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RAPID COMMUNICATIONS PHYSICAL REVIEW A 69, 050301(R) (2004) Geometric manipulation of the quantum states of two-level atoms Mingzhen Tian, Zeb W. Barber, Joe A. Fischer, and Wm. Randall Babbitt Department of Physics, Montana State University, Bozeman, Montana 59717, USA (Received 5 December 2003; published 6 May 2004) Manipulation of the quantum states of two-level atoms has been investigated using laser-controlled geometric phase change, which has the potential to build robust quantum logic gates for quantum computing. For a qubit based on two electronic transition levels of an atom, two basic quantum operations that can make any universal single qubit gate have been designed employing resonant laser pulses. An operation equivalent to a phase gate has been demonstrated using Tm3+ doped in a yttrium aluminum garnet crystal. DOI: 10.1103/PhysRevA.69.050301 PACS number(s): 03.67.Lx, 42.50.Md, 03.65.Vf Two-level systems, such as atoms or spins, have been extensively used as prototypes for the quantum information carrier—qubit, in quantum computing. The building blocks of quantum computation consist of two types of accurately controllable universal operations on a set of two-level systems: (i) any arbitrary unitary operations on a single qubit and (ii) a controlled-NOT operation involving two qubits [1,2]. These operations have been sought through dynamic evolutions of various two-level systems driven by interaction Hamiltonians from external sources, such as laser pulses and magnetic fields. It was also realized in recent years that the whole set of universal quantum operations can be accomplished purely through geometric effects on the wave functions of the qubits that are imparted by the external controls. The geometric operations originate from the geometric phase, a fascinating result of quantum theory, which shows that a quantum system evolving through a cyclic path in the wave function space gains a geometric phase in addition to the Hamiltonian-dependent dynamic phase [3]. One of the attractive features of geometric operation is the potential robustness. The geometric phase solely depends on the amount of the solid angle enclosed by the evolution path. It does not depend on the details of the path, the time spent, the driving Hamiltonian, or the initial and final states of the evolution. Therefore, it is expected that the geometric operations are relatively robust compared to the dynamic ones and are resilient to errors caused by certain types of noises in the driving Hamiltonian. Take, for example, the noises that cause the jitter on the evolution path, but keep the amount of solid angle associated with the path unchanged. Quite a few allgeometric or hybrid (with dynamic operations) quantum operations have been proposed for various physical systems, such as nuclear magnetic resonance (NMR) [4], iontraps [5], neutral atoms [6], optical systems [7], and superconducting nanocircuits [8]. However, the exploration of geometric quantum computation is still in its early stage; experimental demonstrations are rare [8–10] and the theoretical analyses on robustness are still far from complete and conclusive. In this paper, we propose a scheme for universal single qubit operations on two-level atoms composed purely by laser-controlled geometric phase changes. These geometric operations are equivalent to the arbitrary rotations of the Bloch vector of a two-level atom on the Bloch sphere. To eliminate the infidelity of the operation associated with the Hamiltonian-dependent dynamic phase, special paths have 1050-2947/2004/69(5)/050301(4)/$22.50 been designed to eliminate the dynamic phase change. We also present the experimental demonstration of the laserpulse-controlled Bloch vector rotation of rare earth ions doped in a crystal, which is one of the basic Bloch vector rotations equivalent to the single qubit phase gate. The proposed single qubit operations are applicable to two-level atoms. Controlled geometric operations can also be designed in