**4.1.3 Random phase initial condition**

In this simulation, the initial wave function features a constant amplitude and random phase *ψ*(**r**) = *h eiθ*(**r**). Bicubic interpolation is adopted to produce random phase which satisfies periodic boundary condition, c.f. Keys (1981). Under this interpolation, a 2D function 14 Will-be-set-by-IN-TECH

Fig. 8. Evolution of internal energy for winding number 2. Around *t* ∼ 21000 and *t* ∼ 41900, internal energy reaches peaks. Comparing with winding number 1 case, more fluctuation is observed and the peaks are much narrower. This is caused by the high density of number of

(a) |*ψ*|,t=0 (b) t=10500 (c) t=21000 (d) t=31300 (e) t=41900

(f) *ωq*,t=0 (g) t=10500 (h) t=21000 (i) =31300 (j) t=41900

(k) *θ*,t=0 (l) t=10500 (m) t=21000 (n) t=31300 (o) t=41900

Fig. 9. Poincaré recurrence for winding number*n* =2. At the semi-Poincaré time, the wave function approximates the initial condition with shift in the origins. At *t* = 41900, the wave

of vortices increases in the system, the probability for Kelvin-Helmholtz triggered vortex

In this simulation, the initial wave function features a constant amplitude and random phase *ψ*(**r**) = *h eiθ*(**r**). Bicubic interpolation is adopted to produce random phase which satisfies periodic boundary condition, c.f. Keys (1981). Under this interpolation, a 2D function

function bears a similar structure as at *t* = 0, but with more noise.

generation increases. Fig.10 demonstrates such a process.

**4.1.3 Random phase initial condition**

vortices and the frequent annihilation and creation of counter-rotating vortex pairs.

(a) *θ*, t=10500 (b) *θ*, t=10600

Fig. 10. Vortices generation via the Kelvin-Helmholtz instability. The region depicted in these plots is a blow-up of [−256, −128] × [−128, 128]. (a) phase *θ* at t=10500; (b) *θ* at t=10600. At *t* = 10600, a new pair of counter-rotating vortices (pointed out by the black arrows) are generated between neighboring counter-rotating vortices.

defined on a unit square is approximated by polynomials: *p*(*x*, *y*) = 3 ∑ *i*=0 3 ∑ *j*=0 *ai*,*jx<sup>i</sup> yj* . The

coefficients *ai*,*<sup>j</sup>* are determined by the enforced continuity at the corners. Since there are 16 unknown coefficients, 16 equations are needed to determine these *ai*,*j*. Usually one can enforce continuity for *p*(*x*, *y*), *∂<sup>x</sup> p*(*x*, *y*), *∂<sup>y</sup> p*(*x*, *y*), *∂x*,*<sup>y</sup> p*(*x*, *y*). To generate random phase in the domain *L*2, the following procedure is followed:


For a grid of 5122 we consider a randomness level *m* = 8. The random phase wave function is dynamically unstable. Sound waves are immediately emitted and create quantum vortices. Typically these vortices will decay away and the GP system tends to a thermal equilibrium, as demonstrated by Numasato et al. (2010). However, if the initial condition is chosen such that energy ratio *γ* � 1, a Poincaré recurrence emerges. In our simulation, *γ* = 0.00287 and number of iteration is 100000. From the energy evolution plot, Fig.11, the Poincaré recurrence time can be clearly identified by the abrupt energy exchanges (i.e., the spikes).

The first spike in Fig.11 appears around *t* = 21000, with the second at *t* = 41900. These are just the semi-Poincaré and Poincaré times. One thus expects the phase distribution of the wave function at *t* = 0, *t* = 10500, *t* = 21000 and *t* = 41900 to illustrate the recurrence, c.f. Fig.12. What is remarkable is that the randomly distributed vortices suddenly disappear from the system at *TP*/2 and *TP*. At *t* = 41900, the phase distribution is very close to the initial state despite a constant shift in the central region. In Fig.13 we have plotted the density-weighted vorticity at various times: there are no vortices at *t* = 0 or at *TP*/2, (e), although a considerable amount of sound waves.

As energy ratio *γ* increases, the strength of the Poincaré recurrence is weakened by noise. Fig.14 demonstrate how the Poincaré recurrence is lost as *γ* increases. When *γ* = 0.0567, one still can observe the depletion of vortices from the system at *TP*/2 = 21000, however, at *TP* = 41900, the initial condition, which is vortex free, can not be reproduced. For *γ* = 0.133,

(a) *n* = 1, *t* = 0 (b) *t* = 200

Unitary Qubit Lattice Gas Representation of 2D and 3D Quantum Turbulence 255

(c) *t* = 20200 (d) *t* = 20800

(e) *t* = 21000 = *TP*/2 (f) *t* = 21100

independent of whether 2D or 3D.

Fig. 13. The evolution of the vorticity |*ω*(**x**, **t**))|. At a quantum vortex *ω*(**x**) ∼ **ffi**(**x** − **xi**). At *t* = 0, the initial conditions for the wave function *ϕ* = √*ρ exp*(*iθ*) are *ρ* = *const* and *θ* random. Thus there are no quantum vortices at *t* = 0. Very rapidly vortices are born. The vortices are annihilated at *t* = *TP*/2. Grid 5122. *TP* scales as *L*2, diffusion ordering,

Fig. 11. Energy evolution of the GP system with random phase initial condition. (a): evolution of energies (*EK*, *EQ*, *ET*). Internal energy is negligible compared to the kinetic energy *EK*. (b) evolution of incompressible (*red*), *EIC*, and compressible kinetic energy (*blue*), *EC*.

Fig. 12. Poincaré recurrence with random phase initial condition. At *t* = 10500, many randomly distributed vortices can be identified via the branch cuts. At *t* = 21000 and *t* = 41900, no vortices exist in the system since there are no branch cuts. There is an induced phase shift seen in the color scheme of the phase plots at *t* = 0 and *t* = 41900, but the geometric patterns are the same.

no trace of a short Poincaré recurrence can now be found.

It needs to be pointed out that in our simulations, the Poincaré recurrence is characterized by abrupt energy exchange *EK* and *EQ* as well as among the compressible and incompressible components of the kinetic energy, *EC* and *EIC*. Therefore such phenomena can not be analyzed via standard turbulence theories invoking things like inverse cascades....

#### **4.2 Energy cascade in 2D QT**

For the simulations with vortices initially embedded in an Guassian BEC background, a *k*−<sup>3</sup> power law is found ubiquitously in the compressible, incompressible and quantum energy spectra whenever vortices are present in the system. Fig.15 describes the time evolution of the incompressible energy spectrum *<sup>k</sup>sIC* . The linear regression fit is over *<sup>k</sup>* <sup>∈</sup> [50, 100]. Simulation grid 5122.

In the time interval around *t* ∼ 24500 for winding number *n* = 1 embedded vortices, we examine the sudden drop in the spectral exponent *sIC*. In Fig.16, the linear regression fit for the incompressible kinetic energy spectrum is made over the wave number interval *k* ∈ [50, 100]. At *t* = 24400 and *t* = 24600, vortices are present in the system and the spectrum exponent *sIC* ∼ −3. At *t* = 24500, when all the vortices are depleted, the incompressible kinetic energy spectral exponent decreases to *sIC* = −5.828. This could well indicate that the existence in 16 Will-be-set-by-IN-TECH

(a) (b)

(a) *θ*(**r**) at t=0 (b) t=10500 (c) t=21000 (d) t=41900

It needs to be pointed out that in our simulations, the Poincaré recurrence is characterized by abrupt energy exchange *EK* and *EQ* as well as among the compressible and incompressible components of the kinetic energy, *EC* and *EIC*. Therefore such phenomena can not be analyzed

For the simulations with vortices initially embedded in an Guassian BEC background, a *k*−<sup>3</sup> power law is found ubiquitously in the compressible, incompressible and quantum energy spectra whenever vortices are present in the system. Fig.15 describes the time evolution of the incompressible energy spectrum *<sup>k</sup>sIC* . The linear regression fit is over *<sup>k</sup>* <sup>∈</sup> [50, 100]. Simulation

In the time interval around *t* ∼ 24500 for winding number *n* = 1 embedded vortices, we examine the sudden drop in the spectral exponent *sIC*. In Fig.16, the linear regression fit for the incompressible kinetic energy spectrum is made over the wave number interval *k* ∈ [50, 100]. At *t* = 24400 and *t* = 24600, vortices are present in the system and the spectrum exponent *sIC* ∼ −3. At *t* = 24500, when all the vortices are depleted, the incompressible kinetic energy spectral exponent decreases to *sIC* = −5.828. This could well indicate that the existence in

Fig. 12. Poincaré recurrence with random phase initial condition. At *t* = 10500, many randomly distributed vortices can be identified via the branch cuts. At *t* = 21000 and *t* = 41900, no vortices exist in the system since there are no branch cuts. There is an induced phase shift seen in the color scheme of the phase plots at *t* = 0 and *t* = 41900, but the

Fig. 11. Energy evolution of the GP system with random phase initial condition. (a): evolution of energies (*EK*, *EQ*, *ET*). Internal energy is negligible compared to the kinetic energy *EK*. (b) evolution of incompressible (*red*), *EIC*, and compressible kinetic energy (*blue*),

*EC*.

geometric patterns are the same.

**4.2 Energy cascade in 2D QT**

grid 5122.

no trace of a short Poincaré recurrence can now be found.

via standard turbulence theories invoking things like inverse cascades....

Fig. 13. The evolution of the vorticity |*ω*(**x**, **t**))|. At a quantum vortex *ω*(**x**) ∼ **ffi**(**x** − **xi**). At *t* = 0, the initial conditions for the wave function *ϕ* = √*ρ exp*(*iθ*) are *ρ* = *const* and *θ* random. Thus there are no quantum vortices at *t* = 0. Very rapidly vortices are born. The vortices are annihilated at *t* = *TP*/2. Grid 5122. *TP* scales as *L*2, diffusion ordering, independent of whether 2D or 3D.

