Completely scrambled memory for quantum superposition

Although constructing a quantum computation device with multiple qubits is arguably a difficult task, several seconds of coherence time with tens of thousands of quantum particles has been demonstrated with a trapped atomic ensemble. As a practical application, a security-enhanced quantum state memory using atoms has been demonstrated. It was shown that the quantum superposition preserved in an atomic ensemble was scrambled and faithfully descrambled; however, the scrambled phase ambiguity remained at 50%. To overcome this problem, we propose and demonstrate a scheme that achieves 100% phase ambiguity without introducing an extra Ramsey interferometer. Moreover, this scheme can be used as a direct application to keep the choice between two values secret without falsification.

Ramsey interferometer). In addition, we call the first and second pulses of the WRI (SRI) write and read (scramble and retrieve) pulses, respectively. In the demonstration described in the previous report, we employed two Ramsey interferometers with two π/2 pulses, and the scrambled phase ambiguity remained at 50%. To understand the mechanism of the limited phase ambiguity, we track the Bloch vector of an atomic ensemble through the memory scrambling process. First, a superposition state is recorded with a write pulse irradiating on an atomic ensemble that has been initialized in the ground state, and the Bloch vector rotates π/2 radians around the x-axis (Fig. 1a). After irradiating the write pulse, the atomic ensemble experiences a time evolution during a time interval T 1 , and the Bloch vector rotates around the z-axis (Fig. 1b). Next, a scramble pulse is irradiated on the atomic cloud to encrypt the recorded superposition state. By irradiating the scramble pulse, the Bloch vector rotates around the X-axis, forming an angle φ s with the x-axis on the xy-plane. Figure 1c shows the rotation of the Bloch vector in the scrambling process when the phase evolution during T 1 equals π/2 radians and φ s equals 2π/3 radians. Depending on the random phase difference between the WRI and SRI, φ s stochastically takes a value between 0 and 2π radians. Therefore, the Bloch vector is stochastically transferred to a point on a curve on the Bloch sphere. The dashed curve in Fig. 1d shows the calculated stochastic distribution of the Bloch vector (SDBV) after irradiating a scramble pulse.
We prepare for the state distribution measurement by irradiating a read pulse after the SDBV has rotated around the z-axis in a time interval T (Fig. 1e). By irradiating the read pulse, the SDBV rotates π/2 radians around the x-axis (Fig. 1f). A projection of the SDBV onto the xz-plane is plotted in Fig. 1g, which represents a circle 0.5 in diameter. This is an important outcome in terms of understanding the ambiguity of the excitation probability of an atomic cloud. The SDBV along the z-axis is the reason for the range of shot to shot fluctuations in the excitation probability when measuring the scrambled state. Depending on the time interval T, the projected SDBV circle rotates around the origin of the xz-plane, but the size of the distribution along the z-axis is equal to the diameter of the SDBV circle regardless of the T value. Therefore, we can understand the 50% ambiguity with respect to the excitation probability of the Ramsey flop regardless of the T value (Fig. 1h).
To increase the ambiguity of the excitation probability of the scrambled state, the SDBV range along the z-axis should be extended. This can be accomplished by changing the area of the scramble pulse. Figure 1i shows SDBVs calculated with different scramble pulse areas. The SDBVs projected onto the xz-plane after a π/2 radians phase evolution and irradiating the read pulse are plotted in Fig. 1j. The SDBV shows a clear dependence on the scramble pulse area, and it becomes the equator of the Bloch sphere when the scramble pulse area equals π. Therefore, we can expect 100% ambiguity of the excitation probability of the Ramsey flop with π pulse scrambling as plotted in Fig. 1k.
Experimental demonstration. The demonstration was performed with 87 Rb Bose-Einstein condensate trapped below a persistent supercurrent atom chip 10,11 . The experimental setup was the same as that in our previous work 9 , but we improved the coherence time by suppressing radio frequency noise. We used the spin states = − F m , 1 , 1 F and |2, 1〉 of 5 2 S 1/2 for the ground and excited states of the Ramsey interferometer, and we denote them as |g〉 and |e〉, respectively. These two states are coupled through a two-photon stimulated Raman transition realized with a pair of D 1 -line lasers (Fig. 2a) or a pair consisting of microwave (MW) and radio frequency (RF) fields (Fig. 2b). The pair of D 1 -line lasers was employed as a coupling field for the WRI, and the pair consisting of MW and RF fields was similarly employed for the SRI. The roles of the WRI and SRI can be swapped without making any practical difference. The coherence time of both Ramsey interferometers was of the order of seconds, while the pulse duration time for a π/2 pulse was of sub-millisecond order. The retrieve pulse was irradiated when the SRI experienced a (2m + 1)π radians phase evolution, where m is an integer. The parameters used in the experiment are summarized in Table 1. To demonstrate complete scrambling, we employed π pulse scrambling and measured three Ramsey flops with different pulse application sequences, i.e., normal, scrambled, and retrieved Ramsey flops. The pulse sequences are shown in Fig. 2c-e. The Ramsey flops calculated with each condition are plotted in Fig. 2f-h, respectively. In these calculations, we neglected the pulse duration time compared to the time interval T. For both the scrambled (Fig. 2g) and retrieved (Fig. 2h). Ramsey flops, several curves with a phase difference between the WRI and SRI (φ π ∈ [0, 2 ] S ) are plotted in different colors, however, there is no difference between the curves for the retrieved Ramsey flop in Fig. 2h. With these calculations we can estimate the range of the shot to shot fluctuations of the excitation probability that we can expect to observe in the Ramsey flop measurement.
The Ramsey flops obtained experimentally for each condition are plotted in Fig. 2i-k, respectively. We performed five measurements and averaged the results to obtain a single data point with a standard deviation. Before scrambling, a clear Ramsey flop was achieved as plotted in Fig. 2i. When the state was scrambled, the measured standard deviation of the excitation probability was sufficiently large regardless of the T value, and no information remained about the Ramsey flop as shown in Fig. 2j; i.e., the Ramsey flop was completely scrambled. The scrambled Ramsey flop was faithfully descrambled by irradiating the retrieve pulse as shown in Fig. 2k. The slight increase in the standard deviation was probably caused by the inaccuracy of the scramble and retrieve pulses.

