## Abstract

In quantum information science, the phase of a wave function plays an important role in encoding information. Although most experiments in this field rely on dynamic effects to manipulate this information, an alternative approach is to use geometric phase, which has been argued to have potential fault tolerance. We demonstrated the controlled accumulation of a geometric phase, Berry's phase, in a superconducting qubit; we manipulated the qubit geometrically by means of microwave radiation and observed the accumulated phase in an interference experiment. We found excellent agreement with Berry's predictions and also observed a geometry-dependent contribution to dephasing.

When a quantum mechanical system evolves cyclically in time such that it returns to its initial physical state, its wave function can acquire a geometric phase factor in addition to the familiar dynamic phase (*1*, *2*). If the cyclic change of the system is adiabatic, this additional factor is known as Berry's phase (*3*), which, in contrast to the dynamic phase, is independent of energy and time.

In quantum information science (*4*), a prime goal is to use coherent control of quantum systems to process information, accessing a regime of computation unavailable in classical systems. Quantum logic gates based on geometric phases have been demonstrated in both nuclear magnetic resonance (*5*) and ion trap–based quantum information architectures (*6*). Superconducting circuits (*7*, *8*) are a promising solid-state platform for quantum information processing (*9*–*14*), in particular because of their potential scalability. Proposals for the observation of geometric phase in superconducting circuits (*15*–*19*) have existed since shortly after the first coherent quantum effects were demonstrated in these systems (*20*).

Geometric phases are closely linked to the classical concept of parallel transport of a vector on a curved surface. Consider, for example, a tangent vector **v** on the surface of a sphere being transported from the sphere's north pole around the path *P* shown in Fig. 1A, with **v** pointing south at all times. The final state of the vector **v**_{f} is rotated with respect to its initial state **v**_{i} by an angle ϕ equal to the solid angle subtended by the path *P* at the origin. Thus, this angle is dependent on the geometry of the path *P* and is independent of the rate at which it is traversed. As a result, departures from the original path that leave the solid angle unchanged will not modify ϕ. This robustness has been interpreted as a potential fault tolerance when applied to quantum information processing (*5*).

The analogy of the quantum geometric phase with the above classical picture is particularly clear in the case of a two-level system (a qubit) in the presence of a bias field that changes in time. A familiar example is a spin-½ particle in a changing magnetic field. The general Hamiltonian for such a system is *H* = ħ**R · σ**/2, where **σ** = (σ_{x}, σ_{y}, σ_{z}) are the Pauli operators, ħ is Planck's constant divided by 2π, and **R** is the bias field vector, expressed in units of angular frequency. The qubit dynamics is best visualized in the Bloch sphere picture, in which the qubit state **s** continually precesses about the vector **R**, acquiring dynamic phase δ(*t*) at a rate *R* =|**R**| (Fig. 1B). When the direction of **R** is now changed adiabatically in time (i.e., at a rate slower than *R*), the qubit additionally acquires Berry's phase while remaining in the same superposition of eigenstates with respect to the quantization axis **R**. The path followed by **R** in the three-dimensional parameter space of the Hamiltonian (Fig. 1C) is the analog of a path in real space in the classical case. When **R** completes the closed circular path *C*, the geometric phase acquired by an eigenstate is ±Θ_{C}/2 (*3*), where Θ_{C} is the solid angle of the cone subtended by *C* at the origin. The ± sign refers to the opposite phases acquired by the ground or excited state of the qubit, respectively. For the circular path shown in Fig. 1C, the solid angle is given by Θ_{C} = 2π (1 – cos θ), depending only on the cone angle θ.

We describe an experiment carried out on an individual two-level system realized in a superconducting electronic circuit. The qubit is a Cooper-pair box (*21*, *22*) with an energy level separation of ħω_{a} ≈ *h* × 3.7 GHz when biased at charge degeneracy, where it is optimally protected from charge noise (*23*). The qubit is embedded in a one-dimensional microwave transmission line resonator with resonance frequency ω_{r}/2π ≈ 5.4 GHz (Fig. 2A). In this architecture, known as circuit quantum electrodynamics (QED) (*24*, *25*), the qubit is isolated effectively from its electromagnetic environment, leading to a long energy relaxation time of *T*_{1} ≈ 10 μs and a spin-echo phase coherence time of . In addition, the architecture allows for a high-visibility dispersive readout of the qubit state (*26*).

Fast and accurate control of the bias field **R** for this superconducting qubit is achieved through phase and amplitude modulation of microwave radiation coupled to the qubit through the input port of the resonator (Fig. 2A). The qubit Hamiltonian in the presence of such radiation is (1) where ħΩ_{R} is the dipole interaction strength between the qubit and a microwave field of frequency ω_{b} and phase φ_{R}, and *t* is time. Thus, Ω_{R}/2π is the Rabi frequency that results from resonant driving. The above Hamiltonian may be transformed to a frame rotating at the frequency ω_{b} by means of the unitary transformation (2) where *U* = exp(*i*ω_{b}*t*σ_{z}/2), and *U̇* denotes its time derivative. Ignoring terms oscillating at 2ω_{b} (the rotating wave approximation), the transformed Hamiltonian takes the form (3) where Ω_{x} = Ω_{R} cos φ_{R} and Ω_{y} = Ω_{R} sin φ_{R}. This is equivalent to the generic situation depicted in Fig. 1, B and C, where **R** = (Ω_{x}, Ω_{y}, Δ) and Δ = ω_{a} – ω_{b} is the detuning between the qubit transition frequency and the applied microwave frequency. In our experiment, we keep Δ fixed and control the bias field to trace circular paths of different radii Ω_{R}.