(a) t=24400, *θ*(**r**) (b) t=24400, *sIC* = −3.005

Unitary Qubit Lattice Gas Representation of 2D and 3D Quantum Turbulence 257

(c) t=24500 (d) t=24500, *sIC* = −5.828

(e) t=24600 (f) t=24600, *sIC* = −3.357

Fig. 16. Phase plot and the incompressible energy spectrum round *t* ∼ 24500. At *t* = 24400 and *t* = 24600 there are branch cuts and vortices in the BEC with spectral exponent *sIC* ∼ −3. But at *t* = 24500, no branch cuts exist in the phase plot, indicative of no vortices in the system. The incompressible spectrum exhibits a discontinuity in the high-k region with a

the incompressible kinetic energy spectrum of *k*−<sup>3</sup> power law in the high-k region could be the by-product of the spectrum of a topological singularity - at least in 2D QT. It should be remembered that in 2D QT there can be no quantum Kelvin wave cascade since the quantum vortex core is just a point singularity, unlike 3D QT where the vortex core is a line or loop. To examine the spectral exponents of the compressible and incompressible kinetic energies in more detail we now discuss some high grid resolution runs: (a) grid 327682 with random phase initial conditions, and (b) grid 8192<sup>2</sup> with winding number *n* = 6 linear vortices in a

(a) For the 32768<sup>2</sup> run, we choose initial conditions similar to Numasato et al. (2010), with

initially there are no vortices in the system because of the random phase initial condition for

<sup>2</sup>)−1/2 = 33.33 - even though, of course,

strong decrease in the exponent, *sIC* ∼ −5.8.

parameters yielding a 'coherence length' *ξ* = (*a g* |*ψ*0|

uniform BEC background.

the wave function.

Fig. 14. Loss of Poincaré recurrence. (a) evolution of *ET*, *EK*, *EQ* and *EI*, *γ* = 0.0567; (b) evolution of *ET*, *EK*, *EQ* and *EI*, *γ* = 0.133; (c) evolution of enstrophy *Zq*, *γ* = 0.0567; (d) evolution of enstrophy *Zq*, *γ* = 0.133. Depletion of vortices can be identified from the sharp decrease of enstrophy.

Fig. 15. Time evolution of the incompressible kinetic energy spectrum *sIC*. The red horizontal line indicates the *k*−<sup>3</sup> power law. For winding number *n* = 1, there are spikes in the slope with *sIC* >> 3 in many instances. While for winding number *n* = 2 the variation in *sIC* is greatly reduced.

18 Will-be-set-by-IN-TECH

(a) (b)

(c) (d)

(a) winding number *n* = 1, *sIC*(*t*) (b) winding number *n* = 2, *sIC*(*t*)

Fig. 15. Time evolution of the incompressible kinetic energy spectrum *sIC*. The red horizontal line indicates the *k*−<sup>3</sup> power law. For winding number *n* = 1, there are spikes in the slope with *sIC* >> 3 in many instances. While for winding number *n* = 2 the variation in

Fig. 14. Loss of Poincaré recurrence. (a) evolution of *ET*, *EK*, *EQ* and *EI*, *γ* = 0.0567; (b) evolution of *ET*, *EK*, *EQ* and *EI*, *γ* = 0.133; (c) evolution of enstrophy *Zq*, *γ* = 0.0567; (d) evolution of enstrophy *Zq*, *γ* = 0.133. Depletion of vortices can be identified from the sharp

decrease of enstrophy.

*sIC* is greatly reduced.

Fig. 16. Phase plot and the incompressible energy spectrum round *t* ∼ 24500. At *t* = 24400 and *t* = 24600 there are branch cuts and vortices in the BEC with spectral exponent *sIC* ∼ −3. But at *t* = 24500, no branch cuts exist in the phase plot, indicative of no vortices in the system. The incompressible spectrum exhibits a discontinuity in the high-k region with a strong decrease in the exponent, *sIC* ∼ −5.8.

the incompressible kinetic energy spectrum of *k*−<sup>3</sup> power law in the high-k region could be the by-product of the spectrum of a topological singularity - at least in 2D QT. It should be remembered that in 2D QT there can be no quantum Kelvin wave cascade since the quantum vortex core is just a point singularity, unlike 3D QT where the vortex core is a line or loop. To examine the spectral exponents of the compressible and incompressible kinetic energies in more detail we now discuss some high grid resolution runs: (a) grid 327682 with random phase initial conditions, and (b) grid 8192<sup>2</sup> with winding number *n* = 6 linear vortices in a uniform BEC background.

(a) For the 32768<sup>2</sup> run, we choose initial conditions similar to Numasato et al. (2010), with parameters yielding a 'coherence length' *ξ* = (*a g* |*ψ*0| <sup>2</sup>)−1/2 = 33.33 - even though, of course, initially there are no vortices in the system because of the random phase initial condition for the wave function.

(a) (b)

Unitary Qubit Lattice Gas Representation of 2D and 3D Quantum Turbulence 259

(c) (d)

Fig. 18. (a): The incompressible spectrum *Einc*(*k*) (*red*) and compressible spectrum *Ecomp*(*k*) (*blue*) on a 327682 grid at *t* = 8000. The dashed vertical line indicates the location of *k<sup>ξ</sup>* , based on the qualitative notation of the coherence length *ξ*. The encircled dip in the compressible energy propagates towards the lower-k region, resembling a backward propagating pulse.

The major differences between 2D and 3D CT is in the behavior of the vorticity vector. In 2D, the vorticity is always perpendicular to the plane of motion while in 3D the vorticity vector can have arbitrary orientation. Here, in 3D we will employ variants of a set of linear vortices following the Pade approximant methods of Berloff Berloff (2004). For winding number *n* = 1 , using cylindrical polar coordinates (*r*, *φ*, *z*), a linear vortex that lies along the *z*-axis (and

11*a r*2(12 + *a r*2)

<sup>384</sup> <sup>+</sup> *a r*2(<sup>128</sup> <sup>+</sup> <sup>11</sup>*a r*2) <sup>=</sup> *<sup>g</sup>*−1/2 *<sup>ϕ</sup>*0(*r*)*<sup>e</sup>*

*<sup>i</sup>φ*, (45)

(b) *Incompressible* kinetic energy spectrum ain Region I (*k* 0.01*k<sup>ξ</sup>* ) and Region II (0.01*k<sup>ξ</sup> k* 0.1*k<sup>ξ</sup>* ). The spectral exponents are *sIC* = +2.34 (red line) and *sIC* = +0.65 (green line). (c) *Incompressible* energy spectrum in Region II and Region III (0.1*k<sup>ξ</sup> k k<sup>ξ</sup>* ). Spectral exponents are: *sIC* = 0.65 (green line); *sIC* = −4.17 (purple line). (d) *Incompressible* energy spectra in Region III and Region IV (*k<sup>ξ</sup> k*). Spectral exponents are: *sIC* = −4.17

(purple line) and *sIC* = −3.03 (black line).

centered at the origin) is given by

*ϕ*(*r*) = *g*−1/2*e*

*iφ* 

**5. 3D QT**

Fig. 17. (a) The time evolution of the kinetic, quantum, internal and total energies on a 32768<sup>2</sup> grid with random phase initial condition. (b) Evolution of theincompressible energy (red) and compressible energy (blue).

On this 327682 grid we performed a relatively short run to *tmax* = 15000. The evolution of the kinetic, quantum, internal and total energies are shown in Fig.17. Based on the time evolution of the energies, the dynamics can be broadly categorized into two stages: (I) generation of vortices and (II) decay of vortices. In stage (I), the compressible energy decreases rapidly while the incompressible energy increases rapidly. Thus a significant amount of energy in the sound waves is transformed into incompressible energy induced by the rotational motion of vortices. In stage (II), the randomly distributed vortices disappear. The energy of the vortices is transferred into sound waves through vortex-vortex annihilation. Note that in this stage, the only major energy exchange occurs between the incompressible and compressible energies while the quantum and internal energies remain almost constants. The spectra for incompressible and compressible energy at *t* = 8000 is given in Fig.18.

At large-k region (*k* > 3000), a *k*−<sup>3</sup> power law is present which can be interpreted as result of FFT of quantum vortices. It is interesting to notice that at *<sup>k</sup>* <sup>∼</sup> *<sup>k</sup><sup>ξ</sup>* , e.g. *<sup>k</sup>* <sup>∈</sup> [700, 1200], a *<sup>k</sup>*−<sup>4</sup> power law is observed. We sampled the incompressible energy spectra every 50 iteration steps between 6000 < *t* < 10000 within a wave number window *k* ∈ [800, 1200]. The time averaged slope �*sIC*� = −4.145 ± 0.066. This is in good agreement with the results obtained in Horng et al. (2009). This *k*−<sup>4</sup> power law can be interpreted as the result of dissipation of randomly distributed vortices, as suggested in Horng et al. (2009). However, in low-k region (region I and II in Fig.18(b) and Fig. 18(c)) where semi-classical Kolmogorov cascade is expected, we did not observe the *k*−5/3 power law. This could be attributed to the compressibility of quantum fluid.

Finally, we consider the case of 12 vortices of winding number *n* = 6 on an 81922 grid. The wave function is rescaled so that it is not a quasi-eigenstate of the GP Eq. (2). This will lead more quickly to turbulence. The *compressible* kinetic energy spectrum is shown in Fig.19(a) and clearly exhibits a triple cascade: a small-k region with spectral exponent *sC* = −1.83, an intermediate-k region with exponent *sC* = −8.1, and a large-k region with exponent *sC* = −2.85. We will comment on this triple cascade spectrum more when we discuss QT in 3D. In Fig.19(b) we plot the *incompressible* kinetic energy spectrum and notice the dual cascade region: for high-k we see the ubiquitous exponent *sIC* = −2.93 while for lower-k, the exponent becomes similar to the Saffman exponent, *sIC* = −4.0. The *k<sup>ξ</sup>* at which we have this dual spectrum meet is around the join of the steep intermediate range spectrum with the *k*−<sup>3</sup> spectral tail. We shall see this also in 3D QT.

