Multiplicative suppression of decoherence.

Optically addressed quantum spin systems have been explored for a number of applications, with quantum information being one of the most exciting. However, noise fields destroy quantum information, especially in solids, through a process called decoherence. This process limits how well quantum information devices perform. The spin decoherence problem has been studied for decades, and many protocols have been devised to suppress it. All of these protocols have limitations, which drives the search for new and better strategies. On page 1493 of this issue, Miao et al. ([ 1 ][1]) show that in some cases, two substantially different protocols can be combined to get a multiplicative improvement of the coherence time. ![Figure][2] Creating quantum systems The magnitude and phase information of a qubit is illustrated with a pair of oscillating balls. GRAPHIC: C. BICKEL/ SCIENCE To explain how quantum coherence is protected, first defining it in a simple way is useful. A qubit or quantum bit is often depicted as a superposition of logical “0” and “1” states, written as (α|0〉 + β|1〉). Hence, at least until readout, a qubit can be viewed as analog, rather than digital. In addition to the relative magnitude of the 0 and 1 components, the relative phase is also important. This is because quantum states are often based on oscillators. Phase is simply a way of specifying where an oscillator is in its cycle at a particular snapshot in time (see the figure). For example, a vibrating ball might be stalled at its maximal displaced position or be at its maximal speed with momentary zero displacement ([ 2 ][3]). Quantum coherence is then defined as a precise magnitude and phase relationship between two quantum oscillators. Decoherence arises when any noise source alters either the magnitude or phase of one of the qubit's oscillators differently than the other. One popular technique to suppress decoherence is through the use of “decoherence-protected subspaces” (DPSs). Constructing such a subspace from the above two-oscillator system is accomplished by weakly coupling them by adding a spring (see the figure). When the two oscillators are far detuned from each other, the spring has no effect. However, near zero detuning, the spring effectively causes the relative phases to have different energies, and this can act like a barrier to suppress phase decoherence. In fact, certain phases become constants of motion, also called normal modes, of the coupled system. Examples of normal modes are the in- and out-of-phase cases, (|0〉 + |1〉〉) = |+〉 and (|0〉 − |1〉) = |−〉, neglecting normalization. Because these normal modes are so stable, using these as the qubit states makes more sense than the original |0〉 and |1〉. So, the qubit now becomes (α|+〉 + β|−〉) in a DPS. The next question is how to create and use one of these DPSs. Many possible approaches exist, but the examples in Miao et al. are useful to consider. The first DPS involves using a spin-1 system in a low-symmetry crystal. The levels of a spin-1 system in a strong magnetic field are, using the notation from Miao et al. , |+1z〉, |−1z〉, |0〉. The |0〉 state is not affected by the magnetic field and therefore is already one state of a DPS. To construct the other state, the magnetic field is removed. The |+1z〉, |−1z〉 states then naturally become coupled to each other by interactions present in a low-symmetry host, like electric field or strain. This gives |+〉 and |−〉 states analogous to those mentioned above and completes the DPS. The challenge is to construct a second DPS that is sufficiently different from the first such that it results in a multiplicative improvement in coherence time. The authors use a strong alternating current field to resonantly drive a transition between two of the above DPS states. Such a driven two-level system is often modeled by a Bloch vector, which is a three-dimensional representation of the relative magnitude and phase (see the figure). The driving field causes rotation about the axis shown at a rate equal to the Rabi frequency, Ω. Decoherence corresponds to rotation about an orthogonal axis, and so it is suppressed by a large Ω. This process is analogous to a mechanical gyro, where attempts to tilt its rotation axis appear to be resisted by an invisible force. Special points on the Bloch sphere traced out by the Bloch vector correspond to normal modes called “dressed states” (|g, ω〉〉 + |e〉) and (|g, ω〉 − |e〉) that are used to construct the second DPS, denoted as |+1〉 and |−1〉 by the authors. Dressed states are not normally used for DPS states because they are sensitive to the strength of the driving field. However, Miao et al. developed a clever scheme to overcome this problem by driving a direct transition between the dressed states ([ 3 ][4]) to probe and correct the driving field strength. This resulted in a further increase in coherence time of two orders of magnitude. The authors' key accomplishment is that DPSs can be combined to achieve multiplicative decoherence suppression. These demonstrations stimulate the imagination in a way that can lead to breakthroughs. For example, could decoherence be further suppressed if the dressed states themselves were strongly driven by a second resonant field (i.e., “nested” dressed states), possibly with a different Rabi frequency? Classically, this process is analogous to achieving a higher level of noise suppression by independently modulating a signal at two frequencies and demodulating it at the sum or difference frequency. An analogous sum-frequency heterodyne experiment showed quantum-like classical entanglement ([ 4 ][5]), so there may be additional information-processing advantages as a side benefit to the decoherence protection. This leads to the question of whether other schemes might generate a DPS with multiplicative benefit. In Miao et al. , the probing transition between dressed states was electric-dipole, whereas the original resonant drive was magnetic dipole. The possibility of using this to achieve further decoherence suppression should be considered. In Miao et al. , the authors still have one unused DPS state. By upgrading the experimental geometry to access this state, it could be possible to simultaneously implement yet another DPS. This might, for example, be accomplished using a Raman-like scheme. By demonstrating a multiplicative advantage combining two DPSs, Miao et al. open the door to new DPS approaches that may substantially enhance the performance of future quantum information systems. 1. [↵][6]1. K. C. Miao et al ., Science 369, 1493 (2020). [OpenUrl][7][Abstract/FREE Full Text][8] 2. [↵][9]1. P. R. Hemmer, 2. M. G. Prentiss , J. Opt. Soc. Am. B 5, 1613 (1988). [OpenUrl][10] 3. [↵][11]1. A. S. M. Windsor, 2. C. Wei, 3. S. A. Holmstrom, 4. J. P. D. Martin, 5. N. B. Manson , Phys. Rev. Lett. 80, 3045 (1998). [OpenUrl][12] 4. [↵][13]1. K. F. Lee, 2. J. E. Thomas , Phys. Rev. A 69, 052311 (2004). 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