written in the {| f (cid:105) , | e (cid:105) , | g (cid:105)} basis, assuming unit quantum efficiency. Where ˜Ψ( r ) = (cid:113) π σ e − ( r − r ) / σ and Ψ( r ) = (cid:113) π σ e − ( r ) / σ are as defined in the main text. Starting from an initial qutrit state | φ i (cid:105) = a ( e ) i | e (cid:105) + a ( g ) i | g (cid:105) , an initial rotation R † produces a target initial state at a chosen lattitude in the {| e (cid:105) , | f (cid:105)} manifold while maintaining the | g (cid:105) state as a phase reference. Subsequently a sequence of measurements isperformed at axes R † k | e (cid:105) with the last measurement P ge being projective onto the {| g (cid:105) , | e (cid:105)} manifold, leavingus with the final system state | φ f (cid:105) = P ge R (cid:89) k (cid:16) R † k M ( r k ) z R k (cid:17) R † | φ i (cid:105) (4)for a certain series of measurement outcomes { r , r , ..., r k , ... } .Eventually, the geometric phase χ and its contrast c are extracted from the ensemble through the interferencebetween state | g (cid:105) and state | e (cid:105) , with operator A = Σ x − i Σ y = 2 | g (cid:105) (cid:104) e | , where Σ x = , Σ y = − i i , (5)and ce iχ = ˆ Ω (cid:104) φ f | A | φ f (cid:105) . (6)Here Ω represents all the combinations for values of { r k } . Since the operator A already projects | φ f (cid:105) onto the {| g (cid:105) , | e (cid:105)} manifold, we note that the final projective measurement P ge can be skipped in actual experiment. χ (radians) Contrast M ea s u r e m en t s t r eng t h : γ e f τ θ (radians) θ (radians) a b π /2 π π /2 π FIG. 5:
Cancellation of relative dynamical phases.
The observed interference phase (a) and contrast (b) for areference experiment with fixed φ = 0 , showing that the dynamical phase is smooth and effectively canceled throughappropriate rotating frames. Given the initial state | φ i (cid:105) = a ( e ) i | e (cid:105) + a ( g ) i | g (cid:105) , after a sequence of measurements with outcomes { r k } , thesystem is still in pure state | φ f (cid:105) = a ( e ) f ( { r k } ) | e (cid:105) + a ( g ) f ( { r k } ) | g (cid:105) . Since all the rotation operators R k are inthe {| e (cid:105) , | f (cid:105)} manifold, the coefficient of the reference state | g (cid:105) becomes a ( g ) f ( { r k } ) = (cid:89) k Ψ( r k ) a ( g ) i . (7)Using (4) and (6), the extracted geometric phase becomes ce iχ = 2 ˆ Ω (cid:104) φ f | g (cid:105)(cid:104) e | φ f (cid:105) = 2 ˆ Ω a ( g ) ∗ f ( { r k } ) (cid:104) e | φ f (cid:105) = 2 (cid:104) e | ˆ Ω (cid:89) k Ψ ∗ ( r k ) a ( g ) ∗ i | φ f (cid:105) = 2 a ( g ) ∗ i (cid:104) e | ˆ Ω (cid:89) k Ψ ∗ ( r k ) P ge R (cid:89) k (cid:16) R † k M ( r k ) z R k (cid:17) R † | φ i (cid:105) = 2 a ( g ) ∗ i (cid:104) e | ˆ Ω R (cid:89) k (cid:16) R † k Ψ ∗ ( r k ) M ( r k ) z R k (cid:17) R † | φ i (cid:105) . (8)Here we note that the whole ensemble is a weighted average over various measurement outcome sequences { r k } . For weak measurements, a null outcome corresponds to a certain probability distribution of mea-surement readouts r k . The weighting with Ψ( r k ) enforces the correct distribution among the states thatcontribute to the interference.Finally we have ce iχ = 2 a ( g ) ∗ i (cid:104) e | R (cid:89) k (cid:16) R † k ˜ M z R k (cid:17) R † | φ i (cid:105) , (9)with the integrated effective Kraus operator ˜ M z = ˆ Ω Ψ ∗ ( r ) ˜Ψ( r ) 0 00 Ψ( r ) 00 0 Ψ( r ) = ´ Ω Ψ ∗ ( r ) ˜Ψ( r ) 0 00 ´ Ω Ψ ∗ ( r )Ψ( r ) 00 0 ´ Ω Ψ ∗ ( r )Ψ( r ) = ´ Ω Ψ ∗ ( r ) ˜Ψ( r ) 0 00 1 00 0 1 = ˜ M {| e (cid:105) , | f (cid:105)} z ⊕ ˆ I {| g (cid:105)} , (10)where ˜ M {| e (cid:105) , | f (cid:105)} z is the null outcome-path partial measurement operator on the {| e (cid:105) , | f (cid:105)} manifold, and ˆ I {| g (cid:105)} is the identity operator on state | g (cid:105) .Hence, even though the measurement readouts in our setup are continuously distributed, the effective back-action of the measurements corresponds to null-outcome partial measurements in the {| e (cid:105) , | f (cid:105)} manifold.Thence, no explicit postselection of measurement readouts is required. Experimental setup —The experiment comprises a superconducting Transmon circuit embedded in a threedimensional aluminum microwave cavity. The Transmon energies E J /h = 13 . GHz, E C /h = 285 MHz,produce transition frequencies ω ge / π = 5 . GHz, ω ef / π = 4 . GHz. The cavity linewidth κ/ π =0 . MHz. The dispersive interaction between the Transmon and the cavity shifts the cavity frequency fromits bare resonance frequency ω bare / π = 5 . GHz to a state dependent frequency, ω g / π = 5 . GHz, ω e / π = 5 . GHz, and ω f / π = 5 . GHz. Three microwave generators are employed to control andmeasure the system, one generator addresses the Transmon transitions through single sideband modulation,another contributes the measurement at frequency ω f , and a final generator operates at ω barebare