Fig. 18. (a): The incompressible spectrum *Einc*(*k*) (*red*) and compressible spectrum *Ecomp*(*k*) (*blue*) on a 327682 grid at *t* = 8000. The dashed vertical line indicates the location of *k<sup>ξ</sup>* , based on the qualitative notation of the coherence length *ξ*. The encircled dip in the compressible energy propagates towards the lower-k region, resembling a backward propagating pulse. (b) *Incompressible* kinetic energy spectrum ain Region I (*k* 0.01*k<sup>ξ</sup>* ) and Region II (0.01*k<sup>ξ</sup> k* 0.1*k<sup>ξ</sup>* ). The spectral exponents are *sIC* = +2.34 (red line) and *sIC* = +0.65 (green line). (c) *Incompressible* energy spectrum in Region II and Region III (0.1*k<sup>ξ</sup> k k<sup>ξ</sup>* ). Spectral exponents are: *sIC* = 0.65 (green line); *sIC* = −4.17 (purple line). (d) *Incompressible* energy spectra in Region III and Region IV (*k<sup>ξ</sup> k*). Spectral exponents are: *sIC* = −4.17 (purple line) and *sIC* = −3.03 (black line).

#### **5. 3D QT**

20 Will-be-set-by-IN-TECH

(a) (b)

On this 327682 grid we performed a relatively short run to *tmax* = 15000. The evolution of the kinetic, quantum, internal and total energies are shown in Fig.17. Based on the time evolution of the energies, the dynamics can be broadly categorized into two stages: (I) generation of vortices and (II) decay of vortices. In stage (I), the compressible energy decreases rapidly while the incompressible energy increases rapidly. Thus a significant amount of energy in the sound waves is transformed into incompressible energy induced by the rotational motion of vortices. In stage (II), the randomly distributed vortices disappear. The energy of the vortices is transferred into sound waves through vortex-vortex annihilation. Note that in this stage, the only major energy exchange occurs between the incompressible and compressible energies while the quantum and internal energies remain almost constants. The spectra for

At large-k region (*k* > 3000), a *k*−<sup>3</sup> power law is present which can be interpreted as result of FFT of quantum vortices. It is interesting to notice that at *<sup>k</sup>* <sup>∼</sup> *<sup>k</sup><sup>ξ</sup>* , e.g. *<sup>k</sup>* <sup>∈</sup> [700, 1200], a *<sup>k</sup>*−<sup>4</sup> power law is observed. We sampled the incompressible energy spectra every 50 iteration steps between 6000 < *t* < 10000 within a wave number window *k* ∈ [800, 1200]. The time averaged slope �*sIC*� = −4.145 ± 0.066. This is in good agreement with the results obtained in Horng et al. (2009). This *k*−<sup>4</sup> power law can be interpreted as the result of dissipation of randomly distributed vortices, as suggested in Horng et al. (2009). However, in low-k region (region I and II in Fig.18(b) and Fig. 18(c)) where semi-classical Kolmogorov cascade is expected, we did not observe the *k*−5/3 power law. This could be attributed to the compressibility of

Finally, we consider the case of 12 vortices of winding number *n* = 6 on an 81922 grid. The wave function is rescaled so that it is not a quasi-eigenstate of the GP Eq. (2). This will lead more quickly to turbulence. The *compressible* kinetic energy spectrum is shown in Fig.19(a) and clearly exhibits a triple cascade: a small-k region with spectral exponent *sC* = −1.83, an intermediate-k region with exponent *sC* = −8.1, and a large-k region with exponent *sC* = −2.85. We will comment on this triple cascade spectrum more when we discuss QT in 3D. In Fig.19(b) we plot the *incompressible* kinetic energy spectrum and notice the dual cascade region: for high-k we see the ubiquitous exponent *sIC* = −2.93 while for lower-k, the exponent becomes similar to the Saffman exponent, *sIC* = −4.0. The *k<sup>ξ</sup>* at which we have this dual spectrum meet is around the join of the steep intermediate range spectrum with the

Fig. 17. (a) The time evolution of the kinetic, quantum, internal and total energies on a 32768<sup>2</sup> grid with random phase initial condition. (b) Evolution of theincompressible energy

incompressible and compressible energy at *t* = 8000 is given in Fig.18.

(red) and compressible energy (blue).

quantum fluid.

*k*−<sup>3</sup> spectral tail. We shall see this also in 3D QT.

The major differences between 2D and 3D CT is in the behavior of the vorticity vector. In 2D, the vorticity is always perpendicular to the plane of motion while in 3D the vorticity vector can have arbitrary orientation. Here, in 3D we will employ variants of a set of linear vortices following the Pade approximant methods of Berloff Berloff (2004). For winding number *n* = 1 , using cylindrical polar coordinates (*r*, *φ*, *z*), a linear vortex that lies along the *z*-axis (and centered at the origin) is given by

$$\varphi(r) = g^{-1/2} e^{i\phi} \sqrt{\frac{11a \, r^2 (12 + a \, r^2)}{384 + a \, r^2 (128 + 11a \, r^2)}} = g^{-1/2} \, \rho\_0(r) \, e^{i\phi} \,\tag{45}$$

(a) *n* = 5, *t* = 0

Unitary Qubit Lattice Gas Representation of 2D and 3D Quantum Turbulence 261

(b) *n* = 5, *t* = 3000

Fig. 20. The |*ϕ*| isosurfaces very near the vortex core singularity for winding number *n* = 5: (a) t = 0, (b) t = 3000 . Phase information is given on both the vortices and the boundary walls. The winding number *n* = 5 is evident from both the 5-fold periodicity around the each vortex as well as the 5 branch cuts emanating from each branch point on the boundary. By *t* = 3000, (b), the 5-fold degeneracy is removed with what seem like quantum Kelvin waves

on the *n* = 1 cores. Basic phase coding : *φ* = 0 in blue, *φ* = 2*π* in red. Grid 20483

Fig. 19. The (a) compressible kinetic energy spectrum and (b) incompressible kinetic energy spectrum for winding number *n* = 6 vortices in a uniform BEC gas. Grid 81922. Notice the triple cascade region for the compressible energy spectrum and the dual cascade spectrum for the incompressible energy.

with <sup>|</sup>*ϕ*| → 1/√*g*, and <sup>|</sup>*ϕ*0| → 1 as r <sup>→</sup> <sup>∞</sup>, and <sup>|</sup>*ϕ*| ∼ *<sup>r</sup> <sup>a</sup>*/*<sup>g</sup>* as r <sup>→</sup> 0. Eq.(45) is an asymptotic solution of the GP Eq.(2). For this isolated linear vortex, the coherence length, from Eq. (6), *<sup>ξ</sup>* <sup>∼</sup> 4/√*a*. This is one of the reasons for introducing the factor *<sup>a</sup>* into the GP Eq.(2): a small *a* permits resolution of the vortex core in the simulations. If one starts with a periodic set of well-spaced non-overlapping Pade asymptotic vortices (clearly, of course, this will be dependent on the choice of the parameter *a* and the grid size *L* of the lattice) an asymptotic solution of the GP Eq.(2) is simply a product of the shifted *ϕ*0's , weighted by *g*−1/2. The system will evolve slowly into turbulence because this initial state is very weakly unstable. For these wave functions the coherence length is initially fairly well defined. On the other hand, in most of our runs, we just simply rescaled the asymptotic basis vortex function *<sup>ϕ</sup>* <sup>→</sup> *<sup>g</sup>σϕ*, for some *<sup>σ</sup>*. Because the GP Eq.((2) is nonlinear, *<sup>g</sup>σϕ* is no longer an asymptotic solution and the definition of coherence length becomes fuzzy.

In Fig. 20 we show a somewhat complex initial vortex core situation. The initial wave function has winding number *n* = 5 and the positions of the initial line vortices are chosen so that there is considerable overlap of the wave functions around the center of the domain. In this plot we show not only the phase information on the vortex core isosurfaces but also the phase information on the boundary walls. On the vortex isosuface at *t* = 0 one can distinguish the 5 periods around the core. On the boundary walls, the intersection of the cores with the walls gives the location of the 4 branch-point like topological singularities. Emanating from each of these singularities are 5 branch cuts because of the chosen winding number *n* = 5. These branch cuts then join the branch points. Because the *n* = 5 state is energetically unfavorable, the initial state rapidly decays into 5× winding number *n* = 1 vortices, Fig. 20(b). It is tempting to identify the wave structures on the vortices as quantum Kelvin waves. Sound waves can also be identified on the boundaries. Near the center of the lattice, where there was initially considerable overlap of the vortices, many vortex loops have now formed.

#### **5.1 Poincare recurrence for certain classes of initial conditions**

As in 2D, a class of initial conditions will also be found for which the Poincare recurrence time is very short. The definitions of the incompressible and compressible kinetic energy, the quantum energy and the internal energy are immediate generalizations of those given in 22 Will-be-set-by-IN-TECH

(a) compressible kinetic energy, *t* = 230000 (b) incompressible kinetic energy, *t* = 230000

*<sup>a</sup>*/*<sup>g</sup>* as r <sup>→</sup> 0. Eq.(45) is an

Fig. 19. The (a) compressible kinetic energy spectrum and (b) incompressible kinetic energy spectrum for winding number *n* = 6 vortices in a uniform BEC gas. Grid 81922. Notice the triple cascade region for the compressible energy spectrum and the dual cascade spectrum

asymptotic solution of the GP Eq.(2). For this isolated linear vortex, the coherence length, from Eq. (6), *<sup>ξ</sup>* <sup>∼</sup> 4/√*a*. This is one of the reasons for introducing the factor *<sup>a</sup>* into the GP Eq.(2): a small *a* permits resolution of the vortex core in the simulations. If one starts with a periodic set of well-spaced non-overlapping Pade asymptotic vortices (clearly, of course, this will be dependent on the choice of the parameter *a* and the grid size *L* of the lattice) an asymptotic solution of the GP Eq.(2) is simply a product of the shifted *ϕ*0's , weighted by *g*−1/2. The system will evolve slowly into turbulence because this initial state is very weakly unstable. For these wave functions the coherence length is initially fairly well defined. On the other hand, in most of our runs, we just simply rescaled the asymptotic basis vortex function *<sup>ϕ</sup>* <sup>→</sup> *<sup>g</sup>σϕ*, for some *<sup>σ</sup>*. Because the GP Eq.((2) is nonlinear, *<sup>g</sup>σϕ* is no longer an asymptotic

In Fig. 20 we show a somewhat complex initial vortex core situation. The initial wave function has winding number *n* = 5 and the positions of the initial line vortices are chosen so that there is considerable overlap of the wave functions around the center of the domain. In this plot we show not only the phase information on the vortex core isosurfaces but also the phase information on the boundary walls. On the vortex isosuface at *t* = 0 one can distinguish the 5 periods around the core. On the boundary walls, the intersection of the cores with the walls gives the location of the 4 branch-point like topological singularities. Emanating from each of these singularities are 5 branch cuts because of the chosen winding number *n* = 5. These branch cuts then join the branch points. Because the *n* = 5 state is energetically unfavorable, the initial state rapidly decays into 5× winding number *n* = 1 vortices, Fig. 20(b). It is tempting to identify the wave structures on the vortices as quantum Kelvin waves. Sound waves can also be identified on the boundaries. Near the center of the lattice, where there was

initially considerable overlap of the vortices, many vortex loops have now formed.