the presence of some auxiliary levels in the system [11]. A universal controlled-NOT gate involving two qubits can be developed based on these single qubit operations for specified interactions that couple qubits, such as the dipole-dipole interaction [12] or the coupling through photons in a cavity [13]. We consider two electronic levels of an atom as the qubit [as shown in Fig. 1(a)] whose state is usually described with a wave function as 冢 冣 cos 兩 典 = e i␥ e −i  2  sin 2 , where the two basis states, 兩0典 = 共 01 兲 and 兩1典 = 共 10 兲, represent the ground and the excited levels of the electronic transition, respectively. The wave function can also be mapped into a unitary vector 共Bloch vector兲 rជ on the Bloch sphere as shown in Fig. 1共b兲. This mapping is accurate up to an arbitrary FIG. 1. (a) Schematic of an electronic transition of a two-level atom with the wave function 兩典 driven by a laser field ⍀0 cos共Lt + 兲 (b) Bloch vector rជ representing the wave function ជ on the Bloch sphere. 兩典 and its precession about the Rabi vector ⍀ 69 050301-1 ©2004 The American Physical Society RAPID COMMUNICATIONS PHYSICAL REVIEW A 69, 050301(R) (2004) TIAN et al. global phase, ␥, which is insignificant for it does not affect any observables in the closed two-level system. Any pair of orthogonal bases of the system is a pair of vectors pointing in opposite directions, for example, 兩0典 and 兩1典 are represented by the vectors pointing down and up along axis 3̂, respectively. If a Hamiltonian drives a Bloch vector going through a cyclic path, the tip of the Bloch vector traces out a circuit on the Bloch sphere and returns to its initial position. This evolution makes the wave function associated with the Bloch vector gain a geometric phase besides the dynamic phase. The geometric phase solely depends on the solid angle ␣ enclosed by the circuit at the center of the unitary globe, as = −␣ / 2. It does not depend on the shape of the circuit, the driving Hamiltonian, or the initial and the final positions of the Bloch vector. The geometric nature of this phase change, independent of the state of the system and the interaction Hamiltonian, makes it attractive for making robust qubit operations and quantum gates. For an electronic transition of an atom, a laser pulse can drive the evolution of the Bloch vector. The motion of the Bloch vector is governed by the optical Bloch equation, as ជ ⫻ rជ. The Rabi vector ⍀ ជ consists of three compodrជ / dt = ⍀ nents along the three axes of the Bloch sphere as 共−⍀0 cos , −⍀0 sin , ⌬ 兲 where the electric field of the laser pulse is denoted as E共t兲 = ⍀0 cos共Lt + 兲 with frequency L, phase , and amplitude ⍀0, in Rabi frequency. The detuning ⌬ = a − L is the frequency difference between the atomic transition and the laser. The motion of the Bloch vector is a ជ at Rabi frequency ⍀ precession about the torque vector ⍀ = 冑⍀02 + ⌬2 as shown in Fig. 1(b). The amount of rotation is determined by the pulse area, defined as the product of the Rabi frequency and the driving pulse duration. In general, any 2 pulse makes the Bloch vector complete one cycle of rotation about the Rabi vector. The solid angle enclosed by the circuit is determined by the angle between the two vectors. However, one cannot use a single pulse to make an arbitrary pure geometric phase operation, since a cyclic evolution is usually accompanied by a dynamic phase change as well. Only when the Bloch and the Rabi vectors are perpendicular to each other throughout the entire circuit will the dynamic phase of the system remain unchanged [14]. Thus, more than one pulse is required to make up a circuit to enclose an arbitrary geometric phase for a qubit operation and at the same time eliminate the dynamic phase change. A universal single qubit operation is equivalent to a rotation of the Bloch vector, which can be accomplished as U = U3共␦3兲U2共␦2兲U3共␦1兲. The two types of basic rotations, U3 and U2, are about axes 3 and 2 of the Bloch sphere, respectively [1]. ␦i 共i = 1, 2 , and 3兲 denotes the corresponding rotation angle. In