We measure Berry's phase in a Ramsey fringe interference experiment by initially preparing an equal superposition of the qubit ground and excited states, which acquires a relative geometric phase γ_{C} = 2π(1 – cos θ), equal to the total solid angle enclosed by the cone depicted in Fig. 1C, with . As the bias field adiabatically follows the closed path *C*_{±}, the qubit state acquires both a dynamic phase δ(*t*) and a geometric phase γ_{C}, corresponding to a total accumulated phase ϕ = δ(*t*)± γ_{C} (the ± sign denoting path direction), which we extract by performing full quantum-state tomography (*4*). To directly observe only the geometric contribution, we use a spin-echo (*27*) pulse sequence that cancels the dynamic phase, as explained below.

The complete sequence (Fig. 2B) starts by preparing the initial σ_{z} superposition state with a resonant π/2 pulse. Then the path *C*_{–} is traversed, causing the qubit to acquire a phase ϕ_{–} = δ(*t*) – γ_{C}. Applying a resonant spin-echo π pulse to the qubit about an orthogonal axis now inverts the qubit state, effectively inverting the phase ϕ_{–}; traversing the control field path again, but in the opposite direction *C*_{+}, adds an additional phase ϕ_{+} = δ(*t*) + γ_{C}. This results in total in a purely geometric phase ϕ = ϕ_{+} – ϕ_{–} = 2γ_{C} being acquired during the complete sequence, which we denote as *C*_{–+}. Note that unlike the geometric phase, the dynamic phase is insensitive to the path direction and hence is completely canceled.

At the end of the sequence, we extract the phase of the qubit state by means of quantum-state tomography. In our measurement technique (*26*), the *z* component of the qubit Bloch vector 〈σ_{z}〉 is determined by measuring the excited-state population *p*_{e} =(〈σ_{z}〉 + 1)/2. To extract the *x* and *y* components, we apply a resonant π/2 pulse rotating the qubit about either the *x* or *y* axis and then perform the measurement, revealing 〈σ_{y}〉 and 〈σ_{x}〉, respectively. The phase of the quantum state after application of the control sequence is then extracted as ϕ = tan^{–1}(〈σ_{y}〉/〈σ_{x}〉).

In Fig. 3A we show the measured phase ϕ and its dependence on the solid angle of the path for a number of different experiments, all carried out at Δ/2π ≈ 50 MHz, and total pulse sequence time *T* = 500 ns. Three parameters are varied: the path radius Ω_{R} (upper *x* axis), the number *n* of circular loops traversed in each half of the spin-echo sequence, and the direction of traversal of the paths (*C*_{–+} and *C*_{+–}). The measured phase is in all cases seen to be linear in solid angle as Ω_{R} is swept, with a root-mean-square deviation across all data sets of 0.14 rad from the expected lines of slope 2*n*. Thus, all results are in close agreement with the predicted Berry's phase, and it is clear that we are able to accurately control the amount of phase accumulated geometrically. Note also that the dynamic phase is indeed effectively eliminated by the spin echo. Reversing the overall direction of the paths is observed to invert the sign of the phase (Fig. 3A). Traversing the circular paths on either side of the spin-echo pulse in the same direction (*C*_{++}) as a control experiment results in zero measured phase (Fig. 3A).

Observation of a pure Berry's phase requires adiabatic qubit dynamics, which in turn requires the rate of rotation of the bias field direction to be much less than the Larmor rate *R* of the qubit in the rotating frame. For the case of constant cone angle θ, this translates to the requirement that the adiabaticity parameter . If the Hamiltonian is changed nonadiabatically, the qubit state can no longer exactly follow the effective field **R**, and the geometric phase acquired deviates from Berry's phase (*28*). For the experiments here, *A* ≤ 0.04, and deviation of the measured phase from Berry's phase is not discernible. We have also verified experimentally that in this adiabatic limit, the observed phase is independent of the total sequence time *T*.

Figure 3B shows a measurement of the *x* and *y* components of the qubit state from which the Berry's phase is extracted. Interestingly, the visibility of the observed interference pattern is seen to have a dependence on Ω_{R}. Because the data were taken at a fixed total sequence time, this is not due to conventional *T*_{2} dephasing, which is also independently observable as a function of time, but can be explained as due to geometric dephasing, an effect dependent on the geometry of the path (*29*).

In our experiment, dephasing is dominated by low-frequency fluctuations in the qubit transition frequency ω_{a} (and thus Δ) induced by charge noise coupling to the qubit (*30*). The spin-echo pulse sequence effectively cancels the dynamic dephasing due to the low-frequency noise. However, the geometric phase is sensitive to slow fluctuations, which cause the solid angle subtended by the path at the origin to change from one measurement to the next (Fig. 4A). The effect on the geometric phase of such fluctuations in the classical control parameters of the system has been studied theoretically (*29*). In the limit of the fluctuations being slower than the time scale of the spin-echo sequence, the variance of the geometric phase has itself a purely geometric dependence, , where is the variance of the fluctuations in ω_{a} (*29*). In Fig. 4B, we show the observed dependence of the coherence on geometry explicitly by plotting (〈σ_{x}〉^{2} + 〈σ_{y}〉^{2})^{1/2} versus Ω_{R}, which fits well to the expected dependence . This is also in agreement with the raw data in Fig. 3B.

We have observed an important geometric contribution to dephasing that occurs when geometric operations are carried out in the presence of low-frequency fluctuations. In contrast, higher-frequency noise in ω_{a} is expected to have little influence on Berry's phase (provided adiabaticity is maintained), because its effect on the solid angle is averaged out (Fig. 4C). This characteristic robustness of geometric phases to high-frequency noise may be exploitable in the realization of logic gates for quantum computation, although the effect of geometric dephasing due to low-frequency noise must be taken into account.