As in 2D, a class of initial conditions will also be found for which the Poincare recurrence time is very short. The definitions of the incompressible and compressible kinetic energy, the quantum energy and the internal energy are immediate generalizations of those given in

for the incompressible energy.

with <sup>|</sup>*ϕ*| → 1/√*g*, and <sup>|</sup>*ϕ*0| → 1 as r <sup>→</sup> <sup>∞</sup>, and <sup>|</sup>*ϕ*| ∼ *<sup>r</sup>*

solution and the definition of coherence length becomes fuzzy.

**5.1 Poincare recurrence for certain classes of initial conditions**

(a) *n* = 5, *t* = 0

(b) *n* = 5, *t* = 3000

Fig. 20. The |*ϕ*| isosurfaces very near the vortex core singularity for winding number *n* = 5: (a) t = 0, (b) t = 3000 . Phase information is given on both the vortices and the boundary walls. The winding number *n* = 5 is evident from both the 5-fold periodicity around the each vortex as well as the 5 branch cuts emanating from each branch point on the boundary. By *t* = 3000, (b), the 5-fold degeneracy is removed with what seem like quantum Kelvin waves on the *n* = 1 cores. Basic phase coding : *φ* = 0 in blue, *φ* = 2*π* in red. Grid 20483

(a) *n* = 1, *t* = 0 , 48 linear

(e) *n* = 2, *t* = 0 , 48 linear

confluent degeneracy . Phase coding : *φ* = 0 *blue*, *φ* = 2*π red*. Grid 12003

vortices

(b) *n* = 1, *t* = 84000

(f) *n* = 2, *t* = 230000

(c) *n* = 1, *t* = 115000 (d) *n* = 1, *t* = 230000

Unitary Qubit Lattice Gas Representation of 2D and 3D Quantum Turbulence 263

Fig. 21. The evolution of quantum core singularities from an initial set of 48 straight line vortices. (a) winding number *n* = 1 and the corresponding wall phase information at *t* = 0, (b) winding number *n* = 1 isosurface cores at *t* = 84000. (c) winding number *n* = 1 isosurface cores at *t* = 115000 = 0.5*TPoin*. The 2*π* phase changes at the core singularity intersections at the walls is very evident. (d) winding number *n* = 1 isosurface cores at *t* = 230000 = *TPoin* showing only small perturbative changes from the initial state given in (a). (e) winding number *n* = 2 and the corresponding wall phase information at *t* = 0 showing the confluence degeneracy. (f) The corresponding isosurface cores at *t* = 230000 = *TPoin* for winding number *n* = 2. The wall phase information is a simple perturbative change to that at *t* = 0, (e) - but there is much small scale vortex loops that has evolved due to the initial

vortices,

2*D*. As in the 2D GP case, short Poincare recurrence will be found for initial conditions such that *Eint*(0), *Equ*(0) << *Ekin*(0) with *Ecomp*(0) << *Eincomp*(0). These conditions are readily satisfied in 3D by considering localized quantum line vortices so that *<sup>ρ</sup>* <sup>∼</sup> *const*. with <sup>∇</sup>√*<sup>ρ</sup>* <sup>∼</sup> <sup>0</sup> throughout much of the lattice. The evolution of a set of 3 × 16 linear vortices in the 3 planes and are examined for winding numbers *n* = 1 and *n* = 2, Fig. 21. Phase information is shown on the boundary walls: *φ* = 0 - *blue*, *φ* = 2*π* - *red*. First consider the case of winding number *n* = 1. At *t* = 0, the phase information on the straight line vortex cores are clearly identified on their intersection with the boundaries. The corresponding branch cuts join the 48 branch points. A snapshot of the vortex isosurfaces is shown at *t* = 84000 and shows strong vortex entanglement with many vortex loops - basically a snapshot of a quantum turbulence state. However by *t* = 115000 we see a point inversion of the Poincare recurrence of the initial line vortices at *t* = 0, as was also seen in 2D GP flows. The full Poincare recurrence occurs around *t* = 230000, (d). The kinks along the the vortex cores may be quantum Kelvin waves: since one outputs at discrete time intervals, one is not at the exact *TP*.

The robustness of the Poincare recurrence time is further exhibited by considering the evolution of 48 quantum vortices with winding number *n* = 2. The two-fold degeneracy translates into a more complex phase on the boundary walls, as can be seen on comparing (e) with (a) in Fig. 21. At *t* = *TP* = 230000, one sees considerable very small scale vortex loops that have arisen from the splitting of the confluent degeneracy although the overall structure of the line vortices can be seen globally seen. The phase information on the boundaries are only a slight perturbation from those initially. The grid for these runs was 12003.

For winding number *n* = 2 we show the details of the isosurfaces at the semi-Poincare recurrence time *t* = 115000 with phase information on the boundaries, and from a slightly different perspective. Also we show a detailed zoomed-in isoruface plot of the vortex cores at the Poincare recurrence time *t* = 230000 with phase information on the vortices themselves, Fig.22.

As in 2D, the signature of the occurrence of the Poincare recurrence can be seen in the evolution of the kinetic and quantum energies as a function of time, Fig. 23. The total energy is very well conserved throughout this run, *tmax* = 250000, by our unitary algorithm on a grid 12003: *E*TOT = *const*. For the parameters chosen here the internal energy is negligible. Note that the peaks in the kinetic energy are well preserved for vortices with winding number *n* = 1:*E*kin(0) ≈ *E*kin(*t* = 115000) ≈ *E*kin(*t* = 230000) ≈ ··· . However, for line vortices with winding number *n* = 2, there is a gradual decay in the peak in the kinetic energy *E*kin(0) ≥ *E*kin(*t* = 115000) ≥ *E*kin(*t* = 230000) ≥ ··· . This also explains why the Poincare recurrence in the isosurfaces for winding number *n* = 2 is not as clean as for winding number *n* = 1. Also it can be seen that in the evolution of *E*kin the vortex motion is much more turbulent for winding number *n* = 2. Since the internal energy for these runs is so low, the quantum energy evolution is the complement of kinetic energy (*ETOT* = *const*. = *Ekin*(*t*) + *Equ*(*t*) + *Eint*(*t*)). It should be noted that the time evolution of *Ekin*(*t*) and *Equ*(*t*) (and, of course, *Eint*(*t*)) are determined directly from their definitions, Eq.(33) and their sum then gives us the (conserved) total energy.

We note the loss of the semi-Poincaré time as the pixel resolution of the Arnold cat is increased from 74 to 300 × 300 yet find the persistence of the semi-Poincaré time for grid resolution from 512<sup>3</sup> to 12003. Presumably this is because the QLG algorithm strictly obeys diffusion ordering so that *TP* on a 5123 grid occurs at *TP* = 41775 and at *TP* = 230000 on a 1200<sup>3</sup> grid. (Diffusion ordering would give t = 229477). There is no such physics scaling laws in the Arnold cat map. 24 Will-be-set-by-IN-TECH

2*D*. As in the 2D GP case, short Poincare recurrence will be found for initial conditions such that *Eint*(0), *Equ*(0) << *Ekin*(0) with *Ecomp*(0) << *Eincomp*(0). These conditions are readily satisfied in 3D by considering localized quantum line vortices so that *<sup>ρ</sup>* <sup>∼</sup> *const*. with <sup>∇</sup>√*<sup>ρ</sup>* <sup>∼</sup> <sup>0</sup> throughout much of the lattice. The evolution of a set of 3 × 16 linear vortices in the 3 planes and are examined for winding numbers *n* = 1 and *n* = 2, Fig. 21. Phase information is shown on the boundary walls: *φ* = 0 - *blue*, *φ* = 2*π* - *red*. First consider the case of winding number *n* = 1. At *t* = 0, the phase information on the straight line vortex cores are clearly identified on their intersection with the boundaries. The corresponding branch cuts join the 48 branch points. A snapshot of the vortex isosurfaces is shown at *t* = 84000 and shows strong vortex entanglement with many vortex loops - basically a snapshot of a quantum turbulence state. However by *t* = 115000 we see a point inversion of the Poincare recurrence of the initial line vortices at *t* = 0, as was also seen in 2D GP flows. The full Poincare recurrence occurs around *t* = 230000, (d). The kinks along the the vortex cores may be quantum Kelvin waves: since

The robustness of the Poincare recurrence time is further exhibited by considering the evolution of 48 quantum vortices with winding number *n* = 2. The two-fold degeneracy translates into a more complex phase on the boundary walls, as can be seen on comparing (e) with (a) in Fig. 21. At *t* = *TP* = 230000, one sees considerable very small scale vortex loops that have arisen from the splitting of the confluent degeneracy although the overall structure of the line vortices can be seen globally seen. The phase information on the boundaries are

For winding number *n* = 2 we show the details of the isosurfaces at the semi-Poincare recurrence time *t* = 115000 with phase information on the boundaries, and from a slightly different perspective. Also we show a detailed zoomed-in isoruface plot of the vortex cores at the Poincare recurrence time *t* = 230000 with phase information on the vortices themselves,