matrix notation, the two basic rotations can be written as U 3共 ␦ 兲 = 冉 ei␦/2 0 0 e −i␦/2 冊 and U2共␦兲 = 冉 cos共␦/2兲 sin共␦/2兲 − sin共␦/2兲 cos共␦/2兲 冊 FIG. 2. Bloch vector rotation about axis 3̂ driven by two resonant pulses, the corresponding Rabi vectors with relative phase ជ and ⍀ ជ , respectively. (a) The path of Bloch ␦ / 2, are labeled as ⍀ 1 2 vector ជr−3 representing wave function 兩0典 enclosing a solid angle, 2共 − ␦ / 2兲. (b) The path of the Bloch vector rជ3 representing 兩1典 enclosing solid angle, −2共 − ␦ / 2兲. two components in an arbitrary wave function. This rotation can be done with two resonant pulses, with the pulse area of , but with a relative phase difference of ␦ / 2. As shown in Fig. 2共a兲, the two Rabi vectors associated with the two control pulses are all in the 1-2 plane, since the pulses have zero frequency detuning. We set the phase of the first pulse to ជ , assozero and the second to ␦ / 2 so that the Rabi vector ⍀ 1 ciated with the first pulse, points to the −1̂ direction, and ជ 兩 = 兩⍀ ជ 兩, but ⍀ ជ is rotated ␦ / 2 counterclockwise in the 1-2 兩⍀ 2 1 2 plane. We track the motion of a vector rជ = rជ−3 pointing down along −3̂ axis, which corresponds to the ground-state wave ជ by to function 兩0典. Driven by pulse 1, rជ rotates about ⍀ 1 position rជ3. The tip of the vector traces out a path, which is half of the great circle of the Bloch sphere in the 2-3 plane. ជ by back Then, pulse 2 drives this vector rotating about ⍀ 2 to its initial position rជ−3 through a path on another great circle of the globe in a plane including axis 3̂ and angling from axis 2̂ by ␦ / 2. We denote the operation of these two pulses, O3共␦ / 2兲. This operation drives rជ through a cyclic path of two halves of the great circles back to is initial state. The solid angle enclosed by the path is 2共 − ␦ / 2兲. Since the Bloch vector is perpendicular to the driving Rabi vectors on the whole path, the dynamic phase is unchanged. The result of O3共␦ / 2兲 acting on rជ−3 imparts a pure geometric phase −共 − ␦ / 2兲 to the corresponding wave function as O3共␦ / 2兲兩0典 = e−i共−␦/2兲兩0典. Similarly, O3共␦ / 2兲 drives a Bloch vector starting from position rជ3 through a different cyclic path, as shown in Fig. 2共b兲 enclosing a solid angle of −2共 − ␦ / 2兲, while the Bloch vector stays perpendicular to the Rabi vectors over the entire path. This results in a pure geometric phase − ␦ / 2 added to the associated wave function 兩1典 as O3共␦ / 2兲兩1典 = ei共−␦/2兲兩1典. Therefore, we can write O3共␦/2兲 = e−i . These basic rotations can be realized purely by controlled geometric phase change. We discuss U3共␦兲 first. This rotation matrix results in a relative phase change ␦ between the 冉 ei␦/2 0 0 e −i␦/2 冊 as the operation matrix on any arbitrary wave function. Ignoring the global phase factor in front of the matrix we have the two-pulse-controlled operation O3共␦ / 2兲, which is equiva- 050301-2 RAPID COMMUNICATIONS PHYSICAL REVIEW A 69, 050301(R) (2004) GEOMETRIC MANIPULATION OF THE QUANTUM… FIG. 3. Bloch vector rotation about axis 2̂ driven by three resoជ ,⍀ ជ , nant pulses, the corresponding Rabi vectors are labeled as ⍀ 1 2 ជ ជ ជ and ⍀3, respectively. ⍀1 and ⍀3 represent two / 2 pulses with the ជ is a pulse with a relative phase of + ␦ / 2. same phase while ⍀ 2 The path of Bloch vector ជr2 representing wave function, 兩i典 = 共1 / 冑2兲共兩0典 + 兩i典兲 encloses solid angle, ␦ / 2. lent to the state-independent rotation U3共␦兲. The rotation angle ␦ is controlled by the relative phase between the two control pulses. Now we discuss the rotation about axis 2̂, U2共␦兲. This operation changes the relative phase of another pair of the orthogonal basis states, 兩 + i典 = 冉冊 冉 冊 1 1 1 1 and 兩− i典 = 冑2 − i , i 冑2 as U2共␦兲兩 ± i典 = e⫿i␦/2兩 ± i典. The two basis states 兩 + i典 and 兩−i典 correspond to Bloch vectors rជ2 and rជ−2, pointing along axes 2̂ and −2̂, respectively. Therefore, we need to design a pulse sequence to drive the two vectors through their corresponding cyclical paths and to gain the opposite phase of ␦ / 2. These can be done with three resonant pulses; the