Discussion
The crux of the phase ambiguity achieved with this method is the SDBV after irradiating the scramble pulse. 100% phase ambiguity is achieved when the SDBV becomes the equator of the Bloch sphere. This can be achieved with π pulse scrambling when an equal superposition of the ground and excited states is recorded. However, π-pulse scrambling does not always result in the maximum phase ambiguity when the other superposition state is recorded; e.g. when an excited state |e〉 is recorded, π/2 pulse rather than π pulse scrambling results in 100% phase ambiguity. When a general superposition is recorded, it is basically impossible to make the SDBV the equator of the Bloch sphere. We still have room to optimize the area of the scramble pulse depending on the recorded superposition state.  Although there remains a requirement for the state to be recorded, the demonstrated 100% phase ambiguity promises useful applications in security enhancement. As an example, we propose a scheme for recording a choice between two values, e.g., yes or no, and keeping it secret without falsification (Fig. 3).
We assume that the choice "yes" or "no" is mapped to a superposition state |+〉 = | 〉 + | 〉 g e ( ) 1 2 or |−〉 = | 〉 − | 〉 g e ( ) 1 2 by employing a write pulse with a π/2 or 3π/2 pulse, respectively. In the π pulse scrambling process, the Bloch vector of |+〉 is stochastically transferred to a point on the equator of the Bloch sphere and that of |−〉 is stochastically transferred to a point on the same equator. At this point, the initial choice of "yes" or "no" cannot be identified from the measurement of the Bloch vector, and falsification is practically impossible.
The preparation for the state readout is performed as follows. The memory is descrambled by irradiating a retrieve pulse when the SRI phase evolution is equal to (2l + 1)π radians, where l is an integer. Then a read pulse, which has a π/2 pulse area irrespective of the choice at the writing process, is irradiated when the WRI phase evolution is equal to 2kπ radians, where k is an integer. Then we can understand that the choice written on the memory was "yes" or "no" by obtaining 100% or 0% excitation, respectively.
In summary, we investigated the phase ambiguity of a security-enhanced quantum memory and made it clear that the ambiguity is dependent on the area of the scramble pulse. Moreover, we proposed and experimentally demonstrated a scheme designed to achieve 100% phase ambiguity, which is useful for recording a choice between two values without falsification. This technique can be applied to a wide range of two-level quantum systems that can be controlled with Ramsey interferometric methods.

Methods
Atomic ensemble memory for quantum superposition. An atomic ensemble is initially prepared in the ground state, and a quantum state is recorded by irradiating a write pulse. The recorded state is verified by irradiating a read pulse after a specific time interval. As long as the write and read pulses have the same pulse area, and the phase evolution during the time interval is equal to (2n + 1)π radians, where n is an integer, faithful recording can be confirmed by measuring the distribution in the initial state.