As in 2D, the signature of the occurrence of the Poincare recurrence can be seen in the evolution of the kinetic and quantum energies as a function of time, Fig. 23. The total energy is very well conserved throughout this run, *tmax* = 250000, by our unitary algorithm on a grid 12003: *E*TOT = *const*. For the parameters chosen here the internal energy is negligible. Note that the peaks in the kinetic energy are well preserved for vortices with winding number *n* = 1:*E*kin(0) ≈ *E*kin(*t* = 115000) ≈ *E*kin(*t* = 230000) ≈ ··· . However, for line vortices with winding number *n* = 2, there is a gradual decay in the peak in the kinetic energy *E*kin(0) ≥ *E*kin(*t* = 115000) ≥ *E*kin(*t* = 230000) ≥ ··· . This also explains why the Poincare recurrence in the isosurfaces for winding number *n* = 2 is not as clean as for winding number *n* = 1. Also it can be seen that in the evolution of *E*kin the vortex motion is much more turbulent for winding number *n* = 2. Since the internal energy for these runs is so low, the quantum energy evolution is the complement of kinetic energy (*ETOT* = *const*. = *Ekin*(*t*) + *Equ*(*t*) + *Eint*(*t*)). It should be noted that the time evolution of *Ekin*(*t*) and *Equ*(*t*) (and, of course, *Eint*(*t*)) are determined directly from their definitions,

We note the loss of the semi-Poincaré time as the pixel resolution of the Arnold cat is increased from 74 to 300 × 300 yet find the persistence of the semi-Poincaré time for grid resolution from 512<sup>3</sup> to 12003. Presumably this is because the QLG algorithm strictly obeys diffusion ordering so that *TP* on a 5123 grid occurs at *TP* = 41775 and at *TP* = 230000 on a 1200<sup>3</sup> grid. (Diffusion ordering would give t = 229477). There is no such physics scaling laws in the Arnold cat map.

only a slight perturbation from those initially. The grid for these runs was 12003.

one outputs at discrete time intervals, one is not at the exact *TP*.

Eq.(33) and their sum then gives us the (conserved) total energy.

Fig.22.

(a) *n* = 1, *t* = 0 , 48 linear vortices, (b) *n* = 1, *t* = 84000

vortices

Fig. 21. The evolution of quantum core singularities from an initial set of 48 straight line vortices. (a) winding number *n* = 1 and the corresponding wall phase information at *t* = 0, (b) winding number *n* = 1 isosurface cores at *t* = 84000. (c) winding number *n* = 1 isosurface cores at *t* = 115000 = 0.5*TPoin*. The 2*π* phase changes at the core singularity intersections at the walls is very evident. (d) winding number *n* = 1 isosurface cores at *t* = 230000 = *TPoin* showing only small perturbative changes from the initial state given in (a). (e) winding number *n* = 2 and the corresponding wall phase information at *t* = 0 showing the confluence degeneracy. (f) The corresponding isosurface cores at *t* = 230000 = *TPoin* for winding number *n* = 2. The wall phase information is a simple perturbative change to that at *t* = 0, (e) - but there is much small scale vortex loops that has evolved due to the initial confluent degeneracy . Phase coding : *φ* = 0 *blue*, *φ* = 2*π red*. Grid 12003

(a) *n* = 1

Unitary Qubit Lattice Gas Representation of 2D and 3D Quantum Turbulence 265

(b) *n* = 2

Fig. 23. The time evolution of the *E*kin(*t*) (in blue) and *E*qu(*t*) (in red) for 0 ≤ *t* ≤ 250000 for

exhibits a triple cascade *<sup>k</sup>*−*<sup>α</sup>* with *<sup>α</sup>* <sup>∼</sup> 3.7 for small *<sup>k</sup>*, an *<sup>α</sup>* <sup>∼</sup> 6 for the intermediate cascade, and *α* ∼ 3.0 for the large-*k* cascade. At the intermittency, the large-*k* exponent increases to a

To investigate the cause of this intermittent loss of the incompressible *k*−<sup>3</sup> spectrum, we then

One notices that the loss of the *k*−<sup>3</sup> corresponds to the apparent loss of vortex loops, i.e., of vortices. This would be consistent with the assumption that the *incompressible* kinetic energy spectrum of *k*−<sup>3</sup> in the very large-*k* regime is due to the Fourier transform of an isolated vortex Nore et al. (1997). As the vortex loops are reestablished, so is the incompressible *k*−<sup>3</sup> kinetic energy spectrum. An alternative but somewhat more speculative explanation rests on the assumption that the incompressible *k*−<sup>3</sup> spectrum is due to the quantum Kelvin wave cascade on the quantum vortices. As the quantum vortex loop shrink topologically, the Kelvin waves are inhibited and hence the loss of the *k*−<sup>3</sup> spectrum. Moreover, if one looks at the time evolution of the mean kinetic *Ekin*(*t*) and quantum *Equ*(*t*) energies one notices that this loss of the vortex loops occurs around the *t* ∼ 82000 around which the *Ekin*(*t*) has a secondary

(a) Winding Number *n* = 1, and (b) Winding Number *n* = 2. Grid 12003.

noisy *α* ∼ 5.2 as well as a steeped intermediate wave number exponent.

examined the vortex isosurfaces around this time interval, Fig.27.

Fig. 22. The vortex isosurfaces for winding number *n* = 2 at the semi- and full Poincare times: (a) *t* = 115000 with phase information on the boundaries and at a different perspective, (b) *t* = 230000 with phase information on the isosurfaces |*ϕ*| = *const*.. Phase coding : *φ* = 0 *blue*, *φ* = 2*π red*. Grid 12003

### **5.2 Energy spectra on** 1200<sup>3</sup> **grid**

We first determine the incompressible and compressible kinetic energy spectrum for the initial profiles considered in Fig. 24 . Nearly all the kinetic energy initially is incompressible.

Very quickly the spectra tend to quasi-steady state, with a typical *Ecomp*(*k*) and *Einc*(*k*) spectrum as in Fig. 25 For winding number *n* = 1 the incompressible spectrum *Einc*(*k*) exhibits two spectral domains *<sup>k</sup>*−*<sup>α</sup>* : for very large *<sup>k</sup>* (*<sup>k</sup>* <sup>&</sup>gt; 100) the spectral exponent *<sup>α</sup>* <sup>∼</sup> 3.05, while in the intermediate *k* range (15 < *k* < 50) one finds *α* ∼ 5.0. The compressible kinetic energy spectrum *Ecomp*(*k*) exhibits three spectral regions: a very fuzzy large *k* region, preceded by a steep spectral region which then merges into the small *k* region. It is interesting to note that around the wave number *k<sup>ξ</sup>* ∼ 70 at which the steep compressible spectral exponent (typically *α* > 7) joins to the large *k* spectrum, we find the switch over in the incompressible spectral exponents. We have noticed this in basically every simulation we have performed (and grids up to 40963). For vortices with winding number *n* = 2, the kinetic energy spectra are much cleaner, presumably because the vortex entanglements are stronger and hence the QT is stronger. For the incompressible spectrum *Einc*(*k*) one again sees two spectral cascade regions: for *k* > *k<sup>ξ</sup>* the spectral exponent is *α* ∼ 3.07 while for *k* < *k<sup>ξ</sup>* , the exponent *α* ∼ 3.90. For the *compressible* spectrum, we find three spectral energy cacades: for very low *k* (5 < *k* < 30) a Kolmogorov-like cascade with exponent *α* ∼ 1.67, with a steep spectra decay followed for *k* > *k<sup>ξ</sup>* a compressible kinetic energy with exponent *α* ∼ 3.28. (The *total* kinetic energy spectrum has the exponents *α* ∼ 1.64 for low *k*, and *α* ∼ 3.17 for large *k*). The crossover *k<sup>ξ</sup>* ∼ 70 − 90.

Somewhat surprising, we find a time interval during which we loose the incompressible kinetic energy spectrum *Einc*(*k*) <sup>∼</sup> *<sup>k</sup>*−<sup>3</sup> for winding number *<sup>n</sup>* <sup>=</sup> 1 vortices. In particular, in the time interval 81400 < *t* < 84300 - except for very brief transient reestablishment of the *k*−<sup>3</sup> spectrum - we find spectra as shown in Fig. 26. In Figs. 26 (b)-(d) there is a sharp drop in the incompressible energy spectrum for wave numbers *k* > 100, except for a very brief transient recovery around *t* ∼ 83000. There is also a sharp cutoff in the compressible spectrum for *k* > 500. Around these intermittencies, the incompressible kinetic energy spectrum also 26 Will-be-set-by-IN-TECH

(a) *n* = 2, *t* = 115000 (b) *n* = 2, *t* = 230000

Fig. 22. The vortex isosurfaces for winding number *n* = 2 at the semi- and full Poincare times: (a) *t* = 115000 with phase information on the boundaries and at a different

perspective, (b) *t* = 230000 with phase information on the isosurfaces |*ϕ*| = *const*.. Phase

We first determine the incompressible and compressible kinetic energy spectrum for the initial profiles considered in Fig. 24 . Nearly all the kinetic energy initially is incompressible. Very quickly the spectra tend to quasi-steady state, with a typical *Ecomp*(*k*) and *Einc*(*k*) spectrum as in Fig. 25 For winding number *n* = 1 the incompressible spectrum *Einc*(*k*) exhibits two spectral domains *<sup>k</sup>*−*<sup>α</sup>* : for very large *<sup>k</sup>* (*<sup>k</sup>* <sup>&</sup>gt; 100) the spectral exponent *<sup>α</sup>* <sup>∼</sup> 3.05, while in the intermediate *k* range (15 < *k* < 50) one finds *α* ∼ 5.0. The compressible kinetic energy spectrum *Ecomp*(*k*) exhibits three spectral regions: a very fuzzy large *k* region, preceded by a steep spectral region which then merges into the small *k* region. It is interesting to note that around the wave number *k<sup>ξ</sup>* ∼ 70 at which the steep compressible spectral exponent (typically *α* > 7) joins to the large *k* spectrum, we find the switch over in the incompressible spectral exponents. We have noticed this in basically every simulation we have performed (and grids up to 40963). For vortices with winding number *n* = 2, the kinetic energy spectra are much cleaner, presumably because the vortex entanglements are stronger and hence the QT is stronger. For the incompressible spectrum *Einc*(*k*) one again sees two spectral cascade regions: for *k* > *k<sup>ξ</sup>* the spectral exponent is *α* ∼ 3.07 while for *k* < *k<sup>ξ</sup>* , the exponent *α* ∼ 3.90. For the *compressible* spectrum, we find three spectral energy cacades: for very low *k* (5 < *k* < 30) a Kolmogorov-like cascade with exponent *α* ∼ 1.67, with a steep spectra decay followed for *k* > *k<sup>ξ</sup>* a compressible kinetic energy with exponent *α* ∼ 3.28. (The *total* kinetic energy spectrum has the exponents *α* ∼ 1.64 for low *k*, and *α* ∼ 3.17 for large *k*). The crossover

Somewhat surprising, we find a time interval during which we loose the incompressible kinetic energy spectrum *Einc*(*k*) <sup>∼</sup> *<sup>k</sup>*−<sup>3</sup> for winding number *<sup>n</sup>* <sup>=</sup> 1 vortices. In particular, in the time interval 81400 < *t* < 84300 - except for very brief transient reestablishment of the *k*−<sup>3</sup> spectrum - we find spectra as shown in Fig. 26. In Figs. 26 (b)-(d) there is a sharp drop in the incompressible energy spectrum for wave numbers *k* > 100, except for a very brief transient recovery around *t* ∼ 83000. There is also a sharp cutoff in the compressible spectrum for *k* > 500. Around these intermittencies, the incompressible kinetic energy spectrum also

coding : *φ* = 0 *blue*, *φ* = 2*π red*. Grid 12003

**5.2 Energy spectra on** 1200<sup>3</sup> **grid**

*k<sup>ξ</sup>* ∼ 70 − 90.