correជ ,⍀ ជ , and ⍀ ជ in Fig. sponding Rabi vectors are labeled as ⍀ 1 2 3 3. The three pulses have pulse areas of / 2, , and / 2, respectively. The first and third pulses have a zero phase and the second pulse has a relative phase of + ␦ / 2. This makes ជ and ⍀ ជ aligned with −1̂ axis, while ⍀ ជ points in a direc⍀ 1 3 2 tion angled ␦ / 2 from axis 1̂ in the 1-2 plane. We denote the operation of this pulse sequence with O2共␦ / 2兲. Under this operation, a Bloch vector rជ = rជ2, representing wave function ជ by / 2 to a position pointing 兩 + i典. rotates first about ⍀ 1 ជ by to point up down along −3̂ axis, then rotates about ⍀ 2 ជ by / 2 back to the along axis 3̂, and finally rotates about ⍀ 3 initial position. The first and the third segments of the cyclic path are on the great circle in plane 2-3 and the second segment on another great circle in a plane including axis 3̂ rotated counterclockwise from axis 2̂ by ␦ / 2. The solid angle enclosed by the path is ␦. Thus, the wave function 兩 + i典 gains a geometric phase of −␦ / 2 as O2共␦ / 2兲兩 + i典 = e−i␦/2兩 + i典. Under the same operation, a Bloch vector rជ = rជ−2, pointing in the FIG. 4. Regular pulse sequence of a photon echo process. (b) Pulse sequence with two control pulses to perform rotation, U3共␦兲, where the rotation angle ␦ is controlled by the relative phase ␦ / 2 between the two control pulses. The rotation results in the echo field phase change by ␦. (c) Schematic of the experimental setup. opposite direction of rជ2, goes through a cyclic path enclosing a solid angle of −␦ and the corresponding wave function gains a geometric phase as, O2共␦ / 2兲兩−i典 = ei␦/2兩−i典. Therefore, O2共␦ / 2兲 is equivalent to the rotation U2共␦兲 about axis 2. The rotation angle is controlled by the relative phase between the driving pulses. If it is desired to make the basic rotation about axis 1̂, instead of U2, a similar approach can be used to design a resonant three-pulse-driven operation U1 in which the rotation angle, again, is controlled by the relative phase between the driving pulses. In a proof-of-concept experiment we demonstrated the laser-pulse-controlled Bloch vector rotation, U3共␦兲 on Tm3+ doped in yttrium aluminum gamet (YAG) crystal. The resulting quantum states of the ions were measured using photon echoes [15] as the analogue of spin echoes in NMR systems [16]. The rephasing process of the photon echo cancels the dynamic phase differences for atoms at different frequencies. This allows us to observe the pure geometric phase change of an ensemble of inhomogeneously detuned two-level systems. Figure 4 shows the schematics of the pulse sequences and the experimental setup. In an ordinary photon echo process [Fig. 4(a)], pulse 1 (usually a / 2 pulse) excites the atoms from the ground state to their superposition states to create a coherence, which dephases during the time interval between pulses 1 and 2 due to the different resonant frequencies. Pulse 2 (usually a pulse) causes the rephasing of the coherence of all the atoms resulting in a macroscopic field as an echo pulse delayed by after the second pulse. The electric field of the echo is proportional to e−i sin  summing over all excited atoms, where  and are the two angular coordinates of the Bloch vector representing an atomic state at the echo rephasing time. In the pulse sequence of the photon echo process,  usually changes when the atoms are excited by a pulse, and changes between the pulses due to the dephasing and rephasing of the atoms at different detunings. When the echo is generated, the Bloch vectors of all atoms have the same , which was set to zero in our experiment. When the two control pulses, as shown in Fig. 4(b), are applied between the first and second pulses or the second 050301-3 RAPID COMMUNICATIONS PHYSICAL REVIEW A 69, 050301(R) (2004) TIAN et al. and echo pulses to perform the U3共␦兲 operation, an extra phase ␦ is introduced to the coherence between the ground and excited states, which means the echo field gains the same amount of extra phase. We measured this phase by