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A logical phi-bit was generated by replicating the experimental arrangement described in10, the setup consisted of three aluminum rods, approximately 60cm long, arranged in a linear array with a lateral gap of 2mm filled with epoxy. Transducers were used to drive and detect the acoustic field at the ends of the rods. The waveguides are labeled 1, 2, and 3, where waveguide 2 is sandwiched between waveguides 1 and 3. Two driving transducers located on waveguide 1 and waveguide 2 were excited with sinusoidal signals at the primary frequencies ({f}_{1}) and ({f}_{2}), respectively. The third waveguide was not driven. The driving transducers were connected to different function generators and amplifiers and were driven with the same peak-to-peak voltage of 80V. Three detecting transducers were connected to an oscilloscope to measure the voltage (displacement field) generated at the other ends of the waveguides. The detected temporal signals are fast Fourier transformed (FFT) to produce a power spectrum. In the spectral domain, we observed strong peaks associated with the primary frequencies as well as weaker peaks associated with nonlinear vibrational modes supported by the inherently nonlinear system. A logical phi-bit,i, is defined as a nonlinear mode in which frequency can be written as a linear combination of the primary frequencies:

$${F}^{(i)}=p{f}_{1}+q{f}_{2}$$

(1)

where p and q are integer coefficients.

A single logical phi-bit i is characterized by its frequency but also the corresponding relative phases between the acoustic waveguides. The relative phases between waveguides 1 and 2 and between waveguides 1 and 3 at the phi-bit frequency, ({F}^{(i)}), are written as the phase difference ({varphi }_{12}^{(i)}={varphi }_{2}^{(i)}-{varphi }_{1}^{(i)}) and ({varphi }_{13}^{(i)}={varphi }_{3}^{(i)}-{varphi }_{1}^{left(iright)}), respectively. Here, the phase of the nonlinear mode at the end of waveguide 1 is used as a reference.

The two-phase differences serve as degrees of freedom for representing a logical phi-bit as a two-level system that may be characterized as a 21 vector with complex components. Within this representation, a logical phi-bit is analogous to a qubit. Notably, logical phi-bits co-exist in the same physical space, that is, the space of the physical system. Since logical phi-bits are nonlinear vibrational modes generated from the same driving frequencies, their associated phase differences are correlated to each other. Subsequently, the two-level system representations of multiple phi-bits are also strongly correlated and can be simultaneously manipulated by tuning the driving conditions of the system. Effectively, the state of N logical phi-bits may be represented as a 2N1 complex vector spanning an exponentially scaling space (i.e., a Hilbert space). The representation of the N phi-bit state depends on the choice of the basis of the Hilbert space. The strong nonlinear coupling between the phi-bits enables manipulation of the component of this large complex vector in a parallel manner by changing the driving conditions. Furthermore, the components of the large multi-phi-bit state vector can be used to encode information. Parallel manipulation of these components can be used to encrypt the encoded information. The inverse manipulation can also be used to decrypt the encrypted information in an efficient manner.

We illustrate this encryption method in the case of a N=5 phi-bit system. For this, the driving frequencies are taken as ({f}_{1}=62,text{kHz}) and ({f}_{2}=66,text{kHz}). We tune ({f}_{1}) by increments of (Delta nu left(nright)=left(n-1right)*50,text{Hz}) with (nin [text{1,81}]), thus spanning a range of frequency of 0 to 4kHz.

We are considering phi-bits 1 through 5 with corresponding values p and q of (5, 4), (4, 3), (1, 2), (1, 1), and (4, 1), respectively. The phi-bits are selected at random which indicates randomness of the p and q values. In Fig.1a, we report experimentally measured ({varphi }_{12}^{left(iright)}(Delta nu )) and ({varphi }_{13}^{left(iright)}(Delta nu )) for the five selected phi-bits. We note that the phase differences exhibit two different behaviors. The first behavior is associated with a common monotonous variation of the phase differences. The second behavior which is only occurring in the case of phi-bits 4 and 5 takes the form of sharp 180 ((pi)) jumps. The possible physical origin of these two behaviors has been discussed at length in references14. Here we focus on the first behavior by correcting the phase differences by the (pi) jumps to obtain continuous variations for all phi-bits. Moreover, in the case of the first behavior we have shown in15 that the phase differences of the phi-bits relate to the phase differences at the primary frequencies in a linear manner. Indeed, we can write.

$${varphi }_{12}^{left(iright)}left({f}_{1}+Delta nu ,{f}_{2}right)=p{varphi }_{12}left({f}_{1}+Delta nu right)+q{varphi }_{12}left({f}_{2}right)$$

(2a)

$${varphi }_{13}^{left(iright)}left({f}_{1}+Delta nu ,{f}_{2}right)=p{varphi }_{13}left({f}_{1}+Delta nu right)+q{varphi }_{13}left({f}_{2}right)$$

(2b)

where p and q are the same coefficients as in Eq.(1).