Fig. 23. The time evolution of the *E*kin(*t*) (in blue) and *E*qu(*t*) (in red) for 0 ≤ *t* ≤ 250000 for (a) Winding Number *n* = 1, and (b) Winding Number *n* = 2. Grid 12003.

exhibits a triple cascade *<sup>k</sup>*−*<sup>α</sup>* with *<sup>α</sup>* <sup>∼</sup> 3.7 for small *<sup>k</sup>*, an *<sup>α</sup>* <sup>∼</sup> 6 for the intermediate cascade, and *α* ∼ 3.0 for the large-*k* cascade. At the intermittency, the large-*k* exponent increases to a noisy *α* ∼ 5.2 as well as a steeped intermediate wave number exponent.

To investigate the cause of this intermittent loss of the incompressible *k*−<sup>3</sup> spectrum, we then examined the vortex isosurfaces around this time interval, Fig.27.

One notices that the loss of the *k*−<sup>3</sup> corresponds to the apparent loss of vortex loops, i.e., of vortices. This would be consistent with the assumption that the *incompressible* kinetic energy spectrum of *k*−<sup>3</sup> in the very large-*k* regime is due to the Fourier transform of an isolated vortex Nore et al. (1997). As the vortex loops are reestablished, so is the incompressible *k*−<sup>3</sup> kinetic energy spectrum. An alternative but somewhat more speculative explanation rests on the assumption that the incompressible *k*−<sup>3</sup> spectrum is due to the quantum Kelvin wave cascade on the quantum vortices. As the quantum vortex loop shrink topologically, the Kelvin waves are inhibited and hence the loss of the *k*−<sup>3</sup> spectrum. Moreover, if one looks at the time evolution of the mean kinetic *Ekin*(*t*) and quantum *Equ*(*t*) energies one notices that this loss of the vortex loops occurs around the *t* ∼ 82000 around which the *Ekin*(*t*) has a secondary

<sup>5</sup> <sup>10</sup> <sup>50</sup> <sup>100</sup> <sup>500</sup> <sup>1000</sup> <sup>10</sup><sup>5</sup>

(a) *n* = 1 , *t* = 81400

k

83 400

<sup>5</sup> <sup>10</sup> <sup>50</sup> <sup>100</sup> <sup>500</sup> <sup>1000</sup> <sup>10</sup><sup>5</sup>

(c) *n* = 1 , *t* = 83400

<sup>5</sup> <sup>10</sup> <sup>50</sup> <sup>100</sup> <sup>500</sup> <sup>1000</sup> <sup>10</sup><sup>5</sup>

(e) *n* = 1 , *t* = 84200

k

k

84 200

<sup>5</sup> <sup>10</sup> <sup>50</sup> <sup>100</sup> <sup>500</sup> <sup>1000</sup> <sup>10</sup><sup>5</sup>

(b) *n* = 1 , *t* = 82200

k

84 000

<sup>5</sup> <sup>10</sup> <sup>50</sup> <sup>100</sup> <sup>500</sup> <sup>1000</sup> <sup>10</sup><sup>5</sup>

(d) *n* = 1 , *t* = 84000

85 000

<sup>5</sup> <sup>10</sup> <sup>50</sup> <sup>100</sup> <sup>500</sup> <sup>1000</sup> <sup>10</sup><sup>5</sup>

(f) *n* = 1 , *t* = 85000

k

k

82 200

0.001 0.1 10 1000 105

Unitary Qubit Lattice Gas Representation of 2D and 3D Quantum Turbulence 267

0.001 0.1 10 1000 105

0.001 0.1 10 1000 105

(g) *n* = 1 , *t* = 85000

Fig. 26. 7 snapshots for winding number *n* = 1 of the incompressible (*blue dots*) and compressible (*red dots*) kinetic energy spectrum at times (a) t = 81400, (b) t = 82200, (c) t = 83400, (d) t = 84000, (e) t = 84200, (f) t = 85000, and (g) *t* = 115000 = *TP*/2. There is a very brief transient recovery of the *<sup>k</sup>*−<sup>3</sup> spectrum around *<sup>t</sup>* <sup>∼</sup> 83000. Also shown in s the spectrum at the *t* = *TP*/2 and should be compared to the initial spectrum in Fig.24(a) . Grid 12003

81 400

0.001 0.1 10 1000 105

0.001 0.1 10 1000 10<sup>5</sup>

> 0.001 0.1 10 1000 105

Fig. 24. The initial incompressible (*blue*) *Einc*(*k*, 0) and compressible (*red*) *Ecomp*(*k*, 0) kinetic energy spectra for (a) Winding Number *n* = 1, and (b) Winding Number *n* = 2. The linear regression (*blue dashed line*) fit to the incompressible kinetic energy, *Einc*(*k*, 0) <sup>∼</sup> *<sup>k</sup>*−*<sup>α</sup>* is: (a) *α* = 3.15, (b) *α* = 3.30. Grid 12003.

Fig. 25. At t = 80 000, the incompressible (*blue dots*) *Einc*(*k*) and compressible (*red dots*) *Ecomp*(*k*) kinetic energy spectra for (a) Winding Number *n* = 1, and (b) Winding Number *n* = 2. The linear regression (*blue dashed line*) fit to the incompressible kinetic energy, *Einc*(*k*, 0) <sup>∼</sup> *<sup>k</sup>*−*<sup>α</sup>* is: (a) *<sup>α</sup>* <sup>=</sup> 3.05, (b) *<sup>α</sup>* <sup>=</sup> 3.07. The intermediate *<sup>k</sup>* range has incompressible kinetic energy exponents (a) *α* ∼ 5.0, and (b) *α* ∼ 3.9 - (*green dashed line*). The small *k*-range for the compressible energy exhibits a weak Kolmogorov-like spectrum for winding number *<sup>n</sup>* <sup>=</sup> 2, (b), of *<sup>α</sup>* <sup>∼</sup> 1.67 - (*read dashed line*). Grid 12003.

28 Will-be-set-by-IN-TECH

(a) *n* = 1, *t* = 0 (b) *n* = 2, t = 0

Fig. 24. The initial incompressible (*blue*) *Einc*(*k*, 0) and compressible (*red*) *Ecomp*(*k*, 0) kinetic energy spectra for (a) Winding Number *n* = 1, and (b) Winding Number *n* = 2. The linear regression (*blue dashed line*) fit to the incompressible kinetic energy, *Einc*(*k*, 0) <sup>∼</sup> *<sup>k</sup>*−*<sup>α</sup>* is: (a)

(a) *n* = 1, *t* = 80000 (b) *n* = 2, t = 80 000

Fig. 25. At t = 80 000, the incompressible (*blue dots*) *Einc*(*k*) and compressible (*red dots*) *Ecomp*(*k*) kinetic energy spectra for (a) Winding Number *n* = 1, and (b) Winding Number *n* = 2. The linear regression (*blue dashed line*) fit to the incompressible kinetic energy, *Einc*(*k*, 0) <sup>∼</sup> *<sup>k</sup>*−*<sup>α</sup>* is: (a) *<sup>α</sup>* <sup>=</sup> 3.05, (b) *<sup>α</sup>* <sup>=</sup> 3.07. The intermediate *<sup>k</sup>* range has incompressible kinetic energy exponents (a) *α* ∼ 5.0, and (b) *α* ∼ 3.9 - (*green dashed line*). The small *k*-range for the compressible energy exhibits a weak Kolmogorov-like spectrum for winding number

*<sup>n</sup>* <sup>=</sup> 2, (b), of *<sup>α</sup>* <sup>∼</sup> 1.67 - (*read dashed line*). Grid 12003.

*α* = 3.15, (b) *α* = 3.30. Grid 12003.