heterodyning the echo with a reference pulse, whose frequency was shifted by 20 MHz from the echo’s frequency. As shown in Fig. 4(c), the laser source in the setup was a cw Ti:sapphire laser, frequency stabilized to submegahertz using the spectral hole burning technique [17]. All the input pulses including the heterodyning reference (not shown in Fig. 4) were generated by an acoustic optical modulator driven by an arbitrary waveform generator. Pulses 1 and 2 were 500 ns long, delayed by = 7 s. Two control pulses were 200 ns each, applied between pulses 1 and 2. An 11 s long heterodyning reference pulse was added to the sequence right after pulse 2. The relative phase ␦ / 2 of the two control pulses was varied by the arbitrary waveform generator from 0 to with 0.05 increments The phase change of the echo field was calculated from a 409.6-ns long segment of the heterodyned echo signal using fast Fourier transform (FFT). The results are plotted in Fig. 5 (depicted as dots), showing the echo phase change as a function of the control phase. The expected values of the geometric phase change ␦ are also plotted in Fig. 5 as a solid line. The three insets are the heterodyning signal segments at ␦ / 2 = 0 , 0.5, and , respectively. The experimental signals are averaged over 32 shots. We can see a good consistency between the experiment and the theory with a standard deviation of 0.03. We also captured 50 single shots of the echo signal under the same condition, but without the control pulses. The standard deviation of the measurement of the echo phase was calculated to be 0.15. For an average over 32 shot, the standard deviation due to measurement error would be 0.027. Therefore, the errors on the controlled geometric phase change are mainly measurement limited. In summary, we proposed the scheme of universal single qubit quantum operations by pure geometric manipulation of two-level atoms with laser pulses. A complete set of basic Bloch vector rotations (about axes 3̂ and 2̂ in our case) was designed using geometric phase changes, each controlled by the relative phase of a series of resonant laser pulses (two pulses for U3 and three for U2兲. The rotation about axis 3̂, U3, was demonstrated in the experiment with an ensemble of inhomogeneously detuned two-level Tm3+ doped in a YAG crystal. The rotation angle controlled by the relative phase of the two control pulses was measured using heterodyned photon echoes. The experimental results are in good agreement with the theory within measurement-limited errors. The rotation accuracy can be improved by a more stable laser source. The phase measurement technique demonstrated in this paper can be further used to evaluate the operation fidelity when various noises are present on the control pulses, such as frequency jitter, amplitude fluctuation, and pulse duration inaccuracy. The other basic rotation, U2, can also be demonstrated in a similar way. [1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000). [2] D. P. DiVincenzo, Fortschr. Phys. 48, 771 (2000). [3] M. V. Berry, Proc. R. Soc. London, Ser. A, Ser. A 392, 45 (1984); Y. Aharonov and J. Anandam, Phys. Rev. Lett. 58, 1593 (1987). [4] A. Ekert, M. Ericson, P. Hayden, H. Inamori, J. A. Jones, D. K. L. Oi, and V. Vedral, J. Mod. Opt. 47, 2501 (2001); X. Wang, and K. Matsumoto, J. Phys. A 34, L631 (2001). [5] L.-M. Duan, J. I. 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Barber, and Wm. R. Babbitt, J. Lumin. 107, 155 (2004). [15] M. Tian, R. R. Randy, Z. W. Barber, J. A. Fischer, and Wm. R. Babbitt, Phys. Rev. A 67, 011403(R) (2003). [16] D. Suter, K. T. Mueller, and A. Pines, Phys. Rev. Lett. 60, 1218 (1988). [17] N. M. Strickland, P. B. Sellin, Y. Sun, J. L. Carlsten, and R. L. Cone, Phys. Rev. B 62, 1473 (2000). FIG. 5. Phase change of the photon echoes vs the control phase. Experimental results (dots) were calculated from 409.6-ns segments of echo signals (see the three examples shown in the insets) using FFT. The theoretical values, ␦, (solid line) are twice of the control phase ␦ / 2. 050301-4