(a) Experimentally measured ({varphi }_{12}^{left(iright)}(Delta nu )) and ({varphi }_{13}^{left(iright)}(Delta nu )) for the five phi-bits i=1,5. (b) Rescaled and corrected ({varphi }_{12}^{left(iright)}(Delta nu )) and ({varphi }_{13}^{left(iright)}(Delta nu )). See text for details.

In Fig.1b, we rescale the corrected phase differences of the phi-bits by the factor p only, since the ({varphi }_{12}left({f}_{2}right)) and ({varphi }_{13}left({f}_{2}right)) remain constant. The rescaled and corrected phase differences of the five phi-bits behave simultaneously in the same manner. The vertical offset is simply due to the constant ({qvarphi }_{12}left({f}_{2}right)) and ({qvarphi }_{13}left({f}_{2}right)).

Each phi-bit, i, is represented in a two-dimensional Hilbert space, taking the form reminiscent of a Bloch sphere representation:

$$V^{left( i right)} = left( {begin{array}{*{20}c} {sin left( {beta^{left( i right)} } right)} \ {e^{{igamma^{left( i right)} }} cos left( {beta^{left( i right)} } right)} \ end{array} } right).$$

(3)

In that representation, for the sake of simplicity, we assume (gamma^{left( i right)} = 0), and

$$beta^{left( i right)} left( {Delta nu left( n right)} right) = Kleft( {varphi_{12}^{left( i right)} left( {Delta nu left( n right)} right) - varphi_{12}^{left( i right)} left( {Delta nu left( {n = 1} right)} right)} right) + frac{3pi }{4}$$

(4)

To address the offset in the rescaled and corrected phases of Fig.1b we use ({varphi }_{12}^{(i)}left(Delta nu left(n=1right)right)) as origins as well as the reference for encoding information. In Eq.(4), K is a scaling factor to amplify the range spanned by ({varphi }_{12}^{(i)}). For the sake of simplicity, we have made ({beta }^{left(iright)}) a function of ({varphi }_{12}^{(i)}) only, other forms could be defined as function of ({varphi }_{12}^{(i)}) and ({varphi }_{13}^{(i)}) offering an extra degree of freedom in the manipulation of the phi-bit state vector. At the driving frequency ({f}_{1}=62,text{kHz} ,,text{and},, n=1), ({beta }^{left(iright)}=frac{3pi }{4}) and

$$V^{left( i right)} = frac{1}{sqrt 2 }left( {begin{array}{*{20}c} 1 \ { - 1} \ end{array} } right).$$

(5)

Let us consider a system made of five phi-bits, we can define the state of the system as the tensor product of the phi-bits:

$$V = V^{left( 1 right)} otimes V^{left( 2 right)} otimes V^{left( 3 right)} otimes V^{left( 4 right)} otimes V^{left( 5 right)} .$$

(6)

This tensor product evaluated at the reference driving frequency takes the form:

$$V^{T} = frac{1}{{2^{5/2} }}left[ {1,, - 1,,- 1,,1,,- 1,,1,,1,,- 1,,- 1,,1,,1,,- 1,,1,,- 1,,- 1,,1,,- 1,,1,,1,,- 1,,1,,- 1,,- 1,,1,,1,,- 1,,- 1,,1,,- 1,,1,,1,,- 1} right].$$

(7)

We now consider a new representation of the five phi-bit state vector,

$$widehat{V}= frac{left(1+ {V}^{{prime}T}right)}{2}= left[1,,0,,0,,1,,0,,1,,1,,0,,0,,1,,1,,0,,1,,0,,0,,1,,0,,1,,1,,0,,1,,0,,0,,1,,1,,0,,0,,1,,0,,1,,1 0right]$$

(8)

where ({V}^{{prime}T}={{2}^{5/2}V}^{T}).

This new representation of the five phi-bit state corresponds to a change of basis of the Hilbert space. While the vector V is separable, the vector (widehat{V}) is not separable into a tensor product of single phi-bit states. In quantum information science non-separability of multiple qubit states is necessary for establishing quantum correlations between the qubits. In the case of phi-bit based information science, the components of a multi-phi-bit state vector such as Eq.(7) are correlated via the nonlinearity of the physical system. The string of correlated ones and zeros in (widehat{V}) can therefore be employed to encode a message as well as encrypt that message by applying a unitary transformation on (widehat{V}). Here we illustrate this encryption process in the case of a short message encoded using 5 phi-bits. In general, the state vector (widehat{V}) of a N phi-bit system is a 2Nx1 vector with 1 and 0 elements providing sufficient bits for encoding very long messages when N is large. For instance, when N=50 the number of bits available for encoding a message is on the order of 1015. The encryption of a message encoded on this large number of bits would only require a change in the driving condition of the array of waveguides to achieve the encrypting unitary transformation. Such an operation takes approximately a millisecond on the physical system used here, since the speed of sound in the constitutive materials is on the order of a few thousand meters per second.

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Information encoding and encryption in acoustic analogues of qubits | Scientific Reports - Nature.com