Fig. 26. 7 snapshots for winding number *n* = 1 of the incompressible (*blue dots*) and compressible (*red dots*) kinetic energy spectrum at times (a) t = 81400, (b) t = 82200, (c) t = 83400, (d) t = 84000, (e) t = 84200, (f) t = 85000, and (g) *t* = 115000 = *TP*/2. There is a very brief transient recovery of the *<sup>k</sup>*−<sup>3</sup> spectrum around *<sup>t</sup>* <sup>∼</sup> 83000. Also shown in s the spectrum at the *t* = *TP*/2 and should be compared to the initial spectrum in Fig.24(a) . Grid 12003

energy spectra, Fig. 28. Due to an oversight, we did not retain data for the incompressible

Unitary Qubit Lattice Gas Representation of 2D and 3D Quantum Turbulence 269

(a) *Grid* 57603, *<sup>t</sup>* <sup>=</sup> <sup>40000</sup> (b) *Grid* 57603, *small* <sup>−</sup> *<sup>k</sup>*

and compressible components of the kinetic energy spectrum. Both *ETOT*(*k*) and *Equ*(*k*) have basically the same spectral properties. One sees 3 distinct energy cascade regions *k*−*α*: a small-*k* band with *α* ∼ 1.30, an intermediate-*k* band with steep slope *α* ∼ 7.76 and a large-*k*

In Yepez et al. (2009), we tried to identify these 3 regions as the small-*k* classical Kolmogorov casacde, followed by a semi-classical intermediate-*k* band (with non-universal exponent *α*) which is then followed a quantum Kelvin wave cascade for the very large-*k* band. Objections were raised against this interpretation Krstulovic & Brachet (2010); L'vov & Nazarenko (2010) based on (a) the kinetic energy spectrum of a single isolated vortex is *k*−<sup>3</sup> for all *k* (for a straight line vortex all the kinetic energy is incompressible); and (b) using the standard definition of the coherence length *ξ* for the parameters chosen in our simulations, *ξ* > 2000 – i.e., it is claimed that we are investigating the physics of very strong vortex-vortex core overlapping wave functions. We counter that the definition of *ξ* is based on a boundary value solution of the GP equation for an isolated single vortex under the condition that the wave function asymptotes to the background value as one moves away from the vortex core. Our simulations are with periodic boundary conditions and we have no pointwise convergence of our wave function to some nice smooth 'background' value. While it can be argued that one should simply replace the usual background density *ρ*<sup>0</sup> by its spatial average < *ρ* >, the definition now of *ξ* becomes qualitative and does not handle large fluctuations about < *ρ* >. It is clear that we cannot categorically claim that the ubiquitous *k*−<sup>3</sup> spectrum for the large-*k* band is due to quantum Kelvin wave cascade on single vortices - especially as this *k*−<sup>3</sup> spectrum is also seen in our 2D QT simulations and in 2D there are no quantum Kelvin waves since the

Fig. 28. (a) Total kinetic energy spectrum, *ETOT*(*k*) − (*blue dots*), and quantum energy spectrum, *Equ*(*k*) <sup>−</sup> (*red dots*), at *<sup>t</sup>* <sup>=</sup> 40000 for winding number *<sup>n</sup>* <sup>=</sup> 6 on a large 57603 grid. 3 energy cascade regions can be readily distinguished, with both the total and quantum energy spectra being very similar. Dashed curves – linear regression fits. (b) Linear regression fits for different *k*-bands in the small-*k* region: for 30 < *k* < 200 (*green dashed line*), the spectra exponent *α* ∼ 1.30, while for 100 < *k* < 250 (*red dashed line*), the spectral exponent *α* ∼ 1.68.

The standard deviation error is 0.06 in both cases.

band with *α* ∼ 3.00.

Fig. 27. 4 snapshots of the winding number *n* = 1 singular vortex core isosurfaces at times (a) t = 78000, (b) t = 81000, (c) t = 82000, and (d) t = 88000. The phase information (blue: *φ* = 0, red: *φ* = 2*π* on the vortex core singularities clearly shows the 2*π* phase change in circumnavigating the vortex core loops. At *t* = 82000 there is a different morphology in the

<sup>|</sup>*ϕ*<sup>|</sup> - isosurfaces. Grid 12003

peak. As there is another secondary peak in *Ekin*(*t*) around *t* ∼ (82000 + *TP*/2) one expects another transient loss in the *k*−<sup>3</sup> spectrum and in the vortex loops. This is indeed found to occur around 196400 < *t* < 199300. For winding number *n* = 2 vortices, we do not find such intermittent loss of vortex loops or any intermittent loss of the *k*−<sup>3</sup> spectrum. These results are in agreement with those found earlier in 2D QT. Moreover, there is not a similar intermittent loss of vortex loops for 48 linear vortices with winding number *n* = 1 (c.f., Fig. 21(b))

### **5.3 Total kinetic energy spectrum for large grid simulations on** 57603

We have performed simulations on 5760<sup>3</sup> grid using winding number *n* = 6 straight line vortices as initial conditions. By *t* = 40000 one obtains the following *total* kinetic and quantum 30 Will-be-set-by-IN-TECH

(a) *n* = 1 , *t* = 78000 (b) *n* = 1 , *t* = 81000

(c) *n* = 1 , *t* = 82000 (d) *n* = 1 , *t* = 88000

peak. As there is another secondary peak in *Ekin*(*t*) around *t* ∼ (82000 + *TP*/2) one expects another transient loss in the *k*−<sup>3</sup> spectrum and in the vortex loops. This is indeed found to occur around 196400 < *t* < 199300. For winding number *n* = 2 vortices, we do not find such intermittent loss of vortex loops or any intermittent loss of the *k*−<sup>3</sup> spectrum. These results are in agreement with those found earlier in 2D QT. Moreover, there is not a similar intermittent

We have performed simulations on 5760<sup>3</sup> grid using winding number *n* = 6 straight line vortices as initial conditions. By *t* = 40000 one obtains the following *total* kinetic and quantum

loss of vortex loops for 48 linear vortices with winding number *n* = 1 (c.f., Fig. 21(b))

**5.3 Total kinetic energy spectrum for large grid simulations on** 57603

Fig. 27. 4 snapshots of the winding number *n* = 1 singular vortex core isosurfaces at times (a) t = 78000, (b) t = 81000, (c) t = 82000, and (d) t = 88000. The phase information (blue: *φ* = 0, red: *φ* = 2*π* on the vortex core singularities clearly shows the 2*π* phase change in circumnavigating the vortex core loops. At *t* = 82000 there is a different morphology in the

<sup>|</sup>*ϕ*<sup>|</sup> - isosurfaces. Grid 12003

energy spectra, Fig. 28. Due to an oversight, we did not retain data for the incompressible

Fig. 28. (a) Total kinetic energy spectrum, *ETOT*(*k*) − (*blue dots*), and quantum energy spectrum, *Equ*(*k*) <sup>−</sup> (*red dots*), at *<sup>t</sup>* <sup>=</sup> 40000 for winding number *<sup>n</sup>* <sup>=</sup> 6 on a large 57603 grid. 3 energy cascade regions can be readily distinguished, with both the total and quantum energy spectra being very similar. Dashed curves – linear regression fits. (b) Linear regression fits for different *k*-bands in the small-*k* region: for 30 < *k* < 200 (*green dashed line*), the spectra exponent *α* ∼ 1.30, while for 100 < *k* < 250 (*red dashed line*), the spectral exponent *α* ∼ 1.68. The standard deviation error is 0.06 in both cases.

and compressible components of the kinetic energy spectrum. Both *ETOT*(*k*) and *Equ*(*k*) have basically the same spectral properties. One sees 3 distinct energy cascade regions *k*−*α*: a small-*k* band with *α* ∼ 1.30, an intermediate-*k* band with steep slope *α* ∼ 7.76 and a large-*k* band with *α* ∼ 3.00.

In Yepez et al. (2009), we tried to identify these 3 regions as the small-*k* classical Kolmogorov casacde, followed by a semi-classical intermediate-*k* band (with non-universal exponent *α*) which is then followed a quantum Kelvin wave cascade for the very large-*k* band. Objections were raised against this interpretation Krstulovic & Brachet (2010); L'vov & Nazarenko (2010) based on (a) the kinetic energy spectrum of a single isolated vortex is *k*−<sup>3</sup> for all *k* (for a straight line vortex all the kinetic energy is incompressible); and (b) using the standard definition of the coherence length *ξ* for the parameters chosen in our simulations, *ξ* > 2000 – i.e., it is claimed that we are investigating the physics of very strong vortex-vortex core overlapping wave functions. We counter that the definition of *ξ* is based on a boundary value solution of the GP equation for an isolated single vortex under the condition that the wave function asymptotes to the background value as one moves away from the vortex core. Our simulations are with periodic boundary conditions and we have no pointwise convergence of our wave function to some nice smooth 'background' value. While it can be argued that one should simply replace the usual background density *ρ*<sup>0</sup> by its spatial average < *ρ* >, the definition now of *ξ* becomes qualitative and does not handle large fluctuations about < *ρ* >. It is clear that we cannot categorically claim that the ubiquitous *k*−<sup>3</sup> spectrum for the large-*k* band is due to quantum Kelvin wave cascade on single vortices - especially as this *k*−<sup>3</sup> spectrum is also seen in our 2D QT simulations and in 2D there are no quantum Kelvin waves since the

operators that entangle the qubit probabilities and unitary streaming operators that propagate this entanglement throughout the lattice, QLG has a small memory imprint. This permits production runs on relatively few processors: e.g., production runs on 5760<sup>3</sup> grids using just over 10000 cores. Using standard pseudo-spectral codes, the largest grids achieved so far have been 20483 by Machida et al. (2010) - and these codes required the ad hoc addition of dissipative terms to damp out high-*k* modes and for numerical stability. For BEC turbulence this destroys the Hamiltonian structure of the GP Eq. (2) and destroys any Poincare recurrence

Unitary Qubit Lattice Gas Representation of 2D and 3D Quantum Turbulence 271

In its current formulation, it is critical that parameters are so chosen that the mesoscopic QLG algorithm yields diffusion ordering at the macroscopic GP level. This does restrict the choice in the values of kinetic, quantum and internal energies. If parameter choices are made that violate the diffusion ordering then the QLG algorithm will not be simulating the GP Eq. (2). There are various tests for the validity of our QLG solution of the GP equation: the conservation of total energy, and the fact the the Poincare recurrence time scales as *grid*<sup>2</sup> whether in 2D or 3D GP. This replaces the naive thought that the time for QLG phenomena would necessarily scale as *gridD*, where *D* is the dimensionality of the macroscopic problem. On the other hand, standard CFD algorithms have the spatial and time step independent

We have presented significant spectral results for both 2D and 3D QT - although their interpretation is not straightforward and much still needs to be done in this area. Much of the controversy surrounds the significance of the coherence length *ξ* and the *k*−<sup>3</sup> spectrum in the high-*k* region. We believe there is much new physics occurring in our QT simulations even if the *k*−<sup>3</sup> spectrum is attributed to the dominance of the spectrum of an isolated vortex: (a) there is clear evidence in both 2D and 3D QLG of a dual cascade in the incompressible kinetic energy spectrum, with a spectrum of *k*−<sup>4</sup> followed by the *k*−3. This *k*−<sup>4</sup> spectrum has also been seen by Horng et al. (2009) and connection implied with the Saffman spectrum arising from vorticity discontinuities; (b) the triple cascade in the *total* kinetic energy spectrum with the small-*k* regime yielding a quasi-Kolmogorov *k*−5/3 spectrum. It is a bit strange that the QT community is that concerned with achieving the *k*−5/3 energy spectrum in the *incompressible* energy *Einc*(*k*) in the small *k*-region where the quantization of the vortex core circulation becomes unimportant. The dynamics of the GP Eq.(2) is fundamentally compressible. In *compressible* CFD simulations the emphasis changes to the *total* kinetic energy spectrum and that its power law is *k*−5/3. However much work remains to be done on clarifying the role of

Finally, we comment on future directions of QLG. In this chapter we have restricted ourselves to the *scalar* GP equation. This is appropriate for a BEC gas with spin *f* confined in a magnetic well. The spin of the atom is aligned to the magnetic field and so the BEC dynamics is given by just one scalar GP equation. However if this BEC gas is confined in an optical lattice, the spin is no longer constrained and one now must work with 2 *f* + 1 GP equations. These so-called spinor BECs yield an enormous field of future research Ueda & Kawaguchi (2010). Moreover, the quantum vortices of a scalar BEC are necessarily Abelian vortices, but the vortices of spinor BECs can be non-Abelian in structure. Since QT is driven by vortex-vortex interactions, research needs to be performed to ascertain the role played by the non-Abelian in the energy cascades. (These non-Abelian vortices have non-integer multiples of the base circulation - not dissimilar to fractional quantum electron charge in the fractional quantum Hall effect). Other

interesting vortices are skyrmions - used by high energy physicists to model baryons.

phenomena.

which, of course, is quite beneficial.

quantum Kelvin waves on the energy spectrum.

vortex core is just a point singularity. It is possible, however, that with the co-existence of this triple cascade region in the kinetic energy spectrum, and with the small-*k* region exhibiting a quasi-classical Kolmogorov *k*−5/3 spectrum, that this large-*k* band *k*−<sup>3</sup> spectrum could be dominated by the spectrum of a single vortex as we are now at very small scales in the problem.

We have investigated in some detail the 3D QT for winding number *n* = 2 on a 3076<sup>3</sup> grid. One again finds the triple cascade region in the *compressible* kinetic energy spectrum, Fig. 29, (*red squares*), as well as in the quantum energy spectrum (*gold diamonds*). However the

Fig. 29. (a) Energy spectra at *t* = 48000 for winding number *n* = 2. Blue circles incompressible kinetic energy, red squares - compressible kinetic energy, gold diamonds quantum energy. A triple cascade is quite evident in both the quantum and compressible kinetic energy spectra. These two spectra only deviate around the transition from the medium *k* to large *k* cascade, i.e., around *k* ∼ 300. (b) A blow-up of the transition in the *incompressible* kinetic energy pectrum from *<sup>k</sup>*−3.6 to *<sup>k</sup>*−3.0 around *<sup>k</sup>* <sup>∼</sup> 300. Grid 30723

*incompressible* kinetic energy (*blue circles*) has a slight bend in its spectral exponent around the wave number *k<sup>ξ</sup>* ∼ 300 from the large-*k* exponent of *α* ∼ 3.0 for *k* > *k<sup>ξ</sup>* to *α* ∼ 3.6 for *k* < *k<sup>ξ</sup>* . This bend in the incompressible kinetic energy spectrum occurs at the *k<sup>ξ</sup>* where the compressible and quantum energy spectra make their transition from the intermediate-*k* band large spectral exponent to the large-*k* band spectral exponent of *α* ∼ 3. It is interesting to note similar behavior in 2D QT, Fig.18(d) and Fig.19(b). The spectral exponents for *ETOT*(*k*) are: a Kolmogorov *α* ∼ 1.66 for the small-*k* band 15 < *k* < 90, a steep *α* ∼ 8.53 for the intermediate-*k* band 180 < *k* < 280 and exponent *α* ∼ 3.04 in the large-*k* band.

#### **6. Conclusion**

Here we have discussed a novel unitary qubit algorithm for a mesoscopic study of quantum turbulence in a BEC dilute gas. Since it requires just 2 qubits/lattice site with unitary collision 32 Will-be-set-by-IN-TECH

vortex core is just a point singularity. It is possible, however, that with the co-existence of this triple cascade region in the kinetic energy spectrum, and with the small-*k* region exhibiting a quasi-classical Kolmogorov *k*−5/3 spectrum, that this large-*k* band *k*−<sup>3</sup> spectrum could be dominated by the spectrum of a single vortex as we are now at very small scales in the

We have investigated in some detail the 3D QT for winding number *n* = 2 on a 3076<sup>3</sup> grid. One again finds the triple cascade region in the *compressible* kinetic energy spectrum, Fig. 29, (*red squares*), as well as in the quantum energy spectrum (*gold diamonds*). However the

(a) *n* = 1 (b) *n* = 2

*incompressible* kinetic energy (*blue circles*) has a slight bend in its spectral exponent around the wave number *k<sup>ξ</sup>* ∼ 300 from the large-*k* exponent of *α* ∼ 3.0 for *k* > *k<sup>ξ</sup>* to *α* ∼ 3.6 for *k* < *k<sup>ξ</sup>* . This bend in the incompressible kinetic energy spectrum occurs at the *k<sup>ξ</sup>* where the compressible and quantum energy spectra make their transition from the intermediate-*k* band large spectral exponent to the large-*k* band spectral exponent of *α* ∼ 3. It is interesting to note similar behavior in 2D QT, Fig.18(d) and Fig.19(b). The spectral exponents for *ETOT*(*k*) are: a Kolmogorov *α* ∼ 1.66 for the small-*k* band 15 < *k* < 90, a steep *α* ∼ 8.53 for the intermediate-*k*

Here we have discussed a novel unitary qubit algorithm for a mesoscopic study of quantum turbulence in a BEC dilute gas. Since it requires just 2 qubits/lattice site with unitary collision

incompressible kinetic energy, red squares - compressible kinetic energy, gold diamonds quantum energy. A triple cascade is quite evident in both the quantum and compressible kinetic energy spectra. These two spectra only deviate around the transition from the medium *k* to large *k* cascade, i.e., around *k* ∼ 300. (b) A blow-up of the transition in the *incompressible* kinetic energy pectrum from *<sup>k</sup>*−3.6 to *<sup>k</sup>*−3.0 around *<sup>k</sup>* <sup>∼</sup> 300. Grid 30723

Fig. 29. (a) Energy spectra at *t* = 48000 for winding number *n* = 2. Blue circles -

band 180 < *k* < 280 and exponent *α* ∼ 3.04 in the large-*k* band.

**6. Conclusion**

problem.

operators that entangle the qubit probabilities and unitary streaming operators that propagate this entanglement throughout the lattice, QLG has a small memory imprint. This permits production runs on relatively few processors: e.g., production runs on 5760<sup>3</sup> grids using just over 10000 cores. Using standard pseudo-spectral codes, the largest grids achieved so far have been 20483 by Machida et al. (2010) - and these codes required the ad hoc addition of dissipative terms to damp out high-*k* modes and for numerical stability. For BEC turbulence this destroys the Hamiltonian structure of the GP Eq. (2) and destroys any Poincare recurrence phenomena.

In its current formulation, it is critical that parameters are so chosen that the mesoscopic QLG algorithm yields diffusion ordering at the macroscopic GP level. This does restrict the choice in the values of kinetic, quantum and internal energies. If parameter choices are made that violate the diffusion ordering then the QLG algorithm will not be simulating the GP Eq. (2). There are various tests for the validity of our QLG solution of the GP equation: the conservation of total energy, and the fact the the Poincare recurrence time scales as *grid*<sup>2</sup> whether in 2D or 3D GP. This replaces the naive thought that the time for QLG phenomena would necessarily scale as *gridD*, where *D* is the dimensionality of the macroscopic problem. On the other hand, standard CFD algorithms have the spatial and time step independent which, of course, is quite beneficial.

We have presented significant spectral results for both 2D and 3D QT - although their interpretation is not straightforward and much still needs to be done in this area. Much of the controversy surrounds the significance of the coherence length *ξ* and the *k*−<sup>3</sup> spectrum in the high-*k* region. We believe there is much new physics occurring in our QT simulations even if the *k*−<sup>3</sup> spectrum is attributed to the dominance of the spectrum of an isolated vortex: (a) there is clear evidence in both 2D and 3D QLG of a dual cascade in the incompressible kinetic energy spectrum, with a spectrum of *k*−<sup>4</sup> followed by the *k*−3. This *k*−<sup>4</sup> spectrum has also been seen by Horng et al. (2009) and connection implied with the Saffman spectrum arising from vorticity discontinuities; (b) the triple cascade in the *total* kinetic energy spectrum with the small-*k* regime yielding a quasi-Kolmogorov *k*−5/3 spectrum. It is a bit strange that the QT community is that concerned with achieving the *k*−5/3 energy spectrum in the *incompressible* energy *Einc*(*k*) in the small *k*-region where the quantization of the vortex core circulation becomes unimportant. The dynamics of the GP Eq.(2) is fundamentally compressible. In *compressible* CFD simulations the emphasis changes to the *total* kinetic energy spectrum and that its power law is *k*−5/3. However much work remains to be done on clarifying the role of quantum Kelvin waves on the energy spectrum.

Finally, we comment on future directions of QLG. In this chapter we have restricted ourselves to the *scalar* GP equation. This is appropriate for a BEC gas with spin *f* confined in a magnetic well. The spin of the atom is aligned to the magnetic field and so the BEC dynamics is given by just one scalar GP equation. However if this BEC gas is confined in an optical lattice, the spin is no longer constrained and one now must work with 2 *f* + 1 GP equations. These so-called spinor BECs yield an enormous field of future research Ueda & Kawaguchi (2010). Moreover, the quantum vortices of a scalar BEC are necessarily Abelian vortices, but the vortices of spinor BECs can be non-Abelian in structure. Since QT is driven by vortex-vortex interactions, research needs to be performed to ascertain the role played by the non-Abelian in the energy cascades. (These non-Abelian vortices have non-integer multiples of the base circulation - not dissimilar to fractional quantum electron charge in the fractional quantum Hall effect). Other interesting vortices are skyrmions - used by high energy physicists